Question

What is the equation for average power of discrete time periodic signal x(n) with period N in terms of Fourier series coefficient ck?

(a) \(\sum_{k=0}^{N-1}|c_k|\)

(b) \(\sum_{k=0}^{N-1}|c_k|^2\)

(c) \(\sum_{k=0}^N|c_k|^2\)

(d) \(\sum_{k=0}^N|c_k|\)

I have been asked this question by my college director while I was bunking the class.

The query is from Frequency Analysis of Discrete Time Signal in division Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Correct choice is (b) \(\sum_{k=0}^{N-1}|c_k|^2\)

For explanation: We know that Px=\(\frac{1}{N} \sum_{n=0}^{N-1}|x(n)|^2\)

=\(\frac{1}{N} \sum_{n=0}^{N-1}x(n).x^*(n)\)

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

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

=\(\sum_{k=0}^{N-1}|c_k |^2\)