Question

What is the system function of the filter with impulse response hk(n)?

(a) \(\frac{1}{1-W_N^{-k} z^{-1}}\)

(b) \(\frac{1}{1+W_N^{-k} z^{-1}}\)

(c) \(\frac{1}{1-W_N^k z^{-1}}\)

(d) \(\frac{1}{1+W_N^k z^{-1}}\)

I had been asked this question in quiz.

Origin of the question is Linear Filtering Approach to Computation of DFT in portion DFT Efficient Computation – Fast Fourier Transform Algorithms of Digital Signal Processing

Answer 1

Correct choice is (a) \(\frac{1}{1-W_N^{-k} z^{-1}}\)

The explanation is: We know that hk(n)= WN^-kn u(n)

On applying z-transform on both sides, we get

Hk(z)=\(\frac{1}{1-W_N^{-k} z^{-1}}\)