OUTLINE - University of the Ryukyusie.u-ryukyu.ac.jp/~wada/system08/SYSARC2008-3.pdf · Periodic...
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OUTLINE Periodic Signal Fourier series introduction Sinusoids Orthogonality It ti i d t Integration vs inner product 2008/11/10 System Arch 2008 (Fire Tom Wada) 1
Transcript of OUTLINE - University of the Ryukyusie.u-ryukyu.ac.jp/~wada/system08/SYSARC2008-3.pdf · Periodic...
OUTLINE
Periodic SignalFourier series introductionSinusoids OrthogonalityI t ti i d tIntegration vs inner product
2008/11/10 System Arch 2008 (Fire Tom Wada) 1
Consider any wave is sum of simple sin and cosine
Periodic Tc
)(f )(tf10 =a
)cos(31 ta ω=
)sin(11 tb ω−= )(1
)2cos(22 ta ω=
)2sin(22 tb ω= )(2
)3cos(13 ta ω=
)3sin(13 tb ω= )(3
)4cos(24 ta ω=
)4sin(2 tb ω−=
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)4sin(24 tb ω=
Periodic signal is composed of DC + same frequency sinusoid + multiple frequency sinusoids
1
)(tf = cc T
f 122 ππω ==
1)(tf
Frequency = 0 Hz
)sin(1)cos(3 tt ωω −+ Basic frequency fc=1/Tc
)3sin(1)3cos(1)2sin(2)2cos(2
tttt
ωωωω
++++ 2 x fc
)4sin(2)4cos(2)3sin(1)3cos(1tt
ttωωωω
−+++ 3 x fc
4 x fc
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)()( 4 x fc
Spectrum of periodic signalp p g
frequencyf (Hz)
0 fc 2・fc 3・fc 4・fc 5・fc-fc-2・fc-3・fc-4・fc-5・fc 0 fc 2 fc 3 fc 4 fc 5 fcc
There are only n * fc (n=integer) frequencies!y ( g ) q
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Another example (even rectangular pulse)p ( g p )
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Increase the number of sum (1)( )
N=1 N=2N 1
N=3 N=10
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Increase the number of sum (2)( )
N=20 N=50N 20
N=100 N=200
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Fourier
Jean Baptiste Joseph, Baron de FrourierFrance, 1778/Mar/21 – 1830/May/16, y
Fourier Series paper is written in 1807
Even discontinue function (such asrectangular pulse) can be composedrectangular pulse) can be composedof many sinusoids.
Nobody believed the paper at that time.
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Fourier Series
If f(t) ‘s period is Tc cf122 ππω ==If f(t) s period is Tc…c
c Tf
∑∞
++= ))sin()cos(()( tnbtnaatf ωω∑∞
=
++=1
0
22
))sin()cos(()(n
nn tnbtnaatf
ππ
ωω
∑=
++=1
0 ))2sin()2cos(()(n c
nc
n tT
nbtT
naatf ππ
If we use complex exponential…,∞
tjnn ectf ω⋅= ∑
∞
)(
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n −∞=
Anyway, when you see the periodic signal,y y y p gPlease think it is just sum of sinusoids!!!
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How we can divide f(t) into sinusoids?( )
)sin()cos( tnbtna ωω +
FilterPass
)sin()cos( tnbtna nn ωω +
Passnω (Hz)
•Filter is used
•an and bn
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If we integrate in [ 0 to Tc] g [ ]Tc
dttfcT
∫ )( dttf∫0
)(
00 aTdta c
Tc
⋅=∫ 00
0 c∫0)1cos(
01 =∫ dtta
Tc
ω0)1sin( =∫ dttb
Tc
ω 0)1sin(0
1 =∫ dttb ω0)2cos(
02 =∫ dtta
Tc
ω0)1sin(1 =∫ dttb
Tc
ω 0)1sin(0
1∫ dttb ω
)( aTdttfTc
⋅=∫2008/11/10 System Arch 2008 (Fire Tom Wada) 12
00
)( aTdttf c=∫
If we integrate in [ 0 to Tc] (2)g [ ] ( )Tc
dtttfcT
)1cos()( ω×∫ dtttf )1cos()(0
ω×∫0
T21cTa ⋅
0)1cos( tω× ∫
cT
dt0
0000 00
00
•a1 can be computed2008/11/10 System Arch 2008 (Fire Tom Wada) 13
01 p
If we integrate in [ 0 to Tc] (3)g [ ] ( )Tc
dtttfcT
)1cos()( ω×∫ dtttf )1cos()(0
ω×∫0
T 1cTb ⋅
0
)1sin( tω× ∫cT
dt0
21b000 00
00
•b1 can be computed2008/11/10 System Arch 2008 (Fire Tom Wada) 14
01 p
By changing multiplier, each coefficient computedTc
)cos( tnω×cT
)i ( tor×
∫c
dt0
One coefficient
)sin( tnω× 0
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Sinusoidal Orthogonalityg y
m,n: integer, Tc=1/f0
⎧T
)(0
)(2)2cos()2cos(
0 00∫⎪⎩
⎪⎨⎧
≠
==⋅cT c
nm
nmTdttnftmf ππ
Orthogonal
)()2sin()2sin(
)(0
∫⎪⎨⎧ =
⎪⎩ ≠
cT c nmTdttnftmf
nm
ππ
Orthogonal
)(0
)(2)2sin()2sin(
0 00
∫
∫⎪⎩⎨
≠=⋅
T
nmdttnftmf ππ
Orthogonal
0)2sin()2cos(0 00∫ =⋅cT
dttnftmf ππ Orthogonal
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Another Orthogonality (1)g y ( )
Vector inner productVector inner product
)5,2(−=Α
)2,5( −=B
0255)2(BΑ ×+×
θcos0255)2(
BABΑ⋅=
=×+×−=⋅ Orthogonal
Θ＝90 degree
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g
Another Orthogonality (2)g y ( )n dimensional vector
n
bbaaa= L21
)(),,,(Α
nn
n
bababaabb
⋅++⋅+⋅=⋅=
L
L
2211
21 ),,,(BΑ
B
nn bababa +++ 2211BΑ
0=⋅BΑIFTHEN A d B O th lTHEN A and B are Orthogonal.
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is same as the N dim inner product∫cT
dt is same as the N dim inner product,sampledissignalIf
∫ dt0
0 00 )2cos()2cos(
,
⋅∫T
dttnftmf
pgfc ππ
0
1
0 )2cos()2cos(−
=∑N
c TNinfT
Nimf ππ
10
)2cos()2cos(−
=
=∑Ni
inim
NN
ππ
111100
0)2cos()2cos(
=
⋅++⋅+⋅=
∑NN
i
yxyxyxN
nN
m
L
ππ
111100 −− NN yyy
Freq=nω(Hz) sinusoids are Orthogonal each other (n=integer)
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q ( ) g ( g )
Fourier Series Summary y
∑ ++=∞
nn tnbtnaatf1
0 ))sin()cos(()( ωω
∑ ++=∞
=
nn
n
tT
nbtT
naatf 0
1
))2sin()2cos(()( ππ
∫
∑
=
=
cT
n cc
dttfa
TT1
)(1
∫
∫=
cT
c
dttfT
a0
0
2
)(
∫=
T
cn dttntf
Ta
0
)cos()(2 ω
∫=cT
cn dttntf
Tb
0
)sin()(2 ω
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Complex form Fourier Seriesp
ectf tjnω∞
∑ ⋅=)( ectf
T
nn
−∞=
∫
∑1
)(
dtetfT
c tjnT
cn
c ω−∫=0
)(1
c
∫ − dtee tjmT tjnc ωω
⎧ =
∫)(
0
nmT
dtee
T
⎩⎨⎧
≠=
== ∫ −
)(0)(
0
)(
nmnmT
dte cT tmnjc ω
Orthogonal
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⎩ )( g
HW2
⎪⎧ ≤≤ 0
101 Tt
⎪
⎪⎨
<≤=
0
112)(
TtTtf
⎪⎩
<≤− 0021 TtT
[2-1]Compute the complex form Fourier Series coefficient cn for f(x)Series coefficient cn for f(x).[2-2]Draw the Spectrum of f(t) when T0=0.04sec.
2.30
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2.30
