Signals and Systems - Electrical Engineering

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

The signalx 2 (t)has jump discontinuities att=0,t=1, andt=2, and we can think of it as

completely discontinuous so that its continuous component is 0. The derivative is

dx 2 (t)

dt

=δ(t)− 2 δ(t− 1 )+δ(t− 2 )

The area of each of the deltas coincides with the jump in the discontinuities. n

Signal Generation with MATLAB

In the following examples we illustrate how to generate analog signals using MATLAB. This is done by

either approximating continuous-time signals by discrete-time signals or by using the symbolic tool-

box. The functionplotuses an interpolation algorithm that makes the plots of discrete-time signals

look like analog signals.

nExample 1.16

Write a script and the necessary functions to generate a signal,

y(t)= 3 r(t+ 3 )− 6 r(t+ 1 )+ 3 r(t)− 3 u(t− 3 )

Then plot it and verify analytically that the obtained figure is correct.

Solution

We wrote functionsrampandustepto generate ramp and unit-step signals for obtaining a numeric

approximation of the signaly(t). The following script shows how these functions are used to gen-

eratey(t). The arguments oframpdetermine the support of the signal, the slope, and the shift (for

advance, a positive number, and for delay, a negative number). Forustepwe need to provide the

support and the shift.

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

% Example 1.16

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

clear all; clf

Ts = 0.01; t = -5:Ts:5; % support of signal

% ramp with support [-5, 5], slope of 3 and advanced

% (shifted left) with respect to the origin by 3

y1 = ramp(t,3,3);

y2 = ramp(t,-6,1);

y3 = ramp(t,3,0);

% unit-step function with support [-5,5], delayed by 3

y4 = -3 * ustep(t,-3);

y = y1 + y2 + y3 + y4;

plot(t,y,’k’); axis([-5 5 -1 7]); grid

Our functionsrampandustepare as follows.

function y = ramp(t,m,ad)

% ramp generation