Question

If X1(n), x2(n) and x3(m) are three sequences each of length N whose DFTs are given as X1(k), X2(k) and X3(k) respectively and X3(k)=X1(k).X2(k), then what is the expression for x3(m)?

(a) \(\sum_{n=0}^{N-1}x_1 (n) x_2 (m+n)\)

(b) \(\sum_{n=0}^{N-1}x_1 (n) x_2 (m-n)\)

(c) \(\sum_{n=0}^{N-1}x_1 (n) x_2 (m-n)_N \)

(d) \(\sum_{n=0}^{N-1}x_1 (n) x_2 (m+n)_N \)

I got this question during an interview.

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

Answer 1

Correct choice is (c) \(\sum_{n=0}^{N-1}x_1 (n) x_2 (m-n)_N \)

Explanation: If X1(n), x2(n) and x3(m) are three sequences each of length N whose DFTs are given as X1(k), x2(k) and X3(k) respectively and X3(k)=X1(k).X2(k), then according to the multiplication property of DFT we have x3(m) is the circular convolution of X1(n) and x2(n).

That is x3(m) = \(\sum_{n=0}^{N-1}x_1 (n) x_2 (m-n)_N \).