sp first l07
TRANSCRIPT
04/15/23 © 2003, JH McClellan & RW Schafer 1
Signal Processing First
Lecture 7Fourier Series & Spectrum
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READING ASSIGNMENTS
This Lecture: Fourier Series in Ch 3, Sects 3-4, 3-5 & 3-6Fourier Series in Ch 3, Sects 3-4, 3-5 & 3-6
Replaces pp. 62-66 in Ch 3 in DSP First Notation: ak for Fourier Series
Other Reading: Next Lecture: Sampling
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LECTURE OBJECTIVES
ANALYSISANALYSIS via Fourier Series
For PERIODIC signals: x(t+T0) = x(t)
SPECTRUMSPECTRUM from Fourier Series ak is Complex Amplitude for k-th Harmonic
0
0
0
0
)/2(1 )(T
dtetxa tTkjTk
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0 100 250–100–250f (in Hz)
3/7 je 3/7 je2/4 je 2/4 je
10
SPECTRUM DIAGRAM
Recall Complex Amplitude vs. Freq
N
k
tfjk
tfjk
kk eaeaatx1
220)(
kjkk eAa
21
ka
0a
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Harmonic Signal
PERIOD/FREQUENCY of COMPLEX EXPONENTIAL:
tfkj
kkeatx 02)(
0
00
001
or2
2f
TT
f
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Example )3(sin)( 3 ttx
tjtjtjtj ej
ej
ej
ej
tx 9339
88
3
8
3
8)(
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Example
In this case, analysisjust requires picking off the coefficients.
)3(sin)( 3 ttx
tjtjtjtj ej
ej
ej
ej
tx 9339
88
3
8
3
8)(
3k1k1k
3k
ka
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STRATEGY: x(t) ak
ANALYSIS Get representation from the signal Works for PERIODICPERIODIC Signals
Fourier Series Answer is: an INTEGRAL over one period
0
0
0
0)(1
T
dtetxa tkjTk
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SQUARE WAVE EXAMPLE
0–.02 .02 0.04
1
t
x(t)
.01
sec. 04.0for
0
01)(
0
0021
021
T
TtT
Tttx
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FS for a SQUARE WAVE {ak}
)0()(1 0
0
0
)/2(
0
kdtetxT
aT
ktTjk
02.
0
)04./2()04./2(04.
102.
0
)04./2(104.
1 ktjkj
ktjk edtea
kje
kj
kkj
2
)1(1)1(
)2(
1 )(
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DC Coefficient: a0
)0()(1 0
0
0
)/2(
0
kdtetxT
aT
ktTjk
)Area(1
)(1
0000
0
Tdttx
Ta
T
21
02.
0
0 )002(.04.
11
04.
1 dta
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Fourier Coefficients ak
ak is a function of k Complex Amplitude for k-th Harmonic This one doesn’t depend on the period, T0
0
,4,20
,3,11
2
)1(1
21 k
k
kkj
kja
k
k
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Spectrum from Fourier Series
0
,4,20
,3,1
21 k
k
kk
j
ak
)25(2)04.0/(20
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Fourier Series Synthesis
HOW do you APPROXIMATE x(t) ?
Use FINITE number of coefficients
0
0
0
0
)/2(1 )(T
tkTjTk dtetxa
real is )( when* txaa kk tfkj
N
Nkkeatx 02)(
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Fourier Series Synthesis
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Synthesis: 1st & 3rd Harmonics))75(2cos(
3
2))25(2cos(
2
2
1)( 22
ttty
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Synthesis: up to 7th Harmonic)350sin(
7
2)250sin(
5
2)150sin(
3
2)50cos(
2
2
1)( 2 ttttty
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Fourier Synthesis
)3sin(3
2)sin(
2
2
1)( 00 tttxN
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Gibbs’ Phenomenon
Convergence at DISCONTINUITY of x(t) There is always an overshoot 9% for the Square Wave case
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Fourier Series Demos
Fourier Series Java Applet Greg Slabaugh
Interactive
http://users.ece.gatech.edu/~slabaugh/java/fourier/fourier.html
MATLAB GUI: fseriesdemo
http://users.ece.gatech.edu/mcclella/matlabGUIs/index.html
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fseriesdemo GUI