Decomposition of Hadamard Matrices

Transcription

1 Chapter 7 Decomposto of Hadamard Matrces We hae see Chapter that Hadamard s orgal costructo of Hadamard matrces states that the Kroecer product of Hadamard matrces of orders m ad s a Hadamard matrx of order m he multplcate theorem was proposed 9 by Agaa ad Saruhaya [] (see also []) hey demostrated how to multply Wllamso Hadamard matrces order to get a Wllamso Hadamard matrx of order m/ hs result has bee exteded by the followg: Crage et al [3] hey show how to multply four Hadamard matrces of orders m,, p, q order to get a Hadamard matrx of order mpq/6 Agaa [] ad Saruhaya et al [] show how to multply seeral Hadamard matrces of orders,,,,, to get a Hadamard matrx of order ( )/, =,, hey obtaed a smlar result for A(,) type Hadamard matrces ad for Baumert Hall, Plot, ad Geothals Sedel arrays [5] Seberry ad Yamada estgated the multplcate theorem of Hadamard matrces of the geeralzed quatero type usg the M- structure [6] Phoog ad Chag [7] show that the Agaa ad Saruhaya theorem results ca be geeralzed to the case of atpodal parautary (APU) matrces A matrx fucto H(z) s sad to be parautary (PU) f t s utary for all alues of the parameters z, HzH () (/ z) I Oe attracte feature of these matrces s ther eergy preserato propertes that ca aod the ose or error amplfcato problem For further detals of PU matrces ad ther applcatos, we refer the reader to [ 0] A PU matrx s sad to be 79

2 0 Chapter 7 a APU matrx f all of ts coeffcet matrces hae as ther etres For the specal case of costat (memory less) matrces, APU matrces reduce to the well-ow Hadamard matrces he aalyss of the aboe-stated results relates wth soluto of the followg problem Problem [, ]: Let X ad A,,,, be ( 0, ) ad (+, ) matrces of dmesos p p ad q, q respectely, ad pq pq 0(mod ) (a) What codtos must matrces X ad A satsfy for to be a Hadamard matrx of order, ad (b) How are these matrces costructed? H X A, (7) I ths chapter, we deelop methods for costructg matrces X ad A, mag t possble to costruct ew Hadamard matrces ad orthogoal arrays We also preset a classfcato of Hadamard matrces based o ther decomposablty by orthogoal (+, )-ectors We wll preset multplcate theorems of costructo of a ew class of Hadamard matrces ad Baumert-Hall, Plot, ad Geothals-Sedel arrays Partcularly, we wll show that f there be Hadamard matrces of order m, m,, m, the a Hadamard matrx of order ( mm m )/ exsts As a applcato of multplcate theorems, oe may fd a example [ ] 7 Decomposto of Hadamard Matrces by (+, ) Vectors I ths secto, a partcular case of the problem ge aboe s studed, e, the case whe A s (+, )-ectors heorem 7: For matrx H [see (7)] to be a Hadamard matrx of order, t s ecessary ad suffcet that there be ( 0, ) matrces X ad (+, ) matrces A,,,, of dmesos p p ad q q, respectely, satsfyg the followg codtos:

3 Decomposto of Hadmard Matrces pq pq 0(mod ), X X 0,,,,,,, * s Hadamard product, 3 X s(, ) matrx,, XX AA X X AA I,,, 5 X X A A X X A A I, he frst three codtos are edet he two last codtos are otly equalet to codtos HH H H I (7) Now, let us cosder the case where Hadamard matrx H of order ca be represeted as A are (+, ) ectors Note that ay a) H ( ) X ( ) Y, b) H A, (73) where X, Y are ( 0, ) matrces of dmeso ( / ), A are ( 0, ) matrces of dmeso ( / ), ad are the followg four-dmesoal (+, ) ectors: 5 ( ), ( ), 6 ( ), ( ), 3 7 ( ), ( ), ( ), ( ) (7) Here, we ge the examples of decomposto of the followg Hadamard matrces:

4 Chapter 7 H, H (75) We use the followg otatos: w ( ), w ( ), ( ), ( ), 3 ( ), ( ) (76) Example 7: he Hadamard matrx H ad H ca be decomposed as follows: () Va two ectors: H w w, H w w (77) () Va four ectors:

5 Decomposto of Hadmard Matrces H 3, H (7) Now, let us troduce matrces B B 3 A A A A A 7 5 A, A A, 6 B B A A 3 3 A A A A, 5 A 7 6 A (79) heorem 7 [5]: For the exstece of Hadamard matrces of order, the exstece of ( 0, ) matrces B,,,3, of dmeso ( / ) s ecessary ad suffcet, satsfyg the followg codtos: B B 0, B B 0, 3 B B, B B are (, ) matrces, 3 BB I (70) 3, BB0,,,,,3,, 5 BB I,,,,3, Proof: Necessty: Let hae H be a Hadamard matrx of order Accordg to (7), we H A A A (7)

6 Chapter 7 From ths represetato, t follows that A A 0,,,,,,, A A A s a (, ) matrx (7) O the other had, t s ot dffcult to show that the matrx preseted as 3 H ca also be H ( ) B ( ) B, ( ) B ( ) B (73) Now, let us show that matrces B satsfy the codtos of Eq (70) From the represetato (73) ad from Eq (7) ad HH I, the frst three codtos of Eq (70) wll follow Because H s a Hadamard matrx of order, the from the represetato (73), we fd the followg system of matrx equatos: B B B B B B B B I / BBBBBBBB0, BBBBBBBB0, 3 3 BBBBBBBB0; 3 3 BBBBBBBB0, B B B B B B B B I / BBBBBBBB0, 3 3 BBBBBBBB0; 3 3 BBBBBBBB0, 3 3 BBBBBBBB0, 3 3 B B B B B B B B I / BBBBBBBB0; BBBBBBBB0, 3 3 BBBBBBBB0, 3 3 BBBBBBBB0, B B B B B B B B I /,,, (7a) (7b) (7c) (7d)

7 Decomposto of Hadmard Matrces 5 hey are equalet to BB,,,3,, I/ BB0,,,,3, (75) Suffcecy: Let ( 0, ) matrces B,,,3, of dmesos ( / ) satsfy the codtos of Eq (70) We ca drectly erfy that Eq (73) s a Hadamard matrx of order Corollary 7: he (+, ) matrces Q ( B B ), Q ( B B ), Q ( B B ), Q ( B B ) (76) of dmesos satsfy the codtos QQ 0,,,,3,, QQ I,,,3, (77) Corollary 7 [3]: If there be Hadamard matrces of order, m, p, q, the the Hadamard matrx of order mpq/6 also exsts Proof: Accordg to heorem 7, there are ( 0, ) matrces A ad B,,,3, of dmesos m (m / ) ad ( / ), respectely, satsfyg the codtos Eq (70) Itroduce the followg (+, ) matrces of orders m/: X A ( B B ) A ( B B ), Y A ( B B ) A ( B B ) (7) It s easy to show that matrces X, Y satsfy the codtos XY X Y 0, XX YY X X Y Y m I m / (79)

8 6 Chapter 7 Aga, we rewrte matrces X, Y the followg form: X [( ) X ( ) X, ( ) X ( ) X ], 3 Y [( ) Y ( ) Y, ( ) Y ( ) Y ] 3 (70) where X, Y,,,3, are ( 0, ) matrces of dmesos ( m / ) ( m /6) satsfyg the codtos XX X3X YY Y3Y 0, X X, X X, Y Y, Y Y are (, ) matrces, 3 3 XY XY0, XX YY X X YY m I m/ (7) Smlarly to Hadamard matrces of orders p ad q ca be costructed (+, ) matrces P ad Q of orders pq/ wth the codtos of Eq (79) Now, cosder the followg ( 0, ) matrces: P Q P Q Z, W, C X Z Y W,,,3, (7) It s ot dffcult to show that matrces Z ad W satsfy the codtos Z W 0, ZW WZ, pq ZZ Z Z WW W W I pq / (73) Assumg that matrces C of dmeso ( mpq /6) ( mpq / 6) satsfy the codtos of Eq (70)