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# Why is Convolution Useful?

Module by: Richard Baraniuk. E-mail the author

Summary: A module concerning the usefulness of convolution.

Convolution concept provides a much clearer, more intuitive idea of what an LTI system is actually doing [phyisical insight].

## Example 1

What does this system do?

Useful signal decomposition into "signal" and "noise" where "signal" is what you want, or "information" and "noise" is what you don't want, or "no information."

Now by linearity of LTI systems

## Example 2

• Little insight from RCdytd t +yt=ft R C t y t y t f t
• but think in terms of The system "operates on/transforms" f f thru h h. ht=1RCetRCut h t 1 R C t R C u t

• So what does look like? yt=fτhtτd τ y t τ f τ h t τ
• Amount of smoothing is controlled by the "RC time constant" of the circuit Limit cases?

## Example 3

gt=et2σ2 g t t 2 σ 2 "Gaussian" What does this system do?

Causality? DE?

## Example 4

• What happens as T0 T 0
• What about with noise? By linearity: so y= y s + y n y y s y n
• Conclusion:
• Sesitivity to value of T T?

## Example 5

Recall Gaussian gt=et2σ2 g t t 2 σ 2

What about convolution with "Mexican hat function"
• Prove, using linearity, that is equivalent to
• Effects:
• Control over noise sensitivity?

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