If yk(n) is the convolution of the finite duration input sequence

if-yk-n-is-the-convolution-of-the-finite-duration-input-sequence

If yk(n) is the convolution of the finite duration input sequence x(n) of length N, then what is the impulse response of the filter?

A. WN-kn

B. WN-kn u(n)

C. WNkn u(n)

D. None of the mentioned

I got this question during an online interview.

This key question is from Linear Filtering Approach to Computation of DFT in chapter DFT Efficient Computation – Fast Fourier Transform Algorithms of Digital Signal Processing

The correct option is B. WN-kn u(n)

The best I can explain: We know that yk(n)=\(\sum_{m=0}^{N-1} x(m)W_N^{-k(n-m)}\)

The above equation is of the form yk(n)=x(n)*hk(n)

Thus we obtain, hk(n)= WN^-kn u(n).