On some applications of Hadamard matrices

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Metrika (2005) : 1–19DOI 10.1007/s00184-005-0415-y

ORIGINAL ARTI CLE

Jennifer Seberry · Beata J Wysocki · TadeuszA Wysocki

On some applications of Hadamard matrices

Received: / Revised: / Published online: 2005© Springer-Verlag 2005

Abstract Modern communications systems are heavily reliant on statistical tech-niques to recover information in the presence of noise and interference. One of themathematical structures used to achieve this goal is Hadamard matrices. They areused in many different ways and some examples are given. This paper concentrateson code division multiple access systems where Hadamard matrices are used foruser separation. Two older techniques from design and analysis of experimentswhich rely on similar processes are also included. We give a short bibliography(from the thousands produced by a google search) of applications of Hadamardmatrices appearing since the paper of Hedayat and Wallis in 1978 and some appli-cations in telecommunications.

1 Introduction

Hadamard matrices seem such simple matrix structures: they are square, haveentries+1 or−1 and have orthogonal row vectors and orthogonal column vectors.Yet they have been actively studied for over 138 years and still have more secretsto be discovered. In this paper we concentrate on engineering and statistical appli-cations especially those in communications systems, digital image processing andorthogonal spreading sequences for direct sequences spread spectrum code divisionmultiple access.

2 Basic definitions and properties

A square matrix with elements±1 and sizeh, whose distinct row vectors are mutu-ally orthogonal, is referred to as anHadamard matrix of order h. The smallestexamples are:

J. Seberry· B.J. Wysocki· T.A. WysockiUniversity of WollongongNorthfields Av., NSW 2522, Australia

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2 Seberry et al.

(1) ,

(1 11 −

),

1 1 1 1− 1 − 11 1 − −− 1 1 −

where− denotes−1. Such matrices were first studied by Sylvester (1867) whoobserved that ifH is an Hadamard matrix, then

(H HH −H

)

is also an Hadamard matrix. Indeed, using the matrix of order 1, we have:

Lemma 2.1 (Sylvester (1867)) : There is an Hadamard matrix of order 2t for allnon-negative t .

The matrices of order 2t constructed using Sylvester’s construction are usuallyreferred to as Sylvester-Hadamard matrices. The Sylvester-Hadamard matrices areassociated with discrete orthogonal functions called Walsh functions which will bediscussed later in the paper.

Hadamard (1893) gave examples for a few small orders. An example of order12 is as follows:

H12 =

1 1 1 − 1 1 − 1 1 − 1 11 1 1 1 − 1 1 − 1 1 − 11 1 1 1 1 − 1 1 − 1 1 −1 − − 1 1 1 − 1 1 1 − −− 1 − 1 1 1 1 − 1 − 1 −− − 1 1 1 1 1 1 − − − 11 − − 1 − − 1 1 1 − 1 1− 1 − − 1 − 1 1 1 1 − 1− − 1 − − 1 1 1 1 1 1 −1 − − − 1 1 1 − − 1 1 1− 1 − 1 − 1 − 1 − 1 1 1− − 1 1 1 − − − 1 1 1 1

.

Some systematic constructions of Hadamard matrices different to Sylvester’s con-struction will be given in the next section. The following two Lemmas give somebasic properties of Hadamard matrices.

Lemma 2.2 Let Hh be an Hadamard matrix of order h. Then:

(i) HhHth = hIh, where Ih is the identity matrix of order h;

(ii) | detHh| = h12h;

(iii) HhHth = Ht

hHh;(iv) Hadamard matrices may be changed into other Hadamard matrices by per-

muting rows and columns and by multiplying rows and columns by −1.Matrices which can be obtained from one another by these methods arereferred to as H -equivalent (not all Hadamard matrices of the same orderare H -equivalent);

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On some applications of Hadamard matrices 3

(v) every Hadamard matrix is H -equivalent to an Hadamard matrix which hasevery element of its first row and column equal to +1 — matrices of thislatter form are called normalized;

(vi) if H4n is a normalized Hadamard matrix of order 4n, then every row (col-umn), except the first, has 2n minus ones and 2n plus ones in each row(column);

(vii) the order of an Hadamard matrix is 1, 2, or 4n, where n is a positive integer.The first unsolved case is order 668. The previous smallest unsolved case,428, was found in 2004 by Kharaghani and Tayfeh-Rezaie Kharaghani andTayfeh-Rezaie (2004).

Lemma 2.3 (Sylvester) Let H1 and H2 be Hadamard matrices of orders h1 and h2,then the Kronecker product of H1 and H2 is an Hadamard matrix of order h1h2.

Hadamard in (1893) considered matrices with entries on the unit circle, and provedthat these matrices,X = [xij ]n×n, satisfied the following inequality:

| detX|2 ≤n∏

i=1

n∑j=1

|xij |2. (1)

If they satisfied the equality of (1) then they satisfiedXXt = In and had maximaldeterminantn

n2 . However, Hadamard’s name has become associated only with the

matrices having their entries equal to±1.

3 Some useful constructions of Hadamard matrices

Hadamard matrices have a very wide variety of applications in modern commu-nications and statistics. A construction technique of crucial importance in oneapplication may be less significant in another application. There is a close rela-tionship between Hadamard matrices constructed using Sylvester’s technique (seeLemma 1) and Walsh functions that are often used in engineering applications,including communication systems and digital image processing. Apart from Syl-vester’s construction there are several other systematic ways of constructing suchmatrices. Good lists of those techniques can be found in Geramita and Seberry(1979), Hedayat et al. (1999) and the listing of the matrices can be found on thehome pages maintained by Seberry (2004), and Sloane (2004).

In this section, we want to introduce some of those techniques that lead to largefamilies ofH -inequivalent matrices sinceH -inequivalent matrices, and evenH -equivalent matrices, can have quite different computational properties in someof the applications discussed and affect system performance. We present heretwo such construction techniques: Golay-Hadamard matrices, and Williamson-Hadamard matrices. More construction techniques can be found in Geramita andSeberry (1979). In addition, we will give a technique leading to the constructionof pairs of amicable Hadamard matrices from circulant matrices. These amicableHadamard matrices can be used to design complex four-phase orthogonal matriceshaving both real and imaginary components independently orthogonal. Hadamardmatrices can be used also to design error correcting block codes which can cor-rect large numbers of errors in communications. For definitions not given here the

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4 Seberry et al.

interested reader is referred to Seberry and Yamada (1992) or Koukouvinos andKounias (1989).

3.1 Construction of Hadamard matrices using Golay sequences

In Seberry et al. (2002), we have studied the use of Golay complementary sequencesto construct Hadamard matrices for use in direct sequence spread spectrum (DS SS)code division multiple access (CDMA) communication systems. For a pair of Go-lay complementary sequencesS1 andS2, the sum of their aperiodic autocorrelationfunctions equals to zero, except for the zero shift Fan and Darnell (1966):

cS1(τ ) + cS2(τ ) = 0, τ �= 0 (2)

wherecS1(τ ) andcS2(τ ) denote the aperiodic autocorrelation functions Fan andDarnell (1966).

It can be proven Seberry andYamada (1992), Wallis et al. (1972) that if matricesA andB are the circulant matrices created from a pair of Golay complementarysequencesS1 andS2 then the matrix

G =(

A BBt −At

)(3)

is an Hadamard matrix. We write in more detail about circulant matrices and howto construct Hadamard matrices using a circulant core in Section 3.3.

Golay complementary sequences have been defined as bipolar sequences, andone can generate bipolar Golay sequences of all lengthsN , such that

N = 2a10b26c (4)

wherea, b, c are non-negative integers. More recent research by a number ofauthors have relaxed the condition of bipolarity. Another technique, which can beemployed to construct Hadamard matrices from bipolar complementary sequences,is based on the Goethals-Seidel array (1970). In (1989), Koukouvinos and Kou-nias, expanding on the work of Turyn (1974), used two Golay complementarysequences,U andV , of length�1 and two Golay complementary sequences,X, Y ,of length�2 to note that

A = [U, X]B = [U, −X]C = [V, Y ]D = [V, −Y ]

are four complementary sequences of length�1 + �2, and that there is a Golay-Hadamard matrix of order

N = 4(�1 + �2)

constructed from Goethals-Seidel array. This technique produces many moreGolay-Hadamard matrices than the construction given by Eq. (3), Craigen et al.(2001).

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3.2 Williamson-Hadamard construction

Theorem 3.1 [Williamson (1944)] Suppose there exist four symmetric (1, −1)matrices A, B, C, D of order n which satisfy

XY t = YXt, X, Y ∈ {A, B, C, D}.Further, suppose

AAt + BBt + CCt + DDt = 4nIn. (5)

Then, using A, B, C, D in the Williamson array H given by

H =

A B C D−B A −D C−C D A −B−D −C B A

(6)

gives an Hadamard matrix of order 4n.

Williamson (1944) developed number theoretic techniques which allow computersearches for these matrices to be made far more efficiently but still in non polyno-mial time. An extensive list of sequences to generate circulant matrices to fill theWilliamson array, to get Hadamard matrices of orders 12 to 252, can be found inSeberry et al. (2003).

3.3 Amicable Hadamard matrices

We now briefly discuss the construction of pairs of of amicable Hadamard matricesas these matrices can be used to design complex four-phase orthogonal matriceshaving both real and imaginary components independently orthogonal.

Two square matricesM andN of ordern are said to be amicable ifMNt =NMt . SupposeI +S andN are amicable Hadamard matrices of ordern satisfying

(I + S)Nt = N(I + S)t , St = −S, Nt = N (7)

then(

I + S N

−N I + S

)and

(N N

N −N

)are amicable Hadamard matrices of order 2n.

Example 3.1 Let S =(

0 1− 0

)andN =

(1 11 −

), thenI + S andN are amicable

Hadamard matrices of order 2. Hence amicable Hadamard matrices of this typeexist for all orders 2t .

An Hadamard matrix,A, of ordern are said to be constructedusing a circulantcore if it uses a suitably use a bordered matrixC = c(i, j) of ordern−1, satisfyingCCt = nI − J , CJ = J , wherec(i, j) = c(1, j − i + 1) wherej − i + 1 isreduced(mod n − 1). For example

C = 1 1 −

− 1 11 − 1

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is a circulant core and

A =

1 1 1 1− 1 1 −− − 1 1− 1 − 1

is an Hadamard matrix constructed using a circulant core.An Hadamard matrix,B, of ordern is said to be constructedusing a back circu-

lant core if it uses a suitably bordered matrixD = d(i, j) of ordern−1 satisfyingDDt = nI − J , DJ = J , whered(i, j) = d(1, j + i − 1) wherej + i − 1 isreduced(mod n − 1). For example

D = 1 1 −

1 − 1− 1 1

is a back circulant core and

B =

− 1 1 11 1 1 −1 1 − 11 − 1 1

is an Hadamard matrix constructed using a back circulant core.

Theorem 3.2 An Hadamard matrix of order p + 1 can be constructed with circu-lant core and with back circulant core when

1. p ≡ 3(mod 4) is prime Paley (1933);2. p = q(q + 2), where q and q + 2 are both prime Stanton and Sprott (1958),

Whiteman (1962);3. p = 2t − 1, where t is a positive integer Singer (1938);4. p = 4x2 + 27, where p is prime and x a positive integer Hall (1967).

Theorem 3.3 [(Wallis et al., 1972, p300)] An Hadamard matrix of order n con-structed using a circulant core A and an Hadamard matrix of order n constructedusing a back circulant core B are amicable. That is ABt = BAt .

Proof We note that ifX is a circulant core andY is a back circulant core, both ofordern−1, thenXY t = YXt . LetJ be the matrix of all 1s. SupposeXJ = YJ = Y .(This can certainly be done for if, sayXJ = −J , we would merely replaceX byits negative.) Further supposeXXt = YY t = nJ − J , (which is a property ofbeing the core of an Hadamard matrix) then

A =

1 1 · · · 1−... X−

andB =

− 1 · · · 11... Y1

can easily be checked to satisfyABt = BAt and so are amicable Hadamard matri-ces.

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This means there are amicable Hadamard matrices of order 2t∏

(p+1) for anyintegert andp given in Theorem 3.2. Using the amicability property, the followingtheorem follows directly:

Theorem 3.4 If matrices A and B are amicable Hadamard matrices of order n,then a matrix X = A + jB, j2 = −1, is a complex orthogonal matrix, i.e.XXH = 2nIn, where (·)H is the Hermitian conjugate.

This leads to the easy design of complex four-phase orthogonal matrices havingboth real and imaginary components independently orthogonal.

4 Applications of Hadamard matrices in telecommunicationsand signal processing

Hadamard matrices have found many applications in modern telecommunicationsand digital signal processing Yarlagadda and Hershey (1997). In this section, wewill briefly discuss some of those applications in error-control coding Peterson(1961), and to define Walsh functions Walsh (1923) that can be used for signalmodelling Harmuth (1960) and transformation of information in image compres-sion and image encoding Jain (1989). We will also consider the use of Hadamardmatrices to design spreading sequences for application in code division multipleaccess (CDMA) direct sequence (DS) spread spectrum (SS) systems Lam andTantaratana (1994).

4.1 Error-control codes derived from Hadamard matrices

Error correcting capabilities of codes derived from Hadamard matrices were firstpointed out by Plotkin in 1951 Plotkin (1960). This has been later studied byBose and Shrikhande (1959), who have shown the connection between Hadamardmatrices and symmetrical block code designs and later extended by Harmuth in(1960).

Theorem 4.1 [Peterson (1961)] If there exists an n × n Hadamard matrix, thenthere exists a binary code with 2n code vectors, and minimum distance n/2. (Thisis not necessarily a linear code.)

The block code designed from an Hadamard matrixH has the following structure:(

H−H

).

Because the minimum distance for such a code is equal ton/2, codes derived fromHadamard matrices have maximal error correcting capability for a given length ofa codeword. These codes were used in the 1960’s in the Mariner and Voyager spaceprobes to encode information transmitted back to the Earth while the probes vis-ited Mars and the outer planets of the solar system. Due to the enormous distancebetween those planets and the Earth, not only does it take the signal a consider-able amount of time to come to the Earth but the signal is also very weak. Only

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8 Seberry et al.

because of the powerful error-correction capabilities of those codes, was it possibleto decode properly the glorious high quality pictures of Jupiter, Saturn, Uranus,Neptune and their moons. As a matter of fact, application of Hadamard matricesto design powerful error-correcting codes was one of the factors in an increasedinterest in finding new constructions of Hadamard matrices, e.g. Ang et al. (2003),Baumert and Hall (1965), Turyn (1972).

4.2 Walsh functions

There are a couple of formal definitions of Walsh functions introduced in the liter-ature (e.g. Szabatin (1982), Yarlagadda and Hershey (1997)). Here, we give theirdefinition on the interval [0, 1].

Wal(0, t) = x0(t) = 1, 0 ≤ t ≤ 1

Wal(1, t) = x1(t) ={

1, 0 ≤ t < 0.5−1, 0.5 ≤ t ≤ 1

Wal(2, t) = x2(t) ={

1, 0 ≤ t < 0.25 and 0.75 ≤ t ≤ 1−1, 0.25 ≤ t < 0.75

Wal(3, t) = x3(t) ={

1, 0 ≤ t < 0.25 and 0.5 ≤ t < 0.75−1, 0.25 ≤ t < 0.5 and 0.75 ≤ t ≤ 1

(8)

and recursively form = 1, 2, · · · andk = 1, · · · , 2m−1 we have:

Wal(2m−1 + k − 1, t) = xkm(t) (9)

x2k−1m+1 (t) =

{xk

m(2t), 0 ≤ t < 0.5(−1)k+1xk

m(2t − 1), 0.5 ≤ t ≤ 1

xkm+1(t) =

{xk

m(2t), 0 ≤ t < 0.5(−1)kxk

m(2t − 1), 0.5 ≤ t ≤ 1

(10)

These recursive formulae can be used to define Walsh functions for any value ofm andk, and the plots of first eight Walsh functions are presented in Figure 1.The Walsh functions can also be defined using the Sylvester-Hadamard matrices.Using such a matrix of order 2m, H2m , we can define the functionsWal(0, t) toWal(2m−1, t) in the following way. If we divide the interval [0, 1] into 2m intervals:

[0,

1

2m

),

[1

2m,

2

2m

), · · ·

[2m − 1

2m, 1

),

then the Walsh function defined by theith row of a matrixH2m takes the value of‘1’ in the interval [k/2m, (k + 1)/2m] if :

H2m(i, k + 1) = 1 (11)

otherwise it takes the value of ‘−1’.

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Wal(3,t)

t10

Wal(0,t)

t10

t10

t10

t10

t10

t10

t10

Wal(1,t)

Wal(2,t)

Wal(4,t)

Wal(5,t)

Wal(6,t)

Wal(7,t)

Fig. 1 The First 8 Walsh functions

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10 Seberry et al.

The following example illustrates the usefulness of the described method togenerate Walsh functions. The matrixH8 is obtained as:

H8 =(

H4 H4H4 −H4

)=

H2 H2 H2 H2H2 −H2 H2 −H2

H2 H2 −H2 −H2H2 −H2 −H2 H2

=

1 1 1 1 1 1 1 11 − 1 − 1 − 1 −1 1 − − 1 1 − −1 − − 1 1 − − 11 1 1 1 − − − −1 − 1 − − 1 − 11 1 − − − − 1 11 − − 1 − 1 1 −

.

It is easy to see by comparing the rows ofH8 with Fig.1 that the mappingbetweenH8 and the Walsh functions is as follows:

H8 =

1 1 1 1 1 1 1 11 − 1 − 1 − 1 −1 1 − − 1 1 − −1 − − 1 1 − − 11 1 1 1 − − − −1 − 1 − − 1 − 11 1 − − − − 1 11 − − 1 − 1 1 −

Wal(0, t)Wal(7, t)Wal(3, t)Wal(4, t)Wal(1, t)Wal(6, t)Wal(2, t)Wal(5, t)

In general, theith row of the matrixH2m defines thekth Walsh functionWal(k, t)wherek is the number of sign changes in the elements of that row.

The Walsh functions constitute a complete set of orthogonal functions Walsh(1923), which means that every function defined on the interval [0, 1] and havingfinite norm can be expanded into the generalized Fourier series [26] using Walshfunctions.

4.3 Direct sequence spread spectrum CDMA systems

Bipolar spreading sequences derived from Sylvester-Hadamard matrices by simplyconsidering each row of the matrix as a spreading sequence can be used for channelseparation in direct sequence code division multiple access (DS CDMA) systems,e.g. Steele (1999). Because of the connection between Sylvester-Hadamard matri-ces and the Walsh functions described in the previous section, these sequences areoften referred to as Walsh-Hadamard sequences in literature dealing with commu-nication systems. Because of the simplicity of Sylvester’s construction, they areeasy to generate. The spread spectrum signals are of course orthogonal Harmuth(1960) in the case of perfect synchronization, which means that signals spread usingdifferent sequences can be perfectly resolved at the receiver. However, the aperiodic

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Fig. 2 Block diagram of a basic DS CDMA system

cross-correlation between two Walsh-Hadamard sequences can rise considerablyin magnitude if there is a non-zero delay shift between them. Unfortunately, thisis very often the case for up-link (mobile to base station) transmission, due to thedifferences in the corresponding propagation delays.

A block diagram of a conventional Direct Sequence Code Division MultipleAccess (DS CDMA) system Lam and Tantaratana (1994) is shown in Figure 2.The first block represents the data modulation of a carrierc(t). Usually, this is ofthe formc(t) = Acos(ω0t) and the modulation is either binary phase shift keying(BPSK) or quaternary phase shift keying QPSK, however, there is no restrictionplaced either on the waveform or the modulation type.

In the case of an angular modulation, the modulation signal of user 1,ζ1(t), isexpressed as

ζ1(t) = Acos[ω0t + φ1(t) + φ0]

whereA is the amplitude of the modulated signal,ω0 = 2π/T0 is the angularcarrier frequency,T0 is a period of the carrier,φ1(t) represents the informationcarrying phase function andφ0 is an initial value of the phase.

Next, the signalζ1(t) is multiplied by the spreading signalg1(t) of the user 1being the physical realization of the spreading sequence{s(1)

n }. The resulting sig-nalg1(t)ζ1(t) is then transmitted over the radio channel. Simultaneously, all otherusers 2 throughM multiply their signals by their own spreading signals. The signalR(t) intercepted in the receiver antenna, neglecting the different path losses andchannel noise, is given by:

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12 Seberry et al.

R(t) = g1(t − τ1)ζ1(t − τ1)

+g2(t − τ2)ζ2(t − τ2) + · · · + gM(t − τM)ζM(t − τM)

whereτi, i = 1, 2, . . . , M, denotes delays corresponding to different transmissionpaths associated with the useri.

Assuming the receiver is configured to receive message from user 1, the de-spread signalw1(t) is given by

w1(t) = g21(t − τ1)ζ1(t − τ1) + · · · + g1(t − τ1)gM(t − τM)ζM(t − τM)

whereg21(t−τ1)ζ1(t−τ1) is the desired signal and the other terms are the interfering

signals responsible for the multi-access interference (MAI).For general polyphase sequences{s(i)

n } and {s(�)n } of length N , the discrete

aperiodic correlation function is defined as (see Fan and Darnell (1966)):

ci,k(τ ) =

1N

∑N−1−τn=0 s(i)

n [s(�)n+τ ]∗, 0 ≤ τ ≤ N − 1

1N

∑N−1+τn=0 s

(i)n−τ [s(�)

n ]∗, 1 − N ≤ τ < 0

0, |τ | ≥ N

(12)

where [∗] denotes a complex conjugate operation. When{s(i)n } = {s(l)

n }, Eq.(12)defines the discrete aperiodic auto-correlation function.

Another important parameter used to assess the synchronization amicability ofthe spreading sequence{s(i)

n } is a merit factor, or a figure of merit Golay (1982),which specifies the ratio of the energy of autocorrelation function main-lobes tothe energy of the auto-correlation function side-lobes in the form:

F = c(0)i

2∑N−1

τ=1 |ci(τ )|2 (13)

In DS CDMA systems, we want to have the maximum values of aperiodic cross-correlation functions and the maximum values of out-of-phase aperiodic autocor-relation functions as small as possible, while the merit factor as great as possiblefor all of the sequences used.

The bit error rate (BER) in a multiple access environment depends on the mod-ulation technique used, demodulation algorithm, and the signal-to-noise powerratio (SNR) available at the receiver. Pursley (1977) showed that in case of a BPSKasynchronous DS CDMA system, it is possible to express the average SNR at thereceiver output of a correlator receiver of theith user as a function of the averageinterference parameter (AIP) for the otherK users of the system, and the powerof white Gaussian noise present in the channel. The SNR forith user, denoted asSNRi , can be expressed in the form:

SNRi = N0

2Eb

+ 1

6N3

K∑k=1,k �=i

ρk,i

−0.5

(14)

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whereEb is the bit energy,N0 is the one-sided Gaussian noise power spectraldensity, andρk,i is the AIP, defined for a pair of sequences as

ρk,i = 2µk,i(0) + Re{µk,i(1)} (15)

The cross-correlation parametersµk,i(τ ) are defined by:

µk,i = N2N−1∑

n=1−N

ck,i(n)[ck,i(n + τ)]∗. (16)

However, following the derivation in Karkkainen (1992),ρk,i for polyphase se-quences may be well approximated as:

ρk,i ≈ 2N2N−1∑

n=1−N

|ck,i(n)|2 (17)

In order to evaluate the performance of a whole set ofM spreading sequences, theaverage mean-square value of cross-correlation for all sequences in the set, denotedby RCC , was introduced by Oppermann and Vucetic (1997) as a measure of the setcross-correlation performance:

RCC = 1

M(M − 1)

M∑i=1

M∑k=1,k �=i

N−1∑τ=1−N

|ck,i(τ )|2 (18)

A similar measure, denoted byRAC was introduced in Oppermann and Vucetic(1997) for comparing the auto-correlation performance:

RAC = 1

M

M∑i=1

N−1∑τ=1−N,τ �=0

|ci,i(τ )|20 (19)

The measure defined by (19) allows for comparison of the auto-correlation prop-erties of the set of spreading sequences on the same basis as the cross-correlationproperties. It can be used instead of the figures of merit, which have to be calculatedfor the individual sequences.

For DS CDMA applications we want both parametersRCC andRAC to be aslow as possible Oppermann and Vucetic (1997). Because these parameters char-acterize the whole sets of spreading sequences, it is convenient to use them as theoptimization criteria in the design of new sequence sets.

The measures defined by Eqs (18) and (19) are very useful for large sets ofsequences and large number of active users, when the constellation of interfer-ers (i.e. relative delays among the active users and the spreading sequences used)changes randomly for every transmitted information symbol. However, for a morestatic situation, when the constellation of interferers stays constant for the dura-tion of many information symbols, it is also important to consider the worst-casescenarios. This can be accounted for by analyzing the maximum value of peaks inthe aperiodic cross-correlation functions over the whole set of sequences. Hence, anadditional measure to compare the spreading sequence sets needs to be consideredSeberry et al. (2003):

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14 Seberry et al.

– The maximum value of the aperiodic cross-correlation functionsCmax

cmax(τ ) = maxi = 1, . . . , Mk = 1, . . . , M

i �= k

|ci,k(τ )|; τ = (−N + 1), . . . , (N − 1)

Cmax = maxτ {cmax(τ )}(20)

In (2002), Wysocki and Wysocki have shown that by applying the property (iv)of Lemma 2, one can alter aperiodic correlation functions for the whole set ofthe spreading sequences. For example, the sequences derived from the normalizedSylvester-Hadamard matrix of order 32 have the following correlation parameters:

Cmax = 0.9688RAC = 6.5938RCC = 0.7873.

On the other hand, the sequences derived from anH -equivalent matrix obtainedby multiplying the Sylvester-Hadamard matrix of order 32 by a diagonal matrixwith the main diagonal

diag = [11 − −1 − 1 − −1 − − − 1111− −111111111111]

are characterized by the following set of the parameters:

Cmax = 0.4063RAC = 0.8925RCC = 0.9738.

The significant reduction in the value ofCmax translates to the improvement inthe bit error rate for the asynchronous CDMA system utilizing such spreadingsequences.

The interesting research question is whether one can further reduce the valueof Cmax by choosing a different diagonal matrix or by permuting the columns ofthe Hadamard matrix. Theoretically, the minimum value ofCmax can be computedusing the Levenshtein bound Levenshtein (1997), which for the case of bipolarsequences of length 32 equals 0.1410. The bound is, however, obtained withoutassuming orthogonality of sequences for their perfect alignment, and as such canbe considered only as a remote target. To find the absolute answer to the questionof a minimum value ofCmax for the given starting set of sequences, one needsto perform an exhaustive search of all possible diagonal matrices and all possiblepermutation of columns. Unfortunately, even for the modest matrix order of 32 thisis not a trivial task that can be completed, at the current level of computing technol-ogy, in a reasonable time. To date, the value ofCmax = 0.4063 is the lowest valueobtained for sequences derived from the Sylvester-Hadamard matrix of order 32.

In our successive works Seberry et al. (2002), Seberry et al. (2003), Ang et al.(2003), Seberry et al. (2004), we considered application of Hadamard matrices con-structed using different techniques to design spreading sequences for DS CDMAapplications. It is interesting to note that the value ofCmax depends on the con-struction method used to obtain the non-modified Hadamard matrix. For example,in case of sequences derived from the Hadamard matrix constructed using Hall’sdifference set Hall (1956) that value can be as low as 0.3750 Seberry et al. (2004).

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5 Some other applications of Hadamard matrices

5.1 Screening Properties of Hadamard Matrices

An array on two symbols withN rows andk columns is a(N, k, p) screeningdesign if for each choice ofp columns, each of the 2p row vectors appears at leastonce. Screening designs are useful for situations where a large number of factors(q) is examined but only a few(k) of these are expected to be important. Screeningdesigns that arise from Hadamard matrices have traditionally been used for identi-fying main effects only, because of their complex aliasing structures. Without lossof generality we can assume that the first column of a Hadamard matrix containsonly 1’s. Then by removing this column we obtain(N, N −1, p) screening design,with p ≥ 2. Some screening designs of this form were introduced by Plackett andBurman (1946) and they are usually referred to as Plackett-Burman designs.

Further interesting papers on the construction of supersaturated screening de-signs are given by Lin (1993a), Tang and Wu (1997) and Wu (1993).

5.2 Hadamard matrices and optimal weighing designs

Suppose we are givenp objects to be weighed inn weighings with a chemicalbalance (two-pan balance) having no bias. Let:

xij = 1 if the j th object is placed in the left pan in theith weighing,xij = −1 if the j th object is placed in the right pan in theith weighing. Then

then × p matrixX = (xij ) completely characterizes the weighing experiment.Let us denote the true weights of thep objects byw1, w2, . . . , wp, and by

y1, y2, . . . , yn the results ofn weighings (so that the readings indicate that theweight of the left pan exceeds that of the right pan byyi in theith weighing). Then,denoting the column vectors ofw’s andy’s by W andY , respectively, the readingscan be represented by a linear model:

Y = XW + e (21)

wheree is the column vector ofe1, e2, . . . , en andei is the error between observedand expected readings. We assume thate is a random vector with a zero mean anda covariance matrixσ 2I . This is a reasonable assumption in the case where objectsto be weighed have small mass compared to the mass, and hence the inertia, of thebalance itself.

Hotelling (1944) showed that for any weighing design the variancewi cannotbe less thanσ 2/n. Therefore, we shall call a weighing designX optimal, if it esti-mates each of the weights with this minimum varianceσ 2/n. Kiefer’s work Kiefer(1975) shows that a chemical balance weighing designX is optimal if it is ann×pmatrix of ±1 whose columns are orthogonal, and are taken from an Hadamardmatrix.

6 Conclusions

Hadamard matrices have a very wide variety of applications in modern commu-nications and statistics. A construction technique of crucial importance in one

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16 Seberry et al.

application may be less significant in another application. EvenH -equivalent Had-amard matrices possess different properties that can lead to different system perfor-mance. We give a short bibliography of applications of Hadamard matrices, chosenfrom the 44 thousand produced by a web search search, to indicate some furtherapplications that have appeared in the literature since Hedayat and Wallis (1978).

7 Bibliography on Hadamard matrices and their applications

AGAI Agaian S S (1985) Hadamard matrices and their applications, LectureNotes in Mathematics, vol.1168, Springer-Verlag, Springer-Verlag,Berlin-Heidelberg-New York.

ARAY Araya M, Harada M and Kharaghani H (2004) Some Hadamard matricesof order 32 and their binary codes, Journal of Combinatorial Designs, 12(2): 142–146.

BOUS Boussakta S and Holt J GA (1989) Fast algorithm for calculation of bothWalsh-Hadamard and Fourier transforms (FWFTs), Electron. Lett., 25(20): 1352–1353.

EVAN Evangelaras H, Koukouvinos C and Seberry J (2003) Applications ofHadamard matrices, J Telecommunications and IT, 2:2–10.

FAND Fan P and Darnell M (1966) Sequence design for communications appli-cations, John Wiley & Sons, New York.

GEAD Geadah Y A and Corinthios M J (1977) Natural, dyadic and sequencyorder algorithms and processors for the Walsh-Hadamard transforms,IEEE Trans. Comput., C-26: 435–442.

GERA Gerakoulis D P and Ghassemzadeh S S ,System and method for gener-ating orthogonal codes, U.S. patent 6,700,864 (Application to wirelesscommunications).

HAMMa Hammer J and Seberry J (1979) Higher dimensional orthogonal designsand Hadamard matrices II, Congressus Numerantium, 27: 23–29. Pro-ceedings of the Ninth Manitoba Conference on Numerical Mathematics.

HAMMb Hammer J and Seberry J (1981) Higher dimensional orthogonal designsand applications, IEEE Transactions on Information Theory, 27(6): 772–779.

HARA Harada M and Tonchev V D (1995/96) Singly-even self-dual codes andHadamard matrices, In: Cohen G, Giusti M and Mora T (eds.), AAECC-11, Hadamard matrices, In: Lecture Notes in Comput. Sci., vol.948,Springer-Verlag, Berlin-Heidelberg-New York.

HARW Harwit M and Sloane N J A (1979) Hadamard transform optics, Aca-demic Press, Sydney.

KITA Kitajima H and Shimono T (1983) Residual correlation of the Hadamardtransforms of stationary Markov-1 signals, IEEE Trans. Communica-tions, 31 (1): 119–121.

LEEK Lee M H and Kaveh M (1986) Fast Hadamard transform based on a sim-ple matrix factorization, IEEETrans.Acoust., Speech, Signal Processing,ASSP-34 (6): 1666–1667.

MASC Maschietti A (1992) Hyperovals and Hadamard designs, J. Geom., 44:107–116.

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MAVR Mavron V and Tonchev V D (2000) On symmetric nets and generalizedHadamard matrices from affine designs, J. Geometry 67: 180–187.

SAMA Samadi S, Suzukake Y and Iwakura H (1998) On automatic deriva-tion of fast Hadamard transform using genetic programming, In: Proc.1998 IEEE Asia-Pacific Conference on Circuits and Systems, Thailand,pp.327–330.

SEBE Seberry Wallis J (1972) Hadamard matrices, In: Wallis W D, Street AP and Seberry Wallis J, Combinatorics: Room squares, sum-free sets,Hadamard matrices, Springer-Verlag, Berlin-Heidelberg-New York.

YUYI Yu, F T S;Yin, S and Ruffin, B P, (1993) Application of Hadamard trans-form to fiber-optic smart-skin sensing, Microwave and Optical Technol-ogy Letters, 6 (11): 632–634.

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On some applications of Hadamard matrices