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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
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