Question

According to Parseval’s Theorem for non-periodic signal, \(\int_{-∞}^∞|x(t)|^2 dt\).

(a) \(\int_{-∞}^∞|X(F)|^2 dt \)

(b) \(\int_{-∞}^∞|X^* (F)|^2 dt \)

(c) \(\int_{-∞}^∞ X(F).X^*(F) dt \)

(d) All of the mentioned

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

My question comes from Frequency Analysis of Continuous Time Signal in section Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right answer is (d) All of the mentioned

Easiest explanation: Let x(t) be any finite energy signal with Fourier transform X(F). Its energy is

Ex=\(\int_{-∞}^∞|x(t)|^2 dt\)

which in turn, can be expressed in terms of X(F) as follows

Ex=\(\int_{-∞}^∞ x^* (t).x(t)\) dt

=\(\int_{-∞}^∞ x(t) dt[\int_{-∞}^∞X^* (F)e^{-j2πF_0 t} dt]\)

=\(\int_{-∞}^∞ X^* (F) dt[\int_{-∞}^∞ x(t)e^{-j2πF_0 t} dt] \)

\(=\int_{-∞}^∞ |X(F)|^2 dt = \int_{-∞}^∞|X^* (F)|^2  dt = \int_{-∞}^∞X(F).X^* (F) dt\)