Question

If x(n) is real and odd, then what is the IDFT of the given sequence?

(a) \(j \frac{1}{N} \sum_{k=0}^{N-1} x(k) sin⁡\frac{2πkn}{N}\)

(b) \(\frac{1}{N} \sum_{k=0}^{N-1} x(k) cos⁡\frac{2πkn}{N}\)

(c) \(-j \frac{1}{N} \sum_{k=0}^{N-1} x(k) sin⁡\frac{2πkn}{N}\)

(d) None of the mentioned

This question was addressed to me in an online quiz.

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

Answer 1

Right option is (a) \(j \frac{1}{N} \sum_{k=0}^{N-1} x(k) sin⁡\frac{2πkn}{N}\)

The best I can explain: If x(n) is real and odd, that is x(n)=-x(N-n), then XR(k)=0. Hence X(k) is purely imaginary and odd. Since XR(k) reduces to zero, the IDFT reduces to

\(x(n)=j \frac{1}{N} \sum_{k=0}^{N-1} x(k) sin⁡\frac{2πkn}{N}\)