Question

Consider the cascade of the three LTI systems having impulse responses: h-1(t) = e^-tu(t + 3) h_2(t) = rect((1 -1)/2) h_3(t) = delta(t - 3) What is the equivalent impulse response of the resulting system?

Answer 1

Explanation:

There are two ways to find out the equivalent impulse response of the system.

1. Convolution in time domain

2. Simple multiplication in Laplace domain

2nd method is efficient, easy and is less time consuming.

Step 1: Take the Laplace transform of the given three impulse response functions to convert time domain signals into s-domain

Step 2: Once we get signals in s-domain, multiply them algebraically to get the equivalent s-domain response.

Step 3: Take inverse Laplace transform of the equivalent impulse response to convert from s-domain into time domain.

Solution using Matlab:

Step 1: Take Laplace Transform

Ys1 =  1/(s + 1)

Ys2 =  1/s - exp(-s/2)/s

Ys3 =  exp(-3*s)

Step 2: Multiplication in s-domain

Y =  (exp(-(7*s)/2)*(exp(s/2) - 1))/(s*(s + 1))

Step 3: Inverse Laplace Transform (Final Solution in Time Domain)

h =  heaviside(t - 7/2)*(exp(7/2 - t) - 1) - heaviside(t - 3)*(exp(3 - t) - 1)