Question

If two LTI systems with impulse response h1 (t) and h2 (t) and are connected in parallel then output is given by ______

(a) y(t) = x(t) *(h1(t) + h2(t))

(b) y(t) = x(t) + (h1(t) + h2(t))

(c) y(t) = x(t) * (h1(t) h2(t))

(d) y(t) = (x(t) * h1(t)) + h2(t)

I had been asked this question by my school principal while I was bunking the class.

This interesting question is from Properties of the Impulse Response Representation for LTI Systems in chapter Time Domain Representation for LTI Systems of Signals and Systems

Answer 1

The correct option is (a) y(t) = x(t) *(h1(t) + h2(t))

Easiest explanation: The equivalent impulse response of two systems connected in parallel is the sum of individual impulse responses. It is represented as

 y(t) = x(t) * h1(t) + x(t) * h2(t) = x(t) * (h1(t) + h2(t)).