Question

If x(n) and X(k) are an N-point DFT pair, then X(k+N)=?

(a) X(-k)

(b) -X(k)

(c) X(k)

(d) None of the mentioned

I had been asked this question during an online interview.

I would like to ask this question from Properties of DFT in section Discrete Fourier Transform – Properties and Applications of Digital Signal Processing

Answer 1

Correct choice is (c) X(k)

The best explanation: We know that

x(n)=\(\frac{1}{N}\sum_{k=0}^{N-1} x(k)e^{j2πkn/N}\)

Let X(k)=X(k+N)

=>x^1(n)=\(\frac{1}{N} \sum_{k=0}^{N-1}X(k+N)e^{j2πkn/N}\)=x(n)

Therefore, we have X(k)=X(k+N)