Lecture notes in mathematics

1, Erevan 44, USSR Consulting Editor D.Yu, Grigorev Leningrad Branch of the Steklov Mathematical Institute Fontanka 27, 191011 Leningrad, D-11, USSR Mathematics Subject Classification 1

S.S Agaian

Computer Center of the Academy of Sciences

Sevak str 1, Erevan 44, USSR

Consulting Editor

D.Yu, Grigorev

Leningrad Branch of the Steklov Mathematical Institute

Fontanka 27, 191011 Leningrad, D-11, USSR

Mathematics Subject Classification (1980): 05XX; 0 5 B X X

ISBN 3-540-16056-6 Springer-Verlag Berlin Heidelberg New York Tokyo

ISBN 0-387-16056-6 Springer-Verlag New York Heidelberg Berlin Tokyo

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a l s o a p p e a r e d : the w o r k of S c a r p i s (1898) p r o v i n g t h a t if p = 3 ( m o d 4)

or p = l ( m o d 4) is a p r i m e n u m b e r t h e n t h e r e is an H a d a m a r d m a t r i x of

o r d e r p+1 a n d p+3, r e s p e c t i v e l y ; the w o r k of H a d a m a r d (1893) w h e r e

n the f o l l o w i n g r e s u l t in p a r t i c u l a r is stated: if A = { a i , j } i , j = 1 ,

lai,jl ~ M , ai, j are r e a l n u m b e r s for a n y i, j, t h e n the a b s o l u t e v a -

t o r i a l p r o b l e m s is the lack of u n i f i e d m e t h o d s for c o n s t r u c t i o n of H a -

d a m a r d m a t r i c e s of o r d e r 4n for all n The k n o w n m e t h o d s of c o n s -

(2.9)

O n t h e o t h e r h a n d L a g r a n g e t h e o r e m [ 1 2 0 ] s h o w s t h a t e v e r y p o s i t i v e

n u m b e r is r e p r e s e n t a b l e a s t h e s u m o f 4 s q u a r e s o f i n t e g e r s ; m o r e o v e r

a 2) if [qi o ]/2 is odd, then

1!I)i = [m-qi-V(i)-1]o /4, L(2)=i [m+qi+V(i)-1]o /4

I + 2 V ~ I ) + 2 V ~ I ) + 2V~I) = + I

Hence, we have from (2.13) and (2.14

V~ I) + V~I) + V~I = -I,

T H E O R E M 2•5 " Let { W ~ 1 i=I be a W i l l i a m s o n family (Sl,S2, •,Sl,I m,

m) and there is an o r t h o g o n a l d e s i g n of type (Sl,S2, ,s I) of order n

c o n s i s t i n g of e l e m e n t s ~xi, xi~0 Then there e x i s t s an H a d a m a r d m a t r i x

b) there e x i s t s a W i l l i a m s o n family (W , W2, W3, W4, I2m,

P R O P E R T Y 2.3 Let (WI, W2, W3, W4, Im, m) be a W i l l i a m s o n family

Then there e x i s t s a W i l l i a m s o n family (W , W2, W3, W4, I2m,

N o w let us i n t r o d u c e a t h e o r e m about e x i s t e n c e of W i l l i a m s o n fami-

lies special cases of which were p r o v e d in [44]

n-1

T H E O R E M 2.6 • Let {Wi}4 i=I ' Wi = Z Ai, ~ U k , n is an odd number,

k=0

Put

W i : V i ( A I , A 2 , B I , B 2 ) , Wi+ 2 = V i ( A 3 , A 4 , B I , B 2) , Wi+ 4 = V i ( A I , A 2 , B 3 , B 4) , Wi+ 6 : V i ( A 3 , A 4 , B 3 , B 4 ) , i=1,2

Let us show that (WI, W 2 , , W 8 , Imn , mn) is a W i l l i a m s o n family, that

C O R O L L A R Y 2.3 T h e r e e x i s t 8 - s y m m e t r i c W i l l i a m s o n m a t r i c e s of o r d e r (p+1)mn, w h e r e m,n6L, P ~ 1 ( m o d 4 ) i s a p r i m e power

N o t e t h a t t h e r e w e r e c o n s t r u c t e d 8 - W i l l i a m s o n m a t r i c e s of o r d e r s

- q ( p + 1 ) / 2 , q ~ l ( m o d 4), p ~ 1 ( m o d 4) are p r i m e n u m b e r s (Wallis [273]) -7 i+I , 11 7 • i , i=1,2, (Wallis, [219])

l i a m s o n m a t r i c e s N o w let us v e r i f y v a l i d i t y of items 2,3,5 a n d 6 C a l -

S e i d e l a r r a y of o r d e r 4p, then there e x i s t s a BX[4pt] array

For p=1 theore, 2.14 c o i n c i d e s w i t h the C o o p e r - W a l l i s t h e o r e m In-

§ 3 some p r o b l e m s of c o n s t r u c t i o n for H a d a m a r d m a t r i c e s

In t h i s p a r a g r a p h we w i l l g i v e a s u r v e y of g e n e r a l a p p r o a c h e s to the c o n s t r u c t i o n s for c l a s s i c H a d a m a r d m a t r i c e s n a m e l y , G o l a y - T u r y n ,

P l o t k i n a n d W a l l i s a p p r o a c h e s L a t e r t h e s e a p p r o a c h e s w i l l be g e n e r a l i - zed a n d s t r e n t h e n e d , in p a r t i c u l a r a r o r r e l a t i o n b e t w e e n g e n e r a l i z e d

m-j

Z i V i V i + j = 0 , j = 1 , 2 , , m - I (3.1) i=

3 If Nx(j) = 0 f o r e v e r y j, t h e n Px(j) = 0 R e v e r s e s t a t e m e n t is

n o t true

L E M M A 3.2 If there e x i s t 4 - s u p p l e m e n t a r y s e q u e n c e s of l e n g t h n, then t h e r e e x i s t 4 ( t + 1 ) - s u p p l e m e n t a r y s e q u e n c e s of l e n g t h n, t = I , 2 , PROOF F r o m s t a t e m e n t 3.2 a n d lemma 3.1 we h a v e four 4 ( t + 1 ) - d i m e n -

s i o n a l o r t h o g o n a l ( - 1 , + 1 ) - v e c t o r s N o w by s u b s t i t u t i o n ( 4 t + 1 ) - o r t h o g o - nal v e c t o r s for o r t h o g o n a l 4 - d i m e n s i o n a l v e c t o r s of ~ ( 4 , n ) - s e q u e n c e , we get a s e q u e n c e of l e n g t h n One can o b t a i n that the s e q u e n c e r e c e i v e d

w h e r e ai, bi, ci, di, i = 1 , 2 , , m , a r e c o o r d i n a t e s of the v e c t o r s X I,

X 2 , X 3 , X 4 r e s p e c t i v e l y • A c c o r d i n g to t h e o r e m 3.2 t h e y s a t i s f y the c o n d i -

L e t u s p r o v e t h a t m a t r i c e s Xi, Yi' Wi' Zi s a t i s f y t h e c o n d i t i o n s

pQ = Qp P ( B k X B n ) Q T = Q ( B k X B n ) P T

(3.20)

P ' Q 6 { X i ' Yi' Zi' Wi} "

V e r i f y the c o n d i t i o n (3.20) for i=I

X I Y I = X ~ Y o x K 1 + X o Y o x K i K 2 - Y o y o x K 2 K I - Y o X o x K 2 ,

2 T + X ~ X o x K 2 K 1 - X o Y o X K 2 YiX1 = y o X o x K I _ y o y o × K i K 2 T T 2

- c C B T + b d~ ~CA T c a ~BD T - c b .BC T + + b i b i + j c c T b i l+ 3 i l+ 3 - 1 l+ 3 i l+ 3

N O T E 3 6 I G ( t 1 , ~ 1 , , t l ) - m a t r i x is a n H a d a m a r d m a t r i x o f o r d e r

n H a d a m a r d m a t r i x G ( I , 1 , , I ) l e t u s c a l l a m a t r i x g e n e r a t e d D ( n , m ) -

d e c o m p o s i t i o n

> {

( V - 3 ) 2 / 4 , for v ~ 1 ( m o d 4) ( v - 5 ) 2 / 8 , for v ~ 3 ( m o d 4)

n u m b e r of n e w r o w s a n d c o l u m n s for r e d u c t i o n to a H a d a m a r d m a t r i x The p r o b l e m arose: to s y n t h e s i z e an H a d a m a r d m a t r i x f r o m the same

n u m b e r s ~i a n d on t h e s e m a t r i c e s to b e c o m e the s q u a r e m a t r i x

k

H (n+1) = X ~ X x H.(n) ,

{x~i)}m-1 {X 2~ (i)}m-1 {x~i) }m-1

s a t i s f y i n g t h e c o n d i t i o n s of t h e t h e o r e m L e t m be a n o d d n u m b e r

(for t h e v e n m t h e p r o o f is a n a l o g o u s )

R e w r i t e (4.22) in f o r m

x I c {xli) m-1 (i)}m-1 , {x~i)}m-1

}i=0 , X 2 6 iX 2 i=0 '''" Xl £ i=0

r e s p e c t i v e l y

The m a t r i c e s Qi' i = 0 , 1 , , m - 1 c o n s t r u c t e d w i l l s a t i s f y the e q u a - lity Qi = Q m - i ' since X i = Xm_i, i = I , 2 , , m - I So, since Qi are o b t a - ined f r o m A - m a t r i c e s (4.25) can be r e w r i t t e n in f o r m

T h a t is the c o n d i t i o n (4.8) of t h e o r e m 4.1 holds So we o b t a i n t h a t

m a t r i x

m-1

H = I U i × Q i i=0

is the b l o c k - c i r c u l a n t H a d a m a r d matrix The b l o c k - s y m m e t r i c c o n d i t i o n

f o l l o w s f r o m e q u a l i t y

Qi = Q m - i ' i = I , 2 , ° , m - I

The s u f f i c i e n c y is p r o v e d

C O R O L L A R Y 4.1 Let Ixi(J) I = 1, i = I , 2 , , 1 , j = 1 , 2 , , m - 1

I) x 1 ( 0 ) x 1 ( J ) + x 2 ( 0 ) x 2 (j) + + X l ( 0 ) X l (j) = -1/2, l=2p

(4.26) 2) x1(i)x1(J) + x 2 ( i ) x 2 (j) + + x l ( i ) X l (j) = 0, i~j, i=I,2 ,

m-1

L e t a l s o there e x i s t s d e p e n d e n t on 1 p a r a m e t e r s A - m a t r i x Q ( X I , X 2 , .,X I) of o r d e r mk

N o w let us try to find the c o n d i t i o n s a l l o w i n g to p i c k out the mi-

n i m a l n u m b e r of (different) m a t r i c e s f r o m the set of m a t r i c e s g e n e r a t e d

+ X 2 X 5 x K 2 - X 5 X 2 × K 2 + X 4 X 3 x K 2 - X 3 X 4 x K 2 + X 3 X 5 x K 4 - X 5 X 3 x K 4 +

5 + X 4 X 5 x K 3 - X 4 X 5 x K 3 + Z X i X x H i H = 2 [ ( X 2 x Q 2 + x 2 x Q ] +

5 + x5x (Q7 + O~) ] + 4 Z x i X ~ x I 4 = - 4 ( X 2 + X3 + X 4 + x 5) xI 4 +

i=1

+ 4i=]XiXixI4X T = 4 ( - X X i + Z X i X ~ + I k ) x I 4

F r o m (4.29) w e h a v e

H ( n + I ) H T ( n + I) =

k

Y X , X T x H , H T + i=1 i i i i

H i + l , 3 = X I x Hi, 3 - X 2 x Hi, 4 - X 3 x Hi, I + X 4 x Hi, 2 ,

T+ X3X[~ XlX[ X4X~

that is, the n e c e s s i t y is proved

It is easy to prove the sufficiency too

4 L e t k = 2 a n d H 1 , H 2 b e i n c o m p l e t e H a d a m a r d m a t r i c e s o f o r d e r ( m , m / 2 ) T h e n t h e r e l a t i o n s (4.37) a n d (4.45) c a n be r e w r i t t e n in f o r m

- G H ( p i+j Cpi) for a l l i > I, j > 0 (Drake, 1979);

C o n s t r u c t e d s e t s X (k) , m k = 1 , 2 , , q a - k - 1 , s a t i s f y t h e f o l l o -

w i n g c o n d i t i o n s :

k-1 a) X (k) =q U -I X(1)

P r o v e t h a t H ' H ' * = 4 k I 4 k ,

AoAI* ÷ AIAo* = 0 ,

AoAo* + AIAI* = 4kI2k ,

The first equation of system (5.14) follows from AoA{* = AIAo*, sin-

§ 6 Construction of Hi~h-dimensional Hadamard

matr ice s

In this paragraph we will give a review and two approaches of construction of high-dimensional (proper and improper) Hadamard mat- rices First of these approaches is new one and the second extents the known "flat" methods to the high-dimensional case and gives so- lution of Shlochta problem Later we will introduce the notations

of density, weight and surplus for (flat and spatial) Hadamard matri-

F u r t h e r , s i n c e

o (2) (I) = (2) (U) = - _0(2) (U n-1 ) = n

s o t

n-1 (2) (2)

(H) = n Z ~ (Qi)

i=0

By a n a l o g y o n e c a n o b t a i n t h e v a l i d i t y of t h i s r e p r e s e n t a t i o n for

S T A T E M E N T 6.5 It is true

3/8 < pw(i) ([HBn ]) < 5/8, i=2,3

-1/4 < Pa(i) ([HBn ]) < I/4, i=2,3

The p r o o f f o l l o w s f r o m items 14 and 15

is the c o m p l e t e l y p r o p e r cubic g e n e r a l i z e d H a d a m a r d m a t r i x [H(3,3) ] 3

l i n e a r for M = 2 - a n d n o n - l i n e a r for d 2 - d + 4 > 4t (Paley c o n s t r u c t i o n s

a r e u s e d here) a n d t h a t t h i s m e t h o d is a d i r e c t one Some p r a c t i c a l

a n d n o i s e l e s s c o d i n g , f i l t r a t i o n , p a t t e r n r e c o g n i t i o n ( m a t h e m a t i c a l

f o r m u l a t i o n of the p r o b l e m w i l l be g i v e n later), w i t h t h a t a n d in v i e w let us n o t e t h a t in s e v e r a l t h e o r e t i c a l a n d a p p l i e d p r o b l e m s an I m p o r -

}N is some f i x e d o r t h o g o n a l b a s e , ~ is a set of type

k

G k = Z ai~i~oit , i=I

{~i } be a n o r t h o g o n a l base, Yi =< x ' ~ i > ' S(';~) = -~p(Y) in p ( Y ) d y The b a s e m i n i m i z i n g e n t r o p y of a r a n d o m s o u r c e w i t h d e n s i t y - o f - p r o b a -

b i l i t y f u n c t i o n p(x) is the b a s e of e i g e n v e c t o r s of m a t r i x C

O b o v e - m e n t i o n e d f o r m u l a t i o n s of p r o b l e m s are of the f u n d a m e n t a l

t h e o r e t i c a l i m p o r t a n c e a n d f i n d w i d e l y p r a c t i c a l a p p l i c a t i o n s So,

d e c o m p o s i t i o n into s i n g u l a r v a l u e s a n d K a r h u n e n - L o e v e d e c o m p o s i t i o n are u s e d for p r o c e s s i n g of d i g i t a l s i g n a l s m a x i m i z i n g R e l a y r e l a t i o n

a d e q u a t e to p r o b l e m of f i l t r a t i o n of l e g i t i m a t e s i g n a l f r o m the n o i s e [323] In t h e s e p r o b l e m s o p t i m a l by c r i t e r i o n S is the b a s e of e i -

ce are t r i g o n o m e t r i c f u n c t i o n s , L i p s h i t s , H e r m i t i a n f u n c t i o n s , a n d so

on R e c e n t l y m u c h a t t e n t i o n is g i v e n to i n v e s t i g a t i o n (both t h e o r e t i - cal a n d p r a c t i c a l ) of W a l s h - H a d a m a r d , Haar, S l a n t f u n c t i o n s [24] W a l s h -

2 in p o i n t X O of c o n t i n u i t y f(x), S (f;x) c o n v e r g e s to f(X O)

2n k

3 if f(x) is c o n t i n u o u s on (0,1) t h e n S (f;x) c o n v e r g e s to

2n k f(x) u n i f o r m l y in X

k m q(i) = E c ( ( j - I) ~ + i) • Y

Let the rows of H a d a m a r d m a t r i c e s H are e i g e n v e c t o r s of m a t r i -

n (n) Pl -I

ces G(n) = {Gi,j }i,j=0

T H E O R E M 7.10 (Matevosyan, 1984) S u p p o s e that elements, the

b l o c k s G (n) i,j of m a t r i c e s G (n) = {G (n) } PI-1 i,j i,j=0 ' n ~ I of o r d e r

[A] 2 = 1'1 ([W] 3 -[X] 2) (7.31)

or (what is the same)

[A] 2 =il Ai,jll =If kZ0Wijk= Xkjll = II k=0Z YN Xk,j i[ (7,32)

The n u m b e r of o p e r a t i o n s (multiplications) r e q u i r e d for r e a l i z a - tion of fast t r a n s f o r m a t i o n from (7.33) is N 2 1 o g N that is twice as little as for r e a l i z a t i o n of k n o w n t w o - d i m e n s i o n a l fast F o u r i e r trans-

f o r m a t i o n ~65]

2 The a l g o r i t h m of t h r e e - d i m e n s i o n a l g e n e r a l i z e d H a d a m a r d t r a n s -