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Signal Processing First
Lecture 7C
Fourier Series Examples:
Common Periodic Signals
10.11.2020 © 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
dtetxatTkj
Tk
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 5
Harmonic Signal is Periodic
tFkj
k
keatx 02)(
−=
=
( )0
0
0
00
1or
22
FT
TF ===
Sums of Harmonic
complex exponentials
are Periodic signals
PERIOD/FREQUENCY of COMPLEX EXPONENTIAL:
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 6
Recall FWRS
121
01 is Period)/2sin()( TTTttx ==
Absolute value flips the
negative lobes of a sine wave
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 7
FWRS Fourier Integral → {ak}
−
=0
0
0
)/2(
0
)(1
T
ktTj
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
0
00
0
0
0
00
−
−=
−=
−=
=
+−
+−
−−
−−
−−
−
ka
ee
dtej
ee
dteta
k
T
kTjTj
tkTjT
kTjTj
tkTj
T
ktTjtTjtTj
T
T
ktTj
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
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 8
)14(
22 −
−=
kak
6336.0/20 == a
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 9
Spectrum from Fourier Series
Plot a for Full-Wave Rectified Sinusoid
0000 2and/1 FTF ==
)14(
22 −
−=
kak
ka
10.11.2020 © 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
tkTj
Tk dtetxa
real is )( when* txaa kk =−
tFkjN
Nk
keatx 02)(
−=
=
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 11
Reconstruct From Finite Number
of Harmonic Components
Full-Wave Rectified Sinusoid
Hz100
ms10
0
0
=
=
F
T
)/sin()( 0Tttx =
6336.0/20 == a
)14(
22 −
−=
kak
=
−++=N
k
tFkj
k
tFkj
kN eaeaatx1
22
000)(
?)/sin()( to)( is closeHow 0TttxtxN =
10.11.2020 © 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(
4015
403
42
2
1522
152
32
322
2
)116(22
)116(2
)14(2
)14(22
)14(
2
2)14(
2
2
0000
0000
02
2
tNtt
eeee
eeee
etx
ka
N
tjtjtjtj
tjtjtjtj
N
Nk
tjk
kN
kk
−
−−
−
−−
−−−
−−
−−
−−
−=−
−
−
−
−−−−=
+−−−−=
++++=
=
−−
−==
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 13
Reconstruct From Finite Number
of Spectrum Components
Full-Wave Rectified Sinusoid )/sin()( 0Tttx =
Hz100
ms10
0
0
=
=
F
T
6336.0/20 == a
)14(
22 −
−=
kak
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 14
Fourier Series Synthesis
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 15
PULSE WAVE SIGNAL
GENERAL FORM
Defined over one period
=
2/2/0
2/01)(
0Tt
ttx
Nonzero DC value
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 16
Pulse Wave {ak}
k
Tkee
eee
dtea
kj
kTjkTj
kj
kTjkTj
kTj
ktTj
T
ktTj
Tk
)/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)(
0Tt
ttx
−
−=
2/
2/
))(/2(1
0
0
0
0)(
T
T
tkTj
Tk dtetxa
10.11.2020 © 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
tTj
T
dt
dtea
=−==
=
−
−
−
−
PulseWave
,...2,1,0)/sin( 0 == k
k
Tkak
0
0
0
)/sin(lim Note,
Tk
Tk
k
→
→
Double check the DC coefficient:
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 18
PULSE WAVE SPECTRA
2/0T=
4/0T=
16/0T=
10.11.2020 © 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
0
01)(
210
k
k
kkj
kja
Tt
ttx
k
k
,...2,1,0)2/sin()/)2/(sin(
2/ 000 ==== k
k
k
k
TTkaT k
PulseWave starting at t=0
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 20
PULSE WAVE SYNTHESIS
with first 5 Harmonics
2/0T=
4/0T=
16/0T=
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 21
Triangular Wave: Time Domain
Defined over one period
2/2/for/2)( 000 TtTTttx −=
Nonzero DC value
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 22
Triangular Wave {ak}
−
−=
2/
2/
)/2(
0
0
0
0)(1
T
T
ktTj
k dtetxT
a
( ) ( )
( )( ) ( )( )220
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/
2
2/
0
2
2/
2/
)/2(
0
0
)(
|/2|1
k
kTjTkj
kTkk
kTjTkj
T
Tk
ktjtkj
T
T
k
ktjtkj
Tk
T
ktj
T
T
ktj
T
T
T
ktTj
k
ee
eea
dtetdtet
dteTtT
a
+−+−
−
+−+−
−
−−
−
−
−−−=
−=
−+=
=
Triangular Wave
0/2)( Tttx =
00 /2 T =
( )220
000 1 integral indefinite theuse
k
ktjtkjktjedtet
+−−
=
10.11.2020 © 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
k
k
k
ee
eea
k
kk
kTkkT
kj
kT
kj
Tk
k
kjkj
Tk
kjkj
TkT
k
kTjTkj
kTkk
kTjTkj
Tk
k
kk
220
20
14
00 2
=
=
T
T
212/
0 0
0
Period
AreaDC
==
=
T
Ta
10.11.2020 © 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
k
k
k
ak
k
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 25
Triangular Wave Synthesis
Hz50
ms20
0
0
=
=
F
T
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 26
Full-Wave Rectified Sine {ak}
−
=0
0
0
)/2(
0
)(1
T
ktTj
k dtetxT
a
0
00
00
00
0
0
0
0
0
0
0
0
0
00
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
T
tkTj
Tj
T
tkTj
Tj
T
ktTjtTjtTj
T
T
ktTj
TTk
ee
dtedte
dtej
ee
dteta
+−
+−
−−
−−
+−−−
−−
−
−=
−=
−=
=
Full-Wave Rectified Sine
121
0
1
:
)/2sin()(
TTPeriod
Tttx
=
=
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 27
( ) ( )
( ) ( )
( )( ))14(
21)1(
11
11
2
2
)14(
)12(12
)12(
)12(1)12(
)12(1
)12)(/(
)12(21)12)(/(
)12(21
0
))12)(/((2
)12)(/(
0
))12)(/((2
)12)(/(
2
0000
0
00
00
00
0
−
−=−−−=
−−−=
−−−=
−=
−
−−+
+−
+
−−
−
+−
+
−−
−
+−
+−
−−
−−
k
ee
ee
eea
k
k
kk
kj
k
kj
k
TkTj
k
TkTj
k
T
kTjTj
tkTjT
kTjTj
tkTj
k
Full-Wave Rectified Sine {ak}
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 28
Half-Wave Rectified Sine
▪ Signal is positive half cycles of sine wave
▪ HWRS = Half-Wave Recitified Sine
10.11.2020 © 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
0
00
0
00
00
2
)sin(
T
kTjTj
tkTjT
kTjTj
tkTj
T
tkTj
Tj
T
tkTj
Tj
T
ktTjtTjtTj
T
TktTj
TTk
ee
dtedte
dtej
ee
dteta
+−
+−
−−
−−
+−−−
−−
−
−=
−=
−=
=
Half-Wave Rectified Sine
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 30
( ) ( )
( ) ( )
( )( )
==−−−=
−−−=
−−−=
−=
−
−
−
−−+
+−
+
−−
−
+−
+
−−
−
+−
+−
−−
−−
even
1?
odd 0
1)1(
11
11
)1(
1
)1(4
)1(1
)1(
)1(41)1(
)1(41
2/)1)(/2(
)1(412/)1)(/2(
)1(41
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
kj
k
kj
k
TkTj
k
TkTj
k
T
kTjTj
tkTjT
kTjTj
tkTj
k
Half-Wave Rectified Sine {ak}
41j
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 31
Half-Wave Rectified Sine {ak}
▪ Spectrum, assuming 50 Hz is the
fundamental frequency
==
−
− even
1
odd 0
)1(
1
4
2 k
k
k
a
k
j
k
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 32
HWRS Synthesis
Hz50
ms20
0
0
=
=
F
T
10.11.2020 © 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/
10.11.2020 © 2003-2016, JH McClellan & RW Schafer 34
fseriesdemo GUI
