Question

Analytically solve for the response y[t] of the system h[t]=2e‐10t for a pulse input x[t] of width 1, 10 or 100 msec with its amplitude such that its width times its amplitude is equal to .1

a.) For each pulse, solve assuming it is an ideal impulse (i.e. use convolution integral)

b.) solve not making this assumption (i.e. use convolution integral).

c. ) Plot your results Matlab and also calculate the response using the conv command in Matlab.

Answer 1

Script:

clc;close all;clear all;

%solution without the assumption

%For pulse width 1msec

T=0.001;TT=5*T;

t=0:TT/500:TT;

h=5*(exp(-20*t));

x=0.1.*T*(t>=0);

y=conv(x,h);

m=0:TT/500:2*TT;

subplot(221)

plot(m,y);grid;xlabel('t');ylabel('y(t)')

title('Pulse width T=1ms')

%For pulse width 10msec

T=0.01;TT=5*T;

t=0:TT/500:TT;

h=5*(exp(-20*t));

x=0.1.*T*(t>=0);

y=conv(x,h);

m=0:TT/500:2*TT;

subplot(222)

plot(m,y);grid;xlabel('t');ylabel('y(t)')

title('Pulse width T=10ms')

%For pulse width 100msec

T=0.1;TT=5*T;

t=0:TT/500:TT;

h=5*(exp(-20*t));

x=0.1.*T*(t>=0);

y=conv(x,h);

m=0:TT/500:2*TT;

subplot(223)

plot(m,y);grid;xlabel('t');ylabel('y(t)')

title('Pulse width T=100ms')

Plot:

Script

clc;close all;clear all;

%solution with the assumption as impulse input

%For 1msec

T=0.001;TT=5*T;

t=0:TT/500:TT;

h=5*(exp(-20*t));

x=0.1.*T*(t==T);

y=conv(x,h);

m=0:TT/500:2*TT;

subplot(221)

plot(m,y);grid;xlabel('t');ylabel('y(t)')

title('T=1ms')

%For10msec

T=0.01;TT=5*T;

t=0:TT/500:TT;

h=5*(exp(-20*t));

x=0.1.*T*(t==T);

y=conv(x,h);

m=0:TT/500:2*TT;

subplot(222)

plot(m,y);grid;xlabel('t');ylabel('y(t)')

title('T=10ms')

%For100msec

T=0.1;TT=5*T;

t=0:TT/500:TT;

h=5*(exp(-20*t));

x=0.1.*T*(t==T);

y=conv(x,h);

m=0:TT/500:2*TT;

subplot(223)

plot(m,y);grid;xlabel('t');ylabel('y(t)')

title('T=100ms')

Plot: