Digital Signal Processing I 4th Class 2020-2021 by Dr. Abbas Hussien & Dr. Ammar Ghalib
Digital Signal Processing I/
4th
Class/
2020-2021
Dr.
Abbas
Hussien
&
Dr. Ammar
Ghalib
b.
31
Digital Signal Processing I/
4th
Class/
2020-2021
Dr.
Abbas
Hussien
&
Dr. Ammar
Ghalib
Table Lookup
Method:
Matrix
by
Vector
Method:
Linear Convolution
and
Circular Convolution:
Linear
convolution:
Circular
convolution:
Note:
If
both
x
(
n
) and
x
(
n
)
are of
finite length
N and N and defined
on [0
N −1] and
[0
N
−1]
1
2
1
2
1
2
respectively,
the value
of
N needed so
that circular
and linear convolution are
the same on [0
N-
1]
is:
N ≥ N +
N
− 1
1
2
32
Digital Signal Processing I/
4th
Class/
2020-2021
Dr.
Abbas
Hussien
&
Dr. Ammar
Ghalib
Example:
If
x
(
n
) = [1 2 3 2],
and
h(
n
)
= [1 1 2].
Find
y
(
n
) such that
linear and circular
convolution are
the same.
sol.
N
= 4 + 3 – 1 =
6
Then
x
(
n
)
= [ 1 2 3 2 0 0 ]
and
h(
n
) = [ 1 1 2 0 0 0]
x
(
n
) is arranged
in
clockwise direction
,while
h(
n
) is arranged
in the opposite
clockwise direction
(bold
numbers). Each
time,
only
h(
n
)
will
be shifted with the
clockwise direction
to find
y
(
n
).
Note
:
the
reference
point
is
*
and,
the
arrows represent multiplication process. Finally, addition
process is
performed.
Example:
Use
graphical
method to find
circular
convolution of
x
(
n
)=[1
2 2]
and
x
(
n
)=[0
1 2
3].
1
2
sol.
Applying
the
equation
of
circular
convolution
1
2
3
2
0
0
1
0
0
0
2
1
2
3
0
0
1
1
0
0
0
2
2
1
2
3
0
0
1
1
1
0
0
2
0
2
2
3
0
0
1
2
1
1
0
2
0
0
2
3
0
0
1
0
2
1
1
2
0
0
2
3
0
0
1
0
0
2
1
2
1
0
2
3
0
0
1
0
0
0
2
2
1
1
x(n)
h(n)
h(n)
h(n)
h(n)
33
h(n)
h(n)
Digital Signal Processing I/
4th
Class/
2020-2021
Dr.
Abbas
Hussien
&
Dr. Ammar
Ghalib
y
(0)
=
x
(0)
x
(0)
+
x
(1)
x
(3)
+
x
(2)
x
(2) +
x
(3)
x
(1)
=
1(0)
+
2(3)
+
2(2)
+
0(1)
=
10
1
2
1
2
1
2
1
2
and so
on
Deconvolution:
The
digital Deconvolution can
be
performed
by
Iterative
Approach
,
Polynomial
Approach
,
and
Graphical Method
.
In
the
following
subsection, the
polynomial approach will
be
explained.
Polynomial
Approach:
A
long division
process
is
applied between two polynomials. For causal system,
the
remainder is
always
zero
.
If
y
(
n
) = [12 10 14 6]
and
h(
n
) = [4
2]
2
3
Then
y
= 12 + 10
x
+
14
x
+ 6
x
,
and
h = 4 + 2
x
. Applying
long division,
we
obtain
2
i/p = 3 +
x
+ 3
x
.
Then
x
(
n
) = [3 1
3]
34
This content delves into Digital Signal Processing concepts taught in the 4th class of 2020-2021 by Dr. Abbas Hussien and Dr. Ammar Ghalib. It covers topics like Table Lookup Method, Linear Convolution, Circular Convolution, practical examples, and Deconvolution techniques such as Polynomial Approach. The material includes graphical representations, equations, and explanations to aid in understanding the subject.
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Digital Signal Processing I/ 4th Class/ 2020-2021 Dr. Abbas Hussien & Dr. Ammar Ghalib b. 31
Digital Signal Processing I/ 4th Class/ 2020-2021 Dr. Abbas Hussien & Dr. Ammar Ghalib Table LookupMethod: Matrix by Vector Method: Linear Convolution and Circular Convolution: Linear convolution: Circular convolution: Note: If both x (n) and x (n) are of finite length N and N and defined on [0 N 1] and [0 N 1] 1 2 respectively, the value of N needed so that circular and linear convolution are the same on [0 N- 1 2 1 2 1] is: N N + N 1 1 2 32
Digital Signal Processing I/ 4th Class/ 2020-2021 Dr. Abbas Hussien & Dr. AmmarGhalib Example: If x(n) = [1 2 3 2], and h(n) = [1 1 2]. Find y(n) such that linear and circular convolution are the same. sol. N = 4 + 3 1 = 6 Then x(n) = [ 1 2 3 2 0 0 ] and h(n) = [ 1 1 2 0 0 0] x(n) is arranged in clockwise direction ,while h(n) is arranged in the opposite clockwise direction (bold numbers). Each time, only h(n) will be shifted with the clockwise direction to find y(n). Note: the reference point is * and, the arrows represent multiplication process. Finally, addition process is performed. 1 1 0 2 1 0 x(n) h(n) 0 2 0 3 0 2 1 1 1 1 1 2 2 2 2 0 0 0 0 1 1 1 2 0 h(n) h(n) h(n) 1 0 0 2 0 0 3 3 0 0 3 0 0 0 0 2 2 2 1 0 1 0 1 0 2 2 2 0 0 0 2 0 0 1 0 0 h(n) h(n) 1 2 0 0 1 1 3 3 0 0 3 0 1 1 2 2 2 2 Example: Use graphical method to find circular convolution of x (n)=[1 2 2] and x (n)=[0 1 2 3]. 1 2 sol. Applying the equation of circular convolution 33
Digital Signal Processing I/ 4th Class/ 2020-2021 Dr. Abbas Hussien & Dr. Ammar Ghalib y(0) = x (0) x (0) + x (1) x (3) + x (2) x (2) + x (3) x (1) = 1(0) + 2(3) + 2(2) + 0(1) = 10 1 2 1 2 1 2 1 2 and so on Deconvolution: The digital Deconvolution can be performed by Iterative Approach, Polynomial Approach, and Graphical Method. In the following subsection, the polynomial approach will be explained. PolynomialApproach: A long division process is applied between two polynomials. For causal system, the remainder is always zero. If y(n) = [12 10 14 6] and h(n) = [4 2] 2 3 Then y = 12 + 10 x + 14 x + 6 x , and h = 4 + 2 x. Applying long division, we obtain 2 i/p = 3 + x + 3 x . Then x(n) = [3 1 3] 34
