doubly stochastic processing on jacket matrices - IEEE Xplore
681
DOUBLY STOCHASTIC PROCESSING ON JACKET MATRICES Jia Hou, Jinnan Liu, and Moon Ho Lee Institute of Info. & Comm., Chonbuk National University, Chonju, 561-756, Korea. Email: [email protected] and [email protected] ABSTRACT We describe so- called doubly stochastic processing by using a simple factorization scheme on Jacket matrices.. 1. INTRODUCTION The doubly stochastic matrix is with entrywise nonnegative with all rows and columns sum one, and it is a special process in combinatorial theory, and probability [1]. Otherwise, Hadamard matrices are used widely in communication, and signal processing. Motivated by the Hadamard, a generalized form called Jacket has been reported and its applications in image processing and communications have been pointed out [2],[3]. The basic idea of Jacket 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. Recently, [4] gives contributions on mixed-radix representation for Jacket transform, which unifies all Hadamard transforms, and Jacket transforms, and also applicable for any even length vectors. In this paper, we investigate a simple Jacket factorization scheme for doubly stochastic processing, which includes a special orthostochastic case for any even length. 2. JACKET MATRICES AND THEIR PROPERTIES In [3], a basic two by two kernel Jacket matrix is defined as
>J @2
ªa «b T ¬
bº , c »¼
ª a2 b2 abT bc º , (2) « T T 2 2» ab bc ( b ) a ¬ ¼ T b , a c , and the thus we have the solution b orthogonal >J @2 can be written by a b º >J @2 ª« (3) », ¬b a ¼ where a, b are real nonzero elements. Additionally, an inverse property should be hold, which is
where
J ij
unitary or orthogonal, we should have
2>I @2
ªa «b T ¬
b º ªa b T º « » c »¼ ¬b c ¼
___________________________________________ 0-7803-8560-8/04/$20.00©2004IEEE
@
> @
1 ª1 / a 1 / b º , (5) 2 «¬1 / b 1 / a »¼ where we should force a b . Clearly the result is a
>J @21
classical two by two Hadamard matrix
>J @2
ªa a º «a a » , ¬ ¼
a >H @2
> @
(6)
where H 2 is the size two Hadamard matrix. The result shows that the two by two orthogonal Jacket matrix is exist and only is the Hadamard case. Therefore, in several works [4][5], Jacket matrices are from four by four form, which is defined as
>J @4
(1)
T denotes the transpose, and a, b, c are all real nonzero element. By considering a >J @2 is a
>
in J N . It implies that the inverse of the Jacket matrix is the entrywise inverse and transpose of itself. Therefore, (3) should be rewritten by
ª1 1 «1 Z « «1 Z « ¬1 1
1
Z Z 1
1º 1»» , 1» » 1¼
(7)
and its inverse is
where
>J @2 >J @T2
T 1 ( J ij ) 1 , (4) N is the i th row and j th column element
>J @N1
>J @41
1 ª1 «1 1 / Z 1« 4 «1 1 / Z « ¬1 1
1 1/ Z 1/ Z 1
1º 1»» , 1» » 1¼
(8)
n
where Z denotes a weight factor, that can be 2 , j , and other complex numbers. The higher order
682 Jacket matrices can be obtained by using the
c 1/4 u (O 1-(1/Z )O 2 +(1/Z )O 3-O 4 )
2 n , n {2,3,4,...} as >J @N >J @N / 2 >H @2 , (9) where is the kronecker product, and >H @2 is recursive function with
N
1/4 u (O 1-O 2 -O 3 O 4 ) b _ pair 1/4 u (O 1+ZO 2 -ZO 3 - O 4 ) c _ pair 1/4 u (O 1-ZO 2 +ZO 3 - O 4 ) .
d
defined by
>H @2
> @4
ª1 1 º «1 1» . ¬ ¼
The sum of rows and that of columns in P listed by Rows: a b _ pair c _ pair d O1 ;
(10)
Thus, the unitary Jacket matrices only exist when
1
Z
b a d c O1 ; (16) Columns: a b c d O1 ; b _ pair a d c _ pair O1 . (17) It is clearly that the >P @4 is doubly stochastic if O1 1 . By using the recursive function as (9), the higher order probability matrix >P @N also is the
Z * . Since the unitary Jacket matrix needs
>J @N >J @HN
N >I @N ,
(11)
that implies
>J @HN
N >J @N , 1
(12)
where H denotes the Hermitian of a matrix, and * is the conjugate of the element.
doubly stochastic. And we call the set of these doubly stochastic matrices as doubly stochastic Jacket matrices (DSJM), several cases are listed in Table 1. Theorem 2: A square doubly stochastic matrix of the U N $ U N * for some unitary U form P N is said to be orthostochastic [1][7][8]. If U N J N , the resulted matrix is orthostochastic, and we always can find a special matrix is orthostochastic from the matrices set according to the Theorem 1, when O1 1, O 2 ... O N 0.
3. DOUBLY STOCHASTIC PROCESSING ON JACKET MATRICES
> @
An N by N matrix is said to be doubly stochastic if
>P @N >Pij @ , ¦ j Pij
1 and
¦P i
ij
1 for
> @
all row i , and column j . Now, we can write a set of doubly stochastic matrices according to a simple factorization scheme as follows. Theorem 1: Assuming a nonnegative probability matrix P N is a nonnegative diagonalizable matrix
> @
with eigenvalues
O1 , O2 O N ,
>P@4
stochastic matrix if
O2 , O3 ,...
(13) and
are any values which can guarantee
that
>P@N
is nonnegative. The inverse form can be easily written by
>P @N1 >J @N [diag (1 / O1 ,1 / O2 ,1 / O N )]>J @N1 .(14) Proof: Based on the basic matrix (7), we have
>P @4 >J @4 >diag ( O1 ,..., O 4 ) @>J @41 0 º ª1 1 1 1 º ªO1 1 1º ª1 1 » « » «1 Z - Z -1» « O2 » «1 Z - Z -1» « »« » «1 - Z Z -1» «1 - Z Z -1» « O3 »« « »« » O3 ¼ ¬1 -1 - 1 1 ¼ ¬1 -1 - 1 1 ¼ ¬ 0
b c d º ª a «b _ pair a d c_pair » « » , « c_pair d a b_pair » « » c b a ¼ ¬ d where
(15)
a
1/4 u ( O 1 + O 2 + O 3 + O 4 )
b
1/4 u (O 1 +(1/ Z ) O 2 -(1/ Z ) O 3 -O 4 )
1
> @ > @
> @
>U @4 >J @4 , >U @4 $ >U @4 *
Proof: Let
it can be a doubly
>P@N >J @N [diag (O1 , O2 , O N )]>J @N1 , where >J @N should be unitary, O1 1 ,
are
we obtain
1 1 1º 1 1 º ª1 ª1 1 «1 Z Z 1» «1 Z * Z * 1» « »$« » «1 Z Z 1» «1 Z * Z * 1» « » « » 1¼ 1 ¬1 1 1 1 ¼ ¬1 1 1 1 1º ª1 1 1 1º ª1 «1 ZZ * ZZ * 1» «1 1 1 1» » , (18) » « « «1 ZZ * ZZ * 1» «1 1 1 1» » » « « 1 1 1¼ ¬1 1 1 1¼ ¬1 1 1 Z * , ZZ * Z 1 , and $ where
Z
Z
denotes the Hadamard product. Clearly, it is orthostochastic matrix, and its eigenvalues matrix is like as shown in Table 1 (b), where O1 1 ,
O2 O3 O4 0 , then ªO1 0 0 0 º ª1 0 «0 O 0 0 »» ««0 0 2 « « 0 0 O 3 0 » «0 0 » « « ¬ 0 0 0 O 4 ¼ ¬0 0
0 0 0 0
0º 0»» . 0» » 0¼
(19)
683 Similarly the higher order orthostochastic probability matrix according to the recursive function as
ª1 «1 « « « ¬1
>P@N
1 1 1
1º 1»» » » 1¼
b c
0 º ª1 0 »» ««0 » « » « O N ¼ ¬0 N 2n , n t 2 .
0
O2 0
0 0 0
0º 0»» » » 0¼
(20) For the special two by two unitary Jacket matrix, i.e. Hadamard, the probability matrix generated according to Theorem 1 has the same properties as that of the four by four unitary Jacket matrix. In general, the Theorem 1 can not only be applied for
N 2 n , but also be used for N 2n , with n t 3 . Since the unitary 2n u 2n Jacket matrix is based on [4]
ª1 «1 « «1 « «1 «1 « «¬1
>J @6
1
D D2 D5 D4
ª1 «1 « «1 « «1 «1 « ¬«1
1
D D2 D5 D4
D D2 D5 D4 1
1
D4 D2 D2 D4
1
1
1
1
D4 D2 D2 D4
1
1
1 1
1
1 º ªO1 1»» «« 1 »« »« 1» « 1 »« »« 1»¼ «¬
1º 1»» 1» », 1» 1» » 1»¼
(21)
1
D2 D5 D4 D4 D4 D D2 D2
D4 D2 D2 D4
1
1
1º 1»» 1» » 1» 1» » 1¼»
b a
c b
d e
e f
d c
a f
f a
c d
f e
e d
b c
a b
1
O2 O3 O4 O5
º » » » » » » » O 6 »¼
1
1
ªa «d « «e « «b «c « «¬ f where
D2 D5 D4 D4 D4 D D2 D2
D2 D5 D4 D4 D4 D D2 D2
1 1
1
1 . Thus we give factorization as >J @6 >diag (O1 ,..., O6 )@>J @61
>P @6 ª1 «1 « «1 « «1 «1 « «¬1
1
1
D6
where
1
fº c »» b» , » e» d» » a »¼
1 ( O1 DO 2 D 2 O 3 D 5 O 4 D 4 O 5 O 6 ) 6 , 1 (O 1 D 2 O 2 D 4 O 3 D 4 O 4 D 2 O 5 O 6 ) 6 1 ( O1 O 2 O 3 O 4 O 5 O 6 ) 6
d
with
ªO1 «0 « « « ¬0
1 (O1 O2 O6 ) 6 1 (O1 D 5 O2 D 4 O3 DO4 D 2 O5 O6 ) 6 1 (O1 D 4 O2 D 2 O3 D 2 O4 D 4 O5 O6 ) 6
a
(22)
e f
and D
e
jS / 3
(23)
which is the complex roots of
6
D 1 . It is clear that D 2 D 4 1 , and D 1 D 5 1 . Hence, we can find if diagonal value O 1 1 , other diagonal values can chose any values that a,b,c,d,e,f , are nonnegative ,then the probability matrix is a double stochastic process. Similar to Theorem 2, the 2n u 2n with n t 3 Jacket matrix is a unitary matrix, thus we have
>J @2Hnu2 n
(2n u 2n) >J @2 nu2 n . We always can 1
find a generalized doubly stochastic matrix is orthostochastic, if the first eigenvalue O1 1 , and the other eigenvalues equal zero. Proof: Let U 6 J 6 , we obtain
> @ > @ >P@6 >U @6 $ >U @6 *
ª1 «1 « «1 « «1 «1 « ¬«1
ª1 «1 « «1 « «1 «1 « «¬1
1
1
1
D D2 D5 D4
D2 D4 D4 D2
D5 D4 D D2
1
1
1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1
1 º ª1 1
D 4 1»» ««1 D 5 D 2 1 » «1 D 4 »$« D 2 1» «1 D D 4 1 » «1 D 2 1
1 1 1 1 1 1
» « 1¼» ¬«1 1
1º 1»» 1» », 1» 1» » 1»¼
1
1
D4 D2 D2 D4
D D2 D5 D4
1
1
1
1º
D 2 1»» D4 1 » » D 4 1» D2 1 » 1
» 1¼»
(24)
Clearly, it is orthostochastic matrix, and its eigenvalues matrix is likely as shown in Theorem 2, ªO1 º ª1 º « » « » O2 « » « 0 » « » « » . (25) O3 0 « » « » O4 0 « » « » « » « O5 0 » « » « » O6 ¼» ¬« 0¼» ¬«
684 Similarly the higher order orthostochastic probability matrix according to the recursive function is
>P@N
ª1 «1 « « « ¬1
1 1 1
1º 1»» » » 1¼
with
ªO1 «0 « « « ¬0
0 º ª1 O 2 0 »» ««0 » « » « 0 O N ¼ ¬0 N 2n , n t 3 . 0
4u 4
Table.1: Different cases of matrices
0 0 0
0º 0»» , » » 0¼
matrices always have the orthostochastic case if the eigenvalues O1 1 , the others are zeros, however, for the doubly stochastic case they may be any values which could approach the elements in the probability matrix are nonnegative. Generally, the proposed scheme uses a simple matrix factorization method to represent the doubly stochastic, Markov vectors and eigenvalues, and it can be easily applied for stochastic signal processing, Markov random process, Miller coding system [7], and orthogonal design for signal processing such as space time codes pattern design [6], [9], [10],[11]. 5. ACKNOWLEDGMENTS
(26)
doubly stochastic Jacket
This work was supported by University IT Research Center Project, Ministry of Information & Communications,Korean Research Foundation KRF-2003-002-D00241, Korea.
4u 4
ª1 «0 « «0 « ¬0
0 0 0º 1 0 0»» 0 1 0» » 0 0 1¼
ª1 «1 « «1 « ¬1
1 º ª1 2 2 1»» ««0 2 2 1» «0 »« 1 1 1 ¼ ¬0
ª1 « 1 «1 4 «1 « ¬1
1 1 1º 1 1 1»» 1 1 1» » 1 1 1¼
ª1 «1 « «1 « ¬1
1
O1 O2 O3 O 4 1
O1 1 z O 2 O3 O4
O1 1 z O2 z O3 z O4
O1 O4 O 2 O3
ª3 «5 «1 « « 40 « 13 « 40 «1 « ¬ 20 ª «0 «0 « «1 « «¬0
11 80 3 5 1 20 17 80
17 80 1 20 3 5 11 80
1º 20 » 13 » » 40 » 1» 40 » 3» » 5¼
ª1 «1 « «1 « ¬1
1
1 º ª1 2 2 1»» ««0 2 2 1» «0 »« 1 1 1 ¼ ¬0
1
ª1 1 º «0 « 2 2 1»» « 2 2 1» «0 »« 1 1 1 ¼« 0 «¬
3 5 º 0 ª1 1 8 8 » « 0 0 1» «1 2 » 0 0 0» «1 2 5 3 » «1 1 0 ¬ 8 8 »¼
1
1 0 0 0º ª« 1 » 1 0 0» 1 « « 0 1 0» 4 «1 » 0 0 1¼ « ¬«1
1
1
1 1 1 º ª« 0 » 2 1» « « 2 1» «0 »« 1 1 ¼ ¬«0
1 1 2 1 2 1
1 1 2 1 2 1
1º » 1» » 1» » 1 ¼»
1 1 2 1 2 1
1 1 2 1 2 1
1º » 1» » 1» » 1 ¼»
1 0 0 0º ª« » 0 0 0» 1 «1 « 0 0 0» 4 «1 » « 0 0 0¼ «¬1
0 0 0º 7 » ª1 0 0» « 1 10 »1« 4 « 0 0 »4 10 » «1 3» « 0 0 ¬«1 10 »¼
0 0 1 0 2 1 0 2 0 0
0 º ª1 » « 0 » «1 1 » « 0 » 4 «1 » « 1¼» ¬«1
1 1 2 1 2 1
1 1 2 1 2 1
1 1 2 1 2 1
1 1 2 1 2 1
1º » 1» » 1» » 1 ¼»
1º » 1» » 1» » 1 ¼»
4. CONCLUSION
We proposed a novel method to generalize a set of
2 n u 2 n and 2n u 2n matrices named generalized doubly stochastic Jacket matrices, also orthostochastic cases are included. The derivation shows that
2 n u 2 n and 2n u 2n , n t 2 , Jacket
6. REFERENCES [1] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis. Cambridge,MA: Cambridge Univ. Press, UK, 1991. [2] Moon Ho Lee, “The center weighted Hadamard transform,” IEEE Trans .Circuits Syst., vol. CAS–36, no.9, pp. 1247–1249, Sept. 1989. [3] Moon Ho Lee, “A New Reverse Jacket Transform and Its Fast Algorithm,” IEEE Trans. On Circuit and System, vol. 47, no. 1, pp. 39-47. Jan. 2000. [4] Moon Ho Lee ,B. Sundar Rajan and J. Y. Park,” A Generalized Reverse Jacket Transform” IEEE Trans. Circuits Syst.. II, vol. 48, no. 7, pp.684–690 July 2001. [5] W.P. Ma and Moon Ho Lee, “Fast reverse jacket
transform algorithm,” IEE Electronics letters, vol.39, no.18, pp.1312-1313, 2003. [6] A. V. Geramita, and J. Seberry, Orthogonal Designs, Marcel Dekker, Inc. UK, 1979. [7] T. M. Cover, and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, Inc., USA, 1991. [8] A. Papoulis, and S. U. Pillai, Probability, Random Variables and Stochastic Processes, Fourth Edition, Mc-Graw Hill Inc. USA, 2002. [9] Jia Hou, Moon Ho Lee, and Ju Yong Park,
“Matrices analysis of quasi orthogonal space time block codes,” IEEE Communications Letters, vol.7, no.8, pp.385-387, August 2003. [10] B. Vucetic and J. Yuan, Space Time Coding, John Wiley & Sons Ltd. USA, 2003. [11] Ahmed, N., and Rao, K.R., Orthogonal transforms for digital signal processing, Springer-Verlag, New York, 1975.
DOUBLY STOCHASTIC PROCESSING ON JACKET MATRICES Jia Hou, Jinnan Liu, and Moon Ho Lee Institute of Info. & Comm., Chonbuk National University, Chonju, 561-756, Korea. Email: [email protected] and [email protected] ABSTRACT We describe so- called doubly stochastic processing by using a simple factorization scheme on Jacket matrices.. 1. INTRODUCTION The doubly stochastic matrix is with entrywise nonnegative with all rows and columns sum one, and it is a special process in combinatorial theory, and probability [1]. Otherwise, Hadamard matrices are used widely in communication, and signal processing. Motivated by the Hadamard, a generalized form called Jacket has been reported and its applications in image processing and communications have been pointed out [2],[3]. The basic idea of Jacket 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. Recently, [4] gives contributions on mixed-radix representation for Jacket transform, which unifies all Hadamard transforms, and Jacket transforms, and also applicable for any even length vectors. In this paper, we investigate a simple Jacket factorization scheme for doubly stochastic processing, which includes a special orthostochastic case for any even length. 2. JACKET MATRICES AND THEIR PROPERTIES In [3], a basic two by two kernel Jacket matrix is defined as
>J @2
ªa «b T ¬
bº , c »¼
ª a2 b2 abT bc º , (2) « T T 2 2» ab bc ( b ) a ¬ ¼ T b , a c , and the thus we have the solution b orthogonal >J @2 can be written by a b º >J @2 ª« (3) », ¬b a ¼ where a, b are real nonzero elements. Additionally, an inverse property should be hold, which is
where
J ij
unitary or orthogonal, we should have
2>I @2
ªa «b T ¬
b º ªa b T º « » c »¼ ¬b c ¼
___________________________________________ 0-7803-8560-8/04/$20.00©2004IEEE
@
> @
1 ª1 / a 1 / b º , (5) 2 «¬1 / b 1 / a »¼ where we should force a b . Clearly the result is a
>J @21
classical two by two Hadamard matrix
>J @2
ªa a º «a a » , ¬ ¼
a >H @2
> @
(6)
where H 2 is the size two Hadamard matrix. The result shows that the two by two orthogonal Jacket matrix is exist and only is the Hadamard case. Therefore, in several works [4][5], Jacket matrices are from four by four form, which is defined as
>J @4
(1)
T denotes the transpose, and a, b, c are all real nonzero element. By considering a >J @2 is a
>
in J N . It implies that the inverse of the Jacket matrix is the entrywise inverse and transpose of itself. Therefore, (3) should be rewritten by
ª1 1 «1 Z « «1 Z « ¬1 1
1
Z Z 1
1º 1»» , 1» » 1¼
(7)
and its inverse is
where
>J @2 >J @T2
T 1 ( J ij ) 1 , (4) N is the i th row and j th column element
>J @N1
>J @41
1 ª1 «1 1 / Z 1« 4 «1 1 / Z « ¬1 1
1 1/ Z 1/ Z 1
1º 1»» , 1» » 1¼
(8)
n
where Z denotes a weight factor, that can be 2 , j , and other complex numbers. The higher order
682 Jacket matrices can be obtained by using the
c 1/4 u (O 1-(1/Z )O 2 +(1/Z )O 3-O 4 )
2 n , n {2,3,4,...} as >J @N >J @N / 2 >H @2 , (9) where is the kronecker product, and >H @2 is recursive function with
N
1/4 u (O 1-O 2 -O 3 O 4 ) b _ pair 1/4 u (O 1+ZO 2 -ZO 3 - O 4 ) c _ pair 1/4 u (O 1-ZO 2 +ZO 3 - O 4 ) .
d
defined by
>H @2
> @4
ª1 1 º «1 1» . ¬ ¼
The sum of rows and that of columns in P listed by Rows: a b _ pair c _ pair d O1 ;
(10)
Thus, the unitary Jacket matrices only exist when
1
Z
b a d c O1 ; (16) Columns: a b c d O1 ; b _ pair a d c _ pair O1 . (17) It is clearly that the >P @4 is doubly stochastic if O1 1 . By using the recursive function as (9), the higher order probability matrix >P @N also is the
Z * . Since the unitary Jacket matrix needs
>J @N >J @HN
N >I @N ,
(11)
that implies
>J @HN
N >J @N , 1
(12)
where H denotes the Hermitian of a matrix, and * is the conjugate of the element.
doubly stochastic. And we call the set of these doubly stochastic matrices as doubly stochastic Jacket matrices (DSJM), several cases are listed in Table 1. Theorem 2: A square doubly stochastic matrix of the U N $ U N * for some unitary U form P N is said to be orthostochastic [1][7][8]. If U N J N , the resulted matrix is orthostochastic, and we always can find a special matrix is orthostochastic from the matrices set according to the Theorem 1, when O1 1, O 2 ... O N 0.
3. DOUBLY STOCHASTIC PROCESSING ON JACKET MATRICES
> @
An N by N matrix is said to be doubly stochastic if
>P @N >Pij @ , ¦ j Pij
1 and
¦P i
ij
1 for
> @
all row i , and column j . Now, we can write a set of doubly stochastic matrices according to a simple factorization scheme as follows. Theorem 1: Assuming a nonnegative probability matrix P N is a nonnegative diagonalizable matrix
> @
with eigenvalues
O1 , O2 O N ,
>P@4
stochastic matrix if
O2 , O3 ,...
(13) and
are any values which can guarantee
that
>P@N
is nonnegative. The inverse form can be easily written by
>P @N1 >J @N [diag (1 / O1 ,1 / O2 ,1 / O N )]>J @N1 .(14) Proof: Based on the basic matrix (7), we have
>P @4 >J @4 >diag ( O1 ,..., O 4 ) @>J @41 0 º ª1 1 1 1 º ªO1 1 1º ª1 1 » « » «1 Z - Z -1» « O2 » «1 Z - Z -1» « »« » «1 - Z Z -1» «1 - Z Z -1» « O3 »« « »« » O3 ¼ ¬1 -1 - 1 1 ¼ ¬1 -1 - 1 1 ¼ ¬ 0
b c d º ª a «b _ pair a d c_pair » « » , « c_pair d a b_pair » « » c b a ¼ ¬ d where
(15)
a
1/4 u ( O 1 + O 2 + O 3 + O 4 )
b
1/4 u (O 1 +(1/ Z ) O 2 -(1/ Z ) O 3 -O 4 )
1
> @ > @
> @
>U @4 >J @4 , >U @4 $ >U @4 *
Proof: Let
it can be a doubly
>P@N >J @N [diag (O1 , O2 , O N )]>J @N1 , where >J @N should be unitary, O1 1 ,
are
we obtain
1 1 1º 1 1 º ª1 ª1 1 «1 Z Z 1» «1 Z * Z * 1» « »$« » «1 Z Z 1» «1 Z * Z * 1» « » « » 1¼ 1 ¬1 1 1 1 ¼ ¬1 1 1 1 1º ª1 1 1 1º ª1 «1 ZZ * ZZ * 1» «1 1 1 1» » , (18) » « « «1 ZZ * ZZ * 1» «1 1 1 1» » » « « 1 1 1¼ ¬1 1 1 1¼ ¬1 1 1 Z * , ZZ * Z 1 , and $ where
Z
Z
denotes the Hadamard product. Clearly, it is orthostochastic matrix, and its eigenvalues matrix is like as shown in Table 1 (b), where O1 1 ,
O2 O3 O4 0 , then ªO1 0 0 0 º ª1 0 «0 O 0 0 »» ««0 0 2 « « 0 0 O 3 0 » «0 0 » « « ¬ 0 0 0 O 4 ¼ ¬0 0
0 0 0 0
0º 0»» . 0» » 0¼
(19)
683 Similarly the higher order orthostochastic probability matrix according to the recursive function as
ª1 «1 « « « ¬1
>P@N
1 1 1
1º 1»» » » 1¼
b c
0 º ª1 0 »» ««0 » « » « O N ¼ ¬0 N 2n , n t 2 .
0
O2 0
0 0 0
0º 0»» » » 0¼
(20) For the special two by two unitary Jacket matrix, i.e. Hadamard, the probability matrix generated according to Theorem 1 has the same properties as that of the four by four unitary Jacket matrix. In general, the Theorem 1 can not only be applied for
N 2 n , but also be used for N 2n , with n t 3 . Since the unitary 2n u 2n Jacket matrix is based on [4]
ª1 «1 « «1 « «1 «1 « «¬1
>J @6
1
D D2 D5 D4
ª1 «1 « «1 « «1 «1 « ¬«1
1
D D2 D5 D4
D D2 D5 D4 1
1
D4 D2 D2 D4
1
1
1
1
D4 D2 D2 D4
1
1
1 1
1
1 º ªO1 1»» «« 1 »« »« 1» « 1 »« »« 1»¼ «¬
1º 1»» 1» », 1» 1» » 1»¼
(21)
1
D2 D5 D4 D4 D4 D D2 D2
D4 D2 D2 D4
1
1
1º 1»» 1» » 1» 1» » 1¼»
b a
c b
d e
e f
d c
a f
f a
c d
f e
e d
b c
a b
1
O2 O3 O4 O5
º » » » » » » » O 6 »¼
1
1
ªa «d « «e « «b «c « «¬ f where
D2 D5 D4 D4 D4 D D2 D2
D2 D5 D4 D4 D4 D D2 D2
1 1
1
1 . Thus we give factorization as >J @6 >diag (O1 ,..., O6 )@>J @61
>P @6 ª1 «1 « «1 « «1 «1 « «¬1
1
1
D6
where
1
fº c »» b» , » e» d» » a »¼
1 ( O1 DO 2 D 2 O 3 D 5 O 4 D 4 O 5 O 6 ) 6 , 1 (O 1 D 2 O 2 D 4 O 3 D 4 O 4 D 2 O 5 O 6 ) 6 1 ( O1 O 2 O 3 O 4 O 5 O 6 ) 6
d
with
ªO1 «0 « « « ¬0
1 (O1 O2 O6 ) 6 1 (O1 D 5 O2 D 4 O3 DO4 D 2 O5 O6 ) 6 1 (O1 D 4 O2 D 2 O3 D 2 O4 D 4 O5 O6 ) 6
a
(22)
e f
and D
e
jS / 3
(23)
which is the complex roots of
6
D 1 . It is clear that D 2 D 4 1 , and D 1 D 5 1 . Hence, we can find if diagonal value O 1 1 , other diagonal values can chose any values that a,b,c,d,e,f , are nonnegative ,then the probability matrix is a double stochastic process. Similar to Theorem 2, the 2n u 2n with n t 3 Jacket matrix is a unitary matrix, thus we have
>J @2Hnu2 n
(2n u 2n) >J @2 nu2 n . We always can 1
find a generalized doubly stochastic matrix is orthostochastic, if the first eigenvalue O1 1 , and the other eigenvalues equal zero. Proof: Let U 6 J 6 , we obtain
> @ > @ >P@6 >U @6 $ >U @6 *
ª1 «1 « «1 « «1 «1 « ¬«1
ª1 «1 « «1 « «1 «1 « «¬1
1
1
1
D D2 D5 D4
D2 D4 D4 D2
D5 D4 D D2
1
1
1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1
1 º ª1 1
D 4 1»» ««1 D 5 D 2 1 » «1 D 4 »$« D 2 1» «1 D D 4 1 » «1 D 2 1
1 1 1 1 1 1
» « 1¼» ¬«1 1
1º 1»» 1» », 1» 1» » 1»¼
1
1
D4 D2 D2 D4
D D2 D5 D4
1
1
1
1º
D 2 1»» D4 1 » » D 4 1» D2 1 » 1
» 1¼»
(24)
Clearly, it is orthostochastic matrix, and its eigenvalues matrix is likely as shown in Theorem 2, ªO1 º ª1 º « » « » O2 « » « 0 » « » « » . (25) O3 0 « » « » O4 0 « » « » « » « O5 0 » « » « » O6 ¼» ¬« 0¼» ¬«
684 Similarly the higher order orthostochastic probability matrix according to the recursive function is
>P@N
ª1 «1 « « « ¬1
1 1 1
1º 1»» » » 1¼
with
ªO1 «0 « « « ¬0
0 º ª1 O 2 0 »» ««0 » « » « 0 O N ¼ ¬0 N 2n , n t 3 . 0
4u 4
Table.1: Different cases of matrices
0 0 0
0º 0»» , » » 0¼
matrices always have the orthostochastic case if the eigenvalues O1 1 , the others are zeros, however, for the doubly stochastic case they may be any values which could approach the elements in the probability matrix are nonnegative. Generally, the proposed scheme uses a simple matrix factorization method to represent the doubly stochastic, Markov vectors and eigenvalues, and it can be easily applied for stochastic signal processing, Markov random process, Miller coding system [7], and orthogonal design for signal processing such as space time codes pattern design [6], [9], [10],[11]. 5. ACKNOWLEDGMENTS
(26)
doubly stochastic Jacket
This work was supported by University IT Research Center Project, Ministry of Information & Communications,Korean Research Foundation KRF-2003-002-D00241, Korea.
4u 4
ª1 «0 « «0 « ¬0
0 0 0º 1 0 0»» 0 1 0» » 0 0 1¼
ª1 «1 « «1 « ¬1
1 º ª1 2 2 1»» ««0 2 2 1» «0 »« 1 1 1 ¼ ¬0
ª1 « 1 «1 4 «1 « ¬1
1 1 1º 1 1 1»» 1 1 1» » 1 1 1¼
ª1 «1 « «1 « ¬1
1
O1 O2 O3 O 4 1
O1 1 z O 2 O3 O4
O1 1 z O2 z O3 z O4
O1 O4 O 2 O3
ª3 «5 «1 « « 40 « 13 « 40 «1 « ¬ 20 ª «0 «0 « «1 « «¬0
11 80 3 5 1 20 17 80
17 80 1 20 3 5 11 80
1º 20 » 13 » » 40 » 1» 40 » 3» » 5¼
ª1 «1 « «1 « ¬1
1
1 º ª1 2 2 1»» ««0 2 2 1» «0 »« 1 1 1 ¼ ¬0
1
ª1 1 º «0 « 2 2 1»» « 2 2 1» «0 »« 1 1 1 ¼« 0 «¬
3 5 º 0 ª1 1 8 8 » « 0 0 1» «1 2 » 0 0 0» «1 2 5 3 » «1 1 0 ¬ 8 8 »¼
1
1 0 0 0º ª« 1 » 1 0 0» 1 « « 0 1 0» 4 «1 » 0 0 1¼ « ¬«1
1
1
1 1 1 º ª« 0 » 2 1» « « 2 1» «0 »« 1 1 ¼ ¬«0
1 1 2 1 2 1
1 1 2 1 2 1
1º » 1» » 1» » 1 ¼»
1 1 2 1 2 1
1 1 2 1 2 1
1º » 1» » 1» » 1 ¼»
1 0 0 0º ª« » 0 0 0» 1 «1 « 0 0 0» 4 «1 » « 0 0 0¼ «¬1
0 0 0º 7 » ª1 0 0» « 1 10 »1« 4 « 0 0 »4 10 » «1 3» « 0 0 ¬«1 10 »¼
0 0 1 0 2 1 0 2 0 0
0 º ª1 » « 0 » «1 1 » « 0 » 4 «1 » « 1¼» ¬«1
1 1 2 1 2 1
1 1 2 1 2 1
1 1 2 1 2 1
1 1 2 1 2 1
1º » 1» » 1» » 1 ¼»
1º » 1» » 1» » 1 ¼»
4. CONCLUSION
We proposed a novel method to generalize a set of
2 n u 2 n and 2n u 2n matrices named generalized doubly stochastic Jacket matrices, also orthostochastic cases are included. The derivation shows that
2 n u 2 n and 2n u 2n , n t 2 , Jacket
6. REFERENCES [1] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis. Cambridge,MA: Cambridge Univ. Press, UK, 1991. [2] Moon Ho Lee, “The center weighted Hadamard transform,” IEEE Trans .Circuits Syst., vol. CAS–36, no.9, pp. 1247–1249, Sept. 1989. [3] Moon Ho Lee, “A New Reverse Jacket Transform and Its Fast Algorithm,” IEEE Trans. On Circuit and System, vol. 47, no. 1, pp. 39-47. Jan. 2000. [4] Moon Ho Lee ,B. Sundar Rajan and J. Y. Park,” A Generalized Reverse Jacket Transform” IEEE Trans. Circuits Syst.. II, vol. 48, no. 7, pp.684–690 July 2001. [5] W.P. Ma and Moon Ho Lee, “Fast reverse jacket
transform algorithm,” IEE Electronics letters, vol.39, no.18, pp.1312-1313, 2003. [6] A. V. Geramita, and J. Seberry, Orthogonal Designs, Marcel Dekker, Inc. UK, 1979. [7] T. M. Cover, and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, Inc., USA, 1991. [8] A. Papoulis, and S. U. Pillai, Probability, Random Variables and Stochastic Processes, Fourth Edition, Mc-Graw Hill Inc. USA, 2002. [9] Jia Hou, Moon Ho Lee, and Ju Yong Park,
“Matrices analysis of quasi orthogonal space time block codes,” IEEE Communications Letters, vol.7, no.8, pp.385-387, August 2003. [10] B. Vucetic and J. Yuan, Space Time Coding, John Wiley & Sons Ltd. USA, 2003. [11] Ahmed, N., and Rao, K.R., Orthogonal transforms for digital signal processing, Springer-Verlag, New York, 1975.
