Consider $ X(j\omega) $ evaluated according to Equation 4.9:
<img alt="tex:$$X(j\omega) = \int_{-\infty}^\infty x(t)e^{-j \omega t} dt$$" />
and let <img alt="tex:$$x(t)$$" /> denote the signal obtained by using <img alt="tex:$$X(j\omega)$$" /> in the right hand side of Equation 4.8:
<img alt="tex:$$x(t) = (1/(2\pi)) \int_{-\infty}^\infty X(j\omega)e^{j \omega t} d\omega$$" />
If <img alt="tex:$$x(t)$$" /> has finite energy, i.e., if it is square integrable so that Equation 4.11 holds:
<img alt="tex:$$\int_{-\infty}^\infty |x(t)|^2 dt < \infty$$" />
then it is guaranteed that <img alt="tex:$$X(j\omega)$$" /> is finite, i.e, Equation 4.9 converges.
Let <img alt="tex:$$e(t)$$" /> denote the error between <img alt="tex:$$\hat{x}(t)$$" /> and <img alt="tex:$$x(t)$$" />, i.e. <img alt="tex:$$e(t)=\hat{x}(t) - x(t)$$" />,
then Equation 4.12 follows:
<img alt="tex:$$\int_{-\infty}^\infty |e(t)|^2 dt = 0$$" />
Thus if <img alt="tex:$$x(t)$$" /> has finite energy, then, although <img alt="tex:$$x(t)$$" /> and <img alt="tex:$$\hat{x}(t)$$" /> may differ significantly at individual values of <img alt="tex:$$t$$" />, there is no energy in their difference.
From mireille.boutin.1 Fri Oct 12 16:23:04 -0400 2007 From: mireille.boutin.1 Date: Fri, 12 Oct 2007 16:23:04 -0400 Subject: this is not clear Message-ID: <20071012162304-0400@https://engineering.purdue.edu>
why does Equation 4.12 follow???? Can somebody explain?
