Question

Matlab help

1) Given the functions x1()= tu()-tuft-1) and X2(t)=10e-5,11(), do the following: 1. Plotx,(t) and x2(0) using MATLAB 2. Use MATLAB to find and plot x(0=x:@*.x2(t), where * denotes convolution. 3. Find x(t)=x;()*X2(1) by hand using Laplace transforms. 4. Plot the result of part 3 in MATLAB and compare it to that found in part 2.

Answer 1

Find x (t) = x (t) * x 2 (t) Laplace transform of x (t) x_1 (t) = t u (t) - t u (t - 1) = t [u (t) - u (t - 1)] = t, 0 lessthanorequalto t lessthanorequalto 1 X_1 (s) = integral_0^1 x_1 (t) e^- st d t = integral_0^1 t e^- s t dt = [t times - 1/s e^- st]_0^1 - integral_0^1 1 times -1/s e^- st dt = -1/s e^-s - [1/s^2 e^- st]_0^1 = -1/s e^-s - [1/s^2 e^-s - 1/s^2] = -1/s e^-s - 1/s^2 e^-s + 1/s^2 = 1/s^2 - 1/s^2 e

MATLAB:

clc;clear all;close all;
t=0:0.01:5
x1=(t.*(t>=0))-(t.*(t>=1))
x2=10*(exp(-5*t))
y=conv(x1,x2)
m=0:0.01:10
subplot(221)
plot(t,x1,'r','linewidth',3)
xlabel('t');ylabel('x1(t)');title('x1(t)')

subplot(222)
plot(t,x2,'b','linewidth',3)
xlabel('t');ylabel('x2(t)');title('x2(t)')

subplot(223)
plot(m,y/100,'m','linewidth',3)
xlabel('t');ylabel('x(t)');title('x(t)=x1*x2')

%from hand calculation
t=0:0.01:10
y1=((2*t)-0.4 +(0.4*(exp(-5*t)))).*(t>=0)
y2=((2*(t-1))-0.4 +(0.4*(exp(-5*(t-1))))).*(t>=1)
y3=(-2+(2*exp(-5*(t-1)))).*(t>=1)
yy=y1-y2+y3
subplot(224)
plot(t,yy,'g','linewidth',3)
xlabel('t');ylabel('x(t)');title('x(t)=x1*x2')