Question

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

(a) True

(b) False

I had been asked this question in a national level competition.

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

Answer 1

The correct choice is (a) True

Best explanation: We know that the expression for an DFT is given as

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

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

Let n+N=l=>X^1(k)=\(\sum_{l=N}^0 x(l)e^{-j2πkl/N}\)=X(k)

Therefore, we got x(n)=x(n+N)