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htttp://en.wikipedia.org/wiki/Category:Matriceshtttp://en.wikipedia.org/wiki/Jacket:Matrix

http://en.wikipedia.org/wiki/user:leejacket

Linear Fractional

2

3

4

Jacket Basic Concept from Center Weighted Hadamard

.

111112/12/1112/12/11

1111

41,

111112211221

1111

144

WHWH

22/ HWHWH NN

11

112Hwhere

Sparse matrix and its relation to construction of center weighted Hadamard

22/ 2 IWCWHHWC NNNN

NNN WCHN

WH 1

212/

1

21 IWCWC NN 11 NNN HWCNWH

* Moon Ho Lee, “Center Weighted Hadamard Transform” IEEE Trans. on CAS, vol.26, no.9, Sept. 1989* Moon Ho Lee, and Xiao-Dong Zhang,“Fast Block Center Weighted Hadamard Transform” IEEE Trans. On CAS-I, vol.54, no.12, Dec. 2007.

5

nmijnm LJ

Tnmijnm LJ

/11

Jacket Definition: element inverse and transpose

and Simple Inverse

Examples:

111

2J

111

211

2J

1,01 2 where

11111111

1111

4 iiii

J

11111111

1111

411

4 iiii

J

111111111111

1111

4H

2

23

11

111J

2

213

11

111

31

J

1,01 32 where

6

7

Paley constr.

Example:

8

9

10

Where are Jacket matrices?

11

12

0=3

1=4

2=5

13

RMp(1,m)

14

DFT (1822) J. Fourier

DCT(1974)N. Ahmed, K.R. Rao,et.

Jacket(1989)*Moon Ho Lee

Hadamard (1893)J. Hadamard

FormulaNj

N

k

nk

ewNn

wkxnX

/2

1

0

,1...1,0

)()(

22/

2

][][][11

11][

HHH

H

nn

4][][][11111111

1111

][

22/

4

nHJJ

jjjj

J

nn

N

nmk

NC mnmN

)21(

cos2,

1,...,1,0, Nnm

Forward

wwwwF

2

23

11

111

3/2iew

82

cos

86cos

82cos

86cos

58

18

78

38

28

68

68

28

78

58

38

184

21

21

21

21

CCCCCCCCCCCCC

8cos8

iC i

kk

n

kji

1

0)1(

))(( 1212

1

0)1(

nnnn

kk

n

k jjiijiw

11111111

1111

4 wwww

J

Hadamard ghtedCenter Wei :2 Hadamard :1

ww

111111111111

1111

4H

Circle

Inverse

No Limited byCircle

Element-Wise Inverse Block-Wise Inverse Element-Wise Inverse Element-Wise Inverseor

Block-Wise Inverse

3N 4N

3/2ie

3/2ie2ie

12

2113

11

111

31

wwwwF

58

28

78

18

68

58

78

68

38

38

28

18

14

212

12

12

1

21

CCC

CCC

CCC

CCC

C

111111111111

1111

411

4H

11111/1/111/1/11

1111

411

4 wwww

J

11 11

j

j

Re

Im

Re

Im

Re

Im

Re

Im

Kronecker

Size

NNN DFTDFTDFT 2 NNN DCTDCTDCT 2 NNN HHH 2 NNN JJJ 2

n2 n2 nn 4,2 Arbitraryor p :p is prime

15

16

.0 6 1 5 3 2 1 1 0 6 1 5 3 2 16 0 6 1 5 3 25 6 0 6 1 5 34 5 6 0 6 1 52 4 5 6 0 6 16 2 4 5 6 0 61 6 2 4 5 6 0

.1

J8= mod 7, J8’= mod 7.

0 6 6 3 2 4 6 16 0 6 6 3 2 4 61 6 0 6 6 3 2 43 1 6 0 6 6 3 25 3 1 6 0 6 6 34 5 3 1 6 0 6 61 4 5 3 1 6 0 61 1 4 5 3 1 6 0

188 8

1 mod 7 4 : 2 4mod 7 121 mod 7 5 : 3 5mod 7 131 mod 7 2 : 4 2mod 7 141 mod 7 3: 3 5mod 7 151 mod 7 6 : 6 6 mod 7 161 6 mod 7

1 0(:mod 7)J J n I

J8.J8’=(8-1) I mod 7 = 0.

17

Fibonacci polynomials.

18

19

20

EXAMPLE OF

TRANSPOCE

OF MAROV & JACKET:

21

22

10111001

10111001

01110010

11100101

01234567

register3 stage2 stage1 stagemovement

Outputoutputoutputoutput

10111001

10111001

01110010

11100101

01234567

register3 stage2 stage1 stagemovement

Outputoutputoutputoutput

23

Output matrix is Jacket matrix:

and,

…

P0P1

P5

P6

1 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1

1 1 1 1 1 1 1 11

11 1 1 1 1 1 1

111111

P

1

1 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 1

1 1 1 1 1 1 11

1 1 1 1 1 1 1 11

1 1 1

1111

1 1 1 11

P

(1 -1, 0 1)

24

Conclusion

Linear Fractional Jacket Matrices

25

M.H. Lee, The Center Weighted Hadamard Transform, IEEE Trans.1989 AS-36, (9), pp.1247-1249.

S.-R.Lee and M.H.Lee, On the Reverse Jacket Matrix for Weighted Hadamard Transform, 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.

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.

W.P. Ma and M. H. Lee, Fast Reverse Jacket Transform Algorithms, Electronics Letters, vol. 39 no. 18 , 2003.

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.

Moon Ho Lee and Jia Hou, Fast Block Inverse Jacket Transform, IEEE Signal Processing Letters, vol.13. No.8, Aug.2006.

Jia Hou and Moon Ho Lee ,Construction of Dual OVSF Codes with Lower Correlations, IEICE Trans. Fundamentals, Vol.E89-A, No.11 pp 3363-3367, Nov 2006.

Jia Hou , Moon Ho Lee and Kwang Jae Lee,Doubly Stochastic Processing on Jacket Matricess, IEICE Trans. Fundamentals, vol E89-A, no.11, pp 3368-3372, Nov 2006.

Ken Finlayson, Moon Ho Lee, Jennifer Seberry, and Meiko Yamada, Jacket Matrices constructed from Hadamard Matrices and Generalized Hadamard Matrices, Australasian

References

26

27

28

29

30