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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

Figure 1
Figure 1 (fig1.png)
What does this system do?
Figure 2
Figure 2 (fig2.png)

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."

Figure 3
Figure 3 (fig4.png)
Now by linearity of LTI systems
Figure 4
Figure 4 (fig3.png)
Figure 5
Figure 5 (examples1.png)

Example 2

Figure 6: What does this RC circuit do to f⁢t f t ?
Figure 6 (fig5.png)
  • Little insight from R⁢C⁢dy⁢td t +y⁢t=f⁢t R C t y t y t f t
  • but think in terms of
    Figure 7
    Figure 7 (fig6.png)
    The system "operates on/transforms" f f thru h h. h⁢t=1R⁢C⁢e−tR⁢C⁢u⁢t h t 1 R C t R C u t

  • So what does
    Figure 8
    Figure 8 (fig7.png)
    look like? y⁢t=∫−∞∞f⁢τ⁢h⁢t−τd τ y t τ f τ h t τ
    Figure 9
    Figure 9 (fig8.png)
  • Amount of smoothing is controlled by the "RC time constant" of the circuit
    Figure 10
    Figure 10 (fig9.png)
    Limit cases?

Example 3

Figure 11
Figure 11 (fig10.png)
g⁢t=e−t2σ2 g t t 2 σ 2 "Gaussian"
Figure 12
Figure 12 (fig11.png)
What does this system do?

Causality? DE?

Example 4

Figure 13
Figure 13 (fig12.png)
  • What happens as T→0 T 0
  • What about with noise?
    Figure 14
    Figure 14 (fig13.png)
    By linearity:
    Figure 15
    Figure 15 (fig14.png)
    so y= y s + y n y y s y n
    Figure 16
    Figure 16 (fig15.png)
  • Conclusion:
  • Sesitivity to value of T T?

Example 5

Recall Gaussian g⁢t=e−t2σ2 g t t 2 σ 2

Figure 17
Figure 17 (fig16.png)
What about convolution with
Figure 18
Figure 18 (examples.png)
"Mexican hat function"
  • Prove, using linearity, that
    Figure 19
    Figure 19 (fig18.png)
    is equivalent to
    Figure 20
    Figure 20 (fig19.png)
  • Effects:
  • Control over noise sensitivity?

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