Embed Size (px)
Citation preview
1
Prof. Nizamettin AYDIN
http://www.yildiz.edu.tr/~naydin
Digital Signal Processing
2
Lecture 7
Fourier Series & SpectrumFourier Series & Spectrum
Digital Signal Processing
4
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
5
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
6
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
7
Harmonic Signal
PERIOD/FREQUENCY of COMPLEX EXPONENTIAL:
tfkj
kkeatx 02)(
0
00
001
or2
2f
TT
f
8
Example
)3(sin)( 3 ttx
tjtjtjtj ej
ej
ej
ej
tx 9339
88
3
8
3
8)(
9
Example
In this case, analysisjust requires picking off the coefficients.
)3(sin)( 3 ttx tjtjtjtj e
je
je
je
jtx 9339
88
3
8
3
8)(
3k1k1k
3k
ka
10
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
11
FS: Rectified Sine Wave {ak}
)1()(1 0
0
0
)/2(
0
kdtetxT
aT
ktTjk
2/
0))1)(/2((2
)1)(/2(2/
0))1)(/2((2
)1)(/2(
2/
0
)1)(/2(21
2/
0
)1)(/2(21
2/
0
)/2()/2()/2(
1
2/
0
)/2(21
0
00
00
00
0
0
0
0
0
0
0
0
0
00
0
0
0
00
2
)sin(
T
kTjTj
tkTjT
kTjTj
tkTj
TtkTj
Tj
TtkTj
Tj
TktTj
tTjtTj
T
TktTj
TTk
ee
dtedte
dtej
ee
dteta
Half-Wave Rectified Sine
12
even
1?
odd 0
1)1(
11
11
)1(1
)1(4
)1(1
)1()1(4
1)1()1(4
1
2/)1)(/2()1(4
12/)1)(/2()1(4
1
2/
0))1)(/2((2
)1)(/2(2/
0))1)(/2((2
)1)(/2(
2
2
0000
0
00
00
00
0
k
k
k
ee
ee
eea
k
k
k
kk
kjk
kjk
TkTjk
TkTjk
T
kTjTj
tkTjT
kTjTj
tkTj
k
FS: Rectified Sine Wave {ak}
41j
13
SQUARE WAVE EXAMPLE
0–.02 .02 0.04
1
t
x(t)
.01
sec. 04.0for
0
01)(
0
0021
021
T
TtT
Tttx
16
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
17
Spectrum from Fourier Series
0
,4,20
,3,1
21 k
k
kk
j
ak
)25(2)04.0/(20
18
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)(
16Kasim2k11
19
Fourier Series Synthesis
20
Synthesis: 1st & 3rd Harmonics))75(2cos(
3
2))25(2cos(
2
2
1)( 22
ttty
21
Synthesis: up to 7th Harmonic)350sin(
7
2)250sin(
5
2)150sin(
3
2)50cos(
2
2
1)( 2 ttttty
22
Fourier Synthesis
)3sin(3
2)sin(
2
2
1)( 00 tttxN
23
Gibbs’ Phenomenon
• Convergence at DISCONTINUITY of x(t)– There is always an overshoot– 9% for the Square Wave case
24
Fourier Series Demos
• Fourier Series Java Applet– Greg Slabaugh
• Interactive
– http://users.ece.gatech.edu/mcclella/2025/Fsdemo_Slabaugh/fourier.html
• MATLAB GUI: fseriesdemo
– http://users.ece.gatech.edu/mcclella/matlabGUIs/index.html
25
fseriesdemo GUI
