AMI 4622 Digital Signal Processing

Unit 6 Convolution

IntroductionWhat is Convolution?

Convolution describes how the input to a system interacts with the system to produce an output

Convolution is used in DSP to find the system response of a Linear Time-invariant System (LTI), to arbitrary inputs

Everything and anything in DSPs are involved with convolution.

If we know a systems impulse response then we can calculate, using convolution, the output from any possible input signal

This means, if we know the systems impulse, we know everything about the system

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The impulse response

The output of a system owing to the impulse input will not be a corresponding impulse, but will vary with time, passing through a maximum.

Lets see what the response will be to a single impulse input to the system

ConvolutionSingleImpulse.m

Now lets introduce a single delayed impulse input

ConvolutionSingleImpulse(k)

The impulse responseNow lets consider what the response will be to a

series impulses input to the system.ConvolutionSeriesImpulse.m

The impulse responseThis output can only be written in this way if the system

is a linear one.

By inspection it can be seen that the output is obtained multiplying the input sequence by the time reversed impulse response function

This can also be written as:

In this case it can be seen that the output is obtained multiplying the impulse response function by the time reversed input function

The impulse responseThe expression

can be expressed compactly as:

And

can be expressed compactly as:

These may be extended to waveforms of infinite duration:

and

The impulse responseExample 1:

Convolution of a sequence with a unit sample sequence

This can be re-written as:

Since: the above equation reduces to:

or in other words:

Properties of convolutionCommutative Property

Properties of convolutionAssociative Property

Properties of convolutionDistributive Property

Convolution in operationConvolutionAnimation.m

Convolution in operationTabular Method of Convolution Computation

n 0 1 2 3 4 5

x(n) x(0) x(1) x(2) x(3)

h(n) h(0) h(1) h(2)

x(0)h(0) x(1)h(0) x(2)h(0) x(3)h(0)

x(0)h(1) x(1)h(1) x(2)h(1) x(3)h(1)

x(0)h(2) x(1)h(2) x(2)h(2) x(3)h(2)

y(n) y(0) y(1) y(2) y(3) y(4) y(5)

Convolution in operationPolynomial Method

Assume the two sequences are polynomials

Let

and

Corresponding polynomials are now constructed as follows

and

Now multiplying out the two polynomials manually

Thus we have convolved the two sequences X and H