Question

What is the formula for Parseval’s relation in Fourier series expansion?

(a) \( \int_{-l}^l (f(x))^2 dx=l[\frac{a_0^2}{2}+∑_{n=1}^∞(a_n^2+b_n^2 ) ] \)

(b) \( \int_{-l}^l (f(x))^2 dx=l[\frac{a_0^2}{2}+∑_{n=1}^∞(a_n^2 ) ] \)

(c) \( \int_{-l}^l (f(x))^2 dx=l⁄2 [\frac{a_0^2}{2}+∑_{n=1}^∞(a_n^2+b_n^2 ) ] \)

(d) \( l\int_{-l}^l (f(x))^2 dx=[\frac{a_0^2}{2}+∑_{n=1}^∞(a_n^2+b_n^2 ) ] \)

I got this question during an interview.

My question is taken from Fourier Half Range Series in section Fourier Series of Engineering Mathematics

Answer 1

Correct choice is (a) \( \int_{-l}^l (f(x))^2 dx=l[\frac{a_0^2}{2}+∑_{n=1}^∞(a_n^2+b_n^2 ) ] \)

Explanation: The real life significance of Parseval’s relation is to find the energy of the signal in its time period (1 time period). This can be found using the Fourier series coefficients. \(\int_{-l}^l (f(x))^2 dx=l[\frac{a_0^2}{2}+∑_{n=1}^∞(a_n^2+b_n^2 ) ] \)  is the realtion between the function and the Fourier series coefficients.