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Consider <math>X(j\omega)</math> evaluated according to Equation 4.9:
 
Consider <math>X(j\omega)</math> evaluated according to Equation 4.9:
  
<img alt="tex:$$X(j\omega) = \int_{-\infty}^\infty x(t)e^{-j \omega t} dt$$" />
+
<math>X(j\omega) = \int_{-\infty}^\infty x(t)e^{-j \omega t} dt</math>
  
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:
+
and let x(t) denote the signal obtained by using <math>X(j\omega)</math> in the right hand side of Equation 4.8:
  
<img alt="tex:$$x(t) = (1/(2\pi)) \int_{-\infty}^\infty X(j\omega)e^{j \omega t} d\omega$$" />
+
<math>x(t) = (1/(2\pi)) \int_{-\infty}^\infty X(j\omega)e^{j \omega t} d\omega</math>
  
If <img alt="tex:$$x(t)$$" /> has finite energy, i.e., if it is square integrable so that Equation 4.11 holds:
+
If x(t) has finite energy, i.e., if it is square integrable so that Equation 4.11 holds:
  
<img alt="tex:$$\int_{-\infty}^\infty |x(t)|^2 dt < \infty$$" />
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<math>\int_{-\infty}^\infty |x(t)|^2 dt < \infty</math>
  
then it is guaranteed that <img alt="tex:$$X(j\omega)$$" /> is finite, i.e, Equation 4.9 converges.
+
then it is guaranteed that <math>X(j\omega)</math> 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:$$e(t)=\hat{x}(t) - x(t)$$" />,
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Let e(t) denote the error between <math>\hat{x}(t)</math> and x(t), i.e. <math>e(t)=\hat{x}(t) - x(t)</math>,
  then Equation 4.12 follows:
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then Equation 4.12 follows:
  
<img alt="tex:$$\int_{-\infty}^\infty |e(t)|^2 dt = 0$$" />
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<math>\int_{-\infty}^\infty |e(t)|^2 dt = 0</math>
  
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.
+
Thus if x(t) has finite energy, then, although x(t) and <math>\hat{x}(t)</math> may differ significantly at individual values of t, there is no energy in their difference.
  
 
From mireille.boutin.1 Fri Oct 12 16:23:04 -0400 2007
 
From mireille.boutin.1 Fri Oct 12 16:23:04 -0400 2007

Latest revision as of 11:24, 24 March 2008

Consider $ X(j\omega) $ evaluated according to Equation 4.9:

$ X(j\omega) = \int_{-\infty}^\infty x(t)e^{-j \omega t} dt $

and let x(t) denote the signal obtained by using $ X(j\omega) $ in the right hand side of Equation 4.8:

$ x(t) = (1/(2\pi)) \int_{-\infty}^\infty X(j\omega)e^{j \omega t} d\omega $

If x(t) has finite energy, i.e., if it is square integrable so that Equation 4.11 holds:

$ \int_{-\infty}^\infty |x(t)|^2 dt < \infty $

then it is guaranteed that $ X(j\omega) $ is finite, i.e, Equation 4.9 converges.

Let e(t) denote the error between $ \hat{x}(t) $ and x(t), i.e. $ e(t)=\hat{x}(t) - x(t) $, then Equation 4.12 follows:

$ \int_{-\infty}^\infty |e(t)|^2 dt = 0 $

Thus if x(t) has finite energy, then, although x(t) and $ \hat{x}(t) $ may differ significantly at individual values of 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?

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