Signals and Systems - Electrical Engineering

104 C H A P T E R 1: Continuous-Time Signals

nExample 1.24

One way of transmitting a message over the airwaves is to multiply it by a sinusoid of frequency

higher than those in the message, thus changing the frequency content of the signal. The resulting

signal is called an amplitude-modulated (AM) signal: The message changes the amplitude of the

sinusoid. To recover the message from the transmitted signal, one can make the envelope of the

modulated signal be related to the message. Use again therampandustepfunctions to generate a

signaly(t)= 2 r(t+ 2 )− 4 r(t)+r(t− 2 )+r(t− 3 )+u(t− 3 )to modulate a so-called carrier signal

x(t)=sin( 5 πt)to give the AM signalz(t). Obtain a script to generate and plot the AM signal.

Indicate whether the envelope of the AM signal is connected with the message signaly(t).

Solution

The signaly(t)analytically equals

y(t)=



















0 t<− 2

2 r(t+ 2 )= 2 (t+ 2 ) − 2 ≤t< 0

2 r(t+ 2 )− 4 r(t)=− 2 t+ 4 0 ≤t< 2

2 r(t+ 2 )− 4 r(t)+r(t− 2 )=−t+ 2 2 ≤t< 3

2 r(t+ 2 )− 4 r(t)+r(t− 2 )+r(t− 3 )+u(t− 3 )= 0 t≥ 3

The following script is used to generate the message signaly(t), the AM signalz(t), and the cor-

responding plots. The MATLAB functionsoundis used to produce the sound corresponding to

100 z(t). In Figure 1.14 we showz(t)and emphasize the envelope (dashed line) that corresponds

to±y(t).

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

% Example 1.24 --- AM signal

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

t = -5:0.01:5;

x = sin(5 * pi * t);

y1 = ramp(t,2,2);

y2 = ramp(t,-4,0);

y3 = ramp(t,1,-2);

y4 = ramp(t,1,-3);

y5 = ustep(t,-3);

y = y1 + y2 + y3 + y4 + y5;

z = y. * x;

sound(100 * z,1000)

plot(t,z,’k’); hold on

plot(t,y,’r’,t,-y,’r’); axis([-5 5 -5 5]); grid

hold off

xlabel(’t’); ylabel(’z(t)’)