Lab #9:FOURIER SERIES SSUET/QR/114

LAB # 9
FOURIER SERIES
Objective
Fourier analysis plays an important role in communication systems. The main objectives
of this experiment are
1. Generate and plot periodic wave amplitude and magnitude spectrum.
2. Potting magnitude and phase spectrums for periodic square pulse waves.

Theory

Introduction

A Fourier series is an expansion of a periodic function in terms of an infinite sum of sine’s

and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine
functions. The Fourier series is a sum of sine and cosine functions that describes a periodic
signal. It is represented in either the trigonometric form or the exponential form.

The usual Fourier series involving sines and cosines is obtained by taking &
Since these functions form a complete orthogonal system over , the Fourier
series of a function is given by

(1)

 Any periodic signal can be represented by the sum of sine and cosine series.
 For periodic signals, we use the Fourier series to find the frequency components in a signal.
 Any periodic signal can be expressed as:


x(t )  a 0   a n cos n 0 t  bn sin n 0 t where  0  2f 0
N 1

T T T
a0  T1  x(t ) dt , an  T2  x(t ) cos n0t dt and bn  2
T  x(t ) sin n t
0 dt
0 0 0

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Lab #9:FOURIER SERIES SSUET/QR/114

Amplitude and phase

 The exponential Fourier representation of these signals is given as:



 d n e jn0t
T
x (t )  1
 x(t )e
 jn0t
and dn  dt
n   T 0

Example 9.1

 The “half-sinusoidal” waveform shown in the figure below represents a voltage response
of a circuit. Find the Fourier series of this signal and plot the amplitudes for different
frequency components.

2Vm cos(n / 2)
 The final expression is a n   n  1 and for n  1, take the limit tends to 1
 1  n2
for this expression. Also note that bn  0 because the function v (t ) is an even function.

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Lab #9:FOURIER SERIES SSUET/QR/114

 Note that in this example, we are not going to directly use this expression. Instead, we are going
to evaluate this expression.
 Also we are going to see that how bn  0 without using the symmetric properties of the original
signal.
 From you lecture notes, we have seen that : v(t )  Vm cos 5t for  0.1  t  0.1
0 for 0.1  t  0.3
 The following program finds a0  C0 , an  C n and bn  0  n  0 or   .
 Assume Vm  2 .

Program

syms anbnnvtCnC0
%initialization
T0 = 0.4;
f0 = 1/T0;
Vm = 2;

n = [1:5]

v = Vm*cos(2*pi*f0*t) %Actual signal

%a0,bn and an formulas for periodic signals

a0 = (1/T0)*int(v,t,-0.1,0.1)
bn = int ( (2/T0)* v * sin(n*2*pi*f0*t) , t , -0.1,0.1 )
an = int ( (2/T0)* v * cos(n*2*pi*f0*t) , t , -0.1,0.1 )

C0 = a0
Cn = ( (an.^2) + (bn.^2) ).^0.5
Cn_values = [eval(C0), eval(Cn)]

plot([0,n],Cn_values,'r')
hold on
stem([0,n],Cn_values,'k')
hold off

%Another plot with negative values indicating the phases

an_values = [eval(a0), eval(an)]

figure
plot([0,n],an_values,'r')
hold on
stem([0,n],an_values,'k')
hold off

%==== Try different n,f0,Vm values of this example.====

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Lab #9:FOURIER SERIES SSUET/QR/114

clear all

Example 9.2

1. A periodic square wave of amplitude V 0 and period T with pulse duration of  can be
shown below:

2. The Fourier representation of a rectangular pulse is sinc function.

3. It has infinite frequency components. This means that it has infinite BW.
4. When the pulse duration equals to half signal duration, then only odd harmonics
(frequencies) are present.
5. It can be verified that for such waveforms:

V0
dn  sin c( 12 n0 )e  jn0 (t0  / 2)
T
6. It is obvious that d n  d n e  jn .

7. Different spectrum plots can be used to see the behavior of such waveforms in frequency
domain.

8. By adding the Harmonics with their corresponding amplitudes and phases together is
going to give us the original (approximate) square wave. Use the compact form to find
these harmonics:

x(t )  C 0   C n cos( n 0 t   n )
N 1

where Cn  2 d n .

Program

V = 1;
T = 0.5;
Tp = 0.2;
F = 1 / T;

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Lab #9:FOURIER SERIES SSUET/QR/114

To = 0
n = [-10:10]
dn1 = ( ( V * (Tp / T)) * sinc (0.5 * n * 2 * pi * F * Tp) )
dn2 = exp ( -j * n * 2 * pi * F * (To + (0.5 * Tp) ) )
dn = dn1 + dn2
dn_value = ( V * Tp / T ) * sinc ( 0.5 * n * 2 * pi *F * Tp )
dn_mag = abs (dn)
dn_ang = angle (dn)
stem(n,dn_ang);
figure;
stem (n,dn_mag);
hold on
plot (n,dn_mag,'r');
hold off
figure
stem (n,dn_value);
hold on
plot (n,dn_value,'r');
hold off
Cn = 2 * dn_value;
Cn_value = abs (Cn);
figure
stem (n,Cn_value);
hold on
plot (n,Cn_value,'r');
hold off

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Lab #9:FOURIER SERIES SSUET/QR/114

LAB ACTIVITY

Task 1: Calculate the Fourier transform of Sine wave with unit amplitude and frequency.
Consider any constant value for ω and the range of t is 0 ≤ t ≤ 100.

Task 2: Calculate the magnitude and phase response of above signal (i.e. signal in task 1) and
also plot them.

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