Question

If x(n) is a real sequence and X(k) is its N-point DFT, then which of the following is true?

(a) X(N-k)=X(-k)

(b) X(N-k)=X*(k)

(c) X(-k)=X*(k)

(d) All of the mentioned

The question was posed to me during an interview for a job.

My doubt is from Properties of DFT topic in chapter Discrete Fourier Transform – Properties and Applications of Digital Signal Processing

Answer 1

Correct option is (d) All of the mentioned

To elaborate: We know that

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

Now X(N-k)=\(\sum_{n=0}^{N-1} x(n)e^{-j2π(N-k)n/N}\)=X*(k)=X(-k)

Therefore,

X(N-k)=X*(k)=X(-k)