Walsh transform

Seminar Topic
on
Walsh Transform
Presented to: Presented By:
Dr. Amod Kumar Sachin Maithani
Electronics and Communication ME(Regular)
Engineering Department 202604 1

Table of Content
1.Introduction
2. Walsh Transform
3.Example of Walsh Transform
4. Walsh Transform Example in Matlab
4. Application of Walsh Transform
5.Refrences
2

INTRODUCTION
• Remember that the Fourier transform is based on trigonometric terms.
• The Walsh transform consists of basis functions whose values are only 1
and -1.
• They have the form of square waves.
• These functions can be implemented more efficiently in a digital
environment than the exponential basis functions of the Fourier
transform.
3

WALSH TRANSFORM
• Based on Hadamrad Transformation (HT)
• Walsh transform is just a sequence ordered hadamard transform.
• sequence means, the no. of sign changes in a row.
• non-sinusoidal, orthogonal transformation
• It is focus on the count the sign change in the (H matrix).
• WT can be used in many different applications, such as power spectrum analysis, filtering,
processing speech and medical signals, multiplexing and coding in communications, characterizing
non-linear signals, solving non-linear differential equations, and logical design and analysis. 4

1-D Walsh Transform
• We define now the 1-D Walsh transform as follows:
• The above is equivalent to:
• The transform kernel values are obtained from:
 
N1
N x0


(u) 

(1)
n1
i0
W(u)  b (x)bn1i
1 i
f (x)
1 N1
W (u)
n1
bi (x)bn1i (u)
 f (x)(1)i1
N x0
(1)




(1)
T(u,x) T(x,u) 
n1
i1
n1i
 i
n1
bi (x)bn1i (u)
N i0 
b (x)b (u)
1
N
1
5

2-D Walsh Transform
• We define now the 2-D Walsh transform as a straightforward
extension of the 1-D transform:
  
(u)b (y)b


(v) 

(1)
n1
i0
W(u,v)  n1i
b (x)b
N1N1
n1i
1 i
i
f (x, y)
N x0 y0
1 N1N1
W (u,v)
n1
(bi (x)bn1i (u)bi (y)bn1i (v))
  f (x, y)(1)i1
N x0 y0
6

1-D Inverse Walsh Transform
• Base on the last equation of the previous slide we can show that the
Inverse Walsh transform is almost identical to the forward transform!
• The above is again equivalent to
• The array formed by the inverse Walsh matrix is identical to the one
formed by the forward Walsh matrix apart from a multiplicative factor
N.


(u) 

x0
n1
i0
b (x)bn1i
i
 W(u)(1)
N1
f (x) 
N1
n1
bi (x)bn1i (u)
f (x)   W(u)(1)i1
x0
7

2-D Inverse Walsh Transform
• We define now the Inverse 2-D Walsh transform. It is identical to the
forward 2-D Walsh transform!
(u)b (y)b


(v) 
n1
i0
n1i
b (x)bn1i i
i
 W(u,v)(1)
x0 y0
f (x, y) 
N 1 N1
N 1 N 1
f (x, y)
n1
(bi (x)bn1i (u)bi (x)bn1i (u))
 W(u,v)(1)i1
x0 y0
8

Implementation of the 2-D Walsh Transform
• The 2-D Walsh transform is separable and symmetric.
• Therefore it can be implemented as a sequence of two 1-D Walsh transforms, in
a fashion similar to that of the 2-D DFT.
9

H =
1 1 1 1
0 0 0 = 0
For first row
Walsh Matrix Formation
10

1 -1 1 -1
The second row =3
Third row
Fourth row
1 1 -1 -1
0 1 0
1 -1 -1 1
1 0 1
= 1
= 2
1 1 1

12
The Walsh matrix build according to number of sign change
1) 1D F = W . f
2) 2D F = W . f . WT
= W . f . W
3) Symmetric

Example
f = { 1, 2, 0, 3} Find WT
F = =
13

f =
Find WT
1) 2D
2) F = W . f . W
3) Sym.
14
Example
F =

Walsh Transform Example in Matlab
16

Application of Walsh transform
• Speech recognition
• medical and biological
• image processing
• digital holography
17

References:-
• Google Wikipedia
• You tube
• Walsh, J.L. (1923). "A closed set of normal orthogonal functions“. Amer. J. Math. 45 (1): 5–
24. doi:10.2307/2387224. JSTOR 2387224. S2CID 6131655.
18

Walsh transform

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