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Convolution Integral

Module by: Richard Baraniuk. E-mail the author

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Summary: An introduction to the convolution integral.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Combine these facts

Signals can be represented as infinite sums (integrals) of scaled and shifted impulses.
Time invariance of system implies Figure 1.
Linearity of the system implies Figure 2.

**Figure 1**

**Figure 2**

Let's do it!

Derivation of Convolution

Figure 3.

**Figure 3:** h h is LTI.

Step 1

Signal representation in terms of impulses:

ft=∫-∞∞ftδt−τdτ

f t τ f t δ t τ

(1)

Step 2

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

**Figure 4:** fτ f τ is a *constant* with respect to the variable t t, which the system operates on.

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:** ft f t is the input and yt y t is the output.

Alternative Derivation in Terms of Limits and Sums

Figure 6.

**Figure 6:** Call this δ Δ t δ Δ t .

δt δ t put through hh yields ht h t
limΔ→0 δ Δ t Δ 0 δ Δ t put through hh yields ht h t
limΔ→0 δ Δ t−nΔ Δ 0 δ Δ t n Δ put through hh yields limΔ→0ht−nΔ Δ 0 h t n Δ
limΔ→0fnΔδt−nΔΔ Δ 0 f n Δ δ t n Δ Δ put through hh yields limΔ→0fnΔht−nΔ Δ 0 f n Δ h t n Δ
limΔ→0∑fnΔδt−nΔΔ Δ 0 n f n Δ δ t n Δ Δ put through hh yields limΔ→0∑fnΔht−nΔΔ Δ 0 n f n Δ h t n Δ Δ
∫-∞∞fτδt−τdτ τ f τ δ t τ put through hh yields ∫-∞∞fτht−τdτ τ f τ h t τ
ft f t put through hh yields yt y t

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

**Figure 7:** h h is LTI.