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9

9.1

, . , , , , , , , .

, ( ), . . , (.. pixel, ).

50, , . , - 0 1, . , . , / , , ! ! :-) ,, ( ) .

. ( 9.1), . , - (). , , - . , , , . 1. , , - . . -, . , . .

9.1: .

1 .

328

. Ts ( ) . . , fs (fs = 1Ts ), - . , CD , 44100Hz. , 44100 , 144100 ! , ! . ...

: ; ; ; ; ; , bits; Nyquist Shannon, 1928 , 1949 , . Shannon-Nyquist.

(- ) B Hz, ,

fs > 2B, (9.1)

.

Ts 2B(

Ts 2fmax x(nTs) (. ) sinc(), nTs, n Z!!! 9.8 sinc() ( ) ( ) ().

t

... ...

9.8: sinc.

, , - 9.9(). ( ) fs ( ). B fs B. , 9.9(), , . , , , 9.9(). , 9.9(), .

B < fs B fs > 2B (9.18)

Shannon. (- :-) ). . 9.9(), Shannon , . fs B . ( ) . , . fs ...

aliasing 4., 9.10.

Arect( tT

) ATsinc(fT) = ATsinc(tT) Arect

(

f

T

)= Arect

( fT

)(9.15)

4 ...

9. 333

-B B0 f-fs-fs-B -fs+B fs fs+Bfs-B

X(f)

-B B0 f

X(0)

X(0)/Ts

Xs(f)

0 f-fs fs

Xs(f)

-B B fs+Bfs-B-fs-B -fs+B

()

()

()

X(0)/Ts

9.9: :

9.2.3

1. , x(t) , . , X(f), ([B,B]). , , - , ! Haa(f), H(f) , anti-aliasing filter, , . .

2. H(f) ( 9.7). , , , . , . .

3. x(t) . ( ) , . Fourier

(t) 1

rect( t

)(9.19)

0, . ,

334

t0 x

s(t)

= x

[nT

s]

T s

X(0

)/T s

Xs(

f)

-BB

0f

-fs

f sf s

+B

f s-B

-fs-

B-f

s+B

x(t

)

0t

X(0

)

X(f

)

-BB

0f

X(f

)

-BB

0f

-fs

f sf s

+B

f s-B

-fs-

B-f

s+B

X(0

)

X(0

)/T s

Xs(

f)

-BB

0f

-fs

f sf s

+Bf s

-B-f

s-B

-fs+

B

H(f

) = T

srec

t(f/

f s)

T s

t0 x

s(t)

= x

[nT

s]

x(t

)

0t

.....

.

x (t

)

-fs/

2f s

/2

9.10

:

.

9. 335

.

4. Shannon . : x(t) X(f), X(f) [B,B]. y(t) = x3(t), Y (f) 6= 0 [3B, 3B]. , fs > 6B. fs = 2B. ; y(nTs). 9.11. ;

h(t) ____3

y(nTs) = x3(nTs)

9.11: .

5. [B,B] [fs/2, fs/2]; , - - fs/2, . , . :-)

6. Shannon; . - - . , , 20 kHz , - 10 kHz. 2 10 = 20kHz .

, 2025 kHz, fs > 2 {20 25} = 40 50kHz. 20 Hz 20 kHz. 2 20 = 40 kHz . , CD ( Sony 70) 44.1 kHz, 20 kHz , .

9.2.4

.

1:

x(t) = 3 cos(400t) + 5 sin(1200t) + 6 cos(4400t) (9.20)

fs = 4000 Hz x[n]. - , x[n].

: x(t) , fmax = 2200 Hz.

2fmax = 2 2200 = 4400 > fs (9.21)

336

Shannon , . , . :-)

t = nTs = n 14000 sec. t nTs. :

x[n] = 3 cos(400nTs) + 5 sin(1200nTs) + 6 cos(4400nTs) (9.22)

= 3 cos(n

10

)+ 5 sin

(12n40

)+ 6 cos

(44n40

)(9.23)

2:

h(t) =sin(2fct)

fst(9.24)

fM < fc < fs fM fs = 1Ts h[nTs] = [n] n, fc =fs2 .

( Dirac ) :

[n] =

1, n = 0,

0,

(9.25)

:,

h(t) =sin(2fct)

fst=

sin(fst)

(fst)= sinc(fst) (9.26)

fs = 1Ts , t nTs,

h[nTs] = sinc(

fsnTs

)= sinc

( 1Ts

nTs

) h[n] = sinc(n) (9.27)

, sinc(n) =sin(n)

n ,

sin(n) = 0 sin(n) = sin(k) (9.28)

n = k n = k, k Z (9.29)

n, , ,

h[n] =

sinc(0) = 1, n = 0,

sinc(n) = 0, n 6= 0

,

h[n] = [n] (9.30)

3:

x(t) . Fourier X(f), [B,B]. Nyquist :

() x(t)

9. 337

() x(t t0)

() x(t)ej2f0t

() x(t t0) + x(t+ t0)

()dx(t)

dt

() x(t)x(t)

() x(t) x(t)

: ( Nyquist) -. :

() B, Nyquist fs = 2B.

() x(t t0) X(f)ej2ft0 (9.31)

B, Nyquist fs = 2B.

() x(t)ej2f0t X(f f0) (9.32)

f0, f0+B, fs = 2(f0+B).

() x(t t0) + x(t+ t0) X(f)ej2ft0 +X(f)ej2ft0 = 2X(f) cos(2ft0) (9.33)

B, fs = 2B.

() dx(t)

dt j2fX(f) (9.34)

, B, fs = 2B.

() x(t)x(t) X(f) X(f) (9.35)

, , . [B B,B +B] = [2B, 2B]. Nyquist fs = 4B.

() x(t) x(t) X(f)X(f) = X2(f) (9.36)

Nyquist fs = 2B.

4:

Nyquist

() sinc2(100t)

() 1100 sinc2(100t)

() sinc(100t) + 3sinc2(60t)

() sinc(50t) sinc(100t)

: Nyquist Shannon, .

338

2fmax ., . Fourier,

x(t) X(f) = X(t) x(f) (9.37)

sinc() , ( A = 1),

rect( t

T

) Tsinc(fT) (9.38)

T sinc(tT) rect(f

T

)= rect

( fT

)(9.39)

tri( t

T

) Tsinc2(fT) (9.40)

T sinc2(tT) tri(f

T

)= tri

( fT

)(9.41)

f f sinc(), rect() .

()

sinc2(100t) 1100

tri( f

100

)(9.42)

tri() [100, 100] Hz. 100 Hz, Nyquist fs = 200 Hz.

() , fs = 200 Hz.

()

sinc(100t) + 3sinc2(60t) 1100

rect( f

100

)+ 3

1

60tri( f

60

)(9.43)

60 Hz, Nyquist fs = 120 Hz.

()

sinc(50t) sinc(100t) 150

rect( f

50

) 1100

rect( f

100

)(9.44)

25 Hz, Nyquist fs = 50 Hz.