<< Chapter < Page Chapter >> Page >
This module discusses the properties of continuous time convolution.

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 x 1 , x 2 , x 3 the following relationship holds.

x 1 * ( x 2 * x 3 ) = ( x 1 * x 2 ) * x 3

In order to show this, note that

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

proving the relationship as desired through the substitution τ 3 = τ 1 + τ 2 .

Commutativity

The operation of convolution is commutative. That is, for all continuous time signals x 1 , x 2 the following relationship holds.

x 1 * x 2 = x 2 * x 1

In order to show this, note that

( x 1 * x 2 ) ( t ) = - x 1 ( τ 1 ) x 2 ( t - τ 1 ) d τ 1 = - x 1 ( t - τ 2 ) x 2 ( τ 2 ) d τ 2 = ( x 2 * x 1 ) ( t )

proving the relationship as desired through the substitution τ 2 = t - τ 1 .

Distributivity

The operation of convolution is distributive over the operation of addition. That is, for all continuous time signals x 1 , x 2 , x 3 the following relationship holds.

x 1 * ( x 2 + x 3 ) = x 1 * x 2 + x 1 * x 3

In order to show this, note that

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

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 variable results from the fact that for all continuous time signals x 1 , x 2 and scalars a the following relationship holds.

a ( x 1 * x 2 ) = ( a x 1 ) * x 2 = x 1 * ( a x 2 )

In order to show this, note that

( a ( x 1 * x 2 ) ) ( t ) = a - x 1 ( τ ) x 2 ( t - τ ) d τ = - ( a x 1 ( τ ) ) x 2 ( t - τ ) d τ = ( ( a x 1 ) * x 2 ) ( t ) = - x 1 ( τ ) ( a x 2 ( t - τ ) ) d τ = ( x 1 * ( a x 2 ) ) ( t )

proving the relationship as desired.

Conjugation

The operation of convolution has the following property for all continuous time signals x 1 , x 2 .

x 1 * x 2 ¯ = x 1 ¯ * x 2 ¯

In order to show this, note that

( x 1 * x 2 ¯ ) ( t ) = - x 1 ( τ ) x 2 ( t - τ ) d τ ¯ = - x 1 ( τ ) x 2 ( t - τ ) ¯ d τ = - x 1 ¯ ( τ ) x 2 ¯ ( t - τ ) d τ = ( x 1 ¯ * x 2 ¯ ) ( t )

proving the relationship as desired.

Time shift

The operation of convolution has the following property for all continuous time signals x 1 , x 2 where S T is the time shift operator.

S T ( x 1 * x 2 ) = ( S T x 1 ) * x 2 = x 1 * ( S T x 2 )

In order to show this, note that

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

proving the relationship as desired.

Differentiation

The operation of convolution has the following property for all continuous time signals x 1 , x 2 .

d d t ( x 1 * x 2 ) ( t ) = d x 1 d t * x 2 ( t ) = x 1 * d x 2 d t ( t )

In order to show this, note that

d d t ( x 1 * x 2 ) ( t ) = - x 2 ( τ ) d d t x 1 ( t - τ ) d τ = d x 1 d t * x 2 ( t ) = - x 1 ( τ ) d d t x 2 ( t - τ ) d τ = x 1 * d x 2 d t ( t )

proving the relationship as desired.

Impulse convolution

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

x * δ = x

In order to show this, note that

( x * δ ) ( t ) = - x ( τ ) δ ( t - τ ) d τ = x ( t ) - δ ( t - τ ) d τ = x ( t )

proving the relationship as desired.

Width

The operation of convolution has the following property for all continuous time signals x 1 , x 2 where Duration ( x ) gives the duration of a signal x .

Duration ( x 1 * x 2 ) = Duration ( x 1 ) + Duration ( x 2 )

. In order to show this informally, note that ( x 1 * x 2 ) ( t ) is nonzero for all t for which there is a τ such that x 1 ( τ ) x 2 ( t - τ ) is nonzero. When viewing one function as reversed and sliding past the other, it is easy to see that such a τ exists for all t on an interval of length Duration ( x 1 ) + Duration ( x 2 ) . 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.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask