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 ª a2  b2 abT  bc º , (2) « T T 2 2 » We describe so- called doubly stochastic processing ¬ab  bc (b )  a ¼ by using a simple factorization scheme on Jacket T matrices.. thus we have the solution b b , a c , and the

orthogonal >J @2 can be written by 1. INTRODUCTION ªa b º >@J 2 « » , (3) The doubly stochastic is with entrywise ¬b  a¼ nonnegative with all rows and columns sum one, and where a,b are real nonzero elements. Additionally, it is a special process in combinatorial theory, and probability [1]. Otherwise, Hadamard matrices are an inverse property should be hold, which is used widely in communication, and signal processing. 1 1 1 T >@J N >@(J ij ) , (4) Motivated by the Hadamard, a generalized form N called Jacket has been reported and its applications in where is the th row and th column element image processing and communications have been J ij i j pointed out [2],[3]. The basic idea of Jacket was in >@J N . It implies that the inverse of the Jacket motivated by the cloths of Jacket. As our two sided matrix is the entrywise inverse and of itself. Jacket is inside and outside compatible, at least two Therefore, (3) should be rewritten by positions of a Jacket matrix are replaced by their 1/ a 1/ b inverse; these elements are changed in their position 1 1 ª º >@J 2 « » , (5) and are moved, for example, from inside of the 2 ¬1/ b 1/ a¼ middle circle to outside or from to inside without loss where we should force a b . Clearly the result is a of sign. Recently, [4] gives contributions on classical two by two Hadamard matrix mixed-radix representation for Jacket transform, which unifies all Hadamard transforms, and Jacket ªa a º >@J 2 a ˜ >H @2 « » , (6) transforms, and also applicable for any even length ¬a  a¼ vectors. In this paper, we investigate a simple Jacket factorization scheme for doubly stochastic processing, where >@H 2 is the size two Hadamard matrix. The which includes a special orthostochastic case for any result shows that the two by two orthogonal Jacket even length. matrix is exist and only is the Hadamard case. Therefore, in several works [4][5], Jacket matrices are 2. JACKET MATRICES AND THEIR from four by four form, which is defined as PROPERTIES ª1 1 1 1 º «1 Z Z 1» In [3], a basic two by two kernel Jacket matrix is « » >@J 4 , (7) defined as «1 Z Z 1» « » ª a b º 1 1 1 1 J , (1) ¬   ¼ >@2 « T » ¬b  c¼ and its inverse is where T denotes the transpose, and a,b,c are all ª1 1 1 1 º « » real nonzero element. By considering a is a 1 1 1 1/Z 1/Z 1 >@J 2  « » >@J 4 , (8) unitary or orthogonal, we should have 4 «1 1/Z 1/Z 1» T J J 2 I « » >@2 ˜ >@2 >@2 ¬1 1 1 1 ¼ T ª a b ºªa b º where Z denotes a weight factor, that can be 2n ,

« T »« » j , and other complex numbers. The higher order ¬b  c¼¬b  c¼ ______0-7803-8560-8/04/$20.00©2004IEEE 682 Jacket matrices can be obtained by using the c 1/4u(O1-(1/Z)O 2+(1/Z)O 3-O 4) recursive function with N 2n ,n {2,3,4,...} as d 1/4u (O 1-O 2-O 3O 4) >@J N >@J N / 2 >H @2 , (9) b _ pair 1/4 u (O 1+ZO 2-ZO 3- O 4) where is the , and H is >@2 c _ pair 1/4u (O -ZO +ZO - O ) . defined by 1 2 3 4 The sum of rows and that of columns in P are ª1 1 º >@4 >@H 2 . (10) listed by «1 1» ¬ ¼ Rows: a  b _ pair  c _ pair  d O ; Thus, the unitary Jacket matrices only exist when 1 1 b  a  d  c O1 ; (16) Z * . Since the unitary Jacket matrix needs Columns: ; Z a  b  c  d O1 H b _ pair a d c _ pair . (17) >@J N ˜ >@J N N >@I N , (11)    O1 that implies It is clearly that the >@P 4 is doubly stochastic if H 1 J N J , (12) >@N >@N O1 1. By using the recursive function as (9), the where H denotes the Hermitian of a matrix, and * higher order probability matrix >@P N also is the is the conjugate of the element. doubly stochastic. And we call the set of these doubly 3. DOUBLY STOCHASTIC PROCESSING ON stochastic matrices as doubly stochastic Jacket JACKET MATRICES matrices (DSJM), several cases are listed in Table 1. Theorem 2: A square doubly of the for some unitary U An N by N matrix is said to be doubly stochastic if form >@P N >@U N $ >@U N * P P , P 1 and P 1 for is said to be orthostochastic [1][7][8]. If >@N > ij @ ¦ j ij ¦i ij >@U N >@J N , the resulted matrix is orthostochastic, all row i , and column j . Now, we can write a set and we always can find a special matrix is of doubly stochastic matrices according to a simple orthostochastic from the matrices set according to the factorization scheme as follows. Theorem 1, when 1, ... 0 . Theorem 1: Assuming a nonnegative probability O1 O2 ON Proof: Let U J , we obtain matrix >@P N is a nonnegative > @>@4 4 P U U * with eigenvalues O1 ,O2 ON , it can be a doubly >@4 >@4 $ >@4 stochastic matrix if ª1 1 1 1 º ª1 1 1 1 º 1 « » « » >@P N >@J N [diag(O1 ,O2 ,ON )] >@J N , (13) 1 1 1 * * 1 « Z Z  » « Z Z  » where should be unitary, , and $ >@J N O1 1 «1 Z Z 1» «1 Z * Z * 1» O ,O ,... are any values which can guarantee that « » « » 2 3 ¬1 1 1 1 ¼ ¬1 1 1 1 ¼ P is nonnegative. The inverse form can be easily >@N ª1 1 1 1º ª1 1 1 1º written by « » « » 1 1 1 * * 1 1 1 1 1 .(14) ZZ ZZ >@P N >@J N [diag(1/ O1 ,1/ O2 ,1/ ON )] >@J N « » « » , (18) Proof: Based on the basic matrix (7), we have «1 ZZ * ZZ * 1» «1 1 1 1» 1 « » « » >@P 4 >@>J 4 diag (O1 ,..., O 4 ) @>@J 4 ¬1 1 1 1¼ ¬1 1 1 1¼ 1 1 1 1 1 O 0 1 1 1 1 1 1 ª ºª 1 ºª º where Z * , ZZ* Z ˜ 1 , and $ «1 - -1»« O »«1 - -1» « Z Z »« 2 »« Z Z » Z Z denotes the Hadamard product. Clearly, it is «1 -Z Z -1»« O3 »«1 -Z Z -1» « »« »« » orthostochastic matrix, and its eigenvalues matrix is ¬1 -1 -1 1 ¼¬ 0 O3 ¼¬1 -1 -1 1 ¼ like as shown in Table 1 (b), where O1 1 , ª a b c d º O2 O3 O4 0 , then «b _ pair a d c_pair» « » , (15) ªO1 0 0 0 º ª1 0 0 0º « c_pair d a b_pair» « » « » « » 0 O2 0 0 0 0 0 0 ¬ d c b a ¼ « » « » . (19) « 0 0 O 0 » «0 0 0 0» a 1/4 u (O +O +O +O ) 3 where 1 2 3 4 « » « » ¬ 0 0 0 O4 ¼ ¬0 0 0 0¼ b 1/4 u (O 1+(1/Z )O 2 -(1/Z )O 3-O 4 ) 683

Similarly the higher order orthostochastic probability 1 a (O  O  O ) matrix according to the recursive function as 6 1 2 6 ª1 1  1º 1 b (O D 5O D 4O DO D 2O  O ) «1 1  1» 6 1 2 3 4 5 6 P « » >@N 1 4 2 2 4 «   » c (O1 D O2 D O3 D O4 D O5  O6 ) « » 6 1 1 1 ¬  ¼ 1 2 5 4 d (O DO  D O D O  D O  O ) with 6 1 2 3 4 5 6 1 , (23) ªO1 0  0 º ª1 0  0º e (O  D 2 O D 4 O D 4 O D 2 O  O ) 6 1 2 3 4 5 6 « 0 O  0 » «0 0  0» « 2 » « » 1 f (O  O  O  O  O  O ) «     » «   » 6 1 2 3 4 5 6 « » « » jS / 3 and D which is the complex roots of ¬ 0 0  ON ¼ ¬0 0  0¼ e 6 2 4 N 2 n , n t 2 . (20) D 1 . It is clear that D D 1 , and For the special two by two unitary Jacket matrix, i.e. D 1 D 5 1. Hence, we can find if diagonal value Hadamard, the probability matrix generated O 1 , other diagonal values can chose any values according to Theorem 1 has the same properties as 1 that a,b,c,d,e,f , are nonnegative ,then the probability that of the four by four unitary Jacket matrix. In matrix is a double stochastic process. Similar to general, the Theorem 1 can not only be applied for n Theorem 2, the 2n u 2n with n t 3 Jacket N 2 , but also be used for N 2n , with matrix is a , thus we have n t 3 . Since the unitary 2n u 2n Jacket matrix is H 1 . We always can based on [4] >@J 2nu2n (2n u 2n) ˜ >@J 2nu2n find a generalized is ª1 1 1 1 1 1 º orthostochastic, if the first eigenvalue O 1, and «1 D D 2 D 5 D 4 1» 1 « » the other eigenvalues equal zero. 2 4 4 2 «1 D D D D 1 » Proof: Let , we obtain , (21) >U @>@6 J 6 >@J 6 « 5 4 2 » 1 D D D D 1 « » >P@>@>@6 U 6 $ U 6 * 4 2 2 4 «1 D D D D 1 » ª1 1 1 1 1 1 º ª1 1 1 1 1 1 º « » «1 2 5 4 1» «1 5 4 2 1» «1 1 1 1 1 1» « D D D D  » « D D D D  » ¬ ¼ 2 4 4 2 4 2 2 4 6 «1 D D D D 1 » «1 D D D D 1 » where D 1. Thus we give factorization as « 5 4 2 »$« 2 5 4 » 1 «1 D D D D 1» «1 D D D D 1» P J diag(O ,...,O ) J 4 2 2 4 2 4 4 2 >@6 >@>6 1 6 @>@6 «1 D D D D 1 » «1 D D D D 1 » ª1 1 1 1 1 1 ºªO º « » « » 1 «1 1 1 1 1 1» «1 1 1 1 1 1» «1 D D 2 D 5 D 4 1»« O » ¬ ¼ ¬ ¼ « »« 2 » «1 D 2 D 4 D 4 D 2 1 »« O » ª1 1 1 1 1 1º 3 « 5 4 2 »« » « » «1 D D D D 1»« O4 » 1 1 1 1 1 1 4 2 2 4 « » «1 D D D D 1 »« O » « »« 5 » «1 1 1 1 1 1» 1 1 1 1 1 1 O ¬« ¼»¬« 6 ¼» « » , (24) 1 1 1 1 1 1 1 ª1 1 1 1 1 1 º « » «1 2 5 4 1» « » « D D D D  » 1 1 1 1 1 1 2 4 4 2 « » «1 D D D D 1 » 1 1 1 1 1 1 « 5 4 2 » ¬« ¼» «1 D D D D 1» 4 2 2 4 Clearly, it is orthostochastic matrix, and its «1 D D D D 1 » Theorem 2 « » eigenvalues matrix is likely as shown in , ¬«1 1 1 1 1 1¼» ªO1 º ª1 º ªa b c d e f º « » « » O2 0 «d a b e f c» « » « » « » « O3 » « 0 » . (25) «e d a f c b» , (22) « » « » « » O4 0 b c f a d e « » « » « » « O » « 0 » «c f e b a d» « 5 » « » « » ¬« O6 ¼» ¬« 0¼» ¬«f e d c b a¼» where 684 Similarly the higher order orthostochastic probability matrices always have the orthostochastic case if the matrix according to the recursive function is eigenvalues O1 1, the others are zeros, however, ª1 1  1º for the doubly stochastic case they may be any values «1 1  1» which could approach the elements in the probability « » matrix are nonnegative. Generally, the proposed >@P N «   » scheme uses a simple matrix factorization method to « » 1 1 1 represent the doubly stochastic, Markov vectors and ¬  ¼ eigenvalues, and it can be easily applied for with stochastic signal processing, Markov random process, Miller coding system [7], and orthogonal design for ªO1 0  0 º ª1 0  0º « » « » signal processing such as space time codes pattern 0 O2  0 0 0  0 design [6], [9], [10],[11]. « » « » , «     » «   » « » « » ¬ 0 0  ON ¼ ¬0 0  0¼ 5. ACKNOWLEDGMENTS N 2n , n t 3 . (26) This work was supported by University IT Research Center Project, Ministry of Information & Table.1: Different cases of 4 u 4 doubly stochastic Jacket Communications,Korean Research Foundation matrices KRF-2003-002-D00241, Korea.

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