1.

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 mentionedI 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»

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

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

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

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


Discussion

No Comment Found

Related InterviewSolutions