jacket transform

8/10/2019 Jacket Transform

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Jacket TransformHung-Yi Cheng()

E-mail: [email protected]

Graduate Institute of Communication Engineering

National Taiwan University, Taipei, Taiwan, ROC

1. Abstract ---------------- 2

2. Introduction

2-1. What is jacket transform ----------------- 2

2-2. Why is called jacket transform ----------------- 3

3.

The Center Weighted Hadamard Transform

3-1. What is weighted Hadamard transform (WHT) -----------------4

3-2. Fast algorithm for high order WHT -----------------4

3-3. The properties for WHT -----------------5

4. The Reverse Jacket Transform

4-1. General definition of jacket matrix -----------------7

4-2. The properties for reverse jacket transform ------------------9

4-3. Fast algorithm for jacket matrix -----------------10

5. Complex reverse jacket transform

5-1. The weighted Hadamard transform ------------------13

5.2 The complex weighted Hadamard transform ------------------14

5.3 The Complex reverse jacket transform ------------------15

6. The Applying of Jacket Matrices in CDMA ------------------16

7. Conclusions ------------------18

8. Reference ------------------19


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1. Abstract

Recently, an interesting matrix named Jacket was proposed by Prof. Moon Ho Lee

(1989, 2000, 2001, IEEE Trans. CAS), it is extended from Walsh Hadamard (1893),

but includes both orthogonal and non-orthogonal cases. In particular, we can present

the Jacket transform and a simple factor decomposition, which is used to develop

either a fast algorithm or communications for the Jacket transform. The matrix

decomposition is of the form of the matrix products Hadamard matrices and

successively lower order coefficient matrices. This decomposition very clearly leads

to a block circular sparse matrix factorization of the Jacket matrix. The main property

of Jacket is that the inverse matrices of its elements can be obtained very easily and

have a special structure. It is useful to construct space time codes, fast algorithm and

so on.

2. Introduction

2-1 What is Jacket martr ix

Inmathematics,a jacket matrix is asquare matrix [ ] of order n if its entries

are non-zero andreal,complex,or from afinite field,and

where is theidentity matrix,and

(

)

where T denotes thetranspose of the matrix.

In other words, the inverse of a jacket matrix is determined its element-wise or

block-wise inverse. The inverse form which is only from the entrywise inverse na d

transpose. The definition above may also be expressed as:

11 1

1

[ ]

n

m m

xn

n

m

J J

J

J J

11 1

1

1

1/ 1/

[1

]

1/ 1/

n

mxn

m mn

J J

J

JC

J
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Finite_fieldhttp://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Finite_fieldhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Mathematics


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The jacket matrix is a generalization of theHadamard matrix,also it is

aDiagonalblock-wise inverse matrix.

2-2 Why is called jacket matrix

The basic idea was motivated by the cloths of Jacket. As our two sided Jacket is

inside and outside compatible, at least two positions of a Jacket matrix are replaced by

their inverse; these elements are changed in their position and are moved, for example,

from inside of the middle circle to outside or from to inside without loss of sign.
http://en.wikipedia.org/wiki/Hadamard_matrixhttp://en.wikipedia.org/wiki/Diagonalhttp://en.wikipedia.org/wiki/File:Had_otr_jac.pnghttp://en.wikipedia.org/wiki/File:Had_otr_jac.pnghttp://en.wikipedia.org/wiki/Diagonalhttp://en.wikipedia.org/wiki/Hadamard_matrix


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3. The Center Weighted Hadamard Transform

3-1. What is weighted Hadamard transform (WHT)

The center weighted Hadamard transform (WHT) is defined. This transform is

similar to the Hadamard transform (HT) in that it requires only real operations. The

WHT weights the region of mid-spatial frequencies of the signal more than the HT. A

simple factonzation of the weighted Hadamard matrix is used to develop a fast

algorithm for the WHT. The matrix decomposition is of the form of the Kronecker

products of fundamental Hadamard matrices and successively lower order weighted

Hadamard matrices.

Let the Hadamard and the Weighted Hadamard matrices of order 2kN be

denoted by [ ]NH and [ ]NWH .

The lowest order WH matrix is of size (4 * 4) and is defined as follows:

4

1 1 1 1 1 1 1 1 4 0 0 0

1 2 2 1 1 1 1 1 0 6 2 01[ ]

1 2 2 1 1 1 1 1 0 2 6 04

1 1 1 1 1 1 1 1 0 0 0 4

WH

And the inverse of 4[ ]WH

4

1

2 2 2 2 1 1 1 1 4 0 0 0

2 1 1 2 1 1 1 1 0 3 1 01 1[ ]

2 1 1 2 1 1 1 1 0 1 3 08 16

2 2 2 2 1 1 1 1 0 0 0 4

WH

3-2. Fast algorithm for high order WHT

As with the Hadamard matrix, a recursive relation governs the generation of high

order [WH] matrices, i.e.,

/2 2[ ] [ ] [ ]N NWH WH H (1)

Where is the Kronecker product and 2[ ]H is the lowest order Hadamard matrix as

following.

2 1 1[ ]1 1

H


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Then we get the N=8

8 4 2[ ] [ ] [ ]

1 1 1 1

1 2 2 1 1 11 2 2 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 2 2 2 2 1 1

1 1 2 2 2 2 1 1

1 1 2 2 2 2 1 1

1 1 2 2 2 2 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

WH WH H

3-3. The properties for WHT

In order to develop a fast general algorithm for the WHT, define a weighed

coefficient matrix [ ]NRC by

[ ] [ ] [ ]N N NRC H WH (2)

and the [ ]NRC is sparse matrix. We can get the properties with [RC]

2

/2 2 /2 2

/2 2 2

/2 2

[ ] ([ ] [ ] )([ ] [ ] )

([ ] [ ] ) ([ ] [ ] )

[ ] 2[ ]

N N N

N N

N

RC H H WH H

H WH H H

RC I

the inverse for RC matrix, that is

/2 21[ ] [ ] [ ]2

N NRC RC I (3)

Where 2[ ]I is 2x2 identity matrix.

Since 1 1

[ ] [ ]N NH HN

, we find the higher order WH matrix by these fast algorithm.

1 1 1

1[ ] [ ]

[ ] [ ] [ ]

[ ]N N N

N N N

WH HN

WH N R H

RC

C

(4)

For example N=4 as below


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4

4 4

1 1 1 1 1 1 1 1 4 0 0 0

1 2 2 1 1 1 1 1 0 6 2 01[ ]

1 2 2 1 1 1 1 1 0 2 6 04

1 1 1 1 1 1 1 1 0 0 0 41

[ ]4

[ ]H R

H

C

W

The simple recursive relationship in (1) (2) (3) and (4) can be used to formulate a high

order [WH] as the fast algorithm.

As an example for N=8 can be represented as

8

2 4

8 8

4 2

4 4 2

1[ ] [ ] [ ]8

1( [ ] [ ] )([ ] 2[ ] )8

1([ ] [ ] ) 2[ ]

8

WH H RC

H H RC I

RC H H

The higher order of WH can be derived by the lower order of WH and lower order of

Hadamard matrix.

After calculation , we get the 8[ ]WH from the fast algorithm.

8

1 1 1 1 4 0 0 0

1 1 1 1 0 6 2 0 1 11[ ] 2

1 1 1 1 0 2 6 0 1 18

1 1 1 1 0 0 0 4

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 11 1 2 2 2 2 1 1

1 1 2 2 2 2 1 1

1 1 2 2 2 2 1 1

1 1 2 2 2 2 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

WH


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The figure is show the fast algorithm of WH flow.

Using the algebra of Kronecker product, it can be shown that

1 1 1

/2 /2 2

1

/2 / 2 2

1[ ] [ ] [ ] [ ] [ ]

([ ] [ ] 2[ ] )*

1([ ] [ ] [

[ ]

] )2

[ ]

N N N N N N

N N

N N

N

WH WH H N RC H N

H RC H

H RC H

RC

I

Property : Products WH and inverse WH is identity matrix.

4. The Reverse Jacket Transform

4-1. Basical definition of jacket matrix

A WH matrix is a slight modification of a Hadamard matrix. The WH, however,

weights the region of mid-spatial frequencies of the signal more than the Hadamard

transform.The reverse jacket transform are a generalized conventional weighted

Hadamard transform [WH] and Hadamard transform [H].The reverse jacket transform

having geometric structure property.


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Now we consider the 4*4 [WH] matrix

4

1 1 1 1

1 2 2 1[ ]

1 2 2 1

1 1 1 1

WH

,2

1 1[ ]

1 2WH

The upper left 2x2 block subsampling given is 2[ ]WH

. We can find some a regular

recursive geometric structure.

Compare with weighted Hadamard transform, we define several matrices for [RJ]

2 1[ ] Ta b

RJ Mb c

The matrix 1M

consists of three elements (a, b and c), which are all non-zero and

take of 2n , n = 0, 1,2. We define other matrices to get the [RJ].

2

b aM

c b

,

4

c bM

b a

Finally, we obtain 4[ ]RJ

, defined by

1 2

4

2 4

[ ]T

a b b a

M M b c c bRJ

M M b c c b

a b b a

If we assume 1M is invertible, we define the inverse RJ matrix as follows:

21

2

2

( * )[ ]

det([ ] )

c bsignum a cRJ

b aRJ

, where

1, 0

( ) 0, 0

1, 0

x

signum x x

x

And we also define the inverse 4[ ]RJ as follow,

1

4

1/ 1/ 1/ 1/

1/ 1/ 1/ 1/1[ ]

1/ 1/ 1/ 1/41/ 1/ 1/ 1/

a b b a

b c c bRJ

b c c ba b b a


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We can see the elements position in the forward matr ix can be replaced by its

inverse matr ix, and the signs are not changed between the matr ix and its inverse.

The figure is show the key idea for jacket transform.

4-2. The properties for reverse jacket transform

We denote the 2[ ] kRJ

for the reverse jacket matrix with (2k * 2k). There are two

properties to the reverse jacket matrix.

Property 1 :

2[ ] kRJ is a symmetric matrix, i.e.,

2 2[ ] [ ]k k

TRJ RJ

where T denotes transpose.

Property 2 :

If2

[ ] kRJ is orthogonal matrix, we can derive the equation as following

2 2 2[ ] [ ] 2 [ ]k k k

T kRJ RJ I

But is a different case from a non-orthogonal matrix,

2

2

22[ ] [ ] [ ]k k

T kRJ RJ RJ


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Property 3 :

A matrix is called the reverse jacket matrix if

1 *

2 2 2 2[ ] [ ] [ ] [ ]k k k k RJ H RJ H

, where k belongs to integer N. All its components are 2k , n = 0,1,2.

According to the properties, it is easy found to be equivalent to the conditions.

* 1 *

2 2 2 2

1

2 2

2

[ ] [ ] [ ] [ ]

1[ ] [ ]

det([ ] )

k k k k

k k

k

RJ H RJ H

RJ RJRJ

, where k>2

So the inverse reverse jacket transform is a function of jacket transform.

4-3. Fast algorithm for jacket matrix

Now we start to get fast algorithm for the high order of reverse jacket transform. The

fast algorithm of RJ is similar as the weighted Hadamard transform and can be

expressed recursively in terms of Kronecker product.

1 22 2 2

1 1[ ] [ ] [ ] [ ]

1 1K k kRJ RJ H RJ

, where k>2

For example, as 8[ ]RJ is an RJ matrix, then it is

2 2 2 2

2 2 2 2

8

2 2 2 2

2 2 2 2

[ ] [ ] [ ] [ ]

[ ] 2[ ] 2[ ] [ ][ ]

[ ] 2[ ] 2[ ] [ ]

[ ] [ ] [ ] [ ]

H H H H

H H H HRJ

H H H H

H H H H

In similar fashion as weighed Hadamard transform, we note that

1 1 12 2 2

2 22 2

22

[ ] [ ] [ ]

([ ] [ ] )([ ] [ ] )

[ ] 2[ ]

K k k

k k

k

RC H RJ

H H RJ H

RC I

Since 12 2

1[ ] [ ]

2k kk

H H , we have from


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1 1 112 2 2

1[ ] [ ] [ ]

2k k kk

RJ H RC

Therefore, the fast RJ transform is simply the fast Hadamard transform followed by a

spars matrix2

1 [ ]2

kk RC .

For example, a=b=1 and c=2, the RJ is the same as WH

2 2

1 1 1 1 2 11[ ]

1 2 1 1 0 32

a bM RJ

b c

And its inverse matrix

1

22 1 1 1 3 01 1[ ]1 1 1 1 1 23 6

RJ

Also, we have

4

1 1 1 1

1 2 2 1[ ]

1 2 2 1

1 1 1 1

RJ

, 14

2 2 2 2

2 1 1 21[ ]

2 1 1 28

2 2 2 2

RJ

In geometric structure, the outside of 4[ ]RJ is corresponding to inside of

1

4[ ]RJ

and vice versa.

The algorithm for the fast reverse jacket transform is similar as the WH. The FRJT


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can be derived by decomposing [RJ] into a product of sparse matrices given each

rows/columns with only two non-zero elements.

The fast RJ transform for N=8,

8 8 8

4 2 4 2

4 2

1[ ] [ ] [ ]

8

1([ ] [ ] )([ ] 2[ ] )

8

1([ ] 2[ ] )

2

RJ H RC

H H RC I

RJ H

In a similar manner as the fast RJ transform, we get the inverse of [RJ]

1 1

8 4 2 4 2

1

4 2

1[ ] ([ ] [ ] )([ ] [ ] )

2

1[ ] [ ]

2

RJ H H RC I

RJ H

Table I : The inverse reverse jacket transform (forward)


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Table II : The reverse jacket transform

The table I and II show the RJ matrices have a geometric structure.

5.Complex reverse jacket transform

5-1. The weighted Hadamard transform

The Hadamard transform (WHT) and discrete are used widely in signal processing.

Variations of these two transforms called center weighted Hadamard transform

(CWHT) and complex reverse jacket transform (CRJT) have been reported and

their applications .Now we discuss about anther from for WHT in order to derive the

complex weighted Hadamard transform.

For 2rn length real vector 0 1 1( , , , )na a a the transform vector is a n-length

real vector 0 1 1( , , , )nA A A given by

0 0

1 1

1

1 1

[ ]n

n n

A a

A aH

A a


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that is,

,1

0

( 1) j i

n

i

j iA a

, j=1 ~ n-1

Where the modulo-2 inner product is given by

1 1 2 2 0 0,

r r r r j i j i j i j i

where denotes modulo two addition.

The inverse transform is given by

,1

0

1( 1) j i

n

i

j ia An

, j=1 ~ n-1

5-2 The center weighted Hadamard transform

The CWHT is obtained by weighting the center portion of the transform matrix for

2rn length real vector and is given by

1 2 1 2

1) )

0

( (,( 1) ( ) , 0,1,2, , 1r r r r

ni ji jj i

j

i

iA a j n

where is any real numberis the weight. The inverse transform is given by

1 2 1 2

1) )

0

( (, 1

( 1) ( ) , 0,1,2, , 1

r r r r

ni jj i j

i

i

ija A j n

We can know that for 1 the CWHT is same as the WHT. The 4x4 and 8x8 CWHT

matrices corresponding to the weight are

4

1 1 1 1

1 1[ ]

1 1

1 1 1 1

W

and

8

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

W


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5-3 The Complex reverse jacket transform

Both the CWHT and the CRJT include the WHT as a special case. The CRJT uses

the complex number 1j which is a fourth root of unity by replacing the weight

and this CRJT is orthogonal matrix.

The 4x4 and 8x8 CRJT are shown below as

4

1 1 1 1

1 1[ ]

1 1

1 1 1 1

j jC

j j

and

8

1 1 1 1 1 1 1 11 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

j j j j

j j j jC

j j j j

j j j j

The following diagram is shown the different matrix and function with DFT,DCT

Hadamard and Jacket transform.


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6. The Applying of Jacket Matrices in CDMA

The Jacket transform can be an orthogonal matrix with highly practical value for

wireless communication. The computation of the transform of a signal consists of

additions and subtractions of the signal samples. The Jacket transform is also the

non-sinusoidal orthogonal transforms. Using the orthogonal Jacket transform, we can

apply this matrix to sequence generator in code-division multiple access (CDMA)

spread-spectrum communication.

Consider the general 4x4 Jacket matrix is

4[ ]

a b b ab c c b

RJb c c b

a b b a

Its inverse matrix is

1

4

1/ 1/ 1/ 1/

1/ 1/ 1/ 1/1[ ]

1/ 1/ 1/ 1/4

1/ 1/ 1/ 1/

a b b a

b c c bRJ

b c c b

a b b a

and we have known that

1

4 4 4[ ] [ ] [ ]RJ RJ I

At the transmitter, assume we send the massage (0 0 1 1) at the same

time. Choosing the Jacket codes used to modulate the signal these CDMA

systems. Since this matrix is orthogonal and easily to inverse matrix. We can

reconstruct the massage easily with the inverse Jacket transform.

As the figure 1, the stations transmit the signal to the multiplexer which is

composed with Jacket matrix. The TX transmit the data with summing the signals

through the Jacket transform.


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Figure 1: Jacket matrix as the multiplexer in CDMA system.

The signal transmitting from the TX is0 1

2 2 1

2 2 1

0 1

a b b a

b c b c c b

b c b c c b

a b b a

Assume we received the signals without noise and the demultiplexer in CDMA

system is the inverse Jacket transform. The received signal multiplex the each low of

Jacket matrix. Because the Jacket transform is orthogonal matrix, we can get the

message perfectly as following step.

Figure 2: Jacket matrix as the demultiplexer in CDMA system.


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We received the signal (0, -2b+2c, -2b-2c, 0) from TX with Jacket transform.

Through the demultiplexer, the signal became

1 1/ 1/ 1/ 1/ 0

1 1/ 1/ 1 / 1/ 2 21

1 1/ 1 / 1/ 1/ 2 24

1 1/ 1/ 1/ 1/ 0

a b b a

b c c b b c

b c c b b c

a b b a

We can reconstruct the message perfectly with the Jacket transform.

This is very simple example for using the Jacket transform.

7. ConclusionsThe reserve Jacket matrix is a generalized form of the Weighted Hadamard and

Hadamard. The matrix has a recursive structure and symmetric characteristics.

The elements positions in the forward matrix canbe replaced by its inverse matrix,

and the signs are not changed between the matrix and its inverse. The Hadamard

matrix is a special case of the Jacket matrix. The fast transform algorithm is the

matrix decomposition of the Hadamard matrices and succesively lower order reverse

Jacket transform. The [ ]NRC sparse matrix of [ ]NRJ leads to very clear

decomposition. Using the orthogonal Jacket transform, we can apply the matrix towireless communication such as CDMA.


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8. Reference

[1]M.H. Lee, The Center Weighted Hadamard Transform, IEEE Trans.1989 AS-36,

(9),pp.1247-1249.

[2]S.-R.Lee and M.H.Lee, On the Reverse Jacket Matrix for Weighted HadamardTransform, IEEE Trans. on Circuit Syst.II, vol.45.no.1, pp.436-441,Mar.1998.

M.H. Lee, A New Reverse Jacket Transform and its Fast Algorithm, IEEE Trans.

Circuits Syst.-II , vol 47, pp.39-46, 2000.

[3] M.H. Lee and B.S. Rajan, A Generalized Reverse Jacket Transform, IEEE Trans.

Circuits Syst. II, Analog Digit. Signal Process., vol. 48 no.7 pp 684-691, 2001.

J. Hou, M.H. Lee and J.Y. Park, New Polynomial Construction of Jacket

Transform, IEICE Trans. Fundamentals, vol. E86-A no. 3, pp.652-659, 2003.

[4]W.P. Ma and M. H. Lee, Fast Reverse Jacket Transform Algorithms, Electronics

Letters, vol. 39 no. 18 , 2003.

[5]Moon Ho Lee, Ju Yong Park, and Jia Hou,Fast Jacket Transform Algorithm Based

on Simple Matrices Factorization, IEICE Trans. Fundamental, vol.E88-A, no.8,

Aug.2005.

[6]Moon Ho Lee and Jia Hou, Fast Block Inverse Jacket Transform, IEEE Signal

Processing Letters, vol.13. No.8, Aug.2006.

[7] Moon Ho Lee, and Ken.Finlayson, A Simple Element Inverse Jacket Transform

Coding, Information Theory Workshop 2005, ITW 2005, Proc. of IEEE ITW 2005,

28.Aug-1.Sept., New Zealand, also IEEE Signal Processing Letters, vol. 14 no. 5,

May 2007

[8]M. H. Lee,A new reverse jacket transform and its fast algorithmIEEE Trans.

Circuits Syst.II, vol. 47, no.1, pp.39-46, Jan.2000.

jacket transform

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