circular convolution and Linear convolution

DIGITAL SIGNAL
PROCESSING
MODULE- 2
Circular Convolution,
Linear Filtering
By
Ms.J.Shiny Christobel, AP/ECE,
Sri Ramakrishna Institute of Technology, Coimbatore 1

X1(n) is fixed & marked in Anti Clockwise Direction
X2(n) is marked in clockwise direction . But
Rotation in Anti Clockwise Direction
Solution:
2
9-May-24

X1(n)
3
X2(n)
1
-1
3
-2
1
-1
2
3
0
0
y(0) = [1x1] + [-1x0] + [-2x0] + [3x3] + [2x-1]
= 1 + 0 + 0 + 9 - 2 = 8
Rotate x2(n) in anti-
clockwise direction
9-May-24

X1(n)
4
X2(n)
1
-1
3
-2
2
-1
3
0
0
1
y(1) = [1x2] + [-1x1] + [-2x0] + [3x0] + [3x-1]
= 2 - 1 + 0 + 0 - 3 = -2
9-May-24

X1(n)
5
X2(n)
1
-1
3
-2
3
-1
0
0
1
2
y(2) = [1x3] + [-1x2] + [-2x1] + [3x0] + [0x-1]
= 3 - 2 - 2 + 0 + 0 = -1
9-May-24

X1(n)
6
X2(n)
1
-1
3
-2
0
-1
0
1
2
3
y(3) = [1x0] + [-1x3] + [-2x2] + [3x1] + [0x-1]
= 0 - 3 - 4 + 3 + 0 = -4
9-May-24

X1(n)
7
X2(n)
1
-1
-2
0
-1
1
2
3
0
3
y(4) = [1x0] + [-1x0] + [-2x3] + [3x2] + [1x-1]
= 0 + 0 - 6 + 6 - 1 = -1
Ans:
y(n) = { 8, -2, -1, -4, -1 }
9-May-24

Linear Convolution
14
9-May-24

Circular Convolution without zero padding
15
9-May-24

Circular Convolution with zero padding
x(n) = {1,2,3,1} , length of x(n) is L = 4
h(n) = {1,1,1} , length of h(n) is M = 3
if
append (L-1) zeroes at the end of x(n)
append (M-1) zeroes at the end of h(n)
then the given sequence will be
x(n) = {1,2,3,1,0,0}
h(n) = {1,1,1,0,0,0}
16
9-May-24

Use of FFT in Linear Filtering
Methods:
(1)Overlap Add Method
(2)Overlap Save Method
 Fast Convolution
 Linear filtering Long Sequences
20
9-May-24

circular convolution and Linear convolution

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