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DSP First, 2/e
Lecture 7CFourier Series Examples: Common Periodic Signals
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 3
READING ASSIGNMENTS This Lecture:
Appendix C, Section C-2 Various Fourier Series
Pulse Waves Triangular Wave Rectified Sinusoids (also in Ch. 3, Sect. 3-5)
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 4
LECTURE OBJECTIVES Use the Fourier Series Integral
Derive Fourier Series coeffs for common periodic signals
Draw spectrum from the Fourier Series coeffs ak is Complex Amplitude for k-th Harmonic
0
0
0
0
)/2(1 )(T
dtetxa tTkjTk
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 5
Harmonic Signal is Periodic
tFkj
kkeatx 02)(
0
00
001or22F
TT
F
Sums of Harmonic complex exponentials are Periodic signals
PERIOD/FREQUENCY of COMPLEX EXPONENTIAL:
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 6
Recall FWRS12
101 is Period)/2sin()( TTTttx
Absolute value flips thenegative lobes of a sine wave
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 7
FWRS Fourier Integral {ak}
0
0
0
)/2(
0
)(1 TktTj
k dtetxT
a
)14(2
2
)sin(
2
0))12)(/((2
)12)(/(
0))12)(/((2
)12)(/(
0
)/2()/()/(
1
0
)/2(1
0
00
00
00
0
0
000
0
0
0
00
ka
ee
dtejee
dteta
k
T
kTjTj
tkTjT
kTjTj
tkTj
TktTj
tTjtTj
T
TktTj
TTk
Full-Wave Rectified Sine
)/sin()(
:
)/2sin()(
0
121
0
1
Tttx
TTPeriod
Tttx
FWRS Fourier Coeffs: ak
ak is a function of k Complex Amplitude for k-th Harmonic
Does not depend on the period, T0
DC value is Aug 2016 © 2003-2016, JH McClellan & RW Schafer 8
)14(22
kak
6336.0/20 a
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 9
Spectrum from Fourier SeriesPlot a for Full-Wave Rectified Sinusoid
0000 2and/1 FTF
)14(22
kak
ka
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 10
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
tFkjN
Nkkeatx 02)(
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 11
Reconstruct From Finite Number of Harmonic ComponentsFull-Wave Rectified Sinusoid
Hz100ms10
0
0
F
T
)/sin()( 0Tttx
6336.0/20 a)14(
22
kak
N
k
tFkjk
tFkjkN eaeaatx
1
220
00)(
?)/sin()( to)( is closeHow 0TttxtxN
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 12
Full-Wave Rectified Sine {ak}
Plots for N=4 and N=9 are shown next Excellent Approximation for N=9
)cos(...)2cos()cos(...
...
)(
valuedreal is)14(
2
0)14(4
0154
0342
215
2215
232
322
2)116(
22)116(
2)14(
2)14(
22
)14(2
2)14(2
2
0000
0000
02
2
tNtteeee
eeee
etx
ka
N
tjtjtjtj
tjtjtjtj
N
Nk
tjkkN
kk
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 13
Reconstruct From Finite Number of Spectrum ComponentsFull-Wave Rectified Sinusoid )/sin()( 0Tttx
Hz100ms10
0
0
F
T
6336.0/20 a
)14(22
kak
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 14
Fourier Series Synthesis
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 15
PULSE WAVE SIGNALGENERAL FORM
Defined over one period
2/2/0
2/01)(
0Ttt
tx
Nonzero DC value
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 16
Pulse Wave {ak}
kTkee
eee
dtea
kj
kTjkTj
kj
kTjkTj
kTj
ktTj
T
ktTjTk
)/sin(
)(
1
0)2(
)()/()()/(
)2(
)2/()/2()2/()/2(2/
2/)/2(
)/2(1
2/
2/
)/2(1
00
00
0
0
0
0
0
General PulseWave
2/2/0
2/01)(
0Ttt
tx
2/
2/
))(/2(10
0
0
0)(
T
T
tkTjTk dtetxa
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 17
Pulse Wave {ak} = sinc
000
0
0
221
2/
2/
1
2/
2/
)0)(/2(10
1
1
TTT
tTjT
dt
dtea
PulseWave
,...2,1,0)/sin( 0 kkTkak
0
0
0
)/sin(lim Note,Tk
Tkk
Double check the DC coefficient:
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 18
PULSE WAVE SPECTRA2/0T
4/0T
16/0T
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 19
50% duty-cycle (Square) Wave
Thus, ak=0 when k is odd Phase is zero because x(t) is centered at t=0
different from a previous case
0
,4,20
,3,11
2)1(1
001
)(
210 k
k
kkj
kja
Ttt
txk
k
,...2,1,0)2/sin()/)2/(sin(2/ 000 k
kk
kTTkaT k
PulseWave starting at t=0
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 20
PULSE WAVE SYNTHESISwith first 5 Harmonics
2/0T
4/0T
16/0T
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 21
Triangular Wave: Time Domain
Defined over one period
2/2/for/2)( 000 TtTTttx
Nonzero DC value
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 22
Triangular Wave {ak}
2/
2/
)/2(
0
0
0
0)(1 T
T
ktTjk dtetxT
a
22
0
000022
02
022
022
0
00002
0
0
220
002
0
0
220
002
0
0
02
0
0
02
0
0
0
0
12/2/12112/2/2
0
2/
122/
0
12
0
2/
22/
0
2
2/
2/
)/2(0
0
)(
|/2|1
kkTjTkj
kTkkkTjTkj
T
Tkktjtkj
T
T
kktjtkj
Tk
T
ktjT
Tktj
T
T
T
ktTjk
ee
eea
dtetdtet
dteTtT
a
Triangular Wave
0/2)( Tttx
00 /2 T
220
000 1 integral indefinite theusekktjtkjktj edtet
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 23
Triangular Wave {ak}
0,...3,10,...4,2
21
1
)1(1
)1(4111)1(21
12124
12/2/12112/2/2
22
2222
220
20
22220
20
220
20
20
22
220
20
220
20
220
20
220
000022
02
022
022
0
00002
0
kkk
ee
eea
k
kk
kTkkTkj
kTkj
Tk
kkjkj
Tkkjkj
TkT
kkTjTkj
kTkkkTjTkj
Tk
k
kk
220
20
14
00 2
T
T
212/
0 0
0
Period
AreaDC
TTa
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 24
Triangular Wave {ak}
Spectrum, assuming 50 Hz is the fundamental frequency
0,...3,10,...4,2
21
122
kkk
ak
k
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 25
Triangular Wave Synthesis
Hz50ms20
0
0
F
T
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 26
Full-Wave Rectified Sine {ak}
0
0
0
)/2(
0
)(1 TktTj
k dtetxT
a
0
00
00
00
0
0
0
0
0
0
0
0
000
0
0
0
00
0))12)(/((2
)12)(/(
0))12)(/((2
)12)(/(
0
)12)(/(21
0
)12)(/(21
0
)/2()/()/(
1
0
)/2(1
2
)sin(
T
kTjTj
tkTjT
kTjTj
tkTj
TtkTj
Tj
TtkTj
Tj
TktTj
tTjtTj
T
TktTj
TTk
ee
dtedte
dtejee
dteta
Full-Wave Rectified Sine
121
0
1
:
)/2sin()(
TTPeriod
Tttx
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 27
)14(
21)1(
11
11
22
)14()12(12
)12()12(
1)12()12(
1
)12)(/()12(2
1)12)(/()12(2
1
0))12)(/((2
)12)(/(
0))12)(/((2
)12)(/(
2
0000
0
00
00
00
0
k
ee
ee
eea
kkkk
kjk
kjk
TkTjk
TkTjk
T
kTjTj
tkTjT
kTjTj
tkTj
k
Full-Wave Rectified Sine {ak}
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 28
Half-Wave Rectified Sine
Signal is positive half cycles of sine wave HWRS = Half-Wave Recitified Sine
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 29
Half-Wave Rectified Sine {ak})1()(1 0
0
0
)/2(
0
kdtetxT
aT ktTj
k
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
000
0
00
00
2
)sin(
T
kTjTj
tkTjT
kTjTj
tkTj
TtkTj
Tj
TtkTj
Tj
TktTj
tTjtTj
T
T ktTjTTk
ee
dtedte
dtejee
dteta
Half-Wave Rectified Sine
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 30
even 1?
odd 01)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
kkk
ee
ee
eea
k
kkkk
kjk
kjk
TkTjk
TkTjk
T
kTjTj
tkTjT
kTjTj
tkTj
k
Half-Wave Rectified Sine {ak}
41j
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 31
Half-Wave Rectified Sine {ak}
Spectrum, assuming 50 Hz is the fundamental frequency
even
1odd 0
)1(14
2 kkk
a
k
jk
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 32
HWRS Synthesis
Hz50ms20
0
0
F
T
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 33
Fourier Series Demo MATLAB GUI: fseriesdemo
Shows the convergence with more terms One of the demos in: http://dspfirst.gatech.edu/matlab/
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 34
fseriesdemo GUI
