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

Revision as of 00:16, 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 $x(t)*h(t)\equiv \int _{-\infty }^{\infty }x(\lambda )h(t-\lambda )\,dx$

@@ Line 1: / Line 1: @@
 ==Mechanics of the Convolution==
 Remember from the game:
+{| border="1" cellpadding="5" cellspacing="0"
+|-
+|Input
+|LTI System
+|Output
+|Reason
+|-
+|<math> \delta (t) \,\!</math>
+|<math> \Longrightarrow </math>
+|<math> h(t) \,\!</math>
+|Given
+|-
+|<math> \delta (t-\lambda)\,\!</math>
+|<math> \Longrightarrow </math>
+|<math> h(t-\lambda) \,\!</math>
+|Time Invarience
+|-
+|<math> x(\lambda) \delta (t-\lambda)\,\!</math>
+|<math> \Longrightarrow </math>
+|<math> x(\lambda)h(t-\lambda) \,\!</math>
+|Proportionality
+|-
+|<math> x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t-\lambda)\, dx</math>
+|<math> \Longrightarrow </math>
+|<math> \underbrace{\int_{-\infty}^{\infty} x(\lambda)h(t-\lambda)\, dx}_{Convolution Integral}</math>
+|Superposition
+|}
+We will also denote the convolution as <math> x(t) * h(t) \equiv \int_{-\infty}^{\infty} x(\lambda)h(t-\lambda)\, dx</math>