The symbols, 𝑎 and 𝑇, represent positive quantities, and 𝑢(𝑡) is the unit step function. Which one of the following impulse responses is NOT the output of a causal linear time-invariant system?

Q. The symbols, 𝑎 and 𝑇, represent positive quantities, and 𝑢(𝑡) is the unit step function. Which one of the following impulse responses is NOT the output of a causal linear time-invariant system?

(A) 𝑒^+𝑎𝑡𝑢(𝑡)

(B) 𝑒^{−𝑎(𝑡+𝑇)}𝑢(𝑡)

(C) 1 + 𝑒^−𝑎𝑡𝑢(𝑡)

(D) 𝑒^{−𝑎(𝑡−𝑇)}𝑢(𝑡)

Ans: 1 + 𝑒^−𝑎𝑡𝑢(𝑡)

Solution:

a and T represents positive quantities.
u(t) is unit step function.
h(t)=1+e−atu(t), is non-causal
Here ‘1’ is a constant and two sided so the system will be non-causal, because for causal system
h(t)=0,t<0
≠0,t>0