PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 413

h (t) = 0.5e−0.5tu(t)

x(t) = 5cost + 15cos15t y(t)

FIGURE 4.60

Block box system diagram of Example 4.11.

Example 4.11

Let us analyze the block system diagram shown in Figure 4.60, by creating the script fi le

analysis that returns the following plots using numerical techniques:

1. x(t) versus t and |X(w)| versus w


2. h(t) versus t, |H(w)| v e r s u s w (magnitude), and phase of [H(w)] versus w


3. y(t) versus t, |Y(w)| versus w, and phase [Y(w)] versus w

MATLAB Solution

% Script file: analysis

t = -5:0.01:5;

xt = 5*cos(t)+15*cos(15*t);

subplot(3,2,1),plot(t,xt)

title(‘ x(t) vs. t’) ,ylabel(‘ Amplitude’)

w =-30:1:30;

xw =pi*[zeros(1,15) 15 zeros(1,13) 5 0 5 zeros(1,13) 15 zeros(1,15)];

% spectrum of x(t)

%from Table 4.1 transform 6

subplot(3,2,2),stem(w,xw)

title(‘X(w) vs. w ‘);ylabel(‘ Magnitude’);

axis([-30 30 -5 60]);

to = 0;

ut = stepfun(t,to);

h = 0.5*exp(-0.5*t);

ht = h.*ut;

subplot (3,2,3);

plot (t,ht);

title (‘ h(t) vs. t’);ylabel(‘Amplitude’)

axis([-3 5 -.5 1]);

% FT of h(t) is 0.5/(s+0.5), s=jw;

% from Table 4.1

a = [0.5];

b = [1 0.5];

Hw = freqs(a,b,w);

magHw = abs(Hw);

phase = angle(Hw);

subplot(3,2,4),plotyy(w,magHw,w,phase)

title(‘Magnitude & phase of H(w)’);

% Convolution of x(t) and h(t)

yt = conv(xt,ht);

lengyt = length(yt)-1;

n = [0:1:lengyt]-[5*ones(1,lengyt+1)];

subplot(3,2,5),plot(n/100,yt/100)