ISRO EC 2023 Q.16: solution and answer

Official answer: CSource: ISRO's revised and final answer key of 16 February 2024

1. The identity element for convolution is a signal $e(n)$ such that $x(n)\ast e(n)=x(n)$ for every signal $x(n)$.

2. Use the sifting property of the unit impulse: $x(n)\ast \delta (n)={\sum }_{k=-\infty }^{\infty }x(k)\delta (n-k)=x(n)$, because $\delta (n-k)$ is nonzero only at $k=n$.

3. So $\delta (n)$ leaves every signal unchanged. It is the identity for convolution (the system with $h(n)=\delta (n)$ is the identity system).

4. Option (a), the constant $1$: $x(n)\ast 1={\sum }_{k}x(k)$, a constant equal to the sum of all samples, not $x(n)$.

5. Option (b), $0$: $x(n)\ast 0=0$, which wipes out the signal rather than preserving it.

6. Option (d), $u(n)$: $x(n)\ast u(n)={\sum }_{k=-\infty }^{n}x(k)$, the running sum (accumulator), not $x(n)$.

7. Hence the identity for convolution is the impulse $\delta (n)$.