3.14 Parseval's theorem for the fourier transform

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

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

It can be shown that $r_x\left(t\right)$ is an even function of $t$ and that $r_x\left(0\right)=e_x$ (see Exercises). The Fourier transform of $r_x\left(t\right)$ is given by $X\left(j\Omega \right)X{\left(j\Omega \right)}^{*}={\left|X,\left(,j,\Omega ,\right)\right|}^2$ . If follows that

Which is Parseval's theorem for the Fourier transform.