10/08 - Mechanics of Convolution & Fourier Transform

Mechanics of the Convolution

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 $x(t)*h(t)\equiv \int _{-\infty }^{\infty }x(\lambda )h(t-\lambda )\,dx$

Communative Property

$x(t)*h(t)\,\!$	$=\int _{-\infty }^{\infty }x(\lambda )h(t-\lambda )\,d\lambda$	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 $-\lambda =u\,\!$
	$=\int _{-\infty }^{\infty }h(u)x(t-u)\,du$
	$=h(t)*x(t)\,\!$

Example 1

$\delta (t)*x(t)\,\!$	$=\int _{-\infty }^{\infty }\delta (\lambda )x(t-\lambda )\,d\lambda$
	$=x(t)\int _{-\infty }^{\infty }\,d\lambda$
	$=x(t)\,\!$

Example 2

$\overbrace {[u(t)-u(t-1)]} ^{x(t)}*\overbrace {[u(t-1)-u(t-3)]} ^{h(t)}={\begin{cases}0,&t\leq 1\\\int _{0}^{t-1}1\cdot 2\,d\lambda ,&1\leq t\leq 2\\\int _{0}^{1}1\cdot 2\,d\lambda ,&2\leq t\leq 3\\\int _{t-3}^{1}1\cdot 2\,d\lambda ,&3\leq t\leq 4\\0,&t>4\\\end{cases}}={\begin{cases}0,&t\leq 1\\2\,t-2,&1\leq t\leq 2\\2,&2\leq t\leq 3\\-2\,t-4,&3\leq t\leq 4\\0,&t>4\\\end{cases}}$

• In this case, we are doing the FSMI to $h(t)\,\!$
• If $u(t)\,\!$ isn't involved, then you can plug n chug with the integral. The u(t) will change the limits, which can be impractical to evaulate if you have more than 2.
• ? Does it matter which one you FSMI?

Convolution: A visual approach

• Flip: Flip one about the dependant axis
• Shift: The initial flipped function is at $t=0\,\!$. Shift this function for the multiply & integrate
• Multiply: Multiply the two functions
• Add/Integrate: You may need to make multiple equations for different intersections