10/08 - Mechanics of Convolution & Fourier Transform: Difference between revisions

Revision as of 00:23, 14 November 2008

Remember from the game:

Input	LTI System	Output	Reason
$\delta (t)\,\!$	$\Longrightarrow$	$h(t)\,\!$	Given
$\delta (t-\lambda )\,\!$	$\Longrightarrow$	$h(t-\lambda )\,\!$	Time Invarience
$x(\lambda )\delta (t-\lambda )\,\!$	$\Longrightarrow$	$x(\lambda )h(t-\lambda )\,\!$	Proportionality
$x(t)=\int _{-\infty }^{\infty }x(\lambda )\delta (t-\lambda )\,dx$	$\Longrightarrow$	$\underbrace {\int _{-\infty }^{\infty }x(\lambda )h(t-\lambda )\,dx} _{ConvolutionIntegral}$	Superposition

We will also denote the convolution as $x(t)*h(t)\equiv \int _{-\infty }^{\infty }x(\lambda )h(t-\lambda )\,dx$

$x(t)*h(t)\,\!$	$=\int _{-\infty }^{\infty }x(\lambda )h(t-\lambda )\,d\lambda$	Let $t-\lambda =u\,\!$ thus $du=-d\lambda \,\!$
	$=-\int _{\infty }^{-\infty }x(t-u)h(u)\,du$	The order of integration switched due to changing from $-\lambda =u\,\!$

@@ Line 30: / Line 30: @@
 We will also denote the convolution as <math> x(t) * h(t) \equiv \int_{-\infty}^{\infty} x(\lambda)h(t-\lambda)\, dx</math>
+===Communative Property===
+{| border="0" cellpadding="0" cellspacing="0"
+|-
+|<math>x(t) * h(t) \,\!</math>
+|<math>=\int_{-\infty}^{\infty} x(\lambda)h(t-\lambda)\, d\lambda</math>
+|Let <math> t-\lambda = u \,\!</math> thus <math> du = -d\lambda \,\! </math>
+|-
+|
+|<math>=-\int_{\infty}^{-\infty} x(t-u)h(u)\, du</math>
+|The order of integration switched due to changing from <math>-\lambda = u\,\!</math>
+|}