Signals&Systems Laboratory IFE 2010/201
1
Experiment 3.
Fourier Series.
Signal expansion and synthesis.
Introduction
1.1. Fourier series.
Consider a periodic signal x(t) with period T. If some conditions (see Dirichlet
conditions from lecture book) are satisfied, the function (signal) x(t) can be expressed as
series:

xt   a0   an cosn0t  bn sin n0t 
(1)
n 1
2
 2f 0 .
T
Coefficients of the trigonometric Fourier series may be calculated from the following
relations:
where 0 
1T
1 t T
a0   f t  dt   f t  dt
T0
T t
0
(2)
0
2T
2 t T
am   f t  cos m0tdt   f t  cos m0tdt
T0
T t
m  1, 2,...
(3)
2T
2 t T
bm   f t sin m0tdt   f t sin m0tdt
T0
T t
m  1, 2,...
(4)
0
0
0
0
where t0 is any Real number.
Formula (1) can be also expressed in more convenient equivalent form:

f t   A0   An cosn0t   n 
(5)
n1
Cosine terms in (1) and (5) are called harmonic (n-th harmonic has amplitude An , phase
 n and angular frequency n 0 , constant term A0  a0 (2) is the 0-th harmonic. First
harmonic:
y1 t   a1 cos 0t  b1 sin 0t  A1 cos 0t  1 
(6)
is called usually basic harmonic with angular (basic) frequency 0  2f 0 . The plot of
amplitude An i terms of n is called amplitude spectrum and plot of phase as a function of
n is called phase spectrum. Amplitudes and phases may be calculated from the relations:
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An  an2  bn2
(7)
 bn
(8)
an
The periodic signals are often presented in more compact form called exponential
Fourier series:
 n  arctan
xt  

 c~ e
n 
jn0t
(9)
n
where
1 t T
c~n   xt  e  jn t dt
n  1, 2, ... .
T t
Relations between different Fourier series forms are specified as follows:
0
0
(10)
0
1 t T
c~0   f t  dt  a0  A0 .
T t
0
(11)
0
b
1 2
1
c~n 
an  bn2  An ,  n  arg c~n  a tan n
2
2
an
n  1,2,3....
Similarly, the plot of c~n versus n is called the amplitude spectrum and the plot of
 n  arg c~n versus n is called phase spectrum. Note, that following relationships are
valid:
c~n  c~n ,  n   n n  1,2,3....
1.2. How to compute coefficients.
The amplitude and phase spectra, based on form (5) or (10) can be computed by
use of Matlab integration formulas like:
quadl
numerically evaluate integral, adaptive Lobatto quadrature
syntax
q= quadl (fun,a,b) approximates the integral of function fun from a to b, to within
an error of 10E-6 uunction handesing recursive adaptive Lobatto quadrature, fun
is a function handle.
More often, Fourier series computations are performed numerically using the
Discrete Fourier Transform (DFT), which in turn is implemented numerically using an
efficient algorithm known as the Fast Fourier Transform (FFT). Assuming, that
sampling frequency used for discretization of the periodic function interval is equal to fs
(samples per second) and applied DFT transform has N-elements, basic harmonic of the
Fourier series may be calculated from he formula:
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fo 
3
fs
N
.
(12)
Frequence of the next stripes of Fourier series spectrum ae of the form:
.
fi 
if s
N
(13)
Constant term and amplitudes of the harmonice (up to i=N/2) of the Fourier series
having form(5) follows the relations:
Ai  2
Xi
N
, i  1,2,...,
N
2
2
j
N
 mn
X n   xm w , n  0,1,2,..., , w  e N
2
m 0
N 1
(14)
Phases o harmonic are computed from the arguments f Xn.
As an illustration, the following code shows how to use fft approach to obtain Fourier
expansion coefficients. You can study this code and further enhance it to complet yoyr
work.
Useful Matlab functions: fft(x,N), length(), abs, angle, stem figure, xlabel, ylabel, title.
Numerical calculations may be perfored also by Simulink models (available only in
Matlab 5.3 instalation). Exelary scheme is shown in Fig.1
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Fig.1. FFT based algorithm for finding harmonic’s coefficients – Simulink
implemetation.
Fig.3.Block parameters..
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Signals&Systems Laboratory IFE 2010/201
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2. Laboratory work
2.1. PART A
2.1.1.
In Matlab (5.3) go to the command window and run one of the dlivered
demo m-files pointed by your teacher. This will bring up a Graphical User Inerface
(GUI) that can be used to test and demonstrate many concepts and properties of the
Fourier series expansion.
2.1.2.
Try different types of function. Explain why some signals require many
more harmonics than others in order to get match between the Fourier approximation
and original signal?
2.1.3.
Consider the plots of amplitude and phase spectra. Describe how the
symmetry of signal affect the spectrum.
2.1.4.
Observe the Gibbs phenomena. Briefly explain.
2.2. PART B
2.2.1.
Consider the periodic signals given by the teacher (see Fig.2,3,4). Find and
plot amplitude and phase spectra. Use your m–file based on integration formulas or
FFT concept. Tabulate results.
REMARKS:
For the expansion of the given signal x(t), the number f harmonics should be >=8.
Observe, what happened, if the sampling period Ts is not an integer multiplicity
of the signal period T.
Your report should contain plot of the investigated functions, amplitude and phase
spectra, TABLES with coefficients, conclusions.
2.2.2.
Write your own m-script enabling to built up periodic signals from its
harmonics. Attach some examples based on coefficients calculated before.
TABLE
Name of example (type of signal)
Method
Coeff.
Analitcal solution
(book)
A0
A1
Q1
A2
Q2
A3
Q3
A4
Q4
A5
Q5
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Integration based
algorithm
FFT based algorithm
Signals&Systems Laboratory IFE 2010/201
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fT(t)
A
T
a
0
B
Fig.2
gT(t)
A
T
0
a
b
c
B
Fig.3
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f 1 (t)
2
1
0
T
T/2
T/4
-2
f 2 (t)
2
1
0
T/4
T/2
T
t
f 3 (t)
2
1
0
T/4
T
T/2
-2
Fig.4
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t