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Properties of Convolution

Module by: Melissa Selik, Richard Baraniuk. E-mail the authors

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Summary: Examples and definitions of the various properties associated with convolution are descrbied.

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In this module we will study several of the most prevalent properties of convolution. Note that these properties apply to both continuous-time convolution and discrete-time convolution. (Refer back to these two modules if you need a review of convolution). Also, for the proofs of some of the properties, we will be using continuous-time integrals, but we could prove them the same way using the discrete-time summations.

Associativity

Theorem 1: Associative Law

f1t*f2t*f3t=f1t*f2t*f3t f1 t f2 t f3 t f1 t f2 t f3 t (1)

Figure 1: Graphical implication of the associative property of convolution.
Figure 1 (convassoc.png)

Commutativity

Theorem 2: Commutative Law

yt=ft*ht=ht*ft y t f t h t h t f t (2)

Proof

To prove Equation 2, all we need to do is make a simple change of variables in our convolution integral (or sum),

yt=∫-∞∞fτht−τdτ y t τ f τ h t τ (3)
By letting τ=t−τ τ t τ , we can easily show that convolution is commutative:
yt=∫-∞∞ft−τhτdτ=∫-∞∞hτft−τdτ y t τ f t τ h τ τ h τ f t τ (4)
ft*ht=ht*ft f t h t h t f t (5)

Figure 2: The figure shows that either function can be regarded as the system's input while the other is the impulse response.
Figure 2 (convcomm.png)

Distribution

Theorem 3: Distributive Law

f1t*f2t+f3t=f1t*f2t+f1t*f3t f1 t f2 t f3 t f1 t f2 t f1 t f3 t (6)

Proof

The proof of this theorem can be taken directly from the definition of convolution and by using the linearity of the integral.

Figure 3
Figure 3 (convdist.png)

Time Shift

Theorem 4: Shift Property

For ct=ft*ht c t f t h t , then

ct−T=ft−T*ht c t T f t T h t (7)
and
ct−T=ft*ht−T c t T f t h t T (8)

Figure 4: Graphical demonstration of the shift property.
(a)
Figure 4(a) (convts1.png)
(b)
Figure 4(b) (convts2.png)
(c)
Figure 4(c) (convts3.png)

Convolution with an Impulse

Theorem 5: Convolving with Unit Impulse

ft*δt=ft f t δ t f t (9)

Proof

For this proof, we will let δt δt be the unit impulse located at the origin. Using the definition of convolution we start with the convolution integral

ft*δt=∫-∞∞δτft−τdτ f t δ t τ δ τ f t τ (10)
From the definition of the unit impulse, we know that δτ=0 δτ 0 whenever τ≠0 τ 0 . We use this fact to reduce the above equation to the following:
ft*δt=∫-∞∞δτftdτ=ft∫-∞∞δτdτ f t δ t τ δ τ f t f t τ δ τ (11)
The integral of δτ δτ will only have a value when τ=0 τ0 (from the definition of the unit impulse), therefore its integral will equal one. Thus we can simplify the equation to our theorem:
ft*δt=ft f t δ t f t (12)

Figure 5: The figures, and equation above, reveal the identity function of the unit impulse.
(a)
Figure 5(a) (convimp1.png)
(b)
Figure 5(b) (convimp2.png)

Width

In continuous time, if Duration f 1 = T 1 Duration f 1 T 1 and Duration f 2 = T 2 Duration f 2 T 2 , then

Durationf1*f2= T 1 + T 2 Duration f1 f2 T 1 T 2 (13)

Figure 6: In continuous-time, the duration of the convolution result equals the sum of the lengths of each of the two signals that are convolved.
(a)
Figure 6(a) (convwidth1.png)
(b)
Figure 6(b) (convwidth2.png)
(c)
Figure 6(c) (convwidth3.png)

In discrete time, if Duration f 1 = N 1 Duration f 1 N 1 and Duration f 2 = N 2 Duration f 2 N 2 , then

Durationf1*f2= N 1 + N 2 −1 Duration f1 f2 N 1 N 2 1 (14)

Causality

If ff and hh are both causal, then f*h f h is also causal.

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