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MPSP

The document discusses Walsh transforms and related topics like Hadamard transforms. It provides information on calculating Walsh coefficients, properties of Walsh transforms, two-dimensional Walsh transforms, and the relationship between Walsh-ordered and Hadamard-ordered sequences.

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dhaval2430
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171 views32 pages

MPSP

The document discusses Walsh transforms and related topics like Hadamard transforms. It provides information on calculating Walsh coefficients, properties of Walsh transforms, two-dimensional Walsh transforms, and the relationship between Walsh-ordered and Hadamard-ordered sequences.

Uploaded by

dhaval2430
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PPTX, PDF, TXT or read online on Scribd
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Walsh Transform

Here we calculate the matrix of Walsh coefficients


Here we calculate the matrix of Walsh coefficients
Here we calculate the matrix of Walsh coefficients
Here we calculate the matrix of Walsh coefficients
We have
done it
earlier in
different
ways
Symmetry of Walsh
Think about other
transforms that you
know, are they
symmetric?
Two-Dimensional Walsh Transform
Two-dimensional Walsh
Inverse Two-dimensional Walsh
Properties of Walsh Transforms
Here is the separable 2-Dim Inverse Walsh
Example for N=4
even
odd
Discuss the importance of this figure
Hadamard Transform
We will go quickly through this material
since it is very similar to Walsh
separabl
e
Example of calculating Hadamard coefficients
analogous to what was before
Standard Trivial Functions for Hadamard
One change
two changes
2
1
3
0
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
2
1
sequency
changes sign of #
2
1
1 1
1 1
2
1
m transfor Hadamard 1)
2
1 1
1 1
1 1
1
(
(
(
(

(
(

= =
(

H
H H
H H
H H H
H
n n
n n
n n
Discrete Walsh-Hadamard
transform
Now we meet our
old friend in a new
light again!
Walsh) (1923, function Walsh the sampling by generated be can also
order Hadamard or natural
5
2
6
1
4
3
7
0
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
8
1
8
1
2 2
2 2
3

(
(
(
(
(
(
(
(
(
(
(








=
(
(

=
H H
H H
H
sequency
order or Walsh sequency
7
6
5
4
3
2
1
0
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
8
1
sequency
m transfor Hadamard - Walsh
3

(
(
(
(
(
(
(
(
(
(
(

H
{ } { }
algorithm fast 3.
tion multiplica no 2.
. 1
Properties
2 2
2 2
and of tion representa binary the are and and
log , ) , ( where
) 1 )( (
1
) (
) 1 )( (
1
) (
1
1
1
1 0
1
1
1 0
2
1
0
1
0
) , (
1
0
) , (
-

=
= = =
+ + + =
+ + + =
= =
=
=

H H H H
m m m m
k k k k
m k m k
N n m k m k b
k v
N
m u
m u
N
k v
t
n
n
n
n
i i
n
i
i i
N
k
m k b
N
m
m k b

i
(Walsh
ordered)
i(binary)
reverse
order
gray
code
decimal
(Hadamard
ordered)
0
1
2
3
4
5
6
7
000
001
010
011
100
101
110
111
000
100
010
110
001
101
011
111
000
111
011
100
001
110
010
101
0
7
3
4
1
6
2
5
Relationship between Walsh-ordered and
Hadamard-ordered

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