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Parseval's Theorem for the Fourier Transform

Module by: Carlos E. Davila. E-mail the author

In Chapter 2, we looked at a version of Parseval's theorem for the Fourier series. Here, we will look at a similar version of this theorem for the Fourier transform. Recall that the energy of a signal is given by

e x = ∫ - ∞ ∞ x ( t ) 2 d t e x = ∫ - ∞ ∞ x ( t ) 2 d t
(1)

If the energy is finite then x(t)x(t) is an energy signal, as described in Chapter 1. Suppose x(t)x(t) is an energy signal, then the autocorrelation function is defined as

r x ( t ) = x ( t ) * x ( - t ) r x ( t ) = x ( t ) * x ( - t )
(2)

It can be shown that rx(t)rx(t) is an even function of tt and that rx(0)=exrx(0)=ex(see Exercises). The Fourier transform of rx(t)rx(t) is given by X(jΩ)X(jΩ)*=X(jΩ)2X(jΩ)X(jΩ)*=X(jΩ)2. If follows that

e x = 1 2 π ∫ - ∞ ∞ X ( j Ω ) 2 e j Ω t d Ω t = 0 = 1 2 π ∫ - ∞ ∞ X ( j Ω ) 2 d Ω e x = 1 2 π ∫ - ∞ ∞ X ( j Ω ) 2 e j Ω t d Ω t = 0 = 1 2 π ∫ - ∞ ∞ X ( j Ω ) 2 d Ω
(3)

Which is Parseval's theorem for the Fourier transform.

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