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Attia, John Okyere. “Fourier Analysis.”
Electronics and Circuit Analysis using MATLAB.
Ed. John Okyere Attia
Boca Raton: CRC Press LLC, 1999
© 1999 by CRC PRESS LLC
CHAPTER EIGHT
FOURIER ANALYSIS
In this chapter, Fourier analysis will be discussed. Topics covered are Fou-
rier series expansion, Fourier transform, discrete Fourier transform, and fast
Fourier transform. Some applications of Fourier analysis, using MATLAB,
will also be discussed.
8.1 FOURIER SERIES
If a function
g t
( )
is periodic with period T
p
, i.e.,
g t g t T
p
( ) ( )
= ± (8.1)
and in any finite interval
g t
( )
has at most a finite number of discontinuities
and a finite number of maxima and minima (Dirichlets conditions), and in
addition,
g t dt
T
p
( )
< ∞
∫
0
(8.2)
then
g t
( )
can be expressed with series of sinusoids. That is,
g t
a
anw t b nw t
n n
n
( ) cos( ) sin( )
= + +
=
∞
∑
0
0 0
1
2
(8.3)
where
w
T
p
0
2
=
π
(8.4)
and the Fourier coefficients
a
n
and
b
n
are determined by the following equa-
tions.
a
T
g t nw t dt
n
p
t
t T
o
o p
=
+
∫
2
0
( ) cos( )
n
= 0, 1,2, … (8.5)
© 1999 CRC Press LLC© 1999 CRC Press LLC
b
T
g t nw t dt
n
p
t
t T
o
o p
=
+
∫
2
0
( ) sin( )
n
= 0, 1, 2 … (8.6)
Equation (8.3) is called the trigonometric Fourier series. The term
a
0
2
in
Equation (8.3) is the dc component of the series and is the average value of
g t
( )
over a period. The term anw t b nw t
n n
cos( ) sin( )
0 0
+ is called the
n
-
th harmonic. The first harmonic is obtained when
n
= 1. The latter is also
called the fundamental with the fundamental frequency of ω
o
. When n = 2,
we have the second harmonic and so on.
Equation (8.3) can be rewritten as
g t
a
Anw t
n n
n
( ) cos( )
= + +
=
∞
∑
0
0
1
2
Θ (8.7)
where
A a b
n n n
= +
2 2
(8.8)
and
Θ
n
n
n
b
a
= −
−
tan
1
(8.9)
The total power in
g t
( )
is given by the Parseval’s equation:
P
T
g t dt A
A
p
t
t T
dc
n
n
o
o p
= = +
+
=
∞
∫
∑
1
2
2 2
2
1
( )
(8.10)
where
A
a
dc
2
0
2
2
=
(8.11)
The following example shows the synthesis of a square wave using Fourier
series expansion.
© 1999 CRC Press LLC© 1999 CRC Press LLC
Example 8.1
Using Fourier series expansion, a square wave with a period of 2 ms, peak-to-
peak value of 2 volts and average value of zero volt can be expressed as
g t
n
n f t
n
( )
( )
sin[( ) ]
=
−
−
=
∞
∑
4 1
2 1
2 1 2
0
1
π
π
(8.12)
where
f
0
500
= Hz
if
a t
( )
is given as
a t
n
n f t
n
( )
( )
sin[( ) ]
=
−
−
=
∑
4 1
2 1
2 1 2
0
1
12
π
π
(8.13)
Write a MATLAB program to plot
a t
( )
from 0 to 4 ms at intervals of 0.05
ms and to show that
a t
( )
is a good approximation of
g(t
).
Solution
MATLAB Script
% fourier series expansion
f = 500; c = 4/pi; dt = 5.0e-05;
tpts = (4.0e-3/5.0e-5) + 1;
for n = 1: 12
for m = 1: tpts
s1(n,m) = (4/pi)*(1/(2*n - 1))*sin((2*n - 1)*2*pi*f*dt*(m-1));
end
end
for m = 1:tpts
a1 = s1(:,m);
a2(m) = sum(a1);
end
f1 = a2';
t = 0.0:5.0e-5:4.0e-3;
clg
plot(t,f1)
xlabel('Time, s')
© 1999 CRC Press LLC© 1999 CRC Press LLC
ylabel('Amplitude, V')
title('Fourier series expansion')
Figure 8.1 shows the plot of
a t
( )
.
Figure 8.1 Approximation to Square Wave
By using the Euler’s identity, the cosine and sine functions of Equation (8.3)
can be replaced by exponential equivalents, yielding the expression
g t c jnw t
n
n
( ) exp( )
=
=−∞
∞
∑
0
(8.14)
where
c
T
g t jnw t dt
n
p
t
T
p
p
= −
−
∫
1
2
2
0
( ) exp( )
/
/
(8.15)
and
w
T
p
0
2
=
π
© 1999 CRC Press LLC© 1999 CRC Press LLC
Equation (8.14) is termed the exponential Fourier series expansion. The coeffi-
cient
c
n
is related to the coefficients a
n
and b
n
of Equations (8.5) and (8.6)
by the expression
c a b
b
a
n n n
n
n
= + ∠ −
−
1
2
2 2 1
tan ( )
(8.16)
In addition,
c
n
relates to
A
n
and
φ
n
of Equations (8.8) and (8.9) by the rela-
tion
c
A
n
n
n
= ∠Θ
2
(8.17)
The plot of
c
n
versus frequency is termed the discrete amplitude spectrum or
the line spectrum. It provides information on the amplitude spectral compo-
nents of
g t
( ).
A similar plot of ∠c
n
versus frequency is called the dis-
crete phase spectrum and the latter gives information on the phase components
with respect to the frequency of
g t
( )
.
If an input signal
x t
n
( )
x t c jnw t
n n o
( ) exp( )
=
(8.18)
passes through a system with transfer function
H w
( )
, then the output of the
system
y t
n
( )
is
y t H jnw c jnw t
n o n o
( ) ( ) exp( )
= (8.19)
The block diagram of the input/output relation is shown in Figure 8.2.
H(s)x
n
(t) y
n
(t)
Figure 8.2 Input/Output Relationship
However, with an input
x t
( )
consisting of a linear combination of complex
excitations,
© 1999 CRC Press LLC© 1999 CRC Press LLC
x t c jnw t
n
n
n o
( ) exp( )
=
=−∞
∞
∑
(8.20)
the response at the output of the system is
y t H jnw c jnw t
n
n
o n o
( ) ( ) exp( )
=
=−∞
∞
∑
(8.21)
The following two examples show how to use MATLAB to obtain the coeffi-
cients of Fourier series expansion.
Example 8.2
Forthefull-wave rectifier waveform shown in Figure 8.3,theperiod is 0.0333s
and the amplitude is 169.71 Volts.
(a) Write a MATLAB program to obtain the exponential Fourier series
coefficients
c
n
for
n
= 0,1, 2, , 19
(b) Find the dc value.
(c) Plot the amplitude and phase spectrum.
Figure 8.3 Full-wave Rectifier Waveform
© 1999 CRC Press LLC© 1999 CRC Press LLC
Solution
diary ex8_2.dat
% generate the full-wave rectifier waveform
f1 = 60;
inv = 1/f1; inc = 1/(80*f1); tnum = 3*inv;
t = 0:inc:tnum;
g1 = 120*sqrt(2)*sin(2*pi*f1*t);
g = abs(g1);
N = length(g);
%
% obtain the exponential Fourier series coefficients
num = 20;
for i = 1:num
for m = 1:N
cint(m) = exp(-j*2*pi*(i-1)*m/N)*g(m);
end
c(i) = sum(cint)/N;
end
cmag = abs(c);
cphase = angle(c);
%print dc value
disp('dc value of g(t)'); cmag(1)
% plot the magnitude and phase spectrum
f = (0:num-1)*60;
subplot(121), stem(f(1:5),cmag(1:5))
title('Amplitude spectrum')
xlabel('Frequency, Hz')
subplot(122), stem(f(1:5),cphase(1:5))
title('Phase spectrum')
xlabel('Frequency, Hz')
diary
dc value of g(t)
ans =
107.5344
Figure 8.4 shows the magnitude and phase spectra of Figure 8.3.
© 1999 CRC Press LLC© 1999 CRC Press LLC
Figure 8.4 Magnitude and Phase Spectra of a Full-wave
Rectification Waveform
Example 8.3
The periodic signal shown in Figure 8.5 can be expressed as
g t e t
g t g t
t
( )
( ) ( )
= − ≤ <
+ =
−
2
1 1
2
(i) Show that its exponential Fourier series expansion can be expressed as
g t
e e
jn
jn t
n
n
( )
( ) ( )
( )
exp( )
=
− −
+
−
=−∞
∞
∑
1
2 2
2 2
π
π
(8.22)
(ii) Using a MATLAB program, synthesize
g t
( )
using 20 terms, i.e.,
© 1999 CRC Press LLC© 1999 CRC Press LLC
g t
e e
jn
jn t
n
n
( )
( ) ( )
( )
exp( )
∧
−
=−
=
− −
+
∑
1
2 2
2 2
10
10
π
π
0 2 4
t(s)
g(t)
1
Figure 8.5 Periodic Exponential Signal
Solution
(i)
g t c jnw t
n o
n
( ) exp( )
=
=−∞
∞
∑
where
c
T
g t jnw t dt
n
p
T
T
o
p
p
= −
−
∫
1
2
2
( ) exp( )
/
/
and
w
T
o
p
= = =
2 2
2
π π
π
c t jn t dt
n
= − −
−
∫
1
2
2
1
1
exp( ) exp( )
π
c
e e
jn
n
n
=
− −
+
−
( ) ( )
( )
1
2 2
2 2
π
thus
© 1999 CRC Press LLC© 1999 CRC Press LLC
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