CT convolution DT convolution

1. 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

2. 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.

3. 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.

4. 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.

6. Definition
The 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.

7. Applications
In 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.

14. 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)

24. 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)

25. EXAMPLE 1
• Determine the convolution, x3p[n] of the
circular sequences x1p[n] and x2p [n] of
length N=3 as shown below.

26. EXAMPLE 2:
Given are two periodic sequence,
x1[n] = { …..,3,1,2,3,1,2,….} and x2[n] = {…,1,-1,1,1,-1,1,…..}
Find the convolution y[n] for x1[n] and x2[n].