Question

An LTI system is said to be causal if and only if?

(a) Impulse response is non-zero for positive values of n

(b) Impulse response is zero for positive values of n

(c) Impulse response is non-zero for negative values of n

(d) Impulse response is zero for negative values of n

The question was posed to me in an online interview.

The doubt is from Analysis of Discrete time LTI Systems topic in division Discrete Time Signals and Systems of Digital Signal Processing

Answer 1

The correct answer is (d) Impulse response is zero for negative values of n

To explain: Let us consider a LTI system having an output at time n=n0 given by the convolution formula

y(n)=\(\sum_{k=-{\infty}}^{\infty}h(k)x(n_0-k)\)

We split the summation into two intervals.

=>y(n)=\(\sum_{k=-{\infty}}^{-1}h(k)x(n_0-k)+\sum_{k=0}^{\infty}h(k)x(n_0-k)\)

=(h(0)x(n0)+h(1)x(n0-1)+h(2)x(n0-2)+….)+(h(-1)x(n0+1)+h(-2)x(n0+2)+…)

As per the definition of the causality, the output should depend only on the present and past values of the input. So, the coefficients of the terms x(n0+1), x(n0+2)…. should be equal to zero.

that is, h(n)=0 for n<0 .