Question

What is the Fourier series representation of a signal x(n) whose period is N?

(a) \(\sum_{k=0}^{\infty}|c_k|^2\)

(b) \(\sum_{k=-\infty}^{\infty}|c_k|\)

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

(d) \(\sum_{k=-\infty}^{\infty}|c_k|^2\)

This question was addressed to me in quiz.

I need to ask this question 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) \(\sum_{k=-\infty}^{\infty}|c_k|\)

Explanation: The average power of a periodic signal x(t) is given as \(\frac{1}{T_p}\int_{t_0}^{t_0+T_p}|x(t)|^2 dt\)

=\(\frac{1}{T_p}\int_{t_0}^{t_0+T_p} x(t).x^* (t) dt\)

=\(\frac{1}{T_p}\int_{t_0}^{t_0+T_p}x(t).\sum_{k=-∞}^∞ c_k^* e^{-j2πkF_0 t} dt\)

By interchanging the positions of integral and summation and by applying the integration, we get

=\(\sum_{k=-∞}^∞|c_k |^2\)