Signals and Systems - Electrical Engineering

272 C H A P T E R 4: Frequency Analysis: The Fourier Series

nExample 4.12

To illustrate the modulation property using MATLAB consider modulating a sinusoid cos( 20 πt)

with a train of square pulses

x 1 (t)=0.5[1+sign(sin(πt)]

and with a sinusoid

x 2 (t)=cos(πt)

Use our functionfourierseriesto find the Fourier series of the modulated signals and plot their

magnitude line spectra.

Solution

The functionsignis defined as

sign(x(t))=

{

− 1 x(t) < 0

1 x(t)≥ 0

(4.28)

That is, it determines the sign of the signal. Thus, 0.5[1+sign

(

sin(πt)]=u(t)−u(t− 1 )equals

1 for 0≤t≤1, and 0 for 1<t≤2, which corresponds to a period of a train of square

pulses.

The following script allows us to compute the Fourier coefficients of the two modulated signals.

%%%%%%%%%%%%%%%%%

% Example 4.12---Modulation

%%%%%%%%%%%%%%%%

syms t

T0 = 2;

m = heaviside(t)−heaviside(t−T0/2);

m1 = heaviside(t)−heaviside(t−T0);

x = m∗cos(20∗pi∗t);

x1 = m1∗cos(pi∗t)∗cos(20∗pi∗t);

[X, w] = fourierseries(x, T0, 60);

[X1, w1] = fourierseries(x1, T0, 60);

The modulated signals and their corresponding magnitude line spectra are shown in Figure 4.13.

The Fourier coefficients of the modulated signals are now clustered around the frequency 20π.