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Properties of Continuous Time Convolution

By Melissa Selik, Richard Baraniuk, Stephen Kruzick

Introduction

We have already shown the important role that continuous time convolution plays in signal processing. This section provides discussion and proof of some of the important properties of continuous time convolution. Analogous properties can be shown for continuous time circular convolution with trivial modification of the proofs provided except where explicitly noted otherwise.

Continuous Time Convolution Properties

Associativity

The operation of convolution is associative. That is, for all continuous time signals f1,f2,f3f1,f2,f3 the following relationship holds.

f 1 * ( f 2 * f 3 ) = ( f 1 * f 2 ) * f 3 f 1 * ( f 2 * f 3 ) = ( f 1 * f 2 ) * f 3
1

In order to show this, note that

( f 1 * ( f 2 * f 3 ) ) ( t ) = - - f 1 ( τ 1 ) f 2 ( τ 2 ) f 3 ( ( t - τ 1 ) - τ 2 ) d τ 2 d τ 1 = - - f 1 ( τ 1 ) f 2 ( ( τ 1 + τ 2 ) - τ 1 ) f 3 ( t - ( τ 1 + τ 2 ) ) d τ 2 d τ 1 = - - f 1 ( τ 1 ) f 2 ( τ 3 - τ 1 ) f 3 ( t - τ 3 ) d τ 1 d τ 3 = ( ( f 1 * f 2 ) * f 3 ) ( t ) ( f 1 * ( f 2 * f 3 ) ) ( t ) = - - f 1 ( τ 1 ) f 2 ( τ 2 ) f 3 ( ( t - τ 1 ) - τ 2 ) d τ 2 d τ 1 = - - f 1 ( τ 1 ) f 2 ( ( τ 1 + τ 2 ) - τ 1 ) f 3 ( t - ( τ 1 + τ 2 ) ) d τ 2 d τ 1 = - - f 1 ( τ 1 ) f 2 ( τ 3 - τ 1 ) f 3 ( t - τ 3 ) d τ 1 d τ 3 = ( ( f 1 * f 2 ) * f 3 ) ( t )
2

proving the relationship as desired through the substitution τ3=τ1+τ2τ3=τ1+τ2.

Commutativity

The operation of convolution is commutative. That is, for all continuous time signals f1,f2f1,f2 the following relationship holds.

f 1 * f 2 = f 2 * f 1 f 1 * f 2 = f 2 * f 1
3

In order to show this, note that

( f 1 * f 2 ) ( t ) = - f 1 ( τ 1 ) f 2 ( t - τ 1 ) d τ 1 = - f 1 ( t - τ 2 ) f 2 ( τ 2 ) d τ 2 = ( f 2 * f 1 ) ( t ) ( f 1 * f 2 ) ( t ) = - f 1 ( τ 1 ) f 2 ( t - τ 1 ) d τ 1 = - f 1 ( t - τ 2 ) f 2 ( τ 2 ) d τ 2 = ( f 2 * f 1 ) ( t )
4

proving the relationship as desired through the substitution τ2=t-τ1τ2=t-τ1.

Distribitivity

The operation of convolution is distributive over the operation of addition. That is, for all continuous time signals f1,f2,f3f1,f2,f3 the following relationship holds.

f 1 * ( f 2 + f 3 ) = f 1 * f 2 + f 1 * f 3 f 1 * ( f 2 + f 3 ) = f 1 * f 2 + f 1 * f 3
5

In order to show this, note that

( f 1 * ( f 2 + f 3 ) ) ( t ) = - f 1 ( τ ) ( f 2 ( t - τ ) + f 3 ( t - τ ) ) d τ = - f 1 ( τ ) f 2 ( t - τ ) d τ + - f 1 ( τ ) f 3 ( t - τ ) d τ = ( f 1 * f 2 + f 1 * f 3 ) ( t ) ( f 1 * ( f 2 + f 3 ) ) ( t ) = - f 1 ( τ ) ( f 2 ( t - τ ) + f 3 ( t - τ ) ) d τ = - f 1 ( τ ) f 2 ( t - τ ) d τ + - f 1 ( τ ) f 3 ( t - τ ) d τ = ( f 1 * f 2 + f 1 * f 3 ) ( t )
6

proving the relationship as desired.

Multilinearity

The operation of convolution is linear in each of the two function variables. Additivity in each variable results from distributivity of convolution over addition. Homogenity of order one in each varible results from the fact that for all continuous time signals f1,f2f1,f2 and scalars aa the following relationship holds.

a ( f 1 * f 2 ) = ( a f 1 ) * f 2 = f 1 * ( a f 2 ) a ( f 1 * f 2 ) = ( a f 1 ) * f 2 = f 1 * ( a f 2 )
7

In order to show this, note that

( a ( f 1 * f 2 ) ) ( t ) = a - f 1 ( τ ) f 2 ( t - τ ) d τ = - ( a f 1 ( τ ) ) f 2 ( t - τ ) d τ = ( ( a f 1 ) * f 2 ) ( t ) = - f 1 ( τ ) ( a f 2 ( t - τ ) ) d τ = ( f 1 * ( a f 2 ) ) ( t ) ( a ( f 1 * f 2 ) ) ( t ) = a - f 1 ( τ ) f 2 ( t - τ ) d τ = - ( a f 1 ( τ ) ) f 2 ( t - τ ) d τ = ( ( a f 1 ) * f 2 ) ( t ) = - f 1 ( τ ) ( a f 2 ( t - τ ) ) d τ = ( f 1 * ( a f 2 ) ) ( t )
8

proving the relationship as desired.

Conjugation

The operation of convolution has the following property for all continuous time signals f1,f2f1,f2.

f 1 * f 2 ¯ = f 1 ¯ * f 2 ¯ f 1 * f 2 ¯ = f 1 ¯ * f 2 ¯
9

In order to show this, note that

( f 1 * f 2 ¯ ) ( t ) = - f 1 ( τ ) f 2 ( t - τ ) d τ ¯ = - f 1 ( τ ) f 2 ( t - τ ) ¯ d τ = - f 1 ¯ ( τ ) f 2 ¯ ( t - τ ) d τ = ( f 1 ¯ * f 2 ¯ ) ( t ) ( f 1 * f 2 ¯ ) ( t ) = - f 1 ( τ ) f 2 ( t - τ ) d τ ¯ = - f 1 ( τ ) f 2 ( t - τ ) ¯ d τ = - f 1 ¯ ( τ ) f 2 ¯ ( t - τ ) d τ = ( f 1 ¯ * f 2 ¯ ) ( t )
10

proving the relationship as desired.

Time Shift

The operation of convolution has the following property for all continuous time signals f1,f2f1,f2 where STST is the time shift operator.

S T ( f 1 * f 2 ) = ( S T f 1 ) * f 2 = f 1 * ( S T f 2 ) S T ( f 1 * f 2 ) = ( S T f 1 ) * f 2 = f 1 * ( S T f 2 )
11

In order to show this, note that

S T ( f 1 * f 2 ) ( t ) = - f 2 ( τ ) f 1 ( ( t - T ) - τ ) d τ = - f 2 ( τ ) S T f 1 ( t - τ ) d τ = ( ( S T f 1 ) * f 2 ) ( t ) = - f 1 ( τ ) f 2 ( ( t - T ) - τ ) d τ = - f 1 ( τ ) S T f 2 ( t - τ ) d τ = f 1 * ( S T f 2 ) ( t ) S T ( f 1 * f 2 ) ( t ) = - f 2 ( τ ) f 1 ( ( t - T ) - τ ) d τ = - f 2 ( τ ) S T f 1 ( t - τ ) d τ = ( ( S T f 1 ) * f 2 ) ( t ) = - f 1 ( τ ) f 2 ( ( t - T ) - τ ) d τ = - f 1 ( τ ) S T f 2 ( t - τ ) d τ = f 1 * ( S T f 2 ) ( t )
12

proving the relationship as desired.

Differentiation

The operation of convolution has the following property for all continuous time signals f1,f2f1,f2.

d d t ( f 1 * f 2 ) ( t ) = d f 1 d t * f 2 ( t ) = f 1 * d f 2 d t ( t ) d d t ( f 1 * f 2 ) ( t ) = d f 1 d t * f 2 ( t ) = f 1 * d f 2 d t ( t )
13

In order to show this, note that

d d t ( f 1 * f 2 ) ( t ) = - f 2 ( τ ) d d t f 1 ( t - τ ) d τ = d f 1 d t * f 2 ( t ) = - f 1 ( τ ) d d t f 2 ( t - τ ) d τ = f 1 * d f 2 d t ( t ) d d t ( f 1 * f 2 ) ( t ) = - f 2 ( τ ) d d t f 1 ( t - τ ) d τ = d f 1 d t * f 2 ( t ) = - f 1 ( τ ) d d t f 2 ( t - τ ) d τ = f 1 * d f 2 d t ( t )
14

proving the relationship as desired.

Impulse Convolution

The operation of convolution has the following property for all continuous time signals ff where δδ is the Dirac delta funciton.

f * δ = f f * δ = f
15

In order to show this, note that

( f * δ ) ( t ) = - f ( τ ) δ ( t - τ ) d τ = f ( t ) - δ ( t - τ ) d τ = f ( t ) ( f * δ ) ( t ) = - f ( τ ) δ ( t - τ ) d τ = f ( t ) - δ ( t - τ ) d τ = f ( t )
16

proving the relationship as desired.

Width

The operation of convolution has the following property for all continuous time signals f1,f2f1,f2 where Duration(f)Duration(f) gives the duration of a signal ff.

Duration ( f 1 * f 2 ) = Duration ( f 1 ) + Duration ( f 2 ) Duration ( f 1 * f 2 ) = Duration ( f 1 ) + Duration ( f 2 )
17

. In order to show this informally, note that (f1*f2)(t)(f1*f2)(t) is nonzero for all tt for which there is a ττ such that f1(τ)f2(t-τ)f1(τ)f2(t-τ) is nonzero. When viewing one function as reversed and sliding past the other, it is easy to see that such a ττ exists for all tt on an interval of length Duration(f1)+Duration(f2)Duration(f1)+Duration(f2). Note that this is not always true of circular convolution of finite length and periodic signals as there is then a maximum possible duration within a period.

Convolution Properties Summary

As can be seen the operation of continuous time convolution has several important properties that have been listed and proven in this module. With slight modifications to proofs, most of these also extend to continuous time circular convolution as well and the cases in which exceptions occur have been noted above. These identities will be useful to keep in mind as the reader continues to study signals and systems.