This document discusses linear time-invariant (LTI) systems and convolution. It begins by defining LTI systems and convolution for both continuous and discrete time. Convolution is described as a way to construct the output of a system given its impulse response. Applications in digital signal processing and image processing are mentioned. Convolution filtering plays an important role in edge detection and related image processing algorithms. The mathematical definition of discrete time convolution is provided. An example problem calculating outputs for different inputs using convolution is given at the end.

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. COURSE LEARNING
OUTCOME (CLO)
Upon completion of this chapter, students
should be able to:
Express the input-output relationship for
Linear time-invariant (LTI) systems.

3. 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.

4. Applications
In digital signal processing and image
processing applications, the entire input
function is often available for computing
every sample of the output function.
In digital image processing,
convolution filtering plays an
important role in many important
algorithms in edge detection and
related processes.

5. CONVOLUTION
ORIGINAL IMAGE

6. 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.

7. 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.

8. DISCRETE TIME CONVOLUTION
Convolution Sum
Convolution Integral

9. EXAMPLE:
1) The output y(t) of a continous-time LTI
system system is found to be 2e-3t u(t)
when the input x(t) is u(t).
a) Find the input response h(t) of the
system.
b) Find the output y(t) when the input x(t) is
e-tu(t).