Question

The system shown below is formed by connecting two systems in parallel. The impulse responses of the systems are given by: t h, (t) = € 2€ u(t) , h (t) = 2e fu(t) 1) Find the impulse response h(t) of the overall system. 2) Is the overall system stable? h,(t) x vo h(t)

Answer 1

Given,

The impulse response of both the systems are

hy(t) = -4 uſt)

12(t) = 2e-tult)

Laplace transform of the impulse response of both the systems are,

Hy(s) = 1 S + 2

Hz(s) = 3,

(a) Since the systems are connected in parallel hence the impulse response of the overall system can be given by,

h(t) = halt) + halt)

= e-* u(t) + 2e-tult)

(7)n[1 +2071,277 =

(b) The transfer function of the overall system is

H(S) = H (s) + H2(s)

  

1 2 s +2. s +1

  

3s + 5 (5+1)(8 + 2)

The poles of the system can be given by

s+1 = 0 and s+2 = 0

=> s = -1 and s = -2

Since, both the poles of the overall system are negative hence the system will be stable.