B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

52 SIGNALS AND SIGNAL SPACE
c3 =
0.9614
c4 =
0.6282
c5
8.6748e-17
Numerical Computation of Coefficients D n
There are several ways to numerically compute the Fourier series coefficients D n , We will use
MATLAB to show how to use numerical integration in the evaluation of Fourier series.
To carry out a direct numerical integration of Eq. (2.60), the first step is to define the
symbolic expression of the signal g(t) under analysis. We use the triangle function (t) in the
following example.
% (funct_tri .m)
% A standard triangle function of base - 1 to 1
function y = funct_tri (t)
% Usage y = func_tri (t)
% t = input variable i
y= ( (t> -1)-(t>l) ) . * (1-abs (t));
Once the file funct _ tri .m defines the function y = g(t), we can directl y carry
out the necessary integration of Eq. (2.60) for a finite number of Fourier series coefficients
{Dn, n = -N, ... , -1, 0, 1, ... , N}. We provide the following MATLAB program called
FSexample .m to evaluate the Fourier series of t,,(t/2) with period [a, b] (a = -2, b = 2).
In this example, N = 11 is selected. Executing this short program in MATLAB will generate
Fig. 2.22 with both amplitude and angle of D ,, .
% (file name : FSexp_a .m)
% This example shows how to numerically evaluate
% the exponential Fourier series coefficients Dn
% directly .
% The user needs to define a symbolic function
% g(t) . In this example, g(t)= funct_tri (t) .
echo off; clear; elf;
j=sqrt (-1) ; % Define j for complex algebra
b=2 ; a=-2 ;
tol=l .e-5;
T=b-a;
N=l l;
% Determine one signal period
% Set integration error tolerance
% length of the period
% Number of FS coefficients
% on each side of zero frequency
Fi= [-N:N] *2*pi/T; % Set frequency range