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Convolution Integral

Module by: Richard Baraniuk. E-mail the author

Summary: An introduction to the convolution integral.

Combine these facts

  1. Signals can be represented as infinite sums (integrals) of scaled and shifted impulses.
  2. Time invariance of system implies Figure 1.
  3. Linearity of the system implies Figure 2.
Figure 1
Figure 1 (fig1.png)
Figure 2
Figure 2 (fig2.png)
Let's do it!

Derivation of Convolution

Figure 3.

Figure 3: h h is LTI.
Figure 3 (fig3.png)

Step 1

Signal representation in terms of impulses:

f⁢t=∫−∞∞f⁢t⁢δ⁢t−τd τ f t τ f t δ t τ (1)

Step 2

Use the time invariance and scaling part of linearity (Figure 4).

Figure 4: f⁢τ f τ is a constant with respect to the variable t t, which the system operates on.
Figure 4 (fig4.png)

Step 3

Use the additivity part of linearity to add up an infinite number of these terms, one for each τ τ (Figure 5).

Figure 5: f⁢t f t is the input and y⁢t y t is the output.
Figure 5 (fig5.png)

Alternative Derivation in Terms of Limits and Sums

Figure 6.

Figure 6: Call this δ Δ ⁢t δ Δ t .
Figure 6 (fig6.png)
  • δ⁢t δ t put through hh yields h⁢t h t
  • limit   Δ → 0 δ Δ ⁢t Δ 0 δ Δ t put through hh yields h⁢t h t
  • limit   Δ → 0 δ Δ ⁢t−n⁢Δ Δ 0 δ Δ t n Δ put through hh yields limit   Δ → 0 h⁢t−n⁢Δ Δ 0 h t n Δ
  • limit   Δ → 0 f⁢n⁢Δ⁢δ⁢t−n⁢Δ⁢Δ Δ 0 f n Δ δ t n Δ Δ put through hh yields limit   Δ → 0 f⁢n⁢Δ⁢h⁢t−n⁢Δ Δ 0 f n Δ h t n Δ
  • limit   Δ → 0 ∑ n f⁢n⁢Δ⁢δ⁢t−n⁢Δ⁢Δ Δ 0 n f n Δ δ t n Δ Δ put through hh yields limit   Δ → 0 ∑ n f⁢n⁢Δ⁢h⁢t−n⁢Δ⁢Δ Δ 0 n f n Δ h t n Δ Δ
  • ∫−∞∞f⁢τ⁢δ⁢t−τd τ τ f τ δ t τ put through hh yields ∫−∞∞f⁢τ⁢h⁢t−τd τ τ f τ h t τ
  • f⁢t f t put through hh yields y⁢t y t

Any way you look at it, we have Figure 7.

Figure 7: h h is LTI.
Figure 7 (fig7.png)

Convolution Integral for LTI Systems

y⁢t=∫−∞∞f⁢τ⁢h⁢t−τd τ y t τ f τ h t τ (2)

Note:

You will need to memorize Equation 2.

Notation:

f⁢t*h⁢t=y⁢t f t h t y t

Interpretation

y⁢t y t is an infinite sum of shifted "shock responses" each weighted by a signal value.

Note:

This equation holds for all LTI systems!!!

Pictorially: Konvolution (Graphical Interpretation I)

Figure 8. Figure 9.

Figure 8: h⁢t h t is the impulse response.
Figure 8 (fig8.png)
Figure 9: y⁢t y t is the convolution integral.
Figure 9 (fig9.png)

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