Properties of Convolution 2.14 2001/06/13 2007/07/18 12:34:16.019 GMT-5 Melissa Selik mselik@alumni.rice.edu Richard G. Baraniuk richb@rice.edu Richard G. Baraniuk richb@rice.edu Justin Romberg jrom@ece.gatech.edu Michael Haag mjhaag@gmail.com Matthew Hutchinson mhutch@rice.edu Marco Felipe Duarte duarte@rice.edu convolution property signals systems Examples and definitions of the various properties associated with convolution are descrbied. 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 Associative Law f1 t f2 t f3 t f1 t f2 t f3 t

Graphical implication of the associative property of convolution.

Commutativity Commutative Law y t f t h t h t f t To prove , all we need to do is make a simple change of variables in our convolution integral (or sum), y t τ f τ h t τ By letting τ t τ , we can easily show that convolution is commutative: y t τ f t τ h τ τ h τ f t τ f t h t h t f t

The figure shows that either function can be regarded as the system's input while the other is the impulse response.

Distribution Distributive Law f1 t f2 t f3 t f1 t f2 t f1 t f3 t The proof of this theorem can be taken directly from the definition of convolution and by using the linearity of the integral.

Time Shift Shift Property For c t f t h t , then c t T f t T h t and c t T f t h t T

Graphical demonstration of the shift property.

Convolution with an Impulse Convolving with Unit Impulse f t δ t f t For this proof, we will let δt be the unit impulse located at the origin. Using the definition of convolution we start with the convolution integral f t δ t τ δ τ f t τ From the definition of the unit impulse, we know that δτ 0 whenever τ 0 . We use this fact to reduce the above equation to the following: f t δ t τ δ τ f t f t τ δ τ The integral of δτ will only have a value when τ0 (from the definition of the unit impulse), therefore its integral will equal one. Thus we can simplify the equation to our theorem: f t δ t f t

The figures, and equation above, reveal the identity function of the unit impulse.

Width In continuous time, if Duration f 1 T 1 and Duration f 2 T 2 , then Duration f1 f2 T 1 T 2

In continuous-time, the duration of the convolution result equals the sum of the lengths of each of the two signals that are convolved.

In discrete time, if Duration f 1 N 1 and Duration f 2 N 2 , then Duration f1 f2 N 1 N 2 1

Causality If f and h are both causal, then f h is also causal.