Ψηφιακή πεξεργασία Eικόνων και Σηµάτων · Ψηφιακή...

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TYPORAMA

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',(E( !&)!"!+#+ !-0'F !(* / 1997–2003

ISBN: 960–538–224–5

KAO46?H P6OJ=2H: *"H 44/2

Copyright 2003 <45 C2@ !IIBO5 654 ?IJ CJ@ 6?=µJ

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&1('#4, !"(8*(1+µ.,% %'(/.6C8µ%/%, G,,(0.4 16.0*0?,

E08%$@$01C4 '%"%/3"28.04 .................................................................................................................................... 11

1.1 1234567 989:9;<5=>5 =7µ5CJH ....................................................................................................... 14

1.2 TK8J4 =2µBCA@ ............................................................................................................................................... 15

1.2.1 H Q@@J45 C2H =LT@?C2C5H =C5 =7µ5C5 ......................................................................... 16

1.2.2 ,9C5C;J87 =7µ5CJH 58? 5@5IJ<46?–=9–R23456? 654 58? R23456?–=9–5@5IJ<46? ................................................................................................ 20

+K@JR2 9@?C2C5H .......................................................................................................................................... 30

1.3 +7µ5C5 O456;4CJK T;?@JL ..................................................................................................................... 31

1.3.1 U5=46B =7µ5C5 O456;4CJK T;?@JL .................................................................................. 31

1.3.2 +CJ4T94MO94H 8;B:94H ................................................................................................................... 33

+K@JR2 9@?C2C5H .......................................................................................................................................... 36

1.4 +L=C7µ5C5 O456;4CJK T;?@JL ............................................................................................................ 36

1.4.1 &;JL=C467 58?6;4=2 =L=C7µ5CJH .................................................................................. 39

1.4.2 +L@QI4:2 ................................................................................................................................................. 40

+K@JR2 9@?C2C5H .......................................................................................................................................... 48

&7,(H3 1.<%6%9() ....................................................................................................................................................... 48

B0I60($"%<9% 1.<%6%9() ......................................................................................................................................... 49

>'#*(83 %$$601+, #"@, 8/3, .663,012 .................................................................................................... 49

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E08%$@$01C4 '%"%/3"28.04 .................................................................................................................................... 51

2.1 ,9C5=T2µ5C4=µ?H Fourier O456;4CJK T;?@JL ..................................................................... 53

2.1.1 ';4=µJ> ................................................................................................................................................... 53


6 æ H º I A K H E ¶ E • E P °A ™ I A E I K O N ø N K A I ™ H M AT ø N

2.1.2 D9A;7µ5C5 654 4O4?C2C9HCJL µ9C=T2µ5C4=µJK Fourier O456;4CJK T;?@JL ............................................................................................................................ 57

2.1.3 E94<µ5CJI2R>5: µ45 56?µ2 µ5C4B .................................................................................... 62

+K@JR2 !@?C2C5H ......................................................................................................................................... 65

2.2 E456;4C?H µ9C5=T2µ5C4=µ?H Fourier ........................................................................................... 65

2.2.1 ';4=µJ> ................................................................................................................................................... 65

2.2.2 %O4?C2C9H CJL O456;4CJK µ9C5=T2µ5C4=µJK Fourier ......................................... 73

2.2.3 0;5µµ467 =L@QI4:2 µQ=A CJL DFT ................................................................................ 77

2.2.4 ' DFT =9 µJ;37 84@B6A@ ...................................................................................................... 79

+K@JR2 9@?C2C5H .......................................................................................................................................... 82

2.3 T5TKH µ9C5=T2µ5C4=µ?H Fourier (FFT) .................................................................................... 82

+K@JR2 9@?C2C5H .......................................................................................................................................... 89

&7,(H3 1.<%6%9() ....................................................................................................................................................... 90

B0I60($"%<9% 1.<%6%9() ......................................................................................................................................... 91

>'#*(83 %$$601+, #"@, 8/3, .663,012 .................................................................................................... 92

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E08%$@$01C4 '%"%/3"28.04 .................................................................................................................................... 93

3.1 ' ,9C5=T2µ5C4=µ?H–z ........................................................................................................................... 95

3.1.1 ';4=µ?H .................................................................................................................................................. 95

3.1.2 V85;:2 CJL µ9C5=T2µ5C4=µJK–z .................................................................................... 96

3.1.3 %O4?C2C9H CJL µ9C5=T2µ5C4=µJK–z ............................................................................... 103

+K@JR2 !@?C2C5H ...................................................................................................................................... 106

3.2 O 5@C>=C;J3JH µ9C5=T2µ5C4=µ?H–z ......................................................................................... 106

3.2.1 F8JIJ<4=µ?H CJL 5@C>=C;J3JL ,.W. µ9 5@B8CL:2 =9 OL@5µJ=94;B ........................................................................................... 107

3.2.2 F8JIJ<4=µ?H CJL 5@C>=C;J3JL ,.W. µ9 5@B8CL:2 =9 µ9;46B 6IB=µ5C5 .............................................................................................. 108


+K@JR2 9@?C2C5H ....................................................................................................................................... 112

3.3 ' µJ@?8I9L;JH µ9C5=T2µ5C4=µ?H–z ........................................................................................ 113

+K@JR2 9@?C2C5H ....................................................................................................................................... 115

3.4 O µ9C5=T2µ5C4=µ?H–z =C2@ 5@BIL=2 =L=C2µBCA@ .................................................... 115

3.4.1 A4C45C?C2C5 LTI =L=C7µ5CJH ........................................................................................... 115

3.4.2 !L=CBG945 LTI =L=C7µ5CJH ................................................................................................ 116

3.4.3 !:4=M=94H O453J;M@ ................................................................................................................. 117

3.4.4 EJµQH =L=C2µBCA@ O456;4CJK T;?@JL ...................................................................... 119

3.4.5 (8?6;4=2 =LT@?C2C5H =L=C2µBCA@ O456;4CJK T;?@JL ............................. 123

+K@JR2 9@?C2C5H ....................................................................................................................................... 133

&7,(H3 1.<%6%9() .................................................................................................................................................... 133

B0I60($"%<9% 1.<%6%9() ...................................................................................................................................... 135

>'#*(83 %$$601+, #"@, 8/3, .663,012 ................................................................................................. 137

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æËÊÈ·Î¿ Ê›ÏÙÚ· !Â!ÂÚ·ÛÌ¤ÓË˜ ÎÚÔ˘ÛÙÈÎ‹˜ ·!fiÎÚÈÛË˜ (FIR)

&1('#4, !"(8*(1+µ.,% %'(/.6C8µ%/%, G,,(0.4 16.0*0?,

E08%$@$01C4 '%"%/3"28.04 ................................................................................................................................ 139

4.1 X5;56C2;4=C46B CA@ FIR 3>IC;A@ ............................................................................................ 141

4.1.1 0;5µµ467 58?6;4=2 3B=2H ................................................................................................. 143

4.1.2 !L=CBG945 CA@ 3>IC;A@ – *?IJ4 654 µ2O9@46B ................................................ 146

+K@JR2 9@?C2C5H ....................................................................................................................................... 147

4.2 )K8J4 654 8;JO45<;53QH CA@ FIR 3>IC;A@ ......................................................................... 147

+K@JR2 9@?C2C5H ....................................................................................................................................... 149

4.3 MQGJOJ4 =T9O>5=2H FIR 3>IC;A@ ................................................................................................ 149

4.3.1 ,QGJOJH CA@ 85;5GK;A@ .................................................................................................... 149

4.3.2 %O4?C2C9H 654 CK8J4 CA@ 85;5GK;A@ ........................................................................... 153

4.3.3 ,QGJOJH =T9O>5=2H SQIC4=CA@ FIR 3>IC;A@ ..................................................... 156

+K@JR2 9@?C2C5H ....................................................................................................................................... 156

7¶ E P I E X O M E N A


8 æ H º I A K H E ¶ E • E P °A ™ I A E I K O N ø N K A I ™ H M AT ø N

4.4 DQµ5C5 LIJ8J>2=2H FIR 3>IC;A@ .............................................................................................. 157

4.4.1 EJµQH 8;5<µ5CJ8J>2=2H CA@ FIR 3>IC;A@ .......................................................... 157

4.4.2 +3BIµ5 58? C2@ 6SB@C4=2 CA@ =L@C9I9=CM@ ..................................................... 159

&7,(H3 1.<%6%9() .................................................................................................................................................... 161

B0I60($"%<9% 1.<%6%9() ...................................................................................................................................... 162

>'#*(83 %$$601+, #"@, 8/3, .663,012 ................................................................................................. 163

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E08%$@$01C4 '%"%/3"28.04 ................................................................................................................................ 165

5.1 (@5IJ<46B 3>IC;5 ..................................................................................................................................... 168

5.1.1 0@A=CB S5GL89;5CB 5@5IJ<46B 3>IC;5 .................................................................. 169

5.1.2 ,9C5=T2µ5C4=µJ> =LT@?C2C5H 5@5IJ<46M@ 3>IC;A@ .................................. 175

+K@JR2 9@?C2C5H ....................................................................................................................................... 178

5.2 +T9O>5=2 IIR R23456M@ 3>IC;A@ ............................................................................................... 178

5.2.1 0;53467 µQGJOJH ......................................................................................................................... 178

5.2.2 ,QGJOJH 5µ9CBSI2C2H 6;JL=C467H ............................................................................. 179

5.2.3 ,QGJOJH O4<;5µµ46JK µ9C5=T2µ5C4=µJK .............................................................. 184

+K@JR2 9@?C2C5H ....................................................................................................................................... 191

5.3 FIJ8J>2=2 IIR R23456M@ 3>IC;A@ .......................................................................................... 191

5.3.1 *989;5=µQ@2 56;>S945 654 LIJ8J>2=2 IIR R23456M@ 3>IC;A@ ........ 193

+K@JR2 9@?C2C5H ....................................................................................................................................... 195

&7,(H3 1.<%6%9() .................................................................................................................................................... 195

B0I60($"%<9% 1.<%6%9() ...................................................................................................................................... 196

>'#*(83 %$$601+, #"@, 8/3, .663,012 ................................................................................................. 197


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E08%$@$01C4 '%"%/3"28.04 ................................................................................................................................ 199

6.1 H R234567 946?@5 ..................................................................................................................................... 201

6.1.1 # 946?@5 AH O4=O4B=C5CJ =7µ5 ....................................................................................... 201

6.1.2 +CJ4T94MO2 O4=O4B=C5C5 R23456B =7µ5C5 ........................................................... 204

6.1.3 # <9Aµ9C;>5 C2H 946?@5H ...................................................................................................... 205

6.1.4 );?8J4 58?6C2=2H C2H R234567H 946?@5H ............................................................. 208

6.1.5 # 989:9;<5=>5 C2H R234567H 946?@5H ....................................................................... 211

+K@JR2 9@?C2C5H ....................................................................................................................................... 212

6.2 # 3L=4JIJ<>5 C2H ?;5=2H .................................................................................................................. 213

6.2.1 ' 5@G;M84@JH J3G5Iµ?H ...................................................................................................... 213

6.2.2 .AC94@?C2C5, I5µ8;?C2C5 654 5@C>G9=2 ................................................................ 214

6.2.3 X;Mµ5 654 RLTJ3L=46B 354@?µ9@5 ............................................................................ 217

6.2.4 &;4C7;45 8J4?C2C5H C2H 946?@5H .................................................................................... 217

+K@JR2 9@?C2C5H ....................................................................................................................................... 218

6.3 !>O2 8;B:9A@ =C2@ 989:9;<5=>5 C2H R234567H 946?@5H ........................................ 218

6.3.1 )J846QH 8;B:94H ............................................................................................................................ 219

6.3.2 &5GJI46QH 8;B:94H ..................................................................................................................... 222

6.3.3 09Aµ9C;46QH 8;B:94H ................................................................................................................ 224

+K@JR2 9@?C2C5H ....................................................................................................................................... 226

6.4 )J 3B=µ5 C2H 946?@5H ........................................................................................................................... 226

+K@JR2 9@?C2C5H ....................................................................................................................................... 230

&7,(H3 1.<%6%9() .................................................................................................................................................... 231

B0I60($"%<9% 1.<%6%9() ...................................................................................................................................... 231

>'#*(83 %$$601+, #"@, 8/3, .663,012 ................................................................................................. 232

9¶ E P I E X O M E N A


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E08%$@$01C4 '%"%/3"28.04 ................................................................................................................................ 233

7.1 E4=O4B=C5C2 =L@QI4:2 ........................................................................................................................... 236

7.1.1 0;5µµ46?C2C5 ................................................................................................................................. 237

7.1.2 (µ9C5SI2C?C2C5 =C2@ JI>=G2=2 .................................................................................. 237

+K@JR2 9@?C2C5H ....................................................................................................................................... 242

7.2 X5;56C2;4=C46B CA@ O4=O4B=C5CA@ R23456M@ =L=C2µBCA@ ............................. 242

7.2.1 );?8J4 =K@O9=2H CA@ O4=O4B=C5CA@ R23456M@ 3>IC;A@ ........................ 242

7.2.2 E45TA;>=4µ5 O4=O4B=C5C5 R23456B =L=C7µ5C5 .............................................. 245

7.2.3 !L=C5G7 O4=O4B=C5C5 =L=C7µ5C5 ................................................................................ 246

7.3 A8?6;4=2 =C2 =LT@?C2C5 CA@ O4=O4B=C5CA@ R23456M@ =L=C2µBCA@ ...... 247

+K@JR2 9@?C2C5H ....................................................................................................................................... 251

7.4 +T9O>5=2 CA@ O4=O4B=C5CA@ R23456M@ 3>IC;A@ .......................................................... 251

7.4.1 F8JIJ<4=µ?H C2H 6;JL=C467H 58?6;4=2H 58? C2@ 58?6;4=2 =LT@?C2C5H ........................................................................................ 251

7.4.2 +T9O>5=2 O4=O4B=C5CA@ FIR 3>IC;A@ µ9 C2 µQGJOJ CA@ 85;5GK;A@ .......................................................................................... 253

7.4.3 YII9H µQGJOJ4 =T9O>5=2H CA@ O4=O4B=C5CA@ FIR 3>IC;A@ ................... 256

7.4.4 !35;µJ<QH CA@ O4=O4B=C5CA@ FIR 3>IC;A@ ....................................................... 257

+K@JR2 9@?C2C5H ....................................................................................................................................... 259

&7,(H3 1.<%6%9() .................................................................................................................................................... 259

B0I60($"%<9% 1.<%6%9() ...................................................................................................................................... 260

>'#*(83 %$$601+, #"@, 8/3, .663,012 ................................................................................................. 260

A85@C7=94H A=67=9A@ ALCJ5:4JI?<2=2H ......................................................................................... 261


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F?"/3/3 µ./%I63/2 .9,%0 ( E"#,(4 1%0 3 .F%"/3µC,3 µ./%I63/2 3 %1()8/012 '9.83

(&E2µ% 1.1%). M0% .01#,% (<@/($"%<9%) %'(/.6.9 E%"%1/3"08/01# '%"?*.0$µ% 82µ%-

/(4 *7( *0%8/?8.@,. :*+ 3 .F%"/3µC,3 µ./%I63/2 .9,%0 3 <@/.0,#/3/% /34 .01#,%4

1%0 (0 *7( %,.F?"/3/.4 µ./%I63/C4 .9,%0 (0 *7( E@"01C4 8),/./%$µC,.4 (&E2µ% 1.1I).

;C6(4, '%"?*.0$µ% 82µ%/(4 /"0+, *0%8/?8.@, .9,%0 3 %1(6()59% .01#,@, (video),

#'() (0 *7( %,.F?"/3/.4 µ./%I63/C4 .9,%0 E@"01C4 1%0 3 /"9/3 .9,%0 ( E"#,(4 (&E2µ%

1.1$). O .F%"/3µC,3 µ./%I63/2 .9,%0 1%0 .*+ 3 <@/.0,#/3/% /34 1?5. .01#,%4.

;( 1%5C,% %'# /% '%"%'?,@ 82µ%/% .9,%0 %'%"%9/3/( ,% /( .'.F."$%8/(7µ. µ.

8/#E( ,% I.6/0+8()µ. /3, '(0#/3/? /(), ,% .F%$?$()µ. /3 E"280µ3 '63"(<("9% 1%0

yt

x

(5)

(S) (<)

™¯‹Ì· 1.1

!%"%*.9$µ%/%83µ?/@,

(%) (µ069%4 (µ0%4*0?8/%834),

(I) .01#,%4 (*7(*0%8/?8.@,), 1%0

($) %1(6()59%4.01#,@, (/"0+,*0%8/?8.@,)


,% .F%6.9H()µ. 1%/? /( *),%/#, /( 5#")I(, ,% /( %'(531.78()µ. 2 ,% /( µ./%*+-

8()µ.. &/( I0I69( %)/# 5% µ.6./28()µ. ("08µC,.4 %'# /04 '0( 8),3508µC,.4 *0%*0-

1%89.4 .'.F."$%89%4 '() E"380µ('(0(7,/%0 $0% /()4 81('(74 %)/(74.

&’ %)/# /( 1.<?6%0( 5% µ.6./28()µ. /04 I%801C4 C,,(0.4 /@, 83µ?/@, 1%0 8)8/3-

µ?/@, *0%1"0/(7 E"#,(). &/3, .,#/3/% 1.1 5% %,%<."5(7µ. 8/% '6.(,.1/2µ%/% 1%0

µ.0(,.1/2µ%/% /34 H3<0%124 .'.F."$%89%4 83µ?/@,. &/3, .,#/3/% 1.2 5% $,@"9-

8()µ. /% %,%6($01? 1%0 H3<0%1? 82µ%/%, 1%0 /% 8E./01? µ. /3 µ./%/"('2 /@, %,%-

6($01+, 8. H3<0%1?. &/3, .,#/3/% 1.3 5% *(7µ. /% I%801? 82µ%/% *0%1"0/(7 E"#-

,(), 1%5+4 1%0 /04 8/(0E.0+*.04 '"?F.04 '() µ'("(7µ. ,% .<%"µ#8()µ. 8’ %)/?.

;C6(4, 8/3, .,#/3/% 1.4 5% %8E(635(7µ. µ. /% 8)8/2µ%/% *0%1"0/(7 E"#,() 1%0 /3,

'"?F3 /34 $"%µµ0124 8),C60F34.

1 3E # ™ " ° ø ° # ' ! ™ ¶ " ƒ"& % ƒ % ™ ! # ™


1 4 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

1.1 æËÊÈ·Î‹ Â!ÂÍÂÚÁ·Û›· Û‹Ì·ÙÔ˜

# H3<0%12 .'.F."$%89% 82µ%/(4 5=TJI9>C54 µ9 C2@ R234567 5@585;B=C5=2 CA@

=2µBCA@ 654 C2@ 5@BIL=2, C;J8J8J>2=2 654 9:5<A<7 8I2;J3J;4M@ 58? 5LCB, µ9 C2

SJ7G945 R23456M@ 989:9;<5=CM@. *9;48CM=94H 65CB C4H J8J>9H GQIJLµ9 @5 5354-

;Q=JLµ9 CJ@ G?;LSJ 58? Q@5 =7µ5 7 @5 S;JKµ9 CJ µ9C5=T2µ5C4=µ? Fourier 6B8J4-

A@ O9OJµQ@A@ 7 @5 OM=JLµ9 =’ Q@5 =7µ5 µJ;37 84J 65CBII2I2 <45 989:9;<5=>5

654 5@BIL=2 C2H 8I2;J3J;>5H 8JL 9µ89;4QT94, 58JC9IJK@ 85;5O9><µ5C5 C2H R2345-

67H 989:9;<5=>5H =7µ5CJH. (LC7 T;2=4µJ8J49>C54 ?IJ 654 89;4==?C9;J =9 8JIIQH

89;4JTQH 935;µJ<M@ ?8JL 85;5OJ=456B T;2=4µJ8J4JK@C5@ 5@5IJ<46QH µJ;3QH 989-

:9;<5=>5H, 5IIB 654 =9 @Q9H 935;µJ<QH =C4H J8J>9H J4 5@5IJ<46QH µQGJOJ4 9>@54

OK=6JIJ 7 654 5OK@5CJ@ @5 T;2=4µJ8J42GJK@. )J <9<J@?H 5LC? J39>I9C54 =C5 8I9-

J@96C7µ5C5 8JL 85;JL=4BZ94 2 R234567 989:9;<5=>5 =7µ5CJH.

F8B;TJL@ 8JIIJ> I?<J4 <45 CJLH J8J>JLH G5 8;JC4µJK=5µ9 C2@ R234567 989:9;<5-

=>5 9@?H =7µ5CJH Q@5@C4 C2H 5@5IJ<467H. &5CB 8;MC4=CJ I?<J, Q@5 R23456? 8;J-

<;5µµ5C4Z?µ9@J =K=C2µ5 85;JL=4BZ94 µ9<BI2 9L9I4:>5 =C2@ C;J8J8J>2=2 CA@ 8;B-

:9A@ R234567H 989:9;<5=>5H µ9 µ45 58I7 µ9C5C;J87 CJL 8;J<;Bµµ5CJH. ,45 CQCJ45

C;J8J8J>2=2 9@?H 5@5IJ<46JK =L=C7µ5CJH =L@98B<9C54 C2@ 985@5=T9O>5=2 CJL

6L6IMµ5CJH 654 =L@9856?IJLGJ QI9<TJ 654 984S9S5>A=2 (testing and verification)

C2H J;G7H I94CJL;<>5H CJL.

# 56;>S945 (accuracy) 85>Z94 98>=2H 8JIK =8JLO5>J ;?IJ. # 5@JT7 CA@ =CJ4T9>A@

CA@ 5@5IJ<46M@ 6L6IAµBCA@ 65G4=CB OK=6JIJ CJ@ 8;J=O4J;4=µ? C2H 56;>S945H

9@?H 5@5IJ<46JK =L=C7µ5CJH 989:9;<5=>5H. +C2@ 89;>8CA=2 9@?H R23456JK =L=C7-

µ5CJH, J QI9<TJH C2H 84=C?C2C5H CA@ 8;JO45<;53M@ 9>@54 8JIK 84J 9K6JIJH.

)5 R23456B =7µ5C5 58JG269KJ@C54 =9 µ5<@2C46B 7 J8C46B µQ=5 (I.T. µ5<@2C46JKH

7 J8C46JKH O>=6JLH, C54@>9H, 6.B.) TA;>H L8JSBGµ4=2 C2H 84=C?C2CBH CJLH, 8Q;5@

5LC7H 8JL L894=7IG9 =C2 O45O465=>5 µ9C5C;J87H CJLH 58? 5@5IJ<46B =9 R23456B.

PC=4, O>@9C54 2 OL@5C?C2C5 µ9C53J;BH 654 989:9;<5=>5H CQCJ4A@ =2µBCA@ =9 µ2

8;5<µ5C46? T;?@J. !848IQJ@, O>@9C54 2 OL@5C?C2C5 935;µJ<7H 84J 89;>8IJ6A@

5I<J;>GµA@ 989:9;<5=>5H =7µ5CJH. +L@7GAH 2 LIJ8J>2=2 µ5G2µ5C46M@ 8;B:9A@

µ9<BI2H 56;>S945H 9>@54 OK=6JIJ @5 <>@94 =9 =7µ5C5 C5 J8J>5 S;>=6J@C54 =9 5@5-

IJ<467 µJ;37, 8;B<µ5 ?µAH 8JL 9>@54 =L@2G4=µQ@J 654 9K6JIJ @5 <>@94 =9 Q@5

R23456? =7µ5 CJ J8J>J 989:9;<5Z?µ5=C9 µ9 Q@5@ L8JIJ<4=C7 654 µ9 65CBII2IJ

IJ<4=µ46?.

+9 8JIIQH 89;48CM=94H, 2 R234567 989:9;<5=>5 9@?H =7µ5CJH QT94 T5µ2I?C9;J

6?=CJH 58? C2@ 5@C>=CJ4T2 5@5IJ<467. (LC? µ8J;9> @5 J39>I9C54 9>C9 =CJ ?C4 CJ


LI46? (hardware) =7µ9;5 9>@54 3G2@?C9;J 9>C9 =C2@ 9L9I4:>5 8JL 85;QT9C54 I?<A

C2H R234567H LIJ8J>2=2H.

(8JCQI9=µ5 CA@ 8I9J@96C2µBCA@ C2H R234567H 989:9;<5=>5H =7µ5CJH 9>@54 2 O45;-

6MH 5L:5@?µ9@2 T;7=2 C2H =9 ?IJ 654 89;4==?C9;JLH CJµ9>H 935;µJ<M@, ?8AH =C2@

989:9;<5=>5 Jµ4I>5H, =C2 µ9CBOJ=2 =7µ5CJH =9 C2I93A@46B 65@BI45, =C2 =94=µJ-

IJ<>5, =C2 <9A3L=467, =C2@ 45C;467, =C2@ 9:9;9K@2=2 CJL O45=C7µ5CJH, =C2 µ9C9-

A;JIJ<>5, 6.B.

.L=46B, 2 R234567 989:9;<5=>5 =7µ5CJH QT94 654 C5 ?;4B C2H, C5 J8J>5 J39>IJ@C54

=CJLH 89;4J;4=µJKH 8JL C>G9@C54 =C2@ C5TKC2C5 I94CJL;<>5H CA@ µ9C5C;J8QA@ 5@5-

IJ<46JK =7µ5CJH =9 R23456?, 65GMH 654 =CJLH >O4JLH CJLH R23456JKH 989:9;<5=CQH

=7µ5CJH. PC=4, =7µ5C5 µ9 9:54;9C46B µ9<BIJ 9K;JH =LT@JC7CA@, <45 85;BO94<µ5,

=7µ5C5 µ9 9K;JH =LT@JC7CA@ C2H CB:9AH CA@ 100 MHz, L3>=C5@C54 989:9;<5=>5

56?µ5 654 =7µ9;5 µ9 5@5IJ<46QH µ9G?OJLH.

1.2 T‡!ÔÈ ÛËÌ¿ÙˆÓ

)5 =7µ5C5 C5:4@JµJK@C54 =9 OKJ µ9<BI9H 65C2<J;>9H: =C5 82µ%/% 8),.E(74 E"#,()

654 =C5 82µ%/% *0%1"0/(7 E"#,(). +L@7GAH, AH 5@9:B;C2C2 µ9C5SI2C7 T;2=4µJ8J4-

9>C54 J T;?@JH, TA;>H ?µAH @5 58J6I9>9C54 2 5@9:B;C2C2 µ9C5SI2C7 @5 9>@54 6B8J4J

BIIJ 3L=46? µQ<9GJH, ?8AH <45 85;BO94<µ5 2 58?=C5=2, 2 G9;µJ6;5=>5 7 2 8>9=2.

*5;’ ?I5 5LCB QT94 9846;5C7=94 @5 µ4IBµ9 <45 =7µ5C5 O456;4CJK T;?@JL.

+C5 =7µ5C5 8),.E(74 E"#,() (continuous time) 2 5@9:B;C2C2 µ9C5SI2C7 9>@54 =L@9-

T7H, O2I5O7 C5 =7µ5C5 5LCB J;>ZJ@C54 <45 J8J45O78JC9 C4µ7 C2H 5@9:B;C2C2H µ9C5-

SI2C7H. # 9:5;C2µQ@2 µ9C5SI2C7, O2I5O7 CJ 8IBCJH (amplitude) CJL =7µ5CJH, 9>@54

654 5LC7 =L@9T7H. 04’ 5LC? 654 C5 =7µ5C5 5LCB 5@53Q;J@C54 654 AH 82µ%/% 8),.-

E(74 E"#,() 8),.E(74 '6?/()4 7 %,%6($01? =7µ5C5 (+T7µ5 1.25). *5;5O9><µ5C5

CQCJ4A@ =2µBCA@ 9>@54 2 Jµ4I>5 AH =L@B;C2=2 CJL T;?@JL 7 2 5CµJ=354;467 8>9=2

AH =L@B;C2=2 CJL KRJLH. P@5 5@5IJ<46? =7µ5 89;4<;B39C54 58? µ45 =L@B;C2=2

x(t), ?8JL t 8;5<µ5C46?H 5;4Gµ?H.

+C5 =7µ5C5 *0%1"0/(7 E"#,() (discrete time) 2 5@9:B;C2C2 µ9C5SI2C7 9>@54 O456;4-

C7, O2I5O7 C5 =7µ5C5 5LCB J;>ZJ@C54 µ?@J <45 =L<696;4µQ@9H C4µQH C2H 5@9:B;C2C2H

µ9C5SI2C7H. ,9 BII5 I?<45, 2 5@9:B;C2C2 µ9C5SI2C7 85>;@94 C4µQH 58? Q@5 O456;4-

C? =K@JIJ C4µM@. # 9:5;C2µQ@2 µ9C5SI2C7, O2I5O7 CJ 8IBCJH CJL =7µ5CJH, 9>@54

=L@9T7H. 04’ 5LC? 654 C5 =7µ5C5 5LCB 5@53Q;J@C54 654 AH =7µ5C5 *0%1"0/(7 E"#,()

8),.E(74 '6?/()4 (+T7µ5 1.2S). *5;5O9><µ5C5 CQCJ4A@ =2µBCA@ 9>@54 J O9>6C2H

Dow–Jones AH =L@B;C2=2 CJL T;?@JL (I.T. 5@B 2µQ;5) 7 CJ 65CB 6935I7@ 94=?O2-

1 51 . 2 T À ¶ $ # ™ % ª "& ø ¡


1 6 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

µ5 AH =L@B;C2=2 CJL C?8JL O45µJ@7H. +C2 89;>8CA=2 8JL 654 2 9:5;C2µQ@2 µ9C5-

SI2C7 85>;@94 O456;4CQH C4µQH, C?C9 µ4IBµ9 <45 82µ%/% *0%1"0/(7 E"#,() *0%1"0/(7

'6?/()4 7 H3<0%1? =7µ5C5 (+T7µ5 1.2<). P@5 =7µ5 O456;4CJK T;?@JL =LµSJI>Z9C54

=L@7GAH AH x(n), ?8JL n 56Q;54JH. *;?694C54 <45 µ>5 56JIJLG>5 (sequence) 5;4G-

µM@, <4’ 5LC? =LT@B 5@539;?µ5=C9 =CJ =7µ5 5LC? 654 AH %1(6()59%.

*;4@ 8;JTA;7=JLµ9 =C2 µ9IQC2 CA@ S5=46M@ =2µBCA@ O456;4CJK T;?@JL, G5 7C5@

65I? @5 <@A;>=JLµ9 C2@ Q@@J45 C2H =LT@?C2C5H C?=J <45 C5 =7µ5C5 =L@9TJKH T;?-

@JL ?=J 654 <45 C5 =7µ5C5 O456;4CJK T;?@JL.

x(t) x(n) x(n)

t n n

1

3/4

1/2

1/4

–2–1

01

23

–2–1

01

23

™¯‹Ì· 1.2

;7'(0 83µ?/@,:(%) %,%6($01#,

(I) *0%1"0/(7 E"#,(),

(<) R23456?

1.2.1 H ¤ÓÓÔÈ· ÙË˜ Û˘¯ÓfiÙËÙ·˜ ÛÙ· Û‹Ì·Ù·

# Q@@J45 C2H =LT@?C2C5H 9>@54 S5=467 654 <@A=C7 =9 ?IJLH µ5H. )2@ QTJLµ9 =L@5-

@C7=94 =CJ ;5O4J3A@46? OQ6C2 8JL T;2=4µJ8J4JKµ9 7 =CJ =C9;9J3A@46? =K=C2µ5

8JL QTJLµ9 7 =CJ 3>IC;J 8JL 8;Q894 @5 SBIJLµ9 =C2 3ACJ<;53467 µ5H µ2T5@7. (8?

C2 3L=467 <@A;>ZJLµ9 ?C4 2 =LT@?C2C5 =T9C>Z9C54 µ9 Q@5@ CK8J 89;4JO467H 6>@2=2H,

2 J8J>5 J@JµBZ9C54 %"µ(,012 /%6?,/@83, 654 2 J8J>5 89;4<;B39C54 58? 2µ4CJ@J94-

O9>H =L@5;C7=94H. # Q@@J45 C2H =LT@?C2C5H =T9C>Z9C54 Bµ9=5 µ9 C2@ Q@@J45 CJL T;?-

@JL, 53JK 2 O4B=C5=2 5LC7H 9>@54 CJ 5@C>=C;J3J CJL T;?@JL. &5CB =L@Q8945, 2 3K=2

CJL T;?@JL (=L@9T7H 7 O456;4C7) 5@5µQ@JLµ9 @5 982;9BZ94 C2 3K=2 C2H =LT@?C2C5H.

%ª#&$¡$!#¢% ™%ª"&" ™À¡!Ã$À™ Ãƒ$¡$À

,>5 58I7 5;µJ@467 C5IB@CA=2 J;>Z9C54 µ5G2µ5C46B 58? CJ 2µ4CJ@J94OQH =7µ5

=L@9TJKH T;?@JL:

x%(t) = >!cos(Qt + 5), – " < t < " (1.1)

?8JL > CJ '6?/(4 (amplitude) CJL 2µ4CJ@J94OJKH, Q 2 8)E,#/3/% =9 56C>@45 5@B O9L-

C9;?I98CJ (rad/s) 654 5 2 3B=2 =9 56C>@45 (+T7µ5 1.3) # =TQ=2 (1.1) µ8J;9> @5 <;5-

39> 654 AH:

x%(t) = >!cos(2'Ft + 5), – " < t < " (1.2)


?C5@ GQ=JLµ9

Q = 2'F (1.3)

?8JL F 2 =LT@?C2C5 =9 6K6IJLH 5@B O9LC9;?I98CJ 7 hertz (Hz). )J 5@5IJ<46? 5LC?

=7µ5 85;JL=4BZ94 C4H 9:7H 4O4?C2C9H:

• !>@54 89;4JO46?: *;B<µ5C4, <45 J8J45O78JC9 C4µ7 C2H =LT@?C2C5H F, 2 =L@B;C2-

=2 x%(t) 9>@54 89;4JO467, O2I5O7 x%(t + ;p) = x%(t), ?8JL ;p = 1/F 9>@54 2 S5=467

89;>JOJH CJL 2µ4CJ@J94OJKH =7µ5CJH.

• 045 O453J;9C46QH =LT@?C2C9H QTJLµ9 O453J;9C46B =7µ5C5.

• (K:2=2 C2H =LT@?C2C5H F =L@98B<9C54 5@C>=CJ4T2 5K:2=2 CJL ;LGµJK C5IB-

@CA=2H CJL =7µ5CJH, O2I5O7 89;4==?C9;9H 89;>JOJ4 9µ89;4QTJ@C54 =9 Q@5 =L<69-

6;4µQ@J T;J@46? O4B=C2µ5.

1 71 . 2 T À ¶ $ # ™ % ª "& ø ¡

™¯‹Ì· 1.3

!%"?*.0$µ% %,%6($01(7 3µ0/(,(.0*(7482µ%/(4

xa(t)=A cos(2'Ft+5)

Tp=1/F

A

–A

A cos5

t0

[

!"µ#$%&'( )#*#%(*"+"

# =LT@?C2C5 9>@54 58? C2 3K=2 C2H G9C467 8J=?C2C5. (LC? 9>@54 8;J35@QH, 53JK 2

=LT@?C2C5 =9 Q@5 89;4JO46? =7µ5 963;BZ94 CJ@ 5;4Gµ? CA@ 6K6IA@ =C2 µJ@BO5 CJL

T;?@JL. +9 J;4=µQ@9H 89;48CM=94H ?µAH, <45 I?<JLH 9L6JI>5H 58? µ5G2µ5C467H 58?-

R9AH, 5854C9>C54 2 94=5<A<7 5;@2C46M@ =LT@JC7CA@. (LC? <>@9C54 65C5@J2C? 5@

GLµ2GJKµ9 ?C4 CJ 2µ4CJ@J94OQH =7µ5 (=TQ=2 1.1) µ8J;9> @5 <;539> 654 AH:

(1.4)

S5=4Z?µ9@J4 =C2@ C5LC?C2C5 CJL Euler e ± j3 = cos< ± jsin<. *5;5C2;JKµ9, IJ48?@,

x t A t

Ae

Aej t j t

5) ))( ) cos( ( (= + = ++ # +$ $ $% % %

2 2


1 8 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

?C4 CJ 2µ4CJ@J94OQH =7µ5 µ8J;9> @5 8;JQIG94 58? C2@ 8;?=G9=2 OKJ =LZL<M@ µ4<5-

O46M@ 96G9C46M@ =2µBCA@ >=JL 8IBCJLH. )5 µ4<5O46B 96G9C46B =7µ5C5 =L@2G>ZJL-

µ9 @5 C5 85;4=CB@JLµ9 AH O45@K=µ5C5 8B@A =CJ µ4<5O46? 98>89OJ, C5 J8J>5 J@J-

µBZJLµ9 <?8(".4 (phasors). '4 3B=J;9H C2H =TQ=2H (1.4) 89;4=C;Q3J@C54 µ9 <A@45-

6QH =LT@?C2C9H ± Q rad/sec. # G9C467 =LT@?C2C5 5@C4=CJ4T9> =9 JµJ4?µJ;32 89;4-

=C;J37 CJL 3B=J;5 µ9 3J;B 5@C>G9C2 C2H 6>@2=2H CA@ O946CM@ CJL ;JIJ<4JK (5;4-

=C9;?=C;J32 89;4=C;J37). &5CB =L@Q8945, 2 5;@2C467 =LT@?C2C5 5@C4=CJ4T9> =9

JµJ4?µJ;32 89;4=C;J37 CJL 3B=J;5 65CB C2 3J;B C2H 6>@2=2H CA@ O946CM@ CJL

;JIJ<4JK (O9:4?=C;J32 89;4=C;J37).

\8AH IJ48?@ 5@53Q;5µ9, <45 I?<JLH 9L6JI>5H 58? (µ5G2µ5C467H 58?R9AH) G5 T;2=4-

µJ8J4JKµ9 G9C46QH 654 5;@2C46QH =LT@?C2C9H =9 ?I2 C2@ Q6C5=2 5LCJK CJL S4SI>JL. (LC?

=2µ5>@94 ?C4 2 89;4JT7 =LT@JC7CA@ CA@ 5@5IJ<46M@ =2µBCA@ G5 9>@54 – " < F < ".

%ª#&$¡$!#¢% ™%ª"&" ¢#"'ƒ#&$À Ãƒ$¡$À

P@5 2µ4CJ@J94OQH =7µ5 O456;4CJK T;?@JL µ8J;9> @5 963;5=C9> AH:

x(n) = >cos(@n + 5), – " < n < " (1.5)

?8JL n 56Q;542 µ9C5SI2C7, 2 J8J>5 5@C48;J=A89K94 CJ@ 5;4Gµ? (C2 GQ=2) CJL O9><-

µ5CJH, > CJ 8IBCJH CJL =7µ5CJH, @ 2 =LT@?C2C5 CJL =7µ5CJH =9 56C>@45 5@B O9><-

µ5 654 5 2 3B=2 =9 56C>@45 (+T7µ5 1.4). # =TQ=2 (1.5) µ8J;9> @5 <;539> 654 AH:

x(n) = >cos(2'fn + 5), – " < n < " (1.6)

?C5@ GQ=JLµ9

@ = 2'f (1.7)

?8JL f 2 =LT@?C2C5 =9 6K6IJLH 5@B O9><µ5.

x(n)=A cos(@n + 5)

A

–A

0 n

... ...™¯‹Ì· 1.4

!%"?*.0$µ% .,#43µ0/(,(.0*(74

82µ%/(4 *0%1"0/(7E"#,() µ.

@ = '/6 1%0 5 = '/3


+9 5@C>G9=2 µ9 Q@5 2µ4CJ@J94OQH =7µ5 =L@9TJKH T;?@JL, Q@5 2µ4CJ@J94OQH O456;4-

CJK T;?@JL 85;JL=4BZ94 C4H 56?IJLG9H 4O4?C2C9H:

• !>@54 89;4JO46? µ?@J ?C5@ 2 =LT@?C2C5 CJL f 9>@54 ;2C?H 5;4Gµ?H.

• )5 2µ4CJ@J94O7 =7µ5C5 O456;4CJK T;?@JL CA@ J8J>A@ J4 =LT@?C2C9H O453Q;JL@

65CB 56Q;54J 8JII58IB=4J CJL 2' 9>@54 >O45 (C5LC>ZJ@C54).

• ' µQ<4=CJH ;LGµ?H C5IB@CA=2H 9@?H 2µ4CJ@J94OJKH O456;4CJK T;?@JL 984CL<TB-

@9C54 <45 @ = ' (7 @ = –') 7 4=JOK@5µ5 <45 (7 ).

¶·Ú¿‰ÂÈÁÌ· 1.1

$5 58JO94TC9> ?C4 Q@5 2µ4CJ@J94OQH O456;4CJK T;?@JL 9>@54 89;4JO46? µ?@J ?C5@ 2

=LT@?C2CB CJL f 9>@54 ;2C?H 5;4Gµ?H.

=G/0:

P@5 =7µ5 O456;4CJK T;?@JL x(n) 9>@54 89;4JO46? µ9 89;>JOJ R (R > 0) 9B@ 654 µ?@J@

9B@

x(n + N) = x(n) <45 ?I5 C5 n (1.8)

# µ46;?C9;2 C4µ7 CJL R <45 C2@ J8J>5 985I2G9K9C54 2 =TQ=2 5LC7, J@JµBZ9C54 I%80-

12 '."9(*(4 (fundamental period).

045 @5 9>@54 89;4JO46? Q@5 2µ4CJ@J94OQH =7µ5 O456;4CJK T;?@JL µ9 =LT@?C2C5 f, G5

8;Q894 @5 4=TK94 2 =TQ=2 cos(2 'f(N + n) + 5) = cos(2'fn + 5). # =TQ=2 5LC7 5I2-

G9K94 9B@ 654 µ?@J 9B@ L8B;T94 56Q;54JH m CQCJ4JH M=C9 2'fN = 2m' 7 4=JOK@5µ5

, O2I5O7 9B@ 654 µ?@J 9B@ 2 =LT@?C2C5 f µ8J;9> @5 <;539> AH 82I>6J OKJ

56Q;54A@ 5;4GµM@.

¶·Ú¿‰ÂÈÁÌ· 1.2

$5 58JO9>:9C9 ?C4 2µ4CJ@J94O7 =7µ5C5 O456;4CJK T;?@JL, CA@ J8J>A@ J4 =LT@?C2-

C9H O453Q;JL@ 65CB 56Q;54J 8JII58IB=4J CJL 2', C5LC>ZJ@C54.

=G/0:

P=CA CJ =7µ5 x(n) = >cos(@n + 5) µ9 =LT@?C2C5 @, 654 CJ =7µ5 x1(n) = >cos[(@ +

2')n + G] µ9 =LT@?C2C5 @ + 2'. (8JO946@K9C54 8JIK 9K6JI5 ?C4:

x1(n) = >cos[(@ + 2')n + G] = >cos(@n + G+2 'n) = >cos(@n + 5) = x(n)

09@46B, ?I5 C5 2µ4CJ@J94O7 =7µ5C5 O456;4CJK T;?@JL xk(n) = >cos(@kn + 5),

f

mN

=

f = # 1

2 f = 1

2

1 91 . 2 T À ¶ $ # ™ % ª "& ø ¡


2 0 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

k = 0,1,2, … µ9 @k = @ + 2k', –' ] @ ] ' C5LC>ZJ@C54. ,?@J C5 2µ4CJ@J94O7 =7µ5-

C5 O456;4CJK T;?@JL CA@ J8J>A@ J4 =LT@?C2C9H S;>=6J@C54 =C2@ 89;4JT7

–' ] @ ] ' 7 9>@54 O453J;9C46B, O2I5O7 µJ@5O46B.

,8J;JKµ9, 98JµQ@AH, @5 85;5C2;7=JLµ9 C2@ JL=45=C467 O453J;B µ9C5:K CA@ 2µ4-

CJ@J94OM@ O456;4CJK T;?@JL 654 CA@ 2µ4CJ@J94OM@ =L@9TJKH T;?@JL. +C5 2µ4CJ@J-

94O7 =L@9TJKH T;?@JL QTJLµ9 O453J;9C46B =7µ5C5 <45 J8J45O78JC9 =LT@?C2C5 Q (7

F) =C2@ 89;4JT7 –" < Q < " (7 –" < F < "), 9@M =C5 2µ4CJ@J94O7 O456;4CJK T;?-

@JL QTJLµ9 O453J;9C46B =7µ5C5 <45 J8J45O78JC9 =LT@?C2C5 @ (7 f) =C2@ 89;4JT7

–' ] @ ] ' (7 ).

1.2.2 ªÂÙ·ÙÚÔ!‹ Û‹Ì·ÙÔ˜ ·!fi ·Ó·ÏÔÁÈÎfi–ÛÂ–„ËÊÈ·Îfi Î·È ·!fi„ËÊÈ·Îfi–ÛÂ–·Ó·ÏÔÁÈÎfi

)5 89;4==?C9;5 =7µ5C5 8JL 85;JL=4BZJL@ 8;56C46? 9@O453Q;J@, ?8AH <45 85;BO94<µ5

2 Jµ4I>5, C5 S4JIJ<46B =7µ5C5, C5 =94=µ46B =7µ5C5, 6.B., 9>@54 5@5IJ<46B. 045 @5 989-

:9;<5=CJKµ9 5@5IJ<46B =7µ5C5 µ9 R23456B µQ=5, 5854C9>C54 2 µ9C5C;J87 5LCM@ =9

R234567 µJ;37, O2I5O7 2 µ9C5C;J87 CJLH =9 µ>5 56JIJLG>5 5;4GµM@ 8989;5=µQ@2H

56;>S945H. # O45O465=>5 5LC7 J@JµBZ9C54 µ./%/"('2 %,%6($01(7–8.–H3<0%1#

(analog–to–digital conversion, (/D) 654 C5 5@C>=CJ4T5 6L6IMµ5C5 J@JµBZJ@C54 «µ9C5-

C;J89>H 5@5IJ<46JK–=9–R23456?» (analog–to–digital converters, ADCs). # 5@C>-

=C;J32 O45O465=>5 C2H µ9C5C;J87H 9@?H R23456JK =7µ5CJH =9 5@5IJ<46? 9>@54 <@A-

=C7 AH µ./%/"('2 H3<0%1(7–8.–%,%6($01# (digital–to–analog conversion, D/A) 654

<>@9C54 µ9 C2 SJ7G945 6L6IAµBCA@ C5 J8J>5 J@JµBZJ@C54 µ9C5C;J89>H

«R23456JK–=9–5@5IJ<46?» (digital–to–analog converters, DACs).

# O45O465=>5 C2H µ9C5C;J87H 9@?H 5@5IJ<46JK =7µ5CJH =9 R23456? <>@9C54 =9 C;>5

=CBO45, ?8AH O9>T@JLµ9 =CJ +T7µ5 1.5.

# & &1

212

f

# & &1

212

f

M9C5C;J8Q5H A/D

E94<µ5CJI78C2H KS5@C4=C7H KAO46J8J42C7H

A@5IJ<46?[+7µ5

123456?[+7µ5

KS5@C4=µQ@J[+7µ5

+7µ5[E456;4CJK[X;?@JL

x%(t) xq(n)x(n)01011...

™¯‹Ì· 1.5

S%801? /µ2µ%/%.,#4 µ./%/"('C%%,%6($01(7–8.–

H3<0%1#


1. T.0$µ%/(63H9% (sampling): (LC7 9>@54 2 O45O465=>5 µ9C5C;J87H 9@?H =7µ5CJH

=L@9TJKH T;?@JL =9 =7µ5 O456;4CJK T;?@JL, 85>;@J@C5H O9><µ5C5 CJL =7µ5CJH

=L@9TJKH T;?@JL =9 O456;4CQH =C4<µQH CJL T;?@JL. PC=4, 5@ x%(t) 9>@54 2 9>=JOJH

=CJ O94<µ5CJI78C2, C?C9 2 Q:JOJH 5LCJK 9>@54 x%(nT)'x(n), ?8JL ; 2 89;>JOJH

O94<µ5CJI2R>5H.

2. =I?,/083 (quantisation): *;?694C54 <45 C2 O45O465=>5 µ9C5C;J87H 9@?H =7µ5CJH

O456;4CJK T;?@JL =L@9TM@ C4µM@ =9 =7µ5 O456;4CJK T;?@JL O456;4CM@ C4µM@

(R23456?). )J 6BG9 O9><µ5 CJL =7µ5CJH 5@C48;J=A89K9C54 58? µ>5 C4µ7 2 J8J>5

984IQ<9C54 58? Q@5 8989;5=µQ@J =K@JIJ 84G5@M@ C4µM@. # O453J;B µ9C5:K CJL

5;T46JK µ2 6S5@C4=µQ@JL O9><µ5CJH x(n) 654 C2H 6S5@C4=µQ@2H 9:?OJL xq(n) 58J-

C9I9> CJ I9<?µ9@J =3BIµ5 6SB@C4=2H.

3. =@*01('(9383 (coding): &5CB C2 O45O465=>5 C2H 6AO46J8J>2=2H, 6BG9 O456;4C7C4µ7 xq(n) 5@C48;J=A89K9C54 58? Q@5@ 5;4Gµ? 58JC9IJKµ9@J 58? b–bits.

(H 9:9CB=JLµ9 :9TA;4=CB 65GQ@5 58? 5LCB C5 C;>5 =CBO45:

¢!#°ª"&$§%æ#" "¡"§$°#'ø¡ ™%ª"&ø¡

# O94<µ5CJI2R>5 9@?H 5@5IJ<46JK =7µ5CJH x%(t) 984CL<TB@9C54 85>;@J@C5H O9><µ5-C5 5LCJK 5@B ; O9LC9;?I98C5, ?8AH 35>@9C54 =CJ +T7µ5 1.6. # O45O465=>5 5LC789;4<;B39C54 58? C2 =TQ=2:

x(n) = x%(nT), –" < n < " (1.9)

?8JL x(n) 9>@54 CJ =7µ5 O456;4CJK T;?@JL 8JL 8;J6K8C94. )J T;J@46? O4B=C2µ5 ;

µ9C5:K CA@ O45OJT46M@ O94<µBCA@ J@JµBZ9C54 '."9(*(4 *.0$µ%/(63H9%4 654 CJ 5@C>-

=C;J3J CJL = Fs 58JC9I9> CJ ")5µ# *.0$µ%/(63H9%4 (sampling rate) =9 O9><µ5C5

5@B O9LC9;?I98CJ 7 5II4MH C2 8)E,#/3/% *.0$µ%/(63H9%4 (sampling frequency) =9 Hz.

'4 µ9C5SI2CQH T;?@JL t 654 n <45 C5 =7µ5C5 =L@9TJKH T;?@JL 654 O456;4CJK T;?@JL

5@C>=CJ4T5, =L@OQJ@C54 <;5µµ46B µQ=A C2H 89;4?OJL O94<µ5CJI2R>5H ; 7 4=JOK@5-

µ5 µQ=A CJL ;LGµJK O94<µ5CJI2R>5H Fs = AH 9:7H:

(1.10)

!8JµQ@AH, 5@5µQ@JLµ9 @5 L8B;T94 6B8J45 =TQ=2 8JL @5 =L@OQ94 C2 =LT@?C2C5 F (7Q) CA@ 5@5IJ<46M@ =2µBCA@ µ9 C2 =LT@?C2C5 f (7 @) CA@ =2µBCA@ O456;4CJK T;?-@JL. 045 @5 S;JKµ9 5LC7 C2 =TQ=2, :964@JKµ9 58? C2@ (1.9) 654 5@C465G4=CJKµ9 CJ

t nT

nFs

= =

1T

1T

2 11 . 2 T À ¶ $ # ™ % ª "& ø ¡


2 2 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

x%(t) µ9 C2 =L@B;C2=2 CJL 2µ4CJ@J94OJKH =7µ5CJH C2H 9:>=A=2H (1.2). PC=4 QTJLµ9:

x(n) = x%(n;) = >cos(2'FnT + 5) = >cos(2'n + 5) (1.11)

+L<6;>@J@C5H C2@ (1.11) µ9 C2@ 5@C>=CJ4T2 =TQ=2 (1.6) <45 CJ 2µ4CJ@J94OQH =7µ5

O456;4CJK T;?@JL, O4584=CM@JLµ9 ?C4:

f = (1.12)

7 4=JOK@5µ5:

@ = Q; (1.13)

(8? C2 =TQ=2 (1.12) 85;5C2;JKµ9 ?C4 2 =LT@?C2C5 f 9>@54 µ>5 1%,(,01('(03µC,3 7

8E./012 8)E,#/3/% (normalized or relative frequency). &5CB =L@Q8945, $0% ,% '"(8-

*0("98()µ. /3, F Hz, #/%, µ%4 *9,./%0 3 f, '"C'.0 %'%"%9/3/% ,% $,@"9L()µ. /3 8)E,#-

/3/% *.0$µ%/(63H9%4 Fs.

!>O5µ9 =CJ 8;J2<JKµ9@J *5;BO94<µ5 1.2, ?C4 2 89;4JT7 =LT@JC7CA@ F 7 Q CA@ 2µ4-

CJ@J94OM@ =L@9TJKH T;?@JL 9>@54:

–" < F < " 7 –" < Q < " (1.14)

045 C5 2µ4CJ@J94O7 O456;4CJK T;?@JL 9>O5µ9 ?C4 µ?@J J4 =LT@?C2C9H f 7 @ 8JL S;>-

=6J@C54 =CJ O4B=C2µ5:

7 –' ] @ ] ' (1.15) # & &1

212

f

FFs

FFs

A@5IJ<46?[+7µ5

E94<µ5CJI78C2H

+7µ5[E456;4CJK[X;?@JL

x%(t)

x%(t) x%(t)

x(n)=x%(nT)

x(n)=x%(nT)

0 0 1 2 3 4 5 6 7 8 9t n

x(n)

Fs=1/T

T 2T 5T 9T t=nT… … …

™¯‹Ì· 1.6

Pµ(0#µ("<3 *.0$-µ%/(63H9% %,%6(-

$01(7 82µ%/(4.


9>@54 O453J;9C46QH. (@C465G4=CM@C5H =C2@ C9I9LC5>5 =TQ=2 C4H µ9C5SI2CQH f 654 @µ9 C4H 4=JOK@5µQH CJLH 58? C4H =TQ=94H (1.12) 654 (1.13), S;>=6JLµ9 ?C4 2 =LT@?C2-

C5 CJL 2µ4CJ@J94OJKH =L@9TJKH T;?@JL, ?C5@ 85>;@JLµ9 O9><µ5CB CJL µ9 ;LGµ?

Fs = 1/T, G5 8;Q894 @5 S;>=69C54 =C2@ 89;4JT7:

(1.16)

7 (1.17)

*5;5C2;JKµ9, 98JµQ@AH, ?C4 2 S5=467 O453J;B µ9C5:K CA@ =2µBCA@ =L@9TJKH T;?-

@JL 654 O456;4CJK T;?@JL S;>=69C54 =C2@ 89;4JT7 C4µM@ CA@ µ9C5SI2CM@ =LT@?C2-

C5H F 654 f 7 Q 654 @. # 89;4JO467 O94<µ5CJI2R>5 9@?H =7µ5CJH =L@9TJKH T;?@JL

JO2<9> =C2@ 58946?@4=2 C2H 589>;JL 9K;JLH 89;4JT7H =LT@JC7CA@ F (7 Q), =C2@

8989;5=µQ@JL 9K;JLH 89;4JT7 =LT@JC7CA@ f (7 @). &54 53JK 2 µQ<4=C2 =LT@?C2C5

=’ Q@5 =7µ5 O456;4CJK T;?@JL 9>@54 7 @ = ', =L@98B<9C54 ?C4 <45 Q@5 ;LGµ?

O94<µ5CJI2R>5H Fs, 2 5@C>=CJ4T2 µQ<4=C2 C4µ7 C2H F 7 Q G5 4=JKC54 µ9:

7 (1.18)

,9 BII5 I?<45, 2 O94<µ5CJI2R>5 94=B<94 5=B3945, 53JK 2 µQ<4=C2 =LT@?C2C5 9@?H

=7µ5CJH =L@9TJKH T;?@JL, 2 J8J>5 µ8J;9> @5 5@585;5=C5G9> =A=CB, 9>@54 Fmax = Fs/2,

?C5@ I5µSB@J@C54 O9><µ5C5 CJL =7µ5CJH 5LCJK µ9 ;LGµ? Fs = 1/T. *;4@ ?µAH 8;J-

TA;7=JLµ9 =9 6B8J45 85;5O9><µ5C5 8JL G5 µ5H O9>:JL@ C4 =LµS5>@94 ?C5@ J4 =LT@?-

C2C9H CJL 5@5IJ<46JK =7µ5CJH 9>@54 µ9<5IKC9;9H 58? Fs/2, 5H OJKµ9 CJ G9M;2µ5 C2H

O94<µ5CJI2R>5H, CJ J8J>J 585@CB =CJ 9:7H 9;MC2µ5: *J4oH J ;LGµ?H O94<µ5CJI2R>5H

Fs <45 C2 =A=C7 5@585;B=C5=2 9@?H 5@5IJ<46JK =7µ5CJH, CJ J8J>J µ5H O>@9C54; E2I5-

O7, 8?=J =LT@B 8;Q894 @5 85>;@JLµ9 O9><µ5C5 M=C9 @5 QTJLµ9 Q@5 84=C? 5@C><;53J

CJL 5@5IJ<46JK =7µ5CJH; # 58B@C2=2 =’ 5LC? CJ 9;MC2µ5 O?G269 5;T46B 58? CJ@

Nyquist (1928) 654 =C2 =L@QT945 58? CJ@ Shannon (1949) 654 58JC9I9> CJ I9<?µ9@J

,-.*"µ# /-&0µ#%12"34#5 7 ,-.*"µ# %16 Shannon– O45CL8M@9C54 O9 AH 9:7H:

O 8)E,#/3/% Fs, µ. /3, ('(9% 6%µI?,(,/%0 /% *.9$µ%/% .,#4 82µ%/(4, '"C'.0 ,%.9,%0 /()6?E08/(, *0'6?80% %'# /3, )H36#/."3 8)E,#/3/% Fmax '() '."0CE./%0 8/(82µ%, O2I5O7

Fs ( 2Fmax (1.19)

,9 BII5 I?<45, CJ G9M;2µ5 O94<µ5CJI2R>5H µ5H IQ94 8AH <45 @5 µ2 T5G9> 8I2;J-3J;>5 G5 8;Q894 @5 85>;@JLµ9 CJLIBT4=CJ@ OKJ O9><µ5C5 5@B 89;>JOJ (C2H LR2I?-

$max = =) )

FTs

F

FT

smax = =

21

2

f = 1

2

# = # & & =) ) ) )

TF F

Ts s$

# = # & & =1

2 2 21

2TF

FF

Ts s

2 31 . 2 T À ¶ $ # ™ % ª "& ø ¡


2 4 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

C9;2H =LT@?C2C5H CJL =7µ5CJH). 045 85;BO94<µ5, 5@ G9I7=JLµ9 @5 R234J8J47=JL-µ9 Q@5 =7µ5 Jµ4I>5H 654 T;2=4µJ8J4JKµ9 µ46;?3A@J CJ J8J>J I94CJL;<9> <45 =LT@?-C2C9H µ9C5:K 300 Hz 654 3 kHz, C?C9 2 µ46;?C9;2 =LT@?C2C5 O94<µ5CJI2R>5H 8JL8;Q894 @5 T;2=4µJ8J47=JLµ9 9>@54 6 kHz. (H OJKµ9 CM;5 C4 G5 =LµS9> 5@ CJ G9M-;2µ5 O94<µ5CJI2R>5H O9 <>@94 =9S5=C?.

¶·Ú¿‰ÂÈÁÌ· 1.3

E>@J@C54 C5 5@5IJ<46B =7µ5C5 654 .

*J45 C5 =7µ5C5 O456;4CJK T;?@JL 8JL G5 8;J6KRJL@ µ9CB C2 O94<µ5CJI2R>5 5LCM@

µ9 ;LGµ? Fs = 1 Hz;

=G/0:

)5 5@C>=CJ4T5 =7µ5C5 O456;4CJK T;?@JL (56JIJLG>9H) 9>@54:

*5;5C2;JKµ9, 98JµQ@AH, ?C4 C5 OKJ 2µ4CJ@J94O7 =7µ5C5 O9@ :9TA;>ZJL@ µ9CB C2

O94<µ5CJI2R>5 CJLH µ9 ;LGµ? 1 Hz (+T7µ5 1.7).

x n nT n n n

n n n x n

2

1

278

278

74

84 4

24 4

( ) sin sin sin sin

sin sin ( )

= #*+,

-./

= #*+,

-./

= # = # +*+,

-./

=

= # +*+,

-./

= =

) ) ) ) )

) ) )

x n nT n n1 2

18

218 4

( ) sin sin sin= *+,

-./

= *+,

-./

=) ) )

x t t2 2

78

( ) sin= #*+,

-./

) x t t1 2

18

( ) sin= *+,

-./

)

*IBCJH

X;?@JH0sec1 2 3 4 5 6 7 8

F2= Hz_18

F1= Hz_78

_

Fs=1 Hz

™¯‹Ì· 1.7

T.0$µ%/(63H9% %,%6($01+, 83µ?-/@, 8)E,#/3/%4

F1 = Hz,

F2 = Hz

µ. ")5µ# Fs = 1 Hz # 7

8

18


(@ µ5H OM=JL@ µ?@J C5 O9><µ5C5 (6JL6>O9H CJL +T7µ5CJH 1.7) C5 J8J>5 5@C4=CJ4-

TJK@ =C4H C4µQH CJL , L8B;T94 5=B3945 =CJ @5 8JKµ9 5@ 5LCB C5 O9><µ5C5

5@C4=CJ4TJK@ =9 C4µQH CJL 5@5IJ<46JK =7µ5CJH x1(t) 7 x2(t). !894O7 CJ x2(t) O>@94 56;4-

SMH C5 >O45 O9><µ5C5 µ9 CJ x1(t), <45 O94<µ5CJI2R>5 µ9 ;LGµ? Fs = 1 O9><µ5CJH 5@B

O9LC9;?I98CJ (Fs = 1 Hz), IQµ9 ?C4 2 =LT@?C2C5 F2 = Hz 9>@54 Q@5 H.)*C4 %,/9-

$"%<( (alias) C2H =LT@?C2C5H F1 = Hz <45 CJ ;LGµ? O94<µ5CJI2R>5H CJL 1 Hz.

,BI4=C5, 9>@54 =2µ5@C46? @5 OJKµ9 ?C4 2 F2 O9@ 58JC9I9> CJ µJ@5O46? R9LOQH 5@C>-

<;53J C2H F1. *;B<µ5C4, <45 CJ ;LGµ? O94<µ5CJI2R>5H Fs = 1 Hz, 2 =LT@?C2C5

F3 = Hz 9>@54 98>=2H R9LOQH 5@C><;53J C2H F1, ?8AH 654 2 =LT@?C2C5 F4 =

Hz, 654 <9@46B ?I9H J4 =LT@?C2C9H F1 + k1 7 <9@46?C9;5 F1 + kFs, ?8JL k = 1,2,…

+L@98MH, ?I9H 5LCQH J4 =LT@?C2C9H 9>@54 R9LO7 5@C><;535 (aliases) C2H =LT@?C2C5H

F1 = Hz.

09@46B, 2 O94<µ5CJI2R>5 CJL 2µ4CJ@J94OJKH =7µ5CJH =L@9TJKH T;?@JL

x%(t) = >cos(2'F0t + 5) (1.20)

µ9 ;LGµ? O94<µ5CJI2R>5H Fs = 1/T, µ5H O>@94 CJ O456;4CJK T;?@JL =7µ5:

x(n) = >cos(2'f0n + 5) (1.21)

?8JL f0 = F0/Fs 9>@54 2 =T9C467 =LT@?C2C5 CJL 2µ4CJ@J94OJKH. (@ G9A;7=JLµ9 ?C4

, 2 =LT@?C2C5 f0 CJL x(n) S;>=69C54 =C2@ 89;4JT7 , 2

J8J>5 5@C48;J=A89K94 C2 S5=467 89;4JT7 =LT@JC7CA@ CA@ =2µBCA@ O456;4CJK T;?-

@JL. +C2@ 89;>8CA=2 5LC7, 2 58946?@4=2 C2H F0 =C2@ f0 9>@54 Q@5–8;JH–Q@5, 64 QC=4

9>@54 OL@5C7 2 9K;9=2 (5@565C5=69L7) CJL 5@5IJ<46JK =7µ5CJH x%(t) 58? C5 O9><-

µ5C5 x(n).

(8? C2@ BII2 8I9L;B, 9B@ C5 5@5IJ<46B 2µ4CJ@J94O7:

x%(t) = >cos(2'Fkt + 5) (1.22)

?8JL

Fk = F0 + kFs, k = ± 1, ± 2, … (1.23)

# & &1

2120f

# & &F

FFs s

2 20

18

178

98

18

# 7

8

sin

)4

n

2 51 . 2 T À ¶ $ # ™ % ª "& ø ¡


2 6 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

L8J=CJK@ O94<µ5CJI2R>5 µ9 ;LGµ? Fs, 9>@54 35@9;? ?C4 2 =LT@?C2C5 Fk S;>=69C54

96C?H C2H S5=467H 89;4JT7H =LT@JC7CA@ [1]. +L@98MH CJ =7µ5 8JL

8;J7IG9 58? O94<µ5CJI2R>5 4=JKC54 µ9:

(1.24)

CJ J8J>J =Lµ8>8C94 µ9 CJ O456;4CJK T;?@JL =7µ5 C2H =TQ=2H (1.21) 8JL 8;J7IG9 58?

C2@ O94<µ5CJI2R>5 CJL 5@5IJ<46JK =7µ5CJH (1.20). !8JµQ@AH, Q@5H B894;JH 5;4Gµ?H

2µ4CJ@J94OM@ =L@9TJKH T;?@JL 5@C48;J=A89K9C54 58? CJ >O4J =K@JIJ O94<µBCA@

(+T7µ5 1.8). 045 85;BO94<µ5 J4 =LT@?C2C9H (Fk) 2,5 kHz, 5,5 kHz, 8,5 kHz, …, 30,5

kHz, … O9@ :9TA;>ZJL@ 58? C2 =LT@?C2C5 (F0) 500Hz <45 =LT@?C2C5 O94<µ5CJI2-

R>5H (Fs) >=2 µ9 3 kHz.

x n x nT AF kF

Fn 5

A nFF

kn

A f n 5

s

s

s

( ) ( ) cos

cos

cos( )

' = + +*+,

-./

=

= + +*+,

-./

=

= +

5

2

2 2

2

0

0

0

)

) )

)

5

# & &F

FFs s

2 2

[1] # =LT@?C2C5 Fs/2 (7 5@C>=CJ4T5 @ = ') J@JµBZ9C54 8)E,#/3/% %,%*9'6@834 (foldingfrequency).

™¯‹Ì· 1.8

T0%<("./01? 3µ0-/(,01? 82µ%/%

%,/0'"(8@'.7(-,/%0 %'# /% 9*0%

*.9$µ%/%

Y;5, 5@ O>@9C54 CJ =K@JIJ CA@ O94<µBCA@ x(n), L8B;T94 5=B3945 AH 8;JH CJ 8J4o

5@5IJ<46? =7µ5 x%(t) 5@C48;J=A89KJL@ 5LCB C5 O9><µ5C5. ,9 BII5 I?<45, µ8J;JK-

µ9 @5 8JKµ9 ?C4 J4 =LT@?C2C9H Fk = F0 + kFs, k = ± 1, ± 2, … O9@ µ8J;JK@ @5 O45-

6;4GJK@ 58? C2@ =LT@?C2C5 F0 µ9CB C2 O94<µ5CJI2R>5 654 =L@98MH ?I9H 5LCQH 9>@54

R9LO7 5@C><;535 (aliases) C2H F0. )J 354@?µ9@J 5LC? J@JµBZ9C54 <%0,#µ.,( E%µ3-

6(7 (%,.'%"1(74) ")5µ(7 *.0$µ%/(63H9%4 (aliasing), 7 <%0,#µ.,( <%8µ%/0124 .'0-

1?6)H34 (spectral overlap). )J 354@?µ9@J 5LC? µ5H 9>@54 <@A=C? 58? C4H 64@2µ5CJ-

<;5346QH C54@>9H, =C4H J8J>9H 8JIIQH 3J;QH 85;5C2;JKµ9 CJLH C;JTJKH µ>5H Bµ5:5H

@5 89;4=C;Q3J@C54 5@C>G9C5 8;JH C2@ 65C9KGL@=2 6>@2=2H C2H Bµ5:5H. (LC? J39>-

I9C54 =CJ <9<J@?H ?C4 J ;LGµ?H I7R2H CA@ «65;Q» (;LGµ?H O94<µ5CJI2R>5H) 9>@54

µ46;?C9;JH 58’ ?=J 8;Q894 <45 @5 «8;JIBSJLµ9» C2@ 89;4=C;J37 CA@ C;JTM@.


¶·Ú¿‰ÂÈÁÌ· 1.4E>@9C54 CJ 5@5IJ<46? =7µ5 x%(t) = 2cos100't. (5) $5 8;J=O4J;>=9C9 C2@ 9IBT4=C2

=LT@?C2C5 O94<µ5CJI2R>5H 8JL 5854C9>C54, M=C9 @5 58J3K<JLµ9 CJ 354@?µ9@J C2H

T5µ2IJK ;LGµJK O94<µ5CJI2R>5H (aliasing). (S) D9A;7=C9 ?C4 I5µSB@J@C54 O9><µ5C5

CJL =7µ5CJH µ9 ;LGµ? Fs = 200 Hz. *J4J CJ =7µ5 O456;4CJK T;?@JL CJ J8J>J G5 8;J-

6KR94 µ9CB C2 O94<µ5CJI2R>5; (<) !85@5IBS9C9 CJ 9;MC2µ5 (S) <45 ;LGµ? O94<µ5CJ-

I2R>5H Fs = 75 Hz. (O) 045 Fs = 75 Hz, 8J45 2 =LT@?C2C5 F0, ?8JL ,

CJL 2µ4CJ@J94OJKH CJ J8J>J O>@94 C5 >O45 56;4SMH O9><µ5C5 µ9 969>@5 8JL 87;5µ9

=C2@ 89;>8CA=2 (<);

=G/0

(5) # =LT@?C2C5 CJL 5@5IJ<46JK =7µ5CJH 9>@54 F = 50 Hz. Y;5 2 9IBT4=C2 =LT@?-

C2C5 O94<µ5CJI2R>5H 8JL 5854C9>C54, <45 @5 58J3K<JLµ9 CJ 354@?µ9@J aliasing

G5 8;Q894 @5 9>@54 Fs = 2F = 100 Hz.

(S) \C5@ Fs = 200 Hz, C?C9 CJ =7µ5 O456;4CJK T;?@JL 8JL 85>;@JLµ9 9>@54:

(<) ' ;LGµ?H O94<µ5CJI2R>5H CA@ 75 Hz 9>@54 µ46;?C9;JH 5LCJK 8JL 5854C9>C54 <45

@5 58J3K<JLµ9 CJ 354@?µ9@J aliasing. )J =7µ5 O456;4CJK T;?@JL 8JL 85>;@JL-

µ9 9>@54 =C2@ 89;>8CA=2 5LC7:

# =LT@?C2C5 CJL =7µ5CJH 9>@54 f = 1/3, O2I5O7 F/Fs = 1/3 7 F = 25 Hz 53JK

Fs = 75 Hz.

(O) (8? C2 =TQ=2 (1.23) <@A;>ZJLµ9 C2 =LT@?C2C5 Fk = F (8JL =C2@ 8;J694µQ@2

89;>8CA=2 9>@54 50 Hz), C2 =LT@?C2C5 O94<µ5CJI2R>5H Fs (8JL =C2@ 8;J694µQ-

@2 89;>8CA=2 9>@54 75 Hz) 654 Z2CJKµ9 @5 8;J=O4J;>=JLµ9 C2 =LT@?C2C5 F0 C2H

I%80124 '."0(E24 8)E,(/2/@,. PC=4, QTJLµ9 F0 = Fk–kFs = 50–1!75 = –25 Hz.

(E45IQ:5µ9 k = 1 <45C> µ?@J 5LC7 2 C4µ7 O>@94 ). Y;5, <45 CJ

;LGµ? O94<µ5CJI2R>5H CA@ 75 Hz, 2 =LT@?C2C5 F = 50 Hz 9>@54 Q@5 R9LOQH 5@C>-

<;53J C2H =LT@?C2C5H CA@ 25 Hz.

# & &F

FFs s

2 20

x n n n n n( ) cos cos cos cos= = = #*

+,-./

=2100

752

43

2 223

223

) ) ) ) )

x n n n( ) cos cos= =2

100200

22

) )

# & &F

FFs s

2 20

2 71 . 2 T À ¶ $ # ™ % ª "& ø ¡


2 8 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

(8? 5@5IJ<46? =7µ5 =LT@?C2C5H 100 Hz I5µSB@JLµ9 O9><µ5C5 µ9 =LT@?C2C9H (5)

75 Hz 654 (S) 150 Hz. +9 8J49H =LT@?C2C9H G5 5@C4=CJ4TJK@ C5 O9><µ5C5 8JL G5

8;J6KRJL@;

ÕÛÎËÛË "˘ÙÔ·ÍÈÔÏfiÁËÛË˜ 1.1

D9A;7=C9 C5 5@5IJ<46B =7µ5C5 x1(t) = cos(60't), x2(t) = cos(140't), x3(t) =

cos(260't), C5 J8J>5 L3>=C5@C54 JµJ4?µJ;35 O94<µ5CJI2R>5 µ9 =LT@?C2C5 100 Hz.

*J45 9>@54 C5 9:5<?µ9@5 =7µ5C5 O456;4CJK T;?@JL;


' ;LGµ?H O94<µ5CJI2R>5H CJL =7µ5CJH x(t) = sin('t) + 4sin(3't)cos(2't) 9>@54 3

kHz. +9 8J49H =LT@?C2C9H G5 5@C4=CJ4TJK@ C5 O9><µ5C5 8JL G5 8;J6KRJL@;


*;JC9>@9C9 OKJ =LT@?C2C9H J4 J8J>9H @5 O>@JL@ C5 >O45 O9><µ5C5 µ9 5LCB CJL =7µ5-

CJH x(t) C2H B=62=2H 5LCJ5:4JI?<2=2H 1.3, ?C5@ J ;LGµ?H O94<µ5CJI2R>5H 9>@54

3 kHz.

¢Ú·ÛÙËÚÈfiÙËÙ· 1.1

'µ"¡&#™% ™%ª"&ø¡ ™À¡!Ã$À™–¶§"&$À™

=I?,/083 (quantisation) J@JµBZ9C54 2 O45O465=>5 C2H µ9C5C;J87H 9@?H =7µ5CJH O45-

6;4CJK T;?@JL =L@9TJKH–8IBCJLH =9 R23456? =7µ5, 963;BZJ@C5H C2@ C4µ7 6BG9

O9><µ5CJH AH Q@5 5;4Gµ? µ9 8989;5=µQ@J 8I7GJH R23>A@ (5@C> <45 B894;J 8I7GJH

R23>A@ 8JL 5854C9>C54 <45 6BG9 =L@9TJKH–8IBCJLH C4µ7).

)J =3BIµ5 8JL L894=Q;T9C54 58? C2@ 5@585;B=C5=2 CJL =7µ5CJH =L@9TM@–C4µM@

µ9 Q@5 8989;5=µQ@J 8I7GJH O456;4CM@–C4µM@, J@JµBZ9C54 8<?6µ% 1I?,/0834

(quantisation error) 7 5#")I(4 1I?,/0834 (quantisation noise). (@ x(n) 9>@54 C5 O9><-

µ5C5 94=?OJL =CJ@ 6S5@C4=C7 654 xq(n) 2 56JIJLG>5 CA@ 6S5@C4=µQ@A@ O94<µBCA@

C2H 9:?OJL CJL 6S5@C4=C7, C?C9 CJ =3BIµ5 6SB@C4=2H 9>@54 2 56JIJLG>5 eq(n), 2

J8J>5 J;>Z9C54 AH 2 O453J;B C2H 8;5<µ5C467H C4µ7H 58? C2@ 6S5@C4=µQ@2 C4µ7, O2I5-


O7 eq(n) = x(n)–xq(n). +CJ +T7µ5 1.9 35>@9C54 85;5=C5C46B CJ 58JCQI9=µ5 C2H 6SB-

@C4=2H CA@ O94<µBCA@ 9@?H 5@5IJ<46JK =7µ5CJH, 65GMH 654 CJ =3BIµ5 6SB@C4=2H.

045 CJ@ 89;4J;4=µ? 6BG9 O9><µ5CJH =CJ 984GLµ2C? 8I7GJH R23>A@, T;2=4µJ8J47=5µ9 C2

µQGJOJ C2H 8/"($$)6('(93834 (rounding) 654 ?T4 C2H %'(1('24 (truncation). '4 C4µQH C4H

J8J>9H 984C;Q89C54 @5 85>;@94 Q@5 R23456? =7µ5, 58JC9IJK@ C5 I9<?µ9@5 .'9'.*% 1I?-

,/0834 (quantisation levels), 9@M 2 58?=C5=2 µ9C5:K OKJ O45OJT46M@ 9848QOA@ 6SB@C4-

=2H J@JµBZ9C54 I2µ% 1I?,/0834 (quantisation step) 7 *0%1"0/012 01%,#/3/%7 %,?6)83

(resolution). &5CB C2 =C;J<<LIJ8J>2=2, J 6S5@C4=C7H 58JO>O94 =C2 =L@9T7 C4µ7 x(n) C2@

C4µ7 CJL 8I2=4Q=C9;JL 9848QOJL 6SB@C4=2H. PC=4, CJ =3BIµ5 6SB@C4=2H eq(n) 6Lµ5>@9-

C54 =C2@ 89;4JT7 µ9C5:K –E/2 654 E/2, O2I5O7 –E/2 ] eq(n) ] E/2. )J S7µ5 6SB@C4=2H

E J;>Z9C54 AH E = (xmax–xmin)/(L–1), ?8JL xmax, xmin 9>@54 2 µ9<5IKC9;2 654 µ46;?C9;2

C4µ7 CJL x(n) 5@C>=CJ4T5, 654 L CJ 8I7GJH CA@ 9848QOA@ 6SB@C4=2H. # O453J;B xmax–xmin

58JC9I9> C2 *),%µ012 '."0(E2 (dynamic range) CJL =7µ5CJH. +C2@ 89;>8CA=2 CJL +T7-

µ5CJH 1.9 QTJLµ9 xmax = 7, xmin = 0, L = 8 654 B;5 E = 1. *5;5C2;7=C9 ?C4, 5@ 2 OL@5µ4-

67 89;4JT7 CJL =7µ5CJH 9>@54 65GJ;4=µQ@2, C?C9 5K:2=2 CJL 8I7GJLH CA@ 9848QOA@

6SB@C4=2H L, =L@98B<9C54 µ9>A=2 CJL S7µ5CJH 6SB@C4=2H. !8JµQ@AH, CJ =3BIµ5 6SB-

@C4=2H µ94M@9C54, O2I5O7, 5L:B@94 2 56;>S945 CJL 6S5@C4=C7. )5 98>89O5 6SB@C4=2H L

9>@54 =L@B;C2=2 CJL 8I7GJLH b CA@ OL5O46M@ R23>A@ (bits) C2H IQ:2H 8JL T;2=4µJ-

8J4JKµ9 <45 C2@ 5@585;B=C5=2 6BG9 O9><µ5CJH, ?8AH G5 OJKµ9 =CJ 5µQ=AH 98?µ9@J

9OB34J CJL 6AO46J8J42C7. (8JO946@K9C54 ?C4 <45 2µ4CJ@J94O7 =7µ5C5, J I?<JH CJL =7µ5-

CJH 8;JH CJ G?;LSJ 6SB@C4=2H 5L:B@9C54 65CB 89;>8JL 6dB[2] <45 6BG9 9848IQJ@ bit 8JL

8;J=C>G9C54 =CJ µ76JH IQ:2H, O2I5O7 <45 6BG9 O48I5=45=µ? CA@ 9848QOA@ 6SB@C4=2H.

(8? C5 85;58B@A <>@9C54 35@9;? ?C4 2 6SB@C4=2 5@5IJ<46M@ =2µBCA@ JO2<9>

8B@CJC9 =9 58MI945 8I2;J3J;>5H, 9:54C>5H C2H 5=B3945H 8JL 5LC7 94=B<94. *;B<-

µ5C4, 2 6SB@C4=2 9>@54 µ>5 µ2 5@C4=C;98C7 O45O465=>5, 53JK ?I5 C5 O9><µ5C5 =9 58?-

=C5=2 E/2 <K;A 58? Q@5 98>89OJ 6SB@C4=2H, 5@C48;J=A89KJ@C54 58? C2@ >O45 C4µ7.

+L@98MH, O9@ µ8J;JKµ9 8JCQ @5 9:5I9>RJLµ9 CJ G?;LSJ 6SB@C4=2H, 85;B µ?@J @5

CJ@ µ94M=JLµ9 5L:B@J@C5H C5 98>89O5 6SB@C4=2H L.

2 91 . 2 T À ¶ $ # ™ % ª "& ø ¡

[2] F89@GLµ>Z9C54 ?C4 CJ dB (decibel) J;>Z9C54 AH 10 log10(P2/P1) = 20 log10(V2/V1), ?8JL µ9P, V =LµSJI>ZJLµ9 C2@ 4=TK 654 C2@ CB=2 9@?H =7µ5CJH 5@C>=CJ4T5. )J dB 5@C4=CJ4T9>=CJ 1/10 CJL Bell, µJ@BO5 2 J8J>5 J;>=C269 654 65G49;MG269 58? CJ@ Alexander GrahamBell, J J8J>JH 5@56BILR9 ?C4 CJ 5@G;M84@J 5LC> 58J6;>@9C54 IJ<5;4Gµ46B =C4H O453J-;QH 4=TKJH. PC=4, 3dB =2µ5>@94 ?C4 QTJLµ9 O48I5=45=µ? C2H 4=TKJH (P2 = 2P1), 9@M –3dB=2µ5>@94 ?C4 QTJLµ9 L8JO48I5=45=µ? 5LC7H (P2 = P1/2). (@539;?µ9@J4 =C2@ CB=2, 6dB=2µ5>@94 O48I5=45=µ? C2H CB=2H (V2 = 2V1), 9@M –6dB =2µ5>@94 ?C4 QTJLµ9 L8JO48I5-=45=µ? 5LC7H (V2 = V1/2).


3 0 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

'ø¢#'$¶$#%™% &ø¡ 'µ"¡&#™ª!¡ø¡ ¢!#°ª"&ø¡

&5CB C2 O45O465=>5 C2H 6AO46J8J>2=2H =’ Q@5 µ9C5C;J8Q5 5@5IJ<46JK–=9–R23456?,

Q@5H µJ@5O46?H OL5O46?H 5;4Gµ?H 96TA;9>C54 =9 6BG9 98>89OJ 6SB@C4=2H. (@ QTJLµ9

L 98>89O5 6SB@C4=2H, T;945Z?µ5=C9 CJLIBT4=CJ@ L O453J;9C46JKH OL5O46JKH 5;4GµJKH.

,9 Q@5 µ76JH IQ:2H b bits µ8J;JKµ9 @5 QTJLµ9 2b O453J;9C46JKH OL5O46JKH 5;4GµJKH.

Y;5, 8;Q894 2b ( L 7 4=JOK@5µ5 b ( log2L. +CJ 85;BO94<µ5 CJL +T7µ5CJH 1.9 T;2=4-

µJ8J47=5µ9 Q@5@ 6AO46J8J42C7 µ9 b = 3 bits. +CJ 9µ8?;4J L8B;TJL@ O45GQ=4µJ4 µ9C5-

C;J89>H A/D µ9 56;>S945 µQT;4 654 b = 24 bits. X5;56C2;4=C467 9>@54 2 89;>8CA=2 CA@

µJL=46M@ CDs, ?8JL T;2=4µJ8J4JK@C54 µ9C5C;J89>H A/D 56;4S9>5H 16 bits.

™¯‹Ì· 1.9

=I?,/083 1%01@*01('(9383

82µ%/(4

E8>89O5 6SB@C4=2H

B7µ5 6SB@C4=2HT

xq(n)x(n)

x(t)

n0

1

2

3

4

5

6

7

x(n)

T4µQH O94<µBCA@EL5O46QH C4µQH

4100[

110[

111[

011[

100[

101[

011[

101[

001[

010[

101[

6 7 3 4 5 6 5 1 2 5

+3BIµ5

™‡ÓÔ„Ë ÂÓfiÙËÙ·˜

+C2@ 9@?C2C5 5LC7 <@A;>=5µ9 CJLH O4B3J;JLH CK8JLH =2µBCA@. !>O5µ9 C4H O453J;QH

µ9C5:K CA@ 5@5IJ<46M@ =2µBCA@, CA@ =2µBCA@ O456;4CJK T;?@JL 654 CA@ R23456M@

=2µBCA@, 654 5=TJI2G765µ9 O49:JO46B µ9 C2 µ9C5C;J87 9@?H 5@5IJ<46JK =7µ5CJH =9

R23456?. P<4@9 35@9;? ?C4 2 µ9C5C;J87 5LC7 585;C>Z9C54 58? OKJ 6K;49H, 5IIB 5@9-

:B;C2C9H µ9C5:K CJLH, O45O465=>9H: C2 *.0$µ%/(63H9%, 2 J8J>5 QT94 =TQ=2 µ9 CJ 8?=J

=LT@B 85>;@JLµ9 C5 O9><µ5C5, 654 C2 1I?,/083, 2 J8J>5 QT94 =TQ=2 µ9 C2@ 56;>S945

5@585;B=C5=2H CJL 8IBCJLH 6BG9 O9><µ5CJH. ,9 BII5 I?<45, 2 O94<µ5CJI2R>5 QT94

@5 6B@94 µ9 C2 8)E,#/3/% CJL =7µ5CJH, 9@M 2 6SB@C4=2 µ9 CJ '6?/(4 CJL =7µ5CJH.

• )J G9M;2µ5 O94<µ5CJI2R>5H µ5H IQ94 ?C4 2 =LT@?C2C5 O94<µ5CJI2R>5H 8;Q894

@5 9>@54 CJLIBT4=CJ@ O48IB=45 C2H µQ<4=C2H =LT@?C2C5H 8JL 89;4QT9C54 =CJ =7µ5

58? CJ J8J>J GQIJLµ9 @5 IBSJLµ9 O9><µ5C5. (@ 5LC? O9 =LµS5>@94, C?C9 85;JL-


=4BZ9C54 CJ 354@?µ9@J C2H T5µ2IJK ;LGµJK O94<µ5CJI2R>5H (aliasing).

• # R234J8J>2=2 CJL 8IBCJLH 9@?H =7µ5CJH 94=B<94 Q@5 G?;LSJ, CJ I9<?µ9@J

G?;LSJ 6SB@C4=2H, J J8J>JH, ?=J 89;4==?C9;5 98>89O5 6SB@C4=2H T;2=4µJ8J4-

JKµ9, O2I5O7 ?=J 89;4==?C9;5 bits T;2=4µJ8J4JKµ9 <45 C2@ 5@585;B=C5=2 C2H

6BG9 C4µ7H CJL 8IBCJLH, C?=J µ46;?C9;JH <>@9C54.

1.3 ™‹Ì·Ù· ‰È·ÎÚÈÙÔ‡ ¯ÚfiÓÔ˘

+C2@ 9@?C2C5 5LC7 G5 5=TJI2GJKµ9 µ9 C5 =7µ5C5 O456;4CJK T;?@JL. D5 <@A;>=JLµ9

C5 84J S5=46B =7µ5C5 O456;4CJK T;?@JL, 65GMH 654 C4H =CJ4T94MO94H 8;B:94H 8JL

935;µ?ZJ@C54 =9 CQCJ4JL 9>OJLH =7µ5C5. \I5 5LCB G5 58JC9IQ=JL@ C5 9;<5I9>5 C5

585;5>C2C5 <45 C2 µ9IQC2 CA@ =L=C2µBCA@ 654 C2@ 5@BIL=2 CA@ =2µBCA@ 8JL G5

µ5H 585=TJI7=JL@ =9 ?I2 C2@ Q6C5=2 5LCJK CJL S4SI>JL.

1.3.1 µ·ÛÈÎ¿ Û‹Ì·Ù· ‰È·ÎÚÈÙÔ‡ ¯ÚfiÓÔ˘

)5 =7µ5C5 8JL 89;4<;B3J@C54 =C2 =L@QT945 G9A;JK@C54 AH C5 S5=46B (=CJ4T94MO2)

=7µ5C5 O456;4CJK T;?@JL.

5) M(,%*0%9( *.9$µ% (unit sample) 7 µ(,%*0%9% 1"()8/012 %1(6()59% (unit impulse

sequence): !>@54 CJ 8IQJ@ S5=46? =7µ5 O456;4CJK T;?@JL CJ J8J>J J;>Z9C54 AH:

(1.25)

S) M(,%*0%9% I3µ%/012 %1(6()59% (unit step sequence): ';>Z9C54 AH:

(1.26)

<) &/%5."2 %1(6()59% (constant sequence):

x(n) = >, –" < n < " (1.27)

O) D"%µµ012 %1(6()59% (linear sequence):

x(n) = >n, –" < n < " (1.28)

'4 6Lµ5CJµJ;3QH ?IA@ CA@ 85;58B@A =2µBCA@ 35>@J@C54 =C5 +T7µ5C5 1.10

QAH 654 1.13.

9) :15./012 %1(6()59% (exponential sequence):

x(n) = an, –" < n < " (1.29)

# 56JIJLG>5 5LC7 85;JL=4BZ94 4O45>C9;J 9@O453Q;J@. # µJ;37 C2H 9:5;CBC54 58?

u( )

0, <n

n

n= (0

12

1 0

0

,

* n( )

0, = =

3012

1 0

0

, n

n

3 11 . 3 ™ % ª "&" ¢ # " ' ƒ # & $ À Ã ƒ $ ¡ $ À


3 2 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

C2@ C4µ7 CJL a. PC=4, 5@ a 8;5<µ5C46?H 5;4Gµ?H, C?C9 5LC7 9>@54 3G>@JL=5 <45 |a|

< 1 (+T7µ5 1.145,S) 654 5K:JL=5 <45 |a| > 1 (+T7µ5 1.14<,O). (@ a µ4<5O46?H 5;4G-

µ?H, O2I5O7 a = rej@, C?C9 x(n) = rnej@n 7 x(n) = rn [cos(@n) + j sin(@n)]. 045 r =

1 CJ 8;5<µ5C46? 654 35@C5=C46? µQ;JH 9>@54 5@C>=CJ4T5 µ>5 =L@2µ4CJ@467 654 µ>5

2µ4CJ@467 56JIJLG>5 =C5G9;JK 8IBCJLH C2H µJ;37H CJL +T7µ5CJH 1.155. 045 r <

1 QTJLµ9 3G>@JL=9H 2µ4CJ@46QH 56JIJLG>9H C2H µJ;37H CJL +T7µ5CJH 1.15S 654

<45 r > 1 QTJLµ9 5K:JL=9H 2µ4CJ@46QH 56JIJLG>9H C2H µJ;37H CJL +T7µ5CJH 1.15<.

*(n)

1

-2 -1 1 20n

u(n)

1

–2 –1 0 1 2 3n

x(n)

n

A

-3 -2 -1 0 1 2 3

n

n

n

n

x(n)

–2 –1

0 1 2 3–A

A

2A

3A

–2A

n

™¯‹Ì· 1.13: D"%µµ012 %1(6()59%

™¯‹Ì· 1.12: &/%5."2 %1(6()59%

™¯‹Ì· 1.11: S3µ%/012 %1(6()59%

™¯‹Ì· 1.10: ="()8/012 %1(6()59%

™¯‹Ì· 1.14

:15./012 %1(6()59% x(n) = an $0% a '"%$µ%/01# 1%0 (%) 0< a < 1, (I) –1 < a < 0, ($) a > 1 1%0 (*) a < –1

(5)

(S)

(<)

(O)


1.3.2 ™ÙÔÈ¯ÂÈÒ‰ÂÈ˜ !Ú¿ÍÂÈ˜

$§#™£%™% ™%ª"&ø¡ ¢#"'ƒ#&$À Ãƒ$¡$À

# µ5G2µ5C467 89;4<;537 C2H JI>=G2=2H 654 2 65C5@?2=2 5LC7H 9>@54 65>;45H =2µ5-

=>5H. 045 85;BO94<µ5, 2 JI>=G2=2 C2H µJ@5O45>5H 6;JL=C467H 65CB no µJ@BO9H (O9><-

µ5C5) J;>Z9C54 AH:

(1.30)

+CJ +T7µ5 1.16 35>@J@C54 J4 =L@5;C7=94H 2!*(n–3) 654 2!*(n + 3).

4 ( )

, n n

n n

n no

o

o

# ==3

012

1

0

,

3 31 . 3 ™ % ª "&" ¢ # " ' ƒ # & $ À Ã ƒ $ ¡ $ À

n

n

n

(5)

(S)

(<)

™¯‹Ì· 1.15

D"%<012 %,%'%"?8/%83 /() '"%$µ%/0-1(7 2 <%,/%8/01(7 µC"()4 /34 .15./0-124 %1(6()59%4 x(n) = an $0% a µ0$%*0-1# (a = rej@), #'() (%) r = 1, (I) r < 11%0 ($) r > 1


3 4 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

,9 ?µJ4J C;?8J J;>Z9C54 654 2 JI4=G2µQ@2 65CB no µJ@5O45>5 S2µ5C467 56JIJLG>5:

(1.31)

+CJ +T7µ5 1.17 35>@J@C54 85;5O9><µ5C5 JI>=G2=2H µ>5H S2µ5C467H =L@B;C2=2H

65CB OKJ O9><µ5C5 (no = 2).

u( )

0, < o

n nn n

n no

o# =(0

12

1,

2*(n-3) 2*(n+3)

2 2

–1–2 0 1 2 3 4 –4 –3 –2 –1 0 1 2 3

(5) (S)

n n

™¯‹Ì· 1.16

D"%<01C4 '%"%-8/?8.04 /@, µ(,%-*0%9@, %1(6()-

50+, (%) 2*(n–3) 1%0

(I) 2*(n + 3).

Au(n-2) Au(n+2) Au(-n+2)

A A A

n n n-2 -1 0 0 1 2 31 2 3 4 -4 -3 -2 -1 0 1 2 3 4-4 -3 -2 -1

(5) (S) (<)

™¯‹Ì· 1.17

D"%<01C4 '%"%-8/?8.04 /@, I3µ%-/01+, %1(6()50+,

(%) >u(n–2), (I) >u(n + 2) 1%0

($) >u(–n + 2)

*5;5C2;JKµ9 ?C4 2 µ2 µ2O9@467 C4µ7 µ45H 6;JL=C467H S;>=69C54 969> ?8JL CJ ?;4-

=µ5 C2H *(!) <>@9C54 µ2OQ@. \µJ45, µ>5 S2µ5C467 56JIJLG>5 9>@54 µ2 µ2O9@467 <45

969>@9H C4H C4µQH <45 C4H J8J>9H CJ ?;4=µ5 C2H u(!) 9>@54 µ9<5IKC9;J 7 >=J CJL µ2O9-

@?H. 045 85;BO94<µ5, µ>5 6;JL=C467 56JIJLG>5 µ9 8IBCJH O9><µ5CJH 4 =C2 GQ=2 n =

3, 963;BZ9C54 AH 4*(n–3). ,>5 S2µ5C467 56JIJLG>5 8IBCJLH –2 <45 ?I9H C4H G9C46QH

C4µQH CJL n, 65GMH 654 <45 n = 0, 963;BZ9C54 AH x(n) = –2u(n). # 65CJ8C;467 5LC7H

AH 8;JH CJ@ B:J@5 CA@ =L@C9C5<µQ@A@ 9>@54 2 x(–n) = –2u(–n). (LC7 QT94 8IBCJH

–2 <45 ?I9H C4H 5;@2C46QH C4µQH CJL n, 65GMH 654 <45 n = 0. # JI>=G2=2 5LC7H 65CB

4 GQ=94H 8;JH C5 5;4=C9;B G5 µ5H OM=94 C2@ 56JIJLG>5 x(–n + 4) = –2u(–n + 4).

!>µ5=C9 CM;5 =9 GQ=2 @5 OJKµ9 9K6JI5 ?C4 J4 =TQ=94H 8JL =L@OQJL@ C2@ 6;JL=C4-

67 654 C2 S2µ5C467 56JIJLG>5 9>@54 J4 9:7H:

(1.32) u n m

m

n

( ) ( )==#"5 4


(1.33)

09@46B, 2 56JIJLG>5 x(n–n0) 9>@54 Q@5 5@C><;53J C2H x(n) CJ J8J>J QT94 L8J=C9> JI>-

=G2=2. 045 n0 > 0 QTJLµ9 µ45 O9:4B JI>=G2=2 2 J8J>5 4=JOL@5µ9> µ9 65GL=CQ;2=2

(delay) CJL =7µ5CJH, 9@M <45 n0 < 0 QTJLµ9 µ45 5;4=C9;7 JI>=G2=2 2 J8J>5 4=JOL-

@5µ9> µ9 8;J7<2=2 (advance) CJL =7µ5CJH.

°!¡#'% ¶!ƒ#°ƒ"º% "'$§$À£#"™

'8J4JO78JC9 =7µ5 x(n) µ8J;9> @5 <;539> AH BG;J4=µ5 JI4=G2µQ@A@ 6;JL=C46M@

O94<µBCA@ 8JII58I5=45=µQ@A@ µ9 =L@C9I9=CQH SB;JLH:

(1.34)

045 @5 <>@94 5LC? 65C5@J2C?, 5H OJKµ9 CJ +T7µ5 1.18. # >*(n) S;>=69C54 =C2@ 5;T7

CA@ 5:?@A@, 9@M 2 >*(n–m) S;>=69C54 =CJ =2µ9>J n = m. PC=4 2 56JIJLG>5 x(n), µ9

{x(n)} = {… 0, 0, –2, 2, 3, 2, 0, –1, 2, 0, 0, …}, ?8JL µ9 Q@CJ@2 <;537 654 L8J<;Bµ-

µ4=2 =2µ94M@9C54 2 T;J@467 =C4<µ7 n = 0 (=CJ4T9>J 3), µ8J;9> @5 89;4<;539> AH:

x(n) = … –2*(n + 2) + 2*(n + 1) + 3*(n) + 2*(n–1) – *(n–3) + 2*(n–4) + … =

= … + x(–2)*(n + 2) + x(–1)*(n + 1) + x(0)*(n) + x(1)*(n–1) + x(3)*(n–3) +

x(4)*(n–4) + …

654 <9@46B 8;J6K8C94 2 =TQ=2 (1.34). # =TQ=2 5LC7 9>@54 8JIK S5=467 654 G5 µ5H

SJ2G7=94 =C2@ 65C5@?2=2 C2H =L@QI4:2H (convolution), ?8AH G5 OJKµ9 5@5ILC46B

=C2@ 9@?C2C5 1.4.

x n x m n m

m

( ) ( ) ( )= #=#"

+"

5 *

4 ( ) ( ) ( )n u n u n= # #1

3 51 . 3 ™ % ª "&" ¢ # " ' ƒ # & $ À Ã ƒ $ ¡ $ À

A*(n) A*(n-m)

A A

0 0 mn n n

x(n)

-4 -3 -1

-1

0 1 2 4 5 6-2 3

3

2

2 2 2

(5) (S) (<)

™¯‹Ì· 1.18

O µ(,%*0%9%1"()8/012 8/3,'."0$"%<2 ('(0()-*2'(/. 82µ%/(4*0%1"0/(7 E"#,()x(n):(%) >*(n), (I)>*(n–m), ($) x(n)


3 6 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™


+C2@ 9@?C2C5 5LC7 J;>=5µ9 ?I9H C4H S5=46QH 56JIJLG>9H (6;JL=C467, S2µ5C467,

96G9C467) 654 <@A;>=5µ9 C4H =CJ4T94MO94H 8;B:94H 8JL µ8J;JKµ9 @5 QTJLµ9 =’

5LCQH. +C2 =L@QT945, 9>O5µ9 ?C4 J8J4JO78JC9 =7µ5 O456;4CJK T;?@JL µ8J;9> @5

963;5=C9> AH <;5µµ46?H =L@OL5=µ?H µJ@5O45>A@ 6;JL=C46M@.

1.4 ™˘ÛÙ‹Ì·Ù· ‰È·ÎÚÈÙÔ‡ ¯ÚfiÓÔ˘

P@5 =K=C2µ5 O456;4CJK T;?@JL 9>@54 969>@o 8JL OQT9C54 µ>5 9>=JOJ O456;4CJK T;?-

@JL x(n) 654 85;B<94 µ>5 Q:JOJ 98>=2H O456;4CJK T;?@JL y(n) (+T7µ5 1.20). )5

=L=C7µ5C5 µ9 C5 J8J>5 G5 5=TJI2GJKµ9 =CJ S4SI>J 5LC? QTJL@ OKJ S5=46B T5;5-

6C2;4=C46B. !>@54 $"%µµ01? (linear) 654 E"(,01? %µ./?I63/% (time–invariant). D5

5@539;?µ5=C9 =’ 5LCB µ9 CJ@ 5<<I46? ?;J LTI (Linear Time–Invariant) <45 I?<JLH

=LµS5C?C2C5H µ9 C2 O49G@7 S4SI4J<;53>5 654 9L6JI>5H CJL =8JLO5=C7.

7*#µµ&'8 J@JµBZ9C54 Q@5 =K=C2µ5 =CJ J8J>J 4=TK94 2 #*9( %"5 6:;*,-+"5. +L<69-

6;4µQ@5, 9B@ 2 9>=JOJH CJL =L=C7µ5CJH, CJ J8J>J 5;T46B S;4=6?C5@ =9 2;9µ>5[3],

$5 963;B=9C9 CJ@ 85Iµ? O456;4CJK T;?@JL p(n) CJL +T7µ5CJH 1.195 AH =L@OL5-

=µ? S2µ5C46M@ 56JIJLG4M@.

™¯‹Ì· 1.19

&2µ%/% *0%1"0/(7 E"#,().


p(n) g(n) x(n)

1

–2–2

–2–1–5 –4 –3 –1–2

–1 0 1 2 3 4n n n0

0 1 2 3

4

41 2 3 4 5

(5) (S) (<)

$5 S;9>C9 C4H 963;B=94H <45 C5 =7µ5C5 g(n), x(n) CA@ +T2µBCA@ 1.19S 654 1.19<.


[3] (;T467 2;9µ>5 =2µ5>@94 ?C4 =CJ =K=C2µ5 O9@ QT94 935;µJ=C9> 65µ>5 O4Q<9;=2 (9>=JOJH)8;4@ 58? C2 T;J@467 =C4<µ7 n = n0, 65CB C2@ J8J>5 935;µ?=C269 2 9>=JOJH x(n).


58JC9I9>C54 58? Q@5 <;5µµ46? =L@OL5=µ? =2µBCA@, C?C9 2 Q:JOJH CJL =L=C7µ5CJH

(58?6;4=2) G5 4=JKC54 µ9 CJ <;5µµ46? =L@OL5=µ? CA@ 58J6;>=9A@ CA@ 984µQ;JLH

=2µBCA@, =5@ 5LCB @5 9>T5@ 935;µJ=G9> CJ 65GQ@5 TA;4=CB. ,5G2µ5C46B 5LC?

963;BZ9C54 AH 9:7H: 5@ y1(n) 9>@54 2 58?6;4=2 CJL =L=C7µ5CJH =C2@ 9>=JOJ x1(n)

654 y2(n) 9>@54 2 58?6;4=2 5LCJK =C2@ 9>=JOJ x2(n), C?C9 2 58?6;4=2 CJL =L=C7µ5-

CJH =C2@ 9>=JOJ ax1(n) + bx2(n) G5 9>@54 ay1(n) + by2(n), ?8JL a, b =C5G9;QH.

(H 9:9CB=JLµ9 C4H 89;48CM=94H 9@?H <;5µµ46JK 654 9@?H µ2 <;5µµ46JK =L=C7µ5CJH.

P@5 85;BO94<µ5 <;5µµ46JK =L=C7µ5CJH 9>@54 5LC? CJL J8J>JL 2 Q:JOJH 4=JKC54 µ9

y(n) = x(n)–x(n–1). 045 9>=JOJ x1(n), 2 Q:JOJH CJL =L=C7µ5CJH G5 9>@54 y1(n) =

x1(n)–x1(n–1). 045 9>=JOJ x2(n) 2 Q:JOJH CJL =L=C7µ5CJH G5 9>@54 y2(n) =

x2(n)–x2(n–1). (@ CM;5 935;µ?=JLµ9 AH 9>=JOJ x3(n) CJ <;5µµ46? =L@OL5=µ? CA@

OKJ 8;J2<JLµQ@A@ 56JIJLG4M@ 94=?OJL, O2I5O7 x3(n) = a x1(n) + b x2(n), 2 Q:JOJH

y3(n) CJL =L=C7µ5CJH G5 4=JKC54 µ9:

y3(n) = x3(n)–x3(n–1) = [a x1(n) + b x2(n)] – [a x1(n–1) + b x2(n–1)]

= a[x1(n) – x1(n–1)] + b[x2(n) – x2(n–1)] = a y1(n) + b y2(n)

B;5, 4=TK94 2 5;T7 C2H L8Q;G9=2H.

P@5 85;BO94<µ5 µ2 <;5µµ46JK =L=C7µ5CJH 9>@54 969>@J CJ J8J>J 85;B<94 =C2@ Q:JO?

CJL CJ C9C;B<A@J C2H 94=?OJL, O2I5O7 y(n) = [x(n)]2. 045 9>=JOJ x1(n) 2 Q:JOJH CJL

=L=C7µ5CJH G5 9>@54 y1(n) = [x1(n)]2. 045 9>=JOJ x2(n) 2 Q:JOJH CJL =L=C7µ5CJH G5

9>@54 y2(n) = [x2(n)]2. (@ CM;5 935;µJ=C9> =C2@ 9>=JOJ CJ =7µ5 x3(n) = ax1(n) +

bx2(n) 2 Q:JOJH G5 9>@54:

<*1$&'= #µ-%=>2"%1 J@JµBZ9C54 Q@5 =K=C2µ5 CJL J8J>JL 2 =Lµ89;43J;B 654 J4

4O4?C2C9H O9@ 5IIBZJL@ µ9 CJ T;?@J. (LC? =2µ5>@94 ?C4 µ45 T;J@467 JI>=G2=2 C2H

94=?OJL G5 5@C4=CJ4T9> =9 T;J@467 JI>=G2=2 C2H 9:?OJL. ,9 BII5 I?<45, 9B@ y(n)

9>@54 2 Q:JOJH 9@?H T;J@46B 5µ9CBSI2CJL =L=C7µ5CJH <45 9>=JOJ x(n), C?C9 y(n–n0)

G5 9>@54 2 Q:JOJH 5LCJK <45 9>=JOJ x(n–n0).

?6+%#,;5 (stable) J@JµBZ9C54 Q@5 =K=C2µ5 .?, 1%0 µ#,( .?, 1?5. <"%$µC,3 .98(*(4

y n abx n x n

a y n b y n abx n x n ay n by n

x n ax n bx n ax n bx n32 2 2 2

1 2

21

22 1 2 1 2

3 1 2 1 2 2

2

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )= [ ] = [ ] = [ ] + [ ] + =

= + + 3 +

+

3 71 . 4 ™ À ™ & % ª "&" ¢ # " ' ƒ # & $ À Ã ƒ $ ¡ $ À

™¯‹Ì· 1.20

D.,01# *0?$"%µµ%8)8/2µ%/(4 *0%-1"0/(7 E"#,()

x(n) y(n)+K=C2µ5 E456;4CJK X;?@JL


3 8 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

'%"?$.0 µ0% <"%$µC,3 CF(*( (Bounded Input Bounded Output, U%U'). ,9 BII5 I?<45,

Q@5 CQCJ4J =K=C2µ5 µ5H 9:5=35I>Z94 ?C4 ?=J 2 9>=JOJH 85;5µQ@94 3;5<µQ@2 (|x(n)| ] Mx

< "), 2 Q:JOJH O9 G5 5894;>Z9C54 (|y(n)| ] My < ") <45 ?I5 C5 n, ?8JL Mx, My 8989;5-

=µQ@J4 5;4GµJ>. +9 O453J;9C467 89;>8CA=2 CJ =K=C2µ5 J@JµBZ9C54 5=C5GQH (unstable).

@&%&#%8 (causal) =K=C2µ5 9>@54 969>@J CJL J8J>JL 2 Q:JOJH, =9 6BG9 T;J@467 =C4<-

µ7, 9:5;CBC54 µ?@J 58? C4H C4µQH CJL =7µ5CJH 94=?OJL =C2@ C;QTJL=5 T;J@467 =C4<-

µ7 654 =9 8;J2<JKµ9@9H T;J@46QH =C4<µQH. ,9 BII5 I?<45, J4 µ9C5SJIQH =C2@ Q:JOJ

9@?H CQCJ4JL =L=C7µ5CJH 9>@54 58JCQI9=µ5 CA@ µ9C5SJIM@ C2H 94=?OJL.

+C2 =L@QT945, G5 9:9CB=JLµ9 C2 =2µ5=>5 C2H 6;JL=C467H 58?6;4=2H µJ@JO4B=C5-

CJL =L=C7µ5CJH O456;4CJK T;?@JL 654 G5 OJKµ9 ?C4 µ9 C2 SJ7G94B C2H µ8J;JKµ9,

µQ=A C2H 8;B:2H C2H =L@QI4:2H, @5 L8JIJ<>=JLµ9 C2@ Q:JOJ 9@?H <;5µµ46JK =L=C7-

µ5CJH O456;4CJK T;?@JL <45 J8J45O78JC9 9>=JOJ.

¶·Ú¿‰ÂÈÁÌ· 1.5

$5 T5;56C2;>=9C9 C5 =L=C7µ5C5 8JL 89;4<;B3J@C54 58? C4H 98?µ9@9H =TQ=94H 94=?-

OJL–9:?OJL, AH 8;JH C4H 4O4?C2C9H C2H <;5µµ46?C2C5H, C2H 9L=CBG945H,C2H T;J@467H

µ9C5SI2C?C2C5H 654 C2H 54C45C?C2C5H.

5. y(n) = 3x(n)–2x(n–1) <. y(n) = nx(n–2)–x(n + 3)

S. y(n) = x(n) + 2y(n–1) O. y(n) = cos[x(n)]

=G/0:

)5 OKJ 8;MC5 =L=C7µ5C5 9>@54 <;5µµ46B, 53JK 2 Q:JOJH L8JIJ<>Z9C54 AH <;5µµ4-

6?H =L@OL5=µ?H O94<µBCA@ C2H 94=?OJL 654 8;J2<JKµ9@A@ C4µM@ C2H 9:?OJL. )J

C;>CJ =K=C2µ5 9>@54 98>=2H <;5µµ46?, 53JK 4=TK94 2 5;T7 C2H L8Q;G9=2H. )J CQC5;-

CJ 9>@54 µ2 <;5µµ46? =K=C2µ5.

NH 8;JH C2@ 9L=CBG945, CJ O9KC9;J =K=C2µ5 O9@ 9>@54 9L=C5GQH. 045 @5 <>@94 5LC?

65C5@J2C?, 5H G9A;7=JLµ9 ?C4 =C2@ 9>=JOJ CJL =L=C7µ5CJH 935;µ?Z9C54 2 3;5<-

µQ@2 56JIJLG>5 x(n) = C*(n), ?8JL C =C5G9;B. D9A;JKµ9 98>=2H ?C4 CJ =K=C2µ5 S;>-

=69C54 =9 5;T467 2;9µ>5, J8?C9 y(–1) = 0. # 56JIJLG>5 9:?OJL 8JL 85;B<9C54 9>@54:

y(0) = C!*(0) + 2!y(–1) = C!1 + 2!0 = C

y(1) = C!*(1) + 2!y(0) = C!0 + 2!C = 2C

y(2) = C!*(2) + 2!y(1) = C!0 + 2!2C = 22C

…

y(n) = C!*(n) + 2!y(n–1) = C!0 + 2!2n–1C = 2nC


!8JµQ@AH, <>@9C54 35@9;?, ?C4 2 Q:JOJH 9>@54 µ2 3;5<µQ@2 654 CJ =K=C2µ5 9>@54 BIBO

5=C5GQH, 53JK µ>5 3;5<µQ@2 9>=JOJH QT94 AH 58JCQI9=µ5 µ45 µ2 3;5<µQ@2 Q:JOJ.

,9C5SI2C? µ9 CJ T;?@J 9>@54 CJ C;>CJ =K=C2µ5 53JK J =L@C9I9=C7H n O9@ 9>@54 =C5-

G9;?H 5IIB µ9C5SBII9C54 O45;6MH.

)QIJH, CJ C;>CJ =K=C2µ5 O9@ 9>@54 54C45C? 53JK 5854C9> <@M=2 µ9IIJ@C46M@ C4µM@

C2H 94=?OJL.

3 91 . 4 ™ À ™ & % ª "&" ¢ # " ' ƒ # & $ À Ã ƒ $ ¡ $ À

!:9CB=C9 5@ CJ =K=C2µ5 y(n) = 3x(n) + 3 9>@54 <;5µµ46?.


!:9CB=C9 5@ C5 =L=C7µ5C5 y(n) = nx(n) 654 y(n) = x(2n) 9>@54 T;J@46B 5µ9CBSI2C5.


1.4.1 #ÚÔ˘ÛÙÈÎ‹ ·!fiÎÚÈÛË Û˘ÛÙ‹Ì·ÙÔ˜

(H GLµ2GJKµ9 C2 µJ@5O45>5 6;JL=C467 56JIJLG>5 *(n). (LC7 QT94 C4µ7 1 <45 n = 0

654 C4µ7 0 J8JLO78JC9 5IIJK (SI. +T7µ5 1.10). !35;µ?ZJLµ9 5LC? CJ =7µ5 =C2@

9>=JOJ 9@?H LTI =L=C7µ5CJH O456;4CJK T;?@JL, CJ J8J>J 5;T46B 2;9µ9>, O2I5O7

935;µ?ZJLµ9 µ>5 O4Q<9;=2 C2 =C4<µ7 n = 0. )J =7µ5 9:?OJL, CJ J8J>J G5 85;5C2-

;2G9> µ9CB C2 =C4<µ7 n = 0, 9>@54 T5;56C2;4=C46? CJL >O4JL CJL =L=C7µ5CJH. (LC?

CJ =7µ5 9:?OJL 58JC9I9> C2@ '*16+%&'( #:8'*&+", h(n), CJL =L=C7µ5CJH. # 6;JL-

=C467 58?6;4=2 J@JµBZ9C54 654 A6+&'( #:8'*&+" CJL =L=C7µ5CJH. P@5 85;BO94<-

µ5 6;JL=C467H 58?6;4=2H =L=C7µ5CJH O456;4CJK T;?@JL 35>@9C54 =CJ +T7µ5 1.21.

*(n) h(n)

1

n n

LTI[+K=C2µ5[

E456;4CJK X;?@JL

0 1 2 30 1 2 3

(@ 935;µ?Z5µ9 C2 O4Q<9;=2 C2 =C4<µ7 n = 1, C?C9 2 58?6;4=2 CJL =L=C7µ5CJH G5

7C5@ >O45 µ9 C2@ 8;J2<JKµ9@2, 5IIB G5 B;T4Z9 58? C2 =C4<µ7 n = 1, ?8AH 35>@9C54

=CJ +T7µ5 1.225. &54 <9@46B, 5@ 935;µ?Z5µ9 C2@ 6;JL=C467 9>=JOJ C2 T;J@467 =C4<-

µ7 n = m, C?C9 CJ 58JCQI9=µ5 G5 7C5@ 2 >O45 58?6;4=2, 5IIB µ9 5;T7 C2 =C4<µ7 m

(+T7µ5 1.22S). \8AH 65C5I5S5>@JLµ9 5LC? J39>I9C54 =CJ <9<J@?H ?C4 CJ =K=C2µB

µ5H 9>@54 T;J@46B 5µ9CBSI2CJ.

™¯‹Ì· 1.21

="()8/012 %'#-1"083 8)8/2µ%/(4*0%1"0/(7 E"#,()


4 0 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

1.4.2 ™˘Ó¤ÏÈÍË

)>G9C54 =L@98MH CJ 9;MC2µ5: *J45 G5 9>@54 2 58?6;4=2 9@?H =L=C7µ5CJH O456;4CJK

T;?@JL <45 9>=JOJ x(n), 5@ <@A;>ZJLµ9 C2@ 6;JL=C467 CJL 58?6;4=2 h(n);

# 58B@C2=2 =CJ 9;MC2µ5 5LC? O>@9C54 µJ@JI96C46B 58? C2 IQ:2 8),C60F3

(convolution). # Q:JOJH y(n) CJL =L=C7µ5CJH (+T7µ5 1.20) G5 4=JKC54 µ9 C2 =L@Q-

I4:2 C2H 94=?OJL x(n) 654 C2H 6;JL=C467H h(n) CJL =L=C7µ5CJH, 7:

y(n) = x(n) * h(n) (1.35)

?8JL * CJ =KµSJIJ C2H =L@QI4:2H. \µAH C4 9>@54 2 =L@QI4:2 654 8MH L8JIJ<>Z9C54;

P=CA, IJ48?@, ?C4 x(n) 2 9>=JOJH 654 h(n) 2 6;JL=C467 58?6;4=2 =L=C7µ5CJH O45-

6;4CJK T;?@JL. *;4@ 8;JTA;7=JLµ9, 5H GLµ2GJKµ9 C2 =TQ=2 (1.34), 2 J8J>5 µ5H

IQ94 ?C4 Q@5 =7µ5 O456;4CJK T;?@JL µ8J;9> @5 963;5=G9> AH <;5µµ46?H =L@OL5=µ?H

JI4=G2µQ@A@ 6;JL=C46M@. !8>=2H, 5H µ2@ :9T@Bµ9 ?C4 CJ =K=C2µ5 8JL 9:9CBZJLµ9

9>@54 <;5µµ46? (B;5 4=TK94 2 5;T7 C2H L8Q;G9=2H) 654 T;J@46B 5µ9CBSI2CJ. PTJ-

@C5H 5LCB 65CB @JL, µ8J;JKµ9 @5 963;B=JLµ9 C2@ 9>=JOJ x(n) AH:

(1.36)

045 6BG9 µ>5 58? C4H 94=?OJLH x(m)*(n–m), 9>O5µ9 =C2@ 8;J2<JKµ9@2 L8J9@?C2C5

1.4.1, ?C4 2 Q:JOJH G5 4=JKC54 µ9 x(m)h(n–m). "?<A C2H <;5µµ46?C2C5H CJL =L=C7-

µ5CJH, 2 C9I467 Q:JOJH y(n) G5 9>@54 >=2 µ9 CJ BG;J4=µ5 CA@ 984µQ;JLH 58J6;>=9-

A@, O2I5O7:

(1.37) y n x m h n m

m

( ) ( ) ( )= #=#"

"

5

x n x m n m

x n x x n x nm

( ) ( ) ( )

... ( ) ( ) ( ) ( ) ( ) ...

= #

= + # + + # + # +=#"

"

5 4

4 4 4 41 1 0 1 2(n)+ (1) (2)

™¯‹Ì· 1.22

="()8/012 %'#-1"083 8)8/2µ%/(4*0%1"0/(7 E"#,()

$0% .98(*( (%) *(n–1) 1%0

(I) *(n–m)

LTI[+K=C2µ5[

E456;4CJK X;?@JL

LTI[+K=C2µ5[

E456;4CJK X;?@JL

*(n–1) h(n–1)

h(n–m)*(n–m)(5)

(S)

0

0 1 2 m m0 1 2

1 2 3 0 1 2 3n n

n n


(LC7 9>@54 2 =TQ=2 C2H <;5µµ467H =L@QI4:2H.

# O45O465=>5 8JL µ?I4H 89;4<;BR5µ9 O>@9C54 85;5=C5C46B µ9 C2 SJ7G945 CJL 85;5-

O9><µ5CJH CJL +T7µ5CJH 1.23, <45 C2@ 89;>8CA=2 65CB C2@ J8J>5 {x(n)} = {1,2,3,–1}

654 {h(n)} = {1,–1,2}. +C5 +T7µ5C5 1.235,S 35>@J@C54 J4 56JIJLG>9H x(n), h(n). +CJ

5;4=C9;? µQ;JH CA@ +T2µBCA@ 1.23< QAH 1.23=C O>@J@C54 J4 6;JL=C46QH x(m)*(n–m),

9@M =CJ O9:> µQ;JH CA@ >O4A@ =T2µBCA@ 35>@J@C54 J4 5@C>=CJ4T9H 58J6;>=94H CJLH.

)J BG;J4=µ5 CA@ 984µQ;JLH 6;JL=C46M@, CJ J8J>J 58JC9I9> 654 C2@ 58?6;4=2 CJL

=L=C7µ5CJH, 35>@9C54 =CJ +T7µ5 1.23Z.

4 11 . 4 ™ À ™ & % ª "&" ¢ # " ' ƒ # & $ À Ã ƒ $ ¡ $ À

x(n)

12

3

-1n

(5)

h(n)

12

-1n

(S)

x(-1)*(n+1)

x(1)*(n-1) x(1)h(n-1)

x(0)*(n) x(0)h(n)

x(-1)h(n+1)

1 1

-1

2

2 2

4

-2

n

n

n n

n

n

(<)

(O)

(9)

3 3

-3

6

™¯‹Ì· 1.23

D"%µµ012 8),C60F3


4 2 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

¶·Ú¿‰ÂÈÁÌ· 1.6

0;53467 µQGJOJH L8JIJ<4=µJK C2H =L@QI4:2H.

' L8JIJ<4=µ?H C2H =L@QI4:2H CA@ x(n) 654 h(n), =Kµ3A@5 µ9 C2 =TQ=2 1.37, µ5H

L85<J;9K94 C2@ 56?IJLG2 =94;B S2µBCA@:

1. (@5O>8IA=2 (65CJ8C;4=µ?) C2H h(m) <K;A 58? CJ O9><µ5 m = 0, M=C9 @5 µ5H

OM=94 C2@ h(–m).

2. 'I>=G2=2 C2H h(–m) =C2@ 984GLµ2C7 GQ=2 n, M=C9 @5 8B;JLµ9 C2@ h(n–m).

3. F8JIJ<4=µ? CA@ <4@JµQ@A@ x(m)h(n–m), O9><µ5 8;JH O9><µ5, <45 @5 µ5H OM=JL@

C2@ 984GLµ2C7 C4µ7 n.

4. *;?=G9=2 CA@ <4@JµQ@A@ 8JL L8JIJ<>=C265@.

(H 85;56JIJLG7=JLµ9 CJ@ L8JIJ<4=µ? C2H 9:?OJL y(n) 9@?H =L=C7µ5CJH O456;4CJK

T;?@JL µ9 6;JL=C467 58?6;4=2 { h(n)} = {1,2,3} 654 9>=JOJ {x(n)} = {3,4,5,2},

?8AH 5LC? 89;4<;B39C54 =CJ +T7µ5 1.24. '4 56JIJLG>9H x(n) 654 h(n) O9>T@J@C54 =C5

x(2)*(n-2) x(2)h(n-2)

-1 -1-2

n n

(=C)

1

y(n)

7

(Z)

3

-2

11

n

*5;5C2;7=C9 ?C4 CJ µ76JH C2H 58?6;4=2H 9>@54 6 O9><µ5C5. 09@46B, 5@ R1 9>@54 CJ

µ76JH C2H µ>5H 56JIJLG>5H 654 R2 CJ µ76JH C2H BII2H 56JIJLG>5H, C?C9 2 <;5µµ4-

67 =L@QI4:2 5LCM@ O>@94 µ45 @Q5 56JIJLG>5 µ9 µ76JH R1 + R2–1. ' L8JIJ<4=µ?H

C2H =L@QI4:2H OKJ =2µBCA@ O456;4CJK T;?@JL µ9 T5;C> 654 µJIKS4 <>@9C54 =L@7-

GAH µ9 OKJ C;?8JLH. !>C9 <;5346B, ?8AH 89;4<;B39C54 =CJ 85;BO94<µ5 1.6 9>C9 µ9

C2 µQGJOJ C2H JI4=G5>@JL=5H ;BSOJL CJL 85;5O9><µ5CJH 1.7


+T7µ5C5 1.245 654 1.24S 5@C>=CJ4T5. +C5 +T7µ5C5 1.24< O>@J@C54 ?I9H J4 O453J;9-

C46QH GQ=94H C2H h(n–m) <45 n = 0, 1, 2, 3, 4, 5. 045 65G9µ4B 58? C4H GQ=94H 5LCQH L8J-

IJ<>Z9C54 CJ 5@C>=CJ4TJ BG;J4=µ5 <4@JµQ@A@ µ9 C2@ 56JIJLG>5 94=?OJL. (LC? 58J-

C9I9> JL=45=C46B 654 CJ 58JCQI9=µ5 C2H =L@QI4:2H CA@ x(n) 654 h(n), O2I5O7 C2@

56JIJLG>5 9:?OJL y(n) 2 J8J>5 35>@9C54 =CJ +T7µ5 1.24O. +CJ 85;BO94<µB µ5H 2

x(n) QT94 µ76JH 4 654 2 h(n) QT94 µ76JH 3, J8?C9 2 56JIJLG>5 y(n), 8JL 8;J6K8C94,

QT94 µ76JH 4 + 3 – 1 = 6 O94<µBCA@.

4 31 . 4 ™ À ™ & % ª "&" ¢ # " ' ƒ # & $ À Ã ƒ $ ¡ $ À

x(n)

h(-m)

h(2-m)

h(4-m) h(5-m)

h(3-m)

h(1-m)

h(n)

5

0

0

0

0 0

0

1

1

1

1 1

1

1

2

2

2

2 2

2

2

0 1 23

3

3

3 3

3

3

3

n

n

n

n

n

n

n

n

n

(5)

(<)

(O)

(S)

y(0)=x(0)h(0)=3^1=3 y(1)=3^2+4^1=6+4=10

y(2)=3^3+2^4+1^5=9+8+5=22 y(3)=4^3+5^2+2^1=12+10+2=24

y(4)=5^3+2^2=15+4=19 y(5)=2^3=6

y(n)

510152025

{y(n)}={3,10,22,24,19,6}

™¯‹Ì· 1.24

B'(6($08µ#4 /34$"%µµ0124 8),C60-F34 *7( %1(6()-50+, µ. /3 $"%<0-12 µC5(*(.


4 4 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

¶·Ú¿‰ÂÈÁÌ· 1.7

F8JIJ<4=µ?H C2H =L@QI4:2H µ9 C2 µQGJOJ C2H JI4=G5>@JL=5H ;BSOJL.

# O45O465=>5 9>@54 2 >O45 µ9 969>@2 CJL 85;5O9><µ5CJH 1.6, µ9 µ?@2 O453J;B ?C4 5@C>

<45 C4H <;5346QH 85;5=CB=94H CA@ 56JIJLG4M@ T;2=4µJ8J4JKµ9 C4H C4µQH CJLH. )5

S7µ5C5 8JL 56JIJLGJKµ9 9>@54 98>=2H C5 >O45, ?8AH 35>@9C54 654 =CJ +T7µ5 1.25

?8JL O>@9C54 J L8JIJ<4=µ?H C2H =L@QI4:2H CA@ 56JIJLG4M@ x(n), h(n) CJL 85;5-

O9><µ5CJH 1.6.

x(n)

h(n)

h(-m)

h(1-m)

h(2-m)

h(3-m)

h(4-m)

h(5-m)

3 4 5 2

1 2 3

3 2 1

3 2 1

3 2 1

3 2 1

3 2 1

3 2 1

y(0)=3^1=3

y(1)=3^2+4^1=6+4=10

y(2)=3^3+4^2+5^1=9+8+5=22

y(3)=4^3+5^2+2^1=12+10+2=24

y(4)=5^3+2^2=15+4=19

y(5)=2^3=6

™¯‹Ì· 1.25

B'(6($08µ#4 /348),C60F34 µ. /3µC5(*( /34 (60-

85%9,()8%4"?I*().

#¢#$&%&!™ &%™ ™À¡!§#•%™

045 C2@ 8;B:2 C2H =L@QI4:2H 4=TKJL@ J4 4O4?C2C9H: 5@C4µ9C5G9C467, 8;J=9C54;4=C4-

67 654 984µ9;4=C467. )4H 85;5GQCJLµ9 =C2 =L@QT945 µ5Z> µ9 =T?I45 =T9C46B µ9 C2

3L=467 CJLH =2µ5=>5, TA;>H @5 OM=JLµ9 C2@ 58?O94:7 CJLH.

6,()µ'(1H'()*% )I)D(0(1

x(n)*h(n) = h(n)*x(n) (1.38)

!>O5µ9 58? C4H =TQ=94H (1.35) 654 (1.37) ?C4

(1.39)

,9 SB=2 C2@ 5@C4µ9C5G9C467 4O4?C2C5 8JL µ?I4H 5@53Q;5µ9, 2 56JIJLG>5 y(n) µ8J-

;9> @5 963;5=C9> AH

(1.40) y n h n x n h m x n m

m

( ) ( )* ( ) ( ) ( )= = #=#"

"

5

y n x n h n x m h n m

m

( ) ( )* ( ) ( ) ( )= = #=#"

"

5


!8JµQ@AH, J4 ;?IJ4 CA@ 56JIJLG4M@ x(n) 654 h(n) µ8J;JK@ @5 5@C4µ9C5C9GJK@, ?8AH

35>@9C54 =CJ +T7µ5 1.26.

4 51 . 4 ™ À ™ & % ª "&" ¢ # " ' ƒ # & $ À Ã ƒ $ ¡ $ À

x(n) y(n) h(n)h(n) x(n)

y(n)

™¯‹Ì· 1.26

&E3µ%/012 %,%'%-"?8/%83 /34 %,/0-µ./%5./0124 0*0#/3/%4 /348),C60F34.

™¯‹Ì· 1.27

(%) !"(8./%0"0-8/012 0*0#/3/% /348),C60F34, 1%0 (I)8),*)%8µ#4 '"(-8./%0"08/0124 1%0%,/0µ./%5./01240*0#/3/%4

>C#/'(1)C)/()*% )I)D(0(1

[x(n)*h1(n)]*h2(n) = x(n)*[h1(n)*h2(n)] (1.41)

# =T2µ5C467 5@585;B=C5=2 C2H 8;J=9C54;4=C467H 4O4?C2C5H 35>@9C54 =CJ +T7µ5

1.275, ?8JL h(n) = h1(n)*h2(n). !K6JI5 µ8J;JKµ9 @5 <9@469K=JLµ9 C2@ 4O4?C2C5 C2H

8;J=9C54;4=C46?C5H =9 89;4==?C9;5 58? OKJ =L=C7µ5C5, C5 J8J>5 =L@OQJ@C54 O45-

OJT46B (in cascade). PC=4, 2 89;>8CA=2 C2H O45OJT467H =K@O9=2H L =L=C2µBCA@ LTI

µ9 6;JL=C46QH 58J6;>=94H h1(n), h2(n), …, hL(n) 4=JOL@5µ9> µ9 Q@5 LTI =K=C2µ5, CJL

J8J>JL 2 6;JL=C467 58?6;4=2 9>@54 h(n) 654 4=JKC54 µ9

h(n) = h1(n)*h2(n)* … *hL(n) (1.42)

# <9@>69L=2 9>@54 8JIK T;7=4µ2, 4O45>C9;5 ?C5@ C2 OJKµ9 AH 5@C>=C;J32 O45O465-

=>5, O2I5O7, AH O45O465=>5 5@BIL=2H 9@?H LTI =L=C7µ5CJH =9 µ45 O45OJT467 =K@-

O9=2 L8J=L=C2µBCA@.

)QIJH, =CJ +T7µ5 1.27S 35>@9C54 J =L@OL5=µ?H C2H 8;J=9C54;4=C467H 654 C2H 5@C4-

µ9C5G9C467H 4O4?C2C5H. (8? 5LC? <>@9C54 35@9;? ?C4, 2 =94;B µ9 C2@ J8J>5 =L@OQ-

JLµ9 C5 L8J=L=C7µ5C5, O2I5O7 L8JIJ<>ZJLµ9 C2 =L@QI4:2, O9@ QT94 =2µ5=>5.

x(n)

x(n) x(n)y(n) y(n)

x(n)y(n) y(n)h1(n)

h1(n) h1(n)

h2(n)

h2(n) h2(n)

h(n)=[h1(n) * h2(n)

(5)

(S)

-.)µ'C)/()*% 8I)D(0(1

x(n)*[h1(n) + h2(n)] = x(n)*h1(n) + x(n)*h2(n) (1.43)

# 4O4?C2C5 5LC7 µ5H IQ94 ?C4, 9B@ QTJLµ9 OKJ LTI =L=C7µ5C5 µ9 6;JL=C46QH 58J6;>=94H

h1(n) 654 h2(n), =C5 J8J>5 935;µ?ZJLµ9 CJ >O4J =7µ5 94=?OJL x(n), C?C9 CJ BG;J4=µ5 CA@


4 6 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

OKJ 58J6;>=9A@ 9>@54 >O4J µ9 C2@ 58?6;4=2 9@?H BIIJL =L=C7µ5CJH µ9 6;JL=C467 58?-

6;4=2 h(n) = h1(n) + h2(n). E2I5O7, CJ @QJ 5LC? =K=C2µ5 4=JKC54 µ9 CJ@ 85;BII2IJ =L@-

OL5=µ? CA@ OKJ LTI =L=C2µBCA@ (+T7µ5 1.28). 09@46B, 2 85;BII2I2 =K@O9=2 L =L=C2-

µBCA@ µ9 6;JL=C46QH 58J6;>=94H h1(n), h2(n),…, hL(n) =C5 J8J>5 935;µ?Z9C54 2 >O45

9>=JOJH x(n), 4=JOL@5µ9> µ9 Q@5 =K=C2µ5 CJL J8J>JL 2 6;JL=C467 58?6;4=2 4=JKC54 µ9

h(n) = h1(n) + h2(n) + … + hL(n) (1.44)

&54 5@C>=C;J35, 6BG9 LTI =K=C2µ5 µ8J;9> @5 5@5ILG9> =9 L8J=L=C7µ5C5 =L@O9-

O9µQ@5 85;BII2I5.

x(n) x(n)y(n) y(n)

h1(n)

h2(n)

h(n)=[h1(n) + h2(n)

™¯‹Ì· 1.28

:'0µ."08/012 0*0#-/3/%: *7( LTI

8)8/2µ%/% 8),*.-*.µC,% '%"?663-6%, µ'("(7, ,%

%,/01%/%8/%5(7,%'# C,% 878/3µ%

/() ('(9() 31"()8/012 %'#1"0-83 08(7/%0 µ. /(?5"(08µ% /@,

1"()8/01+, /()4.

,QT;4 9OM QTJLµ9 98469@C;M=94 C2 µ9IQC2 µ5H =CJ@ L8JIJ<4=µ? C2H <;5µµ467H

=L@QI4:2H µ9 <;5346? 7 5;4Gµ2C46? C;?8J. D5 9:9CB=JLµ9 CM;5 CJ@ L8JIJ<4=µ?

C2H =L@QI4:2H 56JIJLG4M@ µ9 5@5ILC46? C;?8J. '4 5@5ILC46QH 963;B=94H JO2<JK@

=9 =Lµ89;B=µ5C5 89;4==?C9;J <9@46B <45 C5 =L=C7µ5C5 8JL 9:9CBZJLµ9, 654 QC=4

µ5H 9>@54 84J T;7=4µ9H. (H OJKµ9 IJ48?@ Q@5 =T9C46? 85;BO94<µ5.

¶·Ú¿‰ÂÈÁÌ· 1.8

+C2@ 9>=JOJ 9@?H =L=C7µ5CJH µ9 6;JL=C467 58?6;4=2 h(n) = anu(n) 935;µ?Z9C54 CJ

=7µ5 x(n) = bnu(n), ?8JL a, b <@A=CQH =C5G9;QH 654 a 3 b. $5 L8JIJ<4=C9> 2 Q:JOJH

y(n) CJL =L=C7µ5CJH.

=G/0:

# Q:JOJH y(n) G5 9>@54 CJ 58JCQI9=µ5 C2H =L@QI4:2H C2H 94=?OJL µ9 C2@ 6;JL=C467

58?6;4=2 CJL =L=C7µ5CJH. ,9 SB=2 C2 =TQ=2 (1.40) QTJLµ9:

(DLµ2G9>C9 ?C4 2 u(m) = 0 <45 m < 0 654 2 u(n–m) = 0 <45 m > n).

!894O7 CJ BG;J4=µ5 L8JIJ<>Z9C54 AH 8;JH m, J ?;JH bn µ8J;9> @5 «S<9>» 96C?H CJL

5G;J>=µ5CJH, J8?C9 2 C9I9LC5>5 =TQ=2 <>@9C54:

y n b a b b abn m m n m

m

n

m

n

( ) ( )= =# #

==55 1

00

y n h n x n h m x n m a u m b u n m a b

m

m n m m n m

m

n

m

( ) ( )* ( ) ( ) ( ) ( ) ( )= = # = # ==#"

"# #

==#"

"

5 550


)J 85;58B@A 9>@54 BG;J4=µ5 CA@ n + 1 8;MCA@ ?;A@ <9Aµ9C;467H 8;J?OJL µ9

8;MCJ ?;J 1 654 I?<J ab–1 654 O>@9C54 =9 6I94=C7 µJ;37 AH

?8JL n ( 0. *5;5C2;JKµ9 ?C4 2 Q:JOJH T5;56C2;>Z9C54 C?=J 58? C2@ 9>=JOJ x(n) =

bn, n ( 0, ?=J 654 58? C2@ 6;JL=C467 CJL =L=C7µ5CJH h(n) = an, n ( 0. (LC7 9>@54 µ45

<9@46?C9;2 O458>=CA=2, O2I5O7 2 Q:JOJH G5 89;4QT94 ?;JLH C2H >O45H 5I<9S;467H µJ;-

37H µ9 CJLH ?;JLH C2H 94=?OJL 654 C2H 6;JL=C467H 58?6;4=2H CJL =L=C7µ5CJH.

y n bab

abb

b a bb a

b ab a

bb a

ba

b aa C b C a

nn

nn n

n nn n

bn

an

( ) (( )

) ( )= ##

= ##

= ##

=#

##

= #

# +

#

+ #

+ +

1

1

1 1

1

1

1 1

4 71 . 4 ™ À ™ & % ª "&" ¢ # " ' ƒ # & $ À Ã ƒ $ ¡ $ À

$5 L8JIJ<>=9C9 C2@ Q:JOJ CJL =L=C7µ5CJH CJL *5;5O9><µ5CJH 1.8 µ9 9>=JOJ C2

S2µ5C467 56JIJLG>5 8IBCJLH (.


045 CJ =K=C2µ5 µ9 µJ@5O45>5 6;JL=C467 >=2 µ9 h(n) = (an + bn)u(n), @5 L8JIJ<4-

=C9> 2 Q:JOJH, ?C5@ =’ 5LC? 935;µ?Z9C54 AH 9>=JOJH 2 S2µ5C467 56JIJLG>5 8IB-

CJLH (.


$5 L8JIJ<4=C9> 2 6;JL=C467 58?6;4=2 h(n) CJL =L=C7µ5CJH O456;4CJK T;?@JL CJL

+T7µ5CJH 1.29, ?C5@ h1(n) = *(n) + *(n–1), h2(n) = *(n)– *(n–1), h3(n) =

2*(n), 654 h4(n) = –2( )nu(n).

™¯‹Ì· 1.29

&78/3µ% *0%1"0/(7 E"#,().

12

14

12

12


h1(n)

h2(n)

h3(n)

h4(n)


4 8 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™


+C2@ 9@?C2C5 5LC7 5=TJI2G765µ9 µ9 <;5µµ46B T;J@46B–5µ9CBSI2C5 (LTI) =L=C7-

µ5C5 O456;4CJK T;?@JL 654 µ9I9C7=5µ9 C2@ 58?6;4=7 CJLH =9 O49<Q;=94H C2H 94=?-

OJL. !>O5µ9, ?C4 2 Q:JOJH 6BG9 LTI =L=C7µ5CJH 4=JKC54 µ9 C2 =L@QI4:2 C2H 56J-

IJLG>5H 94=?OJL µ9 C2@ µJ@5O45>5 6;JL=C467 58?6;4=2 CJL =L=C7µ5CJH. 0@A;>-

=5µ9 C4H 4O4?C2C9H C2H =L@QI4:2H (5@C4µ9C5G9C467, 8;J=9C54;4=C467, 984µ9;4=C467)

654 89;4<;BR5µ9 O453J;9C46JKH C;?8JLH L8JIJ<4=µJK C2H C?=J 5;4Gµ2C46B, ?=J

654 5@5ILC46B.

™‡ÓÔ„Ë ÎÂÊ·Ï·›Ô˘

+CJ &93BI54J 5LC? µBG5µ9 ?C4:

# R234567 989:9;<5=>5 =2µBCA@ 85;JL=4BZ94 =C4H µQ;9H µ5H 9@CL8A=456B 8I9J-

@96C7µ5C5 Q@5@C4 C2H 5@C>=CJ4T2H 5@5IJ<467H 989:9;<5=>5H, ?8AH 9L9I4:>5, 5:4J-

84=C>5 654 56;>S945.

&BG9 5@5IJ<46? =7µ5, <45 @5 L8J=C9> 989:9;<5=>5 µ9 Q@5 R23456? =K=C2µ5, 8;Q-

894 8;MC5 @5 µ9C5C;589> =9 R23456? =7µ5. (LC? 984CL<TB@9C54 µ9 65CBII2I2 O94<-

µ5CJI2R>5 CJL 5@5IJ<46JK =7µ5CJH 654 6SB@C4=2 (65GMH 654 6AO46J8J>2=2) CA@

O94<µBCA@.

&5CBII2I2 O94<µ5CJI2R>5 =2µ5>@94 C2 I7R2 CJLIBT4=CJ@ OKJ O94<µBCA@ 5@B

89;>JOJ CJL =7µ5CJH, O2I5O7 Fs ( 2Fmax (G9M;2µ5 O94<µ5CJI2R>5H 7 G9M;2µ5 CJL

Shannon).

!IBCCA=2 CJL =3BIµ5CJH 6SB@C4=2H, 65CB C2@ R234J8J>2=2 9@?H O9><µ5CJH, =L@9-

8B<9C54 89;4==?C9;5 bits <45 C2@ 5@585;B=C5=2 5LCJK.

&BG9 =7µ5 O456;4CJK T;?@JL µ8J;9> @5 963;5=C9> AH BG;J4=µ5 JI4=G2µQ@A@

6;JL=C46M@, 8JII58I5=45=µQ@A@ µ9 65CBII2IJLH =L@C9I9=CQH SB;JLH.

# 58?6;4=2 =L=C7µ5CJH O456;4CJK T;?@JL 4=JKC54 µ9 C2 <;5µµ467 =L@QI4:2 C2H

94=?OJL 654 C2H 6;JL=C467H 58?6;4=2H CJL =L=C7µ5CJH.


BÈ‚ÏÈÔÁÚ·Ê›· ÎÂÊ·Ï·›Ô˘

!§§%¡$°§ø™™%

+. D9JOA;>O2H, &. ,89;µ89;>O2H, :08%$@$2 8/3 -.@"9% &3µ?/@, & &)8/3µ?/@,,

)L8AG7CA 0. E5;O5@?H, (G7@5, 1998.

N.&5IJL8C=>O2H, &2µ%/%, &)8/2µ%/% 1%0 >6$#"05µ(0 , !6O?=94H E>5LIJH,(G7@5,

1993.

•!¡$°§ø™™%

R.I.Damper, Introduction to Discrete–Time Signals and Systems, Chapman & Hall,

1995.

E.C.Ifeachor and B.W. Jervis, Digital Signal Processing: A Practical Approach,

Second Edition, Pearson Education Limited, 2002.

P.A. Lynn and W. Fuerst, Introductory Digital Signal Processing With Computer

Applications, J.Wiley and Sons Ltd, 1989.

S.K.Mitra, Digital Signal Processing: A computer–Based Approach, Second Edition,

McGraw Hill, 2001.

S.J.Orfanidis, Introduction to Signal Processing, Prentice–Hall, 1996.

A.V.Oppenheim and A.S.Willsky, Signals & Systems, Second Edition, Prentice–Hall,

1997.

J.G.Proakis and D.G.Manolakis, Digital Signal Processing: Principles, Algorithms

and Applications, Prentice–Hall, 1996.

R.D.Strum and D.E.Kirk, First Principles of Discrete Systems and Digital Signal

Processing, Addison–Wesley Publishing Company, 1988.

"!fi‰ÔÛË ·ÁÁÏÈÎÒÓ fiÚˆÓ ÛÙËÓ ÂÏÏËÓÈÎ‹

accuracy 56;>S945 (8;B:9A@)

advance 8;J7<2=2

aliasing 354@?µ9@J T5µ2IJK (5@985;6JKH) ;LGµJK O94<µ5CJI2-

R>5H, 35=µ5C467 9846BILR2, 5IIJ>A=2

amplitude 8IBCJH

angular frequency <A@4567 =LT@?C2C5

bounded 3;5<µQ@JH

4 9B I B § I O ° PA º I A K E º A § A I O Y


5 0 K E º A § A I O 1 : E # ™ " ° ø ° % ™ & % ¡ æ % º # " ' % ! ¶ ! • ! ƒ °" ™ # " ™ % ª "&$ ™ ' " # ! # ' $ ¡ " ™

cascade O45OJT467 =K@O9=2

convolution =L@QI4:2

delay 65GL=CQ;2=2

digital signal processing R234567 989:9;<5=>5 =7µ5CJH

dimension O4B=C5=2

discrete O456;4C?H

dynamic range OL@5µ467 89;4JT7

exponential sequence 96G9C467 56JIJLG>5

folding frequency =LT@?C2C5 5@5O>8IA=2H

fundamental period S5=467 89;>JOJH

image processing 989:9;<5=>5 946?@5H

impulse response 6;JL=C467 58?6;4=2

linear <;5µµ46?H

normalised frequency 65@J@46J8J42µQ@2 =LT@?C2C5

periodic 89;4JO46?H

phasor 3B=J;5H

quantisation 6SB@C4=2

quantisation error =3BIµ5 7 G?;LSJH 6SB@C4=2H

relative frequency =T9C467 =LT@?C2C5

resolution 5@BIL=2, O456;4C467 465@?C2C5

rounding =C;J<<LIJ8J>2=2

sinusoidal 2µ4CJ@J94OQH

speech Jµ4I>5

superposition L8Q;G9=2

time invariant T;J@46B 5µ9CBSI2CJH

truncation 58J6J87

unit sample µJ@5O45>J O9><µ5, 6;JL=C467 56JIJLG>5

unit step µJ@5O45>5 S2µ5C467 56JIJLG>5

video 56JIJLG>5 946?@A@


AÓ¿Ï˘ÛË ÛÙÔ !Â‰›Ô ÙË˜ Û˘¯ÓfiÙËÙ·˜

™ÎÔ!fi˜

;( %,/01.9µ.,( /() '%"#,/(4 1.<%6%9() .9,%0 3 '."0$"%<2 /() 82µ%/(4 8/( '.*9(

/34 8)E,#/3/%4. O 5.+"383 /@, 83µ?/@, 1%0 8)8/3µ?/@, *0%1"0/(7 E"#,() %'# /(

'.*9( /34 8)E,#/3/%4, µ%4 *9,.0 µ0% .,/.6+4 *0%<("./012 *),%µ012 1%/%,#3834 1%0

.'.F."$%89%4 /()4. D0% /% 82µ%/% *0%1"0/(7 E"#,(), /% ('(9% µ%4 .,*0%<C"(),, 3

µ./?I%83 8/( '.*9( /34 8)E,#/3/%4 .'0/)$E?,./%0 µ. /3 I(25.0% /() µ./%8E3µ%/0-

8µ(7 Fourier *0%1"0/(7 E"#,(). D0% %1(6()59.4 '.'."%8µC,() µ21()4 E"380µ('(0-

.9/%0 ( *0%1"0/#4 µ./%8E3µ%/08µ#4 Fourier.


M./? /3 µ.6C/3 /() 1.<%6%9() 5% .98/. 8. 5C83 ,%:

• B'(6($9L./. '(8(/01? 1%0 '(0(/01? /( <?8µ% .,#4 82µ%/(4 *0%1"0/(7 E"#,()

• B'(6($9L./. /( µ./%8E3µ%/08µ# Fourier $0% 82µ%/% *0%1"0/(7 E"#,() '.'."%-

8µC,34 *0?"1.0%4

• :1/.6.9/. /3, 1)16012 (6985383 83µ?/@, *0%1"0/(7 E"#,()

• B'(6($9L./. /3, %'#1"083 .,#4 8)8/2µ%/(4 *0%1"0/(7 E"#,() ($"%µµ012 8),C60-

F3) µC8@ /() '.*9() /34 8)E,#/3/%4, I%80L#µ.,(0 8/( *0%1"0/# µ./%8E3µ%/08µ#

Fourier

• B6('(0.9/. C,%, %'# /()4 /%E.94 µ./%8E3µ%/08µ(74 Fourier


• T0%1"0/#4 µ./%8E3µ%/08µ#4 Fourier (DFT)

• T%E74 µ./%8E3µ%/08µ#4 Fourier (FFT)

• =)16012 8),C60F3

• M./%8E3µ%/08µ#4 Fourier

• U?8µ% 82µ%/(4


&/( '"(3$(7µ.,( 1.<?6%0( .9*%µ. #/0 C,% 82µ% *0%1"0/(7 E"#,() µ'(".9 ,% .1<"%-

2! " º#§#$%


5 2 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

8/.9 @4 $"%µµ01#4 8),*)%8µ#4 (60853µC,@, µ(,%*0%9@, 1"()8/01+, {*(n–m)}. &/(

1.<?6%0( %)/#, 5% '%"()80?8()µ. µ0% .,%66%1/012 %,%'%"?8/%83 ('(0%8*2'(/.

%1(6()59%4 @4 8),*)%8µo7 µ0$%*01+, .15./01+, %1(6()50+, /34 µ("<24 {e–j@n},

(1%/’ %,%6($9% µ. /( %,?'/)$µ% 8. 8.0"? Fourier .,#4 '."0(*01(7 %,%6($01(7 82µ%-

/(4). !"#1.0/%0 $0% µ9% '(67 E"280µ3 %,%'%"?8/%83 /@, 83µ?/@, 1%0 /@, 8)8/3-

µ?/@, *0%1"0/(7 E"#,() 8’ C,% µ./%8E3µ%/08µC,( '.*9(. M. ?66% 6#$0%, 8/( 1.<?-

6%0( %)/# 5% *(7µ. '+4 µ'("(7µ. ,% µ./%I(7µ. %'# /( '.*9( /() E"#,() 8/( '.*9(

/34 8)E,#/3/%4 1%0 %,/98/"(<%. =%0 #6% %)/? 5% %<("(7, '?,/(/. /3, '."9'/@83

/@, 83µ?/@, *0%1"0/(7 E"#,().

&/3, .,#/3/% 2.1 '"%$µ%/.)#µ%8/. /( µ./%8E3µ%/08µ# Fourier[1] *0%1"0/(7 E"#,()

(DTFT, Discrete Time Fourier Transform), ( ('(9(4 .<%"µ#L./%0 8. µ3 '."0(*01?

82µ%/% *0%1"0/(7 E"#,() 1%0 '%"()80?L()µ. /04 0*0#/3/C4 /(). M. I?83 /( *0%<(-

"./01# /"#'( 5.+"3834 /@, 83µ?/@, %'# /( '.*9( /34 8)E,#/3/%4, .'%,.F./?L()µ.

/( 5Cµ% /34 *.0$µ%/(63H9%4. &/3 8),CE.0%, µ.6./?µ. /( *0%1"0/# µ./%8E3µ%/08µ#

Fourier (DFT, Discrete Fourier Transform) ( ('(9(4 .<%"µ#L./%0 8. 82µ%/% *0%-

1"0/(7 E"#,() '.'."%8µC,() µ21()4 (.,#/3/% 2.2). ;C6(4, 8/3, .,#/3/% 2.3 '%"()-

80?L()µ. /( I%801# %6$#"05µ( $0% /(, /%E7 )'(6($08µ# /() *0%1"0/(7 µ./%8E3µ%/0-

8µ(7 Fourier (FFT, Fast Fourier Transform).

[1] %=CJ;467 5@5O;Jµ7: ' Jean Baptiste Joseph Fourier (1768–1830) 7C5@ 0BIIJH µ5G2-µ5C46?H 654 3L=46?H. ' Fourier 9;9L@JK=9 C2 O4BOJ=2 C2H G9;µ?C2C5H 654 CJ 180785;JL=>5=9 =CJ %@=C4CJKCJ C2H 05II>5H µ45 9;<5=>5 CJL <45 C2 T;7=2 CA@ 2µ4CJ@46M@6Lµ5CJµJ;3M@ =C2@ 5@585;B=C5=2 CA@ 65C5@JµM@ G9;µJ6;5=>5H. # 9;<5=>5 CJL 5LC7L8J=C7;4Z9 C2@ 5@J;G?OJ:2, <45 C2@ 98JT7 969>@2, B8JR2 ?C4 J8J4JO78JC9 =L@9TQH89;4JO46? =7µ5 G5 µ8J;JK=9 @5 963;5=C9> AH BG;J4=µ5 65CBII2IA@ 2µ4CJ@46M@ 6Lµ5-CJµJ;3M@. (@Bµ9=5 =CJLH 6;4CQH C2H 9;<5=>5H L87;T5@ 654 OKJ 58? CJLH 84J 32µ4=µQ-@JLH µ5G2µ5C46JKH, J Joseph Luis Lagrange (1736–1813) 654 J Pierre Simon de Laplace(1749–1827). &54 9@M J Laplace, ?8AH 654 BIIJ4 6;4CQH, R734=5@ L8Q; C2H O2µJ=>9L-=2H C2H 9;<5=>5H, J Lagrange 5;@7G269, <45C> µQT;4 C2 =C4<µ7 969>@2 O9@ L87;T9 CJ65CBII2IJ µ5G2µ5C46? L8?S5G;J, =CJ J8J>J G5 µ8J;JK=9 @5 =C2;4TC9> 2 =T9C467 58?-O94:2. 045 89@7@C5 T;?@45 J Lagrange 98Qµ9@9 ?C4 µ45 CQCJ45 8;J=Q<<4=2 O9 G5 µ8J;JK=9@5 T;2=4µJ8J42G9> <45 C2@ 5@585;B=C5=2 =2µBCA@ µ9 «<A@>9H», O2I5O7 =2µBCA@ µ95=L@QT949H, ?8AH 9>@54 J4 C9C;5<A@46J> 85IµJ>. )J %@=C4CJKCJ C2H 05II>5H, L8J6K8CJ-@C5H =CJ <?2C;J CJL Lagrange, 58Q;;4R9 C2@ 9;<5=>5 CJL Fourier. )9I46B, 5LC7 O2µJ-=49KC269 µ9CB CJ GB@5CJ CJL Lagrange.


2.1 ªÂÙ·Û¯ËÌ·ÙÈÛÌfi˜ Fourier ‰È·ÎÚÈÙÔ‡ ¯ÚfiÓÔ˘

2.1.1 $ÚÈÛÌÔ›

' µ9C5=T2µ5C4=µ?H Fourier O456;4CJK T;?@JL (DTFT, discrete–time Fourier transform)

9@?H =7µ5CJH O456;4CJK T;?@JL x(n) 9>@54 2 5@585;B=C5=2 CJL =7µ5CJH 5LCJK AH

=L@OL5=µJK µ4<5O46M@ 96G9C46M@ 56JIJLG4M@ C2H µJ;37H {e–j@n}, ?8JL @ µ9C5SI2-

C7, <@A=C7 654 AH (6L6I467) =LT@?C2C5. ' DTFT µ>5H 56JIJLG>5H, .?, )'?"E.0, 9>@54

µJ@5O46?H. # 5;T467 56JIJLG>5 µ8J;9> @5 L8JIJ<4=C9>, ?C5@ µ5H O>@9C54 J DTFT

5LC7H, µ9 C2 SJ7G945 CJL 5@C>=C;J3JL µ9C5=T2µ5C4=µJK Fourier O456;4CJK T;?@JL.

' DTFT X(ej@) C2H 56JIJLG>5H x(n) J;>Z9C54 AH

(2.1)

9@M J 5@C>=C;J3JH µ9C5=T2µ5C4=µ?H Fourier O456;4CJK T;?@JL J;>Z9C54 AH

(2.2)

'4 =TQ=94H (2.1) 654 (2.2) 58JC9IJK@ CJ I9<?µ9@J Z9K<JH µ9C5=T2µ5C4=µM@ Fourier

O456;4CJK T;?@JL. # =TQ=2 (2.1) J@JµBZ9C54 654 .F98@83 %,?6)834, 9@M 2 (2.2) .F9-

8@83 87,5.834. UIQ8JLµ9, IJ48?@, ?C4 µ9% %1(6()59% µ'(".9 ,% .1<"%8/.9 @4 $"%µ-

µ01#4 8),*)%8µ#4 µ0$%*01+, .15./01+, %1(6()50+,. # =L@B;C2=2 X(ej@) 9>@54 µ4<5-

O467 654 µ8J;9> @5 <;539> AH

(2.3)

?8JL Xr(ej@) 654 Xi(e

j@) 9>@54 8;5<µ5C46QH =L@5;C7=94H AH 8;JH @, 654 58JC9IJK@ CJ

8;5<µ5C46? 654 35@C5=C46? µQ;JH C2H X(ej@) 5@C>=CJ4T5. # X(ej@) =9 8JI46QH =L@C9-

C5<µQ@9H <;B39C54

(2.4)

?8JL (2.5)

(2.6)

# 8J=?C2C5 J@JµBZ9C54 8),?"/383 µC/"() 654 2 8J=?C2C5 G(@) 8),?"/3- X e j( )6

5( )6 6

6

6= { } =7

89

:

;<#arg ( ) tan

( )

( )X e

X e

X ej i

j

rj

1

X e X e X ej

rj

ij( ) ( ) ( )6 6 6= +2 2

X e X e ej j j( ) ( )6 6 6= ( )5

X e X e jX ejr

ji

j( ) ( ) ( )6 6 6= +

x n F X e X e e dj j j n( ) { ( )} ( )' =#

#=1 1

26 6 6

)

)

)6

X e F x n x n ej j n

n

( ) { ( )} ( )6 6' = #

=#"

"

5

5 32 . 1 ª ! &" ™ Ã % ª "& # ™ ª $ ™ F O U R I E R ¢ # " ' ƒ # & $ À Ã ƒ $ ¡ $ À


5 4 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

83 <?834. &54 J4 OKJ 5LCQH =L@5;C7=94H 9>@54 8;5<µ5C46QH =L@5;C7=94H CJL @. +9

8JIIQH 935;µJ<QH, J µ9C5=T2µ5C4=µ?H Fourier J@JµBZ9C54 <?8µ% Fourier, 654 65C’

98Q6C5=2 J4 =L@5;C7=94H 654 G(@) 5@53Q;J@C54 AH <?8µ% µC/"()

(magnitude spectrum) 654 <?8µ% <?834 (phase spectrum) 5@C>=CJ4T5.

' DTFT X(ej@) C2H 56JIJLG>5H x(n) 9>@54 µ>5 =L@9T7H =L@B;C2=2 CJL @. !848IQJ@,

9>@54 '."0(*012 @4 '"(4 @ µ. '."9(*( 983 µ. 2', O2I5O7

(2.7)

# C9I9LC5>5 5LC7 =TQ=2 µ8J;9> @5 58JO94TG9> AH 9:7H:

# 89;4JO46?C2C5 9>@54 58JCQI9=µ5 CJL <9<J@?CJH ?C4 C5 O456;4CJK T;?@JL µ4<5O46B

96G9C46B =7µ5C5, C5 J8J>5 O453Q;JL@ =C2 =LT@?C2C5 65CB 8JII58IB=45 CJL 2',

9>@54 µ9C5:K CJLH C5LC?=2µ5.

' DTFT, ?8AH J;>Z9C54 58? C2 =TQ=2 (2.1), 9>@54 Q@5 BG;J4=µ5 B894;A@ ?;A@, CJ

J8J>J 84G5@?@ @5 µ2@ L8B;T94 (@5 µ2 =L<6I>@94) <45 6B8J45 =7µ5C5. (8JO946@K9C54

?C4 2 5@5<65>5 =L@G762 <45 C2@ K85;:2 5LCJK 465@J8J49>C54 ?C5@ 2 56JIJLG>5 x(n)

9>@54 5G;J>=4µ2 65C’ 58?ILC2 C4µ7, O2I5O7:

(2.8)

*;4@ 8;JTA;7=JLµ9 =C4H 4O4?C2C9H CJL DTFT, 5H OJKµ9 J;4=µQ@5 85;5O9><µ5C5

L8JIJ<4=µJK CJL.

¶·Ú¿‰ÂÈÁÌ· 2.1

$5 L8JIJ<4=C9> J DTFT C2H 56JIJLG>5H x(n) = anu(n), ?8JL |a| < 1.

=G/0:

(8? CJ@ J;4=µ? (2.1) QTJLµ9:

x n

n

( ) < "=#"

"

5

X e x n e

x n e e x n e X e

j k j k n

n

j n

n

j kn j n

n

j

( ) ( )

( ) ( ) ( ).

( ) ( )6 ) 6 )

6 ) 6 6

+ +2 2

2

=

= = =

#

=#"

"

#

=#"

"# #

=#"

"

5

5 5

X X(e ) (e( +2j k j6 ) 6= ).

X e j( )@

X e x n e a u n e a e ae

aej j n

n

n j n

n

n j n

n

j nj

n

( ) ( ) ( ) ( )6 6 6 6 66= = = = =

##

=#"

"#

=#"

"#

=

"#

#=

"

5 5 5 50 0

1

1


)J µQC;J 654 2 3B=2 C2H X(ej@) 35>@J@C54 =CJ +T7µ5 2.1. *5;5C2;JKµ9, ?C4 J4

=L@5;C7=94H 5LCQH 9>@54 =L@9T9>H 654 89;4JO46QH AH 8;JH @ µ9 89;>JOJ 2'. !8>=2H,

µ8J;JKµ9 9K6JI5 @5 OJKµ9 ?C4 J DTFT C2H x(n) L8B;T94,

53JK < " (<45 |a| < 1).

x n a u n aan

n

n

n

n

( ) ( )=#"

"

=#"

"

=

"

5 5 5= = =#1

10


™¯‹Ì· 2.1

MC/"( 1%0 <?83/() DTFT /34%1(6()59%4 x(n) =anu(n), |a| < 1 $0%(%) a > 0 1%0 (I) a < 0.

|X(ej@)|

|X(ej@)|

>X(ej@)

1(1-a)

1(1+a)

11+a

11-a

-2'

-2'

-2'

-2'

2'

2'

2'

2'

-'

-'

-'

-'

'

'

'

'

0

0

0

0

>X(ej@)

tan-1

@

@

@

@

(a/ 1-a2)

tan-1 (|a|/ 1-a2)

–tan-1(|a|/ 1-a2)

–tan-1(a/ 1-a2) (5)

(S)


5 6 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

¶·Ú¿‰ÂÈÁÌ· 2.2

$5 L8JIJ<4=C9> J DTFT C2H µJ@5O45>5H 6;JL=C467H 56JIJLG>5H x(n) = *(n).

=G/0:

(8? CJ@ J;4=µ? (2.1) QTJLµ9:

'4 <;5346QH 85;5=CB=94H C2H *(n) 654 CJL µ9C5=T2µ5C4=µJK Fourier 5LC7H 35>@J-

@C54 =CJ +T7µ5 2.2. *5;5C2;JKµ9, ?C4 2 µJ@5O45>5 6;JL=C467 QT94 µ9C5=T2µ5C4=µ?

Fourier J J8J>JH =L@>=C5C54 =C2@ 4=?8J=2 =L@94=3J;B ?IA@ CA@ =LT@JC7CA@.

X e x n e n e ej j n

n

j n

n

j( ) ( )6 6 6 6= = = ! =#

=#"

"#

=#"

"#5 5 *( ) 1 10

F8JIJ<>=C9 CJ@ DTFT C2H µ9C5CJ84=µQ@2H µJ@5O45>5H 6;JL=C467H 56JIJLG>5H

x(n) = *(n–M) 654 =T9O4B=C9 C4H 5@C>=CJ4T9H <;5346QH 85;5=CB=94H.


F8JIJ<>=C9 CJ@ DTFT CJL C9C;5<A@46JK 85IµJK 654 =T9-

O4B=C9 C4H 5@C>=CJ4T9H <;5346QH 85;5=CB=94H. x n

n

n( )

, | |

, | |,=

&>

012

1 2

0 2


1

0 0n @

1

*(n) X(ej@)

-2' -' ' 2'

™¯‹Ì· 2.2

(%) O µ(,%*0%9%1"()8/012 %1(-

6()59% 1%0 (I) ( µ./%8E3µ%/0-8µ#4 Fourier *0%-

1"0/(7 E"#,()%)/24.

' µ9C5=T2µ5C4=µ?H Fourier (DTFT) CA@ 84J <@A=CM@ =2µBCA@ O456;4CJK T;?@JL

O>@9C54 58? CJ@ *>@565 2.1.


¶›Ó·Î·˜ 2.1

M./%8E3µ%/08µ#4 Fourier $,@8/+, %1(6()50+,

!%µ1 J'(1/"0µ1()/µD5

*(n) 1, – " < @ < "

*(n – M) e–j@,

u(n)

%nu(n), |%| < 1

sin(@0n)

cos(@0n)

u(n) – u(n – M)

2.1.2 £ÂˆÚ‹Ì·Ù· Î·È È‰ÈfiÙËÙÂ˜ ÙÔ˘ ÌÂÙÛ¯ËÌ·ÙÈÛÌÔ‡ Fourier ‰È·ÎÚÈÙÔ‡ ¯ÚfiÓÔ˘

' DTFT 85;JL=4BZ94 J;4=µQ@9H 8JIK =2µ5@C46QH 4O4?C2C9H, J4 J8J>9H 9>@54 T;7=4-

µ9H <45 C2@ 5@BIL=2 =2µBCA@ 654 =L=C2µBCA@ O456;4CJK T;?@JL. # 58?O94:7 CJLH,

=C2@ 8I94J@?C2C5 CA@ 89;48CM=9A@, <>@9C54 9K6JI5, ?C5@ 8B;JLµ9 AH 539C2;>5 CJLH

J;4=µJKH (2.1) 654 (2.2). D5 5@53Q;JLµ9 =C2 =L@QT945 C4H 84J <@A=CQH 58? 5LCQH.

,9 G5 =LµSJI>ZJLµ9 CJ Z9K<JH CA@ µ9C5=T2µ5C4=µM@ Fourier

O456;4CJK T;?@JL, ?8AH 5LC? O>@9C54 58? C4H =TQ=94H (2.1) 654 (2.2).

°ƒ"ªª#'$&%&"

!B@ 654

C?C9 (2.9) a x n a x n a X e a X eF j j1 1 2 2 1 1 2 2( ) ( ) ( ) ( )+ ? @A +6 6

x n X eF j2 2( ) ( )? @A 6

x n X eF j1 1( ) ( )? @A 6

x n X eF j( ) ( )? @A 6

sin(sin(

( )/66M / 2)

/ 2)e j M# #1 2

) 6 6 ) 6 6 )[ ( (0 0* *# + + ++

=#"

"

5 2 2k kk

) )]

) ) )4 6 6 4 6 6j

k kk

[ ( (0 0# + # ++=#"

"

5 2 2) )]

2 2) 6 6 )*( 0# +

=#"

"

5 kk

) e j n@0

1

1 # #%e j6

1

12

#+ ##

=#"

"

5ek

jk

6 6 )8 )*(



5 8 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

(LC? =2µ5>@94 ?C4 J µ9C5=T2µ5C4=µ?H Fourier 9>@54 Q@5H <;5µµ46?H µ9C5=T2µ5C4-

=µ?H, 65CBII2IJH <45 C2 µ9IQC2 <;5µµ46M@ =L=C2µBCA@. PC=4, J DTFT CJL <;5µ-

µ46JK =L@OL5=µJK OKJ 7 89;4==JCQ;A@ =2µBCA@ 4=JKC54 µ9 CJ <;5µµ46? =L@OL5-

=µ? CA@ DTFT CJL 6BG9 =7µ5CJH.

$§#™£%™% ™&$ Ãƒ$¡$

!B@

C?C9 (2.10)

6.DI')B0

P=CA g(n) = x(n–n0). (8? CJ@ J;4=µ? (2.1) CJL DTFT QTJLµ9:

DQCJ@C5H q = n–n0, J8?C9 n = q + n0, 2 C9I9LC5>5 =TQ=2 <>@9C54:

.

# 4O4?C2C5 5LC7 =2µ5>@94 ?C4, 9B@ Q@5 =7µ5 µ9C5CJ84=C9> =CJ 89O>J CJL T;?@JL 65CB

n0 O9><µ5C5, C?C9 CJ 3B=µ5 CJL µQC;JL CJL 85;5µQ@94 5@5IIJ>ACJ. (LC? 8JL 5IIB-

Z94 65CB –@n0 9>@54 CJ 3B=µ5 C2H 3B=2H. 0>@9C54 35@9;?, 98JµQ@AH, ?C4 CJ 89;49T?-

µ9@J CA@ =LT@JC7CA@ 9@?H =7µ5CJH 9:5;CBC54 µ?@J@ 58? C2 µJ;37 CJL 654 ?T4 58?

C2 GQ=2 CJL. (8? µ5G2µ5C467 B8JR2, 5LC? 963;BZ9C54 AH 9:7H: 2 JI>=G2=2 (µ9C5-

C?84=2) =CJ 89O>J CJL T;?@JL 65CB n0, 4=JOL@5µ9> µ9 CJ@ 8JII58I5=45=µ? CJL

3B=µ5CJH 98> =CJ 89O>J C2H =LT@?C2C5H.

¶·Ú¿‰ÂÈÁÌ· 2.3

$5 L8JIJ<4=C9> J DTFT C2H 56JIJLG>5H .

=G/0:

+T9O4BZJ@C5H C2@ 56JIJLG>5 g(n), 85;5C2;JKµ9 ?C4 5LC7 QT94 C2@ >O45 µJ;37 µ9 C2@

x(n) C2H Y=62=2H ALCJ5:4JI?<2=2H 2.2, µ?@J 8JL 9>@54 µ9C5CJ84=µQ@2 8;JH C5

O9:4B 65CB OKJ O9><µ5C5, O2I5O7 g(n) = x(n–2). PC=4, µ8J;JKµ9 @5 58J3K<JLµ9 CJ@

L8JIJ<4=µ? CJL µ9C5=T2µ5C4=µJK Fourier G(ej@) µ9 SB=2 CJ@ J;4=µ? (2.1), T;2=4-

g n

n( )

,

,= & <0

12

5IIJK

1 0 5

0

e j n# 6 0

G e x q e e x q e e X ej j q n

q

j n j q j n j

q

( ) ( ) ) ( )( )6 6 6 6 6 6= = =# +

=#"

"# # #

=#"

"

5 50 0 0(

G e g n e x n n ej j n

n

j n

n

( ) ( ) )6 6 6= = ##

=#"

"#

=#"

"

5 5 ( 0

x n n e X eF j n j( ) ( )# ? @A #0

06 6

x n X eF j( ) ( )? @A 6


µJ8J4M@C5H C2@ 4O4?C2C5 C2H µ9C5C?84=2H =CJ T;?@J 65CB n0 = 2. Y;5,

*5;5C2;JKµ9 ?C4 CJ 58JCQI9=µ5 5LC? =Lµ3A@9> µ9 C2@ C9I9LC5>5 =TQ=2 CJL *>@5-

65 2.1 <45 M = 5, 53JK 8;B<µ5C4 2 56JIJLG>5 g(n) <;B39C54 654 AH g(n) =

u(n)–u(n–5).

G e F g n F x n e X e ej j j j( ) { ( )} { ( )} ( )sin( )

sin( )

6 6 6 6

6

6= = # = =# #2

52

2

2 2


F8JIJ<>=C9 CJ@ DTFT C2H x(n)–x(n–1), 5@ <@A;>Z9C9 ?C4 J µ9C5=T2µ5C4=µ?H

Fourier C2H x(n) 9>@54 X(ej@).


F8JIJ<>=C9 CJ@ DTFT C2H 54C45C7H 56JIJLG>5H x(n) = >ancos(@0n + <)u(n), ?8JL

>, a, @0, < 8;5<µ5C46J>.


$§#™£%™% ™&% ™ÀÃ¡$&%&"

!B@

C?C9 (2.11)

+Kµ3A@5 µ9 C2@ 4O4?C2C5 5LC7, J 8JII58I5=45=µ?H µ45H 56JIJLG>5H x(n) 98>

4=JOL@5µ9> µ9 C2@ JI>=G2=2 (µ9C5C?84=2) CJL 3B=µ5CJH X(ej@) 65CB @0. # 4O4?C2-

C5 5LC7 G5 µ5H 35@9> 4O45>C9;5 T;7=4µ2 =C2 O45O465=>5 C2H =T9O>5=2H R23456M@

3>IC;A@ 8JL G5 <@A;>=JLµ9 =CJ &93BI54J 4.

£!øƒ%ª" &%™ ™À¡!§#•%™

!B@ 654

C?C9 (2.12) x n x n x n X e X e X eF j j j( ) ( )* ( ) ( ) ( ) ( )= ? @A = !1 2 1 26 6 6

x n X eF j2 2( ) ( )? @A 6

x n X eF j1 1( ) ( )? @A 6

e j n6 0

e x n X ej n F j6 6 60 0( ) ( )( )? @A #

x n X eF j( ) ( )? @A 6


6 0 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

6.DI')B0:

< 5IIBZJ@C5H C2 =94;B CA@ 5G;J4=µBCA@ 654 GQCJ@C5H 9848IQJ@ q = n–m QTJLµ9 >

)J G9M;2µ5 C2H =L@QI4:2H 9>@54 58? C5 8IQJ@ =2µ5@C46B 9;<5I9>5 =C2@ 5@BIL=2

<;5µµ46M@ =L=C2µBCA@. 'L=45=C46B, µ5H IQ94 ?C4 2 =L@QI4:2 OKJ 56JIJLG4M@ =CJ

89O>J CJL T;?@JL, 4=JOL@5µ9> µ9 CJ <4@?µ9@J CA@ 35=µBCA@ CJLH =CJ 89O>J C2H

=LT@?C2C5H. )J <9<J@?H 5LC? 9>@54 µ9<BI2H =2µ5=>5H =C2@ 5@BIL=2 =2µBCA@ 654

=L=C2µBCA@ 654 =C2@ 65C5@?2=2 CJL C;?8JL µ9 CJ@ J8J>J Q@5 LTI =K=C2µ5 58J-

6;>@9C54 =9 6B8J45 9>=JOJ 8JL 935;µ?Z9C54 =’ 5LC?. DLµ2G9>C9 C2 <;5µµ467 =L@Q-

I4:2, ?8AH C2 <@A;>=5µ9 =C2@ 9@?C2C5 1.4. !>O5µ9 ?C4, 5@ =’ Q@5 =K=C2µ5 µ9 6;JL-

=C467 58?6;4=2 h(n) 935;µJ=C9> CJ =7µ5 x(n), C?C9 2 Q:JOJH CJL =L=C7µ5CJH y(n)

G5 4=JKC54 µ9 C2 <;5µµ467 =L@QI4:2 C2H 94=?OJL 654 C2H 6;JL=C467H 5LCJK, O2I5-

O7 y(n) = x(n)*h(n). !8JµQ@AH, =Kµ3A@5 µ9 CJ G9M;2µ5 C2H =L@QI4:2H, G5 4=TK94

Y(ej@) = X(ej@) H(ej@), ?8JL X(ej@), Y(ej@), H(ej@) 9>@54 J4 µ9C5=T2µ5C4=µJ> Fourier

CA@ 56JIJLG4M@ x(n), y(n), h(n) 5@C>=CJ4T5.

£!øƒ%ª" &$À PARSEVAL

!B@ C?C9

(2.13)

6.DI')B0:

+C2@ 8;J694µQ@2 89;>8CA=2 µ5H =Lµ3Q;94 @5 5;T>=JLµ9 C2@ 58?O94:2 58? CJ O9K-

C9;J µQIJH C2H =TQ=2H. PTJLµ9 IJ48?@:

x n X e dn

j( ) ( )2 2

2

12=#"

"

5 ==)

66

)

x n X eF j( ) ( )? @A 6

= # =

= =

=#"

"#

=#"

"

=#"

"# +

=#"

"

#

=#"

"#

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55 55

55

x m x n m e x m q e

x m e x q e X e X e

j

m q

j q m

m

j m

q

j q

m

j j

1 2 1 2

1 2 1 2

( ) ) ( ) )

( ) ) ( ) ( )

)

n

n (( x (

(

6 6

6 6 6 6

X e x n e x n x n e

x m x n m e

j j n

n

j n

n

m

j n

n

( ) ( ) [ ( )* ( )]

( ) )]

6 6 6

6

= =

= #

#

=#"

"#

=#"

"

=#"

"#

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5 5

55

1 2

1 2[ (


)J G9M;2µ5 CJL Parseval 5@53Q;9C54 =C2 O45C7;2=2 C2H 9@Q;<945H 65CB C2 µ9CB-

S5=2 58? CJ 89O>J CJL T;?@JL =CJ 89O>J C2H =LT@?C2C5H. # 8J=?C2C5

5@53Q;9C54 654 AH <?8µ% .,."$.0%124 ')1,#/3/%4 (energy–density spectrum) CJL

=7µ5CJH x(n).

!6C?H 58? C4H 4O4?C2C9H 8JL 5@53Q;5µ9, L8B;TJL@ 654 8JIIQH BII9H C4H J8J>9H 58IMH

85;5GQCJLµ9 =CJ@ *>@565 2.2.

X e j( )6 2

12

12

12

12

2

2 2 2

2

)6

)6

)6

)6

6

)

6 6

)

6 6

)

6

)

6

X e d X e X e d x n e X e d

x n X e e d x n x n x

j j j j n

n

j

n

j j n

n

( ) ( ) ( ) ( ( ) ) ( )

( ) ( ) ( ) ( ) (

* *

* *

= = =

= =

= = 5=

5 = 5

#

=#"

"

=#"

"#

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nnn

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5


$5 58JO94TC9> 2 4O4?C2C5 CJL 8JII58I5=45=µJK OKJ =2µBCA@ O456;4CJK T;?@JL.


¶›Ó·Î·˜ 2.2

V*0#/3/.4 /() µ./%8E3µ%/08µ(7 Fourier *0%1"0/(7 E"#,() (DTFT)

8I)D(0(1 J0 >'C)#I)*D !%µ1 J'(1/"0µ1()/µD5 Fourier

89;4JO46QH =L@5;C7=94H µ9

89;>JOJ 2'

0;5µµ46?C2C5 a1x1(n) + a2x2(n) a1X1(ej@) + a2X2(e

j@)

,9C5C?84=2 =CJ T;?@J x(n – n0) X(ej@)

,9C5C?84=2 =C2 =LT@?C2C5 x(n)

+LZL<7H 56JIJLG>5 x*(n) X*(e–j@)

&5CJ8C;4=µ?H =CJ T;?@J x( – n) X(e–j@)

!8Q6C5=2 T;?@JL x(n/k) <45 n 8JII58IB=4J CJL k X(ejk@)

+L@QI4:2 x1(n)*x2(n) X1(ej@)X2(e

j@)

X e j( )( )6 6# 0 e j n6 0

e j n# 6 0

X e

X e

j

j

1

2

( )

( )

6

6

BCD

ED

x n

x n1

2

( )

( )

BCE


6 2 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

2.1.3 ¢ÂÈÁÌ·ÙÔÏË„›·: ÌÈ· ·ÎfiÌË Ì·ÙÈ¿

0@A;>=5µ9 C2 O94<µ5CJI2R>5 =C2@ 9@?C2C5 1.2 CJL &935I5>JL 1 654 9>O5µ9 ?C4 5LC7

QT94 @5 6B@94 µ9 C2 µ9C5C;J87 9@?H 5@5IJ<46JK =7µ5CJH =9 R23456?. D5 5=TJI2-

GJKµ9 654 8BI4 µ9 C2 O94<µ5CJI2R>5, 8;J=9<<>ZJ@C5H, ?µAH C2 3J;B 5LC7, CJ ?IJ

GQµ5 58? µ>5 O453J;9C467 J8C467 <A@>5. 0@A;>ZJ@C5H, 8IQJ@, CJ µ9C5=T2µ5C4=µ?

Fourier 654 CJ 89O>J C2H =LT@?C2C5H, G5 9:9CB=JLµ9 C5 =7µ5C5 58? CJ 89O>J C2H

=LT@?C2C5H 654 G5 8;J=85G7=JLµ9 @5 65C5@J7=JLµ9 C2@ 98>O;5=2 8JL QT94 2 O94<-

µ5CJI2R>5 =’ 5LCB. ' =C?TJH µ5H G5 9>@54 654 8BI4 J >O4JH, O2I5O7 J 8;J=O4J;4=µ?H

C2H 65CBII2I2H 89;4?OJL O94<µ5CJI2R>5H ;, QC=4 M=C9, @5 µ2 TB@9C54 8I2;J3J;>5,

654 @5 9>µ5=C9 8B@CJC9 =9 GQ=2 @5 5@565C5=69LB=JLµ9 CJ 5;T46? 5@5IJ<46? =7µ5

58? C5 O9><µ5CB CJL.

P=CA IJ48?@, ?C4 CJ 5@5IJ<46? =7µ5 xa(t) C;J3JOJC9>C54 =C2@ 9>=JOJ 9@?H 4O5@46JK

O94<µ5CJI78C2. # Q:JOJH 5LCJK G5 9>@54 2 56JIJLG>5 x(n), C5 =CJ4T9>5 C2H J8J>5H

5@C4=CJ4TJK@ =C4H µ9C;7=94H CJL 8IBCJLH C2H xa(t) 5@B T;J@46B O45=C7µ5C5 T O9L-

C9;JIQ8CA@

*JII58I5=45=µ?H x1(n)x2(n)

E453?;4=2 =C2 =LT@?C2C5 nx(n)

+LZL<7H =Lµµ9C;>5 <45 x(n) 8;5<µ5C467

8;5<µ5C46B =7µ5C5

+Lµµ9C;>5 <45 8;5<µ5C46B

654 B;C45 =7µ5C5 x(n) 8;5<µ5C467 654 B;C45 X(ej@) 8;5<µ5C467 654 B;C45

+Lµµ9C;>5 <45 8;5<µ5C46B

654 89;4CCB =7µ5C5 x(n) 8;5<µ5C467 654 89;4CC7 X(ej@) 35@C5=C467 654 89;4CC7

D9M;2µ5 CJL Parseval

x n X e dn

j( )8

( )8

2 2

2

12

==#"

"

5 = 6 6

X e X e

X e X e

X e X e

X e X e

X e X e

j j

j j

j j

j j

j j

( ) ( )

Re{ ( )} Re{ ( )}

Im{ ( )} Im{ ( )}

( ) ( )

( ) ( )

*6 6

6 6

6 6

6 6

6 6

=== #

=

> = #>

0

1

DDD

2

DDD

#

#

#

#

#

j

dX ed

j( )6

6

12 1

2

2)%%

)

6 %X e X e dj j( ) ( )(= # )


x(n) = xa(nT) (2.14)

# 85;Bµ9C;JH T 58JC9I9> C2@ '."9(*( *.0$µ%/(63H9%4 654 CJ 5@C>=C;J3J 5LC7H C2

=)E,#/3/% 2 /( ")5µ# *.0$µ%/(63H9%4 (Fs = 1/T), O2I5O7, CJ 8I7GJH CA@ O94<µBCA@

8JL I5µSB@J@C54 =C2 µJ@BO5 CJL T;?@JL.

045 @5 9>µ5=C9 =9 GQ=2 @5 5@565C5=69LB=JLµ9 CJ 5@5IJ<46? =7µ5 xa(t) 58? C2@

56JIJLG>5 x(n) G5 8;Q894 C5 35=µ5C46B T5;56C2;4=C46B CJL 5@5IJ<46JK =7µ5CJH

@5 O45C2;JK@C54 µ9CB C2 O94<µ5CJI2R>5. P=CA Xa(jQ) J µ9C5=T2µ5C4=µ?H Fourier

CJL =7µ5CJH xa(t) 654 X(ej@) J µ9C5=T2µ5C4=µ?H Fourier O456;4CJK T;?@JL C2H 56J-

IJLG>5H x(n). (8JO946@K9C54 ?C4

(2.15)

# =TQ=2 5LC7 O2IM@94 ?C4 CJ 3B=µ5 CJL =7µ5CJH 8JL 8;JQ6LR9 58? C2 O94<µ5CJ-

I2R>5, 58JC9I9>C54 58? 89;4JO46QH 985@5I7R94H CJL 3B=µ5CJH CJL 5@5IJ<46JK

=7µ5CJH. 0;5346B 5LC? 85;4=CB@9C54 =CJ +T7µ5 2.3. +CJ +T7µ5 2.35 35>@9C54 CJ

3B=µ5 CJL 5@5IJ<46JK =7µ5CJH xa(t). # 56;4S7H µJ;37 CJL (C;4<A@467 =C2@ 8;J-

694µQ@2 89;>8CA=2) O9@ QT94 =2µ5=>5. (LC? 8JL QT94 =2µ5=>5, 9>@54 CJ ?C4 8;?694-

C54 <45 =7µ5 89;4J;4=µQ@JL 9K;JLH (band–limited), O2I5O7, ?I9H J4 =LT@?C2C9H

6Lµ5>@J@C54 µ9C5:K –Qmax 654 Qmax (–Qmax ] Q ] Qmax). +CJ +T7µ5 2.3S 35>@9C54

Q@5H 58? CJLH ?;JLH C2H 9:>=A=2H (2.15), 9@M =CJ +T7µ5 2.3< O>@J@C54 ?IJ4 J4 ?;J4

C2H 9:>=A=2H 5LC7H, O2I5O7 CJ 3B=µ5 CJL O94<µ5CJI28C2µQ@JL =7µ5CJH. 045 @5

9>µ5=C9 =9 GQ=2 @5 5@565C5=69LB=JLµ9 CJ 5;T46? =7µ5 58? C5 O9><µ5CB CJL, G5

8;Q894 @5 3;J@C>=JLµ9 M=C9 5LCQH (0 '."0(*01C4 .'%,%62H.04 ,% µ3, 1%67'(), 3 µ9%

/3, ?663. D5 8;Q894 O2I5O7 @5 465@J8J49>C54 2 =L@G762

Qmax; ] ' (2.16)

!894O7 Qmax = 2'Fmax, 2 =L@G762 5LC7 <;B39C54 AH

Fs ( 2Fmax (2.17)

8JL 9>@54 CJ <@A=C? µ5H G9M;2µ5 O94<µ5CJI2R>5H. +C2@ 89;>8CA=2 8JL 2 =L@G762

5LC7 O9@ 465@J8J49>C54, ?8AH 35>@9C54 =CJ +T7µ5 2.3O, C?C9 IQµ9 ?C4 QTJLµ9 CJ 354-

@?µ9@J C2H <%8µ%/0124 .'01?6)H34 (spectral overlap) 7 C2H E%µ36(7 ")5µ(7 *.0$µ%-

/(63H9%4 (aliasing / undersampling). +9 µ45 CQCJ45 89;>8CA=2 2 µJ;37 CJL 3B=µ5-

CJH CJL 5;T46JK =7µ5CJH O9 O45C2;9>C54 =CJ 3B=µ5 CJL O94<µ5CJI28C2µQ@JL =7µ5-

CJH, µ9 =L@Q8945 @5 µ2@ 9>@54 OL@5C7 2 56;4S7H 5@565C5=69L7 CJL 5;T46JK 5@5IJ-

<46JK =7µ5CJH.

X e

TX j

kT

ia

k

( ) ( )6 6 )= #

=#"

"

51 2



6 4 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

D9A;7=5µ9 =C5 8;J2<JKµ9@5 ?C4 CJ 5@5IJ<46? =7µ5 9>@54 89;4J;4=µQ@JL 9K;JLH

=LT@JC7CA@, ?8AH BIIA=C9 85;4=CB@9C54 =CJ +T7µ5 2.35. +C2@ 8;B:2 5LC? O9 =Lµ-

S5>@94 8B@CJC9. !848IQJ@, 5H µ2@ :9T@Bµ9, ?C4 J G?;LSJH 8JL 85;94=OK94 =9 6BG9

=K=C2µ5 96C9>@9C54 =L@7GAH =9 ?IJ CJ 9K;JH CA@ =LT@JC7CA@. PC=4, <45 @5 O45-

=35I>=JLµ9 CJ <9<J@?H ?C4 CJ =7µ5 94=?OJL 9>@54 89;4J;4=µQ@JL 9K;JLH, CJ8JG9-

CJKµ9 8B@CJC9 Q@5 5@5IJ<46? S5GL89;5C? 3>IC;J (analog lowpass filter) 8;4@ 58?

CJ =K=C2µ5 R234567H 989:9;<5=>5H =2µBCA@, <45 @5 58J6?8C94 C4H =LT@?C2C9H 8JL

9>@54 µ9<5IKC9;9H 58? Fs/2. P@5 85;?µJ4J 5@5IJ<46? 3>IC;J CJ8JG9C9>C54 654 =C2@

Q:JOJ CJL R23456JK =L=C7µ5CJH <45 C2 =A=C7 5@565C5=69L7 CJL =7µ5CJH 9:?OJL,

O2I5O7 <45 C2@ 984IJ<7 µ?@J CJL S5=46JK 35=µ5C46JK 89;49TJµQ@JL µ9C5:K –Fs/2

654 Fs/2. PC=4, Q@5 8I7;9H =K=C2µ5 R234567H 989:9;<5=>5H =2µBCA@ 9>@54 ?8AH

5LC? 8JL 35>@9C54 =CJ +T7µ5 2.4.

Xa(Q)

-Qmax Qmax

Q

1

(5)2QmaxT

(S)

@

@

2'k

1

Xa(@ - 2'k)

T

X(e j@)

1/T... ...

-4' 4'-2' 2'

(<)

-QmaxT QmaxT

X(e j@)

1/T

@

-4' 4'-2' ' 2'

(O)

™¯‹Ì· 2.3

O *.0$µ%/(63H9%%'# /( '.*9( /34

8)E,#/3/%4: (%) <?8µ% %,%6(-$01(7 82µ%/(4, (I)

C,%4 %'# /()4#"()4 /() <?8µ%-/(4 /() *.0$µ%/(-63'/3µC,() 82µ%-

/(4, ($) <?8µ%*.0$µ%/(63'/3µC-,() 82µ%/(4 $0%

Fs > 2Fmax, 1%0 (*) <?8µ% *.0$µ%-

/(63'/3µC,()82µ%/(4

$0% Fs < 2Fmax



+C2@ 9@?C2C5 5LC7 5=TJI2G765µ9 µ9 CJ µ9C5=T2µ5C4=µ? Fourier O456;4CJK T;?-

@JL (DTFT). !>O5µ9 ?C4 J µ9C5=T2µ5C4=µ?H 5LC?H 935;µ?Z9C54 <9@46B =9 µ2 89;4J-

O46B =7µ5C5 O456;4CJK T;?@JL B894;JL µ76JLH. O DTFT CQCJ4A@ =2µBCA@ 9>@54

=L@9T7H 654 89;4JO46?H. # <@M=2 CJL µ9C5=T2µ5C4=µJK Fourier µ5H SJ7G2=9 @5

8;J=9<<>=JLµ9 654 8BI4 CJ GQµ5 C2H O94<µ5CJI2R>5H, 5IIB 58? CJ 89O>J C2H =LT@?-

C2C5H C2 3J;B 5LC7, 58J=532@>ZJ@C5H CJ 354@?µ9@J C2H 35=µ5C467H 9846BILR2H,

CJ J8J>J 85;JL=4BZ9C54 ?C5@ J ;LGµ?H I7R2H O94<µBCA@ 9>@54 T5µ2I?C9;JH CJL

5854CJKµ9@JL.

2.2 ¢È·ÎÚÈÙfi˜ ÌÂÙ·Û¯ËÌ·ÙÈÛÌfi˜ Fourier

2.2.1 $ÚÈÛÌÔ›

!>O5µ9 =C2@ 9@?C2C5 2.1 ?C4 J µ9C5=T2µ5C4=µ?H Fourier 9@?H =7µ5CJH O456;4CJK

T;?@JL x(n) 9>@54 2 =L@9T7H =L@B;C2=2 X(ej@). # =L@B;C2=2 5LC7 O9@ 9>@54 9K6JIJ

@5 L8JIJ<4=C9> µ9 C2 T;7=2 9@?H R23456JK 989:9;<5=C7 =7µ5CJH, J J8J>JH 9>@54

=L@7GAH Q@5H <9@46JK =6J8JK L8JIJ<4=C7H 7 Q@5 94O46B =T9O45=µQ@J R23456?

6K6IAµ5. !69>@J 8JL 9>@54 9K6JIJ @5 L8JIJ<4=C9>, 9>@54 O9><µ5C5 CJL 3B=µ5CJH

X(ej@). (LC? 9>@54 CJ 5@C469>µ9@J µ9 CJ J8J>J 5=TJIJKµ5=C9 =C2@ 85;JK=5 9@?C2-

C5, O2I5O7 J L8JIJ<4=µ?H CJL 3B=µ5CJH =9 O456;4CQH =LT@?C2C9H (O456;4C?H µ9C5-

=T2µ5C4=µ?H Fourier), 65GMH 654 J4 4O4?C2C9H 8JL 4=TKJL@ =’ 5LC7 C2@ 89;>8CA=2.

¢#"'ƒ#&$™ ª!&"™Ã%ª"&#™ª$™ FOURIER (DFT)

' µ9C5=T2µ5C4=µ?H Fourier O456;4CJK T;?@JL (DTFT) C2H 8989;5=µQ@JL µ76JLH R

56JIJLG>5H x(n), O2I5O7 x(n) = 0 <45 n < 0 654 n ( N, 4=JKC54 µ9

, 0 ] A ] 28 (2.18)

?8JL C5 ?;45 CJL 5G;J>=µ5CJH 5@C465CJ8C;>ZJL@ CJ <9<J@?H ?C4 x(n) = 0 96C?H CJL X e x n ej j n

n

N

( ) ( )6 6= #

=

#

50

1

6 52 . 2 ¢ # " ' ƒ # & $ ™ ª ! &" ™ Ã % ª "& # ™ ª $ ™ F O U R I E R

™¯‹Ì· 2.4

T0?$"%µµ% I%5µ9*@, .,#4 8)8/2µ%/(4 H3<0%124 .'.F."$%89%4 %,%6($01+, 83µ?/@,

A@5IJ<46?[+7µ5

A@5IJ<46?[+7µ5

A@5IJ<46?[B5GL89;5C?[.>IC;J

A@5IJ<46?[B5GL89;5C?[.>IC;J

123456?H[E89:9;<5=C7H[+7µ5CJH

A/D D/A


6 6 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

O45=C7µ5CJH 0 ] n ] N–1. !B@ IBSJLµ9 O9><µ5C5 C2H =L@9TJKH =L@B;C2=2H X(ej@)

=9 4=58QTJ@C5 O45=C7µ5C5 =C2 =LT@?C2C5 @k = kE@ = k(2'/R), k = 0, 1,…,N–1, (UI.

+T7µ5 2.5), C?C9 G5 8B;JLµ9 C5 O9><µ5C5

(2.19)

(H =2µ94AG9> ?C4, 53JK 2 X(ej@) 9>@54 89;4JO467 µ9 89;>JOJ 2 ', µ?@J C5 O9><µ5C5

C2H S5=467H 89;4JT7H =LT@JC7CA@ 9>@54 585;5>C2C5. !894O7 C5 O9><µ5C5 5LCB 8;J-

6K8CJL@ 58? CJ@ L8JIJ<4=µ? CJL µ9C5=T2µ5C4=µJK Fourier X(ej@) =9 R 4=58QTJL-

=9H O456;4CQH =LT@?C2C9H, 2 =TQ=2 (2.19) J@JµBZ9C54 *0%1"0/#4 µ./%8E3µ%/08µ#4

Fourier (Discrete Fourier Transform, DFT) C2H x(n). (8JO946@K9C54 ?C4 µ8J;JKµ9

@5 5@565C5=69LB=JLµ9 C2@ 56JIJLG>5 x(n) 58? C5 O9><µ5C5 X(k) =C2 =LT@?C2C5,

µ9 SB=2 C2 =TQ=2

(2.20)

2 J8J>5 58JC9I9> CJ@ %,/98/"(<( *0%1"0/# µ./%8E3µ%/08µ# Fourier (inverse DFT,

IDFT).

x n

NX k e n Nj kn N

k

N

( ) ( ) , , , , ,/= = … #=

#

510 1 12

0

1)

X k X

kN

X x n e k Njk N

j kn N

n

N

( ) ) ( ) , , , , ) (2

(e =2 //' = = = … ##

=

#

5) 66 )

)2

0

1

0 1 1

-' 2''0 kT@ T@

X(e j@)

X(kT@)

@

™¯‹Ì· 2.5

T.0$µ%/(63H9%8/3 8)E,#/3/% /()µ./%8E3µ%/08µ(7

Fourier *0%1"0/(7E"#,() (DTFT).

'4 =TQ=94H (2.19) 654 (2.20) 58JC9IJK@ CJ I9<?µ9@J «Z9K<JH CJL DF)». *5;5C2;7-

=C9 ?C4, C?=J 2 56JIJLG>5 x(n), ?=J 654 2 56JIJLG>5 X(k) 9>@54 CJL >O4JL µ76JLH R.

D5 =LµSJI>ZJLµ9 CJ Z9K<JH CJL DFT R–=2µ9>A@ AH . '4 =TQ-

=94H 5LCQH µ8J;JK@ @5 963;5=CJK@ 654 AH

(2.21)

(2.22)

x n

NX W n NN

nk

k

N

( ) ( ) , , , , ,= = … ##

=

#

510 1 1

0

1

k

X x n W k NN

n

n

N

( ( ) , , , , ,k) k= = … #=

#

50

1

0 1 1

x n X kDFTN( ) ( )? @AA


?8JL 9: J;4=µJK

WN = e–j28/N (2.23)

9>@54 2 R8/2 ;>Z5 C2H µJ@BO5H. '4 85;B<J@C9H 58JC9IJK@ C4H µ4<5O46QH 56J-

IJLG>9H SB=2H =C4H J8J>9H =C2;>Z9C54 J DFT 654 J@JµBZJ@C54 '%"?$(,/.4 8/"(<24

(twiddle factors). +CJ +T7µ5 2.6 35>@9C54 2 µ4<5O467 5@585;B=C5=2 CA@ WN 654

5@5O946@K9C54 2 6L6I467 4O4?C2C5 CA@ 85;5<?@CA@ =C;J37H <45 Q@5 DFT 8–=2µ9>-

A@ (N = 8).

# µ9CBS5=2 58? CJ 89O>J CJL T;?@JL =CJ 89O>J C2H =LT@?C2C5H 654 5@C4=C;?3AH,

µ9 C2 SJ7G945 CJL Z9K<JLH CA@ µ9C5=T2µ5C4=µM@ Fourier (9:4=M=94H 2.21 654 2.22),

58946J@>Z9C54 =CJ +T7µ5 2.7.

WNnk

6 72 . 2 ¢ # " ' ƒ # & $ ™ ª ! &" ™ Ã % ª "& # ™ ª $ ™ F O U R I E R

W6[8

W4[8

W5[8 W7[

8

W0[8 =1

W1[8

W2[8 =-W6[8

=-W5[8W3[8 =-W7[8

X(k)=+x(n)WnkN

N-1

n=0

x(n)= +X(k)W-nkN

N-1

k=0

DFT

IDFT

x(n) X(k)

1N

™¯‹Ì· 2.6

!%"?$(,/.4 8/"(<24 $0% C,% DFT 8–83µ.9@, (R = 8).

™¯‹Ì· 2.7

&E3µ%/012 %,%'%"?8/%83 /34 µ./?I%834 %'# /( '.*9(/() E"#,() 8/( '.*9( /34 8)E,#/3/%4 1%0 %,/08/"#<@4 µ.E"283 /() L.7$()4 DFT.

(8JO946@K9C54 8JIK 9K6JI5 58? CJLH J;4=µJKH CJL Z9K<JLH DFT, ?C4 J4 56JIJLG>9H

x(n) 654 X(k) 9>@54 89;4JO46QH µ9 89;>JOJ R, O2I5O7

x(n + N) = x(n) <45 ?I5 C5 n (2.24)

X(k + N) = X(k) <45 ?I5 C5 k (2.25)


6 8 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

¶·Ú¿‰ÂÈÁÌ· 2.4

$5 L8JIJ<4=G9> J O456;4C?H µ9C5=T2µ5C4=µ?H Fourier C2H 56JIJLG>5H {1, 1, 0, 0}.

=G/0:

# 56JIJLG>5 x(n) 58JC9I9>C54 58? CQ==9;5 O9><µ5C5 (R = 4), C5 9:7H: x(0) = 1, x(1)

= 1, x(2) = 0, x(3) = 0. ,9 SB=2 C2 =TQ=2 (2.21) L8JIJ<>ZJLµ9 C4H CQ==9;4H =L@4-

=CM=9H C2H =LT@?C2C5H:

?8JL

\µJ45 58JO946@K9C54 ?C4 X(2) = 0 654 X(3) = 1 + j. *5;5C2;7=C9 ?C4 2 =L@4=CM=5

µ2O9@467H =LT@?C2C5H X(0) 4=JKC54 8B@CJC9 µ9 CJ BG;J4=µ5 ?IA@ CA@ =CJ4T9>A@ C2H

56JIJLG>5H x(n). Y;5 J DFT C2H 56JIJLG>5H {1, 1, 0, 0} 4=JKC54 µ9 C2 µ4<5O467 56J-

IJLG>5 {2, 1–j, 0, 1 + j}. 045 @5 9>@54 84J 65C5@J2C7 2 5@585;B=C5=2 CJL 58JC9IQ-

=µ5CJH, =L@2G>Z9C54 @5 963;BZJLµ9 CJLH µ4<5O46JKH 5;4GµJKH µ9 CJ µQC;J 654 C2

3B=2 CJLH. PC=4, J DFT 8JL µ?I4H L8JIJ<>=5µ9, 963;BZ9C54 AH

.

'4 <;5346QH 85;5=CB=94H C2H 56JIJLG>5H x(n) 654 CJL 5@C>=CJ4TJL DFT X(k), 35>-

@J@C54 =CJ +T7µ5 2.8.

2 2 0 24 4, , , e ej j#0

1D

2D

BCD

ED

) )

W e e j j j

j j

43

2 34

32 3

232

0 1= = = # = # # =# #) ) ) )

cos sin ( )

W e e jj

j42

2 24 1= = = # = #

# #)

) ) )cos sin

W e e j j j

j j

41

24 2

2 20= = = # = # = #

# #) ) ) )cos sin

X x n W x W x W x W x W

x x j x x j j j j

n

n

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 0 1 2 3

0 1 2 1 3 1 1 0 1 0 1

41

401

411

421

431

0

3

= = + + + =

= + ! # + ! # + ! = + ! # + ! # + ! = #=

5

X x n W x W x W x W x W

x x x x

n

n

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

0 0 1 2 3

0 1 2 3 1 1 0 0 2

40

400

410

420

430

0

3

= = + + + =

= + + + = + + + ==

5


'À'§#'% $§#™£%™% "'$§$À£#"™

# 89;4JO467 98Q6C5=2 C2H 8989;5=µQ@JL µ76JLH 56JIJLG>5H x(n), 0 ] n ] N–1, 5@B

R O9><µ5C5, JO2<9> =C2@ 89;4JO467 56JIJLG>5

(2.26)

µ9 S5=467 89;>JOJ R, ?8AH 35>@9C54 =C5 +T7µ5C5 2.95,S <45 R = 6. # JI>=G2=2

(µ9C5C?84=2) C2H 89;4JO467H 56JIJLG>5H xp(n) 65CB n0 O9><µ5C5 8;JH C5 O9:4B, G5

µ5H OM=94 µ>5 BII2 89;4JO467 56JIJLG>5

(2.27)

# 8989;5=µQ@JL µ76JLH 56JIJLG>5

(2.28) x n

x n n Np'' 0

5IIJK( )

( ),

,=

& & #012D

1

0

x n x n n x n n qNp p

q

' ( ) ( ) ( )= # = # #=#"

"

50 0

x n x n qNp ( ) ( )= #=#"

"

5q

6 92 . 2 ¢ # " ' ƒ # & $ ™ ª ! &" ™ Ã % ª "& # ™ ª $ ™ F O U R I E R

x(n)|X(k)| X(k)

0 1

11

2 3 0

0

1

1

2 3

2 3

2F2 F2

n k

k

__- '4

__'4

(5) (S)

™¯‹Ì· 2.8

(%) O %1(6()59%x(n) 1%0 (I) /( µC/"( 1%0 3<?83 /() DFT%)/24.

$5 L8JIJ<4=C9> J DFT C2H µJ@5O45>5H 6;JL=C467H.


$5 L8JIJ<4=C9> J DFT C2H =C5G9;7H 56JIJLG>5H x n

A n N( )

, , , ...,

, = = #0

12

5IIJK

0 1 1

0



7 0 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

=L@OQ9C54 µ9 C2@ 5;T467 56JIJLG>5 x(n) µQ=A C2H 1)160124 (69853834. +CJ +T7µ52.9< 35>@9C54 2 56JIJLG>5 x'p(n) = xp(n–n0) <45 n0 = 2 654 =CJ +T7µ5 2.9O 2 8989-;5=µQ@JL µ76JLH 56JIJLG>5 x'(n) (N = 6).

09@46B, 2 6L6I467 JI>=G2=2 µ45H 56JIJLG>5H µ8J;9> @5 85;5=C5G9> µ9 C2 SJ7G945CA@ L8JIJ>8A@ (modulo) AH

x'(n) = x( < n–n0 > N) (2.29)

?8JL J =LµSJI4=µ?H < m > N O45SBZ9C54 AH m modulo N 654 =2µ5>@94 CJ L8?IJ48JC2H O45>;9=2H CJL m O45 CJL R. 045 n0 > 0 (6L6I467 JI>=G2=2 8;JH C5 O9:4B), 2 =TQ=25LC7 =2µ5>@94 ?C4

(2.30)

# 6L6I467 JI>=G2=2 65CB n0, ?8JL n0 > R, 4=JOL@5µ9> µ9 C2@ 6L6I467 JI>=G2=265CB < n0 > N. 045 CJ 85;BO94<µ5 C2H 56JIJLG>5H x(n) = {1,2,3,4,5,6} CJL +T7µ5CJH2.9, 2 6L6I467 JI>=G2=2 65CB OKJ G5 µ5H OM=94 C2@ 56JIJLG>5 x'(n) = x( < n–2 > 6)= {5,6,1,2,3,4}. ' L8JIJ<4=µ?H C2H 6L6I46B JI4=G2µQ@2H 56JIJLG>5H x'(n) =Kµ3A-@5 µ9 CJ@ J;4=µ? (2.30) QT94 AH 9:7H:

x'(0) = x( < 0–2 > 6) = x(0–2 + 6) = x(4) = 5

x'(1) = x( < 1–2 > 6) = x(1–2 + 6) = x(5) = 6

x'(2) = x( < 2–2 > 6) = x(0) = 1

x'(3) = x( < 3–2 > 6) = x(1) = 2

x'(4) = x( < 4–2 > 6) = x(2) = 3

x'(5) = x( < 5–2 > 6) = x(3) = 4

+Lµ89;5>@JLµ9, 98JµQ@AH, ?C4 3 1)16012 (6985383 µ0%4 %1(6()59%4 R–83µ.9@, 08(-*),%µ.9 µ. /3 $"%µµ012 (6985383 /34 '."0(*0124 /34 .'C1/%834.

P@5H 85;5=C5C46?H C;?8JH, <45 C2@ 65C5@?2=2 C2H 6L6I467H JI>=G2=2H µ45H 8989-;5=µQ@JL µ76JLH 56JIJLG>5H, 9>@54 5LC?H C2H CJ8JGQC2=2H CA@ =CJ4T9>A@ C2H =C2@89;43Q;945 9@?H 6K6IJL, ?8AH 35>@9C54 =CJ +T7µ5 2.99 <45 C2@ 56JIJLG>5 x(n) Q:4=CJ4T9>A@. )5 =CJ4T9>5 CJ8JG9CJK@C54 =9 4=58QTJ@C5 =2µ9>5 =C2@ 89;43Q;945 9@?H6K6IJL 654 µ9 3J;B 5@C>G9C2 C2H 6>@2=2H CA@ O946CM@ CJL A;JIJ<>JL (5;4=C9;?-=C;J35). &L6I467 JI>=G2=2 65CB n0 =2µ5>@94 C2@ JI>=G2=2 ?IA@ CA@ =CJ4T9>A@ 65CBn0 GQ=94H 5;4=C9;?=C;J35 7 O9:4?=C;J35, 5@BIJ<5 µ9 CJ 5@ CJ n0 9>@54 G9C46? 7 5;@2-C46? 5@C>=CJ4T5. +CJ +T7µ5 2.9=C 35>@9C54 2 x( < n–2 > 6), 2 J8J>5 8;JQ6LR9 58? C2@x(n) CJL +T7µ5CJH 2.99 µ9 6L6I467 JI>=G2=2 5;4=C9;?=C;J35 65CB n0 = 2.

x n n

x n n N n n

x n n n n NN( )

( ) , , ,

( ) , , < # > =

# + = ## = #

012

00 0

0 0

0 1 1

1

<45

<45

K

K


7 12 . 2 ¢ # " ' ƒ # & $ ™ ª ! &" ™ Ã % ª "& # ™ ª $ ™ F O U R I E R

x(n)

xp(n)

xp(n–2)

x'(n)=x(<n–2>6)

1

1 1

2

2 2 2

3

3

3

3 3

4

4

4

4

5

5

5

5

-5 -4 -3 -2

-2

-1

-1

0

0

0 1

14

5 6

2 3

3 2

4 5

1

1

1

1

1

2

2

2

2

3

3

3

4

4

4

5

5

6

6

7 8 9

2 3 4 5 6 7 8

5

6

6

6

6

6

0 1 2 3 4 5 n

n

n

n

(5)

(S)

(<)

(O)

(9) (=C)

52

3 4

1 6

+

™¯‹Ì· 2.9

=)16012 (6985383 %1(6()59%4: (%) %"E012 %1(6()59%, (I) '."0(*012 .'C1/%83 /34 %"E0124,($) $"%µµ012 (6985383 /34 '."0(*0124%1(6()59%4,(*) 1)1601? (60853µC,3 %1(6()59%1%/? *7(, (.) 3 %"E012 %1(6()59% 8. 1716(, (8/)1)16012 (6985383 1%/? *7( %"08/."#-8/"(<%.


7 2 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

'À'§#'% ™À¡!§#•%

# 6L6I467 =L@QI4:2 CA@ 56JIJLG4M@ x1(n) 654 x2(n), n = 0,1,…,N–1, J;>Z9C54 AH

(2.31)

*5;5C2;7=C9 ?C4 8;?694C54 <45 µ>5 @Q5 56JIJLG>5 98>=2H µ76JLH R, ?=J O2I5O7 654

CJ µ76JH 65G9µ4BH 58? C4H 5;T46QH 56JIJLG>9H, 654 ?T4 µ76JLH 2R–1, ?8AH G5 =L@Q-

S54@9 =C2@ 89;>8CA=2 C2H <;5µµ467H =L@QI4:2H 5LCM@. )5 S7µ5C5 <45 CJ@ L8JIJ<4-

=µ? C2H 6L6I467H =L@QI4:2H 9>@54 >O45 µ9 C5 CQ==9;5 S7µ5C5 8JL <@A;>=5µ9 65CB

CJ@ L8JIJ<4=µ? C2H <;5µµ467H =L@QI4:2H (SI. 9@?C2C5 1.4): %,/08/"(<2 E"#,()(65CJ-

8C;4=µ?H) C2H µ45H 56JIJLG>5H, (6985383 C2H 65CJ8C;467H 56JIJLG>5H, '(66%'6%-

80%8µ#4 CA@ OKJ 56JIJLG4M@ =2µ9>J 8;JH =2µ9>J 654 ?5"(083 CA@ <4@JµQ@A@. #

S5=467 O453J;B µ9C5:K CA@ OKJ 5LCM@ CK8A@ =L@QI4:2H, 9>@54 ?C4 65CB C2@ 6L6I467

=L@QI4:2 J 65CJ8C;4=µ?H 654 2 JI>=G2=2 <>@J@C54 µ9 6L6I46? C;?8J, L8JIJ<>ZJ@C5H

CJLH O9>6C9H C2H µ45H 56JIJLG>5H µ9 SB=2 C2@ 8;B:2 CJL L8JIJ>8JL (modulo N). +C2@

89;>8CA=2 C2H <;5µµ467H =L@QI4:2H, O9@ L8B;T94 2 8;B:2 CJL L8JIJ>8JL.

,8J;JKµ9 9K6JI5 @5 58JO9>:JLµ9 ?C4 2 4O4?C2C5 C2H 5@C4µ9C5G9C46?C2C5H 4=TK94

654 <45 C2@ 89;>8CA=2 C2H 6L6I467H =L@QI4:2H, O2I5O7

x1(n)Gx2(n) = x2(n)Gx1(n) (2.32)

(LC? =2µ5>@94 ?C4 J8J45O78JC9 58? C4H OKJ 56JIJLG>9H L8J=C9> 65CJ8C;4=µ? 654

JI>=G2=2, CJ 58JCQI9=µ5 C2H 6L6I467H =L@QI4:2H O9@ 5IIBZ94.

(H OJKµ9 =C2 =L@QT945 Q@5 85;BO94<µ5 L8JIJ<4=µJK C2H 6L6I467H =L@QI4:2H OKJ

56JIJLG4M@.

¶·Ú¿‰ÂÈÁÌ· 2.5

$5 L8JIJ<4=G9> 2 6L6I467 =L@QI4:2 CA@ 56JIJLG4M@ x1(n) = {1,2,3}, x2(n) = {4,5,6}.

=G/0:

' L8JIJ<4=µ?H C2H 6L6I467H =L@QI4:2H x3(n) = x1(n) Gx2(n) µ9 SB=2 C2 =TQ=2 (2.31),

µ5H L85<J;9K94 C2@ 6L6I467 JI>=G2=2 C2H 65CJ8C;467H C2H 56JIJLG>5H x2(m). (LC? µ8J-

;9> @5 <>@94 9K6JI5 µ9 C2@ CJ8JGQC2=2 CA@ 56JIJLG4M@ 8B@A =C4H 89;43Q;949H OKJ Jµ?-

69@C;A@ 6K6IA@, ?8AH 35>@9C54 =CJ +T7µ5 2.10. (;T46B J4 OKJ 56JIJLG>9H CJ8JG9-

CJK@C54 8B@A =C4H 89;43Q;949H 5;4=C9;?=C;J35 (+T7µ5 2.105). +C2 =L@QT945, S;>=6JL-

µ9 C2@ 65CJ8C;467 C2H 56JIJLG>5H x2(m), CJ8JG9CM@C5H 5LC7 O9:4?=C;J35, ?8AH 35>-

@9C54 =CJ@ 9=AC9;46? 6K6IJ CJL +T7µ5CJH 2.10S. ' 9:AC9;46?H 6K6IJH 85;5µQ@94 =C5-

G9;?H, 9@M J 9=AC9;46?H 89;4=C;Q39C54 6BG9 3J;B 5;4=C9;?=C;J35 65CB 2'/3 7 65CB

x n x n x m x n m n NN

m

N

1 2 1 2

0

1

0 1 1( ) ( ) ( ) ( ), , , ,G = < # > = #=

#

5 K


2.2.2 %‰ÈfiÙËÙÂ˜ ÙÔ˘ ‰È·ÎÚÈÙÔ‡ ÌÂÙ·Û¯ËÌ·ÙÈÛÌÔ‡ Fourier

' DFT 85;JL=4BZ94 J;4=µQ@9H =2µ5@C46QH 4O4?C2C9H C4H J8J>9H µ8J;JKµ9 @5 5:4J-

8J47=JLµ9 =C4H O4B3J;9H 935;µJ<QH µ5H. ,9;46QH 58? C4H 4O4?C2C9H CJL DFT 9>@54

5@BIJ<9H µ9 C4H 5@C>=CJ4T9H CJL DTFT. YII9H 9>@54 O453J;9C46QH, <9<J@?H 8JL J39>-

I9C54 =CJ 8989;5=µQ@J µ76JH CA@ 56JIJLG4M@ 654 CJL DFT 5LCM@.

7 32 . 2 ¢ # " ' ƒ # & $ ™ ª ! &" ™ Ã % ª "& # ™ ª $ ™ F O U R I E R

2

2 2

25

5

5

56

6

6

6

3

3 3

3

x1(m)

x1(m) x1(m)

x1(m)

x2(<-m>3)

x2(<1-m>3) x2(<2-m>3)

x2(m)

4

4

4

41

1 1

1

(5)

(<) (O)

(9)

(S)

x3(0)=1^4+2^6+3^5=31

x3(1)=1^5+2^4+3^6=31 x3(2)=1^6+2^5+3^4=28

x3(n)=x1(n)Gx2(n)

{x3(n)}={31,31,28}

31 31

28

0 1 2n

™¯‹Ì· 2.10

=)16012 8),C60F3/@, %1(6()50+,{1,2,3} 1%0 {4,5,6}

2'/R <9@46B. ,9 6BG9 89;4=C;J37 65CB 2'/3, L8JIJ<>ZJLµ9 C5 5G;J>=µ5C5 CA@ <4@JµQ-

@A@ =Kµ3A@5 µ9 C2 =TQ=2 (2.31). )J 58JCQI9=µ5 C2H 6L6I467H =L@QI4:2H 35>@9C54 =CJ

+T7µ5 2.109.


7 4 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

°ƒ"ªª#'$&%&"

!B@ 654

C?C9 (2.33)

?8JL a1, a2 8;5<µ5C46QH 7 µ4<5O46QH =C5G9;QH. # 4O4?C2C5 C2H <;5µµ46?C2C5H 58J-

O946@K9C54 8JIK 9K6JI5 58? CJ@ J;4=µ? CJL DFT (=TQ=2 2.21).

'À'§#'% $§#™£%™% ™&$ Ãƒ$¡$

!B@

C?C9 (2.34)

# =TQ=2 5LC7 µ5H O9>T@94 ?C4 J DFT C2H 6L6I46B JI4=G2µQ@2H 56JIJLG>5H QT94 CJ

>O4J µQC;J µ9 C2@ 5;T467, 5IIB O453J;9C467 3B=2.

x n n W X kNDFT

NknN( ) ( )< # > ? @AA0

0


a x n a x n a X k a X kDFTN1 1 2 2 1 1 2 2( ) ( ) ( ) ( )+ ? @AA +

x n X kDFTN2 2( ) ( )? @AA x n X kDFTN

1 1( ) ( )? @AA

$5 L8JIJ<4=G9> J DFT CA@ 56JIJLG4M@ x1(n) = {1, 0, 0, 1} 654 x2(n) = {0, 0, 1, 1}.


'À'§#'% $§#™£%™% ™&% ™ÀÃ¡$&%&"

!B@

C?C9 (2.35)

,9 BII5 I?<45, J 8JII58I5=45=µ?H C2H 56JIJLG>5H x(n) µ9 C2@ 96G9C467 56JIJL-

G>5 , 4=JOL@5µ9> µ9 C2@ 6L6I467 JI>=G2=2 =C2 =LT@?C2C5, 65CB k0 µJ@BO9H,

CJL DFT 5LC7H.

'À'§#'% ™À¡!§#•%

!B@ 654

C?C9 (2.36)

# 6L6I467 =L@QI4:2 OKJ 56JIJLG4M@ =CJ 89O>J CJL T;?@JL, 4=JOL@5µ9> µ9 CJ@ 8JI-

I58I5=45=µ? CA@ 5@C>=CJ4TA@ DFT 5LCM@.

x n x n X k X kDFTN1 2 1 2( ) ( ) ( ) ( )G ? @AA !


1 1( ) ( )? @AA

ej

Nk n

20

)

W x n X k kNk n DFT

NN# ? @AA < # >0

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!B@ 654

C?C9 (2.37)

*;?694C54 JL=45=C46B <45 C2 O4CC7 Q63;5=2 C2H 8;J2<JKµ9@2H 4O4?C2C5H, 654 985-

I2G9K9C54 QC=4 CJ <9<J@?H ?C4 J 8JII58I5=45=µ?H OKJ 56JIJLG4M@ =CJ 89O>J CJL

T;?@JL 5@C4=CJ4T9> =C2@ 6L6I467 =L@QI4:2 CA@ DFT 5LCM@.

£!øƒ%ª" &$À PARSEVAL

!B@

C?C9 (2.38)

)J G9M;2µ5 CJL Parseval 963;BZ94 C2 O45C7;2=2 C2H 9@Q;<945H 65CB C2 µ9CBS5=2

58? CJ 89O>J CJL T;?@JL =CJ 89O>J C2H =LT@?C2C5H. ,9 BII5 I?<45, CJ BG;J4=µ5

CA@ C9C;5<M@A@ CA@ C4µM@ CA@ O94<µBCA@ 4=JKC54 µ9 C2 µQ=2 C4µ7 CA@ C9C;5<M-

@A@ CA@ 35=µ5C46M@ <;5µµM@. # 5@585;B=C5=2 CA@ |X(k)|2 J@JµBZ9C54 <?8µ%

08E7(4 C2H x(n). *5;5C2;JKµ9 ?C4 CJ 3B=µ5 4=TKJH 9:5;CBC54 µ?@J 58? CJ µQC;J

CJL 3B=µ5CJH 654 ?T4 58? C2 3B=2 CJL.

| ( ) | (x n

NX k

k

N

n

N2 2

0

1

0

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#

=

#

55 )


x n x n

NX k X kDFTN

1 2 1 21

( ) ( ) ( ) ( )! ? @AA G


1 1( ) ( )? @AA

7 52 . 2 ¢ # " ' ƒ # & $ ™ ª ! &" ™ Ã % ª "& # ™ ª $ ™ F O U R I E R

$5 985I2G9LC9> 2 4O4?C2C5 C2H 6L6I467H =L@QI4:2H <45 C2@ 89;>8CA=2 CA@ 56J-

IJLG4M@ CJL *5;5O9><µ5CJH 2.5.


$5 985I2G9LC9> CJ G9M;2µ5 CJL Parseval <45 CJ =7µ5 x(n) = {1,2,3}.



7 6 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

' 8I7;2H 65CBIJ<JH CA@ 4O4JC7CA@ CJL DFT 85;JL=4BZ9C54 =CJ@ *>@565 2.3.

¶›Ó·Î·˜ 2.3

V*0#/3/.4 /() DFT

8I)D(0(1 6*#$#KH41 µ%*#K5 7 DFT 7 /0µ'4E,

x1(n) X1(k)

x2(n) X2(k)

0;5µµ46?C2C5 a1x1(n) + a2x2(n) a1X1(k) + a2X2(k)

&L6I467 µ9C5C?84=2

=CJ T;?@J x( < n – n0 > $)

&L6I467 µ9C5C?84=2

=C2 =LT@?C2C5 X( < k – k0 > $)

+LZL<7H 56JIJLG>5 x*(n) X*( < – k > $)

&5CJ8C;4=µ?H =CJ T;?@J x( < – n > $) X*(k)

&L6I467 =L@QI4:2 x1(n) Gx2(n)[1] X1(k)X2(k)

*JII58I5=45=µ?H x1(n)x2(n)

+LZL<7H =Lµµ9C;>5 <45x(n) 8;5<µ5C467

8;5<µ5C46B =7µ5C5

D9M;2µ5 CJL Parseval

[1] x1(n)Gx2(n) = x2( < n – m > N)

x mm

N

1

0

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#

5

x n

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n

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X k X k

X k X k

X k X k

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Im{ ( )} Im{ ( )}

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*= #= #= # #

= #

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0

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N

N

N

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#

5 < # >

W x nk n# 0 ( )

W X kkn0 ( )


2.2.3 °Ú·ÌÌÈÎ‹ Û˘Ó¤ÏÈÍË Ì¤Ûˆ ÙÔ˘ DFT

!>O5µ9 =C2@ 8;J2<JKµ9@2 L8J9@?C2C5 ?C4 CJ <4@?µ9@J OKJ DFT 9>@54 4=JOK@5µJ µ9

C2@ 6L6I467 =L@QI4:2 CA@ 5@C4=CJ>TA@ 56JIJLG4M@ =CJ T;?@J. # 6L6I467 =L@QI4-

:2 ?µAH O9 35>@9C54 @5 µ5H 9>@54 T;7=4µ2 =C2@ 8;B:2, 53JK <45 CJ@ L8JIJ<4=µ? C2H

58?6;4=2H y(n) 9@?H <;5µµ46JK =L=C7µ5CJH, 8;Q894 @5 S;JKµ9 C2 <;5µµ467 =L@Q-

I4:2 C2H 94=?OJL x(n) µ9 C2@ 6;JL=C467 58?6;4=2 h(n) 5LCJK, O2I5O7 y(n) =

x(n)*h(n) (9@?C2C5 1.4). ,BI4=C5, 9B@ L 9>@54 CJ µ76JH C2H 56JIJLG>5H x(n) 654 M

CJ µ76JH C5 56JIJLG>5H h(n), C?C9 CJ µ76JH C2H 56JIJLG>5H y(n) G5 4=JKC54 µ9 L +

M–1. (8? C4H 4O4?C2C9H CJL DTFT (9@?C2C5 2.1), <@A;>ZJLµ9 ?C4

(2.39)

Y;5, 9B@ GQIJLµ9 @5 5@565C5=69LB=JLµ9 C2@ y(n) 58? C5 O9><µ5C5 CJL 3B=µ5CJH

C2H Y(ej@) =9 O456;4CQH =LT@?C2C9H, G5 8;Q894 @5 8B;JLµ9 CJLIBT4=CJ@ L + M–1 O9><-

µ5C5 =C2 =LT@?C2C5. +L@98MH, 5854C9>C54 Q@5H DFT µ76JLH N ( L + M–1, <45 C2@

5@585;B=C5=2 C2H 56JIJLG>5H y(n) =CJ 89O>J CJL T;?@JL. PC=4,

9B@

C?C9 Y(k) = X(k)H(k), k = 0,1,…,N–1 (2.40)

?8JL X(k), H(k) J4 DFT R–=2µ9>A@ CA@ 56JIJLG4M@ x(n), h(n) 5@C>=CJ4T5. !894O7 J4

56JIJLG>9H x(n) 654 h(n) QTJL@ µ76JH µ46;?C9;J 58? R, 8;J=GQCJLµ9 =CJ4T9>5 µ2O9-

@467H C4µ7H =9 65G9µ4B 58? 5LCQH, QC=4 M=C9 CJ µ76JH CJLH @5 <>@94 >=J µ9 R (zero

padding). (LC7 2 5K:2=2 CJL µ76JLH CA@ 56JIJLG4M@ O9@ 982;9BZ94 C5 3B=µ5CB

CJLH X(ej@) 654 Y(ej@), C5 J8J>5 9>@54 =L@9T7. (8IMH, 5LC? 8JL QTJLµ9 984CKT94 µ9 CJ

@5 8B;JLµ9 R O9><µ5C5 =C2 =LT@?C2C5, (L8JIJ<>ZJ@C5H CJLH DFT µ76JLH R), 9>@54

@5 5L:7=JLµ9 CJ 8I7GJH CA@ O94<µBCA@, C5 J8J>5 5@C48;J=A89KJL@ C4H 56JIJLG>9H

5LCQH =CJ 89O>J C2H =LT@?C2C5H, 8Q;5 58? CJ 9IBT4=CJ 8I7GJH (L 654 M 5@C>=CJ4T5).

' R–=2µ9>A@ DFT, ?8JL R = L + M–1, C2H 56JIJLG>5H 9:?OJL y(n), 9>@54 5;69C?H

<45 C2@ 5@585;B=C5=2 C2H y(n) =CJ 89O>J CA@ =LT@JC7CA@. &5CB =L@Q8945, 654 J

8JII58I5=45=µ?H CA@ R–=2µ9>A@ DFT X(k) 654 H(k), =Kµ3A@5 µ9 C2 =TQ=2 (2.40)

654 J 9856?IJLGJH R–=2µ9>A@ %DFT CJL <4@JµQ@JL, G5 µ5H OM=94 C2@ 56JIJLG>5

y(n) (UI. +T7µ5 2.11). &5C_ 5@5IJ<>5, =CJ 89O>J CJL T;?@JL 5LC? =2µ5>@94 8AH 2

R–=2µ9>A@ 6L6I467 =L@QI4:2 CA@ x'(n) 654 h'(n) 4=JOL@5µ9> µ9 C2 <;5µµ467 =L@Q-

I4:2 CA@ x(n) (µ76JLH L) 654 h(n) (µ76JLH M). ,9 BII5 I?<45, %)F?,(,/%4 /( µ21(4

/@, %1(6()50+, x(n) 1%0 h(n) 8. R ('"(85C/(,/%4 8/( /C6(4 1%5.µ0?4 8/(0E.9% µ3*.-

,0124 /0µ24) 1%0 .1/.6+,/%4 /3, 1)16012 8),C60F3 /@, %1(6()50+, '() '"(17'/(),,

Y k Y X H k Njk N

j jk N( ) ) ) , , , , ) (e (e (e=2 / =2 /' = = … #6

6 )6 6

6 ) 0 1 1

y n x n h n Y e X e H eF j j j( ) ( )* ( ) ( ) ( ) ( )= ? @A = !6 6 6

7 72 . 2 ¢ # " ' ƒ # & $ ™ ª ! &" ™ Ã % ª "& # ™ ª $ ™ F O U R I E R


7 8 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

'%9",()µ. /( 9*0( %'(/C6.8µ% µ. .1.9,( '() 5% '%9",%µ., .?, .1/.6(78%µ. /3 $"%µ-

µ012 8),C60F3 /@, %"E01+, %1(6()50+,. !8JµQ@AH, J DFT µ8J;9> @5 T;2=4µJ8J42-

G9> <45 CJ@ L8JIJ<4=µ? C2H <;5µµ467H =L@QI4:2H, 935;µ?ZJ@C5H 5LC?@ =C4H 56J-

IJLG>9H 8JL QTJL@ 65CBII2I5 984µ26L@G9>.

DFT

h'(n)

x'(n)

y(n) = x(n)*h(n)

X(k)

X(k)!H(k)

H(k)DFT

IDFT

¶·Ú¿‰ÂÈÁÌ· 2.6

X;2=4µJ8J47=C9 CJ Z9K<JH CJL DFT <45 CJ@ L8JIJ<4=µ? C2H 9:?OJL 9@?H =L=C7µ5-

CJH µ9 6;JL=C467 58?6;4=2 h(n) = {1, 4, 4, 1} =CJ J8J>J 935;µ?Z9C54 9>=JOJH x(n)

= {5, 4, 3, 2, 1}.

=G/0

# 56JIJLG>5 94=?OJL QT94 µ76JH L = 4 654 2 6;JL=C467 56JIJLG>5 QT94 µ76JH M =

5. # <;5µµ467 =L@QI4:2 CA@ OKJ 5LCM@ 56JIJLG4M@ G5 µ5H OM=94 C2@ 56JIJLG>5

9:?OJL y(n) µ76JLH R = L + M–1 = 8. !8JµQ@AH, CJ µ76JH CJL DFT G5 8;Q894 @5

9>@54 (CJLIBT4=CJ@) JTCM. !84µ26K@JLµ9 C4H 56JIJLG>9H x(n) 654 h(n), M=C9 65G9-

µ4B @5 58J6C7=94 µ76JH >=J µ9 JTCM, 8;J=GQCJ@C5H =CJ4T9>5 µ2O9@467H C4µ7H, 654

=C2 =L@QT945 L8JIJ<>ZJLµ9 CJ@ DFT CA@ @QA@ 56JIJLG4M@. '4 56JIJLG>9H 8JL 8;J-

6K8CJL@ 9>@54 x'(n) = {5, 4, 3, 2, 1, 0, 0, 0} 654 h'(n) = {1, 4, 4, 1, 0, 0, 0, 0}.

' 8–=2µ9>A@ DFT C2H 94=?OJL x'(n) 4=JKC54 µ9 , k = 0, 1,

…, 7 654 98JµQ@AH:

X(0) = 15 X(4) = 3

X(1) = 5,41–j7,24 X(5) = 2,6 + j1,24

X(2) = 3–j2 X(6) = 3 + j2

X(3) = 2,6–j1,24 X(7) = 5,41 + j7,24

' 8–=2µ9>A@ DFT C2H 6;JL=C467H h'(n) 4=JKC54 µ9 , k = 0,

1, …, 7 654 98JµQ@AH: H k h n e j kn

n

( ( )) ' /8= #

=5 2

0

7)

X k x n e j kn

n

( ( )) ' /8= #

=5 2

0

7)

™¯‹Ì· 2.11

D"%µµ012 8),C60-F3 8/( '.*9( /34

8)E,#/3/%4


H(0) = 10 H(4) = 0

H(1) = 3,12–j7,54 H(5) = –1,12–j0,46

H(2) = –3–j3 H(6) = –3 + j3

H(3) = –1,12 + j0,46 H(7) = 3,12 + j7,54

)J <4@?µ9@J CA@ OKJ 5LCM@ DFT =CJ4T9>J 8;JH =CJ4T9>J G5 µ5H OM=94 C2@ Y(k):

Y(0) = 150 Y(4) = 0

Y(1) = –37,68–j63,41 Y(5) = –2,32–j2,59

Y(2) = –15–j3 Y(6) = –15 + j3

Y(3) = –2,32 + j2,59 Y(7) = –37,68 + j63,41

H 58?6;4=2 y(n) 8;J6K8C94 58? CJ@ 8–=2µ9>A@ IDFT C2H Y(k) AH

, n = 0, 1, …, 7 654 C9I46B y(n) = {5,25,39,35,25,15,6,1}.

+2µ94M=C9 ?C4 =CJ >O4J 58JCQI9=µ5 G5 65C5I7:JLµ9 5@ 96C9IQ=JLµ9 C2@ 6L6I467

=L@QI4:2 589LG9>5H =CJ 89O>J CJL T;?@JL, ?8AH Q<4@9 =CJ *5;BO94<µ5 2.5, CJ8JG9-

CM@C5H C4H µ76JLH JTCM =2µ9>A@ 56JIJLG>9H x'(n) 654 h'(n) =C4H 89;43Q;949H 6K6IA@,

89;4=C;Q3J@C5H 654 8;J=GQCJ@C5H C5 <4@?µ9@5.

y n Y k e j kn

k

( ( )) /8==

518

2

0

7)

7 92 . 2 ¢ # " ' ƒ # & $ ™ ª ! &" ™ Ã % ª "& # ™ ª $ ™ F O U R I E R

$5 L8JIJ<>=9C9 C2@ 58?6;4=2 9@?H =L=C7µ5CJH µ9 6;JL=C467 h(n) = {1, 2, 3} <45

9>=JOJ x(n) = {3, 4, 5, 2} T;2=4µJ8J4M@C5H DFT 654 IDFT CA@ JTCM =2µ9>A@.


2.2.4 $ DFT ÛÂ ÌÔÚÊ‹ !ÈÓ¿ÎˆÓ

' DFT R–=2µ9>A@ (9:>=A=2 2.21) µ8J;9> @5 G9A;2G9> 654 AH <;5µµ46?H µ9C5=T2-

µ5C4=µ?H 9@?H O45@K=µ5CJH R–O45=CB=9A@ =CJ T;?@J =9 Q@5 R–O45=CB=9A@ O4B@L-

=µ5 =C2 =LT@?C2C5. P=CA x CJ O4B@L=µ5 CA@ R =CJ4T9>A@ C2H 56JIJLG>5H x(n),

x = [x(0) x(1) … x(N–1)]T (2.41)

?8JL J 96GQC2H ; L8JO2IM@94 CJ@ 5@B=C;J3J 8>@565, X CJ O4B@L=µ5 CA@ R =L@C9-

I9=CM@ X(k)

L = [X(0) X(1) … X(R–1)]T (2.42)


8 0 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

654 W J N H N DFT 8>@565H

(2.43)

?8JL 2 R(8/2 ;>Z5 C2H µJ@BO5H. ' 8>@565H W 9>@54 =Lµµ9C;46?H. ,9

SB=2 5LCJKH CJLH J;4=µJKH, J DFT µ8J;9> @5 963;5=G9> =9 µJ;37 84@B6A@ AH:

X = Wx (2.44)

D9A;JKµ9 ?C4 J 5@C>=C;J3JH CJL W L8B;T94 654 9>@54 J W–1, J8?C9 8JII58I5=4B-

ZJ@C5H 654 C5 OKJ µQI2 C2H =TQ=2H µ9 W–1 85>;@JLµ9

x = W–1X (2.45)

\µAH, 2 =TQ=2 (2.22) CJL IDFT 963;BZ9C54 =9 µJ;37 84@B6A@ AH:

x = (1/N)W*X (2.46)

?8JL W* J =LZL<7H µ4<5O46?H CJL W. +L<6;>@J@C5H C4H OKJ C9I9LC5>9H =TQ=94H µ8J-

;JKµ9 @5 =Lµ89;B@JLµ9 ?C4

W–1 = (1/N)W* (2.47)

654, 65CB =L@Q8945,

WW* = NI (2.48)

?8JL 8 J µJ@5O45>JH 8>@565H O45=CB=9A@ R H R. !8JµQ@AH, J 8>@565H W 9>@54 Q@5H

J;GJ<M@4JH (unitary) 8>@565H.

(8? C4H =TQ=94H 5LCQH <>@9C54 =53QH ?C4 <45 CJ@ L8JIJ<4=µ? CJL DFT 5854CJK@C54

R2 µ4<5O46J> 8JII58I5=45=µJ> 654 R(R–1) µ4<5O46QH 8;J=GQ=94H (I5µSB@J@C5H

L8?R2 CJLH 8JII58I5=45=µJKH 56?µ2 654 µ9 C2 µJ@BO5). \µAH, 53JK J 8>@565H W9>@54 =Lµµ9C;46?H, J4 µ4=QH C4µQH 5LCJK 9>@54 >O49H µ9 C4H BII9H µ4=QH. !848IQJ@, J4

C4µQH 5LCQH 9>@54 J4 R ;>Z9H C2H µJ@BO5H, 654 58? 5LCQH J4 µ4=QH 9>@54 5@C>G9C9H CA@

BIIA@. (LC? µ8J;9> @5 <>@94 35@9;? 5@ <;BRJLµ9 CJ@ 8>@565 W <45 R = 8, ?8AH

35>@9C54 =CJ +T7µ5 2.12. ' 8>@565H 5LC?H 8;JQ6LR9 µ9 C2 SJ7G945 CJL +T7µ5CJH

W eN

jN=

# 28

W =

…………………

7

8

999999999

:

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<<<<<<<<<

#

#

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1 1 1 1

1

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W W W

W W W

W W W

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2.6. *5;5C2;JKµ9 ?C4 L8B;TJL@ µ?@J CQ==9;4H O453J;9C46QH C4µQH =CJ@ 8>@565 5LC?.

(:4J8J4M@C5H, 98JµQ@AH, C2 =Lµµ9C;>5 654 C2@ 89;4JO46?C2C5 CA@ C4µM@ CJL 8>@5-

65, µ8J;JKµ9 @5 585I9>RJLµ9 CJ@ 8I9J@5=µ?. PC=4, µ8J;JKµ9 @5 65C5I7:JLµ9 =9

C9T@46QH L8JIJ<4=µJK CJL DFT µ9 5;69CB I4<?C9;9H 8;B:94H, O2I5O7, 65C5I7<JL-

µ9 =’ Q@5@ C5TK µ9C5=T2µ5C4=µ? Fourier (fast Fourier transform, FFT), ?8AH G5

OJKµ9 =C2@ 9@?C2C5 2.3.

8 12 . 2 ¢ # " ' ƒ # & $ ™ ª ! &" ™ Ã % ª "& # ™ ª $ ™ F O U R I E R

0[

1[

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3[

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aW

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b

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j

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21

¶·Ú¿‰ÂÈÁÌ· 2.7

$5 L8JIJ<4=C9> J DFT C2H 56JIJLG>5H x(n) = {1, 1, 0, 0}.

=G/0

# 56JIJLG>5 x(n) 9>@54 µ76JLH CQ==9;5, J8?C9 8;Q894 @5 L8JIJ<>=JLµ9 CJ@ 8>@565

W <45 R = 4. (:4J8J4M@C5H C2@ 89;4JO46?C2C5 654 C2 =Lµµ9C;>5 ( )

CA@ C4µM@ CJL 8>@565 QTJLµ9

Y;5, X = Wx = .

\8AH 7C5@ 3L=46?, CJ 58JCQI9=µ5 5LC? =Lµ3A@9> µ9 969>@J CJL *5;5O9><µ5CJH 2.4.

2

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40

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42

40

42

43

42

41

1 1 1 1

1

1

1

1 1

=

7

8

99999

:

;

<<<<<

=

7

8

99999

:

;

<<<<<

=

W W W W

W W W W

W W W W

W W W W

W W W

W W W

W W W

11 1

1 1

1 1 1 1

1 1

# ## #

# #

7

8

99999

:

;

<<<<<

j j

j j

W WNk N

Nk+ = #/2


8 2 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™


+C2@ 9@?C2C5 5LC7 <@A;>=5µ9 CJ O456;4C? µ9C5=T2µ5C4=µ? Fourier, O2I5O7 CJ

µ9C5=T2µ5C4=µ? 8JL 935;µ?Z9C54 =9 56JIJLG>9H 8989;5=µQ@JL µ76JLH. (LC?H

9>@54 J µ9C5=T2µ5C4=µ?H 8JL T;2=4µJ8J49>C54 =C2@ 8;B:2 =C2@ 989:9;<5=>5 =2µB-

CA@ µ9 L8JIJ<4=C7. !>O5µ9 ?C4 <45 Q@5 =7µ5 R O94<µBCA@, 8;J6K8CJL@ R O9><µ5-

C5 =C2 =LT@?C2C5. *;?694C54 <45 Q@5 89;4JO46? O456;4C? 3B=µ5. 0@A;>=5µ9 98>-

=2H C2@ Q@@J45 C2H 6L6I467H JI>=G2=2H 9@?H =7µ5CJH. !>O5µ9 ?C4 2 6L6I467 =L@Q-

I4:2 OKJ =2µBCA@ O456;4CJK T;?@JL, µ76JLH L 654 M 5@C>=CJ4T5, µ8J;9> @5 OM=94

CJ >O4J 58JCQI9=µ5 µ9 C2 <;5µµ467 =L@QI4:2 5LCM@, µ?@J 53JK 9896C9>@JLµ9 C5

5;T46B =7µ5C5 µ9 9848IQJ@ =CJ4T9>5 µ2O9@467H C4µ7H, M=C9 @5 58J6C7=JL@ µ76JH

CJLIBT4=CJ@ >=J µ9 L + M–1 CJ 65GQ@5, 654 53JK L8JIJ<>=JLµ9 C2@ 6L6I467 =L@Q-

I4:2 CA@ @QA@, µ9<5IKC9;JL µ76JLH, 56JIJLG4M@. )QIJH, O4584=CM=5µ9 <45 µ>5

56?µ5 3J;B, ?C4 65CB C2 µ9CBS5=2 58? CJ 89O>J CJL T;?@JL =CJ 89O>J C2H =LT@?-

C2C5H 2 9@Q;<945 O45C2;9>C54 (G9M;2µ5 Parseval).

2.3 T·¯‡˜ ÌÂÙ·Û¯ËÌ·ÙÈÛÌfi˜ Fourier (FFT)

' C5TKH µ9C5=T2µ5C4=µ?H Fourier (fast Fourier transform, FFT) O9@ 9>@54 C>8JC9 BIIJ

85;B Q@5H 58JOJC46?H 5I<?;4GµJH <45 CJ@ L8JIJ<4=µ? CJL DFT. +C2@ 8;B:2, O9@

L8B;T94 µ?@J Q@5H 5I<?;4GµJH, 5IIB 8I7GJH 58? O453J;9C46oKH 5I<?;4GµJLH 8JL

984CL<TB@JL@ CJ =6J8? 5LC?. '4 O453J;QH CJLH S;>=6J@C54 6L;>AH =CJ 8I7GJH 654

=CJ 9>OJH CA@ 8;B:9A@ 65GMH 654 =CJ µQ<9GJH C2H 5854CJKµ9@2H µ@7µ2H. \IJ4 ?µAH,

QTJL@ Q@5 6J4@? T5;56C2;4=C46?: E".0?L(,/%0 µ#,( (R/2) log2N µ0$%*01(74 '(66%-

'6%80%8µ(74 $0% /(, )'(6($08µ# .,#4 DFT R–83µ.9@,. !µ9>H G5 5@539;GJKµ9 =CJ@

5I<?;4GµJ CA@ Cooley–Tukey, J J8J>JH 8;JCBG269 CJ 1965 654 J J8J>JH QOA=9 µ>5

8;5<µ5C46B 9@CL8A=4567 MG2=2 =C2@ R234567 989:9;<5=>5 =2µBCA@. ' 5I<?;4G-

µJH 5LC?H µ8J;9> @5 935;µJ=C9> =9 =7µ5C5 58JC9IJKµ9@5 58? R O9><µ5C5, ?8JL R

= 2m, O2I5O7 8;?694C54 <45 Q@5@ 5I<?;4GµJ SB=2H–2 (radix–2), ?8AH 5LC?H J@JµB-

Z9C54. (;T46B, 2 56JIJLG>5 CA@ R =CJ4T9>A@ 94=?OJL x(n) TA;>Z9C54 =9 OKJ 56J-

IJLG>9H µ76JLH R/2 2 65G9µ>5, C4H x(2n) 654 x(2n + 1), 58JC9IJKµ9@9H 58? =CJ4T9>5

µ9 B;C4JLH 654 89;4CCJKH O9>6C9H, 5@C>=CJ4T5. PC=4, :964@M@C5H 58? CJ@ J;4=µ? CJL

DFT (9:>=A=2 2.21) QTJLµ9:

(2.49)

X k x n W x n W x n W

x n W W x n W

Nnk

n

N

Nnk

n

N

Nn k

n

N

Nnk

n

N

Nk

Nnk

n

N

( ) ( ) ( ) ( )

( ) ( )

( )= = + + =

= + +

=

#

=

#

+

=

#

=

#

=

#

5 5 5

5 5

0

12

0

21

2 1

0

21

2

0

21

2

0

21

2 2 1

2 2 1


\µAH J8?C9 2 C9I9LC5>5 =TQ=2 <>@9C54:

7 (2.50)

*5;5C2;JKµ9 ?C4 J L8JIJ<4=µ?H CA@ R =2µ9>A@ CJL X(k) QT94 963;5=C9> AH BG;J4-

=µ5 OKJ DFTs µ9 8I7GJH =2µ9>A@ R/2 J 65GQ@5H, 96 CA@ J8J>A@ J Q@5H 8;Q894 @5

8JII58I5=45=C9> 98> . (LC?H J 8JII58I5=45=µ?H 7C5@ 5@5µ9@?µ9@JH, 5@ =69-

3GJKµ9 ?C4 2 56JIJLG>5 CA@ 89;4CCM@ O946CM@ 9>@54 JI4=G2µQ@2 65CB Q@5 O9><µ5 AH

8;JH C2@ 5;T7 CJL T;?@JL, J8?C9 µ9 SB=2 C2@ 4O4?C2C5 C2H 6L6I467H JI>=G2=2H =CJ

T;?@J, 5LC? =L@98B<9C54 8JII58I5=45=µ? 98> =CJ 89O>J C2H =LT@?C2C5H.

"5µSB@J@C5H L8?R2 µ5H ?C4 654 ,2 =TQ=2 (2.50)

<45 k >=J µ9 k + (R/2) <>@9C54:

(2.51)

&5CB =L@Q8945, QTJLµ9 984CKT94 µQT;4 CJ =2µ9>J 5LC?, @5 963;B=JLµ9 CJ@ 5;T46? DFT

CA@ R =2µ9>A@ AH =L@OL5=µ? OKJ BIIA@ DFTs µ4=JK µ76JLH J 65GQ@5H. (H :5@5-

<;BRJLµ9 C4H OKJ C9I9LC5>9H =TQ=94H, 8;J=QTJ@C5H ?µAH 5LC7 C2 3J;B, ?C4 C5 n, k

6Lµ5>@J@C54 µ9C5:K 0 654 (R/2)–1 654 ?T4 µ9C5:K 0 654 R–1, ?8AH 9>T5µ9 :964@7=94:

(2.52)

(2.53)

?8JL k = 0, 1, …., R/2–1.

' L8JIJ<4=µ?H CA@ C4µM@ 5LCM@ µ8J;9> @5 <>@94 =L@OL5=C46B 96C9IM@C5H µ?@J Q@5

8JII58I5=45=µ?. *5;5=C5C46B, 2 O45O465=>5 5LC7 O>@9C54 µ9 C2 SJ7G945 C2H I9<?-

µ9@2H './%6(7*%4 (butterfly) CJL +T7µ5CJH 2.13. *;?694C54 JL=45=C46B, <45 CJ@ L8J-

IJ<4=µ? 9@?H DFT 2 =2µ9>A@. +CJ 8B@A O9:> B6;J C2H 89C5IJKO5H QTJLµ9 C2@ 8;?-

=G9=2 CA@ 5;4GµM@ 8JL 65C5I7<JL@ =’ 5LC?, 9@M =CJ 6BCA O9:> B6;J 96C9IJKµ9

C2@ 535>;9=2 CA@ >O4A@ 5;4GµM@.

X k

NX k W X kN

k+*+,

-./

= #2 1 2( () )

X k X k W X kNk( ) ( (= +1 2) )

X k

NX k W X kN

k+*+,

-./

= #2 1 2( ) ( )

+

W WN

n kN

Nnk

2

2

2

*+,

-./ = W WN

kN

Nk

+ 2 = #

WNk

WNk

X k X k W X kNk( ( () )+ )= 1 2

X k x n W W x n WNnk

n

N

X k

Nk

Nnk

n

N

X k

( ) ( ) ( )/

(

/= + +=

#

=

#

5 52 2 12

0

21

2

0

21

1 2) ( )1 244 344 1 2444 3444

W e e WNnk

jN

nk jN

nk

Nnk2

22

22

2= = =# #) )

//

8 32 . 3 T" Ã À ™ ª ! &" ™ Ã % ª "& # ™ ª $ ™ F O U R I E R ( F F T )


8 4 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

X1(k)

X2(k)WN

k -1

X(k)=X1(k)+WNkX2(k)

X(k+N/2)=X1(k)–WNkX2(k)

™¯‹Ì· 2.13

S%8012 './%6(7*%2 83µ.9@,

0;5346B, 2 µQT;4 CM;5 5@BIL=2 CJL DFT O>@9C54 =CJ +T7µ5 2.145 <45 C2@ 89;>8CA-

=2 8JL QTJLµ9 JTCM O9><µ5C5 (R = 8). 0>@9C54 =53QH ?C4 J L8JIJ<4=µ?H CJL R–=2µ9>-

A@ DFT QT94 5@5TG9> =CJ@ L8JIJ<4=µ? OKJ DFTs R/2–=2µ9>A@ J 65GQ@5H 654 =CJ@

C9I46? =L@OL5=µ? CA@ 58JC9I9=µBCA@ CJLH. # O45O465=>5 5@BIL=2H 8JL 56JIJL-

G7G269 8;J2<JLµQ@AH µ8J;9> @5 =L@9T4=C9> 654 <45 CJLH OKJ @QJLH DFTs CA@ R/2

=2µ9>A@. PC=4, TA;>ZJLµ9 654 8BI4 =9 B;C45 654 89;4CCB 654 56JIJLGM@C5H C2@ >O45

C56C467 65C5I7<JLµ9 =CJ O4B<;5µµ5 CJL +T7µ5CJH 2.14S. (LC7 2 O45O465=>5 5@B-

IL=2H (58JO965C4=µJK) =L@9T>Z9C54 µQT;4H ?CJL 65C5I7:JLµ9 =CJ@ L8JIJ<4=µ? DFT

2–=2µ9>A@ (+T7µ5 2.14<). (LC? G5 <>@94 µ9CB 58? m =CBO45, ?8JL m = log2R. )J

=L@JI46? O4B<;5µµ5 ;J7H 9@?H FFT JTCM =2µ9>A@ 35>@9C54 =CJ +T7µ5 2.14O.

x(0) X(0)

x(2)

x(4)

x(6)

x(1)

x(3)

x(5)

x(7)

X(1)

X(2)

X(3)

X(4)

X(5)

X(6)

X(7)

X1(0)

X1(1)

X1(2)

X1(3)

X2(0)

X2(4)

X2(2)

X2(3)

3J +CBO4J

-1

-1

-1

-1

W08

W28

W38

W18

(5)

DFT[N =2µ9>A@2

DFT[N =2µ9>A@2

™¯‹Ì· 2.14

T0%$"?µµ%/% "(24FFT (E/+ 83µ.9-@,: (%)–($) /%/"9% *0%*(E01?

8/?*0% %,?'/)F34/() %6$("95µ(),(*) /( 8),(601#*0?$"%µµ% "(24



x(0) X1(0)

x(4)

x(2)

x(6)

x(1)

x(5)

x(3)

x(7)

X1(1)

X1(2)

X1(3)

X2(0)

X2(1)

X2(2)

X2(3)

X11(0)

X11(1)

X12(0)

X12(1)

X21(0)

X21(1)

X22(0)

X22(1)

2J +CBO4J

-1

-1

-1

-1W08

W28

W28

W08

(S)

DFT[N =2µ9>A@4

DFT[N =2µ9>A@4

DFT[N =2µ9>A@4

DFT[N =2µ9>A@4

(<)

x(3)

x(7)

X22(0)

X22(1)-1W0

8

x(1)

x(5)

X21(0)

X21(1)-1W0

8

x(2)

x(6)

X12(0)

X12(1)-1W0

8

x(0) X11(0)

x(4) X11(1)

1J +CBO4J

-1W08


x(0) X(0)

x(4)

x(2)

x(6)

x(1)

x(5)

x(3)

x(7)

X(1)

X(2)

X(3)

X(4)

X(5)

X(6)

X(7)

1o +CBO4J 2o +CBO4J 3o +CBO4J

-1

-1

-1 -1

-1 -1

-1

-1 -1

-1 -1

-1

W08

W08

W08

W08

W08

W08

W08

W28

W28

W28

W38

W18

(O)

8 6 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

(8? CJ +T7µ5 2.14O µ8J;JKµ9 @5 85;5C2;7=JLµ9 ?C4:

# O4BC5:2 CA@ O94<µBCA@ 9:?OJL (=L@C9I9=CM@) 9>@54 65@J@467, O2I5O7 X(0),

X(1),…,X(7), 9@M 2 O4BC5:2 CA@ O94<µBCA@ 94=?OJL (O9OJµQ@A@) 9>@54 µ2 65@J@4-

67, x(0), x(4), x(2), x(6), x(1), x(5), x(3), x(7). # µ2 65@J@467 5LC7 O4BC5:2 CA@ O94<-

µBCA@ 94=?OJL 9>@54 58JCQI9=µ5 C2H =C5O4567H 58J=K@G9=2H CA@ O94<µBCA@ 94=?-

OJL (89O>J T;?@JL), 8JL 935;µ?=5µ9 65CB C2@ 5@B8CL:2 CJL 5I<J;>GµJL. (LC7 2

O45O465=>5 58J=K@G9=2H J@JµBZ9C54 58JO965C4=µ?H =CJ T;?@J (decimation–in–time,

DIT). H O4BC5:2 CA@ =CJ4T9>A@ 94=?OJL O9@ 9>@54 CLT5>5, 5IIB 8;J6K8C94 58? C2@

65@J@467 O4BC5:2 CA@ O94<µBCA@ µ9 5@C4=C;J37 C2H =94;BH CA@ OL5O46M@ R23>A@

CA@ O946CM@ CJLH (bit–reversal). # 8I7;2H 5@C4=CJ>T4=2 CA@ 65@J@46M@ 654 CA@ 5@5-

O45C5<µQ@A@ O946CM@ <45 R = 8 O>@9C54 =CJ@ *>@565 2.4.

™¯‹Ì· 2.14

T0%$"?µµ%/% "(24FFT (E/+ 83µ.9-@,: (%)–($) /%/"9% *0%*(E01?

8/?*0% %,?'/)F34/() %6$("95µ(),(*) /( 8),(601#*0?$"%µµ% "(24



¶›Ó·Î·˜ 2.4

=%,(,012 *0?/%F3 1%0 *0?/%F3 '() '"(17'/.0 %'# %,/08/"(<2 /34 8.0"?4 /@, *)%*0-

1+, H3<9@, $0% R = 8

&5@J@467 O4BC5:2 E4BC5:2 58? 5@C4=C;J37

C2H =94;BH CA@ bits

E965O46?H EL5O46?H EL5O46?H E965O46?H

0 000 000 0

1 001 100 4

2 010 010 2

3 011 110 6

4 100 001 1

5 101 101 5

6 110 011 3

7 111 111 7

P@5 9@O453Q;J@ =CJ4T9>J CJL FFT 8JL µ?I4H 5@58CK:5µ9, 9>@54 2 OL@5C?C2C5 96CQ-

I9=2H CA@ L8JIJ<4=µM@ =C4H >O49H GQ=94H µ@7µ2H (in place). E9@ 5854C9>C54, O2I5-

O7, 9848IQJ@ TM;JH <45 C2@ 58JG769L=2 CA@ 9@O4Bµ9=A@ 58JC9I9=µBCA@. (8? CJ

+T7µ5 2.14, µ8J;JKµ9 @5 85;5C2;7=JLµ9 ?C4 =9 6BG9 =CBO4J, J4 Q:JOJ4 µ8J;JK@ @5

58JG269KJ@C54 =C4H >O49H GQ=94H µ@7µ2H, ?8JL 7C5@ 58JG269LµQ@9H J4 9>=JOJ4 CJL

=C5O>JL 969>@JL.

)QIJH, CJ 8IQJ@ JL=45=C46? =CJ4T9>J CJL FFT 9>@54 2 µ94AµQ@2 L8JIJ<4=C467 8JIL-

8IJ6?C2CB CJL, O2I5O7, CJ =L@JI46? 8I7GJH 8;B:9A@ (8JII58I5=45=µM@ 654 8;J-

=GQ=9A@). 09@469KJ@C5H C2@ 89;>8CA=2 CJL +T7µ5CJH 2.14O µ8J;JKµ9 @5 OJKµ9 ?C4

J L8JIJ<4=µ?H CJL FFT R O94<µBCA@ (R = 2m) <>@9C54 =9 m =CBO45 (m = log2N). +9

6BG9 =CBO4J QTJLµ9 R/2 89C5IJKO9H, O2I5O7, (R/2) log2N 89C5IJKO9H =L@JI46B. (8?

CJ +T7µ5 2.14O O4584=CM@JLµ9 ?C4 <45 CJ@ L8JIJ<4=µ? 6BG9 89C5IJKO5H 5854C9>C54

Q@5H µ4<5O46?H 8JII58I5=45=µ?H 654 OKJ µ4<5O46QH 8;J=GQ=94H. PC=4, J N–=2µ9>A@

FFT 5854C9> (R/2) log2N µ4<5O46JKH 8JII58I5=45=µJKH 654 R log2N µ4<5O46QH 8;J-

=GQ=94H. !>O5µ9 =C2@ 85;B<;53J 2.2.4, ?C4 <45 CJ@ 589LG9>5H L8JIJ<4=µ? CJL

N–=2µ9>A@ DFT, 5854CJK@C54 R2 µ4<5O46J> 8JII58I5=45=µJ> 654 R(N–1) µ4<5O46QH

8;J=GQ=94H. &5CB =L@Q8945, CJ 6Q;OJH =9 8;B:94H 654 B;5 =9 T;?@J L8JIJ<4=µJK, 9>@54

8JIK µ9<BIJ, ?8AH 35>@9C54 =CJ@ *>@565 2.5 <45 O453J;9C46? 8I7GJH O94<µBCA@.


8 8 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™

!@O946C46B 5@53Q;9C54 ?C4 <45 R = 1024 CJ 6Q;OJH 5LC? <>@9C54 µ9<5IKC9;J CJL O45-

6?=45, O2I5O7 J T;?@JH L8JIJ<4=µJK µ94M@9C54 CJLIBT4=CJ@ 65CB OKJ CB:94H µ9<Q-

GJLH.

¶›Ó·Î·˜ 2.5

M0$%*01(9 '(66%'6%80%8µ(9 $0% /(, )'(6($08µ# /() DFT 1%0 /() FFT

DFT FFT "?<JH ,4<5O46M@

N *I7GJH ,4<5O46M@ *I7GJH ,4<5O46M@ *JII58I5=45=µM@

*JII58I5=45=µM@ *JII58I5=45=µM@ (#DFT / #FFT)

2 4 1 4

4 16 4 4

8 64 12 5,3

16 256 32 8,0

32 1 024 80 12,8

64 4 096 192 21,3

128 16 384 448 36,6

256 65 536 1 024 64,0

512 262 144 2 304 113,8

1024 1 048 576 5 120 204,8

2048 4 194 304 11 264 372,4

4096 16 777 216 24 576 682,7

8192 67 108 864 53 248 1 260,3

D5 8;Q894 @5 CJ@4=C9> ?C4 J FFT 5I<?;4GµJH, 8JL µ?I4H 5@58CKTG269, 935;µ?Z9C54

µ?@J <45 8I7GJH O94<µBCA@ 8JL 9>@54 OK@5µ2 CJL OKJ (radix–2). )J <9<J@?H 5LC?

58JC9I9> BIIA=C9 654 CJ@ 89;4J;4=µ? CJL FFT, O2I5O7 ?C4 O9@ µ8J;9> @5 935;µJ-

=C9>, QC=4 ?8AH 85;JL=4B=C269 9OM, <45 CLT5>J 8I7GJH =2µ9>A@ ` ?8AH J DFT.

F8B;TJL@ 8JIIJ> FFT 5I<?;4GµJ4 =C2 O49G@7 S4SI4J<;53>5 J4 J8J>J4 5:4J8J4JK@ C5

O453J;9C46B T5;56C2;4=C46B C2H 94=?OJL 654 984CL<TB@JL@ LR2I?C9;9H 984O?=94H.

PC=4, =L@5@CJKµ9 5I<J;>GµJLH 8JL 935;µ?ZJ@C54 µ?@J <45 8I7GJH =2µ9>A@ 8JL

9>@54 OL@Bµ94H CJL CQ==9;5 (radi x–4), 7 5I<J;>GµJLH µ?@J <45 8;5<µ5C46QH 94=?-

OJLH (real–valued), 7 5I<J;>GµJLH <45 8I7GJH =2µ9>A@ 8JL 9>@54 8;MCJ4 5;4GµJ>


8 9™ Y N O æ H K E º A § A I O Y

(prime length).

# 8;J2<JKµ9@2 5@BIL=2 S5=>=C269 =Co@ %'(*.1%/08µ# 8/( E"#,( (DIT). ,9 85;?-

µJ4J C;?8J G5 µ8J;JK=5µ9 @5 984T94;7=JLµ9 %'(*.1%/08µ# 8/3 8)E,#/3/%

(decimation–in–frequency, DIF) 65C5I7<J@C5H =’ Q@5@ FFT 5I<?;4GµJ

(Sande–Tukey, 1966), J J8J>JH QT94 C2@ >O45 L8JIJ<4=C467 8JIL8IJ6?C2C5 µ9 CJ@

5@C>=CJ4TJ DIT 5I<?;4GµJ. X5;56C2;4=C46? CJL DIF 5I<J;>GµJL 9>@54 ?C4 J4 8JI-

I58I5=45=µJ> =C4H 89C5IJKO9H <>@J@C54 µ9CB C2@ BG;J4=2 CA@ 94=?OA@ 654 ?C4 C5

O9><µ5C5 94=?OJL 9>@54 =9 65@J@467 O4BC5:2, 9@M C5 O9><µ5C5 9:?OJL =9 µ2 65@J-

@467 O4BC5:2 (bit–reversed).

F89@GLµ>Z9C54 CQIJH, ?C4 J FFT O9@ 58JC9I9> 6B8J4J @QJ µ9C5=T2µ5C4=µ? Fourier,

5IIB µ>5 58JOJC467 5I<J;4Gµ467 µQGJOJ <45 CJ@ L8JIJ<4=µ? CJL DFT.

*?=J4 µ4<5O46J> 8JII58I5=45=µJ> 5854CJK@C54 <45 C2@ LIJ8J>2=2 C2H <;5µµ467H

=L@QI4:2H OKJ 56JIJLG4M@ µ76JLH R 2 65G9µ4B, ?8JL R OK@5µ2 CJL OKJ; F8JIJ-

<>=C9 C2 <;5µµ467 =L@QI4:2 (5) 589LG9>5H =CJ 89O>J CJL T;?@JL, (S) µQ=A CJL 89O>-

JL C2H =LT@?C2C5H µ9 T;7=2 CJL DFT 654 (<) µQ=A CJL 89O>JL C2H =LT@?C2C5H µ9

T;7=2 CJL FFT. &5C5=C;M=C9 8>@565 J J8J>JH @5 O9>T@94 CJ 8I7GJH CA@ 8;B:9A@

=L@5;C7=94 CJL 5;4GµJK CA@ =2µ9>A@ R. 045 8J49H C4µQH CJL R µ5H =Lµ3Q;94 @5

L8JIJ<>=JLµ9 C2 <;5µµ467 =L@QI4:2 µQ=A CJL FFT 654 ?T4 589LG9>5H;



+C2@ 9@?C2C5 5LC7 <@A;>=5µ9 CJ@ C5TK µ9C5=T2µ5C4=µ? Fourier. (@58CK:5µ9 CJ@

5I<?;4GµJ 58JO965C4=µJK =CJ T;?@J <45 C2@ 89;>8CA=2 8JL CJ 8I7GJH CA@ =2µ9>-

A@ 9>@54 OK@5µ2 CJL OKJ. !>O5µ9 ?C4 L8B;TJL@ 8JIIJ> 5I<?;4GµJ4 <45 CJ@ C5TK L8J-

IJ<4=µ? CJL DFT, 2 L8JIJ<4=C467 8JIL8IJ6?C2C5 CA@ J8J>A@, 9>@54 C2H CB:9AH

CJL R log2R 654 ?T4 CJL R2, ?8AH =LµS5>@94 µ9 CJ@ 589LG9>5H L8JIJ<4=µ? CJL DFT.


9 0 K E º A § A I O 2 : A ¡ " § À ™ % ™ & $ ¶ ! ¢ # $ & % ™ ™ À Ã ¡ $ & % &" ™


+CJ 693BI54J 5LC? µ9I9C7=5µ9 C5 =7µ5C5 O456;4CJK T;?@JL 58? C2@ 8I9L;B CJL

89O>JL C2H =LT@?C2C5H 654 9>O5µ9 ?C4:

# µ9CBS5=2 58? CJ 89O>J CJL T;?@JL =CJ 89O>J C2H =LT@?C2C5H 984CL<TB@9C54 µ9

CJ µ9C5=T2µ5C4=µ? Fourier O456;4CJK T;?@JL (9:>=A=2 5@BIL=2H), 9@M 2 µ9CB-

S5=2 58? CJ 89O>J C2H =LT@?C2C5H =CJ 89O>J CJL T;?@JL <>@9C54 µ9 CJ@ 5@C>=C;J-

3J µ9C5=T2µ5C4=µ? Fourier O456;4CJK T;?@JL (9:>=A=2 =K@G9=2H).

&5CB C2 µ9CBS5=2 58? CJ 89O>J CJL T;?@JL =CJ 89O>J C2H =LT@?C2C5H 2 9@Q;<945

CJL =7µ5CJH O45C2;9>C54 (G9M;2µ5 Parseval). *;?694C54 =C2@ JL=>5 <45 C2@ 5@5-

85;B=C5=2 C2H >O45H 8I2;J3J;>5H µ9 O453J;9C46? C;?8J.

' DTFT 9@?H =7µ5CJH O456;4CJK T;?@JL 9>@54 µ>5 =L@9T7H 654 89;4JO467 =L@B;C2=2.

' DFT 9@?H =7µ5CJH O456;4CJK T;?@JL 8989;5=µQ@2H O4B;6945H 9>@54 µ>5 89;4J-

O467 56JIJLG>5.

)5TKH µ9C5=T2µ5C4=µ?H Fourier (FFT) 9>@54 6BG9 5I<?;4GµJH J J8J>JH QT94 L8J-

IJ<4=C467 8JIL8IJ6?C2C5 µ46;?C9;2 58? 969>@2 CJL 589LG9>5H L8JIJ<4=µJK CJL

DFT. # =L@7G2H L8JIJ<4=C467 8JIL8IJ6?C2C5 CJL FFT 9>@54 C2H CB:9AH CJL

Nlog2N, 9@M 969>@2 CJL DFT 9>@54 C2H CB:9AH CJL N2.


9 1" ¶$¢$™% "°°§#'ø¡ $ƒø¡ ™&%¡ !§§%¡#'%


!§§%¡$°§ø™™%

+. D9JOA;>O2H, &. ,89;µ89;>O2H, :08%$@$2 8/3 -.@"9% &3µ?/@, & &)8/3µ?/@,,

)L8AG7CA 0. E5;O5@?H, (G7@5,1998.

N.&5IJL8C=>O2H, &2µ%/%, &)8/2µ%/% 1%0 >6$#"05µ(0, !6O?=94H E>5LIJH,

(G7@5,1993.

•!¡$°§ø™™%

D.J.DeFatta, J.G.Lucas, W.S.Hodgkiss, Digital Signal Processing: A System Design

Approach, J. Wiley & Sons, 1988.

P.Duhamel, M.Vetterli, Fast Fourier Transforms: A Tutorial Review and a State of

the Art, Signal Processing, 19, 259, 1990.

E.C.Ifeachor, B.W.Jervis, Digital Signal Processing: A Practical Approach, Second

Edition, Pearson Education Limited, 2002.

P.A.Lynn, W.Fuerst, Introductory Digital Signal Processing With Computer

Applications, J.Wiley & Sons Ltd, 1989.

S.K.Mitra, Digital Signal Processing: A computer–Based Approach, Second Edition,

McGraw Hill, 2001.


A.V.Oppenheim, A.S.Willsky, Signals & Systems, Second Edition, Prentice–Hall,

1997.

J.G.Proakis, D.G.Manolakis, Digital Signal Processing: Principles, Algorithms and

Applications, Prentice–Hall, 1996.

L.R.Rabiner et al, Terminology in Digital Signal Processing, IEEE Trans. on Audio

and Electroacoustics, Vol. AU–20, No. 5, pp. 322–337, Dec. 1972.

S.W.Smith, The Scientist and Engineer’s Guide to Digital Signal Processing, Second

Edition, California Technical Publishing, 1999

R.D.Strum, D.E.Kirk, First Principles of Discrete Systems and Digital Signal




band–limited 89;4J;4=µQ@JL 9K;JLH

bit reversal 5@C4=C;J37 OL5O46M@ R23>A@

circular shift 6L6I467 JI>=G2=2 7 µ9C5C?84=2

decimation in frequency 58JO965C4=µ?H =C2 =LT@?C2C5

decimation in time 58JO965C4=µ?H =CJ T;?@J

discrete Fourier transform O456;4C?H µ9C5=T2µ5C4=µ?H Fourier

domain 89O>J, TM;JH

dual O4CC?H

energy–density spectrum 3B=µ5 9@9;<94567H 8L6@?C2C5H

fast Fourier transform C5TKH µ9C5=T2µ5C4=µ?H Fourier

magnitude spectrum 3B=µ5 µQC;JL

phase spectrum 3B=µ5 3B=2H

radix SB=2

sampling frequency =LT@?C2C5 O94<µ5CJI2R>5H

sampling rate ;LGµ?H O94<µ5CJI2R>5H

shift JI>=G2=2, µ9C5C?84=2

spectral overlap 35=µ5C467 9846BILR2

twiddle factor 85;B<J@C5H =C;J37H

zero padding 8;J=G762 µ2O9@46M@ =CJ4T9>A@


ªÂÙ·Û¯ËÌ·ÙÈÛÌfi˜–z

™ÎÔ!fi˜

P0 µ./%8E3µ%/08µ(9 %'(/.6(7, C,% %'# /% 83µ%,/01#/."% ."$%6.9% 8/3, %,?6)83

83µ?/@, 1%0 LTI 8)8/3µ?/@,. &/( '%"#, =.<?6%0( .08?$()µ. /( µ./%8E3µ%/08µ#–z,

µ.6./(7µ. /04 0*0#/3/C4 /() 1%0 %,%*.01,7()µ. /3 8'()*%0#/3/? /() 8/3, %,?6)83

1%0 8/( E%"%1/3"08µ# *0%1"0/+, LTI 8)8/3µ?/@,.

P µ./%8E3µ%/08µ#4–z '%9L.0 8'()*%9( "#6( 8/3, %,?6)83 83µ?/@, *0%1"0/(7 E"#-

,() 1%0 LTI 8)8/3µ?/@,, %,/98/(0E( µ. .1.9,(, /() µ./%8E3µ%/08µ(7 Laplace $0%

/3, %,?6)83 83µ?/@, 1%0 LTI 8)8/3µ?/@, 8),.E(74 E"#,(). D0% '%"?*.0$µ%, 5%

*(7µ. #/0 3 8),C60F3 *7( 83µ?/@, 8/( E"#,(, 08(*),%µ.9 µ. /(, '(66%'6%80%8µ#

/@, %,/98/(0E@, µ./%8E3µ%/08µ+,–z %)/+,. M9% /C/(0% 0*0#/3/% %'6('(0.9 83µ%-

,/01? /3, %,?6)83 /34 %'#1"0834 .,#4 LTI 8)8/2µ%/(4 $0% *0?<("% 82µ%/%. >1#µ3,

( µ./%8E3µ%/08µ#4–z µ%4 '%"CE.0 /3 *),%/#/3/% E%"%1/3"08µ(7 .,#4 LTI 8)8/2µ%-

/(4 1%0 )'(6($08µ(7 /34 %'#1"0824 /() $0% *0?<("% 82µ%/%, µ. %'62 /('(5C/383

/@, '#6@, 1%0 µ3*.,01+, /() 8/( .'9'.*(–z.


J/%, (6(163"+8./. /3 µ.6C/3 /() 1.<%6%9() %)/(7 5% .98/. 8. 5C83 ,%:

• B'(6($9L./. /( µ./%8E3µ%/08µ#–z ('(0%8*2'(/. '.'."%8µC,34 %1(6()59%4

• B'(6($9L./. /( µ./%8E3µ%/08µ#–z 1%0 ,% '"(8*0("9L./. /3, '."0(E2 87$160834%1(6()50+, %'.9"() µ21()4

• !"(8*0("9L./. /(, %,/98/"(<( µ./%8E3µ%/08µ#–z 8),%"/28.@,

• >'(<%9,.8/. .71(6% 8E./01? µ. /3, %0/0%/#/3/% 1%0 .)8/?5.0% 8)8/3µ?/@, *0%-1"0/(7 E"#,()

• &E.*0?L./. /3 *(µ2 '"%$µ%/('(93834 8)8/3µ?/@, *0%1"0/(7 E"#,()

• B'(6($9L./. /3, %'#1"083 8)E,#/3/%4 8)8/3µ?/@, *0%1"0/(7 E"#,()


• M./%8E3µ%/08µ#4–z

• !."0(E2 87$160834

3! " º#§#$%


9 4 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

• &),?"/383 µ./%<("?4

• :)8/?5.0%

• !"%$µ%/('(9383 8)8/3µ?/@, *0%1"0/(7 E"#,()

• >'#1"083 8)E,#/3/%4

• :F08+8.04 *0%<("+,


P µ./%8E3µ%/08µ#4–z (z–transform) $0% /% 82µ%/% *0%1"0/(7 E"#,() .9,%0 #,/0 1%0

( µ./%8E3µ%/08µ#4 Laplace $0% /% 82µ%/% 8),.E(74 E"#,() (continuous–time). -%

*(7µ. #/0 ( µ./%8E3µ%/08µ#4–z .9,%0 '0( $.,01#4 %'# /( µ./%8E3µ%/08µ# Fourier

*0%1"0/(7 E"#,() (discrete–time Fourier transform, DTFT), %<(7 ( /.6.)/%9(4 %'(-

/.6.9 .0*012 '."9'/@83 /() µ./%8E3µ%/08µ(7–z. &)µ'."%8µ%/01?, ( µ./%8E3µ%/0-

8µ#4–z .9,%0 C,% '(67 08E)"# µ%53µ%/01# ."$%6.9( $0% /3 µ.6C/3 *0%1"0/+, 83µ?-

/@, 1%0 8)8/3µ?/@,. D0% /3, 1%/%,#383 /() =.<%6%9() ( %,%$,+8/34 5% '"C'.0 ,%

CE.0 .F([email protected] µ. /()4 µ0$%*01(74 %"05µ(74, 1%5+4 1%0 µ. /04 8.0"C4 µ. 8/%5."(74

#"()4.

&/3, .,#/3/% 3.1 ("9L()µ. /( µ./%8E3µ%/08µ#–z 1%0 µ.6./(7µ. /04 0*0#/3/C4 /(). &/3

8),CE.0% ("9L()µ. /(, %,/98/"(<( µ./%8E3µ%/08µ#–z, *36%*2 /( µ./%8E3µ%/08µ# (

('(9(4 µ%4 (*3$.9 %'# /( '.*9(–z 8/( '.*9( /() E"#,() (.,#/3/% 3.2). P µ(,#'6.)-

"(4 µ./%8E3µ%/08µ#4–z, ( ('(9(4 E"380µ('(0.9/%0 $0% /3, %,?6)83 8)8/3µ?/@, '()

%"E01? *., I"981(,/%0 8. 3".µ9%, '%"()80?L./%0 8/3, .,#/3/% 3.3. ;C6(4, $,@"9L(-

,/%4 /( µ./%8E3µ%/08µ#–z 1%0 /04 0*0#/3/C4 /(), '"(E@"(7µ. 8/3, %F0('(9383 %)/(7

$0% /3 µ.6C/3 8)8/3µ?/@, *0%1"0/(7 E"#,(). G/80, 8/3, .,#/3/% 3.4 µ.6./(7µ. /3,

%0/0%/#/3/% 1%0 /3, .)8/?5.0% /C/(0@, 8)8/3µ?/@,, /04 *(µC4 '"%$µ%/('(93824 /()4

1%0 /3, %'#1"082 /()4 8/3 8)E,#/3/%.


3.1 $ ªÂÙ·Û¯ËÌ·ÙÈÛÌfi˜–z

3.1.1 $ÚÈÛÌfi˜

' µ9C5=T2µ5C4=µ?H–z (M.Z.) 9@?H =7µ5CJH x(n) O456;4CJK T;?@JL J;>Z9C54 AH:

(3.1)

?8JL z µ>5 µ4<5O467 µ9C5SI2C7. 045 I?<JLH 9L6JI>5H G5 =LµSJI>ZJLµ9 CJ M.Z. C2H

56JIJLG>5H x(n) AH W{x(n)} 654 C2 =TQ=2 µ9C5:K C2H x(n) 654 CJL M.Z. X(z) AH:

(3.2)

' M.Z. ?8AH J;>Z9C54 =C2 =TQ=2 (3.1) J@JµBZ9C54 %µ<9'6.)"(4 M.Z. (bilateral

z–transform), <45C> CJ BG;J4=µ5 96C9>@9C54 58? CJ –" QAH CJ + ". F8B;T94 654 J µ(,#-

'6.)"(4 M.Z. (unilateral z–transform), CJ@ J8J>J G5 9:9CB=JLµ9 5;<?C9;5 (9@?C2C5

3.3), ?8JL CJ BG;J4=µ5 96C9>@9C54 58? CJ 0 QAH CJ + ". ' µJ@?8I9L;JH M.Z. 935;-

µ?Z9C54 =C2@ 5@BIL=2 54C45CM@ =L=C2µBCA@ µ9 µ2 µ2O9@46QH 5;T46QH =L@G769H.

# µ4<5O467 µ9C5SI2C7 z =9 8JI46QH =L@C9C5<µQ@9H 963;BZ9C54 AH:

(3.3)

?8JL r CJ µQC;J C2H z 654 @ 2 <A@>5 C2H z, ?8AH 35>@9C54 =CJ +T7µ5 3.15.

z re j= @

x n X zZ( ) ( )? @A

X z x n z n

n

( ) ( )= #

=#"

+"

5

9 53 . 1 $ ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

Im{z} Im{z}

0

r@

@

z

Re{z}

Re{z}

MJ@5O45>JH[6K6IJH

z=ej@

E8>89OJ - z

1

(5) (S)

™¯‹Ì· 3.1

(%) M0$%*01# .'9'.*(, (I) ( M.Z.#/%, )'(6($9L./%0 $0% /04 /0µC4/() z '() %,21(), 8/( µ(,%*0%9(1716( 08(*),%µ.9 µ. /(, DTFT.

)M;5 µ8J;JKµ9 @5 963;B=JLµ9 C2@ (3.1) AH =L@B;C2=2 CA@ r 654 @, 5@C465G4=CM-

@C5H =’ 5LC7@ C2 µ9C5SI2C7 z 58? C2 =TQ=2 (3.3). PC=4 QTJLµ9:

(3.4)

?8JL µ9 F{g(n)} =LµSJI>ZJLµ9 CJ µ9C5=T2µ5C4=µ? Fourier C2H 56JIJLG>5H g(n). X re x n re x n r e F x n rj j n

n

n j n

n

n( ) ( )( ) ( ) ( )6 6 6= = [ ] = { }#

=#"

+"# #

=#"

+"#5 5


9 6 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

# =TQ=2 (3.4) µ5H O9>T@94 ?C4 J M.Z. C2H 56JIJLG>5H x(n) =CJ =2µ9>J z = rej@ 4=JK-

C54 µ9 CJ µ9C5=T2µ5C4=µ? Fourier O456;4CJK T;?@JL (DTFT) C2H C;J8J8J42µQ@2H

56JIJLG>5H x(n)r–n <45 C2 <A@4567 =LT@?C2C5 @. (@ 2 85;Bµ9C;JH r 984I9<9> >=2 µ9

C2 µJ@BO5, C?C9 2 (3.4) µ5H O>@94 CJ@ DTFT C2H x(n). !848IQJ@, ?C5@ 2 85;Bµ9C;JH

@ µ9C5SBII9C54, C?C9 ?I9H J4 C4µQH C2H µ4<5O467H µ9C5SI2C7H z G5 S;>=6J@C54 8B@A

=CJ µ(,%*0%9( 1716( (unit circle) CJL µ4<5O46JK 9848QOJL, O2I5O7 =CJ@ 6K6IJ 8JL

QT94 6Q@C;J C2@ 5;T7 CA@ 5:?@A@ 654 56C>@5 >=2 µ9 C2 µJ@BO5. # =TQ=2 µ9C5:K CJL

DTFT 654 CJL M.Z. 9>@54 8IQJ@ 8;J35@7H. P DTFT µ0%4 %1(6()59%4 08(7/%0 µ. /(

M.Z. %)/24 $0% /04 /0µC4 /() z '() I"981(,/%0 '?,@ 8/( µ(,%*0%9( 1716( /() µ0$%*0-

1(7 .'0'C*()–z (z–plane) (UI. +T7µ5 3.1S). E2I5O7 <45 r = |z| = 1 2 =TQ=2 (3.4)

65C5I7<94 =’ 5LC7 CJL O456;4CJK T;?@JL µ9C5=T2µ5C4=µJK Fourier:

(3.5)

3.1.2 ⁄!·ÚÍË ÙÔ˘ ªÂÙ·Û¯ËÌ·ÙÈÛÌÔ‡–z

' M.Z., AH OL@5µJ=94;B 589>;A@ ?;A@, µ8J;9> @5 µ2@ L8B;T94, O2I5O7 @5 µ2@ =L<6I>-

@94 <45 ?I9H C4H C4µQH C2H µ4<5O467H µ9C5SI2C7H z. # 89;4JT7 C4µM@ CJL z <45 C4H J8J>9H

J M.Z. X(z) QT94 8989;5=µQ@9H C4µQH 65I9>C54 '."0(E2 87$160834 (*.+.) (region of

convergence, ROC). P=CA, CM;5 ?C4 2 x(n)r–n 9>@54 5G;J>=4µ2 65C’ 58?ILC2 C4µ7. )?C9

(3.6)

# =TQ=2 (3.6) µ5H O9>T@94 ?C4 2 |X(z)| 9>@54 8989;5=µQ@2 (=L<6I>@94) 5@ 2 56JIJL-

G>5 x(n)r–n 9>@54 5G;J>=4µ2 65C’ 58?ILC2 C4µ7 (absolutely summable). 0>@9C54 35@9-

;? ?C4 2 =K<6I4=2 9:5;CBC54 µ?@J 58? CJ r = |z| 654 ?T4 58? CJ @. 09@46B, 4=TK94 ?C4

2 *.+. G5 65GJ;>Z9C54 58? Jµ?69@C;JLH O56CLI>JLH µ9 6Q@C;J C2@ 5;T7 CA@ 5:?-

@A@ CJL µ4<5O46JK 9848QOJL. +C2 <9@467 89;>8CA=2 2 *.+. CJL M.Z. G5 9>@54 C2H

µJ;37H , ?8JL C5 ?;45 R1, R2 9:5;CM@C54 58? C2@ 56JIJLG>5 x(n). (H

8;J=85G7=JLµ9 @5 9:9CB=JLµ9 C4H O453J;9C46QH 89;48CM=94H =K<6I4=2H µ9 C2 SJ7-

G945 J;4=µQ@A@ 85;5O94<µBCA@.

¶·Ú¿‰ÂÈÁÌ· 3.1

$5 L8JIJ<4=G9> 2 *.+. CJL M.Z. C2H O9:4?8I9L;2H (right–sided) 96G9C467H 56JIJL-

G>5H x1(n) = anu(n).

R z R1 2< <| |

X z x n z x n r e x n r e x n rn

n

n j n

n

n j n

n

n

n

( ) ( ) ( ) ( ) ( )= = & = < "#

=#"

+"# #

=#"

+"# #

=#"

+"#

=#"

+"

5 5 5 56 6

X z X e F x n

z e

jj( ) ( ) ( )

== = { }6

6


=G/0

X;2=4µJ8J4M@C5H CJ@ J;4=µ? (3.1) CJL M.Z. QTJLµ9:

(3.7)

# (3.7) 9>@54 BG;J4=µ5 589>;A@ ?;A@ <9Aµ9C;467H 8;J?OJL µ9 I?<J az–1 654 4=JK-

C54 µ9

(3.8)

µ9 C2@ 8;Ja8?G9=2 SQS545 ?C4 |az–1|<1 7 4=JOK@5µ5 |z| > |a|. # 5@4=?C2C5 5LC7 8;J=-

O4J;>Z94 C2@ *.+. 8B@A =Co µ4<5O46? 98>89OJ–z. *;?694C54 <45 ?I5 C5 =2µ9>5 8JL

S;>=6J@C54 =CJ 9:AC9;46? CJL 6K6IJL µ9 56C>@5 |a|, ?8AH 35>@9C54 =CJ +T7µ5 3.25.

O M.Z. C2H x1(n) 9>@54 ;2C7 =L@B;C2=2 (=TQ=2 3.8). &BG9 =L@B;C2=2 5LC7H C2H µJ;-

37H T5;56C2;>Z9C54 58? C5 µ3*.,01? (O2I5O7 C4H ;>Z9H CJL 5;4Gµ2C7) 654 CJLH '#6()4

(O2I5O7 C4H ;>Z9H CJL 85;5@Jµ5=C7). # =L@B;C2=2 X1(z) QT94 Q@5 µ2O9@46? =CJ

=2µ9>J z = 0 654 Q@5 8?IJ =CJ z = a. *5;5C2;JKµ9 98JµQ@AH, ?C4 2 *.+. C2H X1(z)

96C9>@9C54 =9 ?IJ CJ µ4<5O46? 98>89OJ 8JL 9>@54 Q:A 58? CJ@ 6K6IJ J J8J>JH QT94

56C>@5 CJ@ 8?IJ 5LC7H. # *.+. O9@ 89;4QT94 8JCQ 6B8J4J@ 8?IJ.

09@46B, µ45 O9:4?8I9L;2 56JIJLG>5 x(n) 465@J8J49> C2 =L@G762 x(n) = 0 <45 n < N1.

' M.Z. 5LC7H 4=JKC54 µ9:

(3.9)

654 2 *.+. 5LCJK 9>@54 C2H µJ;37H |z| > R1. E2I5O7 2 *.+. 89;4I5µSB@94 ?I9H C4H

/0µC4 /() µ0$%*01(7 .'0'C*()–z '() I"981(,/%0 CF@ %'# /(, 1716( %1/9,%4 |z| = R1,

#'() R1 .9,%0 .1.9,(4 ( '#6(4 /34 X(z) ( ('(9(4 %'CE.0 '."088#/."( %'# /3, %"E2 /@,

%F#,@,. *;J=JT7 G5 8;Q894 @5 OJG9> ?C5@ R1 < 0. +9 µ45 CQCJ45 89;>8CA=2 2 =TQ=2

(3.9) 89;4I5µSB@94 ?;JLH µ9 G9C46QH OL@Bµ94H CJL z. !8JµQ@AH, 2 C4µ7 z = " G5 8;Q-

894 @5 58J6I94=C9> 58? C2@ *.+., 53JK 5LC7 G5 JO2<9> =9 µ2 3;5<µQ@J M.Z.

¶·Ú¿‰ÂÈÁÌ· 3.2

$5 L8JIJ<4=G9> 2 *.+. C2H 5;4=C9;?8I9L;2H (left–sided) 96G9C467H 56JIJLG>5H x2(n)

= – anu( – n – 1).

X z x n z n

n N

( ) ( )= #

=

+"

51

X z

az

zz a1 1

1

1( ) =

#=

##

X z x n z a u n z a z azn

n

n n

n

n n n

nn

1 11

00

( ) ( ) ( ) ( )= = = =#

=#"

+"#

=#"

+"# #

=

+"

=

+"

5 5 55

9 73 . 1 $ ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z


9 8 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

=G/0

(8? CJ@ J;4=µ? (3.1) CJL M.Z. QTJLµ9:

(3.10)

045 |a–1z| < 1 7 4=JOK@5µ5 |z| < |a| CJ BG;J4=µ5 C2H (3.10) =L<6I>@94 O>@J@C5H:

(3.11) X z

a z az

zz a2 1 1

11

1

1

1( ) = #

#=

#=

## #

X z x n z a u n z a z

a z a z

n

n

n n

n

n n

n

n n

n

n

n

2 2

1

1

1

0

1

1

( ) ( ) ( )

( )

= = # # # = # =

= # = #

#

=#"

+"#

=#"

+"#

=#"

#

#

=

+"#

=

+"

5 5 5

5 5

Im ImIm

Re Re Re% %% %-1

(5) (S) (<)

™¯‹Ì· 3.2

!."0(E2 87$160-834 (%) *.F0#'6.)-"34, (I) %"08/."#-'6.)"34, ($) %µ<9-'6.)"34 %1(6()-

59%4

# *.+. C2H X2(z) 9>@54 J4 C4µQH CJL µ4<5O46JK 9848QOJL–z |z| < |a|, O2I5O7 CJ 9=AC9;4-

6? CJL 6K6IJL 56C>@5H |z| = |a| (+T7µ5 3.2S). !>@54 =2µ5@C46? @5 85;5C2;7=JLµ9 ?C4

J4 =L@5;C7=94H X1(z) 654 X2(z) 9>@54 56;4SMH >O49H (UI. =TQ=94H 3.8 654 3.11). # µ?@2

O453J;B S;>=69C54 =C2@ *.+. >)/# 83µ%9,.0 #/0 3 *.+. %'(/.6.9 %,%'#8'%8/( /µ2µ%

/() M.Z. 1%0 '"C'.0 '?,/(/. ,% 1%5("9L./%0. (@ µ5H OJG9> µ?@J J M.Z. TA;>H C2@ 5@C>-

=CJ4T2 *.+. 5LCJK, G5 L8B;T94 5S9S54?C2C5 =CJ@ 8;J=O4J;4=µ? C2H 56JIJLG>5H 8JL

JO7<2=9 =’ 5LC?@, 53JK µ8J;9> @5 L8B;TJL@ 89;4==?C9;9H 58? µ>5 585@C7=94H.

09@46B, µ>5 5;4=C9;?8I9L;2 56JIJLG>5 x(n) 465@J8J49> C2 =L@G762 x(n) = 0 <45 n >

R2. O M.Z. 5LC7H 4=JKC54 µ9:

(3.12) X z x n z n

n

N

( ) ( )= #

=#"5

2


654 2 *.+. 5LCJK 9>@54 C2H µJ;37H |z| < R2, O2I5O7 (0 /0µC4 /() µ0$%*01(7 .'0'C*()–z

'() I"981(,/%0 8/( .8@/."01# /() 1716() |z| = R2, #'() R2 .9,%0 .1.9,(4 ( '#6(4 /34

X(z) ( ('(9(4 %'CE.0 60$#/."( %'# /3, %"E2 /@, %F#,@,. *;J=JT7 G5 8;Q894 @5 OJG9>

?C5@ R2 > 0. +' 5LC7 C2@ 89;>8CA=2 2 =TQ=2 (3.12) 89;4I5µSB@94 ?;JLH µ9 5;@2C4-

6QH OL@Bµ94H CJL z. !8JµQ@AH, 2 C4µ7 z = 0 G5 8;Q894 @5 58J6I94=C9> 58? C2@ *.+.,

53JK 5LC7 G5 JO2<7=94 =9 µ2 3;5<µQ@J M.Z.

¶·Ú¿‰ÂÈÁÌ· 3.3

$5 L8JIJ<4=C9> 2 *.+. C2H 5µ3>8I9L;2H (two–sided) 96G9C467H 56JIJLG>5H

x(n) = a|n|, a > 0.

=G/0

# 56JIJLG>5 x(n) µ8J;9> @5 963;5=C9> AH BG;J4=µ5 OKJ BIIA@ 56JIJLG4M@, µ45H

O9:4?8I9L;2H 654 µ45H 5;4=C9;?8I9L;2H, AH 9:7H:

(3.13)7

(3.14)

(8? C5 OKJ 8;J2<JKµ9@5 85;5O9><µ5C5 <@A;>ZJLµ9 ?C4 J4 M.Z. CA@ 85;58B@A 56J-

IJLG4M@ 9>@54:

<45 (3.15)

<45 (3.16)

' M.Z. X(z) C2H 5µ3>8I9L;2H 56JIJLG>5H 4=JKC54 µ9 CJ BG;J4=µ5 CA@ M.Z. CA@ 984-

µQ;JLH 56JIJLG4M@ (SI. 4O4?C2C5 <;5µµ46?C2C5H =C2@ 98?µ9@2 L8J9@?C2C5), B;5:

(3.17)

7 (3.18)

7

# 5@C>=CJ4T2 *.+. 35>@9C54 =CJ +T7µ5 3.2< 654 4=JKC54 µ9 CJ 6J4@? Cµ7µ5 CA@ OKJ

89;4JTM@ ?8AH C4H <@A;>=5µ9 =C5 85;5O9><µ5C5 3.1 654 3.2 8;J2<JLµQ@AH. :9,%0

<%,."# #/0 $0% ,% )'?"E.0 1(0,# /µ2µ% (.'01?6)H3) µ./%F7 /@, *7( '."0(E+, 87$160-

X z

aa

z

z a z aa z a( )

( )( ), | |= #

# #< <#

#2

111

X z

az a za z a( ) , | |=

##

#< <# # #

#1

1

1

11 1 11

X z X z X z( ) ( ) ( )= +1 2

| |z a< #1

X z

a z2 1 1

1

1( ) = #

# # #

| |z a> X z

az1 1

1

1( ) =

# #

x n x n x n( ) ( ) ( )= +1 2

x n a u n a u nn n( ) ( ) ( )= + # ## 1

9 93 . 1 $ ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z


1 0 0 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

834, 5% '"C'.0 a < 1. 045 a > 1 O9@ L8B;T94 9846BILR2 CA@ 89;4JTM@ =K<6I4=2H 654

QC=4 2 (3.13) O9@ G5 QT94 M.Z., 85;?IJ 8JL J4 984µQ;JLH 56JIJLG>9H 8JL C2@ 585;-

C>ZJL@ (5;4=C9;?8I9L;2 654 O9:4?8I9L;2) 58? µ?@9H CJLH QTJL@!

09@46B, 9B@ L8B;T94 2 *.+. µ45H 5µ3>8I9L;2H 56JIJLG>5H C2H µJ;37H

, ?8JL x1(n), x2(n) O9:4?8I9L;2 654 5;4=C9;?8I9L;2 56JIJLG>5

B894;JL µ76JLH 5@C>=CJ4T5, C?C9 5LC7 58JC9I9>C54 58? C4H C4µQH CJL µ4<5O46JK 984-

8QOJL–z <45 C4H J8J>9H 4=TK94:

(3.20)

*;?694C54 O2I5O7 <45 C,% *%1/760( /() ('(9() 3 µ01"#/."3 %1/9,% R1 '"(8*0("9L.-

/%0 %'# .1.9,(, /(, '#6( /34 X 1(z) ( ('(9(4 %'CE.0 '."088#/."( %'# /3, %"E2 /@,

%F#,@, /() .'0'C*()–z, .,+ 3 µ.$%67/."3 %1/9,% R2 '"(8*0("9L./%0 %'# .1.9,( /(,

'#6( /34 X2(z) ( ('(9(4 %'CE.0 60$#/."( %'# /3, %"E2 /@, %F#,@,. (@ R1 ( R2, C?C9

J4 OKJ 89;4JTQH =K<6I4=2H O9@ 65IK8CJL@ 2 µ>5 C2@ BII2 654 2 C9I467 *.+. C2H X(z)

9>@54 CJ 69@?, O2I5O7 J M.Z. O9@ L8B;T94.

!"µ#$%&'( :#*#%(*"+"

,>5 8989;5=µQ@JL µ76JLH 56JIJLG>5 58JC9I9>C54 58? 8989;5=µQ@J 8I7GJH µ2

µ2O9@46M@ =CJ4T9>A@. P=CA ?C4 µ45 CQCJ45 56JIJLG>5 x(n) 96C9>@9C54 µ9C5:K R1 654

R2, O2I5O7

x(n) = 0 <45 n < N1 7 n > N2 (3.21)

?8JL R1, R2 8989;5µQ@J4 5;4GµJ>. ' M.Z. 5LC7H 4=JKC54 µ9:

(3.22)

!>@54 35@9;? ?C4 2 =94;B 5LC7 =L<6I>@94 <45 6BG9 C4µ7 CJL z, O2I5O7 2 *.+. 9>@54

?IJ CJ µ4<5O46? 98>89OJ–z. *;J=JT7 ?µAH T;94BZ9C54 =C5 9:7H =2µ9>5: !B@ R1 5;@2-

C46?H 654 R2 G9C46?H, C?C9 CJ BG;J4=µ5 89;4I5µSB@94 ?;JLH µ9 5;@2C46QH 654 G9C4-

6QH OL@Bµ94H CJL z. PC=4, ?C5@ z = 0, J4 ?;J4 µ9 5;@2C46QH OL@Bµ94H CJL z C9>@JL@ =CJ

B894;J, 9@M ?C5@ z = ", J4 ?;J4 µ9 G9C46QH OL@Bµ94H CJL z C9>@JL@ =CJ B894;J. !8J-

µQ@AH, =9 µ45 CQCJ45 89;>8CA=2 2 *.+. O9@ 89;4I5µSB@94 C4H C4µQH z = 0 654 z = ".

!B@ R1 ( 0, C?C9 QTJLµ9 µ?@J 5;@2C46QH OL@Bµ94H CJL z =C2@ 9:>=A=2 (3.22) 654,

65CB =L@Q8945, 2 *.+. 89;4I5µSB@94 C2@ C4µ7 z = ". !B@ R2 ] 0, C?C9 QTJLµ9 µ?@J

G9C46QH OL@Bµ94H CJL z =C2@ 9:>=A=2 (3.22) 654 65CB =L@Q8945 2 *.+. 89;4I5µSB@94

C2@ C4µ7 z = 0.

X z x n z n

n N

N

( ) ( )= #

=5

1

2

R z R1 2< <| |

x n x n x n( ) ( ) ( )= +1 2


¶·Ú¿‰ÂÈÁÌ· 3.4

$5 L8JIJ<4=C9> J M.Z. 65G9@?H 58? C5 85;56BCA O456;4CB =7µ5C5 8989;5=µQ@2H

O4B;6945H: (# Q@CJ@2 L8J<;Bµµ4=2 L8JO2IM@94 C2@ T;J@467 =C4<µ7 n = 0).

1 0 13 . 1 $ ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

5. {x1(n)} = {3,4,5,0,1,2}

S. {x2(n)} = {3,4,5,0,1,2}

<. {x3(n)} = {0,0,3,4,5,0,1,2}

O. {x4(n)} = {4,6,5,0,1,2}

9. x5(n) = *(n)

=C. x6(n) = *(n–m), m > 0

Z. x7(n) = *(n + m), m > 0

=G/0

(8? CJ@ J;4=µ? (3.1) CJL M.Z. QTJLµ9:

5. X1(z) = 3 + 4z–1 + 5z–2 + z–4 + 2z–5, µ9 *.+.: ?IJ CJ 98>89OJ–z 96C?H CJL z = 0

S. X2(z) = 3z2 + 4z + 5 + z–2 + 2z–3, µ9 *.+.: ?IJ CJ 98>89OJ–z 96C?H CJL z = 0

654 CJL z = "

<. X3(z) = 3z–2 + 4z–3 + 5z–4 + z–6 + 2z–7, µ9 *.+.: ?IJ CJ 98>89OJ–z 96C?H CJL z = 0

O. X4(z) = 4z2 + 6z + 5 + z–2 + 2z–3, µ9 *.+.: ?IJ CJ 98>89OJ–z 96C?H CJL z = 0

654 CJL z = "

9. X5(z) = 1, µ9 *.+.: ?IJ CJ 98>89OJ–z

=C. X6(z) = z–m, ?8JL m > 0, µ9 *.+.: ?IJ CJ 98>89OJ–z 96C?H CJL z = 0

Z. X7(z) = zm, ?8JL m > 0, µ9 *.+.: ?IJ CJ 98>89OJ–z 96C?H CJL z = "

(8? CJ 85;BO94<µ5 5LC? <>@9C54 35@9;? #/0 3 *.+. /() M.Z. µ9%4 %1(6()59%4 '.'.-

"%8µC,() µ21()4 (*0?"1.0%4), .9,%0 #6( /( .'9'.*(–z, .1/#4 98@4 %'# /% 83µ.9% z =

0 1%0/2 z = ". (LCB C5 =2µ9>5 58J6I9>J@C54 <45C> CJ zm (m > 0) C9>@94 =CJ B894;J <45

z = " 654 CJ z–m C9>@94 =CJ B894;J <45 z = 0.

(8? CJ >O4J 85;BO94<µ5 =L@94O2CJ8J4JKµ9 98>=2H ?C4 58? µ5G2µ5C467 B8JR2 J M.Z.

9>@54 58IMH Q@5H 9@5II56C46?H C;?8JH 5@585;B=C5=2H 9@?H =7µ5CJH. UIQ8JLµ9,

O2I5O7, ?C4 J =L@C9I9=C7H CJL z–n 9>@54 JL=45=C46B 2 C4µ7 CJL =7µ5CJH C2 T;J@467

=C4<µ7 n. ,9 BII5 I?<45, J 96GQC2H CJL z 85;QT94 969>@2 C2@ 8I2;J3J;>5 <45 CJ T;?@J,

2 J8J>5 µ5H 9>@54 585;5>C2C2 <45 CJ@ 8;J=O4J;4=µ? CA@ O94<µBCA@ 9@?H =7µ5CJH.

+9 8JIIQH 89;48CM=94H µ8J;JKµ9 @5 963;B=JLµ9 CJ BG;J4=µ5 µ45H =94;BH 8989;5-

=µQ@JL 7 589>;JL µ76JLH, =9 6I94=C7 µJ;37. +C4H 89;48CM=94H 5LCQH J M.Z. µ5H

85;QT94 µ45 9@5II56C467 =Lµ85<7 µJ;37 5@585;B=C5=2H 9@?H =7µ5CJH, ?8AH 35>-


1 0 2 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

@9C54 654 =CJ@ *>@565 3.1. 8JL 56JIJLG9>.

™À¡%£% (!À°% ª.(.

';4=µQ@5 58? C5 84J =L@2G4=µQ@5 Z9K<2 M.Z. 85;5C>G9@C54 =CJ@ *>@565 3.1. (LCB

G5 µ5H SJ2G7=JL@ 8B;5 8JIK 654 =CJ@ L8JIJ<4=µ? CJL 5@C>=C;J3JL M.Z., 963;B-

ZJ@C5H C2 =L@B;C2=2 X(z) AH <;5µµ46? =L@OL5=µ? 58IJK=C9;A@ =L@5;C7=9A@,

?8AH G5 OJKµ9 =C2@ 9@?C2C5 3.2.

¶›Ó·Î·˜ 3.1

&),253 L.7$3 µ./%8E3µ%/08µ+,–z

!%µ1 J'(1/"0µ1()/µD5–z >'C)#"% !G3*$)/05

*(n) 1 \IJ CJ z

*(n – m) z–m \IJ CJ z, 96C?H z = 0 5@ m > 0

7 z = " 5@ m < 0

u(n) |z| > 1

– u( – n – 1) |z| < 1

anu(n) |z| > |a|

– anu( – n – 1) |z| < |a|

nanu(n) |z| > |a|

– nanu( – n – 1) |z| < |a|

cos(@0n)u(n) |z| > 1

sin(@0n)u(n) |z| > 1

(sin )

( cos )

66

0

0

z

z z

#

# ## +

1

1 21 2

1

1 20

1

01 2

## +

#

# #(cos )

( cos )

66

z

z z

az

az

#

##

1

1 21( )

az

az

#

##

1

1 21( )

1

1 1# #az

1

1 1# #az

1

1 1# #z

1

1 1# #z


(rncos@0n)u(n) |z| > |r|

(rnsin@0n)u(n) |z| > |r|

( sin )

( cos )

r z

r z r z

66

0

0

#

# ## +

1

1 2 21 2

1

1 2

1

1 2 2

## +

#

# #( cos )

( cos )

r z

r z r z

66

0

0

1 0 33 . 1 $ ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

3.1.3 %‰ÈfiÙËÙÂ˜ ÙÔ˘ ÌÂÙ·Û¯ËÌ·ÙÈÛÌÔ‡–z

O M.Z. QT94 J;4=µQ@9H 4O4?C2C9H J4 J8J>9H CJ@ 65G4=CJK@ Q@5 9:54;9C46B T;7=4µJ

9;<5I9>J =C2 µ9IQC2 CA@ =2µBCA@ 654 =L=C2µBCA@ O456;4CJK T;?@JL. +C2@ 85;JK-

=5 L8J9@?C2C5 85;JL=4BZJLµ9 µ9 =L@CJµ>5 C4H 84J T;7=4µ9H 58? 5LCQH 654 ?I9H µ5Z>

C4H 85;5GQCJLµ9 =CJ@ *>@565 3.2.

°ƒ"ªª#'$&%&"

!B@

µ9 *.+. -1

654

µ9 *.+. -2

C?C9

µ9 *.+. CJLIBT4=CJ@ -1I-2 (3.23)

?8JL a, b =C5G9;QH 654 -1I-2 2 CJµ7 CA@ 89;4JTM@ -1 654 -2. # IQ:2 «CJLIBT4=CJ@»

T;2=4µJ8J47G269 <45 C2@ 89;>8CA=2 65CB C2@ J8J>5 J <;5µµ46?H =L@OL5=µ?H 9>@54

CQCJ4JH M=C9 6B8J45 µ2O9@46B 9:JLO9C9;M@JL@ J;4=µQ@JLH 8?IJLH. +9 µ45 CQCJ45

89;>8CA=2 2 *.+. 9>@54 µ9<5IKC9;2 58? C2@ CJµ7 CA@ OKJ 89;4JTM@.

$§#™£%™% ™&$ Ãƒ$¡$

!B@ µ9 *.+. -

C?C9 µ9 *.+. -_ (3.24)

?8JL -_ = - µ9 9@O9T?µ9@J ?µAH C2@ 8;J=G762 7 58?;;4R2 CA@ C4µM@ 0 654 ", I?<A

CJL 85;B<J@C5 z–m.

6.DI')B0:

x n m z X zZ m( ) ( )# ? @A #

x n X zZ( ) ( )? @A

ax n bx n aX z bX zZ1 2 1 2( ) ( ) ( ) ( )+ ? @A +

x n X zZ2 2( ) ( )? @A

x n X zZ1 1( ) ( )? @A

Z x n m x n m z z x n m z z x l z z X zn

n

m n m

n

m l

l

m( ) ( ) ( ) ( ) ( )( )#{ } = # = # = =#

=#"

+"# # #

=#"

+"# #

=#"

+"#5 5 5


1 0 4 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

™À¡!§#•% ™&$ ¶!¢#$ &$À Ãƒ$¡$À

!B@ µ9 *.+. -1

654 µ9 *.+. -2

C?C9 µ9 *.+. CJLIBT4=CJ@ -1I-2 (3.25)

6.DI')B0:

(8? CJ@ J;4=µ? C2H =L@QI4:2H C2H 9@?C2C5H 1.5 CJL &935I5>JL 1 <@A;>ZJLµ9 ?C4

.

(@C465G4=CM@C5H C2 =TQ=2 5LC7 =C2@ (3.1) QTJLµ9:

+CJ@ *>@565 3.2 85;5GQCJLµ9 C4H 84J <@A=CQH 4O4?C2C9H CJL M.Z.

Z x n x n x n x n z x m x n m z

x m x n m z x m z

n

n m

n

n

n

nm

1 2 1 2 1 2

1 2 1

( )* ( ) ( )* ( ) ( ) ( )

( ) ( ) ( )

{ } = [ ] = ! #7

899

:

;<<

=

= #7

899

:

;<<

=

#

=#"

+"

=#"

+"#

=#"

+"

#

=#"

+"

=#"

+"#

5 55

55 mm

m

m

m

X z

x m z X z X z X z

2

1 2 1 2

( )

( ) ( ) ( ) ( )

[ ] =

= =

=#"

+"

#

=#"

+"

5

5

x n x n x m x n m

m

1 2 1 2( )* ( ) ( ) ( )= #=#"

+"

5

x n x n X z X zZ1 2 1 2( )* ( ) ( ) ( )? @A

x n X zZ2 2( ) ( )? @A

x n X zZ1 1( ) ( )? @A

¶›Ó·Î·˜ 3.2

V*0#/3/.4 /() M.Z.

8I)D(0(1 !%µ1 J'(1/"0µ1()/µD5–z >'C)#"% /G3*$)/05

x(n) X(z) P: R1 < |z| < R2

x1(n) X1(z) P1

x2(n) X2(z) P2

0;5µµ46?C2C5 5x1(n) + bx2(n) 5X1(z) + bX2(z) P1IP2 CJLIBT4=CJ@

'I>=G2=2 =CJ T;?@J x(n – m) z–mX(z) P, 96C?H z = 0 5@ m > 0

7 z = " 5@ m < 0

&I4µB6A=2 =CJ 89O>J–z 5nx(n) X(5–1z) |5|R1 < |z| < |5|R2


&5CJ8C;4=µ?H =CJ T;?@J x( – n) X(z–1)

+LZL<>5 x*(n) X*(z*) P

*5;5<M<4=2 =CJ 89O>J–z nx(n) P

+L@QI4:2 x1(n)*x2(n) X1(z)X2(z) P1IP2 CJLIBT4=CJ@

*;MC2 O453J;B x(n) – x(n – 1) (1 – z–1)X(z) )JLIBT4=CJ@ 2 CJµ7 C2H -

654 C2H |z| > 0

D9M;2µ5 5;T467H C4µ7H (@ x(n) = 0 <45 n < 0, C?C9 x(0) = lim ( )z

X z@"

#z

dX zdz( )

1 1

2 1Rz

R<| |<

1 0 53 . 1 $ ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

¶·Ú¿‰ÂÈÁÌ· 3.5

F8JIJ<>=C9 CJ M.Z. CA@ =2µBCA@ x2(n) 654 x3(n) CJL *5;5O9><µ5CJH 3.4, S5=4Z?-

µ9@J4 =CJ M.Z. CJL =7µ5CJH x1(n).

=G/0

E4584=CM@JLµ9 9K6JI5 ?C4 x2(n) = x1(n + 2) 654 x3(n) = x1(n–2). Y;5, µ9 SB=2 C2@

4O4?C2C5 C2H JI>=G2=2H =CJ T;?@J, S;>=6JLµ9 ?C4

X2(z) = z2X1(z) = 3z2 + 4z + 5 + z–2 + 2z–3 654 X3(z) = z–2X1(z) = 3z–2 + 4z–3 + 5z–4 + z–6

+ 2z–7.

(H =2µ94AG9> ?C4 I?<A CJL 8JII58I5=45=µJK 98> z2 2 *.+. CJL X2(z) O9@ 89;4I5µ-

SB@94 CJ =2µ9>J z = ", 5@ 654 5LC? 5@769 =C2@ *.+. CJL X1(z).

)J 85;BO94<µ5 5LC? µ5H O>@94 C2 OL@5C?C2C5 @5 65C5@J7=JLµ9 65IKC9;5 C2@ 4O4?-

C2C5 C2H JI>=G2=2H. *;B<µ5C4, 5@ GLµ2GJKµ9 ?C4 J =L@C9I9=C7H CJL z–n 5@C4=CJ4T9>

=C2@ C4µ7 CJL O9><µ5CJH =C2@ T;J@467 =C4<µ7 n, 65C5I5S5>@JLµ9 5µQ=AH ?C4 2 65GL-

=CQ;2=2 9@?H =7µ5CJH 65CB m (m > 0) O9><µ5C5, O2I5O7 x(n)@x(n–m), 4=JOL@5µ9>

µ9 8JII58I5=45=µ? ?IA@ CA@ ?;A@ CJL M.Z. 98> z–m. ' =L@C9I9=C7H CJL z–n <>@9-

C54 =L@C9I9=C7H CJL z–(n + m).

$5 L8JIJ<>=9C9 C2@ Q:JOJ y(n) CJL *5;5O9><µ5CJH 1.6, 5:4J8J4M@C5H C4H 4O4?C2-

C9H CJL M.Z.



1 0 6 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z


+C2@ 9@?C2C5 5LC7 <@A;>=5µ9 CJ M.Z. X(z) C2H 56JIJLG>5H x(n) 654 9>O5µ9 ?C4:

# *.+. C2H =L@B;C2=2H X(z) 9>@54 =C2 <9@467 89;>8CA=2, Q@5H 6L6I46?H O56CKI4JH

(R1 < |z| < R2) =CJ 98>89OJ–z µ9 6Q@C;J C2@ 5;T7 CA@ 5:?@A@. +9 J;4=µQ@9H 89;4-

8CM=94H R1 = 0, J8?C9 2 *.+. <>@9C54 Q@5H O>=6JH 56C>@5H R2, 9@M =9 BII9H R2 = ",

J8?C9 2 *.+. 96C9>@9C54 =9 ?IJ CJ 98>89OJ |z| > R1.

# *.+. O9@ 89;4I5µSB@94 CJL 8?IJLH C2H =L@B;C2=2H X(z).

!B@ 2 x(n) 9>@54 8989;5=µQ@JL µ76JLH, C?C9 2 *.+. 9>@54 ?IJ CJ 98>89OJ–z, µ9 84G5-

@7 9:5>;9=2 C5 =2µ9>5 z = 0 654 / 7 z = ".

!B@ 2 X(z) 9>@54 ;2C7 654 2 x(n) O9:4?8I9L;2, C?C9 2 *.+. J;>Z9C54 =CJ 98>89OJ–z

96C?H CJL 6K6IJL µ9 56C>@5 CJ@ 8IQJ@ 58Jµ56;L=µQ@J 8?IJ C2H X(z). !B@ 984-

8IQJ@, 2 x(n) 9>@54 54C45C7, C?C9 2 *.+. =Lµ89;4I5µSB@94 654 C2@ C4µ7 z = ".

!B@ 2 X(z) 9>@54 ;2C7 654 2 x(n) 5;4=C9;?8I9L;2, C?C9 2 *.+. J;>Z9C54 =CJ 98>89-

OJ–z 9@C?H CJL 6K6IJL µ9 56C>@5 CJ@ 8I2=4Q=C9;J =CJ 6Q@C;J 8?IJ C2H X(z). !B@,

9848IQJ@, 2 x(n) 9>@54 5@C4–54C45C7 (O2I5O7 5;4=C9;?8I9L;2 654 >=2 µ9 0 <45 n >

0), C?C9 2 *.+. =Lµ89;4I5µSB@94 654 C2@ C4µ7 z = 0.

3.2 O ·ÓÙ›ÛÙÚÔÊÔ˜ ÌÂÙ·Û¯ËÌ·ÙÈÛÌfi˜–z

' 5@C>=C;J3JH M.Z. µ5H SJ2GB @5 L8JIJ<>=JLµ9 CJ =7µ5 O456;4CJK T;?@JL x(n)

?C5@ <@A;>ZJLµ9 CJ M.Z. 5LCJK, X(z). +LµSJI46B <;B3JLµ9 x(n) = Z–1{X(z)}. (8J-

O946@K9C54 ?C4 J 5@C>=C;J3JH M.Z. C2H X(z) O>@9C54 58? C2 =TQ=2:

(3.26)

?8JL c µ45 5;4=C9;?=C;J32 6I94=C7 65µ8KI2 JIJ6I7;A=2H $7"@ %'# /3, %"E2 /@,

%F#,@, 1%0 .,/#4 /34 *.+. /34 X(z). D0% *.*(µC,3 *.+. ( %,/98/"(<(4 M.Z. .9,%0 µ(,%-

*01? ("08µC,(4.

P@5H C;?8JH @5 L8JIJ<>=JLµ9 C2@ x(n) 9>@54 @5 5@58CK:JLµ9 C2@ X(z) =9 OL@5µJ-

=94;B (power–series expansion) AH 8;JH z–1, O2I5O7:

x nj

X z z dzn

c

( )8

( )= #=12

1

$5 L8JIJ<>=9C9 CJ M.Z. CJL =7µ5CJH x n

n N( )

, , , ...,

,= = #0

12

1 0 1 1

0

%66(7



(3.27)

E4584=CM@JLµ9 9K6JI5 ?C4 CJ Z2CJKµ9@J =7µ5 9>@54 J4 =L@C9I9=CQH CJL 8JILA@KµJL.

\C5@ 2 X(z) O>@9C54 =9 6I94=C7 µJ;37, I.T. AH µ45 ;2C7 =L@B;C2=2 8JILA@KµA@, Q@5

CQCJ4J 5@B8CL<µ5 O9@ 9>@54 9K6JIJ @5 L8JIJ<4=C9>. +9 5LCQH C4H 89;48CM=94H, <45 CJ@

L8JIJ<4=µ? C2H 56JIJLG>5H x(n), T;2=4µJ8J4JKµ9 9>C9 C2 µQGJOJ C2H 5@B8CL:2H =9

µ9;46B 6IB=µ5C5 (partial–fraction expansion) 9>C9 C2 µQGJOJ CA@ JIJ6I2;AC46M@ L8J-

IJ>8A@ (residue method). D5 9:9CB=JLµ9 =C5 98?µ9@5 9OB345 CJ@ L8JIJ<4=µ? CJL 5@C>-

=C;J3JL M.Z. µ9 5@B8CL:2 =9 OL@5µJ=94;B 654 µ9 5@B8CL:2 =9 µ9;46B 6IB=µ5C5.

3.2.1 À!ÔÏÔÁÈÛÌfi˜ ÙÔ˘ ·ÓÙ›ÛÙÚÔÊÔ˘ ª.&. ÌÂ ·Ó¿!Ù˘ÍË ÛÂ ‰˘Ó·ÌÔÛÂÈÚ¿

# 5@B8CL:2 µ45H ;2C7H =L@B;C2=2H =9 OL@5µJ=94;B 984CL<TB@9C54 =L@7GAH µ9 8),.-

E2 *0%9".83 (long division). # µQGJOJH 5LC7 O9@ 65C5I7<94 =9 µ45 5@5ILC467 Q63;5-

=2 <45 C2@ x(n). !>@54 µ>5 5;4Gµ2C467 µQGJOJH µ9 C2@ J8J>5 µ8J;JKµ9 @5 L8JIJ<>-

ZJLµ9 Q@5 @QJ =CJ4T9>J C2H x(n) 6BG9 3J;B.

¶·Ú¿‰ÂÈÁÌ· 3.6$5 L8JIJ<4=C9> 2 O9:4?8I9L;2 56JIJLG>5 x(n) C2H J8J>5H J M.Z. 9>@54:

.

=G/0 # =L@9T7H O45>;9=2 CJL 5;4Gµ2C7 µ9 CJ@ 85;5@Jµ5=C7 µ5H O>@94:

1 + 1,6 z–1 – 0,52 z–2 + 0,4 z–3 + …

1 + 0,4z–1 – 0,12z–2 1 + 2 z–1

1 + 0,4z–1 – 0,12z–2

1,6z–1 + 0,12z–2

1,6z–1 + 0,64z–2 – 0,192z–3

– 0,52z–2 + 0,192z–3

– 0,52z–2 – 0,208z–3 + 0,0624z–4

– 0,4z–3 – 0,0624 z–4

…………………

…………………

X z

z

z z( )

, ,= +

+ #

#

# #1 2

1 0 4 0 12

1

1 2

X z x n z x z x z x z x z x zn

n

( ) ( ) .... ( ) ( ) ( ) ( ) ( ) ....= = + # + # + + + +# # #

=#"

+"

5 2 1 0 1 22 1 0 1 2

1 0 73 . 2 O " ¡ & # ™ & ƒ $ º $ ™ ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z


1 0 8 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

!8JµQ@AH 2 X(z) µ8J;9> @5 963;5=C9> AH OL@5µJ=94;B:

X(z) = 1 + 1,6z–1 – 0,52z–2 + 0,4z–3 + ….

!>@54 35@9;? ?C4:

x(0) = 1, x(1) = 1,6, x(2) = – 0,52, x(3) = 0,4,….

+L@9T>ZJ@C5H C2 O45>;9=2 G5 µ8J;JK=5µ9 @5 8B;JLµ9 654 BII5 =CJ4T9>5 C2H x(n).

3.2.2 À!ÔÏÔÁÈÛÌfi˜ ÙÔ˘ ·ÓÙ›ÛÙÚÔÊÔ˘ ª.&. ÌÂ ·Ó¿!Ù˘ÍË ÛÂ ÌÂÚÈÎ¿ ÎÏ¿ÛÌ·Ù·

P=CA X(z) ;2C7 =L@B;C2=2 C2H µJ;37H

(3.28)

D9A;7=5µ9 ?C4 b0 = 1. (LC? ?µAH O9@ 89;4J;>Z94 C2 <9@46?C2C5 C2H Q63;5=2H, 53JK

9B@ b0 3 1, µ8J;JKµ9 @5 O454;Q=JLµ9 5;4Gµ2C7 654 85;J@Jµ5=C7 µ9 b0 654 @5 65C5-

I7:JLµ9 =C2 µJ;37 C2H =TQ=2H (3.28). ,>5 ;2C7 =L@B;C2=2 C2H µJ;37H 5LC7H J@J-

µBZ9C54 65@J@467 (proper) 9B@ bN 3 0 654 M < R, O2I5O7 ?C5@ J 5;4Gµ?H CA@ µ2O9-

@46M@ C2H =L@B;C2=2H 9>@54 µ46;?C9;JH CJL 5;4GµJK CA@ 8?IA@ 5LC7H. !B@ 2 ;2C7

=L@B;C2=2 9>@54 µ2–65@J@467 (M ( R), C?C9 5LC7 µ8J;9> 8B@CJC9 @5 963;5=C9> AH

BG;J4=µ5 9@?H 8JILA@KµJL 654 µ45H 65@J@467H ;2C7H =L@B;C2=2H, 96C9IM@C5H C2

O45>;9=2 CJL C(z) O45 CJL D(z), O2I5O7

(3.29)

?8JL (3.30)

654 (3.31)

' L8JIJ<4=µ?H CJL 5@C>=C;J3JL M.Z. CJL 8JILA@KµJL X0(z) 9>@54 8;J35@7H 654

<>@9C54 589LG9>5H TA;>H 9848IQJ@ 8;B:94H, 58? CJ@ J;4=µ? CJL M.Z. (=TQ=2 3.1).

# =L@B;C2=2 X1(z) 9>@54 65@J@467 ;2C7 =L@B;C2=2. +C?TJH µ5H 9>@54 @5 C2@ 5@5-

8CK:JLµ9 =9 µ9;46B 6IB=µ5C5, O2I5O7 @5 C2@ 963;B=JLµ9 AH BG;J4=µ5 58IM@ 6I5-

=µBCA@ 654 =C2 =L@QT945 @5 L8JIJ<>=JLµ9 CJ@ 5@C>=C;J3J M.Z. 6BG9 6IB=µ5CJH

:9TA;4=CB. "?<A C2H 4O4?C2C5H C2H <;5µµ46?C2C5H, J 5@C>=C;J3JH M.Z. C2H =L@B;-

X z

C zD z1

1( )( )( )

=

X z c c z c z c zM NM N

qq

q

M N

0 0 11

0

( ) ... ( )= + + + =##

# # #

=

#

5 X z

C zD z

c c z c zC zD z

X z X zM NM N( )

( )( )

...( )( )

( ) ( )( )= = + + + + = +##

# #0 1

1 10 1

X z

C zD z

a a z a z

b z b zM

M

NN

( )( )( )

...

...= = + + +

+ + +

# #

# #0 1

1

111


C2=2H X1(z) G5 4=JKC54 µ9 CJ BG;J4=µ5 CA@ 984µQ;JLH 5@C>=C;J3A@ M.Z. CA@ 6I5-

=µBCA@. (H OJKµ9 IJ48?@, 8MH <>@9C54 2 5@B8CL:2 µ45H 65@J@467H ;2C7H =L@B;C2-

=2H =9 µ9;46B 6IB=µ5C5.

P=CA ?C4 2 X(z) 9>@54 65@J@467 ;2C7 =L@B;C2=2 C2H µJ;37H (3.28). 045 @5 58IJ-

8J47=JLµ9 C2@ ?I2 5@BIL=2, 585I9>3JLµ9 C4H 5;@2C46QH OL@Bµ94H CJL z 8JII58I5-

=4BZJ@C5H 5;4Gµ2C7 654 85;J@Jµ5=C7 98> zN, J8?C9 2 =TQ=2 <>@9C54:

(3.32)

!894O7 R > M 2 =L@B;C2=2

(3.33)

9>@54 8B@CJC9 65@J@467. 045 @5 5@58CK:JLµ9 C2@ C9I9LC5>5 =L@B;C2=2 =9 µ9;46B

6IB=µ5C5, S;>=6JLµ9 C4H ;>Z9H p1, p2,…,pN CJL 85;J@Jµ5=C7, O2I5O7 CJLH 8?IJLH

C2H X(z). D5 9:9CB=JLµ9 =C2 =L@QT945 C2@ 89;>8CA=2 8JL J4 8?IJ4 9>@54 ?IJ4 O45-

3J;9C46J> µ9C5:K CJLH, O2I5O7 9>@54 8JII58I?C2C5H 1. +9 µ45 CQCJ45 89;>8CA=2 2

(3.33) <;B39C54 AH

(3.34)

+C?TJH µ5H 9>@54 @5 L8JIJ<>=JLµ9 CJLH =L@C9I9=CQH >1, >2,…,>$. (LC? µ8J;JKµ9

@5 CJ 89CKTJLµ9 8JII58I5=4BZJ@C5H 654 C5 OKJ µQI2 C2H (3.34) µ9 65GQ@5@ 58? CJLH

?;JLH (z–pk), k = 1,2,…,N 654 L8JIJ<>ZJ@C5H C2@ Q63;5=2 8JL 8;J6K8C94 =C4H 5@C>-

=CJ4T9H GQ=94H CA@ 8?IA@ p1, p2,…,pN. PC=4, =C2 <9@467 89;>8CA=2 G5 QTJLµ9

(3.35)

654, 98JµQ@AH, <45 z = pk 2 =TQ=2 5LC7 G5 µ5H OM=94 CJ =L@C9I9=C7 >k, 53JK ?IJ4

J4 BIIJ4 ?;J4 CJL O9LCQ;JL µQIJLH µ2O9@>ZJ@C54. E2I5O7

(3.36)

+2µ94M@9C54 ?C4 J4 =TQ=94H (3.34) 654 (3.36) 4=TKJL@ C?=J <45 8;5<µ5C46JKH, ?=J

654 <45 µ4<5O46JKH 8?IJLH.

Az p X z

zk Nk

k

z pk

= # ==

( ) ( ), ,...,1 2

( ( ( (z p X zz

z p Az p

z p Az p

z p Az p

k k k k N

N

# = ##

+ ##

+ + ##

) ( ) ) )....

)1

1

2

2

X zz

Az p

Az p

Az p

N

N

( )....=

#+

#+ +

#1

1

2

2

X zz

a z a z a z

z b z b

N NM

N M

N NN

( ) ...

...= + + +

+ + +

# # # #

#0

11

2 1

11

X z

a z a z a z

z b z b

N NM

N M

N NN

( )...

...= + + +

+ + +

# #

#0 1

1

11

1 0 93 . 2 O " ¡ & # ™ & ƒ $ º $ ™ ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z


1 1 0 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

# O45O465=>5 C2H 5@B8CL:2H C2H X(z) =9 µ9;46B 6IB=µ5C5 QT94 JIJ6I2;AG9> =CJ

=2µ9>J 5LC?, 53JK QTJLµ9 L8JIJ<>=94 C4H C4µQH >1, >2,…,>$. # X(z) <;B39C54 CM;5 AH

(3.37)

7 O454;M@C5H 5;4Gµ2C7 654 85;J@Jµ5=C7 6BG9 µ9;46JK 6IB=µ5CJH µ9 z QTJLµ9 C2@

C9I467 Q63;5=2 CJL X(z) =9 µJ;37 5;@2C46M@ OL@Bµ9A@ CJL z:

(3.38)

' 5@C>=C;J3JH M.Z. x(n) = Z–1{X(z)} µ8J;9> CM;5 9K6JI5 @5 L8JIJ<4=C9> µ9 CJ@

5@C>=C;J3J M.Z. 6BG9 µ9;46JK 6IB=µ5CJH :9TA;4=CB 654 C9I46B CJ@ <;5µµ46? =L@-

OL5=µ? CJLH. (8? CJ@ *>@565 3.1 µ8J;JKµ9 @5 OJKµ9 ?C4

(3.39)

)J 8J45 58? C4H =TQ=94H G5 984IQ:JLµ9, 9:5;CBC54 58? C2@ 9848IQJ@ 8I2;J3J;>5 8JL

QTJLµ9. 045 85;BO94<µ5, 5@ <@A;>ZJLµ9 ?C4 CJ =7µ5 x(n) 9>@54 54C45C?, C?C9 G5 984-

IQ:JLµ9 C2 O9:4?8I9L;2 56JIJLG>5 (8;MC2 58? C4H =TQ=94H 3.39) 654 J 5@C>=C;J3JH

M.Z. C2H (3.38) G5 4=JKC54 µ9

(3.40)

(H 85;56JIJLG7=JLµ9 C2@ ?I2 O45O465=>5 L8JIJ<4=µJK CJL 5@C>=C;J3JL M.Z. µ45H

65@J@467H ;2C7H =L@B;C2=2H µ9 C2 µQGJOJ C2H 5@B8CL:2H =9 µ9;46B 6IB=µ5C5,

µQ=5 58? CJ 98?µ9@J 85;BO94<µ5.

¶·Ú¿‰ÂÈÁÌ· 3.7

$5 L8JIJ<4=C9> 2 O9:4?8I9L;2 56JIJLG>5 x(n) C2H =L@B;C2=2H CJL *5;5O9><µ5CJH 3.6.

=G/0

&5C5;T7@ 585I9>3JLµ9 C4H 5;@2C46QH OL@Bµ94H CJL z 58? C2 =L@B;C2=2, 8JII5-

8I5=4BZJ@C5H CJ@ 5;4Gµ2C7 654 CJ@ 85;J@Jµ5=C7 µ9 CJ z2. PC=4 QTJLµ9:

.

'4 8?IJ4 C2H X(z) 9>@54 p1 = 0,2 654 p2 = –0,6, J8?C9 2 =TQ=2 5LC7 <;B39C54 =C2

X z

z

z z

z z

z z( )

, , , ,= +

+ #= +

+ #

#

# #1 2

1 0 4 0 12

2

0 4 0 12

1

1 2

2

2

x n A p A p A p u nn nN N

n( ) ( ... ) ( )= + + +1 1 2 2

Z

p z

p u n z p

p u n z pk

kn

k

kn

k

###

=>

# # <

01D

2D1

1

1

1 1{ }

( ) | | | | ( )

( ) | | | | ( )

5@ *.+.

5@ *.+.

:

- :

%0/0%/# 82µ%

µ3 %0/0%/# 82µ%

X z A

p zA

p zA

p zN

N

( ) ....=#

+#

+ +#1

11 2

21 1

1

1

1

1

1

1- - -

X z A

zz p

Az

z pA

zz pN

N

( ) ....=#

+#

+ +#1

12

2


µJ;37 C2H (3.34) AH 9:7H:

!35;µ?ZJ@C5H C2 =TQ=2 (3.36) L8JIJ<>ZJLµ9 C5 >1 654 >2

Y;5 2 X(z) µ8J;9> 8IQJ@ @5 963;5=C9> AH 5@B8CL<µ5 µ9;46M@ 6I5=µBCA@

0@A;>ZJ@C5H ?C4 2 Z2CJKµ9@2 56JIJLG>5 x(n) 9>@54 O9:4?8I9L;2, L8JIJ<>ZJLµ9 CJ@

5@C>=C;J3J M.Z. 58? C2@ 8;MC2 =TQ=2 C2H (3.39) AH

x(n) = 2,75(0,2)nu(n) – 1,75( – 0,6)nu(n)

(LC7 9>@54 2 Z2CJKµ9@2 56JIJLG>5. *5;5C2;JKµ9 ?C4 <45 C4H C4µQH CJL n = 0,1,2,3

85>;@JLµ9 C5 =CJ4T9>5 x(0) = 1, x(1) = 1,6, x(2) = – 0,52, x(3) = 0,4, 984S9S54M@J-

@C5H QC=4 C5 58JC9IQ=µ5C5 CJL *5;5O9><µ5CJH 3.6.

# O45O465=>5 C2H 5@B8CL:2H =9 µ9;46B 6IB=µ5C5 8JL 9:9CB=5µ9 µQT;4 CM;5, 935;-

µ?Z9C54 =C2@ 89;>8CA=2 8JL QTJLµ9 58IJKH 8?IJLH, O2I5O7 8?IJLH O453J;9C46JKH

µ9C5:K CJLH. !B@ 2 X(z) QT94 8?IJLH 8JII58I?C2C5H l, O2I5O7 J 85;J@Jµ5=C7H C2H

89;4QT94 85;B<J@C9H C2H µJ;37H (z–pk)l, C?C9 2 8;J2<JKµ9@2 O45O465=>5 O9@ 4=TK94

?8AH QT94, 654 8;Q894 @5 C;J8J8J42G9>. # 89;>8CA=2 5LC7, ?µAH, 85;966I>@94 58?

CJLH =C?TJLH CJL 85;?@CJH S4SI>JL 654 O9 G5 9:9C5=C9>. \8AH 98>=2H O9 G5 9:9C5-

=C9> 2 89;>8CA=2 L8JIJ<4=µJK CJL 5@C>=C;J3JL M.Z. µ9 T;7=2 CA@ JIJ6I2;AC4-

X z

z z( ) ,

,,

,=

##

+# #2 751

1 0 21 75

1

1 0 61 1

X z

zz

zz

( ) ,,

,,

=#

#+

J2 750 2

1 750 6

X zz z z( )

, ,=

##

+J2 75

0 21 75

0 6, ,

Az X z

zz

z

Az X z

zz

z

z z

z z

10 2 0 2

20 6 0 6

0 2 20 6

2 75

0 6 20 2

1 75

= # = ++

=

= + = +#

= #

= =

=# =#

( , ) ( ),

,

( , ) ( ),

,

, ,

, ,

X zz

zz z

Az

Az

( )( , )( , ) , ,

= +# +

=#

++

20 2 0 6 0 2 0 6

1 2

1 1 13 . 2 O " ¡ & # ™ & ƒ $ º $ ™ ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z


1 1 2 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

6M@ L8JIJ>8A@. +C2 S4SI4J<;53>5 CJL &935I5>JL O>@J@C54 =L<<;Bµµ5C5 C5 J8J>5

5@53Q;J@C54 96C9@MH =C5 GQµ5C5 5LCB.

$5 L8JIJ<>=9C9 C2@ x(n) ?C5@ <@A;>Z9C9 ?C4 X(z) = (z–1)/(z2–1,9z + 0,84) 654 2 *.+.

9>@54 (5) |z| > 1 7 (S) |z| < 0,5.


$5 L8JIJ<>=9C9 CJ@ 5@C>=C;J3J M.Z. C2H X(z) = (6 + z–1)/(1–0,25z–2).



+C2@ 9@?C2C5 5LC7 9>O5µ9 C4H µ9G?OJLH L8JIJ<4=µJK CJL 5@C>=C;J3JL M.Z. +L<69-

6;4µQ@5, 9>O5µ9 ?C4 ?C5@ µ5H O>@9C54 J M.Z. µ45H 56JIJLG>5H, J J8J>JH 9>@54 =L@7-

GAH ;2C7H µJ;37H, µ8J;JKµ9 @5 S;JKµ9 C2@ >O45 C2@ 56JIJLG>5 9>C9 5@58CK==J-

@C5H =9 OL@5µJ=94;B 96C9IM@C5H C2 I9<?µ9@2 =L@9T7 O45>;9=2, 9>C9 5@58CK==J-

@C5H =9 µ9;46B 6IB=µ5C5. # 8;MC2 58? C4H µ9G?OJLH 5LCQH O9@ 65C5I7<94 =9

6B8J45 5@5ILC467 µJ;37 <45 C2@ Z2CJKµ9@2 56JIJLG>5, 654 T;2=4µJ8J49>C54 6L;>AH

<45 CJ@ L8JIJ<4=µ? µ9;46M@ 58? CJLH 8;MCJLH ?;JLH CJL 5@58CK<µ5CJH.


3.3 $ ÌÔÓfi!ÏÂ˘ÚÔ˜ ÌÂÙ·Û¯ËÌ·ÙÈÛÌfi˜–z

O M.Z. 8JL <@A;>=5µ9 µQT;4 CM;5 5@53Q;9C54 =9 =7µ5C5 C5 J8J>5 J;>ZJ@C54 =9 ?IJ

CJ O4B=C2µ5 CJL T;?@JL 58? CJ – " QAH CJ + " ( – " < n < + ") 654 J@JµBZ9C54

5µ3>8I9L;JH M.Z. (bilateral z–transform). !6C?H 5LCJK, L8B;T94 654 J µ(,#'6.)"(4

M.Z. (one–sided or unilateral z–transform), J J8J>JH 9>@54 4O45>C9;5 T;7=4µJH <45

C2@ 5@BIL=2 O456;4CM@ =L=C2µBCA@ C5 J8J>5 5;T46B O9@ S;>=6J@C54 =9 2;9µ>5.

(LCB C5 =L=C7µ5C5 89;4<;B3J@C54 58? 9:4=M=94H O453J;M@ µ9 µ3 µ3*.,01C4 %"E0-

1C4 8),521.4. ,9 BII5 I?<45, Q@5 CQCJ4J =K=C2µ5 5;T46B O9@ S;>=69C54 =9 2;9µ>5,

I?<A O49<Q;=9A@ 8JL 935;µ?=C265@ =’ 5LC? 8;4@ 56?µ5 935;µJ=C9> 2 9>=JOJH x(n)

65CB C2 T;J@467 =C4<µ7 n = 0. '4 5;T46QH C4µQH y(–1), y(–2),…,y(–N) C2H 9:?OJL CJL

=L=C7µ5CJH, J4 J8J>9H J39>IJ@C54 =C4H 8;J2<JKµ9@9H O49<Q;=94H, 58JC9IJK@ C4H 5;T4-

6QH =L@G769H 5LCJK.

' µJ@?8I9L;JH M.Z. C2H 56JIJLG>5H x(n) J;>Z9C54 AH

(3.41)

654 <;B3JLµ9 =L@7GAH

(3.42)

*5;5C2;JKµ9 ?C4 =CJ µJ@?8I9L;J M.Z. CJ BG;J4=µ5 L8JIJ<>Z9C54 µ?@J <45 C4H

µ2–5;@2C46QH C4µQH CJL n, 5@9:B;C2C5 58? CJ 5@ 2 56JIJLG>5 x(n) 9>@54 µ2O9@467 7

µ2 <45 n < 0. E2I5O7 J µJ@?8I9L;JH M.Z. C2H x(n) µ8J;9> @5 G9A;2G9> AH J 5µ3>-

8I9L;JH M.Z. C2H x(n)u(n). !894O7 2 56JIJLG>5 x(n)u(n) 9>@54 8B@CJC9 µ>5 O9:4?-

8I9L;2 56JIJLG>5, 3 *.+. /() X + (z) .9,%0 '?,/(/. /( .F@/."01# .,#4 1716(). *JI-

IQH 58? C4H 4O4?C2C9H CJL µJ@?8I9L;JL M.Z. 9>@54 >O49H µ9 969>@9H CJL 5µ3>8I9L;JL

M.Z., ?8AH J4 4O4?C2C9H C2H <;5µµ46?C2C5H, 6I4µB6A=2H =CJ 98>89OJ–z, 98Q6C5=2H

=CJ T;?@J, =LZL<JKH 56JIJLG>5H 654 85;5<M<4=2H. ,>5 58? C4H 4O4?C2C9H 8JL O45-

3Q;JL@, 654 85;JL=4BZ94 9@O453Q;J@ 9>@54 969>@2 C2H JI>=G2=2H, C2@ J8J>5 654 G5

9:9CB=JLµ9 =C2 =L@QT945.

$§#™£%™% ™&$ Ãƒ$¡$

!B@

C?C9 <45 m > 0 (3.43)

x n m z X z x n zZ m n

n

m

( ) ( ) ( )# ? @A + #7

899

:

;<<

+ # +

=5

1

x n X zZ( ) ( )+

? @A +

x n X z Z x nZ( ) ( ) ( )+

? @A = { }+ +

X z x n z n

n

+ #

=

+"

= 5( ) ( )0

1 1 33 . 3 $ ª $ ¡ $ ¶ § ! À ƒ $ ™ ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z


1 1 4 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

<45 m > 0 (3.44)

(8?O9:2:

# =TQ=2 (3.43) 53J;B C2 O9:4B JI>=G2=2 7 65GL=CQ;2=2 C2H 56JIJLG>5H x(n), 9@M

2 (3.44) C2@ 5;4=C9;7 JI>=G2=2 7 8;J7<2=2 5LC7H. +L<6;>@J@C5H C4H =TQ=94H 5LCQH

µ9 C2@ 5@C>=CJ4T2 =TQ=2 (3.24), <45 CJ@ 5µ3>8I9L;J M.Z., <>@9C54 35@9;? ?C4 65CB

C2 O9:4B JI>=G2=2, @Q5 O9><µ5C5 94=Q;TJ@C54 =CJ O4B=C2µ5 [0, + ") 654 B;5 8;Q894

@5 IBSJL@ 654 5LCB µQ;JH =CJLH L8JIJ<4=µJKH. )5 @Q5 5LCB O9><µ5C5 9>@54 C5 x(–1),

x(–2), …, x(–m). &5CB C2@ 5;4=C9;7 JI>=G2=2, 6B8J45 58? C5 L8B;TJ@C5 O9><µ5C5

S;>=6J@C54 96C?H CJL O45=C7µ5CJH [0, + ") 654 =L@98MH 8;Q894 @5 5354;9GJK@ 58?

CJ =L@JI46? BG;J4=µ5. *;?694C54 <45 C5 O9><µ5C5 x(0), x(1), …, x(m–1).

¶·Ú¿‰ÂÈÁÌ· 3.8

$5 L8JIJ<4=C9> J µJ@?8I9L;JH M.Z. CA@ =2µBCA@ x1(n) = anu(n) 654 x2(n) = x1(n+1).

=G/0

' L8JIJ<4=µ?H CJL X1+(z) <>@9C54 9K6JI5 58? CJ@ J;4=µ? CJL µJ@?8I9L;JL M.Z.

(3.41). YIIA=C9, CJ 58JCQI9=µ5 9>@54 >O4J µ9 969>@J CJL 5µ3>8I9L;JL M.Z. 53JK

8;?694C54 <45 54C45C7 56JIJLG>5 (SI. =TQ=2 3.8). PTJLµ9 IJ48?@:

X1+(z) = X1(z) = 1/(1–az–1) µ9 *.+. |z| > |a|.

045 CJ@ L8JIJ<4=µ? CJL X2+(z) 935;µ?ZJLµ9 CJ G9M;2µ5 C2H JI>=G2=2H (=TQ=2 3.44)

<45 m = 1 654 85>;@JLµ9

X2+(z) = z1[X1

+(z)–x1(0)z0] = z[X1+(z) – 1] = z/(1–az–1) – z

53JK x1(0) = 1.

Z x n m x n m z x l z z x l z

z x l z x l z z x n

n l m

l mn

m l

l m

m l l

ll m

m

+ # # +

=#

"

=

"# #

=#

"

# # #

=

"

=#

##

#{ } = # = =7

899

:

;<<

=

= +7

899

:

;<<

= #

55 5

55

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

0

0

1

zz X zn

n

m

+7

899

:

;<<

+

=5 ( )

1

x n m z X z x n zZ m n

n

m

( ) ( ) ( )+ ? @A #7

899

:

;<<

+ + #

=

#

50

1



+C2@ 9@?C2C5 5LC7 <@A;>=5µ9 CJ µJ@?8I9L;J M.Z., J J8J>JH T;2=4µJ8J49>C54 =C2@

5@BIL=2 O456;4CM@ =L=C2µBCA@, ?C5@ 5LCB O9@ S;>=6J@C54 =9 5;T467 2;9µ>5. '

µJ@?8I9L;JH M.Z. O9@ 89;4QT94 8I2;J3J;>5 <45 CJ =7µ5 =C4H 5;@2C46QH T;J@46QH

=C4<µQH. ' µJ@?8I9L;JH M.Z. CJL =7µ5CJH x(n) C5LC>Z9C54 µ9 CJ@ 5µ3>8I9L;J M.Z.

CJL =7µ5CJH x(n)u(n), CJ J8J>J 9>@54 54C45C?. Y;5, 2 *.+. CJL M.Z. J;>Z9C54 8B@C5

=CJ 9:AC9;46? 9@?H 6K6IJL.

3.4 O ÌÂÙ·Û¯ËÌ·ÙÈÛÌfi˜–z ÛÙËÓ ·Ó¿Ï˘ÛË Û˘ÛÙËÌ¿ÙˆÓ

' M.Z. 85>Z94 8;AC9KJ@C5 ;?IJ =C2 µ9IQC2 LTI =L=C2µBCA@ O456;4CJK T;?@JL. (8?

CJ@ J;4=µ? C2H =L@QI4:2H (=TQ=2 1.37) ?8AH 654 C2@ 4O4?C2C5 C2H =L@QI4:2H (=TQ=2

3.25), <>@9C54 8;J35@QH ?C4 Y(z) = H(z)X(z) 7:

(3.45)

?8JL X(z), Y(z), H(z) 9>@54 J4 M.Z. C2H 94=?OJL x(n), C2H 9:?OJL y(n) 654 C2H 6;JL-

=C467H 58?6;4=2H h(n) CJL =L=C7µ5CJH, 5@C>=CJ4T5. E45<;5µµ5C46B 5LC? 35>@9C54

=CJ +T7µ5 3.3. # =L@B;C2=2 H(z) J@JµBZ9C54 8),?"/383 8)8/2µ%/(4 (system

function) 7 8),?"/383 µ./%<("?4 (transfer function) CJL =L=C7µ5CJH. 045 C4µQH CJL

z 8B@A =CJ µJ@5O45>J 6K6IJ, O2I5O7 <45 z = ej@, 2 H(z) µ5H O>@94 C2@ 58?6;4=2

=LT@?C2C5H (frequency response) CJL =L=C7µ5CJH, )'# /3, '"(Y'#5.83, ICI%0%, #/0

( µ(,%*0%9(4 1716(4 %,21.0 8/3, !.&. /34 H(z). *JIIQH 58? C4H 4O4?C2C9H 9@?H =L=C7-

µ5CJH 9:B<J@C54 65C9LG9>5@ 58? C5 T5;56C2;4=C46B CA@ 8?IA@, CA@ µ2O9@46M@ 654

C2H *.+. C2H =L@B;C2=2H µ9C53J;BH 5LCJK, ?8AH G5 OJKµ9 =C2 =L@QT945.

H z

Y zX z

( )( )( )

=

1 1 53 . 4 O ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z ™ & % ¡ " ¡ " § À ™ % ™ À ™ & % ª "& ø ¡

X(z) Y(z)H(z)

™¯‹Ì· 3.3

D.,01# *0?$"%µµ%8)8/2µ%/(4 *0%-1"0/(7 E"#,()

3.4.1 "ÈÙÈ·ÙfiÙËÙ· LTI Û˘ÛÙ‹Ì·ÙÔ˜

!>O5µ9 =C2@ 9@?C2C5 1.4 ?C4 Q@5 =K=C2µ5 9>@54 54C45C? ?C5@ 2 Q:JO?H CJL 9:5;CB-

C54 µ?@J 58? C4H 85;JK=9H 654 8;J2<JKµ9@9H C4µQH C2H 94=?OJL CJL =L=C7µ5CJH 654

?T4 58? µ9IIJ@C46QH C4µQH 5LC7H. X;2=4µJ8J4M@C5H CJ@ J;4=µ? C2H =L@QI4:2H (9:>-

=A=2 1.17), µ8J;JKµ9 @5 =L@OLB=JLµ9 C2@ 4O4?C2C5 C2H 54C45C?C2C5H µ9 C2@ 6;JL-

=C467 58?6;4=2 9@?H LTI =L=C7µ5CJH. +L<696;4µQ@5, <45 @5 9>@54 54C45C? Q@5 LTI

=K=C2µ5 O456;4CJK T;?@JL, G5 8;Q894 2 y(n) @5 µ2@ 9:5;CBC54 58? C2 x(m) <45 m >


1 1 6 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

n. (8? C2 =TQ=2 (1.17), SIQ8JLµ9 ?C4 <45 @5 4=TK94 5LC?, G5 8;Q894 ?IJ4 J4 =L@C9-

I9=CQH h(n–m) J4 J8J>J4 8JII58I5=4BZJL@ C4H C4µQH CJL x(m) <45 m > n @5 9>@54 >=J4

µ9 µ2OQ@. (LC? =2µ5>@94 ?C4 2 6;JL=C467 58?6;4=2 9@?H 54C45CJK LTI =L=C7µ5CJH

O456;4CJK T;?@JL G5 8;Q894 @5 465@J8J49> C2 =L@G762

h(n) = 0 <45 n < 0 (3.46)

# =L@G762 5LC7, <45 Q@5 54C45C? LTI =K=C2µ5 O456;4CJK T;?@JL, =L@98B<9C54 ?C4

CJ BG;J4=µ5 C2H =L@QI4:2H C2H =TQ=2H (1.17) <>@9C54

(3.47)

654 2 4=JOK@5µ2 =TQ=2 (1.18) <>@9C54

(3.48)

(8? C2 =TQ=2 (3.46) 85;5C2;JKµ9 ?C4 2 h(n) 9>@54 µ45 O9:4?8I9L;2 56JIJLG>5.

(F89@GLµ>Z9C54 ?C4 J M.Z. 5LC7H 4=JKC54 µ9 C2@ H(z), C2 =L@B;C2=2 µ9C53J;BH CJL

=L=C7µ5CJH). Y;5, ?8AH 9>O5µ9 =C2@ 9@?C2C5 3.1, 2 *.+. CJL M.Z. µ45H CQCJ45H 56J-

IJLG>5H G5 9>@54 CJ 9:AC9;46? 9@?H 6K6IJL µ9 6Q@C;J C2@ 5;T7 CA@ 5:?@A@, =Lµ89-

;4I5µS5@JµQ@JL 654 CJL 589>;JL. +C2@ 89;>8CA=2 65CB C2@ J8J>5 2 H(z) 9>@54 ;2C7

=L@B;C2=2, 5LC? =2µ5>@94 ?C4 2 *.+. G5 J;>Z9C54 =CJ 9:AC9;46? CJL 6K6IJL µ9 56C>-

@5 CJ@ 8IQJ@ 58Jµ56;L=µQ@J 8?IJ 5LC7H. !894O7 µBI4=C5, 2 C4µ7 z = " 5@7694 =C2@*.+., =L@98B<9C54 ?C4 ( I%5µ#4 /() '(6)@,7µ() /() %"05µ3/2 /34 H(z) *. 5% '"C'.0

,% .9,%0 µ.$%67/."(4 /() I%5µ(7 /() '%"%,(µ%8/2, #/%, /% '(6)+,)µ% %)/? .1<"%-

8/(7, @4 8),?"/383 /() z.

3.4.2 '˘ÛÙ¿ıÂÈ· LTI Û˘ÛÙ‹Ì·ÙÔ˜

+C2@ 9@?C2C5 1.4 9>O5µ9 ?C4 Q@5 =K=C2µ5 9>@54 9L=C5GQH 9B@ 6BG9 3;5<µQ@2 9>=J-

OJH 85;B<94 µ45 3;5<µQ@2 Q:JOJ (U%U'). (H OJKµ9 CM;5 C4H =L@G769H L8? C4H J8J>-

9H C5 LTI =L=C7µ5C5 O456;4CJK T;?@JL 9>@54 9L=C5G7. (;T>ZJLµ9 G9A;M@C5H ?C4 2

9>=JOJH x(n) 9>@54 3;5<µQ@2 65CB µQC;J, O2I5O7 |x(n)| < B <45 ?I5 C5 n. !B@ 2 9>=J-

OJH 5LC7 935;µJ=C9> =’ Q@5 LTI =K=C2µ5 CJL J8J>JL 2 6;JL=C467 58?6;4=2 9>@54

h(n), C?C9 CJ µQC;J C2H 9:?OJL G5 9>@54

(3.49) | ( ) | | ( ) ( ) |y n h m x n m

m

= #=#"

"

5

y n h m x n m

m

( ) ( ) ( )= #=

"

50

y n x m h n m

m

n

( ) ( ) ( )= #=#"5


!894O7 2 58?ILC2 C4µ7 CJL 5G;J>=µ5CJH 5;4GµM@ 9>@54 8B@CJC9 µ46;?C9;2 7 >=2 CJL

5G;J>=µ5CJH CA@ 58JIKCA@ C4µM@ CA@ 5;4GµM@, 2 =TQ=2 (3.49) <>@9C54

(3.50)

654 53JK |x(n–m)| < B <45 ?I9H C4H C4µQH CA@ m 654 n, 2 =TQ=2 5LC7 65C5I7<94 =C2@

<45 ?I5 C5 n. (3.51)

Y;5, µ8J;JKµ9 @5 =Lµ89;B@JLµ9 ?C4 9B@ 2 6;JL=C467 58?6;4=2 9>@54 5G;J>=4µ2

65C’ 58?ILC2 C4µ7, O2I5O7 9B@

(3.52)

C?C9 2 y(n) 9>@54 3;5<µQ@2 65CB µQC;J 654 65CB =L@Q8945 CJ =K=C2µ5 9>@54 9L=C5-

GQH. # =TQ=2 (3.52) 58JC9I9> C2@ 465@7 =L@G762 <45 C2@ 9L=CBG945 9@?H LTI =L=C7-

µ5CJH O456;4CJK T;?@JL. (8JO946@K9C54 ?C4 5LC7 9>@54 98>=2H, 654 5@5<65>5 =L@-

G762 <45 C2@ 9L=CBG945 9@?H LTI =L=C7µ5CJH O456;4CJK T;?@JL. 0@A;>ZJLµ9 ?µAH

?C4 2 3.52 58JC9I9> 654 C2@ 465@7 =L@G762 <45 @5 L8B;T94 J µ9C5=T2µ5C4=µ?H

Fourier. Y;5, <45 9L=C5G7 =L=C7µ5C5, J µ9C5=T2µ5C4=µ?H Fourier C2H h(n) =L<6I>-

@94 654 65CB =L@Q8945, 2 *.+. C2H H(z) 8;Q894 @5 89;4I5µSB@94 CJ µJ@5O45>J 6K6IJ.

Y;5, C,% LTI 878/3µ% .9,%0 .)8/%5C4 .?, 1%0 µ#,( .?, 3 *.+. /34 8),?"/3834 µ./%-

<("?4 /() H(z) '."06%µI?,.0 /( µ(,%*0%9( 1716(, |z| = 1.

+L@OLBZJ@C5H C5 OKJ 85;58B@A =Lµ89;B=µ5C5, C5 J8J>5 8;JQ6LR5@ 58? C2@ 54C45-

C?C2C5 654 C2@ 9L=CBG945 9@?H LTI =L=C7µ5CJH O456;4CJK T;?@JL, µ8J;JKµ9 @5

65C5I7:JLµ9 =C2@ 465@7 654 C2@ 5@5<65>5 =L@G762 <45 @5 9>@54 Q@5 54C45C? =K=C2-

µ5 654 9L=C5GQH: G,% %0/0%/# LTI 878/3µ% µ. "3/2 8),?"/383 µ./%<("?4 H(z) .9,%0

.)8/%5C4 .?, 1%0 µ#,( .?, #6(0 (0 '#6(0 /34 H(z) I"981(,/%0 8/( .8@/."01# /() µ(,%-

*0%9() 1716(), *36%*2 /( µC/"( /()4 .9,%0 µ01"#/."( /34 µ(,?*(4.

3.4.3 'ÍÈÛÒÛÂÈ˜ ¢È·ÊÔÚÒÓ

,>5 µ9<BI2 65C2<J;>5 =L=C2µBCA@ 9>@54 5LC7 8JL C5 =7µ5C5 94=?OJL–9:?OJL =L@-

OQJ@C54 µ9 µ45 9:>=A=2 O453J;M@ (difference equation) C2H µJ;37H

(3.53) y n x n i b y n ii

i

M

i

i

N

( ) ( ) ( )= # # #= =5 55

0 1

| ( ) |h m

m=#"

"

5 & "

| ( ) | | ( ) |y n B h m

m

&=#"

"

5

| ( ) | | ( ) | | ( ) |y n h m x n m

m

& #=#"

"

5

1 1 73 . 4 O ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z ™ & % ¡ " ¡ " § À ™ % ™ À ™ & % ª "& ø ¡


1 1 8 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

?8JL x(n) 2 56JIJLG>5 94=?OJL, y(n) 2 56JIJLG>5 9:?OJL 654 5i, bi J4 =L@C9I9=CQH

CJL =L=C7µ5CJH. # 9:>=A=2 5LC7 µ5H IQ94 ?C4 2 C;QTJL=5 C4µ7 CJL O9><µ5CJH 9:?-

OJL 8;J6K8C94 AH <;5µµ46?H =L@OL5=µ?H CJL 85;?@CJH 654 CA@ 8;J2<JLµQ@A@ O94<-

µBCA@ C2H 94=?OJL, 65GMH 654 58? 8;J2<JKµ9@9H C4µQH C2H 9:?OJL. # =K@O9=2 µ9C5-

:K C2H 9:>=A=2H 5LC7H 654 C2H =L@B;C2=2H µ9C53J;BH CJL =L=C7µ5CJH µ8J;9> 9K6J-

I5 @5 <>@94 µ9 C2 SJ7G945 CA@ 4O4JC7CA@ C2H <;5µµ46?C2C5H (=TQ=2 3.23) 654 C2H

JI>=G2=2H =CJ T;?@J (=TQ=2 3.24) CJL M.Z. PC=4, 85>;@J@C5H CJ M.Z. 654 CA@ OKJ

µ9IM@ C2H (3.53) QTJLµ9:

(3.54)

7 (3.55)

654, 65CB =L@Q8945:

(3.56)

,9 BII5 I?<45, 2 =L@B;C2=2 µ9C53J;BH 9@?H =L=C7µ5CJH, CJL J8J>JL 2 9:>=A=2

O453J;M@ 9>@54 <;5µµ467 µ9 =C5G9;JKH =L@C9I9=CQH, 9>@54 8B@CJC9 ;2C7. !B@ J4

=L@C9I9=CQH bi CJL 85;5@Jµ5=C7 9>@54 µ2O9@46J> (bi = 0, i = 1, 2, …, N), C?C9 J4

85;58B@A 9:4=M=94H (3.53) 654 (3.56) <>@J@C54 5@C>=CJ4T5:

(3.57)

(3.58)

# Q:JOJH CJL =L=C7µ5CJH y(n) 9:5;CBC54 CM;5 µ?@J 58? C2@ C;QTJL=5 654 C4H 8;J2-

<JKµ9@9H C4µQH C2H 94=?OJL, 654 ?T4 58? C4H 8;J2<JKµ9@9H C4µQH C2H 9:?OJL. "5µ-

SB@J@C5H L8?R2 ?C4 , 65C5I5S5>@JLµ9 ?C4 5i = h(n). E2I5O7, J4

=L@C9I9=CQH 5i =Lµ8>8CJL@ µ9 C2@ 6;JL=C467 58?6;4=2 h(n) CJL =L=C7µ5CJH.)5 LTI

=L=C7µ5C5 5LC7H C2H 65C2<J;>5H J@JµBZJ@C54 =L=C7µ5C5 '.'."%8µC,34 1"()8/0-

124 %'#1"0834 (FIR, Finite Impulse Response), 9894O7 CJ µ76JH C2H 56JIJLG>5H h(n)

9>@54 8989;5=µQ@J. )5 =L=C7µ5C5 5LCB 9>@54 8B@CJC9 9L=C5G7, 53JK 2 6;JL=C467

h n H zZ( ) ( )? @A

H z

Y zX z

zii

i

M

( )( )( )

= = #

=55

0

y n x n ii

i

M

( ) ( )= #=55

0

H zY zX z

z

b z

ii

i

M

ii

i

N( )

( )( )

= =+

#

=

#

=

5

5

%0

1

1

Y z b z X z zii

i

N

ii

i

M

( ) ( )! +*

+,

-

./ = !#

=

#

=5 51

1 0

%

Y z z X z b z Y zi

i

i

M

ii

i

N

( ) ( ) ( )= ##

=

#

=5 5%

0 1


CJLH 58?6;4=2 h(n) 9>@54 8989;5=µQ@JL µ76JLH 654 B;5 5G;J>=4µ2 65C’ 58?ILC2

C4µ7 (SI. =TQ=2 3.52).

)5 =L=C7µ5C5 µ9 =L@B;C2=2 µ9C53J;BH C2H µJ;37H (3.56) =C2@ J8J>5 CJLIBT4=CJ@

Q@5H 58? CJLH =L@C9I9=CQH CJL 85;5@Jµ5=C7 9>@54 O4B3J;JH CJL µ2O9@?H, J@JµB-

ZJ@C54 =L=C7µ5C5 ?'.0"34 1"()8/0124 %'#1"0834 (IIR, Infinite Impulse Response),

9894O7 2 5@C>=CJ4T2 6;JL=C467 58?6;4=2 h(n) 58JC9I9>C54 58? B894;JLH ?;JLH.

(DLµ2G9>C9 C2@ 9@?C2C5 3.2 =C2@ J8J>5 J L8JIJ<4=µ?H CJL 5@C>=C;J3JL M.Z. µ45H

;2C7H =L@B;C2=2H µ9 C2 µQGJOJ C2H 5@B8CL:2H =9 OL@5µJ=94;B 7 =9 µ9;46B 6IB-

=µ5C5, 65CQI2<9 =9 56JIJLG>5 589>;A@ ?;A@). 045 54C45CB =L=C7µ5C5, J4 8?IJ4 C2H

H(z) G5 8;Q894 @5 S;>=6J@C54 9@C?H CJL µJ@5O45>JL 6K6IJL <45 @5 9>@54 9L=C5G7.

,45 O453J;9C467 C5:4@?µ2=2 CA@ =L=C2µBCA@ 9>@54 =9 .'%,%63'/01? (recursive) 7

µ3 .'%,%63'/01? (non recursive) =L=C7µ5C5[1]. !85@5I28C46B 9>@54 C5 =L=C7µ5C5

=C5 J8J>5 2 9:>=A=2 O453J;M@ 963;BZ9C54 µ9 C2 µJ;37 (3.53) µ9 bi 3 0. +’ 5LCB C5

=L=C7µ5C5 2 Q:JOJH y(n) =9 6BG9 T;J@467 =C4<µ7 9:5;CBC54 654 58? C4µQH C2H 9:?-

OJL =9 8;J2<JKµ9@9H T;J@46QH =C4<µQH y(n–1), y(n–2),… PTJLµ9 O2I5O7, 5@5C;J-

3JO?C2=2 (feedback) C4µM@ C2H 9:?OJL =CJ =K=C2µ5. +C5 µ2 985@5I28C46B =L=C7-

µ5C5, 2 9:>=A=2 O453J;M@ 963;BZ9C54 µ9 C2 µJ;37 (3.57), <9<J@?H 8JL =2µ5>@94

8AH 2 Q:JOJH y(n) =9 6BG9 T;J@467 =C4<µ7 9:5;CBC54 µ?@J 58? C2@ C;QTJL=5 654 C4H

8;J2<JKµ9@9H C4µQH C2H 94=?OJL.

3.4.4 ¢ÔÌ¤˜ Û˘ÛÙËÌ¿ÙˆÓ ‰È·ÎÚÈÙÔ‡ ¯ÚfiÓÔ˘

)5 =L=C7µ5C5 O456;4CJK T;?@JL 85;4=CB@J@C54 =LT@B µ9 C2 µJ;37 O45<;5µµBCA@

;J7H. *;?694C54 <45 Q@5 58I? 654 65C5@J2C? C;?8J 5@585;B=C5=2H CA@ 9:4=M=9A@

O453J;M@ 654 CA@ =L@5;C7=9A@ µ9C53J;BH. (H µ9I9C7=JLµ9 5LC?@ CJ@ C;?8J 5@5-

85;B=C5=2H µ9 C2 SJ7G945 C2H 9:>=A=2H O453J;M@ 9@?H 54C45CJK R23456JK 3>IC;JL

O9KC9;2H CB:2H. )J ?C4 CJ =L<696;4µQ@J 3>IC;J 9>@54 O9KC9;2H CB:2H, =2µ5>@94 ?C4

=C2@ 9:>=A=2 (3.53) QTJLµ9 R = 2. )J ?C4 5LC? 9>@54 654 54C45C?, =2µ5>@94 9848IQJ@

?C4 M ] R. Y;5 2 <9@467 µJ;37 C2H 9:>=A=2H O453J;M@ 9@?H CQCJ4JL 3>IC;JL G5 9>@54

y(n) = %0x(n) + 51x(n – 1) + 52x(n – 2) – b1y(n – 1) – b2y(n – 2) (3.59)

654 2 5@C>=CJ4T2 =L@B;C2=2 µ9C53J;BH

(3.60) H z

z z

b z b z( ) = + +

+ +

# #

# #% % %0 1

12

2

11

221

1 1 93 . 4 O ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z ™ & % ¡ " ¡ " § À ™ % ™ À ™ & % ª "& ø ¡

[1] +C2@ 9II2@467 S4SI4J<;53>5 C5 =L=C7µ5C5 5LCB G5 C5 OJKµ9 654 AH %,%*"(µ01? 654 AHµ3 %,%*"(µ01?.


1 2 0 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

045 CJ@ L8JIJ<4=µ? CA@ 9:4=M=9A@ O453J;M@ C2H µJ;37H (3.59), 7 <9@46?C9;5 C2H

µJ;37H (3.53), T;945Z?µ5=C9 8JII58I5=45=CQH, 5G;J4=CQH 654 =CJ4T9>5 65GL=CQ;2-

=2H. '4 5@C>=CJ4T9H =T2µ5C46QH CJLH 5@585;5=CB=94H O>@J@C54 58? CJ +T7µ5 3.4.

)J O4B<;5µµ5 ;J7H 8JL 5@C4=CJ4T9> =C2@ 9:>=A=2 (3.59) O>@9C54 58? CJ +T7µ5 3.55.

)J =KµSJIJ z–1 5@C48;J=A89K94 65GL=CQ;2=2 1 µJ@BO5H T;?@JL (µJ@5O45>5 65GL-

=CQ;2=2, unit delay), <9<J@?H 8JL µ5H 9>@54 7O2 <@A=C? 58? C2@ 4O4?C2C5 C2H JI>-

=G2=2H CJL M.Z. (=TQ=2 3.24).

x(n) x(n) x(n)k

kx(n) x(n-1)x(n) + w(n)

w(n)

z-1

*JII58I5=45=C7H AG;J4=C7H +CJ4T9>J K5GL=CQ;2=2H

x(n)x(n) w(n)y(n)

y(n)%0 %0

%1-b1

%2-b2

%1 -b1

%2 -b2

z-1 z-1

z-1z-1

(5) (S)

z-1

z-1

™¯‹Ì· 3.4

;% I%801? *(µ01?8/(0E.9% /@,

H3<0%1+, 8)8/3-µ?/@,.

™¯‹Ì· 3.5

!"%$µ%/('(9383*(µ24 *.7/."34/?F34: (%) ?µ.83µ("<2 /7'()–1,

(I) 1%,(,012 2?µ.83 µ("<2

/7'()–2

# 8;5<µ5CJ8J>2=2[2] CJL 3>IC;JL O9KC9;2H CB:2H µ9 CJ O4B<;5µµ5 CJL +T7µ5CJH

3.55, 8;J6K8C94 Bµ9=5 58? C2 =TQ=2 (3.59) 654 J@JµBZ9C54 ?µ.83 µ("<2 /7'()–1

(direct form type – 1). *5;5C2;JKµ9 ?C4 µ45 CQCJ45 OJµ7 8;5<µ5CJ8J>2=2H 5854C9>

CQ==9;5 =CJ4T9>5 65GL=CQ;2=2H. P@5H O453J;9C46?H C;?8JH 8;5<µ5CJ8J>2=2H CJL

>O4JL 3>IC;JL 9>@54 J 56?IJLGJH:

[2] ,9 CJ@ ?;J «'"%$µ%/('(9383» (realization) 9@@JJKµ9 CJ@ C;?8J µ9 CJ@ J8J>J 5@585;4-=CJKµ9 654 C9I46B L8JIJ<>ZJLµ9 µ>5 9:>=A=2 O453J;M@ 7 C2@ 5@C>=CJ4T2 =L@B;C2=2µ9C53J;BH. ,9 CJ@ ?;J « )6('(9383» (implementation) 9@@JJKµ9 CJ@ C9I46? L8JIJ<4-=µ? 5LCM@ µ9 C2 SJ7G945 94O46M@ R23456M@ 6L6IAµBCA@ 7 L8JIJ<4=CM@.


(;T>ZJ@C5H 58? C2 =L@B;C2=2 µ9C53J;BH (3.60), QTJLµ9 ?C4:

7 (3.61)

?8JL (3.62)

' 5@C>=C;J3JH M.Z. CA@ C9I9LC5>A@ OKJ =TQ=9A@ (3.61) 654 (3.62) µ5H O>@94 5@C>=CJ4T5:

y(n) = 50w(n) + 51w(n – 1) + 52w(n – 2) (3.63)

w(n) = x(n) – b1w(n – 1) – b2w(n – 2) (3.64)

# OJµ7 8;5<µ5CJ8J>2=2H CA@ C9I9LC5>A@ =TQ=9A@ O>@9C54 58? CJ +T7µ5 3.5S 654

9>@54 <@A=C7 AH ?µ.83 µ("<2 /7'()–27 1%,(,012 µ("<2 8;5<µ5CJ8J>2=2H CJL >O4JL

3>IC;JL. +C2@ 89;>8CA=2 5LC7 5854CJK@C54 µ?@J OKJ =CJ4T9>5 65GL=CQ;2=2H.

+L=C7µ5C5 O456;4CJK T;?@JL CB:2H LR2I?C9;2H C2H O9KC9;2H, 8JL µ?I4H 9:9CB=5µ9,

µ8J;JK@ @5 8;5<µ5CJ8J42GJK@ µ9 C;?8J 5@C>=CJ4TJ 969>@JL CJL +T7µ5CJH 3.5. +C2@

8;B:2 ?µAH, J4 LR2I7H CB:2H =L@5;C7=94H (R > 2) 8;5<µ5CJ8J4JK@C54 AH =L@OL5=µJ>

OJµM@ 8;MC2H 654 / 7 O9KC9;2H CB:2H =9 8.0"? (cascade) 7 '%"?6636% (parallel), ?8AH

35>@9C54 =CJ +T7µ5 3.6. (LC? <>@9C54 <45 @5 µ94AGJK@ C5 =3BIµ5C5 CA@ L8JIJ<4=µM@

65CB C2@ LIJ8J>2=7 CJLH µ9 R23456B =L=C7µ5C5 C5 J8J>5 T;2=4µJ8J4JK@ 8989;5=µQ@2

56;>S945 <45 C2@ 5@585;B=C5=2 CA@ =L@C9I9=CM@ 654 CJ@ L8JIJ<4=µ? CA@ 8;B:9A@.

W z

X z

b z b z( )

( )=+ +# #1 1

12

2

Y z a a z a z W z( ) ( ) ( )= + + !# # 0 11

22

H zY zX z

z zb z b z

Y z z zX z

b z b z

( )( )( )

( )

( ) ( )( )

= = + + !+ +

J

J = + + !+ +

# ## #

# ## #

% % %

% % %

0 11

22

11

22

0 11

22

11

22

1

1

1

1 2 13 . 4 O ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z ™ & % ¡ " ¡ " § À ™ % ™ À ™ & % ª "& ø ¡

x(n)

x(n)

y(n)

y(n)H1(z)

H1(z)

H2(z)

HL(z)

H2(z) HL(z)

(5)

(S)

B0

…

…

™¯‹Ì· 3.6

T(µC4 )H3624/?F34 @4 8),*)%-8µ#4 *(µ+, '"+-/34 2 *.7/."34/?F34 (%) 8. 8.0"?1%0 (I) '%"?6636%


1 2 2 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

045 C2@ '"%$µ%/('(9383 8. 8.0"?(cascade realisation), 2 =L@B;C2=2 µ9C53J;BH H(z)

5@5IK9C54 =9 <4@?µ9@J 85;5<?@CA@ AH:

(3.65)

?8JL L CJ 56Q;54J µQ;JH CJL (R + 1)/2 654 Hi(z) OJµ7 8;MC2H 7 O9KC9;2H CB:2H:

8;MC2H CB:2H (3.66)

O9KC9;2H CB:2H (3.67)

);>5 9>@54 C5 8;JSI7µ5C5 8JL 8;Q894 @5 5@C4µ9CA84=CJK@ 65CB C2@ 8;5<µ5CJ8J>2-

=2 =9 =94;B:

1) µ9 8J4J C;?8J G5 8;Q894 @5 =L@OL5=CJK@ J4 5;4Gµ2CQH µ9 CJLH 85;5@Jµ5=CQH,

µ9CB C2@ 85;5<J@CJ8J>2=2 C2H H(z)

2) µ9 8J45 =94;B G5 8;Q894 J4 984µQ;JLH OJµQH @5 =L@O9GJK@, 654

3) 2 5@B<62 6I4µB6A=2H, O2I5O7 µ9>A=2H CJL 8IBCJLH CJL =7µ5CJH =9 9@O4Bµ9=5

=2µ9>5 C2H OJµ7H, M=C9 5LC? @5 µ2@ 9>@54 JKC9 8JIK µ9<BIJ, JKC9 8JIK µ46;?.

045 C2@ '%"?66363 '"%$µ%/('(9383 (parallel realisation), 2 =L@B;C2=2 µ9C53J;BH

H(z) 5@5IK9C54 =9 BG;J4=µ5 85;5<?@CA@ AH:

(3.68)

?8JL L CJ 56Q;54J µQ;JH CJL (R + 1)/2, U0 = 5,/bN 654

8;MC2H CB:2H (3.69)

O9KC9;2H CB:2H (3.70)

+C2@ 85;BII2I2 8;5<µ5CJ8J>2=2 O9@ 85>Z94 ;?IJ 2 =94;B O45=K@O9=2H CA@ 984µQ-

;JLH OJµM@. !848IQJ@, 2 6I4µB6A=2 9>@54 84J 9K6JI2 L8?G9=2, 53JK µ8J;9> @5 <>@94

<45 6BG9 µ>5 OJµ7 5@9:B;C2C5. )5 µ2O9@46B ?µAH CA@ 984µQ;JLH OJµM@ 9>@54 89;4=-

=?C9;J 9L5>=G2C5 =C5 =3BIµ5C5 6SB@C4=2H CA@ =L@C9I9=CM@.

H z

z

b z b zi

i i

i i

( ) = ++ +

#

# #% %0 1

1

11

221

H z

b zi

i

i

( ) =+ #%0

111

H z B H zi

i

L

( ) ( )= +=

50

1

H z

z z

b z b zi

i i i

i i

( ) = + ++ +

# #

# #% % %0 1

12

2

11

221

H z

z

b zi

i i

i

( ) = ++

#

#% %0 1

1

111

H z H z H z H z H zL i

i

L

( ) ( ) ( )... ( ) ( )= ==

K1 2

1


3.4.5 "!fiÎÚÈÛË Û˘¯ÓfiÙËÙ·˜ Û˘ÛÙËÌ¿ÙˆÓ ‰È·ÎÚÈÙÔ‡ ¯ÚfiÓÔ˘

# =TQ=2 94=?OJL–9:?OJL 9@?H LTI =L=C7µ5CJH O456;4CJK T;?@JL µ9 6;JL=C467 58?-

6;4=2 h(n) O>@9C54 58? C2@ 9:>=A=2 C2H =L@QI4:2H (1.18)

(3.71)

?8JL x(n), y(n) J4 56JIJLG>9H 94=?OJL 654 9:?OJL 5@C>=CJ4T5. (@ 2 56JIJLG>5 94=?-

OJL x(n) 9>@54 96G9C467H µJ;37H

(3.72)

C?C9 2 Q:JOJH y(n) G5 4=JKC54 µ9

(3.73)

2 J8J>5 µ8J;9> @5 <;539> AH

(3.74)

?8JL

(3.75)

# 8J=?C2C5 H(ej@) J@JµBZ9C54 %'#1"083 8)E,#/3/%4 (frequency response) CJL LTI

=L=C7µ5CJH 654 µ5H 85;QT94 µ45 89;4<;537 5LCJK =CJ 89O>J C2H =LT@?C2C5H. (8?

C2@ 9:>=A=2 (3.75) 85;5C2;JKµ9 ?C4 3 O(ej@) %'(/.6.9 ()80%8/01? /( µ./%8E3µ%/08µ#

Fourier *0%1"0/(7 E"#,() (DTFT) /34 1"()8/0124 %'#1"0834 h(n) /() 8)8/2µ%/(4. #

9:>=A=2 (3.74) µ5H IQ94 ?C4 <45 µ45 µ4<5O467 56JIJLG>5 94=?OJL x(n) <A@4567H =LT@?-

C2C5H @, 2 Q:JOJH 9>@54 98>=2H µ45 µ4<5O467 56JIJLG>5 C2H >O45H <A@4567H =LT@?C2-

C5H, 8JII58I5=45=µQ@2 ?µAH µ9 CJ µ4<5O46? 85;B<J@C5 H(ej@). ' 85;B<J@C5H 5LC?H

9:5;CBC54 58? C2 =LT@?C2C5 94=?OJL @ 654 C2@ 6;JL=C467 56JIJLG>5 h(n).

H H(ej@), ?8AH 9>O5µ9 BIIA=C9 654 =C2@ 9@?C2C5 2.1 <45 CJ@ DTFT, 9>@54 <9@46B µ45

µ4<5O467 =L@B;C2=2 CJL @ µ9 89;>JOJ 2' 654 µ8J;9> @5 963;5=C9> AH

(3.76)

?8JL Hr(ej@) 654 Hi(e

j@) 9>@54 8;5<µ5C46QH =L@5;C7=94H AH 8;JH @, 654 58JC9IJK@

CJ 8;5<µ5C46? 654 35@C5=C46? µQ;JH C2H H(ej@) 5@C>=CJ4T5, 654

H e H e jH e H e ej

rj

ij j j( ) ( ) ( ) ( ) ( )6 6 6 6 6= + = L

H e h m ej j m

m

( ) ( )6 6= #

=#"

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5

y n H e ej j n( ) = ( )6 6

y n h m e h m e ej n m

m

j m

m

j n( ) ( ) ( )( )= =*

+,

-

./#

=#"

"#

=#"

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5 56 6 6

x n e nj n( ) = # " < < "6

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m

( ) ( ) ( )= #=#"

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5

1 2 33 . 4 O ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z ™ & % ¡ " ¡ " § À ™ % ™ À ™ & % ª "& ø ¡


1 2 4 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

(3.77)

# 8J=?C2C5 J@JµBZ9C54 %'#1"083 µC/"() (magnitude response) 654 2

8J=?C2C5 -(@) %'#1"083 <?834 (phase response) CJL LTI =L=C7µ5CJH O456;4CJK

T;?@JL. &54 J4 OKJ 5LCQH =L@5;C7=94H 9>@54 8;5<µ5C46QH =L@5;C7=94H CJL @.

# 58?6;4=2 =LT@?C2C5H 9@?H =L=C7µ5CJH O456;4CJK T;?@JL µ8J;9> 9K6JI5 @5 L8J-

IJ<4=C9> 58? C2 =L@B;C2=2 µ9C53J;BH H(z) CJL =L=C7µ5CJH. .L=46B, ?8AH QTJLµ9

7O2 =TJI4B=94, 2 58?6;4=2 =LT@?C2C5H QT94 @?2µ5 µ?@J <45 9L=C5G7 =L=C7µ5C5.

!>O5µ9 ?C4 2 58?6;4=2 =LT@?C2C5H CJL =L=C7µ5CJH O9@ 9>@54 BII2 58? CJ µ9C5-

=T2µ5C4=µ? Fourier C2H 6;JL=C467H CJL 58?6;4=2H. 045 @5 S;JKµ9 CJ µ9C5=T2µ5-

C4=µ? Fourier CJL =L=C7µ5CJH 8;Q894 @5 L8JIJ<>=JLµ9 CJ M.Z. 8B@A =CJ µJ@5O4-

5>J 6K6IJ, O2I5O7 <45 z = ej@. PTJLµ9, 98JµQ@AH, 58? CJ@ J;4=µ? CJL M.Z.:

(3.78)

%O45>C9;J 9@O453Q;J@ 85;JL=4BZ94 J $.@µ./"01#4 2 $"%<01#4 '"(8*0("08µ#4C2H 58?-

6;4=2H =LT@?C2C5H 8JL G5 9:9CB=JLµ9 5µQ=AH CM;5. !>@54 Q@5H 9K6JIJH 654 T;7-

=4µJH C;?8JH <45 @5 L8JIJ<>ZJLµ9 8J4JC46B 6L;>AH, C2@ 58?6;4=2 =LT@?C2C5H =T9-

C46B 58IM@ =L=C2µBCA@ 654 S5=>Z9C54 =CJ O4B<;5µµ5 8?IA@–µ2O9@46M@ C2H =L@B;-

C2=2H µ9C53J;BH H(z). )5LC?T;J@5 5@5O946@K94 654 µ45 3L=467 =2µ5=>5 CA@ 8?IA@

654 µ2O9@46M@ C2H =L@B;C2=2H µ9C53J;BH. +L<696;4µQ@5, 2 ;2C7 =L@B;C2=2 H(z)

C2H =TQ=2H (3.56) <;B39C54 AH:

(3.79)

?8JL z1, z2, …, zM C5 M µ2O9@46B 654 p1, p2, …, pN J4 R 8?IJ4 5LC7H.

045 z = ej@ 2 =TQ=2 5LC7 <>@9C54:

(3.80)

654 963;BZJ@C5H 6BG9 µ4<5O46? 5;4Gµ? CJL 5;4Gµ2C7 654 85;5@Jµ5=C7 =9 8JI46QH

=L@C9C5<µQ@9H QTJLµ9:

H e b e

e z e z e z

e p e p e pj j N M

jM

j j jN

( )( )( )...( )

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01 2

1 2

66

H z

b z z z z z z zz p z p z p

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1 2

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H ej j i

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6 6 1


(3.81)

"5µSB@J@C5H L8?R2 ?C4 6BG9 µ4<5O46?H zi µ8J;9> @5 5@585;5=C5G9> 654 58? Q@5

O4B@L=µ5 =CJ µ4<5O46? 98>89OJ 654 ?C4 J ej@ 9>@54 CJ O4B@L=µ5 µQC;JL >=JL µ9 C2

µJ@BO5 µ9 6Q@C;J C2@ 5;T7 CA@ 5:?@A@ 654 <A@>5H @ µ9 CJ@ B:J@5 CA@ G9C46M@

8;5<µ5C46M@, C?C9 CJ ej@–zi = Ciej%i, i = 1,2,…,M, 5@C48;J=A89K94 CJ O4B@L=µ5 58?

CJ µ2O9@46? zi µQT;4 969>@J CJ =2µ9>J CJL µJ@5O45>JL 6K6IJL, ?8JL 65C5I7<94 CJ ej@

(UI. +T7µ5 3.75). (@C>=CJ4T5 J;>ZJ@C54 654 C5 O45@K=µ5C5 ej@–pl = DlejIl, l =

1,2,…,N, <45 CJLH 8?IJLH C2H =TQ=2H (3.80). *5;5C2;JKµ9 IJ48?@ ?C4 CJ Ci 9>@54 2

58?=C5=2 CJL µ2O9@46JK zi 58? CJ =2µ9>J CJL µJ@5O45>JL 6K6IJL ?8JL 65C5I7<94

CJ O4B@L=µ5 ej@654 Dl 9>@54 2 58?=C5=2 CJL 8?IJL pl 58? CJ >O4J =2µ9>J. '4 3B=94H

%i 654 Il 9>@54 J4 <A@>9H CA@ 5@C4=CJ>TA@ O45@L=µBCA@ µ9 CJ@ B:J@5 CA@ G9C46M@

8;5<µ5C46M@. (LCQH J4 <9Aµ9C;46QH 5@585;5=CB=94H 35>@J@C54 =CJ +T7µ5 3.7S.

H e b e

C e C e C e

D e D e D eVej j N M

j j j

j jN

jj

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( )( )...( )

( )( )...( )( )6 6

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01 2

1 21 2

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I I I

1 2O L

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1 2 53 . 4 O ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z ™ & % ¡ " ¡ " § À ™ % ™ À ™ & % ª "& ø ¡

ej@–pl=DlejIl

ej@–zi=Ciej%i

Im Im

pl

pl

Dl

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ej@

Re Re

MJ@5O45>JH 6K6IJH

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zi

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(5) (S)

™¯‹Ì· 3.7

D"%<01#4 '"(8-*0("08µ#4 /34%'#1"0834 8)E,#-/3/%4.

(8? C2 =TQ=2 (3.81) S;>=6JLµ9 8IQJ@ ?C4 CJ µQC;J C2H 58?6;4=2H =LT@?C2C5H 4=JK-

C54 µ9

(3.82)

654 2 3B=2 C2H µ9

- = >b0 + (N – M)@ + (%1 + %2 + … + %M) – (I1 + I2 + …IR) (3.83)

V b

C C CD D D

M

N

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1 2

1 2


1 2 6 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

# 3B=2 CJL =L@C9I9=C7 b0 9>@54 0 7 ', 5@BIJ<5 µ9 CJ 8;?=2µJ 5LCJK (G9C46? 7

5;@2C46? 5@C>=CJ4T5).

# <9Aµ9C;467 5@585;B=C5=2 9>@54 8JIK T;7=4µ2 =C2@ 65C5@?2=2 CJL C;?8JL µ9

CJ@ J8J>J J4 GQ=94H CA@ 8?IA@ 654 CA@ µ2O9@46M@ 982;9BZJL@ C2@ 58?6;4=2 µQC;JL

654 3B=2H 9@?H =L=C7µ5CJH. 045 85;BO94<µ5, 9B@ Q@5 µ2O9@46? zi S;>=69C54 8B@A

=CJ µJ@5O45>J 6K6IJ, C?C9 CJ µQC;J CJL 5@C>=CJ4TJL O45@K=µ5CJH Ci G5 4=JKC54 µ9

µ2OQ@ <45 @ = >zi, µ9 58JCQI9=µ5 654 CJ µQC;J |H(ej@)| @5 <>@9C54 0 <45 C2 =LT@?-

C2C5 5LC7. \µJ45, ?C5@ Q@5H 8?IJH pl S;>=69C54 8B@A =CJ µJ@5O45>J 6K6IJ, C?C9 CJ

µQC;J CJL 5@C>=CJ4TJL O45@K=µ5CJH Dl G5 4=JKC54 µ9 0 <45 @ = >pl, µ9 58JCQI9=µ5

CJ µQC;J |H(ej@)| @5 <>@9C54 B894;J <45 C2 =LT@?C2C5 5LC7. 0>@9C54 IJ48?@ 35@9;?,

?C4 2 85;JL=>5 9@?H µ2O9@46JK 6J@CB =CJ µJ@5O45>J 6K6IJ 5@5<6BZ94 CJ µQC;J C2H

58?6;4=2H =LT@?C2C5H @5 85>;@94 8JIK µ46;QH C4µQH, <45 C4µQH C2H =LT@?C2C5H 6J@CB

=CJ =L<696;4µQ@J =2µ9>J CJL µJ@5O45>JL 6K6IJL. &5C’ 5@5IJ<>5, 2 85;JL=>5 9@?H

8?IJL 6J@CB =CJ µJ@5O45>J 6K6IJ, 5@5<6BZ94 CJ µQC;J C2H 58?6;4=2H =LT@?C2C5H

@5 85>;@94 8JIK µ9<BI9H C4µQH <45 C4µQH C2H =LT@?C2C5H 6J@CB =CJ =L<696;4µQ@J

=2µ9>J CJL µJ@5O45>JL 6K6IJL. !8JµQ@AH, J4 8?IJ4 9843Q;JL@ CJ 5@C>G9CJ 58JCQ-

I9=µ5 58? C5 µ2O9@46B. DQCJ@C5H Q@5 µ2O9@46? 6J@CB =’ Q@5 8?IJ, 5LC? 9:JLO9C9-

;M@94 C2@ 98>O;5=2 CJL 8?IJL. )J >O4J =LµS5>@94 654 5@ GQ=JLµ9 Q@5 8?IJ 6J@CB

=’ Q@5 µ2O9@46?. (LC? <>@9C54 65C5@J2C? 654 58? C2@ 9:>=A=2 (3.80), ?8JL 35>@9-

C54 ?C4 <45 zi = pl J4 ?;J4 ej@–zi 654 e

j@–pl 58IJ8J4JK@C54.

&5C5I5S5>@JLµ9 IJ48?@ ?C4 2 <9Aµ9C;467 5@585;B=C5=2 CA@ 8?IA@ 654 CA@ µ2O9-

@46M@ C2H =L@B;C2=2H µ9C53J;BH 9@?H =L=C7µ5CJH 9>@54 JL=4MOJLH =2µ5=>5H <45

CJ@ 8;J=O4J;4=µ? C2H 58?6;4=2H =LT@?C2C5H 5LCJK. &54 5@C>G9C5, 9B@ µ5H O>@9C54

2 984GLµ2C7 58?6;4=2 =LT@?C2C5H, µ8J;JKµ9 9K6JI5 @5 8;J=O4J;>=JLµ9 C2 =L@B;-

C2=2 µ9C53J;BH CJL =L=C7µ5CJH 969>@JL 8JL QT94 C2@ 58?6;4=2 5LC7. ,9 BII5

I?<45, µ8J;JKµ9 @5 =T9O4B=JLµ9 CJ 65CBII2IJ 3>IC;J CJ J8J>J 85;JL=4BZ94 C2@

58?6;4=2 =LT@?C2C5H 8JL µ5H 9@O453Q;94. *;4@ ?µAH 8;JTA;7=JLµ9 =9 6B8J45

85;5O9><µ5C5 =T9C46B µ9 CJ@ <9Aµ9C;46? 8;J=O4J;4=µ? C2H 58?6;4=2H =LT@?C2C5H

9@?H =L=C7µ5CJH, G5 7C5@ 65I? @5 6B@JLµ9 µ45 =K@CJµ2 5@53J;B =C5 3>IC;5.

' ?;JH <96/"( (filter) 5@53Q;9C54 =9 Q@5 LTI =K=C2µ5 CJ J8J>J T;2=4µJ8J49>C54 <45

C2@ 984IJ<7 O453?;A@ =LT@JC7CA@ µ9C5:K 5LCM@ 8JL 935;µ?ZJ@C54 =C2@ 9>=JO?

CJL. '4 =LT@?C2C9H 8JL G5 984I9<JK@ 65GJ;>ZJ@C54 58? C2@ 58?6;4=2 =LT@?C2C5H

H(ej@) CJL =L=C7µ5CJH, 2 J8J>5 µ9 C2 =94;B C2H 9:5;CBC54 58? C4H 85;5µQC;JLH CJL

=L=C7µ5CJH, I.T. CJLH =L@C9I9=CQH 5i, bi C2H 9:>=A=2H O453J;M@ (3.53). !8JµQ@AH,

984IQ<J@C5H 65CBII2I5 CJLH =L@C9I9=CQH, µ8J;JKµ9 @5 =T9O4B=JLµ9 3>IC;5 C5

J8J>5 984C;Q8JL@ C2 O4QI9L=2 =2µBCA@ µ9 =L@4=CM=9H =LT@?C2C5H =9 J;4=µQ@9H


89;4JTQH, 9@M 9:5=G9@>ZJL@ C4H =L@4=CM=9H =LT@?C2C5H =9 BII9H 89;4JTQH. 09@46B,

Q@5 LTI =K=C2µ5 C;J8J8J49> CJ 3B=µ5 CJL =7µ5CJH X(ej@) =Kµ3A@5 µ9 C2@ 58?-

6;4=2 H(ej@) 5LCJK, O>@J@C5H =C2@ Q:JOJ Q@5 =7µ5 µ9 3B=µ5 Y(ej@) = H(ej@) X(ej@).

,9 BII5 I?<45, 2 H(ej@) O;5 AH 8),?"/383 I?"()4 (weighting function) <45 C4H O45-

3J;9C46QH =L@4=CM=9H C2H =LT@?C2C5H CJL =7µ5CJH 94=?OJL. (8? 5LC7 C2@ B8JR2,

µ8J;JKµ9 @5 8JKµ9 ?C4 6BG9 LTI =K=C2µ5 µ8J;9> @5 T5;56C2;4=C9> AH Q@5 3>IC;J,

56?µ2 654 5@ 5LC? O9@ 9µ8JO>Z94 C2 O4QI9L=2 =9 6B8J49H 7 =9 ?I9H C4H =LT@?C2C9H.

(LC?H 9>@54 J I?<JH <45 CJ@ J8J>J J4 ?;J4 «LTI =K=C2µ5» 654 «3>IC;J» T;2=4µJ-

8J4JK@C54 J Q@5H 5@C> CJL BIIJL.

)5 3>IC;5 C5:4@JµJK@C54 =9 I%5)'."%/? (lowpass), )H3'."%/? (highpass), L@,('.-

"%/? (bandpass) 654 %'#""0H34 L+,34 (bandstop), 5@BIJ<5 µ9 C5 T5;56C2;4=C46B

CJLH =CJ 89O>J C2H =LT@?C2C5H. # 4O5@467 58?6;4=2 µQC;JL CA@ 3>IC;A@ 5LCM@ 35>-

@9C54 =CJ +T7µ5 3.8. '4 =LT@?C2C9H @c, @1, @2 J@JµBZJ@C54 8)E,#/3/.4 %'(1('24

(cutoff frequencies) 654 J;>ZJL@ C5 ?;45 µ9C5:K CA@ =LT@JC7CA@ 8JL 984C;Q89C54 @5

O4QIGJL@ 58? CJ =K=C2µ5 654 5LCM@ 8JL 58J;;>8CJ@C54. # 89;4JT7 =LT@JC7CA@ =C2@

J8J>5 984C;Q89C54 2 O4QI9L=2 58JC9I9> C2 L+,3 *0C6.)834 (passband), 9@M 2 89;4J-

T7 =LT@JC7CA@ =C2@ J8J>5 O9@ 984C;Q89C54 2 O4QI9L=2 J@JµBZ9C54 L+,3 %'#""0H34

2 L+,3 %'(1('24 (stopband). \8AH 35>@9C54 =CJ +T7µ5 3.8, C5 4O5@46B 3>IC;5 QTJL@

ZM@2 O4QI9L=2H =C5G9;JK 6Q;OJLH, O2I5O7 =C5G9;7H C4µ7H (=L@7GAH >=2H µ9 1) 654

ZM@2 58J6J87H µ2O9@46JK 6Q;OJLH.

+C2@ 8;5<µ5C46?C2C5 ?µAH, 2 µJ;37 9@?H 3>IC;JL 58J6I>@94 58? C4H 4O5@46QH 5LCQH

µJ;3QH, ?8AH <45 85;BO94<µ5 5LC7 CJL +T7µ5CJH 3.9 <45 Q@5 S5GL89;5C? 3>IC;J.

# µ9CBS5=2 58? C2 ZM@2 O4QI9L=2H =C2 ZM@2 58J6J87H O9@ <>@9C54 565;45>5, 654

65GJ;>Z94 QC=4 µ45 @Q5 89;4JT7 =LT@JC7CA@, C2 I9<?µ9@2 L+,3 µ./?I%834 (transition

band). # =LT@?C2C5 @p 5@53Q;9C54 654 AH #"0( L+,34 *0C6.)834 (passband edge),

654 2 =LT@?C2C5 @s AH #"0( L+,34 %'(1('24 (stopband edge). # ZM@2 µ9CBS5=2H

65GJ;>Z9C54 µ9C5:K CA@ =LT@JC7CA@ @p 654 @s. !8>=2H, CJ 6Q;OJH =C4H ZM@9H O4Q-

I9L=2H 654 58J6J87H O9@ 9>@54 =C5G9;?, 5IIB µ9C5SBII9C54. 045 85;BO94<µ5, µ8J-

;9> @5 85;JL=4BZ9C54 6B8J45 6LµBCA=2 (58?6I4=2) <K;A 58? C4H C4µQH 1 654 0 5@C>-

=CJ4T5, 2 J8J>5 =LµSJI>Z9C54 µ9 Op 654 Os <45 C2@ 6BG9 89;4JT7 5@C>=CJ4T5.

1 2 73 . 4 O ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z ™ & % ¡ " ¡ " § À ™ % ™ À ™ & % ª "& ø ¡


1 2 8 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

B

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¶·Ú¿‰ÂÈÁÌ· 3.9

$5 8;J=O4J;4=C9> 2 58?6;4=2 =LT@?C2C5H CJL =L=C7µ5CJH O456;4CJK T;?@JL CJ

J8J>J QT94 =L@B;C2=2 µ9C53J;BH H(z) = (z + 0,5)/(z – 0,5) µ9 |z| > 0,5.

=G/0

045 @5 S;JKµ9 C2@ 58?6;4=2 =LT@?C2C5H CJL =L=C7µ5CJH, 8;Q894 @5 L8JIJ<>=JLµ9

C2 =L@B;C2=2 µ9C53J;BH <45 ?I9H C4H C4µQH |z| = 1 7 z = ej@. PC=4, µ9 SB=2 C2@ 5@B-

IL=2 8JL 8;J2<7G269 QTJLµ9:

?8JL V = C/D 654 - = 5 – S. 0;5346B, J4 L8JIJ<4=µJ> 5LCJ> 5;T>ZJL@ µ9 C2@ CJ8J-

GQC2=2 =CJ µ4<5O46? 98>89OJ–z CJL 8?IJL =CJ =2µ9>J z = 0,5 (=KµSJIJ HH) 654 CJL

µ2O9@46JK =CJ =2µ9>J z = – 0,5 (=KµSJIJ ••), ?8AH 35>@9C54 =CJ +T7µ5 3.105. +C2

=L@QT945, <45 6BG9 <A@>5 @ L8JIJ<>ZJLµ9 CJ µQC;J V 654 C2@ 3B=2 -. )J µQC;J 654

2 3B=2 C2H 58?6;4=2H =LT@?C2C5H CJL =L<696;4µQ@JL =L=C7µ5CJH O9>T@J@C54 =CJ

+T7µ5 3.109 654 3.10=C 5@C>=CJ4T5.

H ee

e

Ce

DeVej

j

j

j

jj( )

,

,@

@

@

a

I= +

# += =0 5

0 5L

1 2 93 . 4 O ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z ™ & % ¡ " ¡ " § À ™ % ™ À ™ & % ª "& ø ¡

0 @p @s '

ZM@2[O4QI9L=2H

ZM@2[µ9CBS5=2H

ZM@2[58J6J87H

|H(ej@)|

1+*p

1-*p

*s

1

@

™¯‹Ì· 3.9

M3 0*%,01# I%5)-'."%/# <96/"(


1 3 0 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

'4 L8JIJ<4=µJ> 5LCJ> 984S9S54M@J@C54 9K6JI5 µ9 C2 SJ7G945 CA@ =T2µBCA@

3.10S–3.10O. PC=4, <45 <A@>5 >=2 µ9 0 (+T7µ5 3.10S) C = 1,5, D = 0,5, 5 = S = 0 654,

65CB =L@Q8945, V = 3 654 - = 0. 045 <A@>5 >=2 µ9 8/2 (+T7µ5 3.10<) C = D, 5 < S

654 B;5 V = 1 654 - < 0. 045 <A@>5 >=2 µ9 ' (+T7µ5 3.10O) C = 0,5, D = 1,5, 5 = S

= ' 654 B;5 V = 1/3 654 - = 0. *5;5C2;JKµ9 ?C4 8;?694C54 <45 Q@5 S5GL89;5C? 3>I-

C;J. !>@54 =2µ5@C46? @5 CJ@>=JLµ9 ?C4 CJ µQC;J C2H 58?6;4=2H =LT@?C2C5H |H(ej@)|

9>@54 =Lµµ9C;467 =L@B;C2=2 <K;A 58? CJ =2µ9>J @ = ', 9@M 2 3B=2 9>@54 5@C4–=Lµ-

µ9C;467 <K;A 58? CJ >O4J =2µ9>J. !848IQJ@, 984S9S54M@9C54 ?C4 2 58?6;4=2 =LT@?-

C2C5H 9>@54 89;4JO467.

3,0

2,5

2,0

1,5

1,0

0,5

0,00,33

0 '/4

'/4 '/2 3'/40

'

'

'/2 3'/4

-5-10-15-20-25-30-35-40-45-50-55

% ej@W@ I

–0,5 –0,5 0,5 –0,5 0,5 –0,5 0,50,5

@=0 @='/2 @='

% I

(5) (S) (<) (O)

(9) (=C)

MQC;J

.B=2

(µJ>;9H)

@

@

™¯‹Ì· 3.10

D.@µ./"01#4 '"(8-*0("08µ#4 /34 %'#-1"0834 8)E,#/3/%4/() !%"%*.9$µ%/(4

3.9: (%)–(*) *0?-$"%µµ%

'#6@,–µ3*.,01+,$0% *0%<("./01C4

8)E,#/3/.4, (.)–(8/)µC/"( 1%0 <?83 /34

%'#1"0834

# 6;JL=C467 58?6;4=2 9@?H 9L=C5GJKH LTI =L=C7µ5CJH O456;4CJK T;?@JL O>@9C54

58? C2 =TQ=2 h(n) = 5nu(n) µ9 5 = 0,5. $5 =T9O45=C9> 2 58?6;4=2 =LT@?C2C5H CJL

=L=C7µ5CJH.



¶·Ú¿‰ÂÈÁÌ· 3.10

$5 =T9O45=C9> R23456? 3>IC;J CJ J8J>J @5 58J6?8C94 C2 =LT@?C2C5 @0.

=G/0

)5 3>IC;5 5LC7H C2H 65C2<J;>5H J@JµBZJ@C54 <96/"% %'(1('24 2 %'#""0H34 8)E,#-

/3/%4 (notch filters). )QCJ45 3>IC;5 9>@54 =L@7GAH 8JIK T;7=4µ5 =9 ?;<5@5 7 =L=C7-

µ5C5 9<<;537H =C5 J8J>5 GQIJLµ9 @5 58J6?RJLµ9 C2 =LT@?C2C5 CA@ 50 Hz C2H

<;5µµ7H C;J3JOJ=>5H 8JL 85;94=OK94 =’ 5LCB.

045 @5 O2µ4JL;<7=JLµ9 Q@5 µ2O9@46? =C2@ 58?6;4=2 =LT@?C2C5H <45 µ45 =L<696;4-

µQ@2 =LT@?C2C5 @0, 58IMH 94=B<JLµ9 Q@5 Z9K<JH =LZL<M@ µ4<5O46M@ µ2O9@46M@ 8B@A

=CJ µJ@5O45>J 6K6IJ 654 =9 µ>5 <A@>5 @0, O2I5O7 z1,2 = 1e ± j@0 (UI. +T7µ5 3.115).

&5CB =L@Q8945, 2 =L@B;C2=2 µ9C53J;BH CJL FIR 3>IC;JL 8JL 8;J6K8C94 9>@54:

+T9O4BZJ@C5H C2@ 58?6;4=2 =LT@?C2C5H CJL =L=C7µ5CJH 5LCJK 85;5C2;JKµ9 ?C4

5LC? µ2O9@>Z94 C2 =LT@?C2C5 @0, C5LC?T;J@5 ?µAH, 9:5=G9@>Z94 =2µ5@C46B 654 C4H

=LT@?C2C9H <K;A 58? C2@ @0. (+CJ +T7µ5 3.115 35>@9C54 2 58?6;4=2 =LT@?C2C5H

<45 @0 = 8/4). 045 @5 µ94M=JLµ9 CJ 9K;JH CA@ =LT@JC7CA@ 8JL 982;9BZJ@C54, G5

8;Q894 @5 65C53K<JLµ9 =9 C9T@46QH =T9O45=µJK FIR 3>IC;A@ µ9<5IKC9;2H CB:2H,

?8AH 5LCQH 85;JL=4BZJ@C54 =CJ 98?µ9@J 693BI54J. ,45 O453J;9C467 8;J=Q<<4=2

9>@54 5LC7 C2H 94=5<A<7H 8?IA@ =C2 =L@B;C2=2 CJL =L=C7µ5CJH. P=CA, IJ48?@, ?C4

CJ8JG9CJKµ9 Q@5 Z9K<JH =LZL<M@ µ4<5O46M@ 8?IA@ =C5 =2µ9>5 p1,2 = re ± j@0,

0 < r < 1, O2I5O7 =C2@ >O45 <A@>5 (=LT@?C2C5) 8JL S;>=6J@C54 654 C5 µ2O9@46B (UI.

+T7µ5 3.11S). )J 58JCQI9=µ5 CA@ 8?IA@ 9>@54 2 94=5<A<7 =L@CJ@4=µJK =C2@ 89;4J-

T7 <K;A 58? C5 µ2O9@46B 654 =L@98MH 2 9IBCCA=2 CJL 9K;JLH CA@ =LT@JC7CA@ 8JL

982;9BZJ@C54. # =L@B;C2=2 µ9C53J;BH CJL =L=C7µ5CJH <>@9C54 CM;5:

# 58?6;4=2 =LT@?C2C5H CJL =L=C7µ5CJH 35>@9C54 =CJ +T7µ5 3.11S <45 @0 = '/4 654

r = 0,85 7 r = 0,95 654 b0 = 1. +L<6;>@J@C5H 5LC7 µ9 C2@ 58?6;4=2 CJL FIR 3>IC;JL

CJL +T7µ5CJH 3.115, 85;5C2;JKµ9 ?C4 CJ 58JCQI9=µ5 C2H 94=5<A<7H CA@ 8?IA@

7C5@ 2 =2µ5@C467 µ9>A=2 CJL 9K;JLH CA@ =LT@JC7CA@ <K;A 58? C2 =LT@?C2C5 @0.

)QIJH, 2 =K<6;4=2 C2H C9I9LC5>5H =L@B;C2=2H µ9C53J;BH µ9 969>@2 C2H =TQ=2H

(3.60) µ5H JO2<9> =C2@ 9K;9=2 CA@ C4µM@ CA@ =L@C9I9=CM@ CJL 3>IC;JL, O2I5O7 a0

H e be z e z

re z re zb

z z

r z r zj

j j

j j( )

( )( )

( )( )

cos

cos@

@ @

@ @= # #

# #= # +

# +

# # #

# # #

# #

# #0

1 1

1 1 00

1 2

01 2 2

1 1

1 1

1 2

1 2

0 0

0 0

66

H e b e z e z b z zj j j( ) ( )( ) ( cos )@ @ @= # # = # +# # # # #0

1 10 0

1 21 1 1 20 0 6

1 3 13 . 4 O ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z ™ & % ¡ " ¡ " § À ™ % ™ À ™ & % ª "& ø ¡


1 3 2 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

= 1, a1 = –2cos@0, a2 = 1, b1 = –2rcos@0, b2 = r2. 045 85;BO94<µ5, =C2@ 89;>8CA=2

8JL @0 = '/4 654 r = 0,85 J4 =L@C9I9=CQH 5LCJ> 9>@54 5@C>=CJ4T5: a0 = 1, a1 = –1,41,

a2 = 1, b1 = –1,2, b2 = 0,72. +L@98MH, 2 8;5<µ5CJ8J>2=2 CJL O9KC9;2H CB:2H 3>I-

C;JL CJ J8J>J 9:9CBZJLµ9, µ8J;9> @5 <>@94 µ9 µ>5 58? C4H Bµ9=9H µJ;3QH CK8JL–1 7

CK8JL–2 CJL +T7µ5CJH 3.5, µ9 C4µQH =L@C9I9=CM@ 5LCQH 8JL µ?I4H L8JIJ<>=5µ9.

E8>89OJ –z E8>89OJ –z

'2 Im '

2 Im

@0 @00 08 Re Re

1,2

1,0

0,8

0,6

0,4

0,2

0

00

0

0 ' '

'

'

-'

'2

'2 '

2

'2

'2

'2

|H(e

j@)|

1,2

1,0

0,8

0,6

0,4

0,2

0

|H(e

j@)|

-(@

)

-

0

'

-'

'2

'2

-(@

)

-

(5) (S)

r=0,95

r=0,95

r=0,85

r=0,85™¯‹Ì· 3.11

T0?$"%µµ% '#6@,– µ3*.,01+, 1%0%'#1"083 8)E,#-/3/%4 (%) /() FIR1%0 (I) /() IIR<96/"() /()

!%"%*.9$µ%/(43.10.



+C2@ 5LC7 9@?C2C5 5=TJI2G765µ9 µ9 C2@ 5@BIL=2 =L=C2µBCA@ S5=4Z?µ9@J4 =CJ@

M.Z. !>O5µ9 ?C4 58? C2 =L@B;C2=2 µ9C53J;BH 9@?H =L=C7µ5CJH µ8J;JKµ9 Bµ9=5

@5 9:B<JLµ9 =Lµ89;B=µ5C5 <45 C2@ 54C45C?C2C5 654 C2@ 9L=CBG945 5LCJK. !>O5µ9

98>=2H ?C4 58? C2 =L@B;C2=2 µ9C53J;BH 9@?H =L=C7µ5CJH µ8J;JKµ9 9K6JI5 @5

8;J=O4J;>=JLµ9 C2@ 5@C>=CJ4T2 9:>=A=2 O453J;M@ 5LCJK, ?8AH 654 CJ 5@C>=C;J-

3J. 0@A;>=5µ9 C4H OJµQH 8;5<µ5CJ8J>2=2H CA@ =L=C2µBCA@ O456;4CJK T;?@JL 654

5@539;G765µ9 =C2@ 5@5<654?C2C5 8;5<µ5CJ8J>2=2H OJµM@ LR2I7H CB:2H µ9 OJµ7

8;MC2H 7 / 654 O9KC9;2H CB:2H =9 =94;B 7 85;BII2I5. )QIJH, µ9I9C7=5µ9 C2@ 58?-

6;4=2 =LT@?C2C5H CA@ =L=C2µBCA@ O456;4CJK T;?@JL 654 5=TJI2G765µ9 µ9 CJ <9A-

µ9C;46? (<;5346?) 8;J=O4J;4=µ? 5LC7H. !>O5µ9 ?C4 2 58?6;4=2 =LT@?C2C5H 8;J-

6K8C94 58? C2 =L@B;C2=2 µ9C53J;BH CJL =L=C7µ5CJH, L8JIJ<>ZJ@C5H 5LC7 8B@A

=CJ µJ@5O45>J 6K6IJ.


+CJ &93BI54J 5LC? 5=TJI2G765µ9 µ9 CJ M.Z. =2µBCA@ O456;4CJK T;?@JL, 654 O45-

84=CM=5µ9 ?C4 8;?694C54 <45 Q@5 4=TL;? µ5G2µ5C46? 9;<5I9>J =C2 µ9IQC2 O456;4-

CM@ =2µBCA@ 654 =L=C2µBCA@, 5@C>=CJ4TJ CJL µ9C5=T2µ5C4=µJK Laplace <45 =7µ5-

C5 654 =L=C7µ5C5 =L@9TJKH–T;?@JL. !>O5µ9 ?C4:

# =L@QI4:2 OKJ 56JIJLG4M@ 4=JOL@5µ9> µ9 CJ@ 8JII58I5=45=µ? CA@ 5@C>=CJ4TA@

M.Z. CJLH.

1 3 3™ À ¡ $æ% '!º"§"#$À

*J45 2 Q:JOJH CJL =L=C7µ5CJH , ?C5@ 2 9>=J-

OJH 9>@54 x(n) = *(n)–(1/3)*(n–1);

y n y n y n x n( ) ( ) ( ) ( )= # # # +5

11

26 6


# =L@B;C2=2 µ9C53J;BH 9@?H 54C45CJK LTI =L=C7µ5CJH O456;4CJK T;?@JL 4=JKC54

µ9 H(z) = 1/(1 + 0,5z–1). !:9CB=C9 5@ CJ =K=C2µ5 9>@54 9L=C5GQH 654 =T9O4B=C9 C2

OJµ7 8;5<µ5CJ8J>2=2H CJL =L=C7µ5CJH 65GMH 654 C2@ 58?6;4=2 =LT@?C2C5H 5LCJK.



1 3 4 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

' 65GJ;4=µ?H C2H *.+. CJL M.Z. 9>@54 8B@CJC9 585;5>C2CJH.

# *.+. 54C45C7H 56JIJLG>5H 96C9>@9C54 =CJ 9:AC9;46? 9@?H 6K6IJL 56C>@5H >=2H µ9

CJ µQC;J CJL 8IQJ@ 58Jµ56;L=µQ@JL 8?IJL CJL M.Z. 58? C2@ 5;T7 CA@ 5:?@A@

CJL µ4<5O46JK 9848QOJL–z.

)5 LTI =L=C7µ5C5, C5 J8J>5 89;4<;B3J@C54 58? 9:4=M=94H O453J;M@ µ9 =C5G9;JKH

=L@C9I9=CQH, QTJL@ ;2CQH =L@5;C7=94H µ9C53J;BH.

' µJ@?8I9L;JH M.Z. µ5H IK@94 CJ 8;?SI2µ5 CJL L8JIJ<4=µJK C2H 58?6;4=2H O45-

6;4CM@ =L=C2µBCA@ µ9 µ2 µ2O9@46QH 5;T46QH =L@G769H.

(8? C2 =L@B;C2=2 µ9C53J;BH H(z) 9@?H =L=C7µ5CJH O456;4CJK T;?@JL, µ8J;JK-

µ9 Bµ9=5 @5 9:B<JLµ9 =Lµ89;B=µ5C5 <45 C2@ 54C45C?C2C5 654 9L=CBG945 CJL

=L=C7µ5CJH. !8>=2H, 9K6JI5 8;J=O4J;>ZJLµ9 C2@ 9:>=A=2 O453J;M@ CJL =L=C7-

µ5CJH 654 C2 OJµ7 8;5<µ5CJ8J>2=2H 5LCJK.

# 58?6;4=2 =LT@?C2C5H 9@?H 9L=C5GJKH =L=C7µ5CJH 8;J=O4J;>Z9C54 58? C2 =L@B;-

C2=2 µ9C53J;BH 5LCJK <45 z = ej@, O2I5O7 <45 C4H C4µQH CJL z 8JL S;>=6J@C54 8B@A

=CJ µJ@5O45>J 6K6IJ. # T;7=2 CJL O45<;Bµµ5CJH 8?IA@–µ2O9@46M@ JO2<9> =9

9K6JIJ <9Aµ9C;46? (<;5346?) 8;J=O4J;4=µ? C2H 58?6;4=2H =LT@?C2C5H 9@?H

=L=C7µ5CJH.



!§§%¡$°§ø™™%

+.D9JOA;>O2H, &.,89;µ89;>O2H, :08%$@$2 8/3 -.@"9% &3µ?/@, & &)8/3µ?/@,,

)L8AG7CA 0. E5;O5@?H,(G7@5, 1998.

N.&5IJL8C=>O2H, &2µ%/%, &)8/2µ%/% 1%0 >6$#"05µ(0, !6O?=94H E>5LIJH, (G7@5, 1993.

•!¡$°§ø™™%

E.C.Ifeachor, B.W. Jervis, Digital Signal Processing: A Practical Approach, Second

Edition, Pearson Education Limited, 2002.

S.K.Mitra, Digital Signal Processing: A computer–Based Approach, Second

Edition,McGraw Hill, 2001.


A.V.Oppenheim, A.S.Willsky, Signals & Systems, Second Edition, Prentice– Hall,

1997.

J.P.Proakis, D.G.Manolakis, Digital Signal Processing: Principles, Algorithms and

Applications, Third Edition, Prentice Hall, 1996.


Processing, Addison–Wesley Publishing Company, 1988 .

$¢%°$™ ¶!ƒ"#&!ƒø ª!§!&%™

+.D9JOA;>O2H, &.,89;µ89;>O2H, :08%$@$2 8/3 -.@"9% &3µ?/@, & &)8/3µ?/@,,

)L8AG7CA 0. E5;O5@?H, 1998.

)J &93BI54J 5 5@53Q;9C54 =9 =7µ5C5 654 =L=C7µ5C5 O456;4CJK T;?@JL. 0>@9C54

5@5ILC467 5@53J;B =C4H 4O4?C2C9H CJL µ9C5=T2µ5C4=µJK–z 654 O>@J@C54 85;5-

O9><µ5C5 654 5=67=94H.

S.K.Mitra, Digital Signal Processing: A computer–Based Approach, Second

Edition,McGraw Hill, 2001.

+CJ &93BI54J 3 <>@9C54 µ45 =T9C46B =K@CJµ2 5@53J;B =CJ µ9C5=T2µ5C4=µ?–z, =C4H

4O4?C2C9H 654 =C4H 935;µJ<QH CJL. *9;4QTJ@C54 8JIIB 85;5O9><µ5C5 C5 J8J>5 LIJ-

8J4JK@C54 =9 Matlab.

A.V.Oppenheim, A.S.Willsky, Signals & Systems, Second Edition, Prentice– Hall,

1997.

1 3 5B # µ § # $ ° ƒ" º # " ' ! º " § " # $ À


1 3 6 K E º A § A I O 3 : ª ! &" ™ Ã % ª "& # ™ ª $ ™ – z

!6C9@7H 5@53J;B =CJ µ9C5=T2µ5C4=µ?–z, µ9 8JIIB 85;5O9><µ5C5 654 µ9 5L=C2;7

µ5G2µ5C467 G9µ9I>A=2 <>@9C54 =CJ &93BI54J 10 CJL S4SI>JL. F8B;TJL@ 98>=2H

8JIIQH 5=67=94H µ9 C4H 585@C7=94H CJLH.



)J &93BI54J 3 CJL S4SI>JL 5@53Q;9C54 =CJ M.Z. !:54;9C467 85;JL=>5=2 µ9 8JIIB

85;5O9><µ5C5 654 5=67=94H.



absolutely summable 5G;J>=4µJH 65C’ 58?ILC2 C4µ7

bandpass filter ZA@JO45S5C? 7 ZA@J89;5C? 3>IC;J

bandstop filter 3>IC;J 58?;;4R2H ZM@2H

bilateral transform 5µ3>8I9L;JH µ9C5=T2µ5C4=µ?H

cascade realisation 8;5<µ5CJ8J>2=2 =9 =94;B

cutoff frequency =LT@?C2C5 58J6J87H

difference equation 9:>=A=2 O453J;M@

feedback 5@BO;5=2, 5@5C;J3JO?C2=2

frequency response 58?6;4=2 =LT@?C2C5H

highpass filter LR289;5C? 3>IC;J

left–sided 5;4=C9;?8I9L;JH

long division =L@9T7H O45>;9=2

lowpass filter S5GL89;5C? 3>IC;J

magnitude response 58?6;4=2 µQC;JL

notch filter 3>IC;o 58J6J87H (58?;;4R2H) =LT@?C2C5H

passband ZM@2 O4QI9L=2H

phase response 58?6;4=2 3B=2H

recursive 985@5I28C46?H, 5@5O;Jµ46?H

right–sided O9:4?8I9L;JH

ring O56CKI4JH

ROC *.+., 89;4JT7 =K<6I4=2H

stopband ZM@2 58?;;4R2H 7 58J6J87H

system function =L@B;C2=2 =L=C7µ5CJH

transfer function =L@B;C2=2 µ9C53J;BH

transition band ZM@2 µ9CBS5=2H

two–sided 5µ3>8I9L;JH

unilateral transform µJ@?8I9L;JH µ9C5=T2µ5C4=µ?H

unit circle µJ@5O45>JH 6K6IJH

unit delay µJ@5O45>5 65GL=CQ;2=2

z–plane 98>89OJ–z

1 3 7" ¶$¢$™% "°°§#'ø¡ $ƒø¡ ™&%¡ !§§%¡#'%


æËÊÈ·Î¿ Ê›ÏÙÚ· !Â!ÂÚ·ÛÌ¤ÓË˜ ÎÚÔ˘ÛÙÈÎ‹˜ ·!fiÎÚÈÛË˜ (FIR)

™ÎÔ!fi˜

&/( 1.<?6%0( %)/# 5% µ.6./28()µ. /% E%"%1/3"08/01? 1%0 /04 0*0#/3/.4 /@, H3<0%-

1+, <96/"@, '.'."%8µC,34 1"()8/0124 %'#1"0834 (Finite Impulse Response – FIR).

:'9834, 5% $,@"98()µ. /04 *7( '0( 83µ%,/01C4 µ.5#*()4 8E.*9%824 /()4 1%0 5% %,%-

678()µ. .1/.,+4 /3 µC5(*( /@, '%"%57"@,. ;% FIR <96/"% E"380µ('(0(7,/%0 .)"7-

/%/% 8/3, H3<0%12 .'.F."$%89% 82µ%/(4 6#$@ *7( '(67 83µ%,/01+, 0*0(/2/@, '()

'%"()80?L(),, 1%0 /04 ('(9.4 .'9834 5% .F./?8()µ. .*+: /3 $"%µµ012 %'#1"083

<?834 1%0 /3, .)8/?5.0%.


J/%, (6(163"+8./. /3 µ.6C/3 /() 1.<%6%9() %)/(7 5% .98/. 8. 5C83 ,%:

• &E.*0?L./. /3 $.,012 *(µ2 .,#4 FIR H3<0%1(7 <96/"()

• D,@"9L./. /% '6.(,.1/2µ%/? /()4, $0% /% ('(9% ?66@8/. 1%0 /% E"380µ('(0(7µ.

• =%5("9L./. /04 '"(*0%$"%<C4 .,#4 FIR <96/"()

• B'(6($9L./. /( µC/"( /34 %'#1"0834 8)E,#/3/%4 .,#4 FIR <96/"()

• =%/%,(.9/. /( "#6( /34 $"%µµ0124 %'#1"0834 <?834 8/3 *0%/2"383 /34 µ("<24

/() 82µ%/(4

• B'(6($9L./. /3, %'#1"083 <?834 8/% <96/"% %)/?

• &E.*0?L./. C,% FIR <96/"( µ. /3 µC5(*( /@, '%"%57"@,

• :F3$.9/. /0 .9,%0 IC6/08/( <96/"( 1%0 8. /0 '6.(,.1/.9 3 µC5(*(4 8E.*9%834 IC6/0-

8/@, FIR <96/"@, 8. 8EC83 µ. .1.9,3 /@, '%"%57"@,

• :'06C$./. /3, %'%0/(7µ.,3 %1"9I.0% /@, 8),/.6.8/+, /() FIR <96/"() $0% ,% .'0-

/.)E5(7, /% .'05)µ3/? E%"%1/3"08/01?


• !.'."%8µC,3 1"()8/012 %'#1"083

• &)µµ./"012 1"()8/012 %'#1"083

4! " º#§#$%


1 4 0 K E º A § A I O 4 : æ % º # " ' " º # § & ƒ" ¶ ! ¶ ! ƒ" ™ ª ! ¡ % ™ ' ƒ $ À ™ & # ' % ™ " ¶ $' ƒ # ™ % ™ ( F I R )

• D"%µµ012 %'#1"083 <?834

• :)8/?5.0% /@, FIR <96/"@,

• MC5(*(4 8E.*9%834 µ. '%"?5)"%

• &),?"/383 '%"%57"()

• U?8µ% '%"%57"()

• U%0,#µ.,( Gibbs

• SC6/08/( <96/"(

• B6('(9383 $"%µµ0124 %'#1"0834 <?834

• =I?,/083 /@, 8),/.6.8/+,


;% $"%µµ01? H3<0%1? <96/"% '() 5% .F./?8()µ. 8/( 1.<?6%0( %)/# '%"()80?L(),

'.'."%8µC,34 E"(,0124 *0?"1.0%4 %'#1"083 8. .98(*( µ(,%*0%9%4 1"(7834. &/3 *0.-

5,2 I0I60($"%<9% /% 8),%,/(7µ. @4 Finite Impulse Response 2 FIR <96/"%. >'(/.-

6(7, ()80%8/01? µ."012 '."9'/@83 /@, <96/"@, ?'.0"34 1"()8/0124 %'#1"0834 (IIR).

>'# /% $"%µµ01? H3<0%1? <96/"%, '"(/0µ(7,/%0 8),25@4 /% FIR C,%,/0 /@, IIR,

6#$@ /34 .)8/?5.0%4 8/3 6.0/()"$9% /()4 1%0 /34 $"%µµ0124 %'#1"0834 '() '%"()-

80?L(), 8/3 <?83. &/( 1.<?6%0( %)/# 5% .F./?8()µ. /% .9*3 1%0 /% E%"%1/3"08/01?

/@, FIR <96/"@, 1%0 5% µ.6./28()µ. /3 µC5(*( 8E.*9%834 /()4 µ. /% '%"?5)"%.

;C6(4, 5% '."0$"?H()µ. *(µC4 )6('(93834 /@, FIR <96/"@,, 1%5+4 1%0 /% '"(-

I62µ%/% '() '"(17'/(), 1%/? /3, )6('(9383 %)/2. >'%"%9/3/( $,@8/01# )'#I%-

5"( %'(/.6.9 3 763 /() '"(3$(7µ.,() 1.<%6%9().

;( 1.<?6%0( %)/# '."06%µI?,.0 /04 %1#6()5.4 .,#/3/.4:

4.1 X%"%1/3"08/01? /@, FIR <96/"@,

4.2 ;7'(0 1%0 '"(*0%$"%<C4 /@, FIR <96/"@,

4.3 MC5(*(0 8E.*9%834 /@, FIR <96/"@,

4.4 -Cµ%/% )6('(93834 FIR <96/"@,


4.1 Ã·Ú·ÎÙËÚÈÛÙÈÎ¿ ÙˆÓ FIR Ê›ÏÙÚˆÓ

# <9@467 OJµ7 CA@ FIR 3>IC;A@ QT94 ?8AH 35>@9C54 =CJ +T7µ5 4.1. (8JC9I9>C54 58?

µ>5 <;5µµ7 65GL=CQ;2=2H (delay line), ?8JL JI4=G5>@JL@ C5 O9><µ5C5 CJL =7µ5CJH

94=?OJL x(n), 654 58? CJLH 8JII58I5=45=CQH h(m). )5 58JC9IQ=µ5C5 CA@ 8JII58I5-

=45=µM@ 8;J=C>G9@C54 <45 @5 OM=JL@ C2@ C9I467 Q:JOJ CJL 3>IC;JL y(n). !>@54 8;J-

35@QH ?C4 2 Q:JOJH y(n) 9>@54 J <;5µµ46?H =L@OL5=µ?H CA@ O94<µBCA@ 94=?OJL x(n):

(4.1)

(@ CJ =7µ5 94=?OJL =CJ FIR 3>IC;J 9>@54 2 µJ@5O45>5 6;JK=2 *(n), C?C9 2 Q:JO?H

CJL (6;JL=C467 58?6;4=2 CJL 3>IC;JL 7 Impulse Response) G5 9>@54 O45OJT46B >=2

µ9 6BG9 Q@5@ 58? CJLH =L@C9I9=CQH h(m). PC=4, QTJLµ9 8989;5=µQ@2 =9 T;J@467

O4B;6945 6;JL=C467 58?6;4=2 h(m) <45 C5 FIR 3>IC;5 (Finite Impulse Response). #

9:>=A=2 O453J;M@ (4.1) 58JC9I9> C2 =TQ=2 94=?OJL–9:?OJL CJL FIR 3>IC;JL. !8>-

=2H :Q;JLµ9 ?C4 <45 C2 =L@B;C2=2 µ9C53J;BH 4=TK94,

(4.2)

' S5Gµ?H N CJL 3>IC;JL T5;56C2;>Z9C54 58? CJ 8I7GJH CA@ ?;A@ C2H 6;JL=C467H

58?6;4=2H (8I7GJH =L@C9I9=CM@ h(m)).

H z h n H z h m z m

m

N

( ) ( ) ( ) ( )P Q = #

=

#

5 0

1

y n h m x n m

m

N

( ) ( ) ( )= #=

#

50

1

1 4 14 . 1 Ã " ƒ" ' & % ƒ # ™ & # ' " & ø ¡ F I R º # § & ƒ ø ¡

*(n)

n

x(n)

h(n)

y(n)

x(n-1) x(n-2) x(n-N+2) x(n-N+1)z-1 z-1 z-1

h(0) h(1) h(2) h(N-2) h(N-1)

+ n

0;5µµ7 65GL=CQ;2=2H

™¯‹Ì· 4.1

T(µ2 FIR H3<0%-1(7 <96/"().

¶·Ú¿‰ÂÈÁÌ· 4.1

045 CJ FIR 3>IC;J CJL +T7µ5CJH 4.2 @5 <;539> 2 9:>=A=2 94=?OJL–9:?OJL 654 2

=L@B;C2=2 µ9C53J;BH. !8>=2H, @5 =T9O45=C9> 2 6;JL=C467 58?6;4=2 CJL 3>IC;JL.



=G/0

)J 3>IC;J CJL +T7µ5CJH 4.2 9>@54 FIR, O4?C4 2 6;JL=C467 CJL 58?6;4=2 9>@54 8989-

;5=µQ@2H T;J@467H O4B;6945H (C;94H ?;J4). &BG9 ?;JH C2H 6;JL=C467H 58?6;4=2H

9>@54 >=JH µ9 1/3, ?=J O2I5O7 J 5@C>=CJ4TJH =L@C9I9=C7H CJL 3>IC;JL. # 9:>=A=2

94=?OJL–9:?OJL (9:>=A=2 O453J;M@) 9>@54 2 56?IJLG2:

# =TQ=2 5LC7 O9>T@94 ?C4 CJ 3>IC;J CJL +T7µ5CJH 4.2 9>@54 Q@5 R23456? =K=C2µ5

µQ=2H C4µ7H (averager). ,9 C2 SJ7G945 C2H =TQ=2H (4.2) 2 =L@B;C2=2 µ9C53J;BH

QT94 AH 9:7H:

H z z z( ) ( ) = + +# #1

31 1 2

y n x n x n x n( ) [ ( ) ( ) ( )]= + # + #1

31 2

*(n)

h(n)

n

n

x(n)

y(n)

x(n-1) x(n-2)

+

z-1 z-1

1/3 1/3 1/3

1/3

™¯‹Ì· 4.2

N3<0%1# <96/"(µC834 /0µ24 µ./".04 #"()4.

$5 =T9O45=C9> Q@5H 58I?H R23456?H O453J;4=C7H (digital differentiator). $5 <;5-

39> 2 =TQ=2 94=?OJL–9:?OJL, 2 =L@B;C2=2 µ9C53J;BH 654 @5 =T9O45=C9> 2 OJµ7

CJL 3>IC;JL. !8>=2H @5 =T9O45=C9> 2 6;JL=C467 CJL 58?6;4=2.


' 58IJK=C9;JH R23456?H O453J;4=C7H G5 QT94 Q:JOJ y(n) C2 O453J;B OKJ O45OJT4-

6M@ O94<µBCA@ C2H 94=?OJL x(n). PC=4 2 =TQ=2 94=?OJL–9:?OJL CJL 3>IC;JL G5 9>@54

# =L@B;C2=2 µ9C53J;BH µ8J;9> @5 L8JIJ<4=C9> 5@ IBSJLµ9 CJ µ9C5=T2µ5C4=µ?–z

654 CA@ OKJ µ9;M@ C2H 8;J2<JKµ9@2H =TQ=2H:

y n x n x n( ) ( ) ( )= # #1


# OJµ7 CJL 3>IC;JL 9>@54 8JIK 58I7 654 QT94 AH 9:7H:

Y z X z z X z

H zY zX z

z

( ) ( ) ( )

( )( )( )

= # J

= = #

#

#

1

11

1 4 34 . 1 Ã " ƒ" ' & % ƒ # ™ & # ' " & ø ¡ F I R º # § & ƒ ø ¡

\C5@ 2 9>=JOJH 9>@54 2 µJ@5O45>5 6;JK=2, 2 Q:JOJH CJL 3>IC;JL (6;JL=C467 58?-

6;4=2) G5 4=JKC54 µ9 CJLH =L@C9I9=CQH C2H 5;T467H =TQ=2H 94=?OJL–9:?OJL, O2I5-

O7 µ9 1 654 –1. # µJ;37 C2H 6;JL=C467H 58?6;4=2H QT94 AH 9:7H:

x(n) x(n-1)z-1

1 –1

y(n)+

1

-1

TS n

4.1.1 °Ú·ÌÌÈÎ‹ ·!fiÎÚÈÛË Ê¿ÛË˜

P@5 3>IC;J QT94 <;5µµ467 58?6;4=2 3B=2H (linear phase response) ?C5@ 2 O453J;B

3B=2H G(@) µ9C5:K 94=?OJL 654 9:?OJL <45 =7µ5 <A@4567H =LT@?C2C5H @, O>@9C54 AH:

7 (4.3)

\8JL 5 654 b =C5G9;QH 8JL 9:5;CM@C54 58? C5 T5;56C2;4=C46B CJL 3>IC;JL. \C5@ J4

5;µJ@46QH =L@4=CM=9H 9@?H =7µ5CJH, O4QIGJL@ 58? =K=C2µ5 8JL QT94 <;5µµ467 58?-

6;4=2 3B=2H =Kµ3A@5 µ9 C2@ (4.3), L8?694@C54 ?I9H =C2@ >O45 T;J@467 65GL=CQ;2-

=2 >=2 µ9 5 sec, µ9 58JCQI9=µ5 @5 µ2@ 65C5=C;Q39C54 2 µJ;37 CJL =7µ5CJH. (LC?

<>@9C54 35@9;? 5@ =693CJKµ9 ?C4 cos(@t + 5) = cos(@t–%@) = cos[@(t–%)]. +CJ +T7µ5

4.35 35>@J@C54 J4 OKJ 5;µJ@46QH =L@4=CM=9H 58? C4H J8J>9H 58JC9I9>C54 Q@5 =7µ5,

8JL QTJL@ µ2O9@467 O453J;B 3B=2H <45 t = 0, 65GMH 654 2 µJ;37 CJL =7µ5CJH. )J

=7µ5 8JL 35>@9C54 =CJ +T7µ5 4.3S 58JC9I9>C54 58? C4H >O49H =L@4=CM=9H, 8JL QTJL@

?µAH L8J=C9> O453J;9C467 T;J@467 65GL=CQ;2=2 2 µ>5 AH 8;JH C2@ BII2, QC=4 M=C9

@5 QTJL@ O453J;B 3B=2H <45 t = 0 >=2 µ9 '. )J 58JCQI9=µ5 9>@54 ?C4 2 µJ;37 CJL

% 6 6( ) = #b a % 6 6( ) = #a



045 @5 QT94 Q@5 FIR 3>IC;J C2@ 4O4?C2C5 C2H <;5µµ467H 58?6;4=2H 3B=2H, G5 8;Q-

894 @5 85;JL=4BZ94 6B8J45H µJ;37H =Lµµ9C;>5 =C2@ 6;JL=C467 58?6;4=2 h(n). PC=4,

Q=CA Q@5 FIR 3>IC;J µ9 N ?;JLH =C2@ 6;JL=C467 58?6;4=2 h(n) (n = 0,…,N–1),

?8JL CJ R 9>@54 89;4CC?H 5;4Gµ?H 654 J4 ?;J4 h(n) 85;JL=4BZJL@ C2 =Lµµ9C;>5:

(4.4)

)?C9, CJ 3>IC;J 5LC? QT94 <;5µµ467 58?6;4=2 3B=2H. (H 8B;JLµ9, <45 85;BO94<µ5, C2@

58?6;4=2 =LT@?C2C5H CJL 3>IC;JL <45 N = 5. DQCJLµ9 =C2@ (4.2) 654 QTJLµ9

(4.5)

H e h m e

h h e h e h e h e

e h e h e h h e h e

e

j j m

m

j j j j

j j j j j

( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

6 6

6 6 6 6

6 6 6 6 6

= =

= + + + +

= + + + +[ ]=

#

=# # # #

# # #

#

50

4

2 3 4

2 2 2

0 1 2 3 4

0 1 2 3 4

jj j j j j

j

h e e h e e h

e h h h

2 2 2

2

0 1 2

2 0 2 2 1 2

6 6 6 6 6

6 6 6

( )( ) ( )( ) ( )

( )cos( ) ( )cos( ) ( )

+ + + +[ ]= + +[ ]

# #

#

z e j= 6

h n h N n( ) ( )= # #1

(5) (S)

1

1

0.5

0

–0.5

–1

1

0.5

0

–0.5

–1

2

–2

–10

0

1

2

–2

–1

0

500 1000 1500

0 500 1000 1500 0 500 1000 1500

0 500 1000 1500

™¯‹Ì· 4.3

T7( %"µ(,01C4'() *9,(), *0%<(-"./012 µ("<2 $0%/( 87,5./( 82µ%%,?6($% µ. /3

µ./%F7 /()4 %"E0-12 *0%<("?

<?834. %) >"E012*0%<("? <?834 0.I) >"E012 *0%<(-

"? <?834 '.

=7µ5CJH QT94 5IIB:94 ;4Z46B. *5;’ ?C4 CJ 5@G;M84@J =K=C2µ5 56J7H O9@ 9>@54 9L5>-

=G2CJ =C4H 35=46QH µ9C5CJ8>=94H CA@ 5;µJ@46M@ 9@?H =7µ5CJH, µ>5 CQCJ45 5II5<7

G5 µ8J;JK=9 @5 QT94 65C5=C;98C46B 58JC9IQ=µ5C5 =9 89;48CM=94H 8JL 9@O453Q;94 2

µJ;37 CJL =7µ5CJH (C2I9846J4@A@>9H, 65;O4J<;B32µ5, 946?@9H 6CI.).


)J µQC;J C2H H(ej@) 9>@54 2 8;5<µ5C467 8J=?C2C5 µQ=5 =C2@ 5<6KI2. # 3B=2 C2H

4=JKC54 µ9 C2 3B=2 8JL O2µ4JL;<9> J ?;JH e–j2@. ' L8JIJ<4=µ?H C2H O>@94

5(@) = –2@. +C2 <9@467 89;>8CA=2 3>IC;A@ µ76JLH R G5 9>@54 .

E8JµQ@AH J =L@C9I9=C7H 5@5IJ<>5H C2H =TQ=2H (4.3) 4=JKC54 µ9 (R–1)/2. +Lµ89;5-

=µ5C46B, Q@5 FIR 3>IC;J µ9 =Lµµ9C;467 6;JL=C467 58?6;4=2 654 89;4CC? 8I7GJH

?;A@ QT94 <;5µµ467 3B=2, 654 ?8AH 9>O5µ9 5LC? QT94 AH =L@Q8945 C2 T;J@467 65GL-

=CQ;2=2 ?IA@ CA@ 5;µJ@46M@ CJL =7µ5CJH 65CB O9><µ5C5.

F8B;TJL@ CQ==9;4H O453J;9C46J> CK8J4 FIR 3>IC;A@ <;5µµ467H 3B=2H, 5@BIJ<5 µ9

CJ 5@ CJ 8I7GJH N CA@ =L@C9I9=CM@ C2H h(n) 9>@54 B;C4J 7 89;4CC? 654 5@ 2 h(n) 9>@54

=Lµµ9C;467 7 5@C4=Lµµ9C;467. +9 ?I9H C4H 89;48CM=94H J L8JIJ<4=µ?H CJL µQC;JL

654 C2H 3B=2H C2H 58?6;4=2H =LT@?C2C5H H(ej@) <>@9C54 ?8AH =CJ 8;J2<JKµ9@J

85;BO94<µ5, 654 58JO946@K9C54 ?C4 2 3B=2 9>@54 <;5µµ467.

¶·Ú¿‰ÂÈÁÌ· 4.2

$5 L8JIJ<4=C9> 2 58?6;4=2 =LT@?C2C5H (µQC;J 654 3B=2) <45 CJ 3>IC;J CJL 85;5-

O9><µ5CJH 4.1.

=G/0

)J µQC;J C2H H(ej@) 9>@54 . # 3B=2 C2H L8JIJ<>Z9C54 58? C2 3B=2

8JL O2µ4JL;<9> J µ4<5O46?H e–j@, 654 2 J8J>5 9>@54 5(@)=–@.

13

2 1[ cos( ) ]6 +

H e h m e e e e e e

e e e e

j j m

m

j j j j j

j j j j

( ) ( ) [ ]

( ) cos( )

6 6 6 6 6 6 6

6 6 6 6 6

= = + + = + +[ ]= + +[ ] = +[ ]

#

=

# # # #

# # #

50

221

31

13

1

13

113

2 1

N #12

5 @ @( ) = # #N 1

2

1 4 54 . 1 Ã " ƒ" ' & % ƒ # ™ & # ' " & ø ¡ F I R º # § & ƒ ø ¡

$5 L8JIJ<4=C9> 2 58?6;4=2 =LT@?C2C5H (µQC;J 654 3B=2) CJL 3>IC;JL 8JL 89;4-

<;B3269 =C2@ O;5=C2;4?C2C5 1/ 693BI54J 4.




# 58?6;4=2 =LT@?C2C5H H(ej@) L8JIJ<>Z9C54 5@ =C2 =TQ=2 GQ=JLµ9

:

)J µQC;J C2H H(ej@) 9>@54:

,

9@M 2 3B=2 4=JKC54 µ9 8/28JL 94=B<94 2 µ4<5O467 µJ@BO5 654 –@/2 58? CJ@ 96G9C4-

6? ?;J:

4.1.2 '˘ÛÙ¿ıÂÈ· ÙˆÓ Ê›ÏÙÚˆÓ – ¶fiÏÔÈ Î·È ÌË‰ÂÓÈÎ¿

+C5 FIR 3>IC;5 2 Q:JOJH 9:5;CBC54 µ?@J 58? C4H C4µQH C2H 94=?OJL x(n–k) µ9

k = 0 ,..., N–1. &5µ>5 65GL=C9;2µQ@2 C4µ7 C2H 9:?OJL y(n–k) O9@ =Lµµ9CQT94 =CJ@

L8JIJ<4=µ? C2H. PC=4 2 6;JL=C467 58?6;4=2 9>@54 8989;5=µQ@2 654 2 =L@B;C2=2

µ9C53J;BH CJL 3>IC;JL H(z) 8JL O>@9C54 58? C2@ (4.2) O9@ QT94 85;J@Jµ5=C7, 98J-

µQ@AH O9@ QT94 8?IJLH 8JL O2µ4JL;<JK@ 5=CBG945. \I5 5LCB QTJL@ AH 58JCQI9=µ5

C2@ 9L=CBG945 CJL 3>IC;JL, 8JL =2µ5>@94 ?C4 <45 J8J4JO78JC9 8989;5=µQ@J =9 8IB-

CJH =7µ5 94=?OJL, 2 Q:JOJH 9>@54 98>=2H 8989;5=µQ@2 =9 µQ<9GJH. # GQ=2 ?8JL 2

H(z) µ2O9@>Z9C54 (µ2O9@46B) µ8J;9> @5 65GJ;4=C9>, QC=4 M=C9 @5 984CL<TB@9C54 58?

CJ 3>IC;J 8JIK 65I7 58?;;4R2 6B8J4A@ 58? C4H 5;µJ@46QH =L@4=CM=9H CJL =7µ5-

CJH.

% 6 ) 6

( ) = #2 2

| ( ) | sinH e j6 6= 2

2

H e e e e e jej j

j j j j( ) [ ] [ sin ]6 6

6 6 6 6 6= # = # =# # # #1 2

22 2 2 2

z e j= 6 H z z( ) = # #1 1

+9 Q@5 FIR 3>IC;J 2 6;JL=C467 58?6;4=2 QT94 CJLH 8Q@C9 ?;JLH C2H >=JLH µ9 0,2.

*J49H 58? C4H 85;56BCA 8;JCB=94H 9>@54 5I2G9>H 654 8J49H R9LO9>H 654 <45C>;

5. )J 3>IC;J T;94BZ9C54 CQ==9;4H µJ@BO9H 65GL=CQ;2=2H <45 C2@ LIJ8J>2=7 CJL.

S. '4 =L@C9I9=CQH CJL 3>IC;JL 9>@54 6.

<. &BG9 =L@C9I9=C7H QT94 C4µ7 0,3.

O. # 65GL=CQ;2=2 3B=2H 8JL O2µ4JL;<9> CJ 3>IC;J 9>@54 2@.

9. 045 =LT@?C2C5 O94<µ5CJI2R>5H 44,1 kHz 2 T;J@467 65GL=CQ;2=2 8JL O2µ4JL;-

<9>C54 =9 6BG9 5;µJ@467 CJL =7µ5CJH 9>@54 2;s = 50,5 µsec.




;% FIR <96/"% '%"()80?L(), 0*0%9/."% E%"%1/3"08/01?, /% ('(9% .9,%0 %'%"%9/3/%

8/04 '."088#/.".4 '."0'/+8.04 H3<0%124 .'.F."$%89%4 82µ%/(4. ;( '"+/( %'# %)/?,

3 .)8/?5.0%, (<.96./%0 8/( $.$(,#4 #/0 3 CF(*(4 y(n) 8/% FIR <96/"% )'(6($9L./%0

µ#,( %'# /0µC4 /34 .08#*() x(n), .'(µC,@4 /% <96/"% *., CE(), 16?*( %,%/"(<(*#-

/3834. ;( ?66( E%"%1/3"08/01#, 3 $"%µµ012 %'#1"083 <?834, 6%µI?,./%0 #/%, 3

1"()8/012 %'#1"083 /() <96/"() .9,%0 8)µµ./"012.

4.2 (‡!ÔÈ Î·È !ÚÔ‰È·ÁÚ·Ê¤˜ ÙˆÓ FIR Ê›ÏÙÚˆÓ

' C;?8JH µ9 CJ@ J8J>J 6L;>AH 65GJ;>Z9C54 5IIB 654 T;2=4µJ8J49>C54 Q@5 R23456?

3>IC;J 9>@54 2 =Lµ89;43J;B CJL =C2 =LT@?C2C5. # =Lµ89;43J;B 5LC7 89;4<;B39-

C54 58? C2@ 58?6;4=2 =C2 =LT@?C2C5 CJL 3>IC;JL, C?=J =9 ?,C4 QT94 =TQ=2 µ9 C2@

98>O;5=7 CJL =CJ 8IBCJH CA@ O453?;A@ 5;µJ@46M@ CJL =7µ5CJH (58?6;4=2 8IB-

CJLH), ?=J 654 =CJ C;?8J µ9 CJ@ J8J>J 982;9BZ94 C2 35=467 =TQ=2 CA@ 5;µJ@46M@

(58?6;4=2 3B=2H). E9OJµQ@JL ?C4 2 58?6;4=2 =C2 =LT@?C2C5 CA@ R23456M@ =L=C2-

µBCA@ 9>@54 89;4JO467, 5;69> @5 65GJ;4=C9> 2 =Lµ89;43J;B CJL R23456JK 3>IC;JL

=C2@ 89;4JT7 58? 0 AH '.

# 58?6;4=2 3B=2H =C5 FIR 3>IC;5 µ8J;9> @5 9>@54 <;5µµ467 5@ 2 6;JL=C467 58?-

6;4=2 h(n) 9>@54 =Lµµ9C;467 7 5@C4=Lµµ9C;467. +C2@ 58?6;4=2 µQC;JL CJL 3>IC;JL

9@O453Q;J@ 85;JL=4BZJL@ C?=J J4 58J6I>=94H 58? C2@ 984GLµ2C7 58?6;4=2 =C4H

ZM@9H O4QI9L=2H 654 58J6J87H, ?=J 654 J4 =LT@?C2C9H 8JL 65GJ;>ZJL@ C5 ?;45 CA@

ZA@M@ 5LCM@. \I5 5LCB C5 T5;56C2;4=C46B 89;4<;B3C265@ =CJ 693BI54J 3, 85;B-

<;53JH 3.4.5 654 4O45>C9;5 =CJ +T7µ5 3.9. +CJ +T7µ5 5LC? µ8J;JKµ9 @5 O456;>@JLµ9

C4H 98?µ9@9H 85;5µQC;JLH:

*p 9>@54 2 58?6I4=2 (58? C2 µJ@BO5) =C2 ZM@2 O4QI9L=2H

*s 9>@54 2 58?6I4=2 (58? CJ µ2OQ@) =C2 ZM@2 58J6J87H

@p 9>@54 CJ ?;4J C2H ZM@2H O4QI9L=2H

@s 9>@54 CJ ?;4J C2H ZM@2H 58J6J87H

2' 9>@54 2 6L6I467 =LT@?C2C5 O94<µ5CJI2R>5H

)J 9K;JH C2H ZM@2H µ9CBS5=2H 9>@54 . +L@7GAH C5 @p 654 @s J;>ZJ-

@C54 AH 6IB=µ5C5 CJL ', 9@M J4 58J6I>=94H Op 654 Os J;>ZJ@C54 =9 dB µQ=A CA@ 8J=J-

C7CA@ 20 log(1 + *p) 654 20 log(*s), 5@C>=CJ4T5. !848IQJ@, AH T5;56C2;4=C46? =T9-

O>5=2H µ8J;9> @5 G9A;2G9> CJ µQ<4=CJ 58JO96C? 8I7GJH N CA@ =L@C9I9=CM@ h(n)

CJL 3>IC;JL. '4 8;J2<JKµ9@9H 8;JO45<;53QH 53J;JK@ C2@ 8;5<µ5C467 =Lµ89;43J-

R6 6 6= #s p

1 4 74 . 2 & À ¶ $ # ' " # ¶ ƒ $ ¢ # " ° ƒ" º ! ™ & ø ¡ F I R º # § & ƒ ø ¡



;B 9@?H FIR 3>IC;JL. \C5@ C5 µ9<QG2 Os, Op 654 E@ C9>@JL@ =CJ µ2OQ@, CJ 5854CJK-

µ9@J 8I7GJH R CA@ =L@C9I9=CM@ C9>@94 =CJ B894;J 654 2 58?6;4=2 CJL 3>IC;JL 8;J-

=9<<>Z94 C2@ 4O5@467 58?6;4=2 (UIQ89 +T7µ5 3.8).

¶·Ú¿‰ÂÈÁÌ· 4.3

'4 8;JO45<;53QH FIR 3>IC;JL 9>@54 0,01 dB 6LµBCA=2 (58?6I4=2) =C2 ZM@2 O4Q-

I9L=2H, –60 dBs =C2 ZM@2 58J6J87H, @s = 0,4' 654 ZM@2 µ9CBS5=2H >=2 µ9 0,2'. $5

9:2<7=9C9 C4 56;4SMH =2µ5>@94 65G9µ4B 58? C4H 8;JO45<;53QH 5LCQH.

=G/0

+Kµ3A@5 µ9 C5 8;J2<JKµ9@5, G5 8;Q894 58? CJ 3>IC;J @5 O4Q;TJ@C54 ?I9H J4 =LT@?-

C2C9H µQT;4 CJ 0,4', 654 @5 58J6?8CJ@C54 ?I9H J4 =LT@?C2C9H 8B@A 58? C5 0,6'. #

ZM@2 µ9CBS5=2H S;>=69C54 58? CJ 0,4' µQT;4 CJ 0,6'. +C2 ZM@2 O4QI9L=2H 2 58?-

6I4=2 58? C2 µJ@BO5 9>@54 Op 654 8;J=O4J;>Z9C54 58? C2 =TQ=2:

20 log(1 + *p) = 0,01J *p = 0,001

+C2 ZM@2 58J6J87H 4=TK94 20 log(*s) = –60J*s = 1/1000. (LC? =2µ5>@94 ?C4 µ?@J CJ Q@5

T4I4J=C? 58? CJ 8IBCJH CA@ =LT@JC7CA@ =C2@ 89;4JT7 5LC7 O4Q;T9C54 58? CJ 3>IC;J.

$5 =T9O45=C9> 2 58?6;4=2 µQC;JL =C2 =LT@?C2C5 4O5@46JK LR289;5CJK 3>IC;JL

µ9 ZM@2 O4QI9L=2H CJ 1/R C2H =LT@?C2C5H Nyquist. # C4µ7 CJL R 9>@54 O467 =5H

984IJ<7.


# =LT@?C2C5 Nyquist 9>@54 >=2 µ9 ' (µ4=? C2H =LT@?C2C5H O94<µ5CJI2R>5H). PC=4, 2

ZM@2 O4QI9L=2H CJL LR289;5CJK 3>IC;JL G5 QT94 9K;JH 8/R. (@ 984IQ:JLµ9 R = 10,

C?C9 2 58?6;4=2 µQC;JL CJL 3>IC;JL =C2 =LT@?C2C5 QT94 ?8AH 35>@9C54 =CJ 98?µ9-

@J +T7µ5.

|H(ej@)|

1

0 ' 2'9'/10



'4 CK8J4 CA@ R23456M@ 3>IC;A@ 8JL µ8J;JK@ @5 =T9O45=CJK@ AH FIR 3>IC;5 8J4-

6>IIJL@ 654 µ8J;JK@ @5 QTJL@ ?I9H C4H µJ;3QH ?8AH S5GL89;5CB, ZA@JO45S5CB 654

LR289;5CB. !8>=2H J4 8;JO45<;53QH CJLH µ8J;9> @5 9>@54 8JIK 5L=C2;QH C?=J =C2

ZM@2 O4QI9L=2H CA@ =LT@JC7CA@, ?=J 654 =C2 ZM@2 58J6J87H.

4.3 M¤ıÔ‰ÔÈ Û¯Â‰›·ÛË˜ ÙˆÓ FIR Ê›ÏÙÚˆÓ

# O45O465=>5 <45 C2 =T9O>5=2 FIR 3>IC;A@ 89;4I5µSB@94 CJ@ 65GJ;4=µ? CA@ 8;J-

O45<;53M@ CJL 3>IC;JL, CJ@ L8JIJ<4=µ? CA@ =L@C9I9=CM@ CJL 3>IC;JL µ9 µ>5 58?

C4H O45GQ=4µ9H µ9G?OJLH, CJ@ 65GJ;4=µ? C2H OJµ7H LIJ8J>2=2H, C2@ 5@BIL=2 =35I-

µBCA@ I?<A CJL 8989;5=µQ@JL µ76JLH CA@ =L@C9I9=CM@ 65GMH 654 C2@ 984IJ<7 <45

LIJ8J>2=2 µ9 IJ<4=µ46? 7 LI46?.

+2µ5@C46? S7µ5 =C2 O45O465=>5 =T9O>5=2H CA@ FIR 3>IC;A@ 9>@54 @5 L8JIJ<4=CJK@

J4 =L@C9I9=CQH h(n), M=C9 CJ 3>IC;J @5 8I2;J> C4H 984GLµ2CQH 8;JO45<;53QH µQC;JL

654 3B=2H =C2 =LT@?C2C5. '4 µQGJOJ4 8JL T;2=4µJ8J4JK@C54 9L;KC5C5 9>@54 6L;>AH

J4 98?µ9@9H C;94H. # µQGJOJH CA@ 85;5GK;A@ (windows method), 2 µQGJOJH =T9-

O>5=2H SQIC4=CA@ 3>IC;A@ (optimal method) 654 2 µQGJOJH C2H O94<µ5CJI2R>5H =C2

=LT@?C2C5 (frequency sampling method). !OM G5 85;JL=4B=JLµ9 C2 µQGJOJ CA@

85;5GK;A@ 8JL 9>@54 58I7. # µQGJOJH =T9O>5=2H SQIC4=CA@ 3>IC;A@ O>@94 C2@ 65IK-

C9;2 OL@5C7 58?6;4=2 =LT@?C2C5H <45 =L<696;4µQ@J 8I7GJH =L@C9I9=CM@. \I9H J4

µQGJOJ4 µ8J;JK@ @5 OM=JL@ FIR 3>IC;5 <;5µµ467H 3B=2H.

4.3.1 ª¤ıÔ‰Ô˜ ÙˆÓ !·Ú·ı‡ÚˆÓ

# µQGJOJH CA@ 85;5GK;A@ 58JC9I9> µ45 =T9C46B 58I7 O45O465=>5 =T9O>5=2H FIR

3>IC;A@. '4 =L@C9I9=CQH hD(n) CJL 4O5@46JK 3>IC;JL L8JIJ<>ZJ@C54 58? CJ@ 5@C>-

=C;J3J µ9C5=T2µ5C4=µ? Fourier C2H 58?6;4=2H =LT@?C2C5H HD(ej@):

1 4 94 . 3 M !£$¢$# ™Ã!¢#"™%™ FIR º#§&ƒø¡

+C2@ 98?µ9@2 8;?C5=2 @5 984IQ:9C9 C2 =A=C7 58B@C2=2.

1. &. C,% L@,(*0%I%/# FIR <96/"( 3 %,+/."3 8)E,#/3/% %'(1('24 .9,%0:

– ,9<5IKC9;2 58? '

– ,46;?C9;2 58? '

– ,9<5IKC9;2 58? 2'




(4.6)

)J <;Bµµ5 D T;2=4µJ8J49>C54 <45 @5 O2IM=94 C2@ 984GLµ2C7 (Desired) 7 4O5@467

=Lµ89;43J;B.

¶·Ú¿‰ÂÈÁÌ· 4.4

$5 =T9O45=C9> Q@5 S5GL89;5C? 3>IC;J µ9 =LT@?C2C5 58J6J87H @c = 8/2 (2µ>=945H

ZM@2H 7 half–band), CJL J8J>JL 2 4O5@467 58?6;4=2 =C2 =LT@?C2C5 O>@9C54 =CJ

+T7µ5 4.45.

=G/0

D9A;JKµ9 ?C4 2 =LT@?C2C5 O94<µ5CJI2R>5H 9>@54 65@J@46J8J42µQ@2 =C2 µJ@BO5

( ). (8? C2 =TQ=2 (4.6) J4 =L@C9I9=CQH h(n) L8JIJ<>ZJ@C54 AH 9:7H:

(4.7)

# <;53467 85;B=C5=2 CA@ =L@C9I9=CM@ O>@9C54 =CJ +T7µ5 4.4S, 58? ?8JL 35>@9-

C54 ?C4 2 6;JL=C467 58?6;4=2 9>@54 =Lµµ9C;467. !8JµQ@AH, CJ 3>IC;J G5 QT94 <;5µ-

µ467 58?6;4=2 3B=2H. (8? C2 =TQ=2 (4.7) 9>@54 35@9;? ?C4 J4 ?;J4 hD(n) 3G>@JL@

=L@9TMH 5IIB, O9@ µ2O9@>ZJ@C54 8JCQ, µ9 58JCQI9=µ5 2 6;JL=C467 58?6;4=2 @5 9>@54

B894;2.

h n H e e d

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nn

nn

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(%). V*%,012 %'#1"083 8)E,#/3-/%4 I%5)'."%/(7 H3<0%1(7 <96-

/"() 3µ98.0%4 L+,34(half–band) 1%0 (I) 3 %,/98/(0E31"()8/012 %'#1"083 (1?5. *.7-

/."(4 8),/.6.8/24, .1/#4

/@, 1.,/"01+,, .9,%0 µ3*.,0-1#4).


# 58J6J87 C2H 6;JL=C467H 58?6;4=2H hD(n) <45 |n|>M, M=C9 5LC7 @5 QT94 8989;5-

=µQ@J µ76JH 654 @5 O45C2;9> C2 =Lµµ9C;>5 C2H, QT94 AH 58JCQI9=µ5 @5 5IIJ4AG9> 2

Z2CJKµ9@2 4O5@467 58?6;4=2 =C2 =LT@?C2C5. # 5IIJ>A=2 5LC7 9>@54 9µ35@7H =CJ

+T7µ5 4.5 ?8JL O9>T@J@C54 J4 58J6;>=94H =C2 =LT@?C2C5 <45 6;JL=C467 58?6;4=2

µ9 13 (+T7µ5 4.55), 25 (+T7µ5 4.5S) 654 B894;JLH ?;JLH (+T7µ5 4.5<). # 5@C>=CJ4-

T2 58?6;4=2 =LT@?C2C5H H(ej@), 9µ35@>Z94 6LµBCA=2 =9 ?I2 C2@ Q6C5=2 C2H ZM@2H

O4QI9L=2H 654 C2H ZM@2H 58J6J87H, 9@M 85;JL=4BZ94 O458IBCL@=2 C2H ZM@2H µ9CB-

S5=2H. # 9µ3B@4=2 C2H 6LµBCA=2H 9>@54 <@A=C7 @4 <%0,#µ.,( Gibbs, 654 I5µSB-

@94 C2 µQ<4=C2 C4µ7 C2H 6J@CB =C4H 5=L@QT949H C2H H(ej@), =C2@ 89;4JT7 C2H ZM@2H

µ9CBS5=2H. # µQ<4=C2 5LC7 C4µ7 6Lµ5>@9C54 89;>8JL =CJ 9% CJL KRJLH C2H 5=L@Q-

T945H 654 85;5µQ@94 =C5G9;7 5@9:B;C2C5 58? CJ µ76JH 2M + 1 C2H 6;JL=C467H 58?-

6;4=2H. 045 C2 ZM@2 C2H 58J6J87H 2 C4µ7 CJL 9% 5@C4=CJ4T9> =9 21 89;>8JL dBs 9:5-

=GQ@4=2, C4µ7 8JL O9@ 9>@54 58JO96C7 <45 8;56C46QH 935;µJ<QH. PC=4, <45 465@J8J4-

2C467 8;J=Q<<4=2 CA@ 4O5@46M@ µJ;3M@ CA@ O4B3o;A@ 3>IC;A@, O9@ =L@4=CBC54 J

589LG9>5H 89;4J;4=µ?H CJL µ76JLH C2H 6;JL=C467H 58?6;4=2H h(n).

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|H1(ej@)|

|H2(ej@)|

|H3(ej@)|

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@c @

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™¯‹Ì· 4.5

>'#1"083 8)E,#-/3/%4 I%5)'."%-/(7 <96/"() µ. (%)13, (I) 25 1%0 ($)?'.0"()4 #"()48/3, 1"()8/012%'#1"083.



P@5H O?64µJH C;?8JH, <45 @5 89CKTJLµ9 465@J8J42C467 8;J=Q<<4=2 =C2@ 58?6;4=2

=LT@?C2C5H 9@?H FIR 3>IC;JL, 9>@54 @5 T;2=4µJ8J42G9> µ45 8989;5=µQ@2 56JIJL-

G>5 =L@C9I9=CM@ w(n), 8JL 65I9>C54 '%"?5)"(, <45 @5 C;J8J8J47=94 CJLH =L@C9I9-

=CQH hD(n) C2H 9:>=A=2H (4.7). # @Q5 6;JL=C467 58?6;4=2 h(n) 8;J6K8C94 AH CJ <4@?-

µ9@J hD(n)w(n). ,9 CJ@ C;?8J 8JL 89;4<;B39C54 =CJ +T7µ5 4.6 9IQ<TJ@C54 C5 8;J-

SI7µ5C5 =K<6I4=2H (354@?µ9@J Gibbs) CJL µ9C5=T2µ5C4=µJK Fourier =C5 =2µ9>5

5=L@QT945H C2H H(ej@). +CJ +T7µ5 4.65 58946J@>Z9C54 2 984GLµ2C7 58?6;4=2 HD(ej@)

CJL 4O5@46JK S5GL89;5CJK 3>IC;JL 2µ>=945H ZM@2H (half–band). (6;4SMH O>8I5 O>@9-

C54 2 589;4?;4=C2 =CJ T;?@J 6;JL=C467 58?6;4=2 hD(n). +CJ 98?µ9@J +T7µ5 4.6S

O>@9C54 2 58?6;4=2 Ht(ej@), ?8AH 8;J6K8C94 58? C2@ 58J6J87 C2H hD(n) <45 M=8.

!OM 9µ35@>Z9C54 65G5;B CJ 354@?µ9@J Gibbs. (6JIJKGAH, =CJ +T7µ5 4.6<. O>@9C54

CJ 85;BGL;J w(n), CJ J8J>J 9>@54 µ2 µ2O9@46? =CJ O4B=C2µ5 654 µ2OQ@

<45 |n| > 8. +CJ 5;4=C9;? µQ;JH O>@9C54 CJ 3B=µ5 CJL 85;5GK;JL W(ej@). +CJ +T7µ5

4.6O O>@9C54 2 C9I467 6;JL=C467 58?6;4=2 h(n) 2 J8J>5 8;J6K8C94 58? CJ <4@?µ9@J

ht(n)w(n), 58? 8JII58I5=45=µ? O2I5O7 =2µ9>J 8;JH =2µ9>J CA@ OKJ T;J@46M@ 56J-

IJLG4M@. # 5@C>=CJ4T2 C9I467 58?6;4=2 H(ej@) 8;J6K8C94 3L=46B 58? C2 =L@QI4:2

C2H Ht(ej@) µ9 C2@ W(ej@). +CJ +T7µ5 5LC? QT94 µ94AG9> 54=G2CB CJ 354@?µ9@J Gibbs,

5IIB QT94 5L:2G9> 2 ZM@2 µ9CBS5=2H CJL 3>IC;JL. H C9I467 6;JL=C467 58?6;4=2

h(n) 8;Q894 @5 JI4=G7=94 =CJ T;?@J <45 @5 <>@94 54C45C7 (causal), 654 98JµQ@AH 3L=4-

6MH 8;5<µ5CJ8J47=4µ2 (2 h(n) 9L;>=69C54 ?I2 =C5 G9C46B n). # T;J@467 5LC7 JI>-

=G2=2 8;56C46B 5@C4=CJ4T9> =C2@ 8;J=G762 C2H <;5µµ467H 58?6;4=2H 3B=2H 8JL

94=B<JL@ C5 FIR 3>IC;5. (LC? <>@9C54 :96BG5;J 58? C2 =TQ=2 (4.5), ?8JL J 6J4@?H

85;B<J@C5H 8JL O2µ4JL;<9> C2 <;5µµ467 58?6;4=2 3B=2H QT94 8;J6KR94 56;4SMH

O4?C4 2 6;JL=C467 58?6;4=2 CJL 3>IC;JL QT94 <>@94 54C45C7.

(:>Z94 @5 =2µ94M=JLµ9 9OM ?C4 2 5K:2=2 CJL 8I7GJLH CA@ =L@C9I9=CM@ 8JL 8;J-

6K8C94 ?C5@ CJ µ76JH CJL 85;5GK;JL 9>@54 µ9<5IKC9;J, QT94 AH 58JCQI9=µ5 @5 <>@94

=C9@?C9;2 2 ZM@2 µ9CBS5=2H, O2I5O7 CJ 3>IC;J 84J 58?CJµJ. (8? C2@ BII2, CJ

9>OJH CJL T;2=4µJ8J4JKµ9@JL 85;5GK;JL G5 982;9B=94 C2@ 6LµBCA=2 =C4H ZM@9H

O4QI9L=2H 654 58J6J87H.

# & &8 8n


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ht(n)

h(n)

w(n)

Ts

n

n

n

n

0 8-8

-8

-8 8

80

1

1

1

1

1

™¯‹Ì· 4.6

D"%<012 '%"?8/%-83 /() /"#'().<%"µ($24 /@,'%"%57"@, 8/38E.*9%83 FIR <96/"@,.

,9 C2 SJ7G945 C2H 9:>=A=2H 4.7, @5 L8JIJ<>=9C9 CJLH =L@C9I9=CQH C2H 6;JL=C4-

67H 58?6;4=2H 4O5@46JK S5GL89;5CJK 3>IC;JL µ9 =LT@?C2C5 58J6J87H @c = '/R

(N–th band 7 R–ZA@46? 3>IC;J). # 984IJ<7 CJL R 9>@54 O467 =5H.


4.3.2 %‰ÈfiÙËÙÂ˜ Î·È Ù‡!ÔÈ ÙˆÓ !·Ú·ı‡ÚˆÓ

# =L@B;C2=2 W(ej@), 8JL 9>@54 CJ 3B=µ5 CJL 85;5GK;JL w(n), 58JC9I9>C54 58? Q@5@

6K;4J IJS? 8JL 89;4QT94 C2@ 89;4==?C9;2 35=µ5C467 9@Q;<945 CJL 85;5GK;JL,

65GMH 654 58? 85;B8I9L;JLH IJSJKH 8JL <9@46B 9I5CCM@J@C54 <;7<J;5 (+T7µ5

4.6<). )J 9K;JH C2H ZM@2H µ9CBS5=2H CJL 3>IC;JL 9:5;CBC54 58? CJ 9K;JH CJL 6K;4-

JL IJSJK C2H W(ej@). +C2@ C9I467 H(ej@) 984O;JK@ 98>=2H J4 85;B8I9L;J4 IJSJ> C2H

W(ej@), 94=B<J@C5H =’ 5LC7 µ46;7 6LµBCA=2 =9 ?I9H C4H =LT@?C2C9H. )J 3B=µ5 µ45H



=L@B;C2=2H 85;5GK;JL 8;Q894 @5 QT94 =C9@? 6K;4J IJS?, J J8J>JH @5 89;4QT94 ?=J

CJ OL@5C?@ 89;4==?C9;2 35=µ5C467 9@Q;<945, µ9 85;B8I9L;JLH IJSJKH 8JL @5 3G>-

@JL@ <;7<J;5. (8? C4H 8IQJ@ <@A=CQH =L@5;C7=94H 85;5GK;A@ 9>@54 J4 56?IJLG9H:

(. )J C9C;5<A@46? 85;BGL;J (rectangular). (@C4=CJ4T9> =C2@ 58I7 58J6J87 C2H 56J-

IJLG>5H hD(n). # 56JIJLG>5 SB;JLH O>@9C54 58? C2 =TQ=2:

(4.8)

U. )J <9@469LµQ@J 85;BGL;J Hamming:

(4.9)

)J 85;BGL;J 5LC? <45 a = 0,54 O>@94 CJ <@A=C? 85;BGL;J Hamming. 045 C2@ C4µ7

5LC7 CJL a, J 69@C;46?H IJS?H C2H W(ej@) 89;4QT94 CJ 99,96% C2H =L@JI467H 35=µ5-

C467H 9@Q;<945H, 9@M 984CL<TB@J@C54 –52 dB 9:5=GQ@4=2 =C2 ZM@2 58?;;4R2H CA@

3>IC;A@ 8JL =T9O4BZJ@C54. # µJ;37 CJL 85;5GK;JL w(n) <45 R = 17 9>@54 5LC7 CJL

+T7µ5CJH 4.6<.

# µQGJOJH =T9O>5=2H CA@ FIR 3>IC;A@ µ9 C5 85;BGL;5, 85;’ ?C4 58I7, O9@ 5>;94 CJ

S5=46? µ94J@Q6C2µ5 CA@ FIR 3>IC;A@, @5 5854CJK@ 8JIIJKH =L@C9I9=CQH 654, 98J-

µQ@AH, 5L:2µQ@9H 5854C7=94H =9 LI46? <45 C2@ LIJ8J>2=7 CJLH. ,9 65CBII2I2

T;7=2 CJL 85;5GK;JL Hamming 654 BIIA@ 85;5GK;A@ QT94 <>@94 OL@5C7 2 µ9>A=2

CJL 5854CJKµ9@JL LI46JK LIJ8J>2=2H CA@ 3>IC;A@.

¶·Ú¿‰ÂÈÁÌ· 4.5

$5 O94TG9> <;5346B 2 98>O;5=2 CJL 8I7GJLH R CA@ =L@C9I9=CM@ CJL 85;5GK;JL

Hamming (654 65CB =L@Q8945 CJL 8I7GJLH CA@ =L@C9I9=CM@ CJL 3>IC;JL) =C5 T5;5-

6C2;4=C46B C2H 58?6;4=2H =LT@?C2C5H 9@?H FIR 3>IC;JL 2µ>=945H ZM@2H.

=G/0

X;2=4µJ8J4M@C5H C4H =TQ=94H (4.7) 654 (4.9), QTJLµ9 <45 CJLH =L@C9I9=CQH h(n) 9@?H

FIR 3>IC;JL 2µ>=945H ZM@2H µ9 C2 T;7=2 CJL 85;5GK;JL Hamming

h n h n w n

nn

nN

Nn

ND R( ) ( ) ( )

sin( / )( / )

, cos(= = + & & #12

22

0 54 0 462 1

21

2)

))

[ , )] --

w n a an

Nn

NR( ) ( )cos(= + # & & #0

1D

2D1

2 12

)) -

N -12

0 ('()*2'(/. %66(7

w nN

nN

R( ) =0 J8JLO78JC9 5IIJK

11

21

2# # & & #0

1D

2D


# =TQ=2 5LC7 G5 T;2=4µJ8J42G9> <45 @5 =T9O45=CJK@ OKJ FIR 3>IC;5, CJ 8;MCJ µ9

R = 17 =L@C9I9=CQH 654 CJ O9KC9;J µ9 R = 29. ' L8JIJ<4=µ?H C2H 58?6;4=2H =LT@?-

C2C5H <45 C5 OKJ 3>IC;5, <>@9C54 ?8AH 56;4SMH 654 =C2 =TQ=2 (4.5). '4 <;5346QH

85;5=CB=94H CJL +T7µ5CJH 4.7 9I73G2=5@ µ9 C2 SJ7G945 CJL MATLAB 654 µ9 935;-

µJ<7 C2H 8;J2<JKµ9@2H O45O465=>5H. )5 OKJ 8;MC5 +T7µ5C5, O9>T@JL@ CJLH =L@C9-

I9=CQH CJL 3>IC;JL 654 C2@ 58?6;4=2 =LT@?C2C5H <45 R = 17. )5 OKJ 98?µ9@5, 85;JL-

=4BZJL@ CJ 3>IC;J µ9 R = 29 =L@C9I9=CQH. (8? C2 =K<6;4=2 CA@ OKJ 58J6;>=9A@ =C2

=LT@?C2C5 <>@9C54 35@9;? ?C4 2 5K:2=2 CA@ =L@C9I9=CM@ CJL 85;5GK;JL JO2<9> =9

84J µ46;7 ZM@2 µ9CBS5=2H (transition band), B;5 654 84J 58?CJµJ 3>IC;J, ?T4 ?µAH

654 =9 µ9<5IKC9;2 9:5=GQ@4=2 =C2 ZM@2 58J6J87H. 045 @5 89CKT94 6BC4 CQCJ4J J =T9-

O45=C7H 8;Q894 @5 5@5Z2C7=94 =C2 S4SI4J<;53>5 O453J;9C46JK CK8JL 85;BGL;J.

1 5 54 . 3 M !£$¢$# ™Ã!¢#"™%™ FIR º#§&ƒø¡

0,6

0,4

0,2

0

0

0

0

–0,2

0,6

0,4

0,2

0

–0,2

–0,2

–0,6

–0,4

0

–0,2

–0,6

–0,4

0 10 20

0 10 20

30

5 15 8

8

h(n)

h(n) |H(ejA)|

|H(ejA)|

™¯‹Ì· 4.7

S%5)'."%/# FIR<96/"( 8E.*0%8µC-,( µ. *C1% .</?1%0 .01(80.,,C%8),/.6.8/C4. &/3*.7/."3 '."9'/@-83 /( <96/"( .9,%0'0( %'#/(µ(.

*J49H 58? C4H 84J 6BCA 8;JCB=94H, 8JL 53J;JK@ =C2 =T9O>5=2 FIR 3>IC;A@ µ9 C2

µQGJOJ CA@ 85;5GK;A@, 9>@54 =A=CQH 654 <45C>;

1. # =L@B;C2=2 CJL 85;5GK;JL 984O;B 589LG9>5H =CJ 3B=µ5 CJL =7µ5CJH.

2. )J C9I46? 3>IC;J QT94 9L;KC9;2 ZM@2 µ9CBS5=2H

3. )J C9C;5<A@46? 85;BGL;J 8JL O>@9C54 58? C2 =TQ=2 4.8 QT94 =5;B@C5 Q@5@ ?;JLH.

4. # 9:5=GQ@4=2 =C2 ZM@2 58J6J87H CJL 3>IC;JL 9:5;CBC54 58? CJ 9>OJH CJL T;2-

=4µJ8J4JKµ9@JL 85;5GK;JL 654 ?T4 58? CJ 8I7GJH CA@ ?;A@ CJL.




,9 C2 SJ7G945 CJL 85;5GK;JL Hamming @5 L8JIJ<4=CJK@ J4 =L@C9I9=CQH CJL 3>I-

C;JL C2H O;5=C2;4?C2C5H 4 /&93BI54J 4 <45 @5 <>@94 FIR 3>IC;J µ9 OQ65 93CB ?;JLH.

,9 65CBII2IJ IJ<4=µ46? @5 =T9O45=C9> 2 58?6;4=2 µQC;JL 654 3B=2H CJL 3>IC;JL.


'4 =L@C9I9=CQH CJL third band 3>IC;JL G5 S;9GJK@ 5@ J4 ?;J4 hD(n) C2H O;5=C2;4?C2C5H4.4 8JII58I5=45=CJK@ µ9 CJLH ?;JLH wR(n) C2H =TQ=2H (4.9) <45 5 = 0,54 654 R = 17.

4.3.3 ª¤ıÔ‰Ô˜ Û¯Â‰›·ÛË˜ ‚¤ÏÙÈÛÙˆÓ FIR Ê›ÏÙÚˆÓ

' ?;JH IC6/08/( FIR <96/"( =2µ5>@94 C2@ 98>C9L:2 CA@ 65IKC9;A@ OL@5CM@ 8;J-O45<;53M@ =C2@ 58?6;4=2 =LT@?C2C5H µ9 CJ µ46;?C9;J 8I7GJH =L@C9I9=CM@. #µQGJOJH =T9O>5=2H SQIC4=CA@ FIR 3>IC;A@ 5@58CKTG269 =C4H 5;TQH C2H O9659C>5HCJL ’70, 654 QT94 6AO46J8J42G9> =9 8;?<;5µµ5 L8JIJ<4=C7 µ9 58JCQI9=µ5 @5 9>@549K6JI2 2 935;µJ<7 C2H. # µQGJOJH 85;QT94 8I7;2 QI9<TJ C2H 6LµBCA=2H =C2 ZM@2O4QI9L=2H 654 58J6J87H, 9@M C5 3>IC;5 8JL 8;J6K8CJL@ 58? 5LC7 9>@54 8B@C5 84JJ46J@Jµ46B =9 8I7GJH =L@C9I9=CM@.


# µQGJOJH =T9O>5=2H SQIC4=CA@ FIR 3>IC;A@ 9>@54 2 65IKC9;2 984IJ<7, ?C5@ 9>@54O45GQ=4µJ CJ 5@C>=CJ4TJ IJ<4=µ46?. # µQGJOJH 85;QT94 8I7;2 QI9<TJ C2H 6LµBCA-=2H =C2 ZM@2 O4QI9L=2H 654 58J6J87H, 9@M C5 3>IC;5 8JL 8;J6K8CJL@ 58? 5LC79>@54 8B@C5 84J J46J@Jµ46B =9 8I7GJH =L@C9I9=CM@. (@C>G9C5, 2 µQGJOJH CA@ 85;5-GK;A@ 9>@54 8JIK 58I7 654 µ8J;9> @5 T;2=4µJ8J42G9> ?C5@ O9@ L8B;T94 CJ 585-;5>C2CJ IJ<4=µ46? <45 C2@ 8;J2<JKµ9@2 µQGJOJ. E>@94 8B@C5 C2@ >O45 6LµBCA=2=C2 ZM@2 µ9CBS5=2H 654 C2 ZM@2 58J6J87H, 9@M O9@ 85;QT94 QI9<TJ =C5 ?;45 CA@ZA@M@. +CJ +T7µ5 4.8 O>@J@C54 3>IC;5 8JL QTJL@ =T9O45=C9> M=C9 @5 QTJL@ C4H >O49H8;JO45<;53QH. # =T9O>5=2 µ9 C5 85;BGL;5 O>@94 9:7@C5 Q@5 =L@C9I9=CQH, 9@M µ9C2 µQGJOJ CA@ SQIC4=CA@ 3>IC;A@ =5;B@C5 Q:4 =L@C9I9=CQH.


1 5 74 . 4 £ ! ª "&" À § $ ¶ $ # % ™ % ™ F I R º # § & ƒ ø ¡

0

0 100 200 300 400 500

0 100 200 300 400 500

–20

–40

–60

–80

0

–20

–40

–60

–80

|H(ej@)|[=9 dBs

|H(ej@)|[=9 dBs

(5)

(S)

+LT@?C2C5

+LT@?C2C5

Hz

Hz

™¯‹Ì· 4.8

(%) >'#1"083 µC/"() FIRH3<0%1(7 <96/"() 8E.*0%8µC,()µ. /( '%"?5)"( Hamming 1%0.F2,/% C,% 8),/.6.8/C4, (I) >'#-1"083 µC/"() IC6/08/() FIRI%5)'."%/(7 <96/"(). ;( %'%0-/(7µ.,( '625(4 /@, 8),/.6.-8/+, .9,%0 8%"?,/% CF0.

4.4 £¤Ì·Ù· ˘ÏÔ!Ô›ËÛË˜ FIR Ê›ÏÙÚˆÓ

4.4.1 ¢ÔÌ¤˜ !Ú·ÁÌ·ÙÔ!Ô›ËÛË˜ ÙˆÓ FIR Ê›ÏÙÚˆÓ

'4 OJµQH 8;5<µ5CJ8J>2=2H CA@ FIR 3>IC;A@ 9>@54 =C2@ JL=>5 =T2µ5C46B O45<;Bµ-

µ5C5 <45 CJLH O453J;9C46JKH, 5IIB G9A;2C46B 4=JOK@5µJLH C;?8JLH 8JL µ8J;9> @5

J;<5@AG9> 2 =L@B;C2=2 µ9C53J;BH H(z). # 84J O45O9OJµQ@2 OJµ7 9>@54 2 Bµ9=2 7

transversal. 045 C2 OJµ7 5LC7, 8JL 85;JL=4BZ9C54 =CJ +T7µ5 4.95, 2 =TQ=2 µ9C5:K

CA@ =2µBCA@ 94=?OJL–9:?OJL CJL 3>IC;JL QT94 AH 9:7H:

(4.10)

+Kµ3A@5 µ9 C2@ 9:>=A=2 5LC7 2 Bµ9=2 OJµ7 5854C9> <45 C2@ 8;5<µ5CJ8J>2=7 C2H:

• R–1 GQ=94H µ@7µ2H <45 C2@ 58JG769L=2 CA@ O94<µBCA@ x(n).

• R GQ=94H µ@7µ2H <45 C2@ 58JG769L=2 CA@ =L@C9I9=CM@ h(m).

• R 8JII58I5=45=µJKH.

• R–1 8;J=GQ=94H.

y n h m x n m

m

N

( ) ( ) ( )= #=

#

50

1



x(n)

y(n)

z-1 z-1 z-1 z-1 z-1 z-1

h(0) h(1) h(2) h(3) h(4) h(5) h(6)

(5)

+

x(n)

y(n)

h(0) h(1) h(2) h(3)

z-1

z-1 z-1 z-1

z-1 z-1

+

(S)

™¯‹Ì· 4.9

P0 *7( 1)"0#/.".4*(µC4 '"%$µ%/(-'(93834 FIR <96-/"@, (%). Zµ.83

(Transversal) 1%0(I) $"%µµ0124

<?834.

);J8J8J>2=2 C2H transversal OJµ7H 58JC9I9> 2 OJµ7 <;5µµ467H 3B=2H. +C2 OJµ7

5LC7 (+T7µ5 4.9S) C5 O9><µ5C5 8JL 8;?694C54 @5 8JII58I5=45=CJK@ µ9 >O4JLH =L@C9-

I9=CQH I?<A C2H =Lµµ9C;467H 6;JL=C467H 58?6;4=2H , 8;MC5

8;J=C>G9@C54 654 µ9CB 8JII58I5=4BZJ@C54. (8JCQI9=µ5 5LCJK 9>@54 @5 5854C9>C54

µ?@J Q@5H 8JII58I5=45=µ?H <45 6BG9 OKJ O9><µ5C5. # 9:>=A=2 94=?OJL–9:?OJL

=C2@ 89;>8CA=2 5LC7 <;B39C54 AH:

(4.11) y n h m x n m x n N m

m

N

( ) ( )[ ( ) ( )]= # + # + +=

# #

50

12

1

1

( ( ) ( ))h n h N n= ± # #1


# OJµ7 C2H <;5µµ467H 3B=2H <45 @5 LIJ8J42G9> 5854C9> CJLH µ4=JKH 8JII58I5=45-

=µJKH =9 =TQ=2 µ9 C2@ Bµ9=2, 654 S;>=694 935;µJ<7 =C5 FIR 3>IC;5 <;5µµ467H

3B=2H (=Lµµ9C;467 6;JL=C467 58?6;4=2).

¶·Ú¿‰ÂÈÁÌ· 4.6

045 CJ FIR 3>IC;J 2µ>=945H ZM@2H CJL 85;5O9><µ5CJH 4.5, @5 L8JIJ<4=CJK@ J4 µJ@B-

O9H 65GL=CQ;2=2H 654 J4 8JII58I5=45=CQH 8JL 5854CJK@C54 <45 C2@ 8;5<µ5CJ8J>2-

=7 CJL µ9 Bµ9=2 OJµ7 654 OJµ7 <;5µµ467H 3B=2H.

=G/0

+Kµ3A@5 µ9 CJ +T7µ5 4.95, Q@5 FIR 3>IC;J µ9 R =L@C9I9=CQH 5854C9> R–1 =CJ4T9>5

65GL=CQ;2=2H, 654 R 8JII58I5=45=CQH <45 C2@ 8;5<µ5CJ8J>2=7 CJL µ9 C2@ Bµ9=2 OJµ7.

+C2@ 89;>8CA=7 µ5H (R = 17 =L@C9I9=CQH) T;94BZJ@C54 OQ65 Q:4 µJ@BO9H 65GL=CQ;2=2H

654 OQ65 93CB 8JII58I5=45=µJ>. (@C>G9C5, <45 C2@ 8;5<µ5CJ8J>2=2 C2H OJµ7H <;5µµ4-

67H 3B=2H, T;94BZJ@C54 (SIQ89 +T7µ5 4.9S) OQ65 Q:4 µJ@BO9H 65GL=CQ;2=2H, O96598CB

8;J=G9CQH 654 9@@Q5 8JII58I5=45=µJ>. *;Q894 @5 =2µ94M=JLµ9, ?C4 =CJ 3>IC;J 5LC?,

µ9 SB=2 C2@ 9:>=A=2 (4.7), 5;69CJ> =L@C9I9=CQH 9>@54 µ2OQ@ 9894O7 CJ 3>IC;J 9>@54 2µ>-

=945H ZM@2H. PC=4, J 5;4Gµ?H CA@ 8JII58I5=45=µM@ 9>@54 56?µ2 µ46;?C9;JH.

4.4.2 ™Ê¿ÏÌ· ·!fi ÙËÓ Î‚¿ÓÙÈÛË ÙˆÓ Û˘ÓÙÂÏÂÛÙÒÓ

# LIJ8J>2=2 CA@ FIR 3>IC;A@ µ9 R23456B 6L6IMµ5C5 (hardware) =L@98B<9C54

8989;5=µQ@2 56;>S945 5@585;B=C5=2H CA@ =L@C9I9=CM@ CJL 3>IC;JL. ' 89;4J;4-

=µ?H =C2@ 56;>S945 CA@ =L@C9I9=CM@ QT94 AH 58JCQI9=µ5 C2@ 58?6I4=2 CA@ T5;5-

6C2;4=C46M@ CJL 3>IC;JL 58? C4H 5;T46QH 8;JO45<;53QH. (H =2µ94AG9> ?C4 µ9CB 58?

CQCJ45 5II5<7 Q@5 3>IC;J 85K94 @5 9>@54 SQIC4=CJ. )J 354@?µ9@J 5@53Q;9C54 =C2

O49G@7 S4SI4J<;53>5 AH <%0,#µ.,( '.'."%8µC,() µ21()4 6CF34 (Finite Wordlength

Effect). +CJ +T7µ5 4.10 µ9 C2 =L@9T?µ9@2 <;5µµ7 O>@9C54 2 58?6;4=2 CJL 3>IC;JL

8JL =T9O4B=C269 5;T46B, 9@M µ9 C2 O45696JµµQ@2 <;5µµ7 O>@9C54 2 58?6;4=2 CJL

>O4JL 3>IC;JL ?C5@ J4 =L@C9I9=CQH CJL QTJL@ 56;>S945 7 bits.

(8JO946@K9C54 ?C4 6BG9 5K:2=2 65CB 1 bit =C2@ 56;>S945 CA@ =L@C9I9=CM@ S9IC4M-

@94 C2@ 58?OJ=2 CJL 3>IC;JL 65CB Q:4 dBs (SIQ89 98>=2H 9@?C2C5 1.2.2). (@ CJ 3>I-

C;J QT94 =T9O45=C9> µ9 G9A;2C467 58?;;4R2 =C2 ZM@2 58J6J87H > dBs, C?C9 5854-

CJK@C54 CJ I4<?C9;J >/6 bits =C2@ OL5O467 5@585;B=C5=2 CA@ =L@C9I9=CM@ <45 @5

µ2@ 5IIJ4AG9> 2 G9A;2C467 58?6;4=2 CJL 3>IC;JL.

1 5 94 . 4 £ ! ª "&" À § $ ¶ $ # % ™ % ™ F I R º # § & ƒ ø ¡



™¯‹Ì· 4.10

>'(/.6C8µ%/% %'#/3, 1I?,/083 /@,8),/.6.8/+,. M.

*0%1.1(µµC,3$"%µµ2 *9,./%0 3%'#1"083 /() <96-

/"() µ./? /3,1I?,/083 /@,

8),/.6.8/+,. (%)>'#1"083 8)E,#-/3/%4, (I) K.'/(-µC".0% /34 L+,34

*0C6.)834.

20

-20

-40

-60

-80

-100

-120

0

0 2 4 6 8 10 12

A8?6;4=2 =9

dB

A8?6;4=2 =9

dB

0,6

0,4

0,2

0,0

-0,2

-0,4

-0,60,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5

+LT@?C2C5 =9 kHz +LT@?C2C5 =9 kHz

|H(ej@)| |H(ej@)|

(5) (S)

P@5 FIR 3>IC;J µ9 31 =L@C9I9=CQH 5854C9> <45 C2@ LIJ8J>2=7 CJL µ9 C2@ Bµ9=2 OJµ7

5. 31 µJ@BO9H 65GL=CQ;2=2H 654 31 8JII58I5=45=µJKH.

S. 30 µJ@BO9H 65GL=CQ;2=2H 654 31 8JII58I5=45=µJKH.

<. 30 µJ@BO9H 65GL=CQ;2=2H 654 30 8JII58I5=45=µJKH.

O. 31 µJ@BO9H 65GL=CQ;2=2H 654 30 8JII58I5=45=µJKH.

*J45 58? C4H 85;58B@A 8;JCB=94H 9>@54 =A=C7;


# G9A;2C467 58J6J87 9@?H FIR 3>IC;JL =C2 ZM@2 58?;;4R2H 9>@54 106 dBs. '4

=L@C9I9=CQH 8JL T;94BZJ@C54 <45 C2@ LIJ8J>2=2 CJL 3>IC;JL 8;Q894 @5 QTJL@ 56;>-

S945 =9 bits: 8, 12, 14, 18.




)5 FIR R23456B 3>IC;5 8JL 5@58CKTG265@ =CJ 693BI54J 5LC? 85;JL=4BZJL@ OKJ

4O45>C9;5 T5;56C2;4=C46B 8JL 9>@54 984GLµ2CB =C2@ R234567 989:9;<5=>5 =7µ5-

CJH. )2 <;5µµ467 58?6;4=2 =C2 3B=2 654 C2@ 9L=CBG945. 0@A;>=5µ9 CJ@ C;?8J 8JL

65GJ;>ZJ@C54 J4 8;JO45<;53QH CA@ 3>IC;A@ 5LCM@ 654 =C2 =L@QT945 µ9I9C7=5µ9

C2 µQGJOJ =T9O>5=2H CA@ FIR 3>IC;A@ µ9 C5 85;BGL;5. )QIJH, µ9 C2 µQGJOJ =T9-

O>5=2H CA@ SQIC4=CA@ FIR 3>IC;A@ µ8J;JKµ9 @5 IBSJLµ9 8JIK 65I7 58?6;4=2

=LT@?C2C5H µ9 CJLH I4<?C9;J OL@5CJKH =L@C9I9=CQH.

1 6 1™ À ¡ $æ% '!º"§"#$À




!§§%¡$°§ø™™%

U. (@5=C5=?8JLIJH, &E.*9%83 (01(,(µ01+, FIR <96/"@, $0% 82µ%/% T0%µ#"<@834

T' E4O56CJ;467 E45C;4S7, *5@984=C7µ4J *5C;M@ 1986

$. &5IJL8C=>O2H, &2µ%/%, &)8/2µ%/% 1%0 >6$#"05µ(0, !6O?=94H E>5LIJH, 1993.

•!¡$°§ø™™%

V. Anastassopoulos and T. Deliyannis, Method to design FIR Hilbert transformers

for delta modulated signals, IEE Proc. G, Circuits, Devices and Systems, 1986,

133, (2), pp. 113–114.

V. Anastassopoulos and T. Deliyannis, Efficient implementation of Nth–band FIR

filters based on a simple window method, IEE Proc. G, Circuits, Devices and

Systems, 1990, 137, (4), pp. 302–308

V. Anastassopoulos, N. Kouvaras and T. Deliyannis, Method for designing half–band

delta–modulation FIR filters, IEE Proc. F, Commun., Radar and Signal Process.,

1985, 132, (1), pp. 13–17

F. Harris, On the use of windows for harmonic analysis with the discrete Fourier

transform, Proc. IEEE, No. 1, pp. 51–83, January 1978.

E.C. Ifeachor and B.W. Jervis, Digital Signal Processing, A Practical Approach,

Second edition, Pearson Education Limited, 2002.

MATLAB, Signal processing toolbox, MathWorks, 1994.

F. Mintzer, On Half–band, Third–band and Nth–band FIR filters and their design,

ASSP 30, No. 5, pp. 734–738, October 1982.

T.W. Parks and J.H. McClellan, A Program for the Design of Linear Phase Finite

Impulse Response Filters, IEEE Trans. Audio Electroacoustics, Vol. AU–20, pp.

195–199, No. 3, Aug. 1972.

L.R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing,

Englewood Cliffs NJ, Prentice–Hall, 1975.

$¢%°$™ ¶!ƒ"#&!ƒø ª!§!&%™

U. (@5=C5=?8JLIJH, >,?6)83 1%0 :'.F."$%89% N3<0%1+, 83µ?/@,, +2µ94M=94H

*5@984=C2µ>JL *5C;M@, 1998.


1 6 3" ¶$¢$™% "°°§#'ø¡ $ƒø¡ ™&%¡ !§§%¡#'%

)J 693BI54J 4 5@53Q;9C54 89;4==?C9;J 5@5ILC46B =C5 FIR 3>IC;5.

MATLAB, Signal processing toolbox, MathWorks, 1994. )J MATLAB µ9 CJ 5@C>-

=CJ4TJ 9<T94;>O4J CJL Toolbox, 58JC9I9> CJ@ 65IKC9;J C;?8J 894;5µ5C4=µJK =C2

=T9O>5=2 CA@ FIR, 65GMH 654 =C2 µ9IQC2 C2H =Lµ89;43J;BH CJLH =9 O4B3J;9H

89;48CM=94H.


Averager &K6IAµ5 µQ=2H C4µ7H

Causal (4C45C?

Delay line 0;5µµ7 65GL=CQ;2=2H

Digital differentiator 123456?H O453J;4=C7H

Finite Impulse Response *989;5=µQ@2 6;JL=C467 58?6;4=2

Frequency sampling method ,QGJOJH O94<µ5CJI2R>5H =C2 =LT@?C2C5

Half band #µ>=945H ZM@2H

Linear Phase Resonse (8?6;4=2 <;5µµ467H 3B=2H

Optimal method UQIC4=C2 µQGJOJH

Pass band WM@2 O4QI9L=2H

Stop band WM@2 58?;;4R2H (58J6J87H)

Third band WM@2H 9@?H C;>CJL

Transversal Yµ9=2

Windows method ,QGJOJH 85;5GK;A@


æËÊÈ·Î¿ Ê›ÏÙÚ· ¿!ÂÈÚË˜ ÎÚÔ˘ÛÙÈÎ‹˜ ·!fiÎÚÈÛË˜ (IIR)

™ÎÔ!fi˜

&/( 1.<?6%0( %)/# µ.6./?µ. /% H3<0%1? <96/"% ?'.0"34 1"()8/0124 %'#1"0834. &/%

<96/"% %)/?, $0% /(, )'(6($08µ# /34 .F#*(), E"380µ('(0(7µ. /0µC4 /34 .08#*(),

1%5+4 1%0 /0µC4 /34 .F#*() 8. '"(3$(7µ.,.4 E"(,01C4 8/0$µC4. :'(µC,@4, %,%<.-

"#µ%8/. 8. .'%,%63'/01? <96/"%. ;% <96/"% %)/? '%"()80?L(), µ01"#/."3 )'(6($0-

8/012 '(6)'6(1#/3/% C,%,/0 /@, %,/98/(0E@, FIR <96/"@,, %66? CE(), µ3 $"%µµ0-

12 %'#1"083 <?834. &/#E(4 µ%4 8/( 1.<?6%0( %)/# .9,%0 3 µ.6C/3 /@, µ.5#*@, 8E.-

*9%834 /@, H3<0%1+, IIR <96/"@, 1%0 /@, /"#'@, )6('(93824 /()4.


J/%, (6(163"+8./. /3 µ.6C/3 /() 1.<%6%9() %)/(7 5% .98/. 8. 5C83 ,%:

• &E.*0?L./. IIR <96/"% µ. I?83 /3 $"%<012 µC5(*(, /3 µC5(*( %µ./?I63/34 1"()-

8/0124 %'#1"0834 1%0 /3 µC5(*( /() *0$"%µµ01(7 µ./%8E3µ%/08µ(7

• &E.*0?L./. ('(0(*2'(/. /7'( <96/"() (I%5)'."%/#, )H3'."%/#, *0C6.)834 L+,34

1%0 %'#""0H34 L+,34)

• M.0+,./. /% 8<?6µ%/% 6#$@ /() '.'."%8µC,() µ21()4 6CF34, '"%$µ%/('(0+,/%4

/% <96/"% @4 *(µC4 E%µ3624 /?F34 8. 8.0"? 2 '%"?6636%


• A,%6($01? <96/"%

• M./%8E3µ%/08µ(9 8)E,#/3/%4

• D"%<012 µC5(*(4 8E.*9%834

• MC5(*(4 %µ./?I63/34 1"()8/0124

• T0$"%µµ01#4 µ./%8E3µ%/08µ#4

• &/"CI6@83 8)E,#/3/%4

• U%0,#µ.,% '.'."%8µC,() µ21()4 6CF34

• O"0%1(9 1716(0

5! " º#§#$%


1 6 6 K E º A § A I O 5 : æ % º # " ' " º # § & ƒ" " ¶ ! # ƒ % ™ ' ƒ $ À ™ & # ' % ™ " ¶ $' ƒ # ™ % ™ ( I I R )


D,@"98%µ. /% <96/"% ?'.0"34 1"()8/0124 %'#1"0834 (IIR, Infinite Impulse Response)

8/3, :,#/3/% 3.4 /() =.<%6%9() 3. :9*%µ. #/0 3 8),?"/383 µ./%<("?4 .,#4 /C/(0-

() <96/"() *9,./%0 %'# /3 8EC83:

(5.1)

#'() ai, bi (0 8),/.6.8/C4 /() <96/"() 1%0 /()6?E08/(, C,%4 %'# /()4 8),/.6.8/C4 bi

.9,%0 *0?<("(4 /() µ3*.,#4. P %"05µ#4 R '"(8*0("9L.0 /3, /?F3 /() <96/"(), /(

('(9( $0% ,% .9,%0 %0/0%/# '"(Y'(5C/.0 #/0 R ( M. D0% ,% .9,%0 C,% /C/(0( <96/"(

.)8/%5C4 5% '"C'.0 #6(0 (0 '#6(0 /() ,% I"981(,/%0 8/( .8@/."01(7 /() µ(,%*0%9()

1716() 8/( .'9'.*(–z. O %,/98/(0E3 .F98@83 *0%<("+, /() <96/"() .9,%0:

(5.2)

#'() x(n) 1%0 y(n) (0 %1(6()59.4 .08#*() 1%0 .F#*(), %,/98/(0E%.

:9*%µ. #/0 $0% /3, )6('(9383 .,#4 <96/"() E".0%L#µ%8/. '(66%'6%80%8/C4, %5"(0-

8/C4 1%0 8/(0E.9% 1%5)8/C"3834 (&E2µ% 3.4), 1%0 %,%<."521%µ. 8/04 *0%<("./01C4

*(µC4 '"%$µ%/('(93824 /() (&E2µ% 3.5), 1%5+4 1%0 8/04 *(µC4 '"%$µ%/('(93834

<96/"@, )H3624 /?F34 @4 8),*)%8µ(7 *(µ+, '"+/34 1%0 *.7/."34 /?F34 8. 8.0"?

2 '%"?6636% (&E2µ% 3.6).

>'# /3 8EC83 (5.2) $9,./%0 <%,."# #/0 /% IIR <96/"% .9,%0 .'%,%63'/01? 2 %,%*"(-

µ01? (recursive), µ. /3, C,,(0% #/0 *.9$µ%/% /34 .F#*() E"380µ('(0(7,/%0 %'# /(

878/3µ% $0% /(, )'(6($08µ# /@, ,C@, /0µ+, /34 .F#*() 8. .'#µ.,.4 E"(,01C4 8/0$-

µC4. ;( $.$(,#4 %)/# '%"()80?L.0 µ.$?6( '6.(,C1/3µ% %'# ?'(H3 )'(6($08/0124

'(6)'6(1#/3/%4, 8. 8EC83 µ. /% FIR <96/"%. M. ?66% 6#$0%, $0% ,% .'0/7E()µ. µ0%

.'05)µ3/2 %'#1"083 E".0%L#µ%8/. 83µ%,/01? 60$#/."()4 8),/.6.8/C4 $0% C,% IIR <96-

/"( 8. 8EC83 µ. /( %,/98/(0E( FIR <96/"(. >'# /3, ?663 '6.)"? #µ@4, )'?"E(), *7(

8(I%"? µ.0(,.1/2µ%/%: (%) /% IIR <96/"% .9,%0 %8/%52, %, (0 8),/.6.8/C4 bi *., CE(),

.'06.$.9 8@8/?, *36%*2 %, (0 '#6(0 /34 8),?"/3834 µ./%<("?4 I"981(,/%0 .1/#4 /()

µ(,%*0%9() 1716() (I6. .,#/3/% 3.4), 1%0 (I) /% IIR <96/"% *., CE(), $"%µµ012 %'#-

1"083 <?834 8/3 L+,3 *0C6.)834, #'@4 CE(), /% µ3 .'%,%63'/01? FIR <96/"% µ.

8)µµ./"012 2 %,/08)µµ./"012 1"()8/012 %'#1"083.

y n a x n i b y n ii

i

M

i

i

N

( ) ( ) ( )= # # #= =5 5

0 1

H z

a z

b z

ii

i

M

ii

i

N( ) =

+

#

=

#

=

5

50

1

1


;( 17"0( %,/01.9µ.,( /() '%"#,/(4 1.<%6%9() .9,%0 3 8E.*9%83 IIR <96/"@,, 2 µ.

?66% 6#$0%, ( )'(6($08µ#4 /@, 8),/.6.8/+, ai, bi, #/%, µ%4 *9,(,/%0 (0 '"(*0%$"%-

<C4 /() <96/"(). KC$(,/%4 «'"(*0%$"%<C4 /() <96/"()», .,,((7µ. 1)"9@4 /3, .'0-

5)µ3/2 %'#1"083 8/3 8)E,#/3/% (&E2µ%/% 3.8 1%0 3.9). ;( 5Cµ% /34 8E.*9%834 /C/(0-

@, <96/"@, /( %,/0µ./@'98%µ. 2*3 8/3, :,#/3/% 3.4 µ. /3, /('(5C/383 /@, '#6@,

1%0 /@, µ3*.,01+, 8/( .'9'.*(–z. >)/2 .9,%0 3 '"+/3 µC5(*(4 8E.*9%834, 3 ('(9%

#µ@4 *., .9,%0 8)8/3µ%/012, .0*01? $0% <96/"% )H3624 /?F34. G/80, .1/#4 %'# %)/2,

8/( '%"#, 1.<?6%0( 5% $,@"98()µ. 1%0 *7( ?66.4 µ.5#*()4: /3 µC5(*( /34 %µ./?-

I63/34 1"()8/0124 %'#1"0834 1%0 /3 µC5(*( /() *0$"%µµ01(7 µ./%8E3µ%/08µ(7. P0

µC5(*(0 %)/C4 I%89L(,/%0 8/3 µ./%/"('2 .,#4 %,%6($01(7 <96/"() 8. H3<0%1# <96-

/"(. M0% /C/(0% '"(8C$$083 E"380µ('(0.9/%0 .)"C@4 $0% '(66(74 6#$()4 1%5+4: (%)

(0 /.E,01C4 8E.*9%834 %,%6($01+, <96/"@, .9,%0 '(67 %,.'/)$µC,.4, (I) 1%/%62$(),

8),25@4 8. 678.04 16.08/24 µ("<24, ($) *0%/95.,/%0 C/(0µ(0 '9,%1.4 $0% /3 8E.*9%83

%,%6($01+, <96/"@,, 1%0 (*) 8. '(66C4 .<%"µ($C4 %'%0/(7,/%0 H3<0%1? <96/"% /%

('(9% ,% '"(8(µ(0+,(), $,@8/? %,%6($01? <96/"%. G/80, %"E9L()µ. /( 8E.*0%8µ#

.,#4 H3<0%1(7 <96/"() %'# /( «%,/98/(0E(» %,%6($01# 1%0 8/3 8),CE.0% 1?,()µ.

/3, 1%/?66363 µ./%/"('2 /() 8. H3<0%1#. D0% /3, 1%67/."3 1%/%,#383 /@, 8E./0-

1+, .,,(0+,, 5% 1?,()µ. µ0% µ01"2 %,%<("? 8/% %,%6($01? <96/"%, 8/3, %µC8@4

.'#µ.,3 .,#/3/%. &/3 8),CE.0%, 5% $,@"98()µ. /04 *0?<(".4 µ.5#*()4 8E.*9%834 /@,

IIR <96/"@, (:,#/3/% 5.2) 1%0 5% 8)L3/28()µ. $0% 5Cµ%/% 8E./01? µ. /3, )6('(93-

82 /()4 (:,#/3/% 5.3).

1 6 7E # ™ " ° ø ° # ' ! ™ ¶ " ƒ"& % ƒ % ™ ! # ™



5.1 "Ó·ÏÔÁÈÎ¿ Ê›ÏÙÚ·

\8AH G5 OJKµ9 =C2@ 98?µ9@2 9@?C2C5, C5 IIR R23456B 3>IC;5 µ8J;JK@ 9K6JI5 @5

=T9O45=CJK@ 5;T>ZJ@C5H 58? Q@5 5@5IJ<46? 3>IC;J 654 65C?84@ T;2=4µJ8J4M@C5H

65CBII2I2 58946?@4=2 CJL 9848QOJL–s =CJ 98>89OJ–z. PC=4, 2 =T9O>5=2 9@?H R2345-

6JK 3>IC;JL 5@B<9C54 JL=45=C46B =C2 =T9O>5=2 CJL 65CBII2IJL 5@5IJ<46JK 3>IC;JL

654 =C2@ µ9CQ894C5 µ9CBS5=2 58? CJ H(s) =CJ H(z), QC=4 M=C9 C5 984GLµ2CB T5;5-

6C2;4=C46B CJL 5@5IJ<46JK 3>IC;JL @5 O45C2;JK@C54 65CB CJ@ 65IKC9;J OL@5C? C;?8J.

# =T9O>5=2 5@5IJ<46M@ 3>IC;A@ 9>@54 Q@5H CJµQ5H <45 CJ@ J8J>J L8B;T94 8IJK=45

S4SI4J<;53>5. +C2@ 85;JK=5 9@?C2C5 G5 89;4<;BRJLµ9 =9 =L@CJµ>5 C5 T5;56C2;4-

=C46B CA@ 8IQJ@ <@A=CM@ S5GL89;5CM@ (lowpass) 5@5IJ<46M@ 3>IC;A@. (6JIJK-

GAH, G5 5@53Q;JLµ9 CJLH µ9C5=T2µ5C4=µJKH =LT@?C2C5H µ9 CJLH J8J>JLH Q@5 5@5-

IJ<46? S5GL89;5C? 3>IC;J µ9C5C;Q89C54 =9 LR289;5C? (highpass), ZA@JO45S5C?

(bandpass) 7 3>IC;J 58?;;4R2H ZM@2H (bandstop).

*;4@ ?µAH 8;JTA;7=JLµ9, 5H OJKµ9 C2 <9@467 µJ;37 C2H 58?6;4=2H =LT@?C2C5H

9@?H 5@5IJ<46JK S5GL89;5CJK 3>IC;JL, M=C9 @5 9:J4694AGJKµ9 µ9 C4H Q@@J49H 654

CJLH =LµSJI4=µJKH. # 58?6;4=2 =LT@?C2C5H 9@?H 5@5IJ<46JK 3>IC;JL (5@5IJ<46JK

=L=C7µ5CJH) 8;J6K8C94 58? C2 =L@B;C2=2 µ9C53J;BH H(s) 5LCJK <45 C4µQH CJL s

8B@A =CJ@ B:J@5 CA@ 35@C5=C46M@ jQ, O2I5O7:

(5.3)

)J µQC;J C2H 65@J@46J8J42µQ@2H 58?6;4=2H =LT@?C2C5H (normalised magnitude

response) 9@?H µ2 4O5@46JK 5@5IJ<46JK S5GL89;5CJK 3>IC;JL 35>@9C54 =CJ +T7µ5

5.1. # µJ;37 C2H 58?6;4=2H 9>@54 85;?µJ45 µ9 969>@2 CJL +T7µ5CJH 3.9 <45 C2@ 89;>-

8CA=2 CA@ R23456M@ 3>IC;A@.

# 58?6;4=2 5LC7 J@JµBZ9C54 «65@J@46J8J42µQ@2», 9894O7 2 µQ<4=C2 C4µ7 CJL µQC;JL

=C2 ZM@2 O4QI9L=2H 9>@54 >=2 µ9 C2 µJ@BO5 (7 >=2 µ9 0 dB). # 6LµBCA=2 (ripple)

=C2 ZM@2 O4QI9L=2H =LµSJI>Z9C54 µ9 654 5@C48;J=A89K94 C2@ 9IBT4=C2 C4µ7

CJL µQC;JL =C2 ZM@2 O4QI9L=2H. # µQ<4=C2 C4µ7 C2H 6LµBCA=2H =C2 ZM@2 58J6J-

87H =LµSJI>Z9C54 µ9 1/>. # C4µ7 5LC7 =9 dB 4=JKC54 µ9 –20 log(1/>). '4 =LT@?C2-

C9H Qp 654 Qs 58JC9IJK@ C4H =LT@?C2C9H =CJ #"0( /34 L+,34 *0C6.)834 (passband edge

frequency) 654 =CJ #"0( /34 L+,34 %'(1('24 (stopband edge frequency), 5@C>=CJ4-

T5. ' I?<JH CA@ OKJ 5LCM@ =LT@JC7CA@ J@JµBZ9C54 6#$(4 µ./?I%834 (transition

ratio) 7 '%"?µ./"(4 .'06.1/01#/3/%4 (selectivity parameter) 654 =LµSJI>Z9C54 =L@7-

1

1 + 92

H j H s

s j( Q

Q) = ( )

=


GAH AH k, O2I5O7 . 045 Q@5 S5GL89;5C? 3>IC;J 4=TK94 k < 1. )QIJH, J I?<JH

J@JµBZ9C54 '%"?µ./"(4 *0%1"0/#/3/%4 (discrimination parameter) 654

=LµSJI>Z9C54 AH k1, O2I5O7 . +L@7GAH 4=TK94 k1<<1.

5.1.1 °ÓˆÛÙ¿ ‚·ı˘!ÂÚ·Ù¿ ·Ó·ÏÔÁÈÎ¿ Ê›ÏÙÚ·

+9 ?I9H C4H 89;48CM=94H CA@ 3>IC;A@ 8JL G5 5@53Q;JLµ9 5µQ=AH CM;5, J =C?TJH

µ5H G5 9>@54 J 9:7H: @5 8;J=9<<>=JLµ9 65CB CJ@ 65IKC9;J OL@5C? C;?8J C2@ 58?-

6;4=2 9@?H 4O5@46JK 3>IC;JL CJ J8J>J QT94 6Q;OJH >=J µ9 1 =9 ?I2 C2 ZM@2 O4QI9L-

=2H, 6Q;OJH >=J µ9 0 =9 ?I2 C2 ZM@2 58J6J87H, 654 2 ZM@2 µ9CBS5=2H 9>@54 µ2O9-

@46JK 9K;JLH, O2I5O7 O9@ L8B;T94 ZM@2 µ9CBS5=2H. )5 84J <@A=CB S5GL89;5CB 5@5-

IJ<46B 3>IC;5 9>@54 C5 56?IJLG5:

M4$(C1 Butterworth. )5 S5GL89;5CB 3>IC;5 Butterworth QTJL@ µ?@J 8?IJLH 654

CJ µQC;J C2H 58?6;4=2H =LT@?C2C5H O>@9C54 58? C2 =TQ=2:

(5.4)

H jN N

( )/ /

Q

Q Q Q Q

=+ ( )*

+-.

=+ ( )*

+-.

1

1

1

12

1 22 2

1 2

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k

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.

A2 1#

k =

Q

Qp

s

1 6 95 . 1 " ¡ " § $ ° # ' " º # § & ƒ"

|H(jQ)|

ZM@2[O4QI9L=2H

ZM@2[58J6J87H

ZM@2[µ9CBS5=2H

1

00

1A

11+.2

Qp Qs

Q

™¯‹Ì· 5.1

=%,(,01('(03µC,3%'#1"083 8)E,#-/3/%4 I%5)'."%-/(7 %,%6($01(7<96/"().



?8JL R 2 CB:2 CJL 3>IC;JL, Qc 2 =LT@?C2C5 58J6J87H (cut–off frequency), Qp 2

=LT@?C2C5 =CJ ?;4J C2H ZM@2H O4QI9L=2H 654 CJ µQC;J C2H 58?6;4=2H

=LT@?C2C5H =CJ ?;4J C2H ZM@2H O4QI9L=2H. *5;5C2;JKµ9 ?C4:

5. 045 Q = Qc CJ µQC;J C2H 58?6;4=2H =LT@?C2C5H 4=JKC54 µ9 , O2I5O7

, 5@9:B;C2C5 58? C2@ C4µ7 CJL R. (LC? =2µ5>@94 ?C4 2 4=TKH

CJL =7µ5CJH L8JO48I5=4BZ9C54 <45 C2 =L<696;4µQ@2 =LT@?C2C5. # C4µ7 5LC7

963;5=µQ@2 =9 decibels (dB) 4=JKC54 µ9 –3, O2I5O7:

.

S. 045 Q = 0 CJ µQC;J C2H 58?6;4=2H 4=JKC54 µ9 1, O2I5O7 µ9 0 dB, 5@9:B;C2C5 58?

C2@ C4µ7 CJL R.

<. !894O7 2 85;B<A<JH CJL µQC;JL C2H 58?6;4=2H 9>@54 8B@CJC9 5;@2C467 <45 G9C4-

6QH C4µQH CJL Q, =L@98B<9C54 ?C4 2 58?6;4=2 =LT@?C2C5H µ94M@9C54 µJ@JCJ@46B

65GMH CJ Q 5L:B@9C54. ,9 BII5 I?<45 <45 0 ] Q1 < Q2.

045 =LT@?C2C5 58J6J87H Qc >=2 µ9 1 rad/sec (Qc = 1) 85>;@JLµ9 58? C2 =TQ=2 (5.4)

CJ J@Jµ5Z?µ9@J «8;AC?CL8J S5GL89;5C? 3>IC;J» (prototype lowpass filter). '4 <;5-

346QH 85;5=CB=94H CJL µQC;JL C2H 58?6;4=2H =LT@?C2C5H CJL 65@J@46J8J42µQ@JL

Butterworth S5GL89;5CJK 3>IC;JL µ9 Qc = 1 (8;AC?CL8JL) <45 O453J;9C46QH C4µQH

CJL R, 35>@9C54 =CJ +T7µ5 5.25. *5;5C2;JKµ9 ?C4 5K:2=2 C2H CB:2H R CJL 3>IC;JL

JO2<9> =9 S9IC>A=2 CA@ ZA@M@ O4QI9L=2H 654 58J6J87H µ9 C5LC?T;J@2 µ9>A=2 C2H

ZM@2H µ9CBS5=2H.

'4 OKJ 85;Bµ9C;J4 J4 J8J>9H T5;56C2;>ZJL@ 8I7;AH Q@5 3>IC;J Butterworth 9>@54 2

=LT@?C2C5 58J6J87H Qc =C5 –3 dB 654 2 CB:2 CJL 3>IC;JL R (SI. =TQ=2 5.4). (LCQH

J4 85;Bµ9C;J4 µ8J;JK@ @5 8;J=O4J;4=CJK@ 58? C4H 8;JO45<;53QH CJL 3>IC;JL, ?8AH

5LCQH 35>@J@C54 =CJ +T7µ5 5.1, O2I5O7 CJ ?;4J C2H ZM@2H O4QI9L=2H Qp, C2@ 9IBT4-

=C2 C4µ7 CJL µQC;JL =C2 ZM@2 O4QI9L=2H , CJ ?;4J C2H ZM@2H 58J6J87H

Qs 654 C2 µQ<4=C2 C4µ7 CJL µQC;JL =C2 ZM@2 58J6J87H 1/>. PC=4, 58? C2@ 9:>=A=2

(5.4) QTJLµ9:

1 1/ + 92

H j H j( ) ( )Q Q2 1<

H j H j

dB( ) log ( ) log .Q Q= =

*+,

-./

= # T #20 201

23 0103 310 10 dB

H j( )Q = 1

2

1

2

1

1 2+ .


(5.55)

(5.5S)

"K@J@C5H CJ =K=C2µ5 5LC? CA@ 9:4=M=9A@ AH 8;JH C2@ CB:2 R CJL 3>IC;JL 65C5-

I7<JLµ9 =C2 =TQ=2:

(5.6)

!:L856JK9C54 ?C4 2 C4µ7 CJL R 8;Q894 @5 9>@54 56Q;54JH 5;4Gµ?H. +C2@ 89;>8CA=2

?8JL 2 C4µ7 8JL L8JIJ<>Z9C54 58? C2@ 85;58B@A =TQ=2 O9@ 9>@54 56Q;54JH, C?C9

5LC7 =C;J<<LI9K9C54 =CJ@ 5µQ=AH µ9<5IKC9;J 56Q;54J.

)QIJH, ?8AH 5@53Q;5µ9 654 =C2@ 5;T7 C2H 85;5<;B3JL, C5 3>IC;5 Butterworth

QTJL@ µ?@J 8?IJLH. (LC? 35>@9C54 58? C2 =TQ=2 (5.4). '4 GQ=94H CA@ 8?IA@ 8B@A

N

A kks p

=#[ ]

( ) =12

1 11

102 2

10

10 1

10

log ( ) /

log /

log ( / )log ( / )

.

Q Q

H jA

s

s cN

( )/

QQ Q

2

2 2

1

1

1=+ ( )

=

H j p

p cN

( )/

QQ Q .

2

2 2

1

1

1

1=

+ ( )=

+

1 7 15 . 1 " ¡ " § $ ° # ' " º # § & ƒ"

MQC;J

MQC;J

MQC;J

MQC;J

1

0,8

0,6

0,4

0,2

0

1

0,8

0,6

0,4

0,2

0

1

0,8

0,6

0,4

0,2

0

1

0,8

0,6

0,4

0,2

00 0,5 1,5 2 32,51

0 0,5 1,5 2 32,51 0 0,5 1,5 2 32,51

0 0,5 1,5 2 32,51

N=2

N=2

N=4

N=4

N=5 N=7 N=3

N=3N=3

N=8

N=10

K5@J@46J8J42µQ@2 +LT@?C2C5

K5@J@46J8J42µQ@2 +LT@?C2C5

K5@J@46J8J42µQ@2 +LT@?C2C5

K5@J@46J8J42µQ@2 +LT@?C2C5

(5)

(S) (O)

(<)

™¯‹Ì· 5.2

>'#1"083 8)E,#-/3/%4 I%5)'."%-/+, %,%6($01+,<96/"@, /7'() (%)Butterworth, (I)Chebyshev /7'()V, ($) Chebyshev/7'() VV 1%0 (*).66.0'/01(7.



=CJ µ4<5O46? 98>89OJ µ8J;JK@ @5 L8JIJ<4=CJK@ AH 9:7H:

(3JK <45 s = jQ, 2 9:>=A=2 (5.4) <;B39C54 AH:

(5.7)

'4 8?IJ4 C2H H(s)H(–s) S;>=6J@C54 8B@A =9 Q@5 6K6IJ 56C>@5H Qc 654 =9 =2µ9>5 8JL

4=58QTJL@ CJ Q@5 58? CJ BIIJ. (LC? 8;J6K8C94 58? C2@ 9:>=A=2 (5.7) µ9 µ2O9@4-

=µ? CJL 85;5@Jµ5=C7 C2H, O2I5O7:

(5.8)

654 =L@98MH

(5.9)

+CJ +T7µ5 5.3 O9>T@J@C54 J4 GQ=94H CA@ 8?IA@ O453?;A@ 3>IC;A@ Butterworth CB:2H

R = 1, R = 2 654 R = 3. '4 8?IJ4 8JL S;>=6J@C54 =CJ 5;4=C9;? 2µ498>89OJ 9>@54 5LCJ>

8JL 5@C4=CJ4TJK@ =C2@ H(s), 9@M 969>@J4 CJL O9:4JK 2µ49848QOJL 5@C4=CJ4TJK@ =C2@

H(–s). F89@GLµ>Z9C54 ?C4 <45 @5 9>@54 Q@5 5@5IJ<46? =K=C2µ5 9L=C5GQH, 8;Q894 J4

8?IJ4 CJL @5 S;>=6J@C54 =CJ 5;4=C9;? 2µ498>89OJ.

s e e q Nq cj j q N= = #+Q 8/ ( )8/ , , ,....2 2 1 2 0 1 1

# = # = = #+se q N

c

N j q N2

21 2 11 0 1 1

Q( ) , , ,..../ ( )8/

H s H ss c

N( ) ( )

/# =

+ #( )1

1 2 2Q

H s H s H j( ) ( ) ( )# = Q2

8 8 8

jQ jQ jQ*?IJH C2H[

H(s)*?IJ4 C2H[

H(s)*?IJ4 C2H[

H(s)

KK6IJH[56C>@5H Qc

KK6IJH[56C>@5H Qc

KK6IJH[56C>@5H Qc

N=1 N=2 N=3

(5) (S) (<)

'/4 '/3

™¯‹Ì· 5.3

-C8.04 /@, '#6@,I%5)'."%/+, <96-/"@, Butterworth/?F34 (%) R = 1,(I) R = 2 1%0 ($)

R = 3.

!>@54 9K6JIJ @5 OJKµ9 ?C4 2 =L@B;C2=2 µ9C53J;BH CA@ 8;AC?CL8A@ (Qc = 1) S5GL-

89;5CM@ 3>IC;A@ Butterworth 8;MC2H, O9KC9;2H 654 C;>C2H CB:2H 9>@54 5@C>=CJ4T5:


, 654 .

¶·Ú¿‰ÂÈÁÌ· 5.1

F8JIJ<>=C9 C2@ CB:2 9@?H S5GL89;5CJK Butterworth 3>IC;JL CJ J8J>J 85;JL=4BZ94

9:5=GQ@4=2 1 dB =CJ 1 kHz 654 40 dB =C5 5 kHz.

=G/0

(8? C4H 8;JO45<;53QH 8JL µ5H O>@J@C54 654 µ9 SB=2 C2 <9@467 µJ;37 C2H 58?6;4-

=2H =LT@?C2C5H 9@?H S5GL89;5CJK 3>IC;JL (+T7µ5 5.1) QTJLµ9 ?C4:

, J8?C9 8;J6K8C94 92 = 0,2589

Fp = 1 kHz, J8?C9 Qp = 2'Fp = 2'1000 rad/s

, J8?C9 >2 = 104

Fs = 5 kHz, J8?C9 Qs = 2'Fs = 2'5000 rad/s.

(@C465G4=CM@C5H C4H C4µQH 5LCQH =C2@ 9:>=A=2 (5.6) S;>=6JLµ9 ?C4:

(3JK 2 CB:2 CJL 3>IC;JL 8;Q894 @5 9>@54 56Q;54JH 5;4Gµ?H, =C;J<<LI9KJLµ9 CJ 58J-

CQI9=µ5 5LC? =CJ@ 5µQ=AH µ9<5IKC9;J 56Q;54J, R = 4.

N

kk

A

s p

= =#( )[ ]

= =log ( / )log ( / )

log /

log ( / )log ( )

log( ),10 1

10

102 2

10

1011

1

238621

2 53 281

.

Q Q

10

14010 2

logA

dB*+,

-./

= #

10

1

1110log

+*+,

-./

= #92

dB

H s

s s s( ) =

+ + +1

2 2 13 2 H s

s s( ) =

+ +1

2 12 H s

s( ) =

+1

1

1 7 35 . 1 " ¡ " § $ ° # ' " º # § & ƒ"

F8JIJ<>=C9 C2@ CB:2 654 CJLH 8?IJLH 9@?H S5GL89;5CJK 3>IC;JL Butterworth CJ

J8J>J 85;JL=4BZ94 9:5=GQ@4=2 3 dB =C5 500 Hz 654 40 dB =C5 1000 Hz.


M4$(C1 Chebyshev. F8B;TJL@ OKJ CK8J4 3>IC;A@ Chebyshev. )5 3>IC;5 CK8JL %

9>@54 3>IC;5 µ?@J µ9 8?IJLH C5 J8J>5 85;JL=4BZJL@ JµJ4?µJ;32 6LµBCA=2 =C2

ZM@2 O4QI9L=2H 654 µJ@JCJ@467 =Lµ89;43J;B =C2 ZM@2 58J6J87H (+T7µ5 5.2S). )J

µQC;J C2H 58?6;4=2H =LT@?C2C5H CA@ S5GL89;5CM@ 3>IC;A@ Chebyshev CK8JL %

O>@9C54 58? C2 =TQ=2:



(5.10)

?8JL 9 2 85;Bµ9C;JH CJL 3>IC;JL 8JL =T9C>Z9C54 µ9 C2@ 6LµBCA=2 =C2 ZM@2 O4Q-

I9L=2H, Qp 2 =LT@?C2C5 =CJ ?;4J C2H ZM@2H O4QI9L=2H 654 ;$(x) 8JILM@LµJ

Chebyshev CB:2H R, CJ J8J>J J;>Z9C54 AH:

(5.11)

)5 3>IC;5 Chebyshev CK8JL %% 9>@54 3>IC;5 µ9 8?IJLH 654 µ2O9@46B, 85;JL=4BZJL@

µJ@JCJ@467 =Lµ89;43J;B =C2 ZM@2 O4QI9L=2H 654 JµJ4?µJ;32 6LµBCA=2 =C2 ZM@2

58J6J87H (+T7µ5 5.2<). )5 µ2O9@46B =C2@ 89;>8CA=2 5LCM@ CA@ 3>IC;A@ S;>=6J-

@C54 8B@A =CJ 35@C5=C46? B:J@5 CJL 9848QOJL–s. )J µQC;J C2H 58?6;4=2H =LT@?-

C2C5H O>@9C54 58? C2 =TQ=2:

5.12)

?8JL ;$(x) 9>@54 654 8BI4 CJ 8JILM@LµJ Chebyshev CB:2H R, 654 Qs 2 =LT@?C2C5

=CJ ?;4J C2H ZM@2H 58J6J87H.

-$$').()*F A4$(C1. )5 9II948C46B 3>IC;5 7 3>IC;5 Cauer 85;JL=4BZJL@ JµJ4?-

µJ;32 6LµBCA=2 C?=J =C2 ZM@2 O4QI9L=2H, ?=J 654 =C2 ZM@2 58J6J87H (+T7µ5

5.2O). )5 3>IC;5 5LCB QTJL@ 8?IJLH 654 µ2O9@46B 654 CJ µQC;J C2H 58?6;4=2H =C2

=LT@?C2C5 O>@9C54 58? C2 =TQ=2:

(5.13)

?8JL UN(x) 2 %56AS45@7 (Jacobian) 9II948C467 =L@B;C2=2 CB:2H R. )5 µ2O9@46B

S;>=6J@C54 8B@A =CJ@ B:J@5 CA@ 35@C5=C46M@ CJL 9848QOJL–s.

)5 9II948C46B 3>IC;5 9>@54 65IKC9;5 CA@ 3>IC;A@ Butterworth 654 Chebyshev, AH

8;JH CJ <9<J@?H ?C4 5854CJK@C54 3>IC;5 µ46;?C9;2H CB:2H, O2I5O7 I4<?C9;J4 =L@C9-

I9=CQH, <45 @5 465@J8J47=JL@ C4H >O49H 8;JO45<;53QH. ,9 BII5 I?<45, <45 O9OJµQ@2

CB:2 CJL 3>IC;JL 654 O9OJµQ@9H 8;JO45<;53QH, Q@5 9II948C46? 3>IC;J G5 85;JL-

=4BZ94 C2@ =C9@?C9;2 ZM@2 µ9CBS5=2H. (8? C2@ BII2 8I9L;B ?µAH, 2 58?6;4=2

3B=2H CA@ 9II948C46M@ 3>IC;A@ 85;JL=4BZ94 89;4==?C9;9H µ2 <;5µµ46?C2C9H =C2

ZM@2 O4QI9L=2H =9 =TQ=2 µ9 C5 5@C>=CJ4T5 3>IC;5 Butterworth 7 Chebyshev.

H jUN

( )Q. Q Q

=+ ( )( )

1

11 2

2p/

H jT TN N

( )/

/Q

. Q Q Q Q=

+ ( ) ( )[ ]( )1

1 2 21 2

2s p s/ /

T x

N x x

N x xN ( )

cos( cos ),

cosh( cosh ),=

01D

2D

#

#

1

1

| |<1

| |>1

H jTN

( )/

Q. Q Q

=+ ( )( )

1

1 21 2

2p/


5.1.2 ªÂÙ·Û¯ËÌ·ÙÈÛÌÔ› Û˘¯ÓfiÙËÙ·˜ ·Ó·ÏÔÁÈÎÒÓ Ê›ÏÙÚˆÓ

)5 3>IC;5 8JL <@A;>=5µ9 =C2@ 8;J2<JKµ9@2 L8J9@?C2C5 7C5@ ?I5 S5GL89;5CB.

(LCB 58JC9IJK@ C2 SB=2 <45 C2 =T9O>5=2 CA@ L8JIJ>8A@ 65C2<J;4M@ 3>IC;A@. 045

@5 =T9O4B=JLµ9 Q@5 LR289;5C?, ZA@JO45S5C? 7 58?;;4R2H ZM@2H 5@5IJ<46? 3>IC;J,

S5=4Z?µ5=C9 =CJ 5@C>=CJ4TJ 5@5IJ<46? S5GL89;5C? (CK8JL Butterworth, Chebyshev

7 !II948C46?) 654 96C9IJKµ9 Q@5 µ9C5=T2µ5C4=µ? =LT@?C2C5H. (LCJKH 56;4SMH CJLH

µ9C5=T2µ5C4=µJKH =LT@?C2C5H G5 <@A;>=JLµ9 =C2@ 85;JK=5 L8J9@?C2C5.

ª!&"&ƒ$¶% µ"£À¶!ƒ"&$À º#§&ƒ$À ™! µ"£À¶!ƒ"&$

!B@ QTJLµ9 Q@5 S5GL89;5C? 5@5IJ<46? 3>IC;J µ9 =LT@?C2C5 Qp =CJ ?;4J C2H ZM@2H

O4QI9L=2H, 654 GQIJLµ9 @5 CJ µ9C5C;QRJLµ9 =’ Q@5 BIIJ S5GL89;5C? 3>IC;J µ9

=LT@?C2C5 Q_p =CJ ?;4J C2H ZM@2H O4QI9L=2H, 96C9IJKµ9 CJ µ9C5=T2µ5C4=µ?:

(S5GL89;5C? =9 S5GL89;5C?) (5.14)

,9 BII5 I?<45, 9B@ Hp(s) 9>@54 2 =L@B;C2=2 µ9C53J;BH CJL 5;T46JK S5GL89;5CJK

3>IC;JL µ9 =LT@?C2C5 Qp =CJ ?;4J C2H ZM@2H O4QI9L=2H, C?C9 2 =L@B;C2=2 µ9C5-

3J;BH HLP(s) CJL @QJL S5GL89;5CJK 3>IC;JL G5 9>@54:

(5.15)

ª!&"&ƒ$¶% µ"£À¶!ƒ"&$À º#§&ƒ$À ™! Àæ%¶!ƒ"&$045 C2 µ9C5C;J87 9@?H S5GL89;5CJK 5@5IJ<46JK 3>IC;JL µ9 =LT@?C2C5 Qp =CJ ?;4J

C2H ZM@2H O4QI9L=2H, =9 LR289;5C? 3>IC;J µ9 =LT@?C2C5 Q_p =CJ ?;4J C2H ZM@2H

O4QI9L=2H, 96C9IJKµ9 CJ µ9C5=T2µ5C4=µ?:

(S5GL89;5C? =9 LR289;5C?) (5.16)

# =L@B;C2=2 µ9C53J;BH HHP(s) CJL LR289;5CJK 3>IC;JL 8;J6K8C94 58? C2@ 5@C>-

=CJ4T2 CJL 5;T46JK S5GL89;5CJK 3>IC;JL =Kµ3A@5 µ9 C2 =TQ=2:

(5.17)

H s H s

ss

HP pp

( ) ( )==

UQ Qp

s

spA @A

UQ Qp

H s H ss s

LP pp

( ) ( )==

UQ

Qp

s spA @A

UQ

Qp

1 7 55 . 1 " ¡ " § $ ° # ' " º # § & ƒ"



ª!&"&ƒ$¶% µ"£À¶!ƒ"&$À º#§&ƒ$À ™! (ø¡$¢#"µ"&$

045 C2 µ9C5C;J87 9@?H S5GL89;5CJK 5@5IJ<46JK 3>IC;JL µ9 =LT@?C2C5 Qp =CJ ?;4J

C2H ZM@2H O4QI9L=2H, =9 ZA@JO45S5C? 3>IC;J µ9 =LT@?C2C9H Ql 654 Qu =CJ 65CMC9;J

654 5@MC9;J ?;4J C2H ZM@2H O4QI9L=2H 5@C>=CJ4T5, 96C9IJKµ9 CJ µ9C5=T2µ5C4=µ?:

(S5GL89;5C? =9 ZA@JO45S5C?) (5.18)

,9 BII5 I?<45, 2 =L@B;C2=2 µ9C53J;BH HBP(s) CJL ZA@JO45S5CJK 3>IC;JL 8;J6K-

8C94 58? C2@ 5@C>=CJ4T2 CJL S5GL89;5CJK 3>IC;JL =Kµ3A@5 µ9 C2 =TQ=2:

(5.19)

(8? C2 =TQ=2 5LC7 <>@9C54 35@9;? ?C4 2 CB:2 CJL ZA@JO45S5CJK 3>IC;JL 8JL 8;J-

6K8C94 9>@54 O48IB=45 C2H CB:2H CJL 5;T46JK S5GL89;5CJK 3>IC;JL.

ª!&"&ƒ$¶% µ"£À¶!ƒ"&$À º#§&ƒ$À ™! "¶$ƒƒ#æ%™ (ø¡%™045 C2 µ9C5C;J87 9@?H S5GL89;5CJK 5@5IJ<46JK 3>IC;JL µ9 =LT@?C2C5 Qp =CJ ?;4J

C2H ZM@2H O4QI9L=2H, =9 3>IC;J 58?;;4R2H ZM@2H µ9 =LT@?C2C9H Ql 654 Qu =C5 ?;45

CA@ ZA@M@ O4QI9L=2H, 96C9IJKµ9 CJ µ9C5=T2µ5C4=µ?:

(S5GL89;5C? =9 58?;;4R2H ZM@2H) (5.20)

# =L@B;C2=2 µ9C53J;BH HBS(s) CJL 3>IC;JL 58?;;4R2H ZM@2H 8;J6K8C94 58? C2@

5@C>=CJ4T2 CJL S5GL89;5CJK 3>IC;JL µ9 SB=2 C2 =TQ=2:

(5.21)

¶·Ú¿‰ÂÈÁÌ· 5.2

$5 µ9C5=T2µ5C4=C9> CJ 8;MC2H CB:2H S5GL89;5C? 3>IC;J Butterworth µ9 =L@B;C2-

=2 µ9C53J;BH H(s) = Qp/(s + Qp), ?8JL Qp = 2' rad/s, =’ Q@5 ZA@JO45S5C? 3>IC;J

µ9 =LT@?C2C9H =C5 ?;45 C2H ZM@2H O4QI9L=2H Ql = ' rad/s 654 Qu = 3' rad/s.

H s H ss

s

s

BS p

pu l

u l

( ) ( ))

=#

=(

+Q

Q Q

Q Q2

s

s

sp

u l

u l

A @A #Q

Q Q

Q Q

(

+

)2

H s H s

sss

BP p

pu l

u l

( ) ( )

( )

=+

#= Q

Q QQ Q

2

s

ssp

u l

u l

A @A +#

QQ Q

Q Q

2

( )


=G/0

1 7 75 . 1 " ¡ " § $ ° # ' " º # § & ƒ"

|H(jQ)| |H(jQ)|

12 1

2

11

0 02' 3''Q Q

Qp Ql Qu

(5) (S)

™¯‹Ì· 5.4

MC/"( /34 %'#1"0-834 8)E,#/3/%4(%) /() %"E01(7I%5)'."%/(7 1%0(I) /() L3/(7µ.-,() L@,(*0%I%/(7%,%6($01(7 <96-/"().

)5 µQC;5 CA@ 58J6;>=9A@ =LT@?C2C5H CJL 5;T46JK S5GL89;5CJK 3>IC;JL

Butterworth 654 CJL Z2CJKµ9@JL ZA@JO45S5CJK 3>IC;JL, 35>@J@C54 =CJ +T7µ5 5.4.

!3?=J@ µ5H O>@9C54 2 =L@B;C2=2 µ9C53J;BH CJL S5GL89;5CJK 3>IC;JL, J L8JIJ<4-

=µ?H C2H =L@B;C2=2H µ9C53J;BH CJL Z2CJKµ9@JL ZA@JO45S5CJK 3>IC;JL <>@9C54

9K6JI5 µ9 SB=2 CJ µ9C5=T2µ5C4=µ? =LT@?C2C5H C2H =TQ=2H (5.18).

Y;5 5@C465G4=CM@C5H =C2 =TQ=2 (5.19) S;>=6JLµ9 C2 Z2CJKµ9@2 =L@B;C2=2 µ9C5-

3J;BH CJL ZA@JO45S5CJK 3>IC;JL:

*5;5C2;JKµ9 ?C4 2 CB:2 CJL 3>IC;JL 8JL 8;JQ6LR9 9>@54 O48IB=45 5LC7H CJL 5;T46JK

S5GL89;5CJK 3>IC;JL. !8>=2H, 9K6JI5 O4584=CM@JLµ9 ?C4 <45 s = 0 QTJLµ9 H(s) = 0.

H s H s

ss

ss

s

s

s sBP ( ) ( )

8

== + =

+*+,

-./

+=

+ +2 2

2 2 2 232

32

2

2 3))

))

) )

s

ss

ss

ssp

u l

u l

= ! +#

= + !#

= +Q

Q QQ Q

2 2 2 2

23

33

( ) ( )) ) )

) ))

# =L@B;C2=2 µ9C53J;BH CJL O9KC9;2H CB:2H 8;AC?CL8JL (Qc = 1) S5GL89;5CJK

3>IC;JL Chebyshev CK8JL % µ9 6LµBCA=2 =C2 ZM@2 O4QI9L=7H CJL >=2 µ9 1 dB,

O>@9C54 58? C2 =TQ=2 Hp(s) = 1/(s2 + 1,098s + 1,103). $5 8;J=O4J;4=C9> 2 =L@B;-

C2=2 µ9C53J;BH CJL 5@C>=CJ4TJL LR289;5CJK 3>IC;JL, CJ J8J>J QT94 =LT@?C2C5

58J6J87H 4 rad/s.





+C2@ 9@?C2C5 5LC7 <@A;>=5µ9 C5 5@5IJ<46B 3>IC;5, C5 J8J>5 9>@54 6L;>AH C9==B-

;A@ CK8A@: Butterworth, Chebyshev I & II 654 !II948C46B. *;?694C54 =C2@ JL=>5,

<45 C4H µ5G2µ5C46QH 969>@9H =L@5;C7=94H J4 J8J>9H, µ9 O453J;9C46? C;?8J 2 65G9-

µ>5, 8;J=9<<>ZJL@ C2@ 65µ8KI2 9@?H 4O5@46JK 3>IC;JL. 0@A;>=5µ9 98>=2H, CJLH

µ9C5=T2µ5C4=µJKH =C2 =LT@?C2C5 CJLH J8J>JLH T;2=4µJ8J4JKµ9 <45 @5 µ8J;Q-

=JLµ9 @5 8B;JLµ9 Q@5 984GLµ2C? S5GL89;5C?, LR289;5C?, ZA@JO45S5C? 7 58?;-

;4R2H ZM@2H 5@5IJ<46? 3>IC;J, QTJ@C5H AH 5@53J;B Q@5 8;AC?CL8J S5GL89;5C?

5@5IJ<46? 3>IC;J.

5.2 ™¯Â‰›·ÛË IIR „ËÊÈ·ÎÒÓ Ê›ÏÙÚˆÓ

\8AH 654 =C2@ 89;>8CA=2 CA@ FIR 3>IC;A@, QC=4 654 9OM, L8B;TJL@ 8JIIQH µQGJ-

OJ4 =T9O>5=2H IIR 3>IC;A@. # 8;MC2 654 84J 58I7 58? 5LCQH, µ5H 9>@54 7O2 <@A=C7

58? CJ &93BI54J 3, 654 9>@54 5LC7 C2H 589LG9>5H CJ8JGQC2=2H CA@ 8?IA@ 654 µ2O9-

@46M@ CJL 3>IC;JL 8B@A =CJ µ4<5O46? 98>89OJ–z, 8;J=O4J;>ZJ@C5H Bµ9=5 C2 =L@B;-

C2=2 µ9C53J;BH 654 65CB =L@Q8945, CJLH =L@C9I9=CQH CJL 3>IC;JL. # µQGJOJH 5LC7

QT94 89;4==?C9;J G9A;2C467 654 96854O9LC467 =2µ5=>5 85;B 8;56C467 5:>5. F8B;-

TJL@ 654 BII9H C9T@46QH, 65CB C4H J8J>9H 2 =T9O>5=2 CJL 3>IC;JL <>@9C54 589LG9>5H

=CJ 89O>J–z, ?8AH 2 µQGJOJH CA@ 8;J=9<<>=9A@ Pade 7 2 µQGJOJH CA@ 9I5T>=CA@

C9C;5<M@A@, C4H J8J>9H ?µAH O9 G5 8;5<µ5C9LCJKµ9.

*9;4==?C9;J =L@2G4=µQ@J 9>@54 @5 T;2=4µJ8J4JKµ9 C9T@46QH J4 J8J>9H µ9C5C;Q8JL@

Q@5 5@5IJ<46? 3>IC;J =9 R23456?. &54 9OM L8B;TJL@ O453J;9C46QH µQGJOJ4 µ9C5-

C;J87H 9@?H 5@5IJ<46JK 3>IC;JL =9 R23456?. !µ9>H G5 5@539;GJKµ9 =C4H OKJ 84J

<@A=CQH 58? 5LCQH, C2 µQGJOJ C2H 5µ9CBSI2C2H 6;JL=C467H 654 C2 µQGJOJ CJL

O4<;5µµ46JK µ9C5=T2µ5C4=µJK.

)QIJH, G5 8;Q894 @5 985@5IBSJLµ9 ?C4 LIJ8J47=4µ5 654 9L=C5G7 IIR 3>IC;5 O9 µ8J-

;9> @5 QTJL@ <;5µµ467 58?6;4=2 3B=2H. PC=4, J4 8;JO45<;53QH CA@ 3>IC;A@ G5 5@5-

3Q;J@C54 µ?@J =CJ µQC;J C2H 58?6;4=2H =C2 =LT@?C2C5. (LC? O9@ =2µ5>@94 ?C4 2

3B=2 O9@ 9>@54 JL=4MOJLH =2µ5=>5H. +2µ5>@94 58IMH 8AH, 53JK µQC;J 654 3B=2

9>@54 5II2IQ@O9C5, O9T?µ5=C9 C2 3B=2 ?8AH 5LC7 8;J6K8C94 58? C2 O45O465=>5 =T9-

O>5=2H. (@ 6B8J45 935;µJ<7 5854C9> <;5µµ467 3B=2, C?C9 G5 8;Q894 @5 =T9O4B=JLµ9

Q@5 FIR 3>IC;J µ9 =Lµµ9C;467 7 5@C4=Lµµ9C;467 6;JL=C467 58?6;4=2.

5.2.1 °Ú·ÊÈÎ‹ Ì¤ıÔ‰Ô˜

)2 <;53467 µQGJOJ =T9O>5=2H 3>IC;A@ C2 <@A;>=5µ9 =CJ &93BI54J 3 (9@?C2C5 3.4).

# µQGJOJH 5LC7 5@53Q;9C54 654 AH $.@µ./"012 7 AH µC5(*(4 8E.*9%834 I%808µC,3


8/()4 '#6()4 1%0 8/% µ3*.,01? . +Kµ3A@5 µ9 C2 µQGJOJ 5LC7, 2 =T9O>5=2 9@?H IIR

R23456JK 3>IC;JL <>@9C54 µ9 65CBII2I2 CJ8JGQC2=2 CA@ 8?IA@ 654 CA@ µ2O9@46M@

8B@A =CJ µJ@5O45>J 6K6IJ =CJ 98>89OJ–z. # CJ8JGQC2=2 9@?H 8?IJL 6J@CB =CJ µJ@5-

O45>J 6K6IJ JO2<9> =9 µ9<BI9H C4µQH <45 C2 =L@B;C2=2 µ9C53J;BH =9 =LT@?C2C9H 8JL

S;>=6J@C54 6J@CB =CJ@ 8?IJ. (@C>=CJ4T5, 2 K85;:2 9@?H µ2O9@46JK 6J@CB 7 654 8B@A

=CJ@ µJ@5O45>J 6K6IJ JO2<9> =9 8JIK µ46;QH C4µQH 7 654 µ2O9@4=µ? C2H =L@B;C2=2H

µ9C53J;BH 654 65C’ 98Q6C5=2 C2H 9:?OJL CJL 3>IC;JL (=L=C7µ5CJH). DLµ2G9>C9 ?C4,

<45 @5 9>@54 9L=C5GQH CJ 3>IC;J 8JL G5 =T9O45=C9>, G5 8;Q894 J4 8?IJ4 CJL @5 S;>-

=6J@C54 =CJ 9=AC9;46? CJL µJ@5O45>JL 6K6IJL. )5 µ2O9@46B µ8J;9> @5 S;>=6J@C54

J8JLO78JC9 =CJ 98>89OJ–z. )QIJH, J4 8?IJ4 654 C5 µ2O9@46B µ8J;9> @5 9>@54 8;5<µ5-

C46J> 7 µ4<5O46J> 5;4GµJ>. +C2@ 89;>8CA=2 ?µAH 8JL 6B8J4J4 58? CJLH 8?IJLH 7 C5

µ2O9@46B 9>@54 µ4<5O46J>, C?C9 5LCJ> G5 8;Q894 @5 9µ35@>ZJ@C54 AH Z9K<2 =LZL<M@.

)J *5;BO94<µ5 3.10 5@53Q;9C54 =CJ =T9O45=µ? 9@?H IIR 3>IC;JL µ9 C2 <;53467

µQGJOJ. (H 8;J=85G7=JLµ9 @5 C2 GLµ2GJKµ9 =T9O4BZJ@C5H C2 3J;B 5LC7 Q@5 58I?

ZA@JO45S5C? 3>IC;J.

1 7 95 . 2 ™ Ã !¢#"™% IIR æ%º#"'ø¡ º#§&ƒø¡

+T9O4B=C9 R23456? 3>IC;J CJ J8J>J @5 9:5=G9@>Z94 C4H =LT@?C2C9H 0 654 ' 654 @5

984C;Q894 C2 O4QI9L=2 CA@ =LT@JC7CA@ <K;A 58? C2 =LT@?C2C5 '/2.


5.2.2 ª¤ıÔ‰Ô˜ ·ÌÂÙ¿‚ÏËÙË˜ ÎÚÔ˘ÛÙÈÎ‹˜

+6J8?H µ5H =Kµ3A@5 µ9 C2 µC5(*( /34 %µ./?I63/34 1"()8/0124 (impulse invariance

method) 9>@54 @5 =T9O4B=JLµ9 Q@5 IIR 3>IC;J CJL J8J>JL 2 µJ@5O45>5 6;JL=C467

58?6;4=2 h(n) @5 4=JOL@5µ9> µ9 C2 O94<µ5CJI28C2µQ@2 6;JL=C467 58?6;4=2 9@?H

65CBII2I5 984I9<µQ@JL 5@5IJ<46JK 3>IC;JL, O2I5O7:

(5.22)

?8JL ; 2 89;>JOJH O94<µ5CJI2R>5H (SI. +T7µ5 5.55, S).

# =L@B;C2=2 µ9C53J;BH H(z) 4=JKC54 µ9 CJ@ µ9C5=T2µ5C4=µ?–z C2H O94<µ5CJI2-

8C2µQ@2H µJ@5O45>5H 6;JL=C467H 58?6;4=2H h(nT), O2I5O7:

H(z) = Z{h(nT)} (5.23)

# 5@5IJ<467 =L@B;C2=2 µ9C53J;BH H(s) 4=JKC54 µ9 CJ@ µ9C5=T2µ5C4=µ? Laplace

C2H 6;JL=C467H 58?6;4=2H h(t).

h n h nT h t

t nT

( ) ( ) ( )' ==

n = 0,1,2,....



+CJ =2µ9>J 5LC? G5 8;Q894 @5 9:9CB=JLµ9 CJ 58JCQI9=µ5 C2H O94<µ5CJI2R>5H C2H

5@5IJ<467H 6;JL=C467H 58?6;4=2H, O2I5O7 CJ 58JCQI9=µ5 C2H =TQ=2H (5.22). (H

GLµ2GJKµ9 ?C4, ?C5@ I5µSB@JLµ9 58? Q@5 5@5IJ<46? =7µ5 O9><µ5C5 µ9 ;LGµ? Fs =

1/; O94<µBCA@ 5@B O9LC9;?I98CJ, CJ 3B=µ5 CJL O94<µ5CJI28C2µQ@JL =7µ5CJH 8JL

8;J6K8C94 9>@54 µ45 89;4JO467 985@BI2R2 CJL 5@5IJ<46JK 3B=µ5CJH, µ9 89;>JOJ Fs.

+L@98MH, 5@ J ;LGµ?H O94<µ5CJI2R>5H 9>@54 µ46;?C9;JH 58? CJ O48IB=4J C2H µQ<4-

=C2H =LT@?C2C5H 8JL L8B;T94 =CJ 5@5IJ<46? 3B=µ5, C?C9 G5 85;JL=45=C9> CJ 354-

@?µ9@J CJL 5@985;6JKH ;LGµJK O94<µ5CJI2R>5H (aliasing), ?8AH 5LC? O9>T@9C54 =CJ

+T7µ5 5.5<. 0>@9C54 35@9;? ?C4 2 89;>JOJH O94<µ5CJI2R>5H ; G5 8;Q894 @5 9>@54

5;69CB µ46;7, <45 @5 58J3K<JLµ9 7 CJLIBT4=CJ@ @5 9I5T4=CJ8J47=JLµ9 CJ 354@?-

µ9@J 5LC?. !8>=2H, 65C5I5S5>@JLµ9 ?C4 2 µQGJOJH C2H 5µ9CBSI2C2H 6;JL=C467H O9@

9>@54 65CBII2I2 <45 C2 =T9O>5=2 LR289;5CM@ 3>IC;A@, 9:54C>5H CJL 354@JµQ@JL

aliasing 8JL 85;JL=4BZ9C54 =CJ 3B=µ5 I?<A C2H O94<µ5CJI2R>5H. 045 CJLH I?<JLH

5LCJKH, 2 µQGJOJH T;2=4µJ8J49>C54 6L;>AH <45 C2@ =T9O>5=2 S5GL89;5CM@ 654 ZA@J-

O45S5CM@ R23456M@ 3>IC;A@.

# 58946?@4=2 CA@ =2µ9>A@ 58? CJ 98>89OJ–s =CJ 98>89OJ–z µ9 C2@ 5µ9CBSI2C2 6;JL-

=C467 58?6;4=2 <>@9C54 µQ=A C2H =TQ=2H:

z = esT (5.24)

045 s = =+ jQ 654 z = rej@ S;>=6JLµ9 ?C4:

r = e=)

h(t)

h1(n)

h2(n)

(5)

(S)

(<)

t

n

n

T1

T2

|H(jQ)|

|H1(ej@)|

|H2(ej@)|

0 Q

@

@

'/T2 '/T1

0 ' 2'

0 ' 2' 3'

™¯‹Ì· 5.5

(%) ="()8/012%'#1"083 1%0

µC/"( <?8µ%/(4%,%6($01(7 <96-/"(), (I) 1%0 ($)

*.0$µ%/(63H9% /341"()8/0124 /()%,%6($01(7 µ.

'."9(*( ;1 < ;2


@ = Q; (5.25)

!8JµQ@AH, <45 = < 0 QTJLµ9 0 < r < 1, 9@M <45 = > 1 QTJLµ9 r > 1. \C5@ = = 0, C?C9

r = 1. (LC? =2µ5>@94 ?C4 CJ 5;4=C9;? 2µ498>89OJ CJL s 58946J@>Z9C54 =CJ 9=AC9;4-

6? CJL µJ@5O45>JL 6K6IJL =CJ z, 9@M CJ O9:> 2µ498>89OJ CJL s 58946J@>Z9C54 =CJ

9:AC9;46? CJL µJ@5O45>JL 6K6IJL =CJ z. !8>=2H, J B:J@5H jQ 58946J@>Z9C54 8B@A

=CJ µJ@5O45>J 6K6IJ CJL 9848QOJL–z. # 58946?@4=2 5LC7 ?µAH O9@ 9>@54 58946?@4-

=2 Q@5 8;JH Q@5. (3JK CJ @ 9>@54 µJ@5O46? =C2@ 89;4JT7 (–', '), 2 58946?@4=2 @

= Q; =L@98B<9C54 ?C4 2 89;4JT7 CA@ =LT@JC7CA@ –'/; ] Q ] '/; 58946J@>Z9C54 =C4H

5@C>=CJ4T9H C4µQH –' ] @ ] '. !848IQJ@, ?µAH, 654 2 89;4JT7 =LT@JC7CA@ '/; ] Q

] 3'/; 58946J@>Z9C54 98>=2H =C2@ >O45 89;4JT7 –' ] @ ] ', 654 <9@46B CJ >O4J =Lµ-

S5>@94 <45 C4H =LT@?C2C9H (2 k–1)8/; ] Q ] (2k + 1)8/;, ?8JL k 56Q;54JH. Y;5, 2

58946?@4=2 C2H 5@5IJ<467H =LT@?C2C5H Q =C2 =LT@?C2C5 @ CJL R23456JK TM;JL

9>@54 µ>5 8JIIB–=9–Q@5 58946?@4=2, <9<J@?H 8JL 5@C465CJ8C;>Z94 CJ 354@?µ9@J CJL

5@985;6JKH ;LGµJK O94<µ5CJI2R>5H. # ?I2 58946?@4=2 CJL 9848QOJL–s =CJ 98>89-

OJ–z 35>@9C54 =CJ +T7µ5 5.6.

1 8 15 . 2 ™ Ã !¢#"™% IIR æ%º#"'ø¡ º#§&ƒø¡

E8>89OJ - z

E8>89OJ - s jQ

MJ@5O45>JH[6K6IJH

3'T

3'T

'T

'T

8

-

-

z=esT™¯‹Ì· 5.6

O %'.01#,083 /().'0'C*()–s 8/(.'9'.*(–z 87µ<@-,% µ. /3 8EC83z = esT. K@"9*.4.7"()4 2'/; ($0%8 < 0) /() .'0'C-*()–s %'.01(,9L(-,/%0 8. 83µ.9% /()µ(,%*0%9()1716() /() .'0'C-*()–z.



¶·Ú¿‰ÂÈÁÌ· 5.3

(@5IJ<46? S5GL89;5C? 3>IC;J QT94 =L@B;C2=2 µ9C53J;BH .

$5 L8JIJ<4=C9> 2 =L@B;C2=2 µ9C53J;BH CJL R23456JK IIR 3>IC;JL CJ J8J>J 8;J-

6K8C94 µ9 C2 µQGJOJ C2H 5µ9CBSI2C2H 6;JL=C467H 58?6;4=2H.

=G/0

5. F8JIJ<>ZJLµ9 C2@ 6;JL=C467 58?6;4=2 CJL 5@5IJ<46JK 3>IC;JL 58? CJ@ 5@C>-

=C;J3J µ9C5=T2µ5C4=µ? Laplace C2H H(s), O2I5O7:

S. "5µSB@JLµ9 O9><µ5C5 58? C2@ h(t) 5@C465G4=CM@C5H CJ t µ9 CJ nT. )5 O9><µ5C5

8JL 8;J6K8CJL@ 5@C4=CJ4TJK@ =C5 =CJ4T9>5 C2H 56JIJLG>5H h(n), =Kµ3A@5 µ9 C2

=TQ=2 (5.22):

<. F8JIJ<>ZJLµ9 C2 =L@B;C2=2 µ9C53J;BH H(z) 58? CJ@ µ9C5=T2µ5C4=µ?–z C2H

h(nT), ?8AH L85<J;9K9C54 58? C2 =TQ=2 (5.23):

(H 8;J=85G7=JLµ9 CM;5 @5 <9@469K=JLµ9 C2 O45O465=>5 =T9O>5=2H 654 C5LC?T;J-

@5 @5 58JO9>:JLµ9 ?C4 2 H(z) µ8J;9> @5 L8JIJ<4=C9> 589LG9>5H 58? C2@ H(s), TA;>H

@5 T;945=C9> @5 L8JIJ<>=JLµ9 C2@ h(t) 654 C2@ h(nT). (H G9A;7=JLµ9 Q@5 5@5IJ<4-

6? 3>IC;J CJ J8J>J QT94 R O453J;9C46JKH 8?IJLH. # =L@B;C2=2 µ9C53J;BH CJL µ8J-

;9> @5 <;539> AH 5@B8CL<µ5 µ9;46M@ 6I5=µBCA@ µ9 C2 µJ;37:

(5.26)

?8JL pF J4 8?IJ4 CJL 5@5IJ<46JK 3>IC;JL 654 CF J4 =L@C9I9=CQH CJL 5@58CK<µ5CJH

=9 µ9;46B 6IB=µ5C5.

# 5@C>=CJ4T2 6;JL=C467 58?6;4=2 G5 9>@54:

(5.27)

h t C e p tN

( ) ==

5 :

:

:

1

H sCN

( ) =#=

5 :

::s p

1

H z Z h nT

e z e z

e e z

e z e zT T

T T

T T( ) ( )

( )

( )( ),= { } =

##

#= #

# ## # # #

# # #

# # # #1

1

1

1 1 11 2 1

2 1

1 2 1 | z |> e -T

h n h t

t nTe enT nT( ) ( )

,=

== ## #2 n = 0,1,2,....

h t L H s L

s sL

s se e tt t( ) ( )

( )( ),= { } =

+ +012

BCE

=+

#+

012

BCE

= # (# # # # #1 1 1 211 2

11

12

0

H s

s s( )

( )( )=

+ +1

1 2


(@ 58? 5LC7 C2@ 6;JL=C467 58?6;4=2 IBSJLµ9 O9><µ5C5 5@B ; O9LC9;?I98C5 (t =

nT), C?C9 G5 8B;JLµ9 C2@ O94<µ5CJI28C2µQ@2 58?6;4=2:

(5.28)

)QIJH, J µ9C5=T2µ5C4=µ?H–z C2H =TQ=2H (5.28) G5 µ5H OM=94 C2@ 984GLµ2C7 =L@B;-

C2=2 µ9C53J;BH O456;4CJK T;?@JL:

(5.29)

*5;5C2;M@C5H C2 O45O465=>5 8JL 89;4<;B39C54 58? C4H =TQ=94H (5.26) QAH (5.29),

O4584=CM@JLµ9 ?C4 C5 OKJ 9@O4Bµ9=5 =CBO45 L8JIJ<4=µJK C2H h(t) 654 C2H h(nT) G5

µ8J;JK=5@ @5 85;5I943GJK@ 654 @5 L8JIJ<>=JLµ9 589LG9>5H C2@ H(z) (=TQ=2 5.29)

58? C2@ H(s) (=TQ=2 5.26) 5@C465G4=CM@C5H CJ s – pF µ9 CJ , O2I5O7:

(5.30)

# =TQ=2 (5.30) 5@53Q;9C54 =C2@ 89;>8CA=2 9@?H 3>IC;JL CJL J8J>JL J4 8?IJ4 9>@54

O453J;9C46J> µ9C5:K CJLH, O2I5O7 QT94 8?IJLH 8JII58I?C2C5H 1. # 89;>8CA=2 3>I-

C;JL µ9 8?IJLH µ9<5IKC9;2H 8JII58I?C2C5H :939K<94 58? CJLH =C?TJLH CJL 85;?-

@CJH 6935I5>JL, 654 O9 G5 µ5H 585=TJI7=94.

¶·Ú¿‰ÂÈÁÌ· 5.4

(8? CJ 5@5IJ<46? 3>IC;J µ9 =L@B;C2=2 µ9C53J;BH H(s) = (s + 0,1)/[(s + 0,1)2 + 9]

@5 8;J=O4J;4=C9> 5@C>=CJ4TJ IIR R23456? 3>IC;J µ9 T;7=2 C2H µ9G?OJL C2H 5µ9CB-

SI2C2H 6;JL=C467H.

=G/0

*5;5C2;JKµ9 ?C4 CJ 5@5IJ<46? 3>IC;J QT94 Q@5 µ2O9@46? =CJ =2µ9>J s = – 0,1 654

Q@5 Z9K<JH =LZL<M@ 8?IA@ pF = – 0,1 ± j3. !63;BZJLµ9 C2@ H(s) AH 5@B8CL<µ5 µ9;4-

6M@ 6I5=µBCA@:

654 L8JIJ<>ZJLµ9 C2@ H(z) µ9 SB=2 C2 =TQ=2 (5.30):

H z

e z e z

e T z

e T z e zj T j T

T

T T( )

, , ( cos )

( cos )( , ) ( , )

,

, ,=

#+

#= #

# +# + # # # #

# #

# # # #0 5

1

0 5

1

1 3

1 2 30 1 3 1 0 1 3 1

0 1 1

0 1 1 0 2 1

H s

s j s j( )

,,

,,

=+ #

++ +

0 50 1 3

0 50 1 3

A @A =# #

=5H z

C

e zp T

N

( ) :

: 1 : 11

H sCN

( ) =#=

5 :

::s p

1

1 :# #e zp T 1

H z Z h nTC

e zp T

N

( ) ( )= { } =# #

=5 :

: 1 : 11

h nT C e p nTN

( ) ==

5 :

:

:

1

1 8 35 . 2 ™ Ã !¢#"™% IIR æ%º#"'ø¡ º#§&ƒø¡



5.2.3 ª¤ıÔ‰Ô˜ ‰ÈÁÚ·ÌÌÈÎÔ‡ ÌÂÙ·Û¯ËÌ·ÙÈÛÌÔ‡

# µQGJOJH C2H 5µ9CBSI2C2H 6;JL=C467H 58?6;4=2H <45 C2 =T9O>5=2 IIR R23456M@

3>IC;A@ 9>@54 65CBII2I2 <45 S5GL89;5CB 654 J;4=µQ@5 ZA@JO45S5CB 3>IC;5. D5

5=TJI2GJKµ9 CM;5 µ9 C2 µQGJOJ CJL O4<;5µµ46JK µ9C5=T2µ5C4=µJK (bilinear

transformation method), 2 J8J>5 CJLIBT4=CJ@ O9@ GQC94 CQCJ4JLH 89;4J;4=µJKH. *;?-

694C54 <45 Q@5@ µ5G2µ5C46? µ9C5=T2µ5C4=µ? CJL 89O>JL–s =CJ 89O>J–z. +Kµ3A@5 µ9

CJ@ µ9C5=T2µ5C4=µ? 5LC?, ?C5@ µ5H O>@9C54 2 =L@B;C2=2 µ9C53J;BH H(s) CJL 5@5-

IJ<46JK 3>IC;JL, µ8J;JKµ9 @5 L8JIJ<>=JLµ9 C2 =L@B;C2=2 µ9C53J;BH H(z) CJL

5@C>=CJ4TJL R23456JK 3>IC;JL, 5@C465G4=CM@C5H CJ s µ9 CJ (z – 1)/(z + 1), O2I5O7:

(5.31)

,9 BII5 I?<45, 2 58946?@4=2 58? CJ 98>89OJ s =CJ 98>89OJ z <>@9C54 µ9 SB=2 C2

=TQ=2[1]:

(5.32)

045 @5 O49;9L@7=JLµ9 C5 T5;56C2;4=C46B CJL O4<;5µµ46JK µ9C5=T2µ5C4=µJK, 5H G9A-

;7=JLµ9 ?C4 z = rej@ 654 s = =+ jQ. )?C9, 2 =TQ=2 (5.32) µ8J;9> @5 963;5=C9> AH:

(5.33)

(8? C2 =TQ=2 5LC7 85;5C2;JKµ9 ?C4, 9B@ r < 1, C?C9 = < 0 654 9B@ r > 1, C?C9 = > 0.

(LC? =2µ5>@94 ?C4 CJ 5;4=C9;? 2µ498>89OJ CJL s 58946J@>Z9C54 =CJ 9=AC9;46? CJL

µJ@5O45>JL 6K6IJL =CJ 98>89OJ–z, 9@M CJ O9:> 2µ498>89OJ CJL s 58946J@>Z9C54 =CJ

sre

re

r

r rj

r

r r

j

j= #

+= #

+ ++

+ +

@

@

Q Q

@

@

@

1

1

1

1 2

2

1 2

2

2 2cos

sin

cos1 244 344 1 244 344

s

zz

= #+

11

H z H s

szz

( ) ( )== #

+11

,9 C2 µQGJOJ C2H 5µ9CBSI2C2H 6;JL=C467H 58?6;4=2H @5 L8JIJ<>=9C9 C2 =L@B;-

C2=2 µ9C53J;BH H(z) R23456JK IIR 3>IC;JL 8JL 5@C4=CJ4T9> =CJ 5@5IJ<46? S5GL-

89;5C? 3>IC;J H(s) = 1/(s + 1)(s2 + s + 1).


[1] # =TQ=2 s = (z – 1)/(z + 1) µ8J;9> @5 <;539> 654 AH sz + s – z + 1 = 0. *5;5C2;JKµ9 ?C42 C9I9LC5>5 =TQ=2 9>@54 <;5µµ467 AH 8;JH s 654 <;5µµ467 AH 8;JH z 7 O4<;5µµ467 AH8;JH s 654 z.


9:AC9;46? CJL µJ@5O45>JL 6K6IJL (SI. +T7µ5 5.7). Y;5, 58? Q@5 9L=C5GQH 5@5IJ-

<46? 3>IC;J 85>;@JLµ9 Q@5 9L=C5GQH R23456? 3>IC;J. \C5@ r = 1, C?C9 8 = 0 654:

(5.34) Q

@= tan2

1 8 55 . 2 ™ Ã !¢#"™% IIR æ%º#"'ø¡ º#§&ƒø¡

E8>89OJ–z

58946?@4=2 CJL[s=j"

58946?@4=2 CJL[s=0

58946?@4=2 CJL[s=–j" MJ@5O45>JH[

6K6IJH

jQE8>89OJ–s

8

Q

0-' -'/2

'/2 '@

Q=tan@2

™¯‹Ì· 5.7

>'.01#,083 /().'0'C*()–s 8/(.'9'.*(–z µC8@/() *0$"%µµ01(7µ./%8E3µ%/08µ(7

™¯‹Ì· 5.8

M3 $"%µµ0128EC83 µ./%F7 /@,%,%6($01+, 1%0H3<0%1+, 8)E,(-/2/@, 6#$@ /()*0$"%µµ01(7 µ./%-8E3µ%/08µ(7

# =TQ=2 µ9C5:K CA@ =LT@JC7CA@ =C5 OKJ 89O>5 s 654 z 35>@9C54 =CJ +T7µ5 5.8. 0>@9-

C54 35@9;? ?C4 ?I2 2 89;4JT7 CA@ =LT@JC7CA@ Q 58946J@>Z9C54 µ?@J µ>5 3J;B =C2@

89;4JT7 – ' ] @ ] '. *;?694C54 98JµQ@AH, <45 µ45 Q@5–8;JH–Q@5 58946?@4=2. # 5894-

6?@4=2 ?µAH, 5LC7 9>@54 µ2 <;5µµ467. *5;5C2;9>C54, O2I5O7, µ45 85;5µ?;3A=2 7

8/"CI6@83 CA@ =LT@JC7CA@ (frequency warping), 9:54C>5H C2H µ2 <;5µµ46?C2C5H C2H



=L@B;C2=2H C2H 9358CJµQ@2H. (LC7 2 =C;QSIA=2 CA@ =LT@JC7CA@ G5 8;Q894, ?8AH

G5 OJKµ9 =C2 =L@QT945, @5 I5µSB@9C54 L8?R2 65CB C2@ =T9O>5=2 9@?H IIR 3>IC;JL.

&E#60% 8E./01? µ. /(, *0$"%µµ01# µ./%8E3µ%/08µ#:

1. +9 J;4=µQ@5 S4SI>5 J O4<;5µµ46?H µ9C5=T2µ5C4=µ?H J;>Z9C54 AH ,

?8JL ; 2 89;>JOJH O94<µ5CJI2R>5H. (8JO946@K9C54 ?C4 J =L@C9I9=C7H O9@

9>@54 585;5>C2CJH, 53JK 585I9>39C54 65CB C2 O45O465=>5 5@C4=CBGµ4=2H C2H =C;Q-

SIA=2H, ?8AH G5 OJKµ9 =CJ 85;BO94<µ5 8JL 56JIJLG9>.

2. 045 C5 S5GL89;5CB 654 LR289;5CB 3>IC;5, 2 CB:2 C2H H(z) 8JL 8;J6K8C94, 9>@54

>O45 µ9 C2@ CB:2 C2H H(s). 045 85;BO94<µ5, 58? Q@5 O9KC9;2H CB:2H S5GL89;5C?

5@5IJ<46? 3>IC;J G5 8B;JLµ9 Q@5 O9KC9;2H CB:2H S5GL89;5C? R23456? 3>IC;J.

\µAH, <45 C5 ZA@JO45S5CB 3>IC;5 654 C5 3>IC;5 58?;;4R2H ZM@2H 5LC? O9@ 4=TK-

94. +C2@ 89;>8CA=2 5LC7, 2 CB:2 C2H H(z) 9>@54 O48IB=45 C2H CB:2H C2H H(s).

3. !L=C5G7 5@5IJ<46B 3>IC;5 JO2<JK@ =9 9L=C5G7 R23456B 3>IC;5.

¶·Ú¿‰ÂÈÁÌ· 5.5

)J 84J 58I? 5@5IJ<46? S5GL89;5C? 3>IC;J 58JC9I9>C54 58? Q@5@ 5@C4=CBC2 654 Q@5@

8L6@AC7. *;?694C54 <45 CJ <@A=C? «RC 6K6IAµ5», 2 Q:JOJH CJL J8J>JL I5µSB@9-

C54 58? C5 B6;5 CJL 8L6@AC7 (+T7µ5 5.95). ,9 C2 µQGJOJ CJL O4<;5µµ46JK µ9C5-

=T2µ5C4=µJK, @5 =T9O45=C9> CJ 5@C>=CJ4TJ R23456? 3>IC;J, CJL J8J>JL 2 =LT@?C2-

C5 58J6J87H @5 9>@54 30 Hz, ?C5@ 2 =LT@?C2C5 O94<µ5CJI2R>5H 9>@54 150 Hz.

=G/0

# =LT@?C2C5 58J6J87H CJL 984GLµ2CJK R23456JK 3>IC;JL 9>@54 (UI. =TQ=2 1.13)

. # =LT@?C2C5 5LC7 CJL 9848QOJL– z

9>@54 2 5@C>=CJ4T2 C2H =LT@?C2C5H CJL 9848QOJL–s

(UI. =TQ=2 5.34). ' L8JIJ<4=µ?H 8JL µ?I4H 96C9IQ=5µ9 58JC9I9> C2 I9<?µ9@2 %,/0-

8/?5µ083 8/"CI6@834 (prewarping). !8JµQ@AH, 5LC? 8JL T;94BZ9C54 @5 =T9O4B=JL-

µ9 9>@54 Q@5 5@5IJ<46? S5GL89;5C? 3>IC;J µ9 =LT@?C2C5 58J6J87H . )J 8;AC?-

CL8J (Qc = 1) 5@5IJ<46? S5GL89;5C? 3>IC;J CJ J8J>J 5@C4=CJ4T9> =CJ 58I? RC

6K6IAµ5 QT94 =L@B;C2=2 µ9C53J;BH Hp(s) = 1/(s + 1). !6C9IJKµ9, 98JµQ@AH, Q@5@

UQc

U = *

+,-./

= *+,

-./

=Q@ '

c tan tan,

,c

20 4

20 73

@ Q ' ' ' 'c c= = = = =T F

Fcs

21

2 301

15025

0 4,

2T

s

Tzz

= #+

2 11


µ9C5=T2µ5C4=µ? =LT@?C2C5H 58? S5GL89;5C? =9 S5GL89;5C? (=TQ=2 5.15) <45 @5

L8JIJ<>=JLµ9 CJ 5@5IJ<46? S5GL89;5C? 3>IC;J CJ J8J>J QT94 Q_c = 0,73.

0@A;>ZJ@C5H C2 =L@B;C2=2 µ9C53J;BH CJL 5@5IJ<46JK S5GL89;5CJK 3>IC;JL, 935;-

µ?ZJLµ9 CJ@ O4<;5µµ46? µ9C5=T2µ5C4=µ? (=TQ=2 5.31) <45 @5 8;J=O4J;>=JLµ9 C2@

5@C>=CJ4T2 =L@B;C2=2 µ9C53J;BH CJL 984GLµ2CJK R23456JK 3>IC;JL:

*5;5C2;JKµ9 ?C4 8;?694C54 <45 Q@5 R23456? 3>IC;J 8;MC2H CB:2H, ?8AH 654 CJ 5@C>-

=CJ4TJ 5@5IJ<46? 3>IC;J. # 9:>=A=2 O453J;M@ CJL 3>IC;JL 8JL 8;J6K8C94 9>@54

y(n) = 0,42[x(n) + x(n – 1)] + 0,16y(n – 1),

9@M 2 OJµ7 8;5<µ5CJ8J>2=7H CJL 35>@9C54 =CJ +T7µ5 5.9S.

H z H s

szz

zz

z

zLP( ) ( ) ,

,

, ( )

,

== #

+=

#+

+= +

#

#

#11

0 7311

0 73

0 42 1

1 0 16

1

1

H s H ss s s s

LP p( ) ( )

,

,,

==

U=

+=

+11

0 731

0 730 73

Qc

1 8 75 . 2 ™ Ã !¢#"™% IIR æ%º#"'ø¡ º#§&ƒø¡

x(n)y(n)

0,42

0,42 0,16z-1 z-1

(5) (S)

R

x(t) y(t)C

™¯‹Ì· 5.9

(%) >,%6($01#I%5)'."%/# <96-/"( '"+/34 /?F34,(I) H3<0%1# <96-/"( '"+/34 /?F34/() !%"%*.9$µ%-/(4 5.5.

# =L@B;C2=2 µ9C53J;BH CJL 8;AC?CL8JL 5@5IJ<46JK S5GL89;5CJK 3>IC;JL 8;M-

C2H CB:2H 9>@54 Hp(s) = 1/(s + 1). +T9O4B=C9 CJ 5@C>=CJ4TJ LR289;5C? IIR R2345-

6? 3>IC;J, CJ J8J>J @5 QT94 =LT@?C2C5 58J6J87H 30 Hz <45 =LT@?C2C5 O94<µ5CJ-

I2R>5H 150 Hz.


*;4@ 8;JTA;7=JLµ9 =CJ =T9O45=µ? IIR 3>IC;A@ CB:2H µ9<5IKC9;2H C2H 8;MC2H, G5

7C5@ 65I? @5 =L=C2µ5CJ8J47=JLµ9 C2 O45O465=>5 =T9O>5=2H µ9 C2 µQGJOJ CJL

O4<;5µµ46JK µ9C5=T2µ5C4=µJK. # O45O465=>5 5LC7, ?8AH 3B@269 58? CJ *5;BO94<µ5

5.5, 58JC9I9>C54 58? C5 9:7H =CBO45:

1. !84IQ<JLµ9 CJ 8;AC?CL8J (Qc = 1) 5@5IJ<46? S5GL89;5C? 3>IC;J.



2. ,9C5=T2µ5C>ZJLµ9 5LC? =CJ S5GL89;5C? 7 CJ LR289;5C? 7 CJ ZA@JO45S5C? 7

CJ 58?;;4R2H ZM@2H 3>IC;J CJ J8J>J QT94 T5;56C2;4=C46QH =LT@?C2C9H Q', 8JL

5@C4=CJ4TJK@ =C4H 984GLµ2CQH =LT@?C2C9H @ CJL R23456JK 3>IC;JL.

3. !35;µ?ZJLµ9 CJ@ O4<;5µµ46? µ9C5=T2µ5C4=µ? =C2 =L@B;C2=2 µ9C53J;BH CJL

5@5IJ<46JK 3>IC;JL 8JL µ?I4H L8JIJ<>=5µ9.

¶·Ú¿‰ÂÈÁÌ· 5.6

,9 C2 µQGJOJ CJL O4<;5µµ46JK µ9C5=T2µ5C4=µJK @5 =T9O45=C9> S5GL89;5C? IIR

R23456? 3>IC;J µ9 =LT@?C2C5 58J6J87H 150 Hz <45 ;LGµ? O94<µ5CJI2R>5H 1,5 kHz.

# =T9O>5=2 @5 S5=4=C9> =CJ 8;AC?CL8J S5GL89;5C? 5@5IJ<46? 3>IC;J O9KC9;2H

CB:2H CK8JL Butterworth.

=G/0

045 CJ@ 8;J=O4J;4=µ? C2H =L@B;C2=2H µ9C53J;BH H(z) CJL 984GLµ2CJK IIR R2345-

6JK 3>IC;JL, 56JIJLGJKµ9 C5 85;58B@A =CBO45 =T9O>5=2H:

1. !84IQ<JLµ9 CJ 8;AC?CL8J (Qc = 1) 5@5IJ<46? S5GL89;5C? 3>IC;J. +C2@ 8;J694µQ-

@2 89;>8CA=2 9>@54 CJ O9KC9;2H CB:2H Butterworth 3>IC;J

2. ,9C5=T2µ5C>ZJLµ9 CJ 8;AC?CL8J S5GL89;5C? =9 S5GL89;5C? µ9 =LT@?C2C5 58J-

6J87H Q[c = tan(@c/2) = tan(Qc;/2) = tan(2'150(1/1500)/2) = tan(0,1') = 0,32

935;µ?ZJ@C5H CJ@ µ9C5=T2µ5C4=µ? =LT@?C2C5H C2H =TQ=2H (5.15):

H s

s sp ( ) =

+ +1

2 12

H s H ss s s

ss s s s

LP pc

c

( ) ( )

, ,, ,

, ,

==

U= =

=*+,

-./

+ *+,

-./

+

=+ +

$$

10 32 0 32

1

0 322

0 321

1

9 47 4 35 12 2

3. !35;µ?ZJLµ9 CJ@ O4<;5µµ46? µ9C5=T2µ5C4=µ? =C2@ C9I9LC5>5 =TQ=2:

)J Z2CJKµ9@J R23456? 3>IC;J 9>@54 O9KC9;2H CB:2H, ?8AH 654 CJ 5@5IJ<46? 58? CJ

J8J>J 5;T>=5µ9 C2 =T9O>5=2, 654 QT94 9:>=A=2 O453J;M@:

y(n) = 0,07[x(n) + 2x(n – 1) + x(n – 2)] + 1,14y(n – 1) – 0,41y(n–2).

H z H s

szz z

zzz

z z

z zLP( ) ( )

, ,

, ( )

, ,

== #

+=

#+

*+,

-./

+ #+

*+,

-./

+

= + +# +

# #

# #11

1

9 4711

4 3511

1

0 07 1 2

1 1 14 0 412

1 2

1 2


¶·Ú¿‰ÂÈÁÌ· 5.7

$5 =T9O4B=9C9 S5GL89;5C? IIR R23456? 3>IC;J CJ J8J>J @5 9:5=G9@>Z94 µJ@JCJ@4-

6B C4H =LT@?C2C9H. # ZM@2 O4QI9L=2H CJL 3>IC;JL @5 96C9>@9C54 µQT;4 C5 125 Hz,

9@M 2 9:5=GQ@4=2 =C2 ZM@2 5LC7 @5 µ2@ L89;S5>@94 C5 0,5 dB. # µQ<4=C2 9:5=GQ-

@4=2 =CJ ?;4J C2H ZM@2H 58J6J87H @5 9>@54 15 dB, 9@M 2 =LT@?C2C5 =CJ ?;4J C2H

ZM@2H 58J6J87H @5 9>@54 275 Hz. )J ?IJ =K=C2µ5 I94CJL;<9> µ9 =LT@?C2C5 O94<-

µ5CJI2R>5H 1 kHz. # =T9O>5=2 @5 <>@94 µ9 SB=2 CJ@ O4<;5µµ46? µ9C5=T2µ5C4=µ?.

=G/0

(8? C4H 8;JO45<;53QH 8JL µ5H O>@J@C54 65C5I5S5>@JLµ9 ?C4 8;?694C54 <45 6B8J4J

S5GL89;5C? 3>IC;J CK8JL Butterworth (53JK 8;Q894 @5 85;JL=4BZ94 µJ@JCJ@467

9:5=GQ@2=2 <45 ?I9H C4H =LT@?C2C9H). E9@ <@A;>ZJLµ9 ?µAH C2@ CB:2 CJL 3>IC;JL.

(LC7 µ8J;JKµ9 @5 C2@ 8;J=O4J;>=JLµ9 58? C4H C4µQH CA@ =LT@JC7CA@ 654 9:5=G9-

@>=9A@ 8JL µ5H O>@J@C54. )J 5@5IJ<46? S5GL89;5C? 3>IC;J 8JL Z2CBµ9 @5 8;J=-

O4J;>=JLµ9 654 =CJ J8J>J G5 935;µ?=JLµ9 CJ@ O4<;5µµ46? µ9C5=T2µ5C4=µ?, 35>-

@9C54 =CJ +T7µ5 5.10. PTJLµ9 <4’ 5LC? CJ 3>IC;J:

1 8 95 . 2 ™ Ã !¢#"™% IIR æ%º#"'ø¡ º#§&ƒø¡

+T9O4B=C9 Q@5 ZA@JO45S5C? IIR R23456? 3>IC;J O9KC9;2H CB:2H CJL J8J>JL 2 ZM@2

O4QI9L=2H @5 6Lµ5>@9C54 µ9C5:K 200 Hz 654 300 Hz, ?C5@ 2 =LT@?C2C5 O94<µ5CJ-

I2R>5H 9>@54 2 kHz. # =T9O>5=2 CJL 3>IC;JL @5 S5=4=C9> =C2 µQGJOJ CJL O4<;5µ-

µ46JK µ9C5=T2µ5C4=µJK, 935;µJ=µQ@2 =9 5@5IJ<46? 3>IC;J Butterworth.


|H(jQ)|

QQ'p Q's

1A

11+.2

™¯‹Ì· 5.10

;( %"E01# %,%6(-$01# <96/"( %'# /(('(9( 5% '"(17-H.0 /( L3/(7µ.,(I%5)'."%/# IIRH3<0%1# <96/"(.



# CB:2 CJL 3>IC;JL µ8J;9> CM;5 @5 8;J=O4J;4=C9> µ9 C2 SJ7G945 C2H =TQ=2H (5.6):

Y;5 CJ 5@5IJ<46? S5GL89;5C? 3>IC;J 8;Q894 @5 9>@54 C;>C2H CB:2H (R = 3).

(8? CJ =2µ9>J 5LC? 8IQJ@, µ8J;JKµ9 @5 56JIJLG7=JLµ9 C5 C;>5 =CBO45 =T9O>5=2H

9@?H R23456JK IIR 3>IC;JL, ?8AH C5 <@A;>=5µ9 8;J2<JLµQ@AH:

1. !84IQ<JLµ9 CJ 8;AC?CL8J S5GL89;5C? 3>IC;J.

!>O5µ9 ?C4 CJ 8;AC?CL8J S5GL89;5C? 3>IC;J G5 8;Q894 @5 9>@54 CK8JL

Butterworth C;>C2H CB:2H. # =L@B;C2=2 µ9C53J;BH 5LCJK 9>@54 (SI. 9@?C2C5 5.1)

Hp(s) = 1/(s3 + 2s2 + 2s + 1).

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C5 LR2I7H CB:2H IIR R23456B 3>IC;5, G5 8;Q894 @5 LIJ8J4JK@C54 AH =L@OL5=µ?H

OJµM@ O9KC9;2H CB:2H, <45 C2@ 9I5T4=CJ8J>2=2 CA@ 85;58B@A =35IµBCA@.


+CJ 693BI54J 5LC? µ9I9C7=5µ9 C5 R23456B 3>IC;5 B894;2H 6;JL=C467H 58?6;4-

=2H (IIR). # µ9IQC2 5LC7 53J;JK=9 6L;>AH C2 =T9O>5=2 654 C2@ LIJ8J>2=2 CQCJ4-

A@ 3>IC;A@. !>O5µ9 ?C4 58? ?I9H C4H µ9G?OJLH =T9O>5=2H IIR R23456M@ 3>IC;A@,

2 8IQJ@ =L=C2µ5C467 654 <9@467, O2I5O7 2 8IQJ@ 65CBII2I2 <45 J8J4JO78JC9 CK8J

3>IC;JL, 9>@54 2 µQGJOJH CJL O4<;5µµ46JK µ9C5=T2µ5C4=µJK. ,9 C2 µQGJOJ 5LC7

984CL<TB@9C54 µ45 Q@5–8;JH–Q@5, 5IIB µ2 <;5µµ467, 58946?@4=2 CJL 35@C5=C46JK

B:J@5 CJL 9848QOJL–s =C2@ 89;43Q;945 CJL µJ@5O45>JL 6K6IJL CJL 9848QOJL– z.

!8>=2H, µ9 C2 µQGJOJ 5LC7 58? 9L=C5G7 5@5IJ<46B 3>IC;5 65C5I7<JLµ9 =9 9L=C5-

G7 R23456B 3>IC;5.

# 5@B<62 =T9O>5=2H 3>IC;A@ µ9 C2 µQGJOJ CJL O4<;5µµ46JK µ9C5=T2µ5C4=µJK µBH

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=5µ9 C5 3>IC;5 Butterworth, C5 3>IC;5 Chebyshev 654 C5 !II948C46B 3>IC;5, 65GMH

654 CJLH 5@5IJ<46JKH µ9C5=T2µ5C4=µJKH =LT@?C2C5H, M=C9 58? Q@5 8;AC?CL8J

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1 9 5™ À ¡ $æ% '!º"§"#$À



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=T9O4B=5µ9. !>O5µ9 ?C4, 6L;>AH I?<A C2H 8989;5=µQ@2H 56;>S945H, CJ R23456?

=K=C2µ5 µ8J;9> @5 µ2 =Lµ89;43Q;9C54 ?8AH 5;T46B 9>T9 =T9O45=C9>. P@5 58? C5

S5=46B µQC;5 8JL 8;Q894 @5 IBSJLµ9, <45 @5 9I5T4=CJ8J47=JLµ9 C5 =3BIµ5C5

I?<A 8989;5=µQ@2H 56;4S9>5H, 9>@54 @5 LIJ8J47=JLµ9 C5 LR2I7H CB:2H 3>IC;5 AH

=L@OL5=µ? (=9 =94;B 7 85;BII2I5) OJµM@ O9KC9;2H CB:2H.


•!¡$°§ø™™%

E.C.Ifeachor, B.W. Jervis, Digital Signal Processing: A Practical Approach, Second

edition, Pearson Education Limited, 2002.

S.K.Mitra, Digital Signal Processing: A computer–Based Approach, McGraw Hill,

2001.





$¢%°$™ ¶!ƒ"#&!ƒø ª!§!&%™



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=2H IIR R23456M@ 3>IC;A@. !8>=2H <>@9C54 µ45 9:54;9C467 85;JL=>5=2 CA@ 5@5-

IJ<46M@ 3>IC;A@.


Processing, Addison–Wesley Publishing Company, 1988 .

,45 96C9C5µQ@2 5@53J;B =C5 5@5IJ<46B 3>IC;5 654 =C4H µ9G?OJLH =T9O>5=2H IIR 3>I-

C;A@, µ9 8JIIB 85;5O9><µ5C5, <>@9C54 =CJ OQ65CJ 693BI54J 5LCJK CJL S4SI>JL.



Aliasing .54@?µ9@J T5µ2IJK (5@985;6JKH) ;LGµJK

O94<µ5CJI2R>5H

Analog filter (@5IJ<46? 3>IC;J

Bandpass filter WA@JO45S5C? 3>IC;J

Bandstop filter .>IC;J 58?;;4R2H ZM@2H

Bilinear transformation E4<;5µµ46?H µ9C5=T2µ5C4=µ?H

Cascade realisation *;5<µ5CJ8J>2=2 =9 =94;B

Computational complexity F8JIJ<4=C467 8JIL8IJ6?C2C5

Elliptic !II948C46?H

Finite wordlength effects .54@?µ9@5 8989;5=µQ@JL µ76JLH IQ:2H

FIR, Finite Impulse Response *989;5=µQ@2 6;JL=C467 58?6;4=2

Frequency warping +C;QSIA=2 =LT@?C2C5H

Highpass filter FR289;5C? 3>IC;J

IIR, Infinite Impulse Response Y894;2 6;JL=C467 58?6;4=2

Implementation FIJ8J>2=2

Impulse invariance method ,QGJOJH 5µ9CBSI2C2H 6;JL=C467H

Limit cycles ';456J> 6K6IJ4

Lowpass filter U5GL89;5C? 3>IC;J

Parallel realisation *5;BII2I2 8;5<µ5CJ8J>2=2

Prewarping (@C4=CBGµ4=2 =C;QSIA=2H

Prototype *;AC?CL8J

Realisation structure EJµ7 8;5<µ5CJ8J>2=2H

Recursive !85@5I28C46?H, 5@5O;Jµ46?H

Rounding +C;J<<KI9L=2

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f(x,y)@fq(n1,n2)


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(6.4)

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2 µQ<4=C2 9L54=G2=>5 S;>=69C54 =C4H µ9=5>9H TA;46QH =LT@?C2C9H (SIQ89 +T7µ5 6.13).

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64@JL, 8;B=4@JL 654 µ8I9 (Red–Green–Blue, RGB).

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RLTJ3L=4JIJ<>5 C2H ?;5=2H 89;4<;B3J@C54 96C9@MH =C2 S4SI4J<;53>5. # 5@BIL=2

85;?µJ4A@ 354@JµQ@A@ µ8J;9> @5 SJ2G7=94 =2µ5@C46B =C2@ 5@B8CL:2 654 C2 S9I-

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CJµ2 µ9C5SJI7 C2H 5µ5K;A=2H. PC=4, =C2@ 89;>8CA=2 CJL C9C;5<M@JL Q@CJ@J 5;µJ-

@46? 89;49T?µ9@J 85;JL=4BZ9C54 =9 O49LGK@=94H 6BG9C9H =C4H 8I9L;QH CJL. )J >O4J

56;4SMH =LµS5>@94 =CJ C;><A@J, 9@M =CJ@ 6K6IJ QTJLµ9 5;µJ@46? 89;49T?µ9@J 8;JH

?I9H C4H 65C9LGK@=94H 53JK 2 58?CJµ2 µ9C5SJI7 C2H 5µ5K;A=2H =LµS5>@94 6L6I4-

6B 65CB µ76JH C2H 89;43Q;94BH CJL. )QIJH, =CJ =T7µ5 µ9 C4H J;4Z?@C49H <;5µµQH QTJL-

µ9 µ9C5SJI7 C2H 5µ5K;A=2H 654 µBI4=C5 89;4JO467 µ?@J 65CB C2@ 65C56?;L32 O49K-

GL@=2. )J 5@C>=CJ4TJ 3B=µ5 9>@54 <;5µµ46? 65CB C2 O49KGL@=2 5LC7.

5 S < O

9 =C Z 2

E46?@5

MQC;J

3B=µ5CJH

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&/3, '"+/3 8.0"?'%"()80?L(,/%0$.@µ./"01? 8E2-

µ%/%, .,+ 8/3,*.7/."3 8.0"? /(µC/"( /@, %,/9-8/(0E@, <%8µ?-/@,. ;% <?8µ%/%

CE(), C,/(,(%"µ(,01# '."0.E#-

µ.,( 8. *0.)57,-8.04 #'() 8/3,

%,/98/(0E3 .01#,%)'?"E.0 %'#/(µ3

µ./%I(62 /34%µ%7"@834.

)QIJH, 8;Q894 @5 984=2µB@JLµ9 C2 =2µ5=>5 8JL QT94 2 3B=2 <45 C2@ 946?@5. +CJ

+T7µ5 6.24 85;JL=4BZJ@C54 CJ µQC;J 654 2 3B=2 CJL 3B=µ5CJH <45 CJ >O4J =T7µ5

(Q@5 C9C;B<A@J) 8JL 9>@54 CJ8JG9C2µQ@J =9 OKJ O453J;9C46QH GQ=94H C2H 946?@5H.

)J µQC;J CJL 3B=µ5CJH 654 <45 C4H OKJ 946?@9H 9>@54 CJ >O4J, 9@M 2 3B=2 9>@54 O45-

3J;9C467. )J <9<J@?H 5LC? 58JC9I9> Q@O94:2 ?C4 2 3B=2 89;4QT94 =2µ5@C467 8I2;J-

3J;>5 <45 C2 GQ=2 9@?H =T7µ5CJH µQ=5 =C2@ 946?@5. !8>=2H 8;Q894 @5 =2µ94AG9> ?C4

2 946?@5 6;5CB C5 <9@46B C2H T5;56C2;4=C46B 5@ CJ µQC;J CJL 3B=µ5CJH µ9C5SI2-

G9>, 5IIB 2 3B=2 O45C2;2G9> 5@5IIJ>AC2. PC=4 2 5@5=K@G9=2 946?@5H, T;2=4µJ-

8J4M@C5H C2@ 8I2;J3J;>5 C2H 3B=2H CJL 3B=µ5C?H C2H 654 µQC;J 58? 3B=µ5 J8J4-

5=O78JC9 3L=467H =62@7H, O>@94 465@J8J42C467 5@585;B=C5=2 C2H 946?@5H ?8AH

35>@9C54 =CJ +T7µ5 6.25. +Lµ89;5=µ5C46B, G5 8;Q894 @5 CJ@>=JLµ9 ?C4 =C2@ R2345-

67 989:9;<5=>5 946?@5H µ9 O4B3J;5 <;5µµ46B 3>IC;5, 5LCB G5 8;Q894 @5 µ2@ 982-

;9BZJL@ C2 35=467 =TQ=2 CA@ TA;46M@ 5;µJ@46M@ C2H 946?@5H. )QCJ45 3>IC;5 9>@54


C5 O4=O4B=C5C5 FIR, 8JL G5 85;JL=4B=JLµ9 =CJ 98?µ9@J 693BI54J. )5 FIR 3>IC;5

AH <@A=C? (&93BI54J 4) 9>@54 OL@5C? @5 OM=JL@ <;5µµ467 58?6;4=2 3B=2H µ9 58J-

CQI9=µ5 @5 O45C2;JK@ C2 35=467 =TQ=2 CA@ 5;µJ@46M@ CJL =7µ5CJH 654 98JµQ@AH,

C2 µJ;37 CJL.

2 2 96 . 4 &$ º"™ª" &%™ !#'$¡"™

™¯‹Ì· 6.24

O '63"(<("9% $0%/3 5C83 /() /./"%-$+,() I"981./%08/3 <?83 /()<?8µ%/(4 /34 .01#-,%4 %<(7 %)/2 *0%-<("('(0.9/%0, 1%0#E0 8/( µC/"( '()'%"%µC,.0 /( 9*0(.

5 S <

O 9 =C

™¯‹Ì· 6.25

O <?83 /()<?8µ%/(4 µ0%4.01#,%4 .9,%01%5("08/012 8/3,%,%87,5.83 /34.01#,%4. ;( µC/"(/() <?8µ%/(4 '%9-L.0 *.)/.".7(,/%"#6(.

MQC;J[3B=µ5CJH[3ACJ<;B3JL

.B=2[3B=µ5CJH[8;J=M8JL

+A@C>=C;J3JH DFT[OKJ O45=CB=9A@

MQC;J[3B=µ5CJH[8;J=M8JL

.B=2[3B=µ5CJH[3ACJ<;B3JL

+A@C>=C;J3JH DFT[OKJ O45=CB=9A@

E46?@5 MQC;J 3B=µ5CJH .B=2 3B=µ5CJH


2 3 0 K E º A § A I O 6 : E #™"°ø°% ™&%¡ æ%º#"'% !#'$¡"

!6C?H 58? CJ O4=O4B=C5CJ µ9C5=T2µ5C4=µ? Fourier, =C2@ 989:9;<5=>5 C2H R2345-

67H 946?@5H T;2=4µJ8J4JK@C54 654 8I7GJH BIIA@ µ9C5=T2µ5C4=µM@ µ9 9@O453Q-

;JL=9H 4O4?C2C9H. ' =2µ5@C46?C9;JH 9: 5LCM@ 9>@54 J O4=O4B=C5CJH µ9C5=T2µ5C4-

=µ?H =L@2µ4C?@JL (Discreet Cosine Transform – DCT). '4 8JIK 65IQH 4O4?C2C9H CJL

DCT CJ@ 65G4=CJK@ 8JIK T;7=4µJ =9 O45O465=>9H =Lµ8>9=2H 946?@5H. (LC? J39>I9-

C54 =CJ <9<J@?H ?C4 85;JL=4BZ94 =L<6Q@C;A=2 C2H 9@Q;<945H CJL =7µ5CJH =9 I><JLH

µ?@J =L@C9I9=CQH =CJ@ µ9C5=T2µ5C4=µQ@J TM;J (µ9 8;Ja8JGQ=94H 3L=46B). ' DCT

µ8J;9> @5 LIJ8J42G9> C5TKC5C5 µQ=A CJL FFT, 7 654 589LG9>5H µ9 C9T@46QH 58JO9-

65C4=µJK (decimation) 5@BIJ<9H 969>@A@ CJL FFT 8JL <@A;>=5µ9 =CJ &93BI54J 2.

$5 =T9O4B=9C9 65CB 8;J=Q<<4=2 C2 µJ;37 CJL µQC;JL CJL 3B=µ5CJH 65@J@46JK

8JIL<M@JL.


!OM G5 G9A;7=JLµ9 Q@5 65@J@46? 9:B<A@J ?8AH 35>@9C54 =C2@ 946?@5 CJL +T7µ5-

CJH 6.26. *9;4µQ@JLµ9 CJ µQC;J CJL 3B=µ5CJH @5 QT94 8IJK=4J 5;µJ@46? 89;49T?-

µ9@J =9 65C9LGK@=94H 6BG9C9H =C4H 8I9L;QH CJL, O9OJµQ@JL ?C4 969> 85;5C2;JKµ9

58?CJµ2 µ9C5SJI7 C2H 5µ5K;A=2H.

™¯‹Ì· 6.26

;( <?8µ% /34.01#,%4 /() .F%-$+,() 5% .9,%0

'6(780( 8. %"µ(-,01C4 8),08/+8.48. 1%/.)57,8.04

1?5./.4 8/04 '6.)-"C4 /().


+C2@ 85;B<;53J 5LC7 µ9I9C7=5µ9 8J4JC46B CJ 3B=µ5 C2H R234567H 946?@5H. (3JK

Q<4@9 94=5<A<7 =CJ O4=O4B=C5CJ O456;4C? µ9C5=T2µ5C4=µ? Fourier, =C2 =L@QT945 9>O5-

µ9 C5 T5;56C2;4=C46B CJL µQC;JL 654 C2H 3B=2H O453?;A@ 946?@A@ 654 =LZ2C7=5µ9

C2 =8JLO54?C2C5 C2H 3B=2H =C2 µJ;37 654 C2 GQ=2 CJL 89;49T?µ9@JL C2H 946?@5H.



+CJ 693BI54J 5LC? <@A;>=5µ9 C5 58IB O4=O4B=C5C5 R23456B =7µ5C5 654 9>O5µ9 ?C4

2 R234567 946?@5 58JC9I9> Q@5 CQCJ4J =7µ5. 0@A;>=5µ9 98>=2H, 58IB =CJ4T9>5 58?

C2@ R234567 946?@5, ?8AH 9>@54 2 <9Aµ9C;>5 C2H 654 J4 O45O465=>9H 8JL µ8J;JK@ @5

96C9I9=CJK@ 8B@A =9 5LC7. 0@A;>=5µ9 C5 =2µ5@C46?C9;5 T5;56C2;4=C46B C2H

5@G;M84@2H ?;5=2H. )5:4@Jµ7=5µ9 C5 9>O2 CA@ 8;B:9A@ 8JL µ8J;JK@ @5 96C9I9-

=CJK@ =C2@ R234567 946?@5. )QIJH, µ9I9C7=5µ9 C5 T5;56C2;4=C46B CJL 3B=µ5CJH

C2H 946?@5H, ?8AH 5LCB 9µ35@>ZJ@C54 C?=J =CJ µQC;J CJL, ?=J 654 =C2 3B=2 CJL.


!§§%¡$°§ø™™%

%. *7C5H, N3<0%12 .'.F."$%89% :01#,%4, D9==5IJ@>62 1996.

!. +5<6;4MC2H 654 +. D9JOA;>O2H, >,?6)83 :01#,%4 1%0 >,%$,+"083 !"(/7'@,,

!II2@46? (@J46C? *5@984=C7µ4J (L8? Q6OJ=2).

•!¡$°§ø™™%

M.D. Levine, Vision in Man and Machine, McGraw–Hill, 1985.

A.K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, 1989.

R.C. Gonzalez and R.E. Woods, Digital Image Processing, Addison–Wesley, New

York, 1993.

R. Klette and P. Zamperoni, Handbook of Image Processing Operators, John Wiley

and Sons, Chichester, 1996.

J.C. Russ, The Image Processing Handbook, CRC Press, Springer, IEEE Press, 1999.

K.R. Castleman, Digital Image Processing, Prentice Hall, 1996

$¢%°$™ ¶!ƒ"#&!ƒø ª!§!&%™

%. *7C5H, N3<0%12 .'.F."$%89% :01#,%4, D9==5IJ@>62 1996.

Pµ35=2 µ8J;9> @5 OJG9> =C5 693BI545 1,2,4 654 6

2 3 1B # µ § # $ ° ƒ" º # " ' ! º " § " # $ À


2 3 2 K E º A § A I O 6 : E #™"°ø°% ™&%¡ æ%º#"'% !#'$¡"


Brightness "5µ8;?C2C5

Geometrical operation 09Aµ9C;467 8;B:2

Global operation &5GJI467 8;B:2

Hue (8?T;A=2

Image compression +Lµ8>9=2 946?@5H

Image acquisition (8?6C2=2 946?@5H

Image restoration (8J65CB=C5=2 C2H 946?@5H

Local operation )J8467 8;B:2

Luminance .AC94@?C2C5

Matching )5>;45=µ5

Mirroring &5G;98C4=µ?H

MTF +L@B;C2=2 ,9C53J;BH E45µ?;3A=2H

Picture element 7 pixel 946J@J=CJ4T9>J

Pixels per inch !46J@J=CJ4T9>5 5@B >@C=5

Point operation *;B:2 =2µ9>JL

Retina (µ34SI2=C;J94O7H

RGB 7 Red–Green–Blue &?664@J–*;B=4@J–,8I9

Rotation +C;J37

Saturation &J;9=µ?H

Scanner +5;AC7H 946?@5H

Translation ,9C53J;B 7 µ9C5C?84=2


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™ÎÔ!fi˜

&/( 1.<?6%0( %)/# 5% $,@"98()µ. /% *08*0?8/%/% H3<0%1? <96/"%

(Two–dimensional digital filters) 1%0 /04 0*0#/3/C4 /()4. O µ./?I%83 %'# /% µ(,(-

*0?8/%/% 8/% *08*0?8/%/% <96/"% *., .9,%0 '(67 .71(63, '%"#/0 (0 C,,(0.4 .9,%0 8.

$.,01C4 $"%µµC4 (0 9*0.4. P 1)"0#/."(4 6#$(4 .9,%0 #/0 (0 *08*0?8/%/.4 8),%"/28.04,

#'@4 3 *08*0?8/%/3 1"()8/012 %'#1"083 1%0 3 *08*0?8/%/3 %'#1"083 8)E,#/3/%4,

.9,%0 *)81(6#/.".4 8/3 µ%53µ%/012 /()4 .'.F."$%89% %'# /04 %,/98/(0E.4 µ(,(*0?-

8/%/.4. &1('#4 /() 1.<%6%9() .9,%0 ,% $,@"98()µ. /% 17"0% E%"%1/3"08/01? /@,

*08*0?8/%/@, <96/"@, '() CE(), µ.$?63 83µ%89% 8/3, H3<0%12 .'.F."$%89% /34

.01#,%4. G/80, 5% $,@"98()µ. 1)"9@4 /% *08*0?8/%/% FIR <96/"% 1%0 /04 .<%"µ(-

$C4 /()4.


J/%, (6(163"+8./. /3 µ.6C/3 /() 1.<%6%9() %)/(7 5% .98/. 8. 5C83 ,%:

• &E.*0?L./. /3 µ("<2 *08*0?8/%/() 82µ%/(4 2 /3, 1"()8/012 %'#1"083 *08*0?-

8/%/() <96/"()

• B6('(0.9/. *08*0?8/%/3 8),C60F3 µ. 81('# ,% )'(6($98./. /3, CF(*( *08*0?8/%-

/() H3<0%1(7 <96/"()

• >,%$,@"9L./. %, C,% *08*0?8/%/( <96/"( µ'(".9 ,% )6('(035.9 @4 8),*)%8µ#4

*7( µ(,(*0?8/%/@, <96/"@, (*0%E@"080µ#/3/%)

• &E.*0?L./. C,% %'6# *08*0?8/%/( FIR <96/"(

• T.9E,./. %, C,% *08*0?8/%/( <96/"( CE.0 %'#1"083 µ3*.,0124 <?834 (zero phase

response)


• >µ./%I63/#/3/% 8/3, (6985383 2 µ./%/#'083

• >"E2 /34 )'C"5.834

• >,0E,.)/C4 %1µ+,

• >,/98/"(<% '"(I62µ%/%

7! " º#§#$%


2 3 4 K E º A § A I O 7 : ¢ # ™ ¢ # " ™ &"&" æ % º # " ' " ° ƒ" ª ª # ' " ™ À ™ & % ª "&"

• D"%µµ01? 8)8/2µ%/%

• T0%1"0/C4 E@"01C4 µ./%I63/C4

• T0%E@"980µ.4 8),%"/28.04

• T0%E@"980µ% <96/"%

• T08*0?8/%/% FIR <96/"%

• T08*0?8/%/3 %'#1"083 8)E,#/3/%4

• T08*0?8/%/3 8),C60F3

• :)8/%52 *08*0?8/%/% <96/"%

• MC5(*(4 /@, '%"%57"@,

• M3*.,012 %'#1"083 <?834

• !%"?66363 87,*.83

• !."0(*012 %'#1"083 8)E,#/3/%4

• &)µµ./"012 1"()8/012 %'#1"083

• &7,*.83 8. 8.0"?

• U96/"( µC834 /0µ24


&/( =.<?6%0( 6 $,@"98%µ. /% I%801? E%"%1/3"08/01? /34 H3<0%124 .01#,%4 1%0 /(,

/"#'( µ. /(, ('(9( /3, %,/06%µI?,./%0 1%0 µ'(".9 ,% /3, .'.F."$%8/.9 ( ?,5"@'(4.

T.*(µC,() #/0 3 H3<0%12 .01#,% .9,%0 C,% *08*0?8/%/( 82µ% '() .1<"?L.0 /04 µ./%-

I(6C4 /34 %µ%7"@834 8/( .'9'.*( (x,y), 3 .'.F."$%89% /34 µ'(".9 ,% $9,.0 µ. /% *08-

*0?8/%/% H3<0%1? <96/"%. ;% <96/"% %)/? E%"%1/3"9L(,/%0 %'# /3, 1"()8/012 /()4

%'#1"083. O 1"()8/012 %'#1"083 .9,%0 *08*0?8/%/3 *0%1"0/2 8),?"/383 1%0 (0 0*0#-

/3/C4 /34 .'3".?L(), 1%0 1%5("9L(), /( 82µ% .F#*() %'# /( *08*0?8/%/( <96/"(.

:'.0*2 3 .01#,% .9,%0 '(67 .)%9853/3 8/3, %66%$2 /34 <%80124 8EC834 µ./%F7 /@,

%"µ(,01+, /34, 5% '"C'.0 1%/? /3, H3<0%12 /34 .'.F."$%89%, /( *08*0?8/%/( <96-

/"( ,% µ3, %66?L.0 /3 8EC83 %)/2. ;C/(0% <96/"% CE(), 8)µµ./"012 1"()8/012 %'#-

1"083 1%0 1%6(7,/%0 <96/"% µ3*.,0124 <?834. &/( 1.<?6%0( %)/# 5% $,@"98()µ. /(,

/"#'( 8E.*9%834 %'6+, *08*0?8/%/@, FIR <96/"@, '() CE(), /04 '%"%'?,@ 0*0#-

/3/.4. ;( 1.<?6%0( 5% /( (6(163"+8()µ. 1?,(,/%4 %'62 %,%<("? 8. I%801C4 .<%"-

µ($C4 /@, *08*0?8/%/@, FIR <96/"@, 8/3, H3<0%12 .'.F."$%89% /34 .01#,%4. &. #63


/3, C1/%83 /() 1.<%6%9() (0 C,,(0.4 H3<0%1? <96/"% 1%0 H3<0%1? 8)8/2µ%/% .,%6-

6?88(,/%0. >'%"%9/3/3 $,+83 $0% /( 1.<?6%0( %)/# %'(/.6.9 3 763 /@, K.<%6%9@,

3, 4 1%0 6 /() I0I69(). !."088#/.".4 '63"(<("9.4 ( %,%$,+8/34 µ'(".9 ,% I".0 8/04

%,%<("C4 /34 .663,0124 1%0 /34 F.,#$6@8834 I0I60($"%<9%4.

;( 1.<?6%0( %)/# '."06%µI?,.0 /04 %1#6()5.4 .,#/3/.4:

7.1. T08*0?8/%/3 8),C60F3.

7.2. X%"%1/3"08/01? /@, *08*0?8/%/@, H3<0%1+, 8)8/3µ?/@,.

7.3. >'#1"083 8/3 8)E,#/3/% /@, *08*0?8/%/@, H3<0%1+, 8)8/3µ?/@,.

7.4. &E.*9%83 /@, *08*0?8/%/@, H3<0%1+, <96/"@,.

2 3 5E # ™ " ° ø ° # ' ! ™ ¶ " ƒ"& % ƒ % ™ ! # ™


2 3 6 K E º A § A I O 7 : ¢ # ™ ¢ # " ™ &"&" æ % º # " ' " ° ƒ" ª ª # ' " ™ À ™ & % ª "&"

7.1 ¢ÈÛ‰È¿ÛÙ·ÙË Û˘Ó¤ÏÈÍË

P@5 <;5µµ46? R23456? =K=C2µ5 JL=45=C46B µ9C5=T2µ5C>Z94 Q@5 R23456? =7µ5 x(n)

=9 Q@5 @QJ y(n) (+T7µ5 7.15). # 8;B:2 8JL 96C9I9> 9>@54 2 =L@QI4:2 C2H 6;JL=C467H

58?6;4=2H h(n) 8JL T5;56C2;>Z94 CJ =K=C2µ5 µ9 CJ =7µ5 94=?OJL x(n):

(7.1)

,9 CJ@ >O4J C;?8J Q@5 O4=O4B=C5CJ R23456? =7µ5 x(n1,n2) µ9C5=T2µ5C>Z9C54 58?

Q@5 O4=O4B=C5CJ <;5µµ46? R23456? =K=C2µ5 =9 Q@5 @QJ O4=O4B=C5CJ R23456? =7µ5

y(n1,n2) ?8AH 85;JL=4BZ9C54 =CJ +T7µ5 7.1S. &54 9OM 2 8;B:2 8JL 96C9I9> CJ O4=-

O4B=C5CJ <;5µµ46? =K=C2µ5 9>@54 2 =L@QI4:2 CJL =7µ5CJH 94=?OJL x(n1,n2) µ9 C2

O4=O4B=C5C2 6;JL=C467 CJL 58?6;4=2 h(n1,n2):

(7.2)

y n n x n n h n n x n r n r h r rr r

( , ) ( , ) ( , ) ( , ) ( , )1 2 1 2 1 2 1 1 2 2 1 2

1 2

= WW = # #=#"

"

=#"

"

5 5

y n x n h n x n r h r

r

( ) ( ) ( ) ( ) ( )= W = #=#"

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5

x(n) x(n1,n2) y(n1,n2)y(n)h(n) h(n1,n2)

(5) (S)

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T08*0?8/%/()H3<0%1(7 8)8/2-

µ%/(4.

+C5 µJ@JO4B=C5C5 R23456B 3>IC;5 2 O456;4C7 µ9C5SI2C7 n 5@C4=CJ4TJK=9 6L;>AH

=CJ T;?@J, 53JK C5 µJ@JO4B=C5C5 R23456B =7µ5C5 9>@54 =L@7GAH =7µ5C5 T;?@JL.

(@C>G9C5, =C5 O4=O4B=C5C5 R23456B 3>IC;5 J4 5@9:B;C2C9H µ9C5SI2CQH n1 654 n2

9>@54 =L@7GAH O456;4CQH TA;46QH µ9C5SI2CQH 8JL 5@C4=CJ4TJK@ =C4H O49LGK@=94H x

654 y. ' 6L;4?C9;JH I?<JH <45 C2@ 984IJ<7 5LC7 9>@54 ?C4 J4 R23456QH 946?@9H G9A-

;JK@C54 O4=O4B=C5C5 O456;4CB =7µ5C5 =CJ 98>89OJ (x,y).

# =TQ=2 (7.2) 8JL µ5H 89;4<;B394 C2 O4=O4B=C5C2 =L@QI4:2 58JC9I9> C2 =TQ=2 94=?-

OJL–9:?OJL CJL <;5µµ46JK R23456JK =L=C7µ5CJH, 53JK =L@OQ94 C5 O9><µ5C5 C2H

94=?OJL x(n) µ9 5LCB C2H 9:?OJL y(n). # O4=O4B=C5C2 =L@QI4:2 9>@54 Bµ9=5 =L@O9-

O9µQ@2 µ9 C2@ 4O4?C2C5 C2H <;5µµ46?C2C5H 654 5LC7 C2H 5µ9C5SI2C?C2C5H =CJ T;?@J.

+C2@ =L@QT945 G5 9:2<7=JLµ9 CJ@ C;?8J µ9 CJ@ J8J>J J4 OKJ 5LCQH 4O4?C2C9H µ5H

JO2<JK@ =C2 O4=O4B=C5C2 =L@QI4:2.


7.1.1 °Ú·ÌÌÈÎfiÙËÙ·

X;2=4µJ8J4M@C5H C4H 8;B:94H C2H 8;?=G9=2H, CJL 8JII58I5=45=µJK µ9 =C5G9;B 654

C2H JI>=G2=2H, 9>@54 OL@5C? @5 963;B=JLµ9 6BG9 O4=O4B=C5CJ =7µ5 x(n1,n2) AH

<;5µµ46? =L@OL5=µ? JI4=G2µQ@A@ O4=O4B=C5CA@ µJ@5O45>A@ 6;JL=C46M@ *(n1,n2):

(7.3)

+C2@ C9I9LC5>5 =TQ=2 2 *(n1–l1,n2–l2) 9>@54 2 O4=O4B=C5C2 µJ@5O45>5 6;JL=C467 µ9C5-

CJ84=µQ@2 =C2 GQ=2 n1 = l1 654 n2 = l2 654 J4 C4µQH x(l1,l2) µ8J;JK@ @5 G9A;2GJK@ AH

J4 5@C>=CJ4TJ4 8JII58I5=45=CQH.

\8AH 7O2 5@58CK:5µ9 =CJ &93BI54J 3, <45 C5 <;5µµ46B =L=C7µ5C5 (linear systems)

4=TK94 2 5;T7 C2H L8Q;G9=2H. (LC? =2µ5>@94, ?C4 5@ GQIJLµ9 @5 L8JIJ<>=JLµ9 C2@

58?6;4=2 CJL <;5µµ46JK =L=C7µ5CJH =9 Q@5 =K@G9CJ =7µ5 8JL 58JC9I9>C54 58? CJ

<;5µµ46? =L@OL5=µ? 984µQ;JLH 58IM@ =2µBCA@, 5;69> @5 IBSJLµ9 CJ@ 5@C>=CJ4TJ

<;5µµ46? =L@OL5=µ? CA@ 58J6;>=9A@ CJL =L=C7µ5CJH =C5 58IB 5LCB =7µ5C5. PC=4,

5@ 2 =Lµ89;43J;B CJL =L=C7µ5CJH =LµSJI4=C9> µ9 L(.) 654 CJ =7µ5 94=?OJL =9 5LC?

9>@54 CJ =7µ5 C2H =TQ=2H (7.3) C?C9 2 Q:JOJH 9>@54:

(7.4)

045 Q@5 <;5µµ46? =K=C2µ5, 2 (7.4) µ8J;9> @5 :5@5<;539> AH

(7.5)

?8JL 9>@54 2 58?6;4=2 CJL =L=C7µ5CJH =C2 O4=O4B=C5C2 6;JL=C467 8JL S;>-

=69C54 =C2 GQ=2 (l1,l2).

7.1.2 "ÌÂÙ·‚ÏËÙfiÙËÙ· ÛÙËÓ ÔÏ›ÛıËÛË

P@5 3>IC;J 65I9>C54 %µ./?I63/( 8/3, (6985383 (shift invariant) 5@ =L<696;4µQ@2

µ9C5C?84=2 CJL =7µ5CJH 94=?OJL 8;J65I9> C2@ >O45 µ9C5C?84=2 =CJ =7µ5 C2H 9:?-

OJL. PC=4, 5@

(7.6) y n n T x n n( , ) [ ( , )]1 2 1 2=

hl l1 2,

y n n x l l L n l n l

x l l h n n

l l

l

l l

l

( , ) ( , ) [ ( , )]

( , ) ( , ),

1 2 1 2 1 1 2 2

1 2 1 2

1 2

1

1 2

2

= # #

=

=#"

"

=#"

"

=#"

"

=#"

"

5 5

5 5

4

y n n L x l l n l n ll l

( , ) ( , ) ( , )1 2 1 2 1 1 2 2

1 2

= # #7

899

:

;<<=#"

"

=#"

"

5 5 4

x n n x l l n l n ll l

( , ) ( , ) ( , )1 2 1 2 1 1 2 2

1 2

= # #=#"

"

=#"

"

5 5 4

2 3 77 . 1 ¢ # ™ ¢ # " ™ &"& % ™ À ¡ ! § # • %


2 3 8 K E º A § A I O 7 : ¢ # ™ ¢ # " ™ &"&" æ % º # " ' " ° ƒ" ª ª # ' " ™ À ™ & % ª "&"

<45 5µ9CBSI2CJ =C2@ JI>=G2=2 3>IC;J 4=TK94

(7.7)

# 5µ9C5SI2C?C2C5 =C2@ JI>=G2=2 654 2 <;5µµ46?C2C5 9@?H 3>IC;JL 9>@54 4O4?C2C9H

5@9:B;C2C9H µ9C5:K CJLH.

E9OJµQ@JL ?C4 2 5µ9C5SI2C?C2C5 =C2@ JI>=G2=2 =L@98B<9C54

, <45 Q@5 <;5µµ46? 3>IC;J 8JL QT94 654 C4H OKJ 4O4?C2-

C9H 2 =TQ=2 (7.5) <;B39C54 AH

(7.8)

# =TQ=2 (7.8) 89;4<;B394 C2 O4=O4B=C5C2 =L@QI4:2 (two–dimensional convolution).

&5GJ;>Z94 ?C4 CJ =7µ5 9:?OJL y(n1,n2) µ8J;9> @5 S;9G9>, 5@ 2 6;JL=C467 58?6;4=2

h(n1,n2) CJL O4=O4B=C5CJL R23456JK 3>IC;JL µ9C5CJ84=C9> 65CBII2I5 h(n1–l1,n2–l2)

654 8JII58I5=45=C9> 6BG9 3J;B µ9 CJ 5@C>=CJ4TJ O9><µ5 CJL =7µ5CJH x(l1,l2). +CJ

CQIJH 5G;J>ZJLµ9 ?I5 C5 <4@?µ9@5 <45 @5 IBSJLµ9 CJ y(n1,n2). # =TQ=2 (7.8) µ8J;9>

4=JOK@5µ5 @5 <;539> 654 AH

(7.9)

# O4=O4B=C5C2 =L@QI4:2 G5 =LµSJI>Z9C54 =CJ 9:7H µ9 O48I? W:

7 (7.10)

¶·Ú¿‰ÂÈÁÌ· 7.1

# 6;JL=C467 58?6;4=2 h(n1,n2) O4=O4B=C5CJL R23456JK 3>IC;JL QT94 ?8AH 35>@9-

C54 =CJ +T7µ5 7.25, µ9 ?IJLH CJLH =L@C9I9=CQH >=JLH µ9 1/3. $5 S;9G9> CJ =7µ5

9:?OJL y(n1,n2) 5@ <45 CJ =7µ5 94=?OJL x(n1,n2) QTJLµ9 x(0,0) = x(1,0) = x(0,1) =

x(1,1) = 1 654 0 =9 ?I9H C4H BII9H GQ=94H (n1,n2) (+T7µ5 7.2S).

y h x= WW y x h= WW

y n n h l l x n l n ll l

( , ) ( , ) ( , )1 2 1 2 1 1 2 2

1 2

= # #=#"

"

=#"

"

5 5

y n n x l l h n l n ll l

( , ) ( , ) ( , )1 2 1 2 1 1 2 2

1 2

= # #=#"

"

=#"

"

5 5

h n n h n l n ll l1 2 1 2 1 1 2 2( , ) ( , )= # #

T x n l n l y n l n l[ ( , )] ( , )1 1 2 2 1 1 2 2# # = # #

(5) (S)

h(n1,n2) x(n1,n2)n2 n2

n1n1

™¯‹Ì· 7.2

%) O 1"()8/012%'#1"083 h(n1,n2)/() <96/"() /()!%"%*.9$µ%/(4

7.1 1%0 I) /( 82µ%.08#*() x(n1,n2).


=G/0

045 @5 L8JIJ<>=JLµ9 CJ =7µ5 9:?OJL G5 T;2=4µJ8J47=JLµ9 C2 =TQ=2 (7.8). +Kµ-

3A@5 µ9 5LC7, 2 Q:JOJH µ8J;9> @5 =T2µ5C4=C9> 5@ IBSJLµ9 5@C><;535 C2H h(l1,l2)

µ9C5CJ84=µQ@5 65CBII2I5 =9 GQ=94H (n1,n2) ?8JL L8B;TJL@ 6J4@QH GQ=94H µ9 CJLH

?;JLH CJL =7µ5CJH 94=?OJL x(l1,l2). )J BG;J4=µ5 CA@ <4@JµQ@A@ CA@ ?;A@ µ9 C4H

6J4@QH GQ=94H G5 OM=94 CJLH ?;JLH C2H =7µ5CJH 9:?OJL y(n1,n2). # O45O465=>5 85;JL-

=4BZ9C54 =CJ +T7µ5 7.3.

2 3 97 . 1 ¢ # ™ ¢ # " ™ &"& % ™ À ¡ ! § # • %

x(l1,l2)

x(l1,l2)

x(l1,l2) x(l1,l2)

x(l1,l2) x(l1,l2)

x(l1,l2) x(l1,l2)l2

l2 l2 l2

l2n2

l2

l2 l2

l1 l1 l1

l1 l1 l1

l1 l1n1

11

11

1

32

2

h(–l1,–l2)

h(–l1,1–l2)

h(–l1,2–l2) h(1–l1,2–l2)

h(1–l1,1–l2) h(2–l1,1–l2)

h(1–l1,–l2) h(2–l1,–l2)

y(0,0)=1/3

y(0,1)=2/3

y(0,2)=1/3 y(1,2)=1/3

3y(n1,n2)

y(2,1)=1/3y(1,1)=1

y(1,0)=2/3 y(2,0)=1/3

™¯‹Ì· 7.3

B'(6($08µ#4 /34.F#*() y(n1,n2) µ.)6('(9383 /348),C60F34 '()'."0$"?<./%0 %'#/3 8EC83 (7.8).M#,( /( $0,#µ.,(/@, #"@, x(l1,l2)1%0 h(l1,l2) '()8)µ'9'/(), 1%0.9,%0 83µ.0@µC,(0µ. 1716(, 8),.0-8<C"(), 8/(, /.60-1# 8E3µ%/08µ# /34y(n1,n2).

¶·Ú¿‰ÂÈÁÌ· 7.2

123456? 3>IC;J OKJ O45=CB=9A@ QT94 6;JL=C467 58?6;4=2 h(n1,n2) CQCJ45, M=C9

h(0,0) = 0,5, h(1,0) = h(–1,0) = h(0,1) = h(0,–1) = 0,125 654 0 J8JLO78JC9 5IIJK.

$5 =T9O45=C9> 2 6;JL=C467 58?6;4=2 654 @5 S;9G9> CJ =7µ5 9:?OJL y(n1,n2) 5@ CJ

=7µ5 94=?OJL x(n1,n2) 9>@54 x(0,0) = 1, x(–1,0) = x(1,0) = 0,5.

=G/0

# 6;JL=C467 58?6;4=2 h(n1,n2) CJL 3>IC;JL 5IIB 654 CJ =7µ5 94=?OJL x(n1,n2) O>@J-

@C54 =CJ +T7µ5 7.4. 045 @5 L8JIJ<>=JLµ9 CJ =7µ5 9:?OJL y(n1,n2) G5 T;2=4µJ8J47-

=JLµ9 C2 =TQ=2 (7.8). +Kµ3A@5 µ9 5LC7 2 Q:JOJH y(n1,n2) µ8J;9> @5 =T2µ5C4=C9> 5@


2 4 0 K E º A § A I O 7 : ¢ # ™ ¢ # " ™ &"&" æ % º # " ' " ° ƒ" ª ª # ' " ™ À ™ & % ª "&"

IBSJLµ9 5@C><;535 C2H h(n1,n2) =9 GQ=94H (n1,n2) ?8JL CJ =7µ5 94=?OJL x(n1,n2) 9>@54

µ2 µ2O9@46?. )5 5@C><;535 5LCB G5 8JII58I5=45=CJK@ µ9 C2@ 5@C>=CJ4T2 C4µ7 CJL

=7µ5CJH 654 C5 58JC9IQ=µ5C5 G5 8;J=C9GJK@. # O45O465=>5 85;JL=4BZ9C54 =CJ

+T7µ5 7.5.

h(n1,n2) x(n1,n2)n2 n2

n1 n1

x(l1,l2)

x(l1,l2)

x(l1,l2)

x(l1,l2) x(l1,l2)

x(l1,l2)

x(l1,l2)

x(l1,l2)

l2

l2

l2

l2

l2

l2

l2

l2

l1

l1

l1

l1

l1

l1

l1

l1

h(–1–l1,–1–l2)

h(2–l1,–l2)h(1–l1,–l2)

h(1–l1,–1–l2)

h(–1–l1,–l2)

h(–l1,–1–l2)

h(–l1,–l2)

y(–1,–1)=0,0625

y(1,–1)=0,0625

y(–1,0)=0,375

y(1,0)=0,375 y(2 ,0)=0,375

y(0,0)=0,625

y(–2,0)=0,0625

y(0,–1)=0,125

h(–2–l1,–l2)

™¯‹Ì· 7.4

O 1"()8/012 %'#-1"083 h(n1,n2) /()<96/"() /() '%"%-*.9$µ%/(4 7.2 1%0/( 82µ% .08#*()

x(n1,n2)


™¯‹Ì· 7.5

B'(6($08µ#4 /34.F#*() y(n1,n2) µ.)6('(9383 /348),C60F34 '()'."0$"?<./%0 %'#/3 8EC83 (7.8).

2 4 17 . 1 ¢ # ™ ¢ # " ™ &"& % ™ À ¡ ! § # • %

x(l1,l2)

x(l1,l2)x(l1,l2)

y(1,1)=0,0625

h(1–l1,1–l2)l2

l2 l2

l1

l1 l1

16y(n1,n2)10

66

11 2

21

1

11

n1

n2

h(–1–l1,1–l2) h(–l1,1–l2)

y(–1,1)=0,0625 y(0,1)=0,125

$5 =T9O4B=9C9 C2@ 6;JL=C467 58?6;4=2 O4=O4B=C5CJL R23456JK 3>IC;JL ?C5@

5LC7 QT94 C;94H ?;JLH 654 9>@54 =Lµµ9C;467 <K;A 58? C2 GQ=2 (n1,n2) = (0,0).


DQIJLµ9 @5 L8JIJ<>=JLµ9 C2@ Q:JOJ y(n1,n2) O4=O4B=C5CJL R23456JK 3>IC;JL µ9

6;JL=C467 58?6;4=2 h(n1,n2) CQCJ45, M=C9 h(0,0) = 0,5, h(0,1) = h(0,–1) = 0,25 654

0 J8JLO78JC9 5IIJK. +Kµ3A@5 µ9 C2 =TQ=2 (7.9) CJ =7µ5 94=?OJL 8JL 35>@9C54

=CJ +T7µ5 7.7 8;Q894 @5 L8J=C9> 5, 10, 15 7 20 µ9C5CJ8>=94H 8;4@ 58? C2@ C9I467

BG;J4=2 CA@ <4@JµQ@A@;


# 6;JL=C467 58?6;4=2 G5 9>@54 =Lµµ9C;467 µ9 CJ@ Q@5@ ?;J @5 S;>=69C54 =C2 GQ=2

(n1,n2) = (0,0) 654 CJLH BIIJLH OKJ 965CQ;AG9@ 5LCJK 654 >=JLH µ9C5:K CJLH. ,9

C;94H µ?@J ?;JLH 2 =Lµµ9C;467 6;JL=C467 58?6;4=2 µ8J;9> @5 QT94 O4B3J;9H µJ;-

3QH (=L@JI46B 4). ,>5 58? 5LCQH O>@9C54 =CJ +T7µ5 7.6.

h(n1,n2)n2

n1

™¯‹Ì· 7.6

&)µµ./"012 *08-*0?8/%/3 1"()8/0-12 %'#1"083 µ./".04 #"()4.


2 4 2 K E º A § A I O 7 : ¢ # ™ ¢ # " ™ &"&" æ % º # " ' " ° ƒ" ª ª # ' " ™ À ™ & % ª "&"

™¯‹Ì· 7.7

;( 82µ% .08#*()x(n1,n2) $0% /3,

Z81383 >)/(%F0(-6#$3834 7.1.

x(n1,n2)n2

n1


+C2@ 9@?C2C5 5LC7 <@A;>=5µ9 C5 <;5µµ46B, 5µ9CBSI2C5 =C2@ JI>=G2=2 O4=O4B-

=C5C5 R23456B 3>IC;5. ' C;?8JH O;B=2H CA@ 3>IC;A@ 5LCM@ 65GJ;>Z9C54 58? C2

O4=O4B=C5C2 6;JL=C467 CJLH 58?6;4=2. *4J =L<696;4µQ@5, CJ =7µ5 9:?OJL L8J-

IJ<>Z9C54 AH 2 O4=O4B=C5C2 =L@QI4:2 C2H 6;JL=C467H 58?6;4=2H 654 CJL =7µ5CJH

94=?OJL. # O4=O4B=C5C2 =L@QI4:2 9>@54 58JCQI9=µ5 C2H <;5µµ46?C2C5H 654 C2H 5µ9-

C5SI2C?C2C5H CA@ 3>IC;A@ =C2@ JI>=G2=2.

7.2 Ã·Ú·ÎÙËÚÈÛÙÈÎ¿ ÙˆÓ ‰ÈÛ‰È¿ÛÙ·ÙˆÓ „ËÊÈ·ÎÒÓ Û˘ÛÙËÌ¿ÙˆÓ

7.2.1 (Úfi!ÔÈ Û‡Ó‰ÂÛË˜ ÙˆÓ ‰ÈÛ‰È¿ÛÙ·ÙˆÓ „ËÊÈ·ÎÒÓ Ê›ÏÙÚˆÓ

# 8;B:2 C2H O4=O4B=C5C2H =L@QI4:2H QT94 =2µ5@C46QH 4O4?C2C9H J4 J8J>9H 9>@54 T;7-

=4µ9H =C2@ 5@BIL=2 CA@ O4=O4B=C5CA@, <;5µµ46M@ 5µ9CBSI2CA@ =C2@ JI>=G2=2

=L=C2µBCA@. PC=4, 5@ OKJ =L=C7µ5C5 µ9 6;JL=C46QH 58J6;>=94H h(n1,n2) 654 g(n1,n2)

9>@54 =L@O9O9µQ@5 =9 =94;B (cascade), ?8AH 35>@9C54 =CJ +T7µ5 7.8, C?C9 2 C9I467

Q:JOJH y(n1,n2) µ8J;9> @5 L8JIJ<4=C9> T;2=4µJ8J4M@C5H CJ =7µ5 94=?OJL x(n1,n2),

CJ 9@O4Bµ9=J =7µ5 w(n1,n2) 654 CJ <9<J@?H ?C4 6BG9 Q@5 3>IC;J LIJ8J49> O4=O4B=C5C2

=L@QI4:2:

(7.11)

E9OJµQ@JL ?C4 <45 C2 O4=O4B=C5C2 =L@QI4:2 4=TK94 2 8;J=9C54;4=C467 4O4?C2C5, 2

=TQ=2 (7.11) µ8J;9> @5 <;539>

(7.12)

(8? C2 =TQ=2 7.12 8;J6K8C94 ?C4 C5 OKJ =9 =94;B 3>IC;5 4=JOL@5µJK@ µ9 Q@5 C;>CJ

8JL QT94 6;JL=C467 58?6;4=2 5LC7 8JL 8;J6K8C94 58? C2 =L@QI4:2 CA@ 984µQ;JLH

y x h g x h= WW WW = WW( ) XYZ4

w x h

y w g x h g

= WW= WW = WW WW( )


6;JL=C46M@ 58J6;>=9A@. !8>=2H, O9OJµQ@JL ?C4 <45 C2 O4=O4B=C5C2 =L@QI4:2 4=TK-

94 2 5@C4µ9C5G9C467 4O4?C2C5

,

2 =L@JI467 =Lµ89;43J;B CJL O4=O4B=C5CJL 3>IC;JL O9@ 5IIBZ94 5@ 935;µJ=C9> =CJ

=7µ5 x(n1,n2) 8;MC5 CJ 3>IC;J g 654 µ9CB CJ h.

h g g hWW = WW

2 4 37 . 2 Ã " ƒ" ' & % ƒ # ™ & # ' " & ø ¡ ¢ # ™ ¢ # " ™ &"& ø ¡ æ % º # " ' ø ¡ ™ À ™ & % ª "& ø ¡

x(n1,n2) h(n1,n2) g(n1,n2)w(n1,n2) y(n1,n2)

x(n1,n2) y(n1,n2)

h(n1,n2)p1(n1,n2)

p2(n1,n2)g(n1,n2)

™¯‹Ì· 7.8

T08*0?8/%/% $"%µ-µ01? <96/"% 8),*.-*.µC,% 8. 8.0"?. O1"()8/012 %'#1"0-83 /() 8),*)%8µ(7'"(17'/.0 %'# /3*08*0?8/%/3 8),C-60F3 h**g. O %66%-$2 /34 8.0"?4 /@,*7( <96/"@, *9,.05.@"3/01? /( 9*0('?,/% %'(/C6.8µ%.

™¯‹Ì· 7.9

;( '%"?6636(878/3µ%, *08*0?-8/%/@,, $"%µµ0-1+, 1%0 %µ./?-I63/@, 8/3, (69-85383 H3<0%1+,<96/"@, 08(*),%-µ.9 µ. <96/"( '()CE.0 1"()8/012%'#1"083h08(*(n1,n2) =h(n1,n2) +g(n1,n2).

+C2@ 89;>8CA=2 8JL QTJLµ9 85;BII2I2 =L@O9=µJIJ<>5 OKJ O4=O4B=C5CA@, <;5µµ4-

6M@ 5µ9CBSI2CA@ =C2@ JI>=G2=2 R23456M@ 3>IC;A@, ?8AH =CJ +T7µ5 7.9, 2 Q:J-

OJH y(n1,n2) O>@9C54 AH

(7.13)

# =TQ=2 (7.13) O2IM@94 ?C4, 5@ OKJ O4=O4B=C5C5, <;5µµ46B 654 5µ9CBSI2C5 =C2@

JI>=G2=2 R23456B =L=C7µ5C5 =L@O9GJK@ 85;BII2I5, 4=JOL@5µJK@ µ9 Q@5 C;>CJ

8JL QT94 6;JL=C467 58?6;4=2 CJ BG;J4=µ5 CA@ 984µQ;JLH 6;JL=C46M@ 58J6;>=9A@.

y n n p n n p n n

x h x g

x h g

x h

( , ) ( , ) ( , )

( )

1 2 1 1 2 2 1 2= + == WW + WW == WW + == WW

XYZ4

¶·Ú¿‰ÂÈÁÌ· 7.3

045 OKJ O4=O4B=C5C5, <;5µµ46B, 5µ9CBSI2C5 =C2@ JI>=G2=2 3>IC;5 J4 6;JL=C46QH

58J6;>=94H h(n1,n2) 654 g(n1,n2) QTJL@ ?8AH 85;JL=4BZJ@C54 =CJ +T7µ5 7.10. $5 =T9-

O45=C9> 2 6;JL=C467 58?6;4=2 CJL 3>IC;JL 8JL 8;J6K8C94 5@ C5 3>IC;5 5LCB =L@-

O9GJK@ 5) =9 =94;B 654 S) 85;BII2I5.


2 4 4 K E º A § A I O 7 : ¢ # ™ ¢ # " ™ &"&" æ % º # " ' " ° ƒ" ª ª # ' " ™ À ™ & % ª "&"

=G/0

+C2@ 89;>8CA=2 =K@O9=2H =9 =94;B 2 6;JL=C467 58?6;4=2 CJL =L@JI46JK 3>IC;JL

O>@9C54 58? C2 =L@QI4:2 CA@ 6;JL=C46M@ 58J6;>=9A@ CJL +T7µ5CJH 7.10:

h08(* = h**g

045 @5 6B@JLµ9 5LC?@ CJ@ L8JIJ<4=µ? G5 µ9C5CJ8>=JLµ9 C2@ h(n1,n2) =9 ?I9H 969>-

@9H C4H GQ=94H 8JL QT94 µ9 C2@ g(n1,n2) =Kµ8CA=2 6B8J4A@ ?;A@. # O45O465=>5 9>@54

85;?µJ45 µ9 C2 =L@QI4:2 8JL 89;4<;B39C54 =CJ +T7µ5 7.5. )J C9I46? 58JCQI9=µ5

35>@9C54 =CJ +T7µ5 7.11.

h(n1,n2) g(n1,n2)n2 n2

n1 n1

0.21/3

15h08(*(n1,n2) n2

n1

1

1

11

11

11

3 2

2

h(n1,n2) g(n1,n2)n2 n2 n2

n1 n1 n1

h08(*(n1,n2)

0.2 0.21/3

8/15

™¯‹Ì· 7.10

="()8/01C4 %'(1"9-8.04 *08*0?8/%/@,,$"%µµ01+,, %µ./?-I63/@, 8/3, (69-85383 H3<0%1+,

<96/"@,.

™¯‹Ì· 7.12

O 1"()8/012 %'#1"083/() '%"?6636() 8),-*)%8µ(7 /@, <96/"@,h 1%0 g '"(17'/.0 %'#/3, %'62 ?5"(083 /@,*7( 1"()8/01+, %'(-

1"98.@,.

™¯‹Ì· 7.11

O 1"()8/012 %'#1"083 /() ., 8.0"?8),*)%8µ(7 /@, <96/"@, h 1%0 g

'"(17'/.0 %'# /3 8),C60F3 /@, *7(1"()8/01+, %'(1"98.@,.

+C2@ 89;>8CA=2 C2H =K@O9=2H CA@ OKJ 3>IC;A@ 85;BII2I5, 2 4=JOK@5µ2 6;JL=C4-

67 58?6;4=2 8;J6K8C94 58? C2@ 58I7 BG;J4=2 CA@ OKJ 984µQ;JLH 6;JL=C46M@ 58J-

6;>=9A@ =Kµ3A@5 µ9 C2 =TQ=2 (7.13) 654 35>@9C54 =CJ +T7µ5 7.12.

*J45 9>@54 2 6;JL=C467 58?6;4=2 h08(* O4=O4B=C5CJL <;5µµ46JK 3>IC;JL 8JL 8;J-

6K8C94 58? CJ@ 85;BII2IJ =L@OL5=µ? CJL O4=O4B=C5CJL 3>IC;JL C2H E;5=C2;4?-

C2C5H 7.1, µ9 O4=O4B=C5CJ 3>IC;J 8JL QT94 6;JL=C467 58?6;4=2 g(n1,n2) =

0,5[*(n1–1,n2) + *(n1 + 1,n2)];



# 4=JOK@5µ2 6;JL=C467 58?6;4=2 G5 8;J6KR94 58? C2@ 8;?=G9=2 CA@ OKJ 6;JL-

=C46M@ 58J6;>=9A@ ?8AH 35>@9C54 =CJ +T7µ5 7.13. +CJ =T7µ5 5LC? O>@9C54 2 =Lµ-

µ9C;467 h(n1,n2) C2H E;5=C2;4?C2C5H 7.1, 2 g(n1,n2) 654 2 h08(*(n1,n2).

2 4 57 . 2 Ã " ƒ" ' & % ƒ # ™ & # ' " & ø ¡ ¢ # ™ ¢ # " ™ &"& ø ¡ æ % º # " ' ø ¡ ™ À ™ & % ª "& ø ¡

+CJ +T7µ5 7.14 O>@J@C54 J4 6;JL=C46QH 58J6;>=94H h(n1,n2) 654 g(n1,n2) OKJ 3>I-

C;A@. # h08(*(n1,n2), 8JL O>@9C54 O>8I5 8;J6K8C94 58? CJ@ 85;BII2IJ 7 CJ@ 9@ =94;B

=L@OL5=µ? CJLH;


h(n1,n2) g(n1,n2)n2 n2 n2

n1 n1 n1

h08(*(n1,n2)

™¯‹Ì· 7.13

O 1"()8/012 %'#-1"083 /() '%"?6-636() 8),*)%8µ(7/@, <96/"@, h 1%0g '"(17'/.0 %'#/3, %'62 ?5"(083/@, *7( 1"()8/0-1+, %'(1"98.@,.

h(n1,n2) g(n1,n2)h08(*(n1,n2)n2 n2

n1 n1 n1

n2

1/3 1/3 1/9

™¯‹Ì· 7.14

="()8/01C4 %'(-1"98.04 *08*0?8/%-/@, $"%µµ01+,%µ./?I63/@, 8/3,(6985383 H3<0%-1+, <96/"@,.

7.2.2 ¢È·¯ˆÚ›ÛÈÌ· ‰ÈÛ‰È¿ÛÙ·Ù· „ËÊÈ·Î¿ Û˘ÛÙ‹Ì·Ù·

P@5 *0%E@"980µ( (separable) O4=O4B=C5CJ =K=C2µ5 QT94 6;JL=C467 58?6;4=2 h(n1,n2)

CQCJ45 M=C9 @5 µ8J;9> @5 <;539> AH <4@?µ9@J OKJ µJ@JO4B=C5CA@ 56JIJLG4M@:

(7.14)

+C2@ 89;>8CA=2 5LC7 2 Q:JOJH y(n1,n2) CJL =L=C7µ5CJH <45 9>=JOJ x(n1,n2) µ8J;9>

@5 <;539>:

(7.15)

y n n x n l n l h l h l

h l x n l n l h l

l

( , ) ( , ) ( ) ( )

( ) ( , ) ( )

1 2 1 1 2 2 1 1 2 2

1 1 1 1 2 2 2 2

2

= # #

= # #

=#"

"

=#"

"

=#"

"

=#"

"

55

5 5

l

l l

1

1 2

h n n h n h n( , ) ( ) ( )1 2 1 1 2 2= !


2 4 6 K E º A § A I O 7 : ¢ # ™ ¢ # " ™ &"&" æ % º # " ' " ° ƒ" ª ª # ' " ™ À ™ & % ª "&"

)J <4@?µ9@J =CJ O9KC9;J BG;J4=µ5 C2H C9I9LC5>5H =TQ=2H 5@C48;J=A89K94 Q@5 O4=-

O4B=C5CJ 8>@565 5;4GµM@. !B@ GQ=JLµ9

(7.16)

2 =TQ=2 (7.15) <>@9C54

(7.17)

# O45O465=>5 5LC7 O2IM@94 ?C4 µ8J;JKµ9 @5 L8JIJ<>=JLµ9 8;MC5 C2@ 56JIJLG>5

f(n1,n2) 8;5<µ5CJ8J4M@C5H µJ@JO4B=C5C2 =L@QI4:2 =C4H =C7I9H CJL 8>@565 x(n1,n2)

(n1 = =C5G9;?). +C2 =L@QT945 L8JIJ<>ZJLµ9 C4H µJ@JO4B=C5C9H =L@9I>:94H =C4H <;5µ-

µQH C2H f(n1,n2) (n2 = =C5G9;?) =Kµ3A@5 µ9 C2 =TQ=2 (7.17).

¶·Ú¿‰ÂÈÁÌ· 7.4

$5 58JO94TG9> ?C4 2 6;JL=C467 58?6;4=2 8JL 89;4<;B39C54 58? C2 =TQ=2

(7.18)

5@C4=CJ4T9> =9 O45TA;>=4µJ O4=O4B=C5CJ =K=C2µ5.

=G/0

# O4=O4B=C5C2 6;JL=C467 58?6;4=2 h(n1,n2) µ8J;9> @5 <;539> AH

(7.19)

O2I5O7, AH <4@?µ9@J OKJ µJ@JO4B=C5CA@ 6;JL=C46M@ 58J6;>=9A@. !8JµQ@AH, 9>@54

O45TA;>=4µ2.

7.2.3 '˘ÛÙ·ı‹ ‰ÈÛ‰È¿ÛÙ·Ù· Û˘ÛÙ‹Ì·Ù·

\8AH 654 =C2@ 89;>8CA=2 CA@ µJ@JO4B=C5CA@ 3>IC;A@, C5 O4=O4B=C5C5 =L=C7µ5-

C5 8JL µ8J;JK@ @5 T;2=4µJ8J42GJK@ 9>@54 µ?@J C5 9L=C5G7. P@5 CQCJ4J 3>IC;J O>@94

8989;5=µQ@2 Q:JOJ <45 6BG9 8989;5=µQ@2 =9 8IBCJH 9>=JOJ. # 5@5<65>5 654 465@7

=L@G762 <45 @5 9>@54 9L=C5GQH Q@5 O4=O4B=C5CJ, <;5µµ46?, 5µ9CBSI2CJ =C2@ JI>-

=G2=2 R23456? 3>IC;J 9>@54 2 56?IJLG2:

h n n

n n n nh n h n( , ) exp( ) exp( )exp( ) ( ) ( )1 2

12

22

12

22

1 1 2 25 5 5= # + = # # =

h n nn n

n n( , )

exp( ) ,1 2

12

22

1 255 5

0

= # + & &01D

2D

J8JLO78JC9 5IIJK

y n n h l f n l n( , ) ( ) ( , )1 2 1 1 1 1 2= #=#"

"

5l1

f n n x n n l h ll

( , ) ( , ) ( )1 2 1 2 2 2 2

2

= #=#"

"

5


(7.20)

?8JL 2 h(n1,n2) 9>@54 2 6;JL=C467 58?6;4=2 CJL 3>IC;JL.

h n nn

( , )1 2

2 =#"

"

=#"

"

55 = < "n1

N

2 4 77 . 3 A ¶ $' ƒ # ™ % ™ & % ™ À Ã ¡ $ & % &" & ø ¡ ¢ # ™ ¢ # " ™ &"& ø ¡ æ % º # " ' ø ¡ ™ À ™ & % ª "& ø ¡

# 6;JL=C467 58?6;4=2 8JL O>@9C54 =CJ *5;BO94<µ5 7.4 5@C4=CJ4T9> =9 9L=C5GQH

7 5=C5GQH O4=O4B=C5CJ 3>IC;J 654 <45C>;


7.3 A!fiÎÚÈÛË ÛÙË Û˘¯ÓfiÙËÙ· ÙˆÓ ‰ÈÛ‰È¿ÛÙ·ÙˆÓ „ËÊÈ·ÎÒÓ Û˘ÛÙËÌ¿ÙˆÓ

+C4H 8;J2<JKµ9@9H 85;5<;B3JLH 9:2<7=5µ9 CJ@ C;?8J µ9 CJ@ J8J>J L8JIJ<>Z9C54

2 Q:JOJH y(n1,n2) 9@?H O4=O4B=C5CJL R23456JK =L=C7µ5CJH. *;JH CJKCJ 9>@54 585-

;5>C2CJ @5 8;5<µ5CJ8J42G9> 2 8;B:2 C2H O4=O4B=C5C2H =L@QI4:2H µ9C5:K CJL =7µ5-

CJH 94=?OJL x(n1,n2) 654 C2H 6;JL=C467H 58?6;4=2H h(n1,n2) CJL =L=C7µ5CJH. +C2@

JL=>5 2 6;JL=C467 58?6;4=2 h(n1,n2) 65GJ;>Z94 8I7;AH C2@ 58?6;4=2 CJL =L=C7-

µ5CJH <45 6BG9 (n1,n2). +C2 =L@QT945 G5 O9>:JLµ9 CJ@ C;?8J 8JL 2 =L@B;C2=2 h(n1,n2)

65GJ;>Z94 C2 =Lµ89;43J;B CJL 3>IC;JL =C4H TA;46QH =LT@?C2C9H @1 654 @2, 98JµQ-

@AH G5 982;9BZ94 654 CJ 3B=µ5 CJL =7µ5CJH 94=?OJL (?8AH 5LC? CJ <@A;>=5µ9 =C2@

!@?C2C5 6.4).

045 @5 8;JTA;7=JLµ9 =C2@ 5@BIL=2 C2H =Lµ89;43J;BH CJL O4=O4B=C5CJL =L=C7-

µ5CJH =C2 =LT@?C2C5 G5 T;2=4µJ8J47=JLµ9 CJ =7µ5

)J =7µ5 5LC? 5@C4=CJ4T9> =9 Q@5 µ4<5O46? O4=O4B=C5CJ 2µ4CJ@J94OQH µ9 @1 654 @2

8;5<µ5C46JKH 5;4GµJKH 8JL 5@C4=CJ4TJK@ =C4H TA;46QH =LT@?C2C9H @1 654 @2 5@C>-

=CJ4T5. # Q:JOJH y(n1,n2) =Kµ3A@5 µ9 C2 =TQ=2 C2H =L@QI4:2H (7.9) QT94 AH 9:7H:

(7.21)

# C9I9LC5>5 =TQ=2 O9>T@94 ?C4 CJ =7µ5 9:?OJL y(n1,n2) 58JC9I9>C54 58? C4H >O49H

y n n e h r r

e e h r r

e H

j n r j n r

rr

j n j n j r j r

rr

j n j n

( , ) ( , )

( , )

(

( ) ( )1 2 1 2

1 2

1 1 1 2 2 2

21

1 1 2 2 1 1 2 2

21

1 1 2 2

=

=

=

# + #

=#"

"

=#"

"

+ # #

=#"

"

=#"

"

+

55

55

6 6

6 6 6 6

6 6 6

11 2, )6

x n n e j n j n( , )1 21 1 2 2= +6 6


2 4 8 K E º A § A I O 7 : ¢ # ™ ¢ # " ™ &"&" æ % º # " ' " ° ƒ" ª ª # ' " ™ À ™ & % ª "&"

=LT@?C2C9H µ9 CJ =7µ5 94=?OJL x(n1,n2), 5IIB CJ 8IBCJH CJLH 654 2 3B=2 CJLH QTJL@

L8J=C9> C2@ 98>O;5=2 C2H =L@B;C2=2H H(@1,@2). # =L@B;C2=2 5LC7, ?8AH 654 =C2@

89;>8CA=2 CA@ µJ@JO4B=C5CA@ =2µBCA@ CJL &935I5>JL 3, 65I9>C54 «58?6;4=2

=LT@?C2C5H CJL 3>IC;JL» 654 8;J35@MH 4=JKC54 µ9:

(7.22)

)J µQC;J |H(@1,@2)| C2H =L@B;C2=2H 65GJ;>Z94 CJ 8MH G5 982;95=CJK@ J4 =LT@?C2-

C9H CJL =7µ5CJH 58? CJ O4=O4B=C5CJ =K=C2µ5. 045 C4H =LT@?C2C9H (@1,@2) ?8JL

|H(@1,@2)|@0, J4 5@C>=CJ4T9H =LT@?C2C9H CJL =7µ5CJH 94=?OJL 58J6?8CJ@C54.

'L=45=C46B 2 =TQ=2 (7.22) O2IM@94 ?C4 2 58?6;4=2 =LT@?C2C5H CJL =L=C7µ5CJH

9>@54 J O4=O4B=C5CJH O456;4C?H µ9C5=T2µ5C4=µ?H Fourier C2H 6;JL=C467H 58?6;4=2H

CJL =L=C7µ5CJH.

# 58?6;4=2 =LT@?C2C5H H(@1,@2) 9>@54 89;4JO467 =L@B;C2=2 CA@ =LT@JC7CA@ @1

654 @2 µ9 89;>JOJ 2':

(7.23)

# 89;4JO46?C2C5 C2H 58?6;4=2H CA@ R23456M@ =L=C2µBCA@ 9>@54 5@5µ9@?µ9@2

I?<A C2H R234567H 3K=2H 5LCM@. bO45 =Lµ89;43J;B QT94 654 CJ 3B=µ5 CA@ O4=-

O4B=C5CA@ R23456M@ =2µBCA@ (!@?C2C5 6.4). (@BIJ<2 O9 =Lµ89;43J;B 85;JL-

=4BZJL@ 654 C5 µJ@JO4B=C5C5 R23456B =7µ5C5 654 =L=C7µ5C5 (&93BI545 1 QAH 3).

¶·Ú¿‰ÂÈÁÌ· 7.5

$5 L8JIJ<4=C9> CJ µQC;J C2H 58?6;4=2H =C2 =LT@?C2C5 CJL O4=O4B=C5CJL R2345-

6JK =L=C7µ5CJH µ9 6;JL=C467 58?6;4=2

h(n1,n2) = *(n1 + 1,n2) + *(n1–1,n2) + *(n1,n2)

=G/0

E9OJµQ@JL ?C4 2 6;JL=C467 58?6;4=2 QT94 µ?@J C;94H ?;JLH >=JLH µ9 C2 µJ@BO5 =C4H

GQ=94H (n1,n2): (–1,0), (1,0) 654 (0,0), 2 =TQ=2 (7.22) <;B39C54

(7.24)

# 58?6;4=2 =LT@?C2C5H 8JL 89;4<;B39C54 58? C2 =TQ=2 (7.24) 9>@54 8;5<µ5C467,

H e e ej j( , )

cos( )

6 66

6 61 2

0

1

1 1

1 2

= + + == +

+ #

H H

H H

( , ) ( , )

( , ) ( , )

6 ) 6 6 66 6 ) 6 6

1 2 1 2

1 2 1 2

2

2

+ =+ =

H e h r rj r j r

rr

( , ) ( , )6 6 6 61 2 1 2

1 1 2 2

21

= # #

=#"

"

=#"

"

55


98JµQ@AH O9@ 5IIBZ94 C2 3B=2 CA@ 5;µJ@46M@ =L@4=CA=M@ CJL =7µ5CJH, µ9C5SBI-

I9C54 O9 µ?@J =C2@ 65C9KGL@=2 C2H @1. X;2=4µJ8J4M@C5H 65CBII2IJ IJ<4=µ46? =T9-

O>5=2H (I.T. MATLAB), µ8J;JKµ9 @5 OJKµ9 ?C4 2 =L@B;C2=2 |H(@1,@2)| QT94 C2

µJ;37 8JL O9>T@9C54 =CJ +T7µ5 7.15.

2 4 97 . 3 A ¶ $' ƒ # ™ % ™ & % ™ À Ã ¡ $ & % &" & ø ¡ ¢ # ™ ¢ # " ™ &"& ø ¡ æ % º # " ' ø ¡ ™ À ™ & % ª "& ø ¡

™¯‹Ì· 7.15

O µ("<2 /348),?"/3834|H(@1,@2)| =|1+2cos(@1)|,#'@4 CE.0 63<5.9µ. /( MATLAB.

$5 L8JIJ<4=C9> CJ µQC;J C2H 58?6;4=2H =C2 =LT@?C2C5 O4=O4B=C5CJL R23456JK

3>IC;JL µQ=2H C4µ7H C2H 984IJ<7H =5H.


P@5 <96/"( µC834 /0µ24 L8JIJ<>Z94 C2 µQ=2 C4µ7 J;4=µQ@A@ O94<µBCA@ CJL =7µ5CJH

94=?OJL x(n1,n2) 8JL S;>=6J@C54 <K;A 58? C2 GQ=2 (n1,n2), 654 O>@94 CJ 58JCQI9=µ5

=CJ O9><µ5 CJL =7µ5CJH 9:?OJL y(n1,n2) 8JL S;>=69C54 =C2@ 5@C>=CJ4T2 GQ=2. !894-

O7 2 O45O465=>5 5LC7 985@5I5µSB@9C54, 65GMH µ9C564@JKµ5=C9 =9 ?I9H C4H GQ=94H

(n1,n2), CJ 3>IC;J 65I9>C54 654 <96/"( /() 10,(7µ.,() µC8(). *;?694C54 <45 Q@5 S5GL-

+LT@?C2C5 @2

+LT@?C2C5 @1

|H(@1,@2)|


2 5 0 K E º A § A I O 7 : ¢ # ™ ¢ # " ™ &"&" æ % º # " ' " ° ƒ" ª ª # ' " ™ À ™ & % ª "&"

89;5C? 3>IC;J CJL J8J>JL J4 =LT@?C2C9H 58J6J87H 9:5;CM@C54 58? CJ 8I7GJH CA@

O94<µBCA@ 8JL G5 I23GJK@ 58? CJ =7µ5 94=?OJL. NH S5GL89;5C? 3>IC;J, 58J6?-

8C94 C4H LR2IQH =LT@?C2C9H, «I945>@94» (smooths) CJ G?;LSJ, 5IIB 8;J65I9> G?IA-

=2 (blurring) =C4H 56µQH CA@ 946?@A@.

P@5 58I? 3>IC;J µQ=2H C4µ7H 9>@54 5LC? 8JL 89;4<;B3269 =CJ *5;BO94<µ5 7.5. +’ 5LC?

6BG9 C4µ7 CJL =7µ5CJH 9:?OJL y(n1,n2) L8JIJ<>Z9C54 58? C;94H C4µQH CJL =7µ5CJH 94=?-

OJL, 5LCQH <45 C4H J8J>9H 2 6;JL=C467 58?6;4=2 CJL 3>IC;JL h(n1,n2) 9>@54 µ2 µ2O9@467.

P@5 BIIJ 3>IC;J µQ=2H C4µ7H 9>@54 5LC? CJL J8J>JL 2 6;JL=C467 58?6;4=2 QT94 AH 9:7H:

&BG9 O9><µ5 CJL =7µ5CJH 9:?OJL L8JIJ<>Z9C54 58? C2 µQ=2 C4µ7 CA@ 8Q@C9 O94<µBCA@

CJL =7µ5CJH 94=?OJL x(n1,n2), x(n1–1,n2), x(n1 + 1,n2), x(n1,n2–1) 654 x(n1,n2 + 1). +Kµ-

3A@5 µ9 C2@ 9:>=A=2 (7.22) CJ µQC;J C2H 58?6;4=2H =LT@?C2C5H H(@1,@2) QT94 AH 9:7H:

(7.25)

E9OJµQ@JL ?C4 2 C9I9LC5>5 8J=?C2C5 9>@54 8;5<µ5C467 CJ µQC;J C2H 9>@54 2 58?IL-

C2 C4µ7 C2H, 2 J8J>5 <;5346B QT94 ?8AH 35>@9C54 =CJ +T7µ5 7.16.

H e e e e ej j j j( , )

cos cos

6 6

6 6

6 6 6 61 2

0

1 2

1515

1 2 2

1 1 2 2= + + + +[ ] =

= + +[ ]

# #

h n n n n n n n n n n n n( , ) [ ( , ) ( , ) ( , ) ( , ) ( , )]1 2 1 2 1 2 1 2 1 2 1 2

15

1 1 1 1= + # + + + # + +4 4 4 4 4

™¯‹Ì· 7.16

O µ("<2 /348),?"/3834

|H(@1,@2)| =|0,2{1 + 2cos(@1)

+ 2cos(@2)}|#'@4 CE.0 63<5.9

µ. /( MATLAB.

+LT@?C2C5 @2

+LT@?C2C5 @1

|H(@1,@2)|



+C2@ 9@?C2C5 5LC7 9>O5µ9 ?C4 2 6;JL=C467 58?6;4=2 9@?H O4=O4B=C5CJL R23456JK

3>IC;JL 65GJ;>Z94 654 C2@ 58?6;4=7 CJL =C2 =LT@?C2C5. \C5@ 2 6;JL=C467 58?-

6;4=2 9>@54 =Lµµ9C;467, 2 58?6;4=2 =C2 =LT@?C2C5 9>@54 8;5<µ5C467 654 O9@ 982-

;9BZ94 C2 3B=2 CA@ =L@4=CA=M@ CJL =7µ5CJH. +C2@ 89;>8CA=2 5LC7 C5 3>IC;5

65IJK@C54 «3>IC;5 µ2O9@467H 3B=2H» (zero phase filters).

7.4 ™¯Â‰›·ÛË ‰ÈÛ‰È¿ÛÙ·ÙˆÓ „ËÊÈ·ÎÒÓ Ê›ÏÙÚˆÓ

# =T9O>5=2 CA@ O4=O4B=C5CA@ R23456M@ 3>IC;A@ 9>@54 OL=6JI?C9;2 58? 5LC7 CA@

µJ@JO4B=C5CA@. !8>=2H, 2 LIJ8J>2=7 CJLH 9>@54 8JIK8IJ62 <45C> =C4H 89;4==?C9;9H

89;48CM=94H J4 O4=O4B=C5C9H =L@5;C7=94H O9@ µ8J;JK@ @5 85;5<J@CJ8J42GJK@, QC=4

M=C9 @5 JO2<7=JL@ =9 58IJ8J42µQ@9H OJµQH.

+C2@ 85;JK=5 9@?C2C5 G5 8;J=O4J;>=JLµ9 5;T46B C2@ 6;JL=C467 58?6;4=2 9@?H

O4=O4B=C5CJL R23456JK 3>IC;JL 58? C4H 984GLµ2CQH 8;JO45<;53QH C2H 58?6;4=7H

CJL H(@1,@2) =C4H =LT@?C2C9H @1 654 @2. +C2 =L@QT945 G5 <@A;>=JLµ9 µ>5 58? C4H

µ9G?OJLH =T9O>5=2H CA@ O4=O4B=C5CA@ FIR 3>IC;A@, 5LC7 CA@ 85;5GK;A@. D5 6I9>-

=JLµ9 C2@ 9@?C2C5 µ9 5@53J;B =9 935;µJ<QH CA@ O4=O4B=C5CA@ R23456M@ 3>IC;A@

=C2@ 989:9;<5=>5 C2H 946?@5H.

7.4.1 À!ÔÏÔÁÈÛÌfi˜ ÙË˜ ÎÚÔ˘ÛÙÈÎ‹˜ ·!fiÎÚÈÛË˜ ·!fi ÙËÓ ·!fiÎÚÈÛËÛ˘¯ÓfiÙËÙ·˜

+Kµ3A@5 µ9 C2 =TQ=2 (7.22) 2 58?6;4=2 =LT@?C2C5H H(@1,@2) 9@?H O4=O4B=C5CJL

R23456JK 3>IC;JL 58JC9I9> CJ O4=O4B=C5CJ µ9C5=T2µ5C4=µ? Fourier C2H 6;JL=C4-

2 5 17 . 4 ™ Ã ! ¢ # " ™ % ¢ # ™ ¢ # " ™ &"& ø ¡ æ % º # " ' ø ¡ º # § & ƒ ø ¡

!>O5µ9 =C5 8;J2<JKµ9@5 ?C4 9>@54 OL@5C? 2 58?6;4=2 =LT@?C2C5H CJL O4=O4B=C5-

CJL 3>IC;JL @5 9>@54 8;5<µ5C467. +C2@ 89;>8CA=2 5LC7 CJ 3>IC;J O9@ 5IIJ4M@94

C2 3B=2 CA@ 5;µJ@46M@ CJL =7µ5CJH (3>IC;J µ2O9@467H 3B=2H). *J4J 58? C5 98?-

µ9@5 T5;56C2;4=C46B 8;Q894, 65CB C2 <@Mµ2 =5H, @5 QT94 2 6;JL=C467 58?6;4=2

CJL 3>IC;JL <45 @5 9>@54 5LC? µ2O9@467H 3B=2H;

5. '4 ?;J4 C2H 6;JL=C467H 58?6;4=2H @5 9>@54 ?IJ4 >=J4.

S. # 6;JL=C467 58?6;4=2 CJL 3>IC;JL @5 9>@54 =Lµµ9C;467.

<. # 6;JL=C467 58?6;4=2 @5 9>@54 O45TA;>=4µ2.



2 5 2 K E º A § A I O 7 : ¢ # ™ ¢ # " ™ &"&" æ % º # " ' " ° ƒ" ª ª # ' " ™ À ™ & % ª "&"

67H CJL 58?6;4=2H h(n1,n2). ,9 BII5 I?<45, 2 h(n1,n2) 9>@54 J O4=O4B=C5CJH 5@C>-

=C;J3JH µ9C5=T2µ5C4=µ?H Fourier C2H H(@1,@2):

(7.26)

# JIJ6I7;A=2 LIJ8J49>C54 =9 µ>5 89;>JOJ C2H =L@B;C2=2H H(@1,@2) =C4H 65C9L-

GK@=94H @1 654 @2.

¶·Ú¿‰ÂÈÁÌ· 7.6

$5 L8JIJ<4=C9> 2 6;JL=C467 58?6;4=2 4O5@46JK S5GL89;5CJK 3>IC;JL CJL J8J>JL

2 58?6;4=2 =LT@?C2C5H H(@1,@2) QT94 AH 9:7H:

(7.27)

=G/0

# 58?6;4=2 =C2 =LT@?C2C5 CJL 3>IC;JL G5 QT94 C2 µJ;37 8JL 35>@9C54 =CJ +T7µ5 7.17.

Ha b

( , ),

6 66 ) 6 )

1 21 21

0=

& < & <01D

2D

J8JLO78JC9 5IIJK

h n n H e d dj n j n( , ) ( , )1 2 2 1 2 1 21

41 1 2 2= +

##==)

6 6 6 66 6

)

)

)

)

@2

@1

'

'-'

-'

a

-a

b-b

™¯‹Ì· 7.17

>'#1"083 8)E,#-/3/%4 0*%,01(7

I%5)'."%/(7 <96-/"() /./"%$@,0124µ("<24. &/3 810%-8µC,3 '."0(E2 3H(@1,@2) .9,%0 1

1%0 CF@ %'# %)/2,0. O %'#1"083.9,%0 '."0(*012

('."9(*(4 2') 1%0.'%,%6%µI?,./%0

8/( .'9'.*((@1,@2) '"(4 #6.4/04 1%/.)57,8.04.

,9 SB=2 C2 =TQ=2 (7.26) 2 6;JL=C467 58?6;4=2 h(n1,n2) G5 4=JKC54 µ9:

(7.28)

h n n e d d

e d e d

ann

bnn

a

aj n j n

b

b

j n

a

aj n

b

b

( , )

sin( ) sin( )

1 2 2 1 2

1 2

1

1

2

2

1

4

12

12

1 1 2 2

1 1 2 2

=

=

=

#

+

#

# #

= =

= =

)6 6

)6

)6

) )

6 6

6 6


*;Q894 @5 =2µ94M=JLµ9 9OM ?C4 2 6;JL=C467 58?6;4=2 h(n1,n2), 8JL L8JIJ<>=C269,

9>@54 O45TA;>=4µ2, O9OJµQ@JL ?C4 µ8J;9> @5 <;539> AH <4@?µ9@J h1(n1)h2(n2). # µJ@J-

O4B=C5C2 6;JL=C467 58?6;4=2 h(n) 9>@54 C2H >O45H µJ;37H µ9 5LC7 8JL =L@5@C7=5-

µ9 =C2 =T9O>5=2 CA@ µJ@JO4B=C5CA@ 3>IC;A@ {=TQ=2 (4.7)} 654 8;J6K8C94 µ9 CJ@

>O4J 56;4SMH C;?8J 8JL <@A;>=5µ9 =C2@ !@?C2C5 4.3. # µJ;37 C2H 58?6;4=2H

=LT@?C2C5H H(@1,@2) µ8J;9> @5 8J46>I94 5@BIJ<5 µ9 C4H 5854C7=94H CJL =T9O45=C7.

\C5@ GQIJLµ9 CJ 3>IC;J @5 85;JL=4BZ94 6L6I467 =Lµµ9C;>5, O2I5O7 @5 =Lµ89;4-

3Q;9C54 8;JH ?I9H C4H 65C9LGK@=94H µ9 CJ@ >O4J C;?8J, C?C9 2 =645=µQ@2 89;4JT7 CJL

+T7µ5CJH 7.17 9>@54 6K6IJH 654 2 IK=2 <45 C2@ h(n1,n2) 9>@54 89;4==?C9;J 8JIK-

8IJ62.

7.4.2 ™¯Â‰›·ÛË ‰ÈÛ‰È¿ÛÙ·ÙˆÓ FIR Ê›ÏÙÚˆÓ ÌÂ ÙË Ì¤ıÔ‰Ô ÙˆÓ !·Ú·ı‡ÚˆÓ

)5 O4=O4B=C5C5 FIR 3>IC;5, =9 5@C>G9=2 µ9 C5 IIR, QTJL@ 8989;5=µQ@2 6;JL=C467

58?6;4=2. (LC? QT94 AH 58JCQI9=µ5 C5 FIR 3>IC;5 @5 9>@54 8B@C5 9L=C5G7, 53JK <4’

5LCB 4=TK94 2 =TQ=2 (7.20). P@5 BIIJ 98>=2H =2µ5@C46? 8I9J@Q6C2µ5 CA@ O4=O4B-

=C5CA@ FIR 3>IC;A@ 9>@54 ?C4 µ8J;JK@ @5 =T9O45=CJK@ M=C9 2 58?6;4=2 =LT@?C2-

CBH CJLH @5 9>@54 8;5<µ5C467, µ9 58JCQI9=µ5 @5 µ2@ 8;J65IJK@ µ9C5SJI7 =C2 3B=2

CA@ O453?;A@ =LT@JC7CA@ CJL =7µ5CJH. 045 CJ I?<J 5LC? 65IJK@C54 «3>IC;5 µ2O9-

@467H 3B=2H» 654 T;94BZ9C54 8;JH CJKCJ @5 QTJL@ =Lµµ9C;467 6;JL=C467 58?6;4=2:

(7.29)

)4H 8989;5=µQ@9H 5LCQH 6;JL=C46QH 58J6;>=94H C4H 89CLT5>@JLµ9 5@ 58J6?RJLµ9

C2@ 589;4?;4=C2 =9 µ76JH O4=O4B=C5C2 6;JL=C467 58?6;4=2 8JL I5µSB@JLµ9 µ9 CJ@

L8JIJ<4=µ? C2H =TQ=2H (7.28). # 58J6J87 C2H 5;T467H 6;JL=C467H 58?6;4=2H

hD(n1,n2) 8JL 5@C4=CJ4T9> =C4H 4O5@46QH 7 984GLµ2CQH 8;JO45<;53QH µ8J;9> @5 <>@94

5@ 8JII58I5=4B=JLµ9 CJLH ?;JLH C2H hD(n1,n2), Q@5@ 8;JH Q@5@, µ9 CJLH ?;JLH C2H

O4=O4B=C5C2H O456;4C7H =L@B;C2=2H w(n1,n2) 8JL 65I9>C54 '%"?5)"( 654 2 J8J>5

QT94 8989;5=µQ@J µ76JH

(7.30)

# O4=O4B=C5C2 O456;4C7 =L@B;C2=2 w(n1,n2) QT94 8989;5=µQ@J µ76JH 654 8;Q894 @5

9>@54 =Lµµ9C;467, =Kµ3A@5 µ9 C2 =TQ=2

, (7.31)

<45 @5 O45C2;2G9> 2 4O4?C2C5 C2H µ2O9@467H 3B=2H =CJ C9I46? 3>IC;J. # C9I467 58?-

6;4=2 =LT@?C2C5H H(@1,@2) 8;J6K8C94 58? C2 O4=O4B=C5C2 =L@QI4:2 =C2 =LT@?C2-

w n n w n n( , ) ( , )1 2 1 2= # #

h n n h n n w n nD( , ) ( , ) ( , )1 2 1 2 1 2= !

h n n h n n( , ) ( , )1 2 1 2= # #

2 5 37 . 4 ™ Ã ! ¢ # " ™ % ¢ # ™ ¢ # " ™ &"& ø ¡ æ % º # " ' ø ¡ º # § & ƒ ø ¡


2 5 4 K E º A § A I O 7 : ¢ # ™ ¢ # " ™ &"&" æ % º # " ' " ° ƒ" ª ª # ' " ™ À ™ & % ª "&"

C5 C2H 5;T467H 984GLµ2C7H 58?6;4=2H HD(@1,@2) µ9 CJ 3B=µ5 W(@1,@2) CJL 85;5-

GK;JL w(n1,n2), 85;?µJ45 µ9 C2 µJ@JO4B=C5C2 89;>8CA=2 (!@?C2C5 3/ &93BI54J 4).

# 984IJ<7 C2H =L@B;C2=2H CJL 85;5GK;JL 9>@54 65GJ;4=C467 <45 C2@ C9I467 µJ;37

C2H 58?6;4=2H CJL O4=O4B=C5CJL 3>IC;JL. \8AH 654 =C2 µJ@JO4B=C5C2 89;>8CA=2,

CJ 85;BGL;J G5 8;Q894 @5 QT94 65CBII2I2 µJ;37 <45 @5 58J39K<9C54 CJ 354@?µ9@J

Gibbs 654 J S5Gµ?H C2H 9:5=GQ@4=2H =C2 ZM@2 58?;;4R2H CJL 3>IC;JL @5 9>@54 465-

@J8J42C46?H. 045 CJ I?<J 5LC? O9@ T;2=4µJ8J49>C54 CJ C9C;5<A@46? 85;BGL;J, CJ

J8J>J 9:BIIJL 5@C4=CJ4T9> =C2@ 58I7 58J6J87 C2H 5;T467H 6;JL=C467H 58?6;4=2H

hD(n1,n2), 654 98JµQ@AH, =L@OQ9C54 Bµ9=5 µ9 C2 O2µ4JL;<>5 CJL 354@JµQ@JL Gibbs.

,8J;JK@ ?µAH @5 T;2=4µJ8J42GJK@ ?I9H J4 =L@5;C7=94H 85;5GK;A@ 8JL L8B;TJL@

=C2 S4SI4J<;53>5, 5@BIJ<5 6BG9 3J;B µ9 C4H C9I46QH 5854C7=94H <45 CJ 3>IC;J. '

84J 9K6JIJH C;?8JH <45 @5 O2µ4JL;<2G9> 2 O4=O4B=C5C2 R234567 =L@B;C2=2 w(n1,n2)

9>@54 58? CJ <4@?µ9@J OKJ µJ@JO4B=C5CA@ 85;5GK;A@:

(7.32)

¶·Ú¿‰ÂÈÁÌ· 7.7

$5 =T9O45=C9> O4=O4B=C5CJ FIR 3>IC;J µ9 6;JL=C467 58?6;4=2 8JL @5 QT94 =L@JI46B

11 H 11 ?;JLH. # 984GLµ2C7 58?6;4=2 =C2 =LT@?C2C5 HD(@1,@2) G5 8;Q894 @5 9>@54:

(7.33)

$5 <>@94 T;7=2 CJL µJ@JO4B=C5CJL 85;5GK;JL CJL Hamming.

=G/0

# 984GLµ2C7 58?6;4=2 CJL 3>IC;JL =C4H =LT@?C2C9H @1 654@2, 2 J8J>5 89;4<;B39-

C54 58? C2 =TQ=2 (7.33) QT94 <;5346B ?8AH 35>@9C54 =CJ +T7µ5 7.18.

HD ( , )/ , /

6 66 ) 6 )

1 21 21 2 2

0=

& &01D

2D

J8JLO78JC9 5IIJK

w n n w n w n( , ) ( ) ( )1 2 1 2= !

@2

@1

'

'/2

-'/2

-'/2 '/2

-'

-' '

™¯‹Ì· 7.18

>'#1"083 8)E,#-/3/%4 0*%,01(7

I%5)'."%/(7 <96-/"() /./"%$@,0124µ("<24. &/3 810%-8µC,3 '."0(E2 3H(@1,@2) .9,%0 11%0 CF@ %'# %)/2

0. O %'#1"083.9,%0 '."0(*012

('."9(*(4 2') 1%0.'%,%6%µI?,./%0

8/( .'9'.*((@1,@2) '"(4 #6.4/04 1%/.)57,8.04.


# 4O5@467 6;JL=C467 58?6;4=2 G5 L8JIJ<4=C9> µ9 SB=2 C2 =TQ=2 (7.26) AH J 5@C>-

=C;J3JH µ9C5=T2µ5C4=µ?H Fourier C2H 58?6;4=2H =LT@?C2C5H HD(@1,@2):

(7.34)

)J µJ@JO4B=C5CJ 85;BGL;J Hamming O>@9C54 58? C2 =TQ=2

(7.35)

045 R = 5 CJ µJ@JO4B=C5CJ 85;BGL;J w(n) QT94 =L@JI46B 11 ?;JLH. PC=4, CJ O4=-

O4B=C5CJ w(n1,n2) 8JL G5 8;J6KR94 58? C2 =TQ=2 (7.32), G5 QT94 =L@JI46B 11 H 11

= 121 ?;JLH. # C9I467 6;JL=C467 58?6;4=2, 8JL 85;JL=4BZ9C54 =CJ +T7µ5 7.19,

L8JIJ<>Z9C54 AH

(3JK 2 6;JL=C467 58?6;4=2 9>@54 <@A=C7, 2 58?6;4=2 =LT@?C2C5H H(@1,@2) CJL

C9I46JK 3>IC;JL G5 L8JIJ<4=C9> ?8AH 56;4SMH =CJ *5;BO94<µ5 7.5. # 58?6;4=2

5LC7 O>@9C54 =CJ +T7µ5 7.20.

h n n h n n w n nD( , ) ( , ) ( , )1 2 1 2 1 2= !

w n

nN

N n N( ) , , cos( )= + & &0 54 0 46)

–

h n n e d d

e d e d

n

n

n

Dj n j n

j n j n

( , )

sin( ) sin( )

/

/

/

/

/

/

/

/

1 2 2 1 2

2

2

2

2

1

2

2

2

2

2

1

1

2

1

4

12

12

14

2

2

2

1 1 2 2

1 1 2 2

=

=

=

+ +

##

+

#

+

#

==

= =

)6 6

)6

)6

)

)

)

6 6

)

)

)

)

6

)

)6

)

)

))n2

2

2 5 57 . 4 ™ Ã ! ¢ # " ™ % ¢ # ™ ¢ # " ™ &"& ø ¡ æ % º # " ' ø ¡ º # § & ƒ ø ¡

™¯‹Ì· 7.19

O /.6012 1"()8/0-12 %'#1"083 /()*08*0?8/%/() FIR<96/"() /()!%"%*.9$µ%/(47.7, µ. 11 H 11 =121 #"()4.

h(n1,n2)

n1n2


2 5 6 K E º A § A I O 7 : ¢ # ™ ¢ # " ™ &"&" æ % º # " ' " ° ƒ" ª ª # ' " ™ À ™ & % ª "&"

7.4.3 ÕÏÏÂ˜ Ì¤ıÔ‰ÔÈ Û¯Â‰›·ÛË˜ ÙˆÓ ‰ÈÛ‰È¿ÛÙ·ÙˆÓ FIR Ê›ÏÙÚˆÓ

# µQGJOJH =T9O>5=2H CA@ O4=O4B=C5CA@ FIR 3>IC;A@ µ9 C5 85;BGL;5 8JL 85;JL-

=4B=C269 8;J2<JKµ9@5, 9>@54 2 58IJK=C9;2 O45O465=>5 =T9O>5=2H 8JL µ8J;9> @5

S;94 J 5@5<@M=C2H =C2 S4SI4J<;53>5. ,9 C2 µQGJOJ 5LC7 9>@54 OL@5C? @5 =T9O45-

=CJK@ 3>IC;5 6L6I46MH =Lµµ9C;46B =C4H =LT@?C2C9H, 654 ?T4 µ?@J µ9 J;GJ<M@45

H(@1,@2) ?8AH 5LCQH 8JL µ9I9C7G265@ =C5 *5;5O9><µ5C5 7.6 654 7.7. YII9H µQGJ-

OJ4 =T9O>5=2H CA@ FIR 3>IC;A@ 9>@54 J4 56?IJLG9H:

• MC5(*(4 8E.*9%834 IC6/08/@, *08*0?8/%/@, FIR <96/"@,. # µQGJOJH 8;JCBG2-

69 58? CJ@ McClellan 654 58JO>O94 O4=O4B=C5C5 FIR 3>IC;5 µ9 C4H 65IKC9;9H 8;J-

O45<;53QH <45 C2@ H(@1,@2), ?C5@ CJ 8I7GJH CA@ ?;A@ C2H 6;JL=C467H 58?6;4-

=2H 9>@54 =L<696;4µQ@J. *;?694C54 JL=45=C46B, <45 5I<?;4GµJ S9IC4=CJ8J>2=2H

8JL 58JC9I9> C2 O4=O4B=C5C2 Q6OJ=2 C2H 5@C>=CJ4T2H µ9G?OJL <45 C5 µJ@JO4B-

=C5C5 FIR 3>IC;5. # µQGJOJH 5854C9> LR2I7 L8JIJ<4=C467 4=TK <45 3>IC;5 µ9=5>-

A@ 8;JO45<;53M@ (I.T., <45 CJ@ 8;J=O4J;4=µ? 6;JL=C467H 58?6;4=2H µ9 33 H 33

= 1089 ?;JLH).

• &E.*9%83 /@, *08*0?8/%/@, FIR <96/"@, E"380µ('(0+,/%4 µ./%8E3µ%/08µ(74.,9

C2 µQGJOJ 5LC7, C2@ J8J>5 98>=2H 8;?C94@9 J McClellan, 984CL<TB@9C54 J =T9-

O45=µ?H O4=O4B=C5CA@ FIR 3>IC;A@ µ9 58?6;4=2 =LT@?C2C5H 6L6I46B =Lµµ9-

™¯‹Ì· 7.20

;( µC/"( /34 %'#-1"0834 8)E,#/3/%4

|H(@1,@2)| /()*08*0?8/%/() FIR<96/"() /() '%"%-

*.9$µ%/(4 7.7.

+LT@?C2C5 @2

+LT@?C2C5 @1

|H(@1,@2)|


C;467 (6L;>AH =C4H T5µ2IQH =LT@?C2C9H). )5 O4=O4B=C5C5 FIR 3>IC;5 8;J6K8CJL@

58? 65CBII2IJ µ9C5=T2µ5C4=µ? µJ@JO4B=C5CA@ FIR 3>IC;A@. (@ C5 µJ@JO4B-

=C5C5 FIR 3>IC;5 9>@54 SQIC4=C5, C?C9 654 C5 O4=O4B=C5C5 8JL 8;J6K8CJL@ 9>@54

65CB 8;J=Q<<4=2 SQIC4=C5.

'IJ6I2;M@J@C5H C2 =LZ7C2=2 <45 C2 =T9O>5=2 CA@ O4=O4B=C5CA@ 3>IC;A@ G5 8;Q-

894 @5 5@539;GJKµ9 <45 I><J =C5 O4=O4B=C5C5 IIR 3>IC;5. \8AH 9>@54 <@A=C? 58?

C5 693BI545 4 654 5 C5 µJ@JO4B=C5C5 %IR 3>IC;5 =9 =TQ=2 µ9 C5 µJ@JO4B=C5C5 FIR

85;JL=4BZJL@ C5 56?IJLG5 OKJ µ94J@96C7µ5C5:

• E9@ 9>@54 8B@C5 9L=C5G7 654 <4’ 5LC? CJ I?<J T;94BZ9C54 8;J=JT7 =C2@ CJ8JGQ-

C2=2 CA@ 8?IA@ CJL 3>IC;JL.

• E9@ QTJL@ <;5µµ467 58?6;4=2 3B=2H.

)5 >O45 56;4SMH µ94J@96C7µ5C5 85;JL=4BZJL@ 654 C5 O4=O4B=C5C5 IIR 3>IC;5. !84-

8;?=G9C5, J4 O4=O4B=C5C9H =L@5;C7=94H µ8J;JK@ @5 9I9<TGJK@ AH 8;JH C2@ 9L=CB-

G94B CJLH 5;69CB OK=6JI5 (9>@54 <9@46B OK=6JI2 2 85;5<J@CJ8J>2=2 654 J 65GJ;4-

=µ?H CA@ 8?IA@), 9@M 2 QII94R2 <;5µµ467H 3B=2H 9>@54 65C5=C;J3467 <45 C2@ 946?-

@5, ?8AH 5@53Q;5µ9 =C2@ !@?C2C5 6.4. 045 CJLH OKJ 5LCJKH I?<JLH 58J39K<9C54 2

T;2=4µJ8J>2=2 CA@ O4=O4B=C5CA@ IIR 3>IC;A@ =C2@ R234567 989:9;<5=>5 946?@5H,

85;?IJ 8JL µ8J;JK@ @5 89CKTJL@ C2@ >O45 58?6;4=2 =C2 =LT@?C2C5 µ9 I4<?C9;JLH

=L@C9I9=CQH.

7.4.4 'Ê·ÚÌÔÁ¤˜ ÙˆÓ ‰ÈÛ‰È¿ÛÙ·ÙˆÓ FIR Ê›ÏÙÚˆÓ

'4 935;µJ<QH CA@ O4=O4B=C5CA@ FIR 3>IC;A@, 6L;>AH =C2@ 989:9;<5=>5 C2H R2345-

67H 946?@5H, 9>@54 8J46>I9H O9OJµQ@2H C2H 4O4?C2C5H CA@ 3>IC;A@ 5LCM@ @5 µ2@ 5IIB-

ZJL@ C2 35=467 =TQ=2 CA@ 5;µJ@46M@ CJL =7µ5CJH. ,9C5:K CA@ 935;µJ<M@ 5LCM@

µ8J;JKµ9 @5 5@53Q;JLµ9 984I96C46B C4H 56?IJLG9H:

• !:5<A<7 CJL GJ;KSJL Gauss µ9 C2 SJ7G945 S5GL89;5CJK FIR 3>IC;JL (SI.

+T7µ5 7.21).

• (@>T@9L=2 CA@ 56µM@ µ45H 946?@5H (edge detection) µ9 C2 SJ7G945 LR289;5CJK

FIR 3>IC;JL (UI. +T7µ5 7.22).

• X;7=2 CA@ FIR 3>IC;A@ =C2@ 58J65CB=C5=2 C2H 946?@5H 58? =L=C2µ5C46B

=3BIµ5C5 (image restoration) 654 <9@46?C9;5 =C2@ 89;>8CA=2 98>IL=2H 5@C>-

=C;J3A@ 8;JSI2µBCA@.

• X;7=2 CA@ FIR 3>IC;A@ =C2 O45O465=>5 C2H =K@G9=2H CJµM@ 5@C4694µQ@JL 58?

8;JSJI46QH 946?@9H (5:J@467 CJµJ<;53>5).

2 5 77 . 4 ™ Ã ! ¢ # " ™ % ¢ # ™ ¢ # " ™ &"& ø ¡ æ % º # " ' ø ¡ º # § & ƒ ø ¡


2 5 8 K E º A § A I O 7 : ¢ # ™ ¢ # " ™ &"&" æ % º # " ' " ° ƒ" ª ª # ' " ™ À ™ & % ª "&"

*;Q894 @5 =2µ94AG9> ?C4 C5 FIR 3>IC;5 AH <;5µµ46B =L=C7µ5C5 QTJL@ 654 µ94J@9-

6C7µ5C5. )J 6L;4?C9;J 58? 5LCB 9>@54 2 G?IA=2 8JL 8;J65I9> =C4H 56µQH C2H 946?-

@5H, ?C5@ 8;J=85GJKµ9 @5 9:B<JLµ9 G?;LSJ (SIQ89 +T7µ5 7.21<).

™¯‹Ì· 7.21

%) O %"E012 .01#-,%, I) 3 .01#,% µ.5#")I( Gauss. $) O µ.9@83 /()5("7I() µ. /3

I(25.0% *08*0?8/%/() FIR<96/"() '"(1%6.9

%,.'057µ3/35#6@83

/@, %1µ+, /34.01#,%4.

™¯‹Ì· 7.22

O E"283 )H0'."%-/(7 FIR <96/"()I(35? 8/3, %'(-µ#,@83 1%0 .,9-

8E)83 /@, %1µ+,/34 .01#,%4.

5 S

<



+C2@ 85;B<;53J 5LC7 5@539;G765µ9 =9 C;?8JLH =T9O>5=2H CA@ O4=O4B=C5CA@ FIR

3>IC;A@. (@5ILC46B, <@A;>=5µ9 C2 µQGJOJ =T9O>5=2H µ9 C2 T;7=2 CA@ 85;5GK-

;A@. )5 FIR 3>IC;5 9>@54 C5 <;5µµ46B 3>IC;5 8JL T;2=4µJ8J4JK@C54 89;4==?C9;J.

)J <9<J@?H 5LC? J39>I9C54 =C2@ 4O4?C2C5 8JL QTJL@ @5 µ2@ 5IIBZJL@ C2 35=467

=TQ=2 CA@ 5;µJ@46M@ CJL =7µ5CJH (3>IC;5 µ2O9@467H 3B=2H), 5@ =T9O45=CJK@ QC=4

M=C9 @5 85;JL=4BZJL@ =Lµµ9C;467 6;JL=C467 58?6;4=2.


+CJ 693BI54J 5LC? <@A;>=5µ9 CJ@ C;?8J 8JL 984O;JK@ C5 <;5µµ46B, 5µ9CBSI2C5

=C2@ JI>=G2=2 O4=O4B=C5C5 R23456B 3>IC;5 8B@A =C5 O4=O4B=C5C5 =7µ5C5. +C2@

JL=>5 J4 OKJ 5LCQH 4O4?C2C9H, <;5µµ46?C2C5 654 5µ9C5SI2C?C2C5 =C2@ JI>=G2=2,

JO2<JK@ =C2@ 8;B:2 C2H O4=O4B=C5C2H <;5µµ467H =L@QI4:2H. )5 3>IC;5 5LCB T5;5-

6C2;>ZJ@C54 58? C2@ 6;JL=C467 CJLH 58?6;4=2, 8JL 9>@54 µ45 O4=O4B=C5C2 O456;4-

C7 =L@B;C2=2 654 65GJ;>Z94 C2@ 58?6;4=2 CJL 3>IC;JL =C4H TA;46QH =LT@?C2C9H

@1 654 @2. )QIJH, <@A;>=5µ9 µ>5 58? C4H µ9G?OJLH =T9O>5=2H CA@ O4=O4B=C5CA@

FIR 3>IC;A@. )5 FIR 3>IC;5 9>@54 65CBII2I5 =C2@ 989:9;<5=>5 C2H R234567H 946?-

@5H O4?C4 O9@ 5IIBZJL@ C4H 35=46QH =TQ=94H CA@ 5;µJ@46M@ CJL 3B=µ5CJH C2H 946?-

@5H (3>IC;5 µ2O9@467H 3B=2H), 98JµQ@AH O9@ 65C5=C;Q3JL@ C2 µJ;37 C2H 946?@5H.

2 5 9™ À ¡ $æ% '!º"§"#$À


2 6 0 K E º A § A I O 7 : ¢ # ™ ¢ # " ™ &"&" æ % º # " ' " ° ƒ" ª ª # ' " ™ À ™ & % ª "&"


!§§%¡$°§ø™™%

%. *7C5H, N3<0%12 .'.F."$%89% :01#,%4, D9==5IJ@>62 1996.

•!¡$°§ø™™%

K.R. Castleman, Digital Image Processing, Prentice Hall, 1996

D.E. Dudgeon and R.M. Mersereau, Multidimensional Digital Signal Processing,

Prentice–Hall, 1984.

A.K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, 1989.

R. Klette and P. Zamperoni, Handbook of Image Processing Operators, John Wiley

and Sons, Chichester, 1996.

J.S. Lim, Two–Dimensional Signal and Image Processing, Prentice Hall, 1990.

MATLAB, Image Processing Toolbox, MathWorks.

$¢%°$™ ¶!ƒ"#&!ƒø ª!§!&%™

%. *7C5H, N3<0%12 .'.F."$%89% :01#,%4, D9==5IJ@>62 1996.

Pµ35=2 µ8J;9> @5 OJG9> =C5 693BI545 3 654 5.


Cascade =9 =94;B

Convolution +L@QI4:2

Edge detection (@>T@9L=2 56µM@

Image restoration (8J65CB=C5=2 C2H 946?@5H

Linear systems 0;5µµ46B =L=C7µ5C5

Separable E45TA;>=4µJH

Shift–Invariant (µ9CBSI2CJ =C2@ JI>=G2=2

Two–dimensional E4=O4B=C5CJH

Zero phase response (8?6;4=2 µ2O9@467H 3B=2H


A!·ÓÙ‹ÛÂÈ˜ AÛÎ‹ÛÂˆÓ A˘ÙÔ·ÍÈÔÏfiÁËÛË˜

1.1

0@A;>ZJ@C5H, 8IQJ@, CJ G9M;2µ5 O94<µ5CJI2R>5H, µ8J;JKµ9 9K6JI5 @5 O4584=CM-

=JLµ9 5@ G5 85;JL=45=C9> CJ 354@?µ9@J C2H T5µ2IJK ;LGµJK O94<µ5CJI2R>5H. &54

5@ 85;JL=45=C9>, C?C9 µ8J;JKµ9 µ9 SB=2 C2 =TQ=2 F0 = Fk–kFs, @5 L8JIJ<>=JLµ9

C2 =LT@?C2C5 =C2@ J8J>5 5@C4=CJ4TJK@ C5 O9><µ5C5 8JL 87;5µ9. PC=4, 2 =LT@?C2-

C5 CJL 5@5IJ<46JK =7µ5CJH 9>@54 F = 100 Hz. 045 @5 5@C48;J=A89LG9> =A=CB CJ

=7µ5 5LC? =CJ@ R23456? TM;J 8;Q894 @5 IBSJLµ9 O9><µ5CB CJL µ9 =LT@?C2C5 CJL-

IBT4=CJ@ 2F = 2!100 = 200 Hz. Y;5, 654 =C4H OKJ 89;48CM=94H O9 G5 58J3K<JLµ9 CJ

354@?µ9@J C2H T5µ2IJK (5@985;6JKH) ;LGµJK O94<µ5CJI2R>5H (354@?µ9@J 35=µ5-

C467H 9846BILR2H).

(5) 045 Fs = 75 Hz C5 O9><µ5C5 8JL G5 8B;JLµ9 G5 5@C4=CJ4TJK@ =C2 O94<µ5CJI2-

R>5 CJL 2µ4CJ@J94OJKH C2H S5=467H 89;4JT7H =LT@JC7CA@ (–Fs/2 ] F0 ] Fs/2) µ9

=LT@?C2C5 >=2 8;JH F( = Fk–kFs = 100–1!75 = 25 Hz.

(S) 045 Fs = 150 Hz C5 O9><µ5C5 8JL G5 8B;JLµ9 G5 5@C4=CJ4TJK@ =C2 O94<µ5CJI2-

R>5 CJL 2µ4CJ@J94OJKH C2H S5=467H 89;4JT7H =LT@JC7CA@ (–Fs/2 ] F0 ] Fs/2) µ9

=LT@?C2C5 >=2 8;JH F0 = Fk–kFs = 100–1!150 = –50 Hz.

(LC7 7C5@ µ>5 9K6JI2 B=62=2. (@ C5 65C53Q;5C9, =2µ5>@94 8AH QT9C9 65C5@J7=94

CJ 354@?µ9@J C2H T5µ2IJK (5@985;6JKH) ;LGµJK O94<µ5CJI2R>5H. +L@9T>=C9 =C4H

98?µ9@9H 5=67=94H. (@ ?T4, C?C9 985@5IBS9C9 C5 *5;5O9><µ5C5 1.3 654 1.4 654 :5@5-

8;J=85G7=C9. X;94BZ9C54 6B8J4JH T;?@JH µQT;4 @5 9:J4694AG9>C9 µ9 C4H 4O454C9;?-

C2C9H CA@ =2µBCA@ 8JL 8;JQ;TJ@C54 58? O94<µ5CJI2R>5.

1.2

F8B;TJL@ OKJ C;?8J4 <45 @5 IK@JLµ9 5LCJK CJL 9>OJLH C4H 5=67=94H, O2I5O7 C4H

5=67=94H =C4H J8J>9H µ5H Z2C9>C54 @5 OM=JLµ9 C5 =7µ5C5 O456;4CJK T;?@JL 654 ?T4

µ?@J C4H =LT@?C2CQH CJLH.

' 8;MCJH 9>@54 5LC?H 65CB CJ@ J8J>J, :964@M@C5H 58? CJ@ J;4=µ? C2H O94<µ5CJI2-

R>5H, 5@C465G4=CJKµ9 C2 µ9C5SI2C7 t µ9 nT, 654 96C9IJKµ9 C4H ?8J49H C;4<A@Jµ9-

C;46QH 8;B:94H.

' O9KC9;JH 9>@54 969>@JH 65CB CJ@ J8J>J L8JIJ<>ZJLµ9 8;MC5 C4H =LT@?C2C9H 8JL

G5 8;J6KRJL@ 58? C2 O94<µ5CJI2R>5 (?8AH 6B@5µ9 654 =C2@ B=62=2 5LCJ5:4JI?-

<2=2H 1.1), 654 µ9CB 8;J=O4J;>ZJLµ9 C4H 5@C>=CJ4T9H =TQ=94H <45 C5 =7µ5C5 O456;4-

CJK T;?@JL. \8J4J@ 58? CJLH C;?8JLH 654 5@ 984IQ:9C9, G5 8;Q894 @5 S;9>C9 CJ >O4J


2 6 2 æ H º I A K H E ¶ E • E P °A ™ I A E I K O N ø N K A I ™ H M AT ø N

58JCQI9=µ5, CJ J8J>J =C2@ 8;J694µQ@2 89;>8CA=2 9>@54 x1(n) = x2(n) = x3(n) =

cos(0,6'n). (H C5 8B;JLµ9 ?µAH 58? C2@ 5;T7.

# 89;>JOJH O94<µ5CJI2R>5H 4=JKC54 µ9 T = 1/Fs = 1/100 Hz = 0,01 sec. Y;5, J4 56J-

IJLG>9H 8JL G5 8;J6KRJL@ G5 9>@54 J4 9:7H:

x1(n) = cos(60'nT) = cos(0,6'n)

x2(n) = cos(140'nT) = cos(1,4'n) = cos[(2'–0,6')n] = cos(0,6'n)

x3(n) = cos(260'nT) = cos(2,6'n) = cos[(2' + 0,6')n] = cos(0,6'n)

+Lµ89;5>@JLµ9, 98JµQ@AH, ?C4 2 O94<µ5CJI2R>5 CA@ C;4M@ 5@5IJ<46M@ =2µBCA@,

O453J;9C467H =LT@?C2C5H CJ 65GQ@5, JO7<2=9 C9I46B =CJ >O4J =7µ5 O456;4CJK T;?-

@JL cos(0,6'n).

\8AH 5@53Q;5µ9 8;J2<JLµQ@AH, =CJ >O4J 58JCQI9=µ5 65C5I7<JLµ9 5@ 935;µ?-

=JLµ9 C2@ =TQ=2 F0 = Fk–kFs <45 65G9µ4B 58? C4H =LT@?C2C9H. E2I5O7, 2 =LT@?C2-

C5 CJL =7µ5CJH x1(t) 9>@54 F1 = 30 Hz 654, 65CB =L@Q8945, O9 G5 85;JL=45=C9> CJ

354@?µ9@J C2H T5µ2IJK ;LGµJK O94<µ5CJI2R>5H, 53JK F1 < Fs/2 = 50 Hz. Y;5, x1(n)

= cos[2'(F1/Fs)n] = cos[2'(30/100)n] = cos(0,6'n). # =LT@?C2C5 CJL =7µ5CJH x2(t)

9>@54 F2 = Fk = 70 Hz 654 C5 O9><µ5C5 8JL G5 8B;JLµ9 G5 5@C4=CJ4TJK@ =C2 O94<µ5-

CJI2R>5 9@?H 2µ4CJ@J94OJKH =LT@?C2C5H F0 = Fk–kFs = 70–100 = –30Hz. Y;5 x2(n)

= cos[2'(F2/Fs)n] = cos[2'(–30/100)n] = cos(–0,6'n) = cos(0.6'n).)QIJH, 2 =LT@?-

C2C5 CJL =7µ5CJH x3(t) 9>@54 F3 = 130Hz 654 C5 O9><µ5C5 8JL G5 8B;JLµ9 G5 5@C4-

=CJ4TJK@ =C2 O94<µ5CJI2R>5 9@?H 2µ4CJ@J94OJKH =LT@?C2C5H F0 = Fk–kFs = 130–100

= 30 Hz. Y;5 x3(n) = x1(n). +L@98MH, 654 C5 C;>5 5@5IJ<46B x1(t), x2(t), x3(t) =7µ5-

C5 5@C48;J=A89KJ@C54 58? C5 >O45 O9><µ5C5 µ9CB C2 O94<µ5CJI2R>5 CJLH µ9 ;LGµ?

100Hz.

(@ 654 9=9>H 65C5I7:5C9 =CJ >O4J 58JCQI9=µ5, C?C9 8;JTA;BC9 8JIK 65IB. +L@9T>-

=C9 µ9 C2@ 98?µ9@2 B=62=2 5LCJ5:4JI?<2=2H. (@ O9@ C5 65C53Q;5C9, µ9I9C7=C9

654 8BI4 C5 *5;5O9><µ5C5 1.3 654 1.4 654 :5@58;J=85G7=C9.

1.3

+C2@ B=62=2 5LC7 G5 8;Q894 @5 8;J=Q:9C9 CJ <4@?µ9@J CJL 2µ4C?@JL µ9 CJ =L@2µ>-

CJ@J. (LC? 9>@54 CJ =L@2G4=µQ@J IBGJH 8JL <>@9C54 =C2@ 89;>8CA=2 5LC7. *5;5C2-

;7=C9 ?C4 =’ ?I9H C4H 8;J2<JKµ9@9H 5=67=94H 9>T5µ9 µ?@J 5G;J>=µ5C5 (7 O453J;QH)

2µ4CJ@J94OM@, 654 9:9CBZJ@C5H 6BG9 ?;J CJL 5G;J>=µ5CJH TA;4=CB, µ8J;JK=5µ9 @5

8;J=O4J;>=JLµ9 C2 =LT@?C2C5 CJL 6BG9 2µ4CJ@J94OJKH. 045 @5 963;B=JLµ9 CJ <4@?-

µ9@J C2H B=62=2H 5LC7H AH BG;J4=µ5 2µ4CJ@J94OM@, G5 T;2=4µJ8J47=JLµ9 C2@


=TQ=2 2sin>cosB = sin (> + B) + sin(>–B). !8JµQ@AH, QTJLµ9:

x(t) = sin('t) + 4sin(3't)cos(2't) = sin('t) + 2[sin(3't + 2't) + sin(3't–2't)] =

sin('t) + 2sin(5't) + 2sin('t) = 3sin('t) + 2sin(5't) = 3sin(2' t) +

2 sin(2' t)

UIQ8JLµ9 ?C4 CJ 5@5IJ<46? =7µ5 x(t) 58JC9I9>C54 58? OKJ =LT@?C2C9H,

C2@ F1 = kHz 654 C2@ F2 = kHz.

# 9IBT4=C2 =LT@?C2C5 O94<µ5CJI2R>5H <45 C2@ F1 9>@54 2F1 = 2(1/2) = 1 kHz, 9@M

<45 C2@ F2 9>@54 2F2 = 2(5/2) = 5 kHz. +L@98MH, I5µSB@J@C5H O9><µ5C5 CJL =7µ5-

CJH µ9 ;LGµ? 3 kHz, <45 C2@ F1 O9 G5 L8B;:94 8;?SI2µ5 654 C5 O9><µ5C5 8JL G5

8B;JLµ9 G5 5@C4=CJ4TJK@ =_ 5LC7. 045 CJ 2µ4CJ@J94OQH ?µAH µ9 =LT@?C2C5 F2, G5

85;JL=45=C9> 8;?SI2µ5, 9:54C>5H C2H T5µ2IJK ;LGµJK O94<µ5CJI2R>5H, 53JK <45

@5 CJ 5@585;5=C7=JLµ9 =A=CB G5 Q8;989 @5 I5µSB@JLµ9 O9><µ5C5 µ9 =LT@?C2C5

CJLIBT4=CJ@ 5 kHz. PC=4, C5 O9><µ5C5 8JL G5 8;J6KRJL@ G5 5@C4=CJ4TJK@ =’ Q@5

2µ4CJ@J94OQH =LT@?C2C5H

F0 = Fk–kFs = –k3 = – kHz, ?8JL Fk = F2 = kHz 654 k = 1,

53JK 2 C4µ7 F0 8JL 8;JQ6LR9 5@7694 =CJ O4B=C2µ5 [–Fs/2, Fs/2]. )9I46B, AH =Lµ8Q-

;5=µ5 8;J6K8C94 ?C4 C5 O9><µ5C5 8JL G5 8B;JLµ9 58? C2 O94<µ5CJI2R>5 CJL x(t),

G5 5@C4=CJ4TJK@ =’ Q@5 654 µ?@J 2µ4CJ@J94OQH =LT@?C2C5H kHz.

!84=2µ5>@9C54 ?C4 =CJ >O4J 58JCQI9=µ5 G5 9>T5µ9 65C5I7:94 5@ 9;<5Z?µ5=C5@ µ?@J

µ9 C2 =TQ=2 x(t) = 3sin('t) + 2sin(5't), 654 5@C465G4=CJK=5µ9 t = nT, ?8JL T = 1/Fs

= 1/3 msec. !85I2G9K=C9 CJ.

1.4

!>@54 9K6JIJ @5 OJKµ9 ?C4 2 56JIJLG>5 p(n) µ8J;9> @5 963;5=C9> AH O453J;B OKJ

S2µ5C46M@ 56JIJLG4M@, 58? C4H J8J>9H 2 µ>5 9>@54 JI4=G2µQ@2 AH 8;JH C2@ BII2

65CB 4 µJ@BO9H (O9><µ5C5). PTJLµ9, O2I5O7, p(n) = u(n)–u(n–4), ?8AH 35>@9C54 654

=CJ +T7µ5 1.30. (@ 8BI4 O9@ CJ =693C765C9 QC=4 654 6B@5C9 T;7=2 C2H µJ@5O45>5H

6;JL=C467H *(n), C?C9 G5 65C5I7:5C9 =CJ =A=C? 58JCQI9=µ5, 5IIB I><J 84J 98>8J-

@5. *5;5C2;9>C9, O2I5O7, ?C4 p(n) = *(n) + *(n–1) + *(n–2) + *(n–3). (IIB, µ9 SB=2

C2 =TQ=2 (1.33) QTJLµ9

12

52

12

52

52

12

52

12

2 6 3A ¶ " ¡ & % ™ ! # ™ A ™ ' % ™ ! ø ¡ AÀ & $ " • # $ § $ ° % ™ % ™



*(n) = u(n) – u(n–1)

*(n–1) = u(n–1) – u(n–2)

*(n–2) = u(n–2) – u(n–3)

*(n–3) = u(n–3) – u(n–4)

*;J=GQCJ@C5H 65CB µQI2 65C5I7<JLµ9 654 8BI4 =CJ >O4J 58JCQI9=µ5 p(n) =

u(n)–u(n–4).

p(n) u(n) u(n–4)

1 1 1

-2 -1 0 1 2 3 -2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4n n n

= –

™¯‹Ì· 1.30

*5Iµ?H O456;4CJKT;?@JL AH O453J-;B S2µ5C46M@56JIJLG4M@

# 58B@C2=2 5LC7 9>@54 9K6JI2 <45 969>@J@ 8JL QT94 65C5@J7=94 C2@ Q@@J45 C2H JI>-

=G2=2H =2µBCA@ 8;JH C5 5;4=C9;B (8;J7<2=2) 7 C5 O9:4B (65GL=CQ;2=2). (@ O9@

C5 65C53Q;5C9 µ2@ 58J<J2C9LC9>C9. !>@54 Q@@J49H 8JL 35>@J@C54 9K6JI9H, 5IIB

=LT@B 8;J65IJK@ =K<TL=2. *4G5@?@ @5 µ89;O9LC9>C9 ?C5@ =5H Z2C2G9> @5 =T9O4B-

=9C9 µ>5 56JIJLG>5 =5@ C2@ u(n–4). UIQ8J@C5H CJ µ9>J@ (–) µ8;J=CB 58? CJ 4, =L@7-

GAH @Jµ>ZJLµ9 ?C4 CJ =7µ5 S;>=69C54 =C5 5;@2C46B. *;J=JT7 =’ 5LC?! X;94BZ9C54

9:J469>A=2 µ9 C4H Q@@J49H 5LCQH 654 @5 C4H 65C5@J7=JLµ9. )J =2µ9>J–6I94O> 9>@54

969> ?8JL µ2O9@>Z9C54 2 85;B=C5=2 =C2@ 85;Q@G9=2. +C2@ 8;J694µQ@2 89;>8CA=2

5LC? =LµS5>@94 <45 n–4 = 0, O2I5O7 <45 n = 4. +L@98MH, 2 56JIJLG>5 (CJ =7µ5) 5;T>-

Z94 58? CJ =2µ9>J n = 4. &54 53JK CJ n 9>@54 G9C46?, CJ =7µ5 G5 =L@9T>Z94 8;JH µ9<5-

IKC9;9H C4µQH, O2I5O7 8;JH C5 O9:4B. P@5 CQCJ4J =7µ5 IQµ9 ?C4 QT94 L8J=C9> 65GL-

=CQ;2=2 65CB 4 µJ@BO9H T;?@JL. )J 5@ Q@5 =7µ5 QT94 L8J=C9> 65GL=CQ;2=2 (delay)

7 8;J7<2=2 (advance) 9:5;CBC54 58? CJ 5@ 5LC? S;>=69C54 µ9CB 7 8;4@ 58? CJ 0

(µ2OQ@). (H µ2@ :9T@Bµ9 ?C4 2 µ9C5SI2C7 n =L@7GAH 5@53Q;9C54 =CJ T;?@J. PC=4,

I5µSB@J@C5H AH 5@53J;B C2 T;J@467 =C4<µ7 n = 0, ?,C4 =LµS5>@94 µ9CB CJ n = 0

=2µ5>@94 ?C4 9>@54 T;J@46B 65GL=C9;2µQ@J, 9@M ?,C4 =LµS5>@94 8;4@ CJ n = 0 =2µ5>-

@94 ?C4 T;J@46B 8;J2<9>C54.

1.5

+’ ?I9H 5LCQH C4H 89;48CM=94H, S5=4Z?µ5=C9 =CJLH 5@C>=CJ4TJLH J;4=µJKH. (@ 654

35>@J@C54 C?=J 9K6JIJ4 654 65C5@J2CJ> ?C5@ CJLH O45SBZJLµ9, 9@CJKCJ4H 2 935;-

µJ<7 CJLH µ8J;9> 8JIIQH 3J;QH @5 µ5H 3Q;@94 6B8J49H OL=6JI>9H. (H 935;µ?=JLµ9

CJ@ J;4=µ? C2H <;5µµ46?C2C5H <45 C2@ B=62=2 5LC7.


045 9>=JOJ x1(n), 2 Q:JOJH CJL =L=C7µ5CJH G5 9>@54 y1(n) = 3x1(n) + 3. 045 9>=JOJ

x2(n), 2 Q:JOJH CJL =L=C7µ5CJH G5 9>@54 y2(n) = 3x2(n) + 3. )QIJH, <45 9>=JOJ x3(n)

= ax1(n) + bx2(n), 2 Q:JOJH G5 4=JKC54 µ9 y3(n) = 3x3(n) + 3 = 3[ax1(n) + bx2(n)] +

3 = 3ax1(n) + 3bx2(n) + 3 = 3ax1(n) + 3bx2(n) + 3 + 3a–3a + 3b–3b = a[3x1(n) + 3]

+ b[3x2(n) + 3] + 3(1–a–b) = ay1(n) + by2(n) + 3(1–a–b) 3 a y1(n) + b y2(n). Y;5, CJ

=K=C2µ5 9>@54 µ2 <;5µµ46?.

(@ 65C5I7:5C9 =CJ =A=C? 58JCQI9=µ5, C?C9 =5H 5:>ZJL@ =L<T5;2C7;45. cC5@ =T9-

C46B OK=6JI2 2 58?O94:2, 53JK Q8;989 @5 O2µ4JL;<7=JLµ9 C5 y1(n), y2(n) =C2@ C9I4-

67 Q63;5=2 8;J=G5354;M@C5H 65CBII2IJLH ?;JLH. (@ O9@ C5 65C53Q;5C9, µ9I9C7-

=C9 654 8BI4 C2@ Q@@J45 C2H <;5µµ46?C2C5H =C2@ 9@?C2C5 1.4 654 :5@58;J=85G7=C9.

1.6

(3JK CJ =K=C2µ5 y(n) = nx(n) QT94 Q@5 =L@C9I9=C7 T;J@46B µ9C5S5II?µ9@J, 89;4-

µQ@JLµ9 5LC? @5 µ2@ 9>@54 T;J@46B 5µ9CBSI2CJ. *;B<µ5C4, 5@ 935;µ?=JLµ9 =C2@

9>=JOJ C2@ 65GL=C9;2µQ@2 65CB n0 56JIJLG>5 x(n–n0), 2 Q:JOJH G5 9>@54 yd(n) = n

x(n–n0). (8? C2@ BII2 8I9L;B, 5@ 65GL=C9;7=JLµ9 C2@ Q:JOJ y(n) 65CB n0, 5@C465-

G4=CM@C5H ?8JL n CJ n–n0, C?C9 2 Q:JOJH G5 9>@54 y(n–n0) = (n–n0)x(n–n0) 3 yd(n).

+L@98MH CJ =K=C2µ5 O9@ 9>@54 T;J@46B 5µ9CBSI2CJ.

%O45>C9;J 9@O453Q;J@ 85;JL=4BZ94 CJ =K=C2µ5 y(n) = x(2n), <45C> =C2@ JL=>5 8;?-

694C54 <45 Q@5 =K=C2µ5 )'(*.0$µ%/(63H9%4. # Q:JOJH 9@?H CQCJ4JL =L=C7µ5CJH «6;5-

CB94» C5 B;C45 µ?@J O9><µ5C5 C2H 56JIJLG>5H 94=?OJL. (@ IJ48?@ 65GL=C9;7=JLµ9

C2@ 9>=JOJ 65CB n0 O9><µ5C5, O2I5O7 5@ 935;µ?=JLµ9 =CJ =K=C2µ5 C2@ x(n–n0),

C?C9 2 Q:JOJH G5 9>@54 yd(n) = x(2n–n0). (@ CM;5 65GL=C9;7=JLµ9 C2@ Q:JOJ 65CB

n0 µJ@BO9H, <9<J@?H 8JL 963;BZ9C54 µ5G2µ5C46B µ9 5@C465CB=C5=2 CJL n µ9 CJ n–n0,

C?C9 2 Q:JOJH G5 4=JKC54 µ9 y(n–n0) = x{2(n–n0)} = x(2n–2n0) 3 x(2n–n0) = yd(n).

0>@9C54 35@9;?, 98JµQ@AH, ?C4 Q@5 CQCJ4J =K=C2µ5 O9@ 9>@54 T;J@46B 5µ9CBSI2CJ.

(LC? µ8J;9> 9K6JI5 @5 <>@94 65C5@J2C? µ9 CJ 56?IJLGJ 85;BO94<µ5. P=CA ?C4 2

56JIJLG>5 94=?OJL 9>@54 {x(n)} = {x0, x1, x2, x3, x4, x5, x6, …}. )?C9 2 56JIJLG>5 9:?-

OJL G5 4=JKC54 µ9 {y(n)} = {x(2n)} = {x0, x2, x4, x6, …}. &5GL=C9;JKµ9 C2@ 9>=JOJ

65CB µ>5 µJ@BO5 (n0 = 1), O2I5O7 65CB Q@5 O9><µ5, J8?C9 5LC7 <>@9C54 {x(n–1)} =

{0, x0, x1, x2, x3, x4, x5, x6, …}. # 56JIJLG>5 9:?OJL G5 9>@54 CM;5 >=2 µ9 {y(n)} =

{x(2n–1)} = {0, x1, x3, x5, …}. *5;5C2;JKµ9, 98JµQ@AH, ?C4 5LC7 O9@ 4=JKC54 µ9 C2@

8;J2<JKµ9@2 56JIJLG>5 9:?OJL 2 J8J>5 QT94 L8J=C9> 65GL=CQ;2=2 65CB µ>5 µJ@B-

O5, 5IIB 8;?694C54 <45 µ>5 9@C9IMH O453J;9C467 56JIJLG>5, 984S9S54M@J@C5H QC=4

?C4 CJ =K=C2µ5 O9@ 9>@54 T;J@46B 5µ9CBSI2CJ.

(@ 585@C7=5C9 =A=CB, C?C9 =5H 5:>ZJL@ 8;5<µ5C46B =L<T5;2C7;45. (@ 8BI4 O9@

2 6 5A ¶ " ¡ & % ™ ! # ™ A ™ ' % ™ ! ø ¡ AÀ & $ " • # $ § $ ° % ™ % ™



65C53Q;5C9 @5 585@C7=9C9 =A=CB, µ2@ 58J<J2C9LC9>C9. 0>@9C54 35@9;?, <45 µ45

56?µ2 3J;B, ?C4 354@Jµ9@46B 8;?694C54 <45 58IQH Q@@J49H, J4 J8J>9H ?µAH 85;JL-

=4BZJL@ 5;69C7 OL=6JI>5 =C2@ 65C5@?2=7 CJLH, ?8AH 654 =C2@ 65C5@?2=2 C2H O45-

O465=>5H 58?O94:7H CJLH. E9>C9 :5@B CJ@ J;4=µ? C2H T;J@467H 5µ9C5SI2C?C2C5H 9@?H

=L=C7µ5CJH 654 985@5IBS9C9 C2@ 8;J=8BG94B =5H.

1.7

(LC7 C2@ B=62=2 µ8J;JKµ9 9K6JI5 @5 C2 IK=JLµ9, 53JK QTJLµ9 7O2 5@C4µ9CA8>-

=94 C2@ 84J OK=6JI2 89;>8CA=2 CJL *5;5O9><µ5CJH 1.8. (H OJKµ9 C2 O45O465=>5 654

8BI4, L8JIJ<>ZJ@C5H C2 =L@QI4:2 y(n) = h(n)*x(n). ,5H O>@9C54 ?C4 h(n) = anu(n) 654

x(n) = >u(n). Y;5

7

?8JL n ( 0 654 C0, C1, =C5G9;QH 9:5;CMµ9@9H 58? CJ 8IBCJH C2H 94=?OJL > 654 C2@

85;Bµ9C;J a C2H µJ@5O45>5H 6;JL=C467H. 045 @5 9:J4694AG9>C9 µ9 5LCJK CJL 9>OJLH

C4H 5=67=94H, 985@5IBS9C9 CJLH L8JIJ<4=µJKH <45 C2@ 89;>8CA=2 y(n) = x(n)*h(n).

1.8

F8B;TJL@ CJLIBT4=CJ@ OKJ C;?8J4 <45 @5 IK=9C9 C2@ B=62=2 5LC7. ' Q@5H 9>@54 J

<@A=C?H C;?8JH, 8JL T;2=4µJ8J47=5µ9 =CJ *5;BO94<µ5 1.8 654 =C2@ B=62=2 5LCJ-

5:4JI?<2=2H 1.7, 65CB CJ@ J8J>J L8JIJ<>ZJLµ9 C2 Z2CJKµ9@2 =L@QI4:2 µ9 SB=2 CJLH

J;4=µJKH (1.39) 7 (1.40), ?8JL h(n) = (an + bn)u(n) 654 x(n) = >u(n). P@5H O9KC9;JH

C;?8JH 9>@54 969>@JH 65CB CJ@ J8J>J, T;2=4µJ8J4M@C5H C2@ 984µ9;4=C467 4O4?C2C5

C2H =L@QI4:2H 654 5:4J8J4M@C5H CJ 58JCQI9=µ5 C2H B=62=2H 5LCJ5:4JI?<2=2H 1.7,

QTJLµ9:

h(n) = (an + bn)u(n) = anu(n) + bnu(n) = h1(n) + h2(n). !8JµQ@AH

y(n) = h(n)*x(n) = [h1(n) + h2(n)]*x(n) = h1(n)*x(n) + h2(n)*x(n).

)J 58JCQI9=µ5 h1(n)*x(n) µ5H 9>@54 7O2 <@A=C? 58? C2@ B=62=2 5LCJ5:4JI?<2=2H

1.7 654 4=JKC54 µ9 . &5C’ 5@5IJ<>5, CJ 58JCQI9=µ5 C2H

Aa

Aaa

a C C an n

1 1 0 1#+

#= +

y n A

aa

Aa

Aaa

a C C an

n n( ) ( )= ##

=#

+#

= ++1

1 1 1

1

0 1

y n h m x n m a u m Au n m a A A a

m

m m

m

nm

m

n

m

( ) ( ) ( ) ( ) ( )= # = # = ==#"

"

= ==#"

"

5 5 550 0


=L@QI4:2H h2(n)*x(n) G5 4=JKC54 µ9 .

+L@98MH, 2 Q:JOJH CJL =L=C7µ5CJH G5 9>@54: y(n) = C0 + C1an + C2 + C3b

n.

(@ L8JIJ<>=5C9 CJ =A=C? 58JCQI9=µ5 µ9 6B8J4J@ 58? 5LCJKH CJLH C;?8JLH, C?C9

=5H 5:>ZJL@ =L<T5;2C7;45. (@ O9@ C5 65C53Q;5C9, µ9I9C7=C9 654 8BI4 CJ *5;B-

O94<µ5 1.8 654 C2@ _A=62=2 ALCJ5:4JI?<2=2H 1.7 654 :5@58;J=85G7=C9. !>@54 GQµ5

T;?@JL CJ @5 9:J4694AG9>C9 µ9 C4H 5=67=94H 5LCJK CJL 9>OJLH.

1.9

!>@54 35@9;? ?C4 G5 8;Q894 @5 5:4J8J47=JLµ9 C4H 4O4?C2C9H C2H =L@QI4:2H, 8JL <@A-

;>=5µ9 =C2@ 9@?C2C5 1.4, <45 @5 L8JIJ<>=JLµ9 C2@ =L@JI467 6;JL=C467 58?6;4=2

CJL =L=C7µ5CJH. (8? CJ +T7µ5 1.29 SIQ8JLµ9 ?C4 h(n) = h1(n) + h2(n)*[h3(n) + h4(n)]

= h1(n) + h2(n)*h3(n) + h2(n)*h4(n). (IIB

h2(n)*h3(n) = [ *(n)– *(n–1)]*2*(n) = *(n)*2*(n)– *(n–1)*2*(n)

= *(n)– *(n–1)

h2(n)*h4(n) = [ *(n)– *(n–1)]*[–2( )nu(n)]

= *(n)*[–2( )nu(n)]– *(n–1)*[–2( )nu(n)]

= –( )nu(n) + ( )n–1u(n–1) = –( )n[u(n)–u(n–1)]

= –( )n*(n) = –*(n).

Y;5, h(n) = [*(n) + *(n–1)] + [*(n)– *(n–1)] + [–*(n)] = *(n)

# OL=6JI>5 =’ 5LC7 C2@ B=62=2 O9 S;>=69C54 =C2 T;7=2 CA@ 4O4JC7CA@ C2H =L@Q-

I4:2H, 5IIB =CJ@ L8JIJ<4=µ? C2H =L@QI4:2H 9@?H =7µ5CJH µ9 C2@ 6;JL=C467 56J-

IJLG>5. (8? CJ@ J;4=µ? C2H =L@QI4:2H (=TQ=2 1.37), µ8J;JKµ9 9K6JI5 @5 O4584-

=CM=JLµ9 ?C4 *(n)*g(n) = g(n), *(n–m)*g(n) = g(n–m). (LC? BIIA=C9 CJ 9>O5µ9 654

=CJ +T7µ5 1.22.

PTJ@C5H ?I5 5LCB 65CB @JL, 8;J=85G7=C9 @5 IK=9C9 C2@ B=62=2 T;2=4µJ8J4M@C5H

?µAH C2@ 8;MC2 4=?C2C5 <45 C2@ h(n), O2I5O7 h(n) = h1(n) + h2(n)*[h3(n) + h4(n)].

12

12

12

12

12

12

12

12

14

12

12

12

14

12

12

14

12

14

12

Ab

Abb

b C C bn n

1 1 2 3#+

#= +

2 6 7A ¶ " ¡ & % ™ ! # ™ A ™ ' % ™ ! ø ¡ AÀ & $ " • # $ § $ ° % ™ % ™



2.1

# B=62=2 5LC7 9>@54 9K6JIJ @5 ILG9>, 5@ GLµ2GJKµ9 ?C4 *(n–M) = 1 µ?@J <45 n =

M, 654 9;<5=CJKµ9 ?8AH =CJ *5;BO94<µ5 2.2. PC=4, 58? CJ@ J;4=µ? (2.1) QTJLµ9:

*5;5C2;JKµ9 ?C4 CJ µQC;J C2H =L@B;C2=2H 8JL L8JIJ<>=5µ9 9>@54 654 8BI4 >=J µ9 1,

53JK ,

9@M 2 3B=2 9>@54 5@BIJ<2 C2H =LT@?C2C5H, O2I5O7 > X(ej@) = –@M.

'4 5@C>=CJ4T9H <;5346QH 85;5=CB=94H O>@J@C54 =CJ +T7µ5 2.15.

X e e jj j( ) cos( ) sin( cos ( (6 6 6 6 6 6= = # =# O O O O O) )+ sin )22

X e x n e n M e ej j

n

j

n

j M( ) ( )6 6 6 6= = # =#

=#"

"#

=#"

"#5 5n nO( )

* (n–M)

0

1

0

0M n

(5)

|X(ej@)|

>X(e j@)

-2' -'

1

' 2' @

@

(S)

™¯‹Ì· 2.15

(5) # 56JIJLG>5*(n–M) 654 (S) CJ µQC;J 654 2 3B=2 CJLµ9C5=T2µ5C4=µJKFourier 5LC7H.

(@ 65C5I7:5C9 =CJ =A=C? 58JCQI9=µ5, µ8;BSJ =5H. (@ ?T4, :5@58;J=85G7=C9. EKJ

9>@54 C5 =2µ9>5 C5 J8J>5 84G5@?@ O9@ QT9C9 65C5@J7=94: (5) C2 µJ@5O45>5 6;JL=C4-

67 *(n–M) 2 J8J>5 9>@54 85@CJK >=2 µ9 µ2OQ@, 96C?H CJL =2µ9>JL n = M, ?8JL QT94

C2@ C4µ7 1 (UI. 9@?C2C5 1.3), 654 (S) CJLH J;4=µJKH CJLH =T9C46JKH µ9 CJLH µ4<5O4-

6JKH 5;4GµJKH. '4 J;4=µJ> (2.3) QAH (2.6) G5 =5H SJ2G7=JL@ =’ 5LC?. E9>C9 CJLH 654

8;J=85G7=C9 654 8BI4 @5 IK=9C9 µ?@J4 =5H C2@ B=62=2.

2.2

DLµ2G9>C9 ?C4 µ9 SB=2 C2 =TQ=2 1.34 J 85Iµ?H x(n) <;B39C54:

. x n x m n m n n n n n

m

( ) ( ) ( ) )= # = ! + + + # #[ ]=#"

"

5 4 4 4 4 4 41 2 1 1 2( ( )+ ( )+ ( )+ ( )


!8JµQ@AH, 58? CJ@ J;4=µ? (2.1) CJL DTFT QTJLµ9:

'4 <;5346QH 85;5=CB=94H CJL 85IµJK x(n) <45 R1 = 2 654 CJL DTFT 5LCJK 35>@J-

@C54 =CJ +T7µ5 2.16.

X e x n e n n n n n e

e e e e e e e

j j

n

j

n

j j j j j j j

( ) ( ) )

( ) (

6 6 6

6 6 6 6 6 6 6

4 4 4 4 4= = ! + + + # #[ ] =

= ! + + + +[ ] = ! + + +

#

=#"

"#

=#"

"

# # #

5 5n n

2 0 2

1 ( ( )+ ( )+ ( )+ ( )

2 1 1 2

1 1 1 ee ej j2 2

1 2 2 2

6 6

6 6

+[ ] =

= + +

# )

cos( ) cos( )

2 6 9A ¶ " ¡ & % ™ ! # ™ A ™ ' % ™ ! ø ¡ AÀ & $ " • # $ § $ ° % ™ % ™

™¯‹Ì· 2.16

(5) # 56JIJLG>5x(n) C2H B=62=2H5LCJ5:4JI?<2=2H2.2 <45 R1 = 2 654 (S) J DTFT 5LC7H.

x[n]

1

-N1 N10

0

n

X(ej@)

-2' 2'-' ' @

(%)

(I)

D5 µ8J;JK=5µ9 @5 5@C4µ9CA8>=JLµ9 CJ@ L8JIJ<4=µ? CJL 5G;J>=µ5CJH <9@46?C9;5,

AH <9Aµ9C;467 8;?JOJ, O2I5O7:

< ?8JL m = n + 2 J n = m–2 >

# C9I9LC5>5 =L@B;C2=2 58JC9I9> JL=45=C46B C2 O456;4CJK T;?@JL 89;>8CA=2 C2H

=

5

2 2

5

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(@ 65C5I7:5C9 =C2 =A=C7 Q63;5=2, =5H 5:>ZJL@ =L<T5;2C7;45. PT9C9 65C5@J7=94

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:5@58;J=85G7=C9.

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= sin[@(R1 + 1/2)]/sin(@/2).

2.3

' L8JIJ<4=µ?H CJL DTFT C2H g(n) <>@9C54 8JIK 9K6JI5 9B@ S5=4=CJKµ9 =C4H 4O4?-

C2C9H C2H $"%µµ01#/3/%4 654 C2H (69853834 8/( E"#,( CJL µ9C5=T2µ5C4=µJK Fourier

O456;4CJK T;?@JL. PC=4 QTJLµ9:

• 9: J;4=µJK.

• I?<A C2H 4O4?C2C5H C2H JI>=G2=2H =CJ T;?@J 65CB

n0 = 1 (SI. *>@565 2.2)

• x(n)–x(n–1) X(ej@)–e–j@X(ej@) = (1–e–j@)X(ej@) I?<A CA@ OKJ 8;J2<JKµ9-

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2.4

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F? @A

x n e X eF j j( ) ( )# ? @A #1 6 6

x n X eF j( ) ( )? @A 6


x(n) = >ancos(@0n + <)u(n) = >an

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2.5

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. PTJLµ9 IJ48?@:

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=T2µ5C4=µM@ Fourier X1(ej@) 654 X2(e

j@). (LC7 2 =TQ=2 58JC9I9> C2 O4CC7 Q63;5=2

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OL@5µ9> µ9 CJ@ 8JII58I5=45=µ? CA@ µ9C5=T2µ5C4=µM@ Fourier 5LCM@.

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=7

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2.6

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X x n W n WNn

n

N

Nn

n

N

N N NN( ) ( ) ( ) ...

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= ! + + + + ==

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=

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0

1

0

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kNn


2.7

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, k = 0, 1, 2, 3, …, R–1

045 k=0 8;J6K8C94:

045 kd0 8;J6K8C94: , k = 1, 2, 3, …, R–1

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<45 k = 1, 2, 3, …, R–1

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µ2O9@?H. Y;5, C9I46B:

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CJL µ9C5=T2µ5C4=µJK Fourier (duality property).

X k Ae

e

Ae

e

jN

k N

jN

k

j k

jk

N

( ) = ! #

#

= ! #

#=

# *+,

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-./

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#

1

1

1

1

0

2

2

2

2

)

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jN

nk

n

Nj

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n

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# #

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#

5 52

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n

N

n

N

n

N

(0 1 10

0

1

0

1

0

1

) = ! = ! = = !#

=

#

=

#

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5 5 5 X k x n W A eN

nk

n

Nj

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n

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( ( )) = ! = !=

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5 50

1 2

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1 )

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x(n)

X(k)

A

AN

1 2 1 20N-1 N-1

™¯‹Ì· 2.18

# =C5G9;7 56J-IJLG>5 8IBCJLH >654 J DFT 5LC7H

(@ O9@ 65C53Q;5C9 @5 IK=9C9 C2@ B=62=2 5LC7 µ2@ 58J<J2C9LG9>C9. E9@ 9>@54 58?

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=µ5CJH µ45H 8989;5=µQ@2H <9Aµ9C;467H =94;BH. *;J=85G7=C9 @5 C2@ 65C5@J7=9C9



654 @5 985@5IBS9C9 C2 IK=2 µ?@J4 =5H.

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<45 C2@ 56JIJLG>5 x(n) = {1, 1, 0, 0}, ?8JL S;QG269 ?C4

.

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654

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(@ S;765C9 CJ =A=C? 58JCQI9=µ5, C?C9 µ8;BSJ =5H. (LC? =2µ5>@94 ?C4 QT9C9 65C5-

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654 µ9 CJLH OKJ C;?8JLH. (;T>=C9 C2 O45O465=>5 µ9 SB=2 CJ@ J;4=µ? (2.21) 654 =L@9-

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3B@269 84J 9K6JIJH;

2.9

' DFT C2H x1(n) = {1, 2, 3} 4=JKC54 µ9 X1(k) = {6, –1,5+j0,866, –1,5–j0,866}.

' DFT C2H x2(n) = {4, 5, 6} 4=JKC54 µ9 X2(k) = {15, –1,5+j0,866, –1,5–j0,866}.

' DFT C2H x3(n) = x1(n)Gx2(n) = {31, 31, 28} 4=JKC54 µ9 X3(k) = {90, 1,5–j2,598,

1,5+j2,598}.

*5;5C2;JKµ9 ?C4 2 56JIJLG>5 X3(k) 4=JKC54 µ9 CJ <4@?µ9@J CA@ X1(k) 654 X2(k) =CJ4-

T9>J 8;JH =CJ4T9>J.

(@ 65C5I7:5C9 =CJ =A=C? 58JCQI9=µ5, C?C9 =L<T5;2C7;45. PT9C9 65C5@J7=94 CJ@

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8;B:9A@ =9 µ4<5O46JKH 5;4GµJKH. +C2@ 5@C>G9C2 89;>8CA=2, 8;J=85G7=C9 654 8BI4.

D5 O9>C9 ?C4 9>@54 GQµ5 9:J469>A=2H µ9 CJLH µ4<5O46JKH.

X k W X kNk

22( () )= #

X k W X kNk

1( () )= #

X k e ej j

( ) , , ,8 8

=01D

2D

BCD

ED

#2 2 0 24 4


™¯‹Ì· 2.19

)5 =7µ5C5 (5)QTJL@ CJ >O4JµQC;J (S),5IIB O453J;9C4673B=2 (<)

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|X(k)|=|X1(k)|=|X2(k)|

2 22

0 1 2 3k

x(n)

X(k)

X1(k)

X2(k)

1

11

1

00

1 2 3

0

00

1

1 1

2 1 23

3

01 2 1 23

3

1

2 3n k

k

k

n

n

'/4

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3'/4

-3'/4

-'/4

-'/4

x1(n)=x(<n+1>4)

x2(n)=x(<n+2>4)

(S)

(5)

(<)



2.10X;2=4µJ8J4M@C5H CJ@ J;4=µ? (2.21) CJL DFT S;>=6JLµ9 ?C4

!8JµQ@AH

654 .

+L@98MH, CJ G9M;2µ5 CJL Parseval 985I2G9K9C54.

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DFT (2.21). !4O46B <45 µ46;QH C4µQH CJL R, 9>@54 GQµ5 58IM@ 8;B:9A@ µ9 µ4<5O46JKH

5;4GµJKH.

2.11

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2, 0, 0, 0, 0}. )J µ76JH R 9>@54 µ9<5IKC9;J CJL µ76JLH C2H 56JIJLG>5H 8JL 8;J6K-

8C94 58? C2 <;5µµ467 =L@QI4:2 CA@ h(n), x(n). ,9 BII5 I?<45, N > L + M–1 = 4 +

3–1 = 6, ?8JL L, M, L + M–1 C5 µ762 CA@ 56JIJLG4M@ h(n), x(n) 654 y(n) =

h(n)*x(n), 5@C>=CJ4T5.

F8JIJ<>ZJLµ9 CJ@ DFT CA@ h'(n), x'(n) µ9 SB=2 C2 =TQ=2 (2.21) 654 S;>=6JLµ9 5@C>-

=CJ4T5:

H(k) = {6, 2,41–j4,41, –2–j2, –0,41+j1,6, 2, –0,41–j1,6, –2+j2, 2,41+j4,41}

X(k) = {14, 4,41–j9,24, –2–j2, 1,6+j0,76, 2, 1,6–j0,76, –2+j2, 4,41+j9,24}.

)J <4@?µ9@J CA@ H(k) 654 X(k) G5 µ5H OM=94 C2@ Y(k), O2I5O7, Y(k) = X(k)H(k) =

{84, –30,14–j41,8, j8, –1,86+j2,2, 4, –1,86–j2,2, –j8, –30,14+j41,8}. 045 @5

S;JKµ9 C2@ y(n) G5 8;Q894 @5 L8JIJ<>=JLµ9 CJ@ IDFT C2H Y(k) =Kµ3A@5 µ9 C2 =TQ=2

(2.22), <45 R = 8, O2I5O7

, n = 0,1,2,…,7. y n Y k e j kn

k

( ) ( ) /==

518

2 8

0

7)

13

13

36 3 313

42 142

X kk

( ( ))=0

2

= ! + + = ! =5

| ( ) |x nn

2 2 2 2

0

2

1 2 3 1 4 9 14= + + = + + ==

5

X k j j( ) , , .= # + # #01D

2D

BCD

ED6

32

32

32

32


)J 58JCQI9=µ5 8JL 8;J6K8C94 9>@54: y(n) = {3, 10, 22, 24, 19, 6, 0, 0}.

*5;5C2;JKµ9 ?C4 C5 OKJ C9I9LC5>5 =CJ4T9>5 C2H 56JIJLG>5H y(n) 9>@54 µ2O9@46B,

?8AH BIIA=C9 7C5@ 5@5µ9@?µ9@J, 53JK 2 <;5µµ467 =L@QI4:2 CA@ x(n), h(n) G5 8;Q-

894 @5 QT94 µ76JH Q:4. )J 58JCQI9=µ5 5LC? =Lµ3A@9> µ9 969>@J CJL *5;5O9><µ5CJH

1.6, =CJ J8J>J L8JIJ<>=5µ9 654 8BI4 C2@ >O45 <;5µµ467 =L@QI4:2, 5IIB 589LG9>5H

=CJ 89O>J CJL T;?@JL.

(@ 65C53Q;5C9 @5 IK=9C9 =A=CB C2@ B=62=2 5LC7, C?C9 =5H 5:>ZJL@ 94I46;4@B =L<-

T5;2C7;45. '4 L8JIJ<4=µJ> <45 µ9<BI9H C4µQH CJL R 9>@54 98>8J@J4 654 9K6JI5 µ8J-

;9> @5 <>@JL@ IBG2. +C4H µQ;9H µ5H, 85;?µJ4J4 L8JIJ<4=µJ> <>@J@C54 µ9 C2 SJ7G945

O453?;A@ 8;J<;5µµ5C4=C46M@ 9;<5I9>A@, ?8AH <45 85;BO94<µ5, CJL MATLAB.

PC=4, 2 Qµ35=2 O>@9C54 =C2@ 65C5@?2=2 C2H 4O4?C2C5H 5LC7H 65G95LC7@ 654 =C2 O45-

O465=>5 L8JIJ<4=µJK CJL 58JC9IQ=µ5CJH.

2.12

(5) 045 CJ@ 589LG9>5H L8JIJ<4=µ? C2H <;5µµ467H =L@QI4:2H OKJ 56JIJLG4M@ 8;5<-

µ5C46M@ 5;4GµM@ µ76JLH R 2 65G9µ4B, 5854CJK@C54 R2 8;5<µ5C46J> 8JII58I5-

=45=µJ> (UI. =TQ=2 1.39). )J 58JCQI9=µ5 C2H <;5µµ467H =L@QI4:2H G5 9>@54 µ45

56JIJLG>5 µ76JLH 2R–1.

(S) ' L8JIJ<4=µ?H C2H <;5µµ467H =L@QI4:2H µQ=A CJL 89O>JL C2H =LT@?C2C5H <>@9-

C54 =Kµ3A@5 µ9 CJ +T7µ5 2.11. '4 OKJ 56JIJLG>9H 8;Q894 @5 984µ26L@GJK@ µ9

C2@ 8;J=G762 =CJ4T9>A@ µ2O9@467H C4µ7H, M=C9 @5 58J6C7=JL@ µ76JH CJLIBT4-

=CJ@ >=J 8;JH 2R–1. P=CA C9I46B, ?C4 CJ µ76JH 6BG9 56JIJLG>5H <>@9C54 >=J

8;JH 2R. ' L8JIJ<4=µ?H CJL DFT µ45H 56JIJLG>5H 2R =CJ4T9>A@ 5854C9> (2R)2

= 4R2 µ4<5O46JKH 8JII58I5=45=µJKH. +Kµ3A@5 µ9 CJ +T7µ5 2.11, 5854C9>C54 J

L8JIJ<4=µ?H 2 DFT, J 8JII58I5=45=µ?H CA@ 2R =CJ4T9>A@ CA@ DFTs 8JL L8J-

IJ<>=5µ9 654 CQIJH, 1 IDFT 2R =2µ9>A@. (LC? 4=JOL@5µ9> µ9 3DFTs 654 2R

µ4<5O46JKH 8JII58I5=45=µJKH, 7 =L@JI46B µ912R2 + 2RV12R2 µ4<5O46JKH 8JI-

I58I5=45=µJKH. !894O7 6BG9 µ4<5O46?H 8JII58I5=45=µ?H T;94BZ9C54 CQ==9;4H

8;5<µ5C46JKH 8JII58I5=45=µJKH <45 @5 8;5<µ5CJ8J42G9>, J =L@JI46?H 5;4Gµ?H

CA@ 8;5<µ5C46M@ 8JII58I5=45=µM@ <>@9C54 48R2. (LC?H J 5;4Gµ?H 9>@54 65CB

8JIK µ9<5IKC9;JH 969>@JL 8JL 9>O5µ9 =C2@ 89;>8CA=2 (5) CJL L8JIJ<4=µJK C2H

<;5µµ467H =L@QI4:2H 589LG9>5H =CJ 89O>J CJL T;?@JL. Y;5, 2 T;7=2 CJL DFT

6;>@9C54 AH µ2 =Lµ3Q;JL=5.

(<) (H OJKµ9 CM;5 C2@ 89;>8CA=2 65CB C2@ J8J>5, <45 CJ@ L8JIJ<4=µ? CJL DFT T;2-

=4µJ8J4JKµ9 Q@5@ C5TK 5I<?;4GµJ Fourier (FFT) SB=2H 2. )J 8I7GJH CA@ µ4<5-

2 7 7A ¶ " ¡ & % ™ ! # ™ A ™ ' % ™ ! ø ¡ AÀ & $ " • # $ § $ ° % ™ % ™



O46M@ 8JII58I5=45=µM@ <45 µ45 56JIJLG>5 2R =CJ4T9>A@ 9>@54 (2R/2) log22N =

N log22N. 045 CJ@ L8JIJ<4=µ? C2H <;5µµ467H =L@QI4:2H =Kµ3A@5 µ9 CJ +T7µ5

2.11 5854CJK@C54 3N log22N + 2R µ4<5O46J> 8JII58I5=45=µJ> 7 12N log22N +

8R 8;5<µ5C46J> 8JII58I5=45=µJ>. &5C5<;B3J@C5H CJ 8I7GJH CA@ 8;5<µ5C46M@

8JII58I5=45=µM@ <45 O453J;9C46B R =CJ@ *>@565 2.6, 85;5C2;JKµ9 ?C4 <45 R

> 64 µ5H =Lµ3Q;94 @5 L8JIJ<>=JLµ9 C2 <;5µµ467 =L@QI4:2 µQ=A CJL 89O>JL C2H

=LT@?C2C5H (µ9 T;7=2 CJL FFT) 654 ?T4 589LG9>5H =CJ 89O>J CJL T;?@JL.

¶›Ó·Î·˜ 2.6

!625(4 '"%$µ%/01+, '(66%'6%80%8µ+, $0% /(, )'(6($08µ# /34 $"%µµ0124 8),C60-

F34 *7( %1(6()50+, R 8/(0E.9@,.

*I7GJH =CJ4T9>A@ 6BG9

56JIJLG>5H

(R)

(89LG9>5H L8JIJ<4=µ?H

C2H <;5µµ467H =L@QI4:2H

=CJ 89O>J CJL T;?@JL

(R2)

F8JIJ<4=µ?H C2H <;5µµ4-

67H =L@QI4:2H =CJ 89O>J

C2H =LT@?C2C5H µQ=A

CJL FFT

(12N log22N + 8R)

4 16 176

8 64 448

16 256 1088

32 1024 2560

64 4096 5888

128 16384 13312

256 65536 29696

512 262144 65536

1024 1048576 143360

(@ 65C53Q;5C9 @5 585@C7=9C9 =A=CB, C?C9 =5H 5:>ZJL@ =L<T5;2C7;45. PT9C9 65C5-

@J7=94 C?=J C5 8I9J@96C7µ5C5 C2H T;7=2H CJL FFT ?=J 654 C5 =T9C46B µ9 CJ@ L8J-

IJ<4=µ? C2H =L@QI4:2H =CJ T;?@J, 7 µQ=A CJL 89O>JL C2H =LT@?C2C5H. (@ O9@ µ8J;Q-

=5C9 @5 IK=9C9 C2@ B=62=2 5LC7, C?C9 µ9I9C7=C9 654 8BI4 C4H 9@?C2C9H 2.2.3 654 2.3.


3.1

,>5 58? C4H 84J =2µ5@C46QH 4O4?C2C9H CJL M.Z. 9>@54 5LC7 C2H =L@QI4:2H. (H OJKµ9

C2@ 9L6JI>5 8JL µ5H 85;QT94 L8JIJ<>ZJ@C5H C2 =L@QI4:2 y(n) = h(n)*x(n), ?8JL h(n)

= {1, 2, 3} 654 x(n) = {3, 4, 5, 2}. U;>=6JLµ9 CJLH M.Z. CA@ =L@5;C7=9A@ 654 L8J-

IJ<>ZJLµ9 C2@ Y(z) = H(z)X(z). H y(n) G5 8;J6KR94 58? CJ@ 5@C>=C;J3J M.Z. C2H

H(z). PTJLµ9 98JµQ@AH:

H(z) = 1 + 2z–1 + 3z–2 X(z) = 3 + 4z–1 + 5z–2 + 2z–3

*JII58I5=4BZJ@C5H 65CB µQI2 C5 8JILM@Lµ5 5LCB 85>;@JLµ9 C2@ Y(z):

Y(z) = 3 + 10z–1 + 22z–2 + 24z–3 + 19z–4 + 6z–5

H Y(z) 963;BZ9C54 AH OL@5µJ=94;B 654 =L@98MH J4 =L@C9I9=CQH 5LC7H 58JC9IJK@

=C2@ 8;B:2 654 C5 O9><µ5C5 9:?OJL y(n):

y(n) = {3, 10, 22, 24, 19, 6}

*5;5C2;JKµ9 ?C4 CJ 58JCQI9=µ5 5LC? =Lµ3A@9> µ9 C5 5@C>=CJ4T5 CA@ *5;5O94<-

µBCA@ 6 654 7 CJL &935I5>JL 1.

3.2

' L8JIJ<4=µ?H CJL M.Z. CJL =7µ5CJH x(n), CJ J8J>J 9>@54 8989;5=µQ@2H O4B;6945H,

µ8J;9> @5 <>@94 589LG9>5H 58? CJ@ J;4=µ? (3.1). # *.+. CJL M.Z. G5 9>@54 ?IJ CJ 98>-

89OJ–z 96C?H C2H C4µ7H z = 0, 53JK 2 x(n) 9>@54 O9:4?8I9L;2 56JIJLG>5 8989;5=µQ-

@JL µ76JLH µ9 n ( 0. (LC?H J C;?8JH L8JIJ<4=µJK 537@9C54 AH B=62=2 =CJ@ 5@5-

<@M=C2. +2µ94M=C9 µ?@J ?C4 <45 z = 1 J X(z) = N, 53JK X(z) = 1 + z–1 + … + z–(N–1).

!@5II56C46B, J >O4JH L8JIJ<4=µ?H G5 µ8J;JK=9 @5 <>@94 µ9 SB=2 C4H 4O4?C2C9H C2H

<;5µµ46?C2C5H 654 C2H JI>=G2=2H. *5;5C2;7=C9 ?C4 CJ =7µ5 x(n) µ8J;9> @5 963;5-

=C9> AH <;5µµ46?H =L@OL5=µ?H OKJ µJ@5O45>A@ S2µ5C46M@ 56JIJLG4M@ B894;JL

µ76JLH, O2I5O7 x(n) = u(n)–u(n–N) (UI. B=62=2 5LCJ5:4JI?<2=2H 1.1). +L@98MH

Z{x(n)} = Z{u(n)}–Z{u(n–N)}

7 X(z) = Z{u(n)}–z–NZ{u(n)} = (1–z–N)Z{u(n)} = (1–z–N) , |z| > 1

?8JL 58? CJ@ *>@565 3.1, T;2=4µJ8J47=5µ9 ?C4 Z{u(n)} = µ9 *.+. |z| > 1.

(:>Z94 @5 =2µ94M=JLµ9 ?C4 2 *.+. CJL X(z) 9>@54, ?8AH 7O2 5@53Q;5µ9, JI?6I2;J

CJ 98>89OJ–z 96C?H C2H C4µ7H z = 0. ,9 BII5 I?<45, 9B@ J <;5µµ46?H =L@OL5=µ?H

8JIIM@ =2µBCA@ QT94 AH 58JCQI9=µ5 Q@5 =7µ5 8989;5=µQ@2H O4B;6945H, C?C9 2 *.+.

1

1 1# #z

1

1

1

11 1#= #

##

#

#z

z

z

N

2 7 9A ¶ " ¡ & % ™ ! # ™ A ™ ' % ™ ! ø ¡ AÀ & $ " • # $ § $ ° % ™ % ™



CJL M.Z. 65GJ;>Z9C54 58J6I94=C46B 654 µ?@J 58? C2 3K=2 CJK 8989;5=µQ@2H O4B;-

6945H =7µ5CJH 654 ?T4 58? C4H *.+. CA@ 984µQ;JLH M.Z. (LC? JL=45=C46B 9>@54 =L@Q-

8945 CJL <9<J@?CJH ?C4 8?IJ4 585I9>3J@C54 58? 5@C>=CJ4T5 µ2O9@46B, ?8AH 9>@54 2

89;>8CA=2 CJL 85;5O9><µ5CJH.

3.3

(5) (3JK 2 *.+. 9>@54 CJ 9:AC9;46? 9@?H 6K6IJL, =Lµ89;5>@JLµ9 ?C4 CJ =7µ5 x(n) G5

9>@54 54C45C?. Y;5 RBT@JLµ9 <45 µ45 OL@5µJ=94;B µ9 5;@2C46QH OL@Bµ94H CJL z. !6C9-

IJKµ9 IJ48?@ C2 =L@9T7 O45>;9=2 CJL 5;4Gµ2C7 µ9 CJ@ 85;J@Jµ5=C7 654 QTJLµ9:

z2–1,9z + 0,84

z–1 + 0,9 z–2 + 0,87 z–3 + ….[

z–1[

z–1,9 + 0,84z–1[

e 0,9–0,846z–1[

e 0,9–1,71z–1 + 0,756z–2[

ee 0,87z–1 – 0,756z–2[

ee 0,87z–1 – 1,653z–2 + 0,7308z–3[

eee 0,897z–2 – 0,7308 z–3[

eeee …………[

!8JµQ@AH 2 X(z) µ8J;9> @5 963;5=C9> AH OL@5µJ=94;B:

X(z) = z–1 + 0,9z–2 + 0,87z–3 + ….

&5CB =L@Q8945, µ8J;JKµ9 Bµ9=5 @5 =Lµ89;B@JLµ9 ?C4

{x(n)} = {0, 1, 0,9, 0,87,…}

*9;4==?C9;5 O9><µ5C5 C2H x(n) µ8J;JKµ9 @5 QTJLµ9 =L@9T>ZJ@C5H C2 O45>;9=2.

+2µ94M=C9 ?C4 <45 C2 O45O465=>5 C2H =L@9TJKH O45>;9=2H <;BR5µ9 CJ@ 5;4Gµ2C7 654

CJ@ 85;J@Jµ5=C7 C2H X(z) 65CB C4H <59,()8.4 *),?µ.04 CJL z.

(S) +C2@ 89;>8CA=2 5LC7 2 *.+. CJL M.Z. X(z) J;>Z9C54 =CJ 9=AC9;46? 9@?H 6K6IJL.

!8JµQ@AH, CJ 5@C>=CJ4TJ =7µ5 x(n) G5 9>@54 5@C4–54C45C? (anticausal). Y;5 RBT@JL-

µ9 <45 µ45 OL@5µJ=94;B µ9 G9C46QH OL@Bµ94H CJL z. !6C9IJKµ9 654 8BI4 C2 =L@9T7

O45>;9=2, O45CB==J@C5H ?µAH C2@ 3J;B 5LC7 5;4Gµ2C7 654 85;J@Jµ5=C7 65CB C4H

%7F()8.4 *),?µ.04 CJL z. PTJLµ9 IJ48?@:


Y;5 X(z) = –1,1905–1,5023z–1,9807z2 + …

654 65CB =L@Q8945 {x(n)} = {…,–1,9807, –1,5023,–1,1905},

O2I5O7 x(–2) = –1,9807, x(–1) = –1,5023, x(0) = –1,1905.

3.4

# O45O465=>5 C2H 5@B8CL:2H =9 µ9;46B 6IB=µ5C5 9>@54 85;?µJ45 969>@2H CJL *5;5-

O9><µ5CJH 3.7. (LC? 8JL 8;Q894 @5 O49;9L@7=JLµ9 96C9@Q=C9;5 9OM 9>@54 CJ GQµ5

C2H Z2CJKµ9@2H 56JIJLG>5H, 53JK O9@ µ5H O>@9C54 CJ 9B@ 8;?694C54 <45 54C45C? 7

µ2–54C45C? =7µ5. (;T>ZJLµ9 IJ48?@ 58? C2@ 5@B8CL:2 =9 µ9;46B 6IB=µ5C5 654

=L@9T>ZJLµ9 µ9 C2 µ9IQC2 CA@ *.+. C2H X(z).

654 98JµQ@AH

'4 8?IJ4 C2H X(z) S;QG265@ >=J4 µ9 p1 = 0,5 654 p2 = –0,5. F8JIJ<>ZJLµ9 CM;5 C5

>1, >2.

Y;5, 654 X z

z z( )

, ,=

#+

+# #41 0 5

21 0 51 1

1 1

X zz z z( )

, ,=

#+

+4 20 5 0 5

A zX z

zz

z

A zX z

zz

z

z z

z z

10 5 0 5

20 5 0 5

0 56 1

0 54

0 56 1

0 52

= # = ++

=

= + = +#

=

= =

=# =#

( , )( )

,

( , )( )

,

, ,

, ,

X zz

zz z

Az

Az

( )( , )( , ) , ,

= +# +

=#

++

6 10 5 0 5 0 5 0 5

1 2

X z

z

z

z z

z( )

, ,= +

#= +

#

#

#6

1 0 25

6

0 25

1

2

2

2

2 8 1A ¶ " ¡ & % ™ ! # ™ A ™ ' % ™ ! ø ¡ AÀ & $ " • # $ § $ ° % ™ % ™

0,84–1,9z + z2

–1,1905 – 1,5023 z + 1,9807 z2 + …[

–1 + z[

–1 + 2,2619z–1,1905z2[

e –1,2619z + 1,1905z2[

e –1,2619z + 2,8543z2–1,5023z3[

ee –1,6638z2 + 1,5023z3[

eee ………[



,QT;4 CJ =2µ9>J 5LC?, 84=C9KJLµ9 8AH 3CB=5C9 =T9C46B 9K6JI5. (@ ?T4, 8;J=85-

G7=C9 654 8BI4 56JIJLGM@C5H CJ *5;BO94<µ5 3.7. !69>@J 8JL >=AH O9 =693C765C9

@5 6B@9C9, 9>@54 µ>5 µ46;7 O49;9K@2=2 =T9C46B µ9 C4H O453J;9C46QH 89;4JTQH =K<6I4-

=2H CJL M.Z., M=C9 @5 8;JC9>@9C9 C2@ 5@C>=CJ4T2 56JIJLG>5.

'4 OKJ 8?IJ4 S;>=6J@C54 =C5 =2µ9>5 ± 0,5 654 QTJL@ CJ >O4J µQC;J. !8JµQ@AH TA;>-

ZJL@ CJ 98>89OJ =9 OKJ *.+. C2@ -1 96C?H CJL 6K6IJL |z| = 0,5 654 C2@ -2 9@C?H CJL

>O4JL 6K6IJL. 045 C2@ *.+. -1 (|z| > 0,5) 654 C5 OKJ µ9;46B 6IB=µ5C5 5@C4=CJ4TJK@

=9 O9:4?8I9L;9H 56JIJLG>9H O>@J@C5H:

x(n) = 4(0,5)nu(n) + 2(–0,5)nu(n).

!894O7 2 *.+. -1 89;4I5µSB@94 CJ µJ@5O45>J 6K6IJ, CJ =7µ5 x(n) G5 9>@54 9L=C5GQH.

045 C2@ *.+. -2 (|z| < 0,5) 654 C5 OKJ µ9;46B 6IB=µ5C5 5@C4=CJ4TJK@ =9 5;4=C9;?-

8I9L;9H 56JIJLG>9H O>@J@C5H:

x(n) = –4(0,5)nu(–n–1)–2(–0,5)nu(–n–1)

+C2@ 89;>8CA=2 5LC7 CJ =7µ5 x(n) 9>@54 5=C5GQH, 53JK J µJ@5O45>JH 6K6IJH O9@

89;4QT9C54 =C2@ *.+.

3.5

F8JIJ<>ZJLµ9 C2 =L@B;C2=2 µ9C53J;BH 654 58? 5LC7 µ8J;JKµ9 9K6JI5 @5 8;J=-

O4J;>=JLµ9 <9Aµ9C;46B C2@ 58?6;4=2 =LT@?C2C5H.

µ9 *.+. |z| > 0,5

' 8?IJH z = 0,5 8;J=O4J;>Z94 654 C2@ *.+. C2H H(z), 2 J8J>5 J;>Z9C54 =CJ 9:AC9;4-

6? CJL 6K6IJL 56C>@5H |z| = 0,5. ' µJ@5O45>JH 6K6IJH 9µ89;4QT9C54 =C2@ *.+. 654

98JµQ@AH, CJ =K=C2µ5 9>@54 9L=C5GQH. # 58?6;4=2 =LT@?C2C5H µ8J;9> @5 L8JIJ<4-

=C9> 58? C2@ H(z) <45 z = ej@. PTJLµ9:

H z Z h n

z( ) { ( )}= =

# #1

1 10,5

P1P2

0 0,5-0,5™¯‹Ì· 3.12

!."0(EC4 87$160-834 8/(

.'9'.*(–z.


?8JL V = C/D = 1/D 654 - = 5–S. # =L@B;C2=2 QT94 Q@5 8?IJ =CJ =2µ9>J z = 0,5.

' <9Aµ9C;46?H 8;J=O4J;4=µ?H C2H 58?6;4=2H =LT@?C2C5H <>@9C54 µ9 C2 SJ7G945 CA@

O45<;5µµBCA@ CJL +T7µ5CJH 3.135–O. 045 O453J;9C46QH <A@>9H @, O2I5O7 <45 O45-

3J;9C46QH =LT@?C2C9H, L8JIJ<>ZJLµ9 CJ µQC;J V 654 C2@ 3B=2 - C2H 58?6;4=2H

=LT@?C2C5H. '4 5@C>=CJ4T9H <;5346QH 85;5=CB=94H O>@J@C54 58? CJ +T. 3.139–=C.

*5;5C2;JKµ9 ?C4 8;?694C54 <45 Q@5 =K=C2µ5 CJ J8J>J 984C;Q894 =C4H T5µ2IQH =LT@?-

C2C9H @5 «89;B=JL@» 58? 5LC?, 9@M 9:5=G9@9> C4H LR2IQH =LT@?C2C9H. E2I5O7, 8;?-

694C54 <45 Q@5 S5GL89;5C? 3>IC;J.

H e

e

e

e

Ce

DeVej

j

j

j

j

jj( )

, ,6 =

#=

#= =#

1

1 0 5 0 5@

@

@

%

IL

2 8 3A ¶ " ¡ & % ™ ! # ™ A ™ ' % ™ ! ø ¡ AÀ & $ " • # $ § $ ° % ™ % ™

0,000,250,500,751,001,25

2,00

0 ''/4 '/2 3'/4

0 ''/4 '/2 3'/4

-30

-25

-20

-15

-10

-5

0,5 0,50,50,5

@=0 @=% @='/2 @='

.B=2

MQC;J

(5) (S) (<) (O)

(9) (=C)

% I

™¯‹Ì· 3.13

D.@µ./"01#4'"(8*0("08µ#4 /34%'#1"0834 8)E,#-/3/%4: (%)–(*) *0?-$"%µµ% '#6@, –µ3*.,01+, $0% *0%-<("./01C4 8)E,#-/3/.4, (.)–(8/)µC/"( 1%0 <?83/34 %'#1"0834.

3.6

045 @5 585@C7=JLµ9 =CJ 9;MC2µ5 5LC? G5 8;Q894 @5 OJKµ9 CJ =K=C2µ5 58? C2@

8I9L;B CJL M.Z. 5LCJK. ,?@J QC=4 G5 µ8J;Q=JLµ9 9K6JI5 @5 CJ 5@5IK=JLµ9 654

@5 8;J=O4J;>=JLµ9 C2 =Lµ89;43J;B CJL.

(8? C2@ 9:>=A=2 O453J;M@ 8JL µ5H O>@9C54 µ8J;JKµ9 @5 L8JIJ<>=JLµ9 C2 =L@B;-

C2=2 µ9C53J;BH CJL =L=C7µ5CJH H(z). (@5ILC46B QTJLµ9:



654

7

7

*5;5C2;JKµ9 ?C4 CJ =K=C2µ5 5LC? QT94 OKJ 8?IJLH =C5 =2µ9>5 z = 1/2 654 z = 1/3.

+C2@ 9>=JOJ CJL =L=C7µ5CJH 935;µ?Z9C54 CJ =7µ5 x(n) = *(n)–(1/3)*(n–1). ' M.Z.

5LCJK 9>@54 X(z) = 1–(1/3)z–1. UIQ8JLµ9 98JµQ@AH, ?C4 CJ =7µ5 94=?OJL 89;4QT94 Q@5

µ2O9@46? =CJ =2µ9>J z = 1/3. )J µ2O9@46? 5LC? 9:JLO9C9;M@94 CJ@ 8?IJ z = 1/3 CJL

=L=C7µ5CJH. E2I5O7, Y(z) = H(z)X(z) 7 .

&5CB =L@Q8945, 2 58?6;4=2 CJL =L=C7µ5CJH <45 C2 =L<696;4µQ@2 9>=JOJ x(n) G5

4=JKC54 µ9 .

3.7

!>O5µ9 ?C4, <@A;>ZJ@C5H C2 =L@B;C2=2 µ9C53J;BH 9@?H =L=C7µ5CJH, µ8J;JKµ9

Bµ9=5 @5 9IQ<:JLµ9 C2@ BIBO 9L=CBG945 5LCJK, @5 =T9O4B=JLµ9 C2 OJµ7 CJL, 8JL

G5 µ5H JO2<7=94 C9I46B =C2@ 84J 65CBII2I2 LIJ8J>2=2 654 @5 S;JKµ9 C2@ 58?6;4-

=2 =LT@?C2C5H CJL =L=C7µ5CJH. (H 9:9CB=JLµ9 C2 =L@B;C2=2 µ9C53J;BH CJL 54C45-

CJK =L=C7µ5CJH H(z) = 1/(1 + 0,5z–1) = z/(z + 0,5). UIQ8JLµ9 ?C4 5LC7 QT94 Q@5 µ2O9-

@46? =CJ =2µ9>J z = 0 654 Q@5@ 8?IJ =CJ =2µ9>J z = –0,5. (3JK J 8?IJH S;>=69C54

9@C?H CJL µJ@5O45>JL 6K6IJL, CJ =K=C2µ5 9>@54 9L=C5GQH. (8? C2@ H(z) µ8J;JKµ9

9K6JI5 @5 8;J=O4J;>=JLµ9 µ9 CJ <9Aµ9C;46? (<;5346?) C;?8J C2@ 58?6;4=2 =LT@?-

C2C5H. )5 O45<;Bµµ5C5 8?IA@–µ2O9@46M@ <45 C4H 8IQJ@ T5;56C2;4=C46QH 89;48CM-

=94H, O2I5O7 @ = 0, @ = 8/2 654 @ = ' 35>@J@C54 =CJ +T7µ5 3.145–<. H 5@C>=CJ4T2

58?6;4=2 =LT@?C2C5H (µQC;J 654 3B=2) O>@9C54 58? CJ +T7µ5 3.14O–9. *5;5C2;JKµ9

?C4 CJ =K=C2µ5 5LC? 9:5=G9@>Z94 C4H T5µ2IQH 654 «9L@J9>» C4H LR2IQH =LT@?C2C9H.

Y;5, CJ =K=C2µ5 9>@54 Q@5 LR289;5C? 3>IC;J.

# OJµ7 8;5<µ5CJ8J>2=2H CJL =L=C7µ5CJH 58JC9I9> JL=45=C46B C2@ Bµ9=2 58946?-

y n Z Y z u n

n

( ) { ( )} ( )= = *+,

-./

#1 12

Y zz

( )( )

=# #

1

112

1

H zY zX z

z z z z( )

( )( )

( )( )= =

# +=

# ## # # #

1

156

16

1

112

113

1 2 1 1

Y z z Y z z Y z X z( ) ( ) ( ) ( )= # +# #5 11 2

6 6

Z y n Z y n Z y n Z x n{ ( )} { ( )} { ( )} { ( )}= # # # +5

11

26 6

y n y n y n x n( ) ( ) ( ) ( )= # # # +5

11

26 6


@4=2 C2H 9:>=A=2H O453J;M@ 5LCJK (+T7µ5 3.15). # 9:>=A=2 O453J;M@ 8;J6K8C94

9K6JI5 58? C2 =L@B;C2=2 µ9C53J;BH, O2I5O7:

*5>;@J@C5H CJ@ 5@C>=C;J3J M.Z. 654 I5µSB@J@C5H L8?R2 C4H 4O4?C2C9H C2H <;5µµ4-

6?C2C5H 654 C2H JI>=G2=2H, 65C5I7<JLµ9 =C2 =TQ=2: y(n) = x(n)–0,5y(n–1).

H z

Y zX z z

Y z z X z Y z X z z Y z( )( )( ) .

( )( . ) ( ) ( ) ( ) . ( )= =+

J + = J = ### #1

1 0 51 0 5 0 5

11 1

2 8 5A ¶ " ¡ & % ™ ! # ™ A ™ ' % ™ ! ø ¡ AÀ & $ " • # $ § $ ° % ™ % ™

0,5 0,5 0,5

@=0 @='/2 @='

(5) (S) (<)

MQC;J

.B=2

0,000,250,500,751,001,251,501,752,00

0 ''/4 '/2 3'/4 0 ''/4 '/2 3'/4

(O) (9)

0

5

10

15

20

25

30

x(n) y(n)+

–0,5z–1

™¯‹Ì· 3.14

D.@µ./"01#4'"(8*0("08µ#4 /34%'#1"0834 8)E,#-/3/%4: (%)–($) *0?-$"%µµ% '#6@, –µ3*.,01+, $0% *0%-<("./01C4 8)E,#-/3/.4, (*)–(.)µC/"( 1%0 <?83/34 %'#1"0834.

™¯‹Ì· 3.15

T(µ2 '"%$µ%/('(938348)8/2µ%/(4.



(:>Z94 @5 =2µ94M=JLµ9 9OM C5 9:7H:

• )J =K=C2µ5 8JL µ?I4H 9:9CB=5µ9 9>@54 Q@5 3>IC;J 8;MC2H CB:2H. (DLµ2G9>C9 ?C4

J S5Gµ?H CJL 8JILA@KµJL CJL 85;5@Jµ5=C7 C2H H(z) 65GJ;>Z94 654 C2@ CB:2 CJL

=L=C7µ5CJH).

• )J 3>IC;J 9>@54 985@5I28C46?, 53JK O9><µ5C5 C2H 9:?OJL 5@5C;J3JOJCJK@C54

=CJ =K=C2µ5.

• +L<6;>@J@C5H C2@ H(z) CJL =L=C7µ5CJH 5LCJK µ9 969>@2 C2H B=62=2H 5LCJ5:4J-

I?<2=2H 3.5, 85;5C2;JKµ9 ?C4 2 µ?@2 O453J;B 8JL L8B;T94 9>@54 =CJ 8;?=2µJ

CJL =L@C9I9=C7 C2H z–1 CJL 85;5@Jµ5=C7, O2I5O7 =CJ@ 8?IJ C2H =L@B;C2=2H.

(LC7 2 O453J;B QT94 AH =L@Q8945 @5 5IIB:94 ;4Z46B 2 =Lµ89;43J;B CJL =L=C7-

µ5CJH 654 58? S5GL89;5C? 3>IC;J =C2 µ>5 89;>8CA=2 @5 <>@94 LR289;5C? =C2@

BII2.

4.1

5. +A=C?.

S. "BGJH (5).

<. "BGJH ()4µ7 =L@C9I9=C7 0,2)

O. +A=C?.

9. "BGJH (&5GL=CQ;2=2 2;s = 1/44100 = 45,3 µsec)

(@ QT9C9 585@C7=94 =A=CB, C?C9 µ8;BSJ =5H! PT9C9 65C5IBS94 C4H S5=46QH Q@@J49H

=C5 FIR 3>IC;5, C5 J8J>5 µ9 C2 =94;B CJLH 58JC9IJK@ JL=45=C46? 9;<5I9>J =C2@

R234567 989:9;<5=>5 =7µ5CJH.

(@ QT9C9 µ9;46QH 58? C4H 585@C7=94H IBGJH µ2@ 58J<J2C9LC9>C9. E45SB=C9 8BI4 C2@

9@?C2C5 4.1, O>@J@C5H Qµ35=2 =C2@ L8J9@?C2C5 4.1.1.

4.2

# =LT@?C2C5 58J6J87H 8;Q894 @5 9>@54 µ46;?C9;2 58? C2@ ', 53JK 2 89;4JT7 9@O45-

3Q;J@CJH =C5 R23456B 3>IC;5 9>@54 [–', 8].

(@ 585@C7=5C9 =A=CB, =L<T5;2C7;45. PT9C9 65C5@J7=94 OKJ 8;B<µ5C5. )J@ C;?8J

8JL 9@9;<JK@ =C4H =LT@?C2C9H C5 O4B3J;5 3>IC;5 (LR289;5CB, S5GL89;5CB 6CI.),

65GMH 98>=2H 654 CJ <9<J@?H ?C4 2 58?6;4=7 CJLH =C2 =LT@?C2C5 9>@54 89;4JO467 µ9

89;>JOJ 2'.

+C2@ 89;>8CA=2 8JL QT9C9 6B@94 IBGJH, G5 8;Q894 @5 9=C4B=9C9 C2@ 8;J=JT7 =5H


C?=J =CJ@ C;?8J J;4=µJK CA@ 8;JO45<;53M@ 9@?H 3>IC;JL (9@?C2C5 4.2), ?=J 654

=CJ 693BI54J 3 9@?C2C5 3.4.5 ?8JL 9:2<9>C54 2 89;4JO46?C2C5 C2H 58?6;4=2H =LT@?-

C2C5H CA@ R23456M@ 3>IC;A@.

4.3

1. "BGJH.

2. +A=C?.

3. "BGJH.

4. +A=C?.

(@ QT9C9 585@C7=94 =A=CB µ8;BSJ =5H! PT9C9 65C5IBS94 CJ@ C;?8J 8JL I94CJL;-

<JK@ C5 85;BGL;5 =C2 O45O465=>5 =T9O>5=2H CA@ FIR 3>IC;A@. (LC? 9>@54 8JIK

=2µ5@C46? <45C> µ9 5@BIJ<J C;?8J C5 85;BGL;5 I94CJL;<JK@ 654 =9 BII9H 89;4-

8CM=94H R234567H 989:9;<5=>5H =7µ5CJH (O9@ 5@53Q;J@C54 =CJ S4SI>J 5LC?).

(@ QT9C9 585@C7=94 IBGJH, C?C9 OM=C9 89;4==?C9;2 8;J=JT7 =C2@ 9@?C2C5 4.3 654

6L;>AH =CJ =T7µ5 4.6., 8JL 89;4<;B394 CJ@ C;?8J 98>O;5=2H CJL 3B=µ5CJH CJL

85;5GK;JL =C2@ C9I467 58?6;4=2 CJL 3>IC;JL.

4.4

+A=C7 58B@C2=2 9>@54 2 S, O4?C4 J4 µJ@BO9H 65GL=CQ;2=2H =C2@ Bµ9=2 OJµ7 LIJ-

8J>2=2H CA@ FIR 3>IC;A@ 9>@54 65CB µ>5 I4<?C9;9H 58? CJLH 8JII58I5=45=CQH.

(@ 585@C7=5C9 =A=CB =2µ5>@94 ?C4 65C5@J7=5C9 CJLH OKJ O453J;9C46JKH C;?8JLH

LIJ8J>2=2H CA@ FIR 3>IC;A@ 8JL 89;4<;B3J@C54 =C2@ L8J9@?C2C5 4.4.1.

+9 5@C>G9C2 89;>8CA=2 OM=C9 89;4==?C9;2 8;J=JT7 =CJ =T7µ5 4.9.

4.5

)5 bits 8JL 5854CJK@C54 =C2@ 5@585;B=C5=2 CA@ =L@C9I9=CM@ 8;Q894 @5 9>@54

89;4==?C9;5 58? 106/6 = 17,6. Y;5 T;94BZJ@C54 18 bits.

(@ =C2@ B=62=2 5LC7 585@C7=5C9 =A=CB, 5LC? =2µ5>@94 ?C4 µ8J;9>C9 @5 9µS5GK-

@9C9 =C2@ =T9O>5=2 CA@ FIR 3>IC;A@, 53JK QT9C9 C2@ 5>=G2=2 CJL 8?=J 2 56;>S945

=C2@ 5@585;B=C5=2 CA@ =L@C9I9=CM@ CJL 3>IC;JL G5 982;9B=94 C2@ 58?6;4=7 CJL.

(@ O9@ 585@C7=5C9 =A=CB, 8;J=85G7=C9 8BI4 =C2@ L8J9@?C2C5 4.4.2.

2 8 7A ¶ " ¡ & % ™ ! # ™ A ™ ' % ™ ! ø ¡ AÀ & $ " • # $ § $ ° % ™ % ™



5.1,9 SB=2 CJLH =LµSJI4=µJKH CJL +T7µ5CJH 5.1 654 C5 O9OJµQ@5 C2H B=62=2H S;>-

=6JLµ9 ?C4:

, J8?C9 8;J6K8C94 ?C4 .2 = 1.

)J 58JCQI9=µ5 5LC? 7C5@ 5@5µ9@?µ9@J, 53JK C5 –3 dB 5@C4=CJ4TJK@ =C2 =LT@?C2-

C5 Qc, <45 C2@ J8J>5 :Q;JLµ9 ?C4 CJ µQC;J C2H 58?6;4=2H 4=JKC54 µ9 .

Fp = 500 Hz, J8?C9 Qp = 2'Fp = 2'500 rad/sec.

+2µ94M=C9 ?C4 =C2@ 8;J694µQ@2 89;>8CA=2 Qp'Qc.

, J8?C9 >2 = 104

Fs = 1000 Hz, J8?C9 Qs = 2'Fs = 2'1000 rad/sec.

(@C465G4=CM@C5H =C2@ 9:>=A=2 (5.6) L8JIJ<>ZJLµ9 C2@ CB:2 CJL 3>IC;JL:

Y;5 CJ 3>IC;J CJ J8J>J 8I2;J> C4H 8;JO45<;53QH 9>@54 QSOJµ2H CB:2H (R = 7).

'4 8?IJ4 CJL 3>IC;JL L8JIJ<>ZJ@C54 58? C2 =TQ=2 (5.9)

.

(@ QT9C9 L8JIJ<>=94 C2 =A=C7 CB:2 CJL 3>IC;JL Butterworth, 65GMH 654 CJLH 8?IJLH

CJL, C?C9 =L<T5;2C7;45. !>=C9 =9 GQ=2 @5 L8JIJ<>Z9C9 Q@5 58? C5 S5=46?C9;5 5@5IJ-

<46B 3>IC;5. 'L=45=C46B, QT9C9 65C5@J7=94 C2 =2µ5=>5 CA@ O453?;A@ 85;5µQC;A@,

<9<J@?H 8JL G5 =5H O49L6JIK@94 =CJLH L8JIJ<4=µJKH <45 C5 98?µ9@5 5@5IJ<46B 3>IC;5.

+9 5@C>G9C2 89;>8CA=2, µ2@ 58J<J2C9LG9>C9. ,9I9C7=C9 654 8BI4 C2@ L8J9@?C2C5

5.1.1 654 =C2 =L@QT945 O45SB=C9 µ9 8;J=JT7 CJ *5;BO94<µ5 5.1.

5.2# =L@B;C2=2 µ9C53J;BH CJL 8;AC?CL8JL 5@5IJ<46JK S5GL89;5CJK 3>IC;JL, 8JL

µ5H O>@9C54, 5@C48;J=A89K94 Q@5 3>IC;J µ9 =LT@?C2C5 58J6J87H Qc = 1 rad/s, O2I.

Qp = Qc = 1. (LC? 8JL µ5H Z2C9>C54 9>@54 @5 =T9O4B=JLµ9 654 8BI4 Q@5 3>IC;J

s e e qqj j q= =+1000 0 1 62 2 1 14) ) )/ ( ) / , , ,....

N

kk

A

s p

= =#( )[ ]

= # =log ( / )log ( / )

log /

log ( / )log ( )

log ( ),10 1

10

102 2

10

104

10

11

1

210 1

2 26 64

.

Q Q

10

14010 2

logA

*+,

-./

= # dB

1

2

10

1

1310 2

log+

*+,

-./

= #.

dB


Chebyshev CK8JL %, CJ J8J>J ?µAH @5 9>@54 LR289;5C? 654 @5 QT94 =LT@?C2C5 58J-

6J87H . )J 3>IC;J 5LC? 8;J6K8C94 9K6JI5 µ9 5@C465CB=C5=2 C2H

µ9C5SI2C7H s CJL 8;AC?CL8JL S5GL89;5CJK 3>IC;JL µ9 , ?8AH L85-

<J;9K9C54 58? C2 =TQ=2 (5.16). Y;5 2 Z2CJKµ9@2 =L@B;C2=2 µ9C53J;BH CJL LR2-

89;5CJK 3>IC;JL Chebyshev G5 9>@54:

*5;5C2;JKµ9 ?C4 CJ LR289;5C? 3>IC;J 8JL 8;JQ6LR9 9>@54 >O45H CB:2H µ9 CJ 5;T46?

S5GL89;5C? 3>IC;J, O2I5O7 O9KC9;2H CB:2H. )J 3>IC;J 5LC? 85;JL=4BZ94 Q@5 µ2O9@4-

6? <45 s = 0. ,9 BII5 I?<45, 9K6JI5 µ8J;JKµ9 @5 O4584=CM=JLµ9 ?C4 <45 Q = 0 2 58?-

6;4=2 9>@54 H(jQ) = 0, <9<J@?H 8JL =Lµ3A@9> µ9 C2@ LR289;5C7 4O4?C2C5 CJL 3>IC;JL.

+C2@ 89;>8CA=2 8JL QT9C9 585@C7=94 =A=CB, µ8;BSJ =5H, <45C> 65C5@J7=5C9 CJ µ2T5-

@4=µ? µ9C5=T2µ5C4=µJK 58? CJ@ Q@5 CK8J 5@5IJ<46JK 3>IC;JL =CJ@ BIIJ@. (LC? 9>@54

8JIK T;7=4µJ 654 585;5>C2CJ <45 C2@ ?I2 =T9O>5=2 CA@ R23456M@ IIR 3>IC;A@. (@

O9@ S;765C9 C2 =A=C7 58B@C2=2, 8;J=85G7=C9 8BI4 T;2=4µJ8J4M@C5H µ9 8;J=JT7

C4H =TQ=94H (5.16) 654 (5.17). *;J=85G7=C9 98>=2H @5 65C5IBS9C9 CJ *5;BO94<µ5 5.2.

5.3

)J µQC;J C2H 58?6;4=2H =LT@?C2C5H CJL 3>IC;JL 8;Q894 @5 QT94 C2 µJ;37 CJL +T7-

µ5CJH 5.125. (3JK 2 58?6;4=2 G5 8;Q894 =T9O?@ @5 µ2O9@>Z9C54 <45 @ = 0 654 @ =

', =L@98B<9C54 ?C4 =C5 =2µ9>5 5LCB G5 8;Q894 @5 L8B;TJL@ 6B8J45 µ2O9@46B.

H s H s

ss

s s ss

s s

s

s s

HP p( ) ( )

, ,

, ,, ,

==

=+ + =

=

=*+,

-./

+ *+,

-./

+

=+ +

41

1 098 1 1034

1

41 098

41 103

1 103 4 392 16

2

2

2

2

Q Qpp s sU =/ /4 U = U =Q Qp s 4 rad / s

2 8 9A ¶ " ¡ & % ™ ! # ™ A ™ ' % ™ ! ø ¡ AÀ & $ " • # $ § $ ° % ™ % ™

'/2(5) (S)

|H(ej@)|

0 @'

0,9 0,9

P1

P2

Re

Im

™¯‹Ì· 5.12

(%) :'05)µ3/2%'#1"083 /()L@,(*0%I%/(7H3<0%1(7 <96-/"(), (I) '#6(01%0 µ3*.,01? 8/(.'9'.*(–z



P=CA ?C4 C5 µ2O9@46B S;>=6J@C54 =C4H GQ=94H z = 0,9 654 z = – 0,9 (+T7µ5 5.12S).

(8? C2@ BII2 8I9L;B, <45 @ = 8/2 2 58?6;4=2 85;JL=4BZ94 µQ<4=CJ 654 B;5 2

=L@B;C2=2 µ9C53J;BH G5 8;Q894 @5 QT94 6B8J4J@ 8?IJ =CJ =2µ9>J 5LC?. P=CA ?C4

J 8?IJH 5LC?H S;>=69C54 =CJ =2µ9>J , O2I5O7 9@C?H CJL µJ@5O45>JL

6K6IJL, M=C9 CJ 3>IC;J @5 9>@54 9L=C5GQH (=2µ9>J \1 =CJ +T7µ5 5.12S). (3JK J

8?IJH 9>@54 µ4<5O46?H, G5 8;Q894 @5 L8B;T94 654 J =LZL<7H 5LCJK

(=2µ9>J -2 =CJ +T7µ5 5.12S), M=C9 J4 C9I46J> =L@C9I9=CQH CJL 3>IC;JL 8JL G5 8;J-

6KR94, @5 9>@54 8;5<µ5C46J>. Y;5 2 =L@B;C2=2 µ9C53J;BH CJL 3>IC;JL G5 9>@54:

+L@98MH, J4 Z2CJKµ9@J4 =L@C9I9=CQH CJL 3>IC;JL 9>@54: a0 = 1, a1 = 0, a2 = – 0,81,

b1 = 0, b2 = 0,81. # 9:>=A=2 O453J;M@ CJL 984GLµ2CJK 3>IC;JL G5 4=JKC54 µ9 (SI.

=TQ=2 3.59): y(n) = x(n) – 0,81x(n – 2) – 0,81y(n – 2).

(@ QT9C9 L8JIJ<>=94 =A=CB CJ ZA@JO45S5C? R23456? 3>IC;J, C?C9 =L<T5;2C7;45.

PT9C9 65C5@J7=94 C2 T;2=4µ?C2C5 CJL µJ@5O45>JL 6K6IJL 654 CJL µ4<5O46JK 984-

8QOJL. (LC? G5 =5H 35@9> T;7=4µJ 654 =C4H L8?IJ489H µ9G?OJLH =T9O>5=2H R2345-

6M@ IIR 3>IC;A@. (@ O9@ L8JIJ<>=5C9 =A=CB CJ 3>IC;J, :5@58;J=85G7=C9 53JK

µ9I9C7=9C9 8;MC5 CJ *5;BO94<µ5 3.10.

5.4

# =L@B;C2=2 H(s) QT94 C;94H O453J;9C46JKH 8?IJLH p1 = – 1, p2 = – 0,5 + j0,87, p3 =

– 0,5 – j0,87. !63;BZJLµ9 C2@ H(s) AH 5@B8CL<µ5 µ9;46M@ 6I5=µBCA@:

654 S;>=6JLµ9 ?C4:

C1 = 1, C2 = 0,6e–j2,6, C3 = 0,6ej2,6

F8JIJ<>ZJLµ9 C2@ H(z) µ9 SB=2 C2 =TQ=2 (5.30), 5@C465G4=CM@C5H CJ s – pF µ9 CJ

, O2I5O7:1 1# #e zp T\

H s

Cs

Cs j

Cs j

( ), , , ,

=+

++ #

++ +

1 2 3

1 0 5 0 87 0 5 0 87

H za a z a z

b z b z

a z a z a

z b z b

z z z zz p z p

z z

z e z e

zj j

( )( )( )( )( )

( , )( , )

( , )( , )8/ 8/

= + ++ +

= + ++ +

= # ## #

=

= # +# #

= #

# #

# #

#

0 11

22

11

22

02

1 22

1 2

1 2

1 2

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2

1

0 9 0 9

0 9 0 9

00 81

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1 0 81

1 0 812

2

2

,

,

,

,z

z

z+= #

+

#

#

z ej

=#

0 9 2,)

z ej

= 0 9 2,)


!6C9IM@C5H C4H 8;B:94H 65C5I7<JLµ9 =C2 =L@B;C2=2 µ9C53J;BH CJL R23456JK

3>IC;JL:

?8JL a0 = – 2e–0,5Tcos(0,87T) + e–T + 1,15e–0,5Tcos(5'/6 + 0,87;)

a1 = e–T + 1,15e–1,5Tcos(5'/6 + 0,87T)

b1 = – e–T – 2e–0,5Tcos(0,87T)

b2 = e–T + 2e–1,5Tcos(0,87T)

b3 = – e–2T

(@ QT9C9 S;94 C2 =A=C7 =L@B;C2=2 µ9C53J;BH CJL R23456JK 3>IC;JL, C?C9 5:>Z9-

C9 =L<T5;2C7;45, <45C> 2 µQGJOJH C2H 5µ9CBSI2C2H 6;JL=C467H 58JC9I9> µ>5 =T9C4-

6B 8JIK8IJ62 O45O465=>5 µ9CBS5=2H 58? CJ@ 5@5IJ<46? =CJ@ R23456? TM;J.

(@ O9@ S;765C9 C2 =A=C7 =L@B;C2=2 µ9C53J;BH C?C9 8;J=85G7=C9 8BI4. X;94BZ9-

C54 984µJ@7. DLµ2G9>C9 ?C4 8;Q894 8;MC5 @5 5@58CK:9C9 C2@ H(s) =9 µ9;46B 6IB-

=µ5C5 µ9 O453J;9C46JKH 8?IJLH =Kµ3A@5 µ9 C2 =TQ=2 (5.26) 654 =C2 =L@QT945 @5

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µ9I9C7=9C9 654 8BI4 CJ *5;BO94<µ5 5.4.

5.5

DQIJLµ9 @5 =T9O4B=JLµ9 Q@5 LR289;5C? 3>IC;J CJL J8J>JL 2 =LT@?C2C5 58J6J87H

@5 9>@54 30 Hz, ?C5@ J ;LGµ?H O94<µ5CJI2R>5H 9>@54 150 Hz. # O45O465=>5 =T9O>5-

=2H, ?8AH 654 J4 =LT@?C2C9H @c 654 Q'c 9>@54 >O49H µ9 969>@9H CJL *5;5O9><µ5CJH 5.5.

)J µ?@J 8JL G5 8;Q894 @5 5IIB:94 9>@54 J µ9C5=T2µ5C4=µ?H =LT@?C2C5H, J J8J>JH

=C2@ 8;J694µQ@2 89;>8CA=2 G5 9>@54 58? S5GL89;5C? =9 LR289;5C? 5@5IJ<46? 3>I-

C;J, ?8AH µ5H L85<J;9K94 2 =TQ=2 (5.17), O2I5O7:

(@C465G4=CM@C5H =C2@ C9I9LC5>5 =TQ=2 CJ s µ9 CJ (z – 1)/(z + 1) 65C5I7<JLµ9 =C2

H s H s

ss s s s

ss

HP pc

( ) ( ), , ,

== U = ! U =

=+

=+Q Q Qc c1 0 73

10 73

10 73

H z

a z a z

z b z b z b( ) = +

+ + +0

21

31

22 3

H z

C

e z

C

e z

C

e zT j T j T( )

( , , ) ( , , )=

#+

#+

## # # + # # # #1

12

0 5 0 87 13

0 5 0 87 11 1 1

2 9 1A ¶ " ¡ & % ™ ! # ™ A ™ ' % ™ ! ø ¡ AÀ & $ " • # $ § $ ° % ™ % ™



Z2CJKµ9@2 =L@B;C2=2 µ9C53J;BH CJL LR289;5CJK R23456JK 3>IC;JL:

# 9:>=A=2 O453J;M@ CJL 3>IC;JL 9>@54 y(n) = 0,58[x(n) – x(n – 1)] – 0,16y(n – 1).

*;?694C54 654 8BI4 <45 Q@5 R23456? 3>IC;J 8;MC2H CB:2H, ?8AH 654 CJ 5@C>=CJ4TJ

5@5IJ<46?, =CJ J8J>J S5=>=5µ9 C2 =T9O>5=7 µ5H. # OJµ7 8;5<µ5CJ8J>2=7H CJL O9>-

T@9C54 =CJ +T7µ5 5.13. *5;5C2;JKµ9 ?C4 2 6K;45 O453J;B µ9 CJ S5GL89;5C? 3>I-

C;J CJL +T7µ5CJH 5.9S S;>=69C54 =C5 8;?=2µ5 CA@ =L@C9I9=CM@ CJL 3>IC;JL.

H z H s

szz

zz

zz

zz

z

zHP( ) ( )

,, ,

, ( )

,

== #

+=

#+

#+

+= #

+= #

+

#

#11

11

11

0 73

11 73 0 27

0 58 1

1 0 16

1

1

x(n) y(n)0,58

0,58 -0,16z-1 z-1

™¯‹Ì· 5.13

T(µ2 '"%$µ%/(-'(93834 /() '"+-/34 /?F.@4 )H3'.-"%/(7 <96/"() /34Z813834 >)/(%-F0(6#$3834 5.5

(@ 585@C7=5C9 =A=CB =C2@ B=62=2 5LC7, =2µ5>@94 ?C4 65C5IBS5C9 C2@ 84J O45O9-

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µ9I9C7=C9 :5@B CJ *5;BO94<µ5 5.5 654 8;J=Q:C9 CJ@ µ9C5=T2µ5C4=µ? 8JL G5 6B@9-

C9 5;T46B 58? CJ S5GL89;5C? 5@5IJ<46? =9 LR289;5C? 5@5IJ<46? 3>IC;J µ9 SB=2

C4H =TQ=94H (5.16) 654 (5.17).

5.6

045 @5 =T9O4B=JLµ9 CJ Z2CJKµ9@J ZA@JO45S5C? 3>IC;J, 56JIJLGJKµ9 C5 C;>5 =CB-

O45 =T9O>5=2H C2H 9@?C2C5H 5.2.3:

1. !84IQ<JLµ9 CJ 8;AC?CL8J 5@5IJ<46? S5GL89;5C? 3>IC;J. +C2@ B=62=2 5LC7 µ5H

Z2C9>C54 @5 =T9O4B=JLµ9 Q@5 ZA@JO45S5C? 3>IC;J O9KC9;2H CB:2H. ,9 SB=2 CJ

O9KC9;J =T?I4J C2H L8J9@?C2C5H 5.2.3, 65C5I5S5>@JLµ9 ?C4 CJ 5@5IJ<46? S5GL-

89;5C? Butterworth 3>IC;J 58? CJ J8J>J G5 8;JQIG94 CJ R23456? 3>IC;J, G5 8;Q-

894 @5 9>@54 8;MC2H CB:2H, O2I5O7 C2H µJ;37H Hl(s) = 1/(s + 1).

2. ,9C5=T2µ5C>ZJLµ9 CJ 8;AC?CL8J S5GL89;5C? 5@5IJ<46? 3>IC;J =9 ZA@JO45S5-

C? µ9 =LT@?C2C9H Q_l 654 Q_u =C5 ?;45 C2H ZM@2H O4QI9L=2H.

Q_l = tan(@l/2) = tan(Ql;/2) = tan(2'200(1/2000)/2) = tan(0,1') = 0,32

Q_u = tan(@u/2) = tan(Qu;/2) = tan(2'300(1/2000)/2) = tan(0,15') = 0,51.


' µ9C5=T2µ5C4=µ?H =LT@?C2C5H =CJ@ 5@5IJ<46? TM;J 58? S5GL89;5C? =9 ZA@J-

O45S5C? 984CL<TB@9C54 µ9 SB=2 C2 =TQ=2 (5.19) 654 µ5H O>@94:

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ZA@JO45S5CJK 5@5IJ<46JK 3>IC;JL, 8JL µ?I4H L8JIJ<>=5µ9, 5@C465G4=CM@C5H

=C2@ C9I9LC5>5 =TQ=2 ?8JL s CJ (z – 1)/(z + 1)

H s H ss

s

s

ss

ss

s ss

ss

s

s

s s

BP p

c

( ) ( ), ,

( , , ),

,

,,

,,

,

,

== + U U

U # U( ) = + !#

= +!

=+ = + =

+*+,

-./

+=

+

QQ Q

Q Ql u

u l

2 2 2

22 2

0 32 0 510 51 0 32

0 170 19

11 0 17

0 19

1

0 170 19

1

0 19

0 19 ++ 0 17,

2 9 3A ¶ " ¡ & % ™ ! # ™ A ™ ' % ™ ! ø ¡ AÀ & $ " • # $ § $ ° % ™ % ™

H z H s

szz

ss s

szz

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,

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), ,

== #

+

=+ + = #

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#+

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=

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=

11

0 190 19 0 17 1

1

0 1911

11

0 1911

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2 =, (

,00 14 1

1 1 22 0 72

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1 2

, ( ), ,

## +

#

# #z

z z

# 5@C>=CJ4T2 9:>=A=2 O453J;M@ CJL IIR R23456JK 3>IC;JL 9>@54:

y(n) = 0,14[x(n) – x(n – 2)] + 1,22y(n – 1) – 0,72y(n – 2).

+5H 5:>ZJL@ =L<T5;2C7;45 5@ 585@C7=5C9 =C2@ B=62=2 5LC7. *;?694C54 <45 =T9C4-

6B OK=6JI2 B=62=2, 53JK 2 O45O465=>5 5854C9> C2 µ9C5C;J87 58? 5@5IJ<46? S5GL-

89;5C? 3>IC;J 8;MC2H CB:2H =9 5@5IJ<46? ZA@JO45S5C? 3>IC;J O9KC9;2H CB:2H.

+C2@ 89;>8CA=2 8JL O9@ C5 65C53Q;5C9, µ2@ 58J<J2C9LG9>C9. &5C5;T7@ O45SB=C9

654 8BI4 CJ *5;BO94<µ5 5.2. +C2 =L@QT945, 98469@C;M=C9 C2 8;J=JT7 =5H =C4H =TQ-

=94H (5.18) 654 (5.19) 654 =C5 =T?I45 8JL C4H =L@JO9KJL@. D5 O4584=CM=9C9 ?C4 =9

O9KC9;2H CB:2H ZA@JO45S5C? 3>IC;J 65C5I7<JLµ9 58? 8;MC2H CB:2H S5GL89;5C?

3>IC;J. )5 L8?IJ485 S7µ5C5 9>@54 C5 >O45 µ9 969>@5 8JL 9>O5µ9 =C5 8;J2<JKµ9@5

85;5O9><µ5C5. /5@58;J=85G7=C9…

6.1

# 5=8;?µ5L;2 946?@5 T;94BZ9C54 =L@JI46B 128 H 128 = 16.384 bytes µ@7µ2H. (@C>-

G9C5, 2 Q<T;Aµ2 5854C9> 3 bytes/946J@J=CJ4T9>J, O2I5O7 =L@JI46B 512 H 512 H 3 =



1.537.536 bytes. )J C9I9LC5>J µQ<9GJH 9>@54 48 3J;QH µ9<5IKC9;J 58? CJ 8;MCJ.

!8JµQ@AH, 2 =A=C7 58B@C2=2 9>@54 2 C;>C2.

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65C5@J7=94 C2@ 4=JOL@5µ>5 946J@J=CJ4T9>JL 654 byte, 65GMH 98>=2H CJ <9<J@?H ?C4

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G7=94 @5 8;JTA;7=9C9 µ9 9LTQ;945 =C4H 8;B:94H 8B@A =C2@ 946?@5.

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5;4Gµ? CA@ bytes 8JL T;94BZJ@C54 =C2@ 5=8;?µ5L;2 654 C2@ Q<T;Aµ2 946?@5. b=AH

969> 9>@54 CJ IBGJH =5H.

6.2

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(@ QT9C9 L8JIJ<>=94 C2 =A=C7 58B@C2=2, S;>=69=C9 =9 8JIK 65I? O;?µJ, O9OJµQ-

@JL ?C4 G5 65C5@J7=9C9 TA;>H 8;?SI2µ5 CJ@ C;?8J LIJ8J>2=2H C2H O4=O4B=C5C2H

=L@QI4:2H 8JL 56JIJLG9> =CJ &93BI54J 7. *;?694C54 <45 CJ8467 8;B:2 µ9 65GJ;4-

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6.3

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µ9C5SBII9C54 2 =L@C9C5<µQ@2 n2 7 y.


7.1

&5C5;T7@ <;B3JLµ9 8BI4 C2 =TQ=2 (7.9)

(7.9)

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?8JL QTJL@ 6B8J45 9846BILR2.

7.2

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7.3

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=µQ@J 8I7GJH ?;A@ (n1 = n2 = 5). +L@98MH 465@J8J49>C54 2 =TQ=2 (7.20), 53JK CJ

BG;J4=µ5 CA@ ?;A@ 5LCM@ G5 9>@54 Q@5H 8989;5=µQ@JH 5;4Gµ?H (< ").

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7.4

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y n n h l l x n l n ll l

( , ) ( , ) ( , )1 2 1 2 1 1 2 2

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5 5

2 9 5A ¶ " ¡ & % ™ ! # ™ A ™ ' % ™ ! ø ¡ AÀ & $ " • # $ § $ ° % ™ % ™



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*5;BO94<µ5 7.5 654 C2 E;5=C2;4?C2C5 7.3. !8JµQ@AH 2 =A=C7 58B@C2=2 9>@54 2 S.

+A=C7 58B@C2=7 =5H =2µ5>@94 ?C4 QT9C9 65C5@J7=94 CJ@ C;?8J O2µ4JL;<>5H 3>IC;A@

µ2O9@467H 3B=2H. (LC? G5 =5H 9>@54 8JIK T;7=4µJ O9OJµQ@JL ?C4 =C2@ R234567 989-

:9;<5=>5 946?@5H T;2=4µJ8J4JKµ9 6L;>AH 3>IC;5 <;5µµ467H 3B=2H.

(@ 6B@5C9 IBGJH, 8;J=85G7=C9 8BI4 O45SBZJ@C5H C2@ 8;J2<JKµ9@2 O;5=C2;4?C2C5

654 C2@ L8J9@?C2C5 4.1.1.

A!·ÓÙ‹ÛÂÈ˜ ¢Ú·ÛÙËÚÈÔÙ‹ÙˆÓ

4.4

D5 L8JIJ<>=JLµ9 CJLH =L@C9I9=CQH CJL 3>IC;JL =C2@ 89;>8CA=2 8JL R = 3 (third–band

3>IC;J). +C2@ 89;>8CA=2 5LC7 4=TK94 @c = 8/3 654 58? C2@ 9:>=A=2 4.7 QTJLµ9:

# 6;JL=C467 58?6;4=2 hD 9>@54 =Lµµ9C;467 <K;A 58? CJ n = 0, QT94 B894;JLH ?;JLH

654 6BG9 C;>CJH ?;JH 9>@54 µ2O9@46?H (n 8JII58IB=4J CJL 3).

h n

nn

nnD

c c

c

c( )sin( ) sin( / )

( / )/= = A @AAA=6 6

) 6)

)6 )

3 1

33

3


Ψηφιακή πεξεργασία Eικόνων και Σηµάτων · Ψηφιακή...

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