Question

Find the transfer functions of the following systems and also determine which systems are causal

(a) h(t) = e^-at u(t+2), a > 0
(b) h(t) = e^-a(t). a<0
(c) h(t) = e^-a(,-0)11(1-0),
(d) h(t)=2t/(1+12)
(e) h(t) - sinc(at a>0
(f) ht) sinc[at o]u(t), a > 0, t02 0 a> 0

Answer 1

The transfer functions and causality of several systems are determined and explained.

Step-by-step explanation:

(a) To find the transfer function, we need to take the Laplace transform of the given function h(t). The Laplace transform of e^-at is 1/(s+a), and the Laplace transform of u(t+2) is e^-2s/(s). Therefore, the Laplace transform of h(t) = e^-at u(t+2) is H(s) = 1/(s+a) * e^-2s/(s).

(b) To determine if a system is causal, we need to check if the impulse response is non-zero for negative times. For the given system h(t) = e^-a(t), the impulse response is e^-a(t), which is non-zero for t > 0. Therefore, the system is causal.

(c) For the given system h(t) = e^-a(,-0)11(1-0), the impulse response is e^-a(,-0)11(1-0). This impulse response is non-zero for t < 0, and therefore, the system is not causal.