LINEAR TIME-INVARIANT

SYSTEM

1) RESPONSE OF A CONTINOUS-TIME LTI SYSTEM

2) CONVOLUTION CT

3) RESPONSE OF DISCRETE-TIME LTI SYSTEM

4) CONVOLUTION DT

Convolution

Convolution is the most important and

fundamental concept in signal processing and

analysis. By using convolution, we can construct

the output of system for any arbitrary input

signal, if we know the impulse response of

system.

How is it possible that knowing only impulse

response of system can determine the output for

any given input signal? We will find out the

meaning of convolution.

INTRODUCTION CONVOLUTION

� Convolution is a mathematical way of combining two signals to form a third signal.

� Convolution is a formal mathematical operation, just as multiplication, addition, and integration. Addition takes two numbers and produces a third number, while convolution takes two signals and produces a third signal.

� First, the input signal can be decomposed into a set of impulses, each of which can be viewed as a scaled and shifted delta function.

� Second, the output resulting from each impulse is a scaled and shifted version of the impulse response.

� Third, the overall output signal can be found by adding these scaled and shifted impulse responses.

DefinitionThe mathematical definition of convolution in

discrete time domain is(We will discuss in discrete time domain only.)

where x[n] is input signal, h[n] is impulse

response, and y[n] is output. * denotes

convolution. Notice that we multiply the terms

of x[k] by the terms of a time-shifted h[n] and

add them up.

ApplicationsIn digital signal processing and image processing applications, the entire input function is often available for computing

every sample of the output function.

Convolution amplifies or attenuates each frequency component of the input

independently of the other components.

In digital image processing, convolution filtering plays an important role in many important algorithms in edge detection

and related processes.

Procedure for evaluating convolution

1) Folding (flip)to obtain x2(-k)

2) Shifting to obtain x2(n-k)

3) Multiplication to obtain the product

sequence x1(k).x2(n-k)

Procedure for evaluating convolution

1) Folding (flip)to obtain x2(-k)

2) Shifting to obtain x2(n-k)

3) Multiplication to obtain the product

sequence x1(k).x2(n-k)

4) Summation to obtain y(k)

EXAMPLE 1

• Determine the convolution, x3p[n] of the

circular sequences x1p[n] and x2p [n] of

length N=3 as shown below.

EXAMPLE 2:

Given are two periodic sequence,

Find the convolution y[n] for x1[n] and x2[n].

x1[n] = { …..,3,1,2,3,1,2,….} and x2[n] = {…,1,-1,1,1,-1,1,…..}