Presentation on convolution sum dsp.pptx

CONVOLUTION SUM
COURSETEACHER: DR. MD. ZAMIL SULTAN
PROFESSOR & CHAIRMAN
DEPT. OF ELECTRICAL AND ELECTRONIC ENGINEERING, HSTU
COURSE CODE : EEE 353
COURSETITLE : DIGITAL SIGNAL PROCESSING

PRESENTED BY
• Anika Tahsin Orny
Student ID: 2002210
• Labaid Ferdous
Student ID: 2002222
• Tuhin Shuvro Mondol
Student ID: 2002231
• Md. Miju Islam
Student ID: 2002247
• Mehedi Hasan Mahim
Student ID: 2002261

Presentation
Overview
1. Convolution
2. Convolution for Continuous and Discrete
Signal
3. Equation derivation of Convolution Sum
4. Mathematical Examples:
a. Based on Equations
b. Based on NumericalValues
5. Convolution By MATLAB
6. Advantages and Limitations of Convolution

1. CONVOLUTION
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.
Convolution is used in the mathematics of many fields, such as probability and statistics. In linear systems, convolution
is used to describe the relationship between three signals of interest: the input signal, the impulse response, and the
output signal.
So, Convolution is a mathematical way of combining two signals to form a third signal in digital signal
processing.
Mathematically,
δ[n] h[n]
Input Output or response
x(n) = y(n) =
x[n] y[n]
x[n]*h[n]= y[n]
So for a given two finite length sequences x(n) and h(n). their linear convolution is defined as
y(n)=x(n)*h(n)
=
System
System
Linear System h[n]

2. CONVOLUTION FOR CONTINUOUS AND DISCRETE SIGNAL
x(t) h(t) y(t)
x(n) h(n) y(n)
For continuous Signal convolution of x(t) and h(t)
y(t)= x(t)* h(t)
=
=
For Discrete convolution of x(n) and h(n)
y(n)= x(n)* h(n)
=
=
LTI system

Hence continuous time convolution of x(t) and h(t)
y(t)= x(t)* h(t) = =
- For each value of t, we compute a different (possibly) infinite
integral.
And for Discrete convolution of x(n) and h(n)
y(n)= x(n)* h(n)= =
So discrete time definition is the continuous – time definition with
integral replaced by summation.

3. EQUATION DERIVATION OF CONVOLUTION SUM
According to the shifting property of signals, any signal can be expressed as a combination of weighted and shifted impulse
signal, i.e.
x(n) = ……………. (1)
Therefore, the impulse response of the system is,
y(n)= T[x(n)] = ……………. (2)
Or, y(n)= ……………. (3)
The expression of Eqn. (3) is known as convolution sum. The convolution sum can be represented symmetrically as,
y(n)= x(n)* h(n) ……………. (4)
Also
y(n)= = ……………. (5)
So, linear invariant system (LTI) is completely characterized by Its impulse response.

4. MATHEMATICAL EXAMPLES:
a. Based on Equations:
Recall that the impulse response for a discrete time echoing feedback system with gain a is
h[n]= ……………….(1)
and consider the response to an input signal that is another exponential
x[n]= ……………….(2)
We know that the output for this input is given by the convolution of the impulse response with the input signal
y(n)= x(n)* h(n) ……………….(3)
We would like to compute this operation by beginning in a way that minimizes the algebraic complexity of the expression. However, in this case, each
possible choice is equally simple. Thus, we would like to compute
……………….(4)
The step functions can be used to further simplify this sum. Therefore,
y[n]=0 ……………….(5)
For n<0 and
……………….(6)
For n≥0. Hence, provided ab≠ 1, we have that
y
Geometric Series for Sum,
=

Presentation on convolution sum dsp.pptx

b. Based on NumericalValues
Problem: h(n)= {1,2,1, -1} and x(n)= {1,2,3,1} compute y(n)=h(n)*x(n)
Solution: we know y(n)=h(n)*x(n)
y(n)= ……………….. (1)
1.When n=0;
y (0) =
x(k)= {1,2,3,1}
h(-k) = {-1,1,2,1}
y (0) =
=0+2+2+0
=4

2.When n=1
y (1) =
x(k)= {1,2,3,1}
h(1-k) = {-1,1,2,1}
y (1) =
=0+1+4+3+0
=8
3.When n=2
y(2)=
x(k)= {1,2,3,1}
h(2-k)={-1,1,2,1}
y(2)=
=-1+2+6+1
=8
4. When n=3
y(3)=
x(k)= {1,2,3,1}
h(3-k)={0}
y(3)=
=-2+3+2
=3
5. When n=4
y(4)=
x(k)= {1,2,3,1}
h(4-k)={0,0}
y(1)=
=-3+1
=-2

6. When n=5
y(5)=
x(k)= {1,2,3,1}
h(5-k)={0}
y(5)=
= -1
7. When n=6
y(6)=
x(k)= {1,2,3,1}
h(6-k)={0}
y(6)=
=0
8. When n=7
y(7)=
x(k)= {1,2,3,1}
h(7-k)={0}
y(7)=
=0
9. When n=-1
y(-1)=
x(k)= {1,2,3,1}
h(-1-k)={-1,1,2 ,1}
y(6)=
=1
10.When n=-2
y(-2)=
x(k)= {1,2,3,1}
h(-2-k)={-1,1,2 ,1,0}
y(-2)=
=0
11.When n=-3
y(-3)=
x(k)= {1,2,3,1}
h(-3-k)={-1,1,2 ,1,0,0}
y (-3) =
=0
y(n)= {………0,0,0,1,4,8,8,3, -2, -1,0,0,………}

5. CONVOLUTION BY MATLAB
MATLAB Script:
clc;
clear all;
close all;
n1 = 0: 1 : 3;
x1 = [ 1 2 3 1];
n2 = -1:1:2;
h1 = [ 1 2 1 -1];
Y = conv (x1, h1);
n3 = -1:1:5;
figure(1);
subplot(3,1,1);
stem(n1, x1,'blue','linewidth',2.5);
title('Input [x(n)]');
subplot(3,1,2);
stem(n2, h1,'green','linewidth',2.5);
title('Impulse Response [h(n)]');
subplot(3,1,3);
stem(n3, Y,'red','linewidth',2.5);
title('output [y(n)]');

6.ADVANTAGES AND LIMITATIONS OF CONVOLUTION
The advantages of convolution sum in digital signal processing include:
- Flexibility in signal manipulation.
- Efficiency with algorithms like FFT.
- Linearity and superposition principle.
- Frequency analysis through time-frequency domain relationship.
- Mathematical representation for linear time-invariant systems.
While convolution in digital signal processing offers numerous advantages, it
also has some limitations:
- Computational complexity.
- Finite-length effects.
- Memory requirements.
- Frequency-domain limitations.
- Challenges in implementation.

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