Question

What is the variance of the output DFT coefficients |X(k)|?

(a) \(\frac{1}{N}\)

(b) \(\frac{1}{2N}\)

(c) \(\frac{1}{3N}\)

(d) \(\frac{1}{4N}\)

This question was addressed to me in my homework.

This intriguing question comes from Quantization Effects in the Computation of DFT topic in section DFT Efficient Computation – Fast Fourier Transform Algorithms of Digital Signal Processing

Answer 1

Correct choice is (c) \(\frac{1}{3N}\)

To explain: We know that the variance of the signal sequence is (2/N)^2/12=\(\frac{1}{3N^2}\)

Now the variance of the output DFT coefficients |X(k)|=N.\(\frac{1}{3N^2} = \frac{1}{3N}\).