with S. Datta

17 Pages·2008·0.41 MB·English

AdvComputMath(2010) 32:191–207
DOI10.1007/s10444-008-9100-9
Construction of infinite unimodular
sequences with zero autocorrelation
JohnJ.Benedetto·SomantikaDatta
Received:13September2007/Accepted:25March2008 /
Publishedonline:18 September 2008
©SpringerScience+BusinessMedia,LLC2008
Abstract Unimodular waveforms x are constructed on the integers with the
propertythattheautocorrelationof xisoneattheoriginandzeroelsewhere.
There are three different constructions: exponentials of the form e2πinαθ,
sequences taken from roots of unity, and sequences constructed from the
elements of real Hadamard matrices. The first is expected and elementary
and the second is based on the construction of Wiener. The third is the
most intricate and is really one of a family of distinct but structurally similar
waveforms.Anaturalerrorestimateproblemisposedforthelastconstruction.
Theanalyticsolutionisnotasusefulasthesimulationsbecauseoftheinherent
countingproblemsintheconstruction.
Keywords Infiniteunimodularsequences·Zeroautocorrelation·
Hadamardmatrices
MathematicsSubjectClassification(2000) 42-XX(FourierAnalysis)
1Introduction
1.1Background
LetRbetherealnumbers,letZbetheintegers,andsetT=R/Z.Ageneral
problemistocharacterizethefamilyofpositiveboundedRadonmeasures F,
CommunicatedbyYueshengXu.
J.J.Benedetto(B)·S.Datta
DepartmentofMathematics,UniversityofMaryland,CollegePark,MD20742,USA
e-mail:[email protected]
192 J.J.Benedetto,S.Datta
whose inverse Fourier transforms are the autocorrelations of bounded func-
tionsorwaveforms x.Aspecialcaseiswhen F ≡1onTand xisunimodular
onZ.ThestatementthatF ≡1isthesameassayingthattheautocorrelationof
xvanishesexceptat0,whereittakesthevalue1.Thisisthesettingwithwhich
wedeal.Infactweshallconstructthreesuchclassesofwaveformsx.Thefirst
constructionuseselementaryanalysis,thesecondgeneralizesaconstructionof
Wiener,andthethirdisbasedonHadamardmatrices.Thislastconstructionis
themostintricate.
The aforementioned problem, which our constructions address, is central
in the general area of waveform design, and it is particularly relevant in
severalapplicationsintheareasofradarandcommunications.Intheformer,
the waveforms x can play a role in effective target recognition, see, e.g.,
[1, 4, 5, 8–11, 13]; and in the latter they are used to address synchronization
issuesincellular(phone)accesstechnologies,especiallycodedivisionmultiple
access,e.g.,[14,15].Theradarandcommunicationmethodscombineinrecent
advancedmultifunctionRFsystems.
Inradartherearetwomainreasonsthatthewaveformsxshouldbeunimod-
ular,thatis,haveconstantamplitude.First,atransmittercanoperateatpeak
powerifxhasconstantpeakamplitude-thesystemdoesnothavetodealwith
thesurpriseofgreaterthanexpectedamplitudes.Second,amplitudevariations
duringtransmissionduetoadditivenoisecanbetheoreticallyeliminated.The
zeroautocorrelationpropertyensuresminimuminterferencebetweensignals
sharingthesamechannel.
1.2Notation
Weshallusethestandardnotationfromharmonicanalysis,e.g.,[3,12].C(Td)
isthespaceofcomplexvaluedcontinuousfunctionsonTd =Rd/Zd,andA(Td)
is the subspace of absolutely convergent Fourier series. M(Td) is the space
of Radon measures on Td, i.e., M(Td) is the dual space of the Banach space
C(Td) taken with the sup norm. For any positive integer N, we denote the
d−dimensionalsquareinZd by S(N),and,so,by S(N)weshallmean
S(N)={m=(m ,m ,··· ,m )∈Zd :−N (cid:2)m (cid:2) N,i=1,··· ,d}.
1 2 d i
Also,fork∈Zd,k=(k ,··· ,k ),
1 d
(cid:2) (cid:2)N (cid:2)N (cid:2)N
x[k+m]x[m]= ··· x[k+m]x[m].
m∈S(N) m1=−Nm2=−N md=−N
Definition1.1 Theautocorrelation A :Zd →Cofx:Zd →Cisdefinedas
x
(cid:2)
1
∀k∈Zd, A [k]= lim x[k+m]x[m].
x N→∞(2N+1)d
m∈S(N)
Unimodularsequenceswithzeroautocorrelation 193
If (cid:3)F ∈ A(Td) we write Fˇ = f ={fk}, i.e., Fˇ[k]= fk, and, for all k∈Zd,
f = F(γ)e2πik·γdγ. There is an analogous definition for μˇ where μ∈
k Td
M(Td).
1.3Perspective
InthesettingofR,wehavethefollowingtheoremduetoWienerandWintner
[17],whichwaslaterextendedtoRd in[2,6].
Theorem1.2 Let μ be a bounded positive Radon measure on R. There is a
constructiblefunction f ∈ L∞(R)whoseautocorrelation A existsforallt∈R,
loc f
and A =μˇ onR,i.e.,
f
(cid:4) (cid:4)
1 T
∀t∈R, lim f(t+x)f(x)dx= e2πitxdμ(x).
T→∞2T −T R
On Zd the following version of the Wiener–Wintner theorem can be
obtained.
Theorem1.3 Letμ∈ A(Td)bepositiveonTd.Thereisaconstructiblefunction
x:Zd →Csuchthat
(cid:2)
1
∀k∈Zd, A [k]= lim x[k+m]x[m]=μˇ[k].
x N→∞(2N+1)d
m∈S(N)
ThoughtheWiener–Wintnertheoremgivestheconstructionofthefunction
xitdoesnotensureboundednessofx,whereasourdesireistoconstructcodes
xthathaveconstantamplitude.
1.4Outline
The paper is divided in the following way. In Section 2 we show that the
unimodular sequence x[n]=e2πinαθ, where α ∈N\{1} and θ is irrational,
has zero autocorrelation on Z\{0}. In Section 3 we construct polyphase
sequencesfromrootsofunitywhichalsohavezeroautocorrelationonZ\{0}.
InSection4weconstructzeroautocorrelationsequencesfromrealHadamard
matrices. The methods of Sections 3 and 4 are actually more general than
stated.Forexample,wecaninvokethesametechniqueofconstruction,butuse
adifferentpermutationoftherootsofunityoradifferentarrangementofrows
of the Hadamardmatrices. In this way we still obtainunimodularwaveforms
withzeroautocorrelation.
194 J.J.Benedetto,S.Datta
2Sequencesoftheforme2πinαθ, α ∈NNNN\{1}andθ irrational
For t∈R, let [t] denote the integral part of t, that is, [t] is the largest integer
lessthanorequaltot;andlet{t}=t−[t]bethefractionalpartoft.Iistheunit
interval[0,1).Foragivensequence(t ),n=1,2,...,inR, A(E;N)denotes
n
thenumberofpoints{t },1(cid:2)n(cid:2) N,thatliein E⊆[0,1).
n
Definition2.1 Thesequence(t ),n=1,2,...,isuniformlydistributedmodulo
n
1(u.d.mod1)inRif
A([a,b);N)
lim =b −a.
N→∞ N
forallintervals[a,b)⊆ I.
ThefollowingistheclassicalWeylcriterionforuniformdistribution[7].
Theorem2.2
a. Thesequence(t ),n=1,2,··· ,isu.d.mod1ifandonlyif
n
(cid:2)N
1
∀h∈Z\{0}, lim e2πihtn =0. (2.1)
N→∞ N
n=1
b. Let p(t)=αmtm+αm−1tm−1+···+α0, m>1, be a polynomial with
realcoefficients,andletatleastoneofthecoefficientsα with j>0beirra-
j
tional.Thenthesequence(p(n)),n=1,2,··· ,isu.d.mod1.
Theorem2.3 Letx:Z→Cbedefinedby
x[n]=e2πinαθ, θ ∈/ Q, α ∈N\{1}. (2.2)
Then
(cid:5)
1,ifk=0
A [k]=
x 0,ifk(cid:8)=0.
Proof Theautocorrelationofxatkis
Ax[k]= Nl→im∞2eN2πi+kαθ1 (cid:2)N e2πiθ((α1)mα−1k+(α2)mα−2k2+···+(α−α1)mkα−1). (2.3)
m=−N
(cid:6) (cid:7) (cid:6) (cid:7) (cid:6) (cid:7)
Let p(t)=θ α ktα−1+θ α k2tα−2+···+θ α kα−1t. Since θ ∈/ Q we can
1 2 α−1
apply Theorem 2.2b when k(cid:8)=0 to say that the sequence (p(n)) is u.d.
mod 1. Therefore, according to Theorem 2.2a, taking h=1 and t = p(n) in
n
(2.1),therightsideof(2.3)iszeroifk(cid:8)=0.Ifk=0then A [0]=1. (cid:9)(cid:10)
x
ItiselementarytoprovetheanalogueofTheorem2.3forthemoregeneral
caseofcrosscorrelation.
Unimodularsequenceswithzeroautocorrelation 195
It is often necessary to co(cid:8)nstruct unimodular waveforms on Zd. Let n=
(n ,n ,···n )∈Zd and|n|= n2+n2+···+n2.Wedefine
1 2 d 1 2 d
∀n∈Zd, x[n]=e2πi|n|αθ, θ ∈/ Q, α ∈N. (2.4)
Theorem2.4 Ifα =2then,forxdefinedin(2.4),
(cid:5)
0ifk(cid:8)=0
A [k]=
x 1ifk=0.
Proof
(cid:2)N (cid:2)N
1
A [k]= lim ··· e2πi|m+k|αθe−2πi|m|αθ. (2.5)
x N→∞(2N+1)d
m1=−N md=−N
Ifα =2thentherightsideof(2.5)is
e2πi|k|2θ (cid:2)N (cid:2)N
lim ··· e4πi(m1k1+···+mdkd)θ
N→∞(2N+1)d
m1=−N md=−N
⎛ ⎞ ⎛ ⎞
(cid:2)N (cid:2)N
= lim e2πi|k|2θ⎝ 1 e4πim1k1θ⎠···⎝ 1 e4πimdkdθ⎠
N→∞ 2N+1 2N+1
m1=−N md=−N
(cid:5)
0ifk(cid:8)=0
=
1ifk=0.
(cid:9)(cid:10)
Anotherwaytodefineunimodularwaveformsx:Zd →Cisasfollows.Let
n=(n ,n ,··· ,n )∈Zd andletθ =(θ ,··· ,θ )havethepropertythatnone
1 2 d 1 d
oftheθ ,··· ,θ isinQ.Definenα =(nα,nα,··· ,nα)andlet
1 d 1 2 d
x[n]=e2πinα·θ. (2.6)
ByacalculationsimilartotheproofofTheorem2.4,weobtainthefollowing
results.
Theorem2.5 Ifα =2then,forxdefinedin(2.6),
(cid:5)
0ifk(cid:8)=0
A [k]=
x 1ifk=0.
Theorem2.6 Define the waveform x:Zd →C as follows: for α ∈N\{1} and
θ ∈R\Q, let y:Z→C be defined by y[n]=e2πinαθ, and for n=(n ,n ,
1 2
196 J.J.Benedetto,S.Datta
...,n )∈Zd let x[n]= x(n ,...,n )= y[n ], where j is fixed. Then, for any
d 1 d j
givenk=(k ,k ,...,k ),
1 2 d
(cid:5)
0ifk (cid:8)=0
A [k]= j
x 1ifk =0.
j
Here the autocorrelation is 1 not just at the origin but at all points on the
hyperplanek =0.
j
3Constructionofpolyphasesequenceswithzeroautocorrelation
In [16] Wiener showed probabilistically that there exist sequences whose
autocorrelationsarezeroeverywhereexceptattheorigin.Healsoconstructed
an explicit example of such a sequence x={x[m]} of ±1s. We go back to
Wiener’s construction and obtain an analogous polyphase construction for n
rootsofunity.
Example3.1 Letn=3.Weshalldefine x:Z→Csuchthattherangeof xis
{w0,w1,w2}, where w0 =1,w1 =ei23π, and w2 =ei43π. x is defined on N in the
followingway.
[w ,w ,w ], repeated 30 =1 times, giving the first 3 terms of x on N, i.e.,
0 1 2
x[1]=w ,x[2]=w ,andx[3]=w ;then
0 1 2
[w , w ; w , w ; w , w ; w , w ; w , w ; w , w ; w , w ; w , w ; w , w ],
0 0 0 1 0 2 1 0 1 1 1 2 2 0 2 1 2 2
repeated 31 =3 times, so that these 18=2·32 terms repeated thrice give
thenext54termsofxonN,i.e.,x[4]=w ,...,x[57]=w ;then
0 2
[w ,w ,w ;w ,w ,w ;w ,w ,w ;w ,w ,w ;w ,w ,w ;w ,w ,w ;w ,w ,
0 0 0 0 0 1 0 0 2 0 1 0 0 1 1 0 1 2 0 2
w ;w ,w ,w ;w ,w ,w ;w ,w ,w ;w ,w ,w ;w ,w ,w ;...;w ,w ,w ]
0 0 2 1 0 2 2 1 0 0 1 0 1 1 0 2 2 2 2
repeated32 =9times,sothatthese81termsrepeated9timesgivethenext
729termsofxonN,i.e.,x[58]=w ,...,x[786]=w .
0 2
WecontinuethisproceduretocompletethedefinitionofxonN.Letx[0]=
1and,forallk∈N,let x[−k]= x[k].Thenumberofelementsineachrowis
anintegeroftheform j3jandeachrowisrepeated3j−1times.
Definition3.2 If n>3, we shall define x:Z→C, analogous to Exam-
ple 3.1. Thus, x takes the values w : j=0,...,n−1, where w =1,w =
j 0 1
ei2nπ,...,wn−1 =ei2π(nn−1),andxisdefinedonNinthefollowingway.
[w0,w1,...,wn−1], repeated n0 =1 times; then [w0,w0;w0,w1;...;
w ,w ;w ,w ;w ,w ;...;w ,w ;...;w ,w ,w ,w ,...,w ,w ]
0 n−1 1 0 1 1 1 n−1 n−1 0 n−1 1 n−1 n−1
repeated n1 =n times; then [w0,w0,w0;w0,w0,w1;...;w0,w0,wn−1;...;
wn−1,wn−1,w0;...;wn−1,wn−1,wn−1]repeatedn2times.
WecontinuethisproceduretocompletethedefinitionofxonN.Letx[0]=
1and,forallk∈N,letx[−k]= x[k].Thenumberofelementsineachofthese
rowsisanintegeroftheform jnjandeachsuchrowisrepeatednj−1times.
Unimodularsequenceswithzeroautocorrelation 197
Lemma3.3 Letn(cid:3)2anddefinex:Z→CasinDefinition3.2.Fix p∈N.Let
s beafixedorderedsequenceofwsoflength p.Then,
p j
Numberoftimess appearsinthefirst Ntermsofx 1
lim p = . (3.1)
N→∞ N np
Proof
i. Inordertoverify(3.1)webeginbynotingthatinaparticularrowhaving
jnjelements,where p(cid:2) j,anyfinitesequenceoflength poccursasoften
as any other finite sequence of the same length. We can prove this by
inductionon p.
Consequently,insucharow,thefractionoftimesaparticular p-tuples
p
occursis 1 .
np
If we wish to calculate the relative frequency of the occurrence of s
p
among the first N terms we might have to stop in the middle of some
row.Assuch,wewrite
N =1·n1·n0+2·n2·n1+...+(p−1)np−1np−2+ pnpnp−1
+...+MnMnM−1+ P(M+1)nM+1+Q,
where0(cid:2) P<nMand0(cid:2) Q<(M+1)nM+1.HereM→∞asN →∞.
Thus,
(cid:2)p−1 (cid:2)M
N = jnjnj−1+ jnjnj−1+ P(M+1)nM+1+Q= S +S +Q,
1 2
j=1 j=p
where
(cid:2)M
S = jn2j−1+ P(M+1)nM+1.
2
j=p
ii. Let us denote the number of occurrences of a particular p-tuple s in a
p
row of length 1·n1 by n , the number of occurrences in a row of length
1
2·n2byn ,andsoon.Notethattheserowsarerepeatedn0,n1,··· times,
2
respectively.Therefore,
Numberoftimess appearsinthefirst Ntermsofx
p
lim
N→∞ N
(cid:13)
np−1.np nM−1.nM PnM+1
= lim S2 +···+ S2 + S2
N→∞ 1+ S1+Q 1+ S1+Q 1+ S1+Q
S2 S2 (cid:14) S2
Repetitionsinthelast Q
+ . (3.2)
N
198 J.J.Benedetto,S.Datta
Weshallshowinpartiiithat S1+Q →0asN →∞.Assumingthisfact,we
S2
have Q →0asN →∞;and,hence,thelasttermontherightside(3.2)is
N
zero.Thus,therightsideof(3.2)is
np−1np +npnp+1 +···+nM−1nM + PnM+1
S S S S
2 2 2 2
= np−1·np+np·np+1+···+nM−1nM+ PnM+1 = 1 ,
S np
2
wheretheequalityfollowsfromourassertioninthebeginningoftheproof
thatinaparticularrowhaving jnjelements,forsome j,where p(cid:2) j,any
finitesequenceoflength poccursasoftenasanyotherfinitesequenceof
thesamelength.Thus,wehaveshownthat
Numberoftimess appearsinthefirst Ntermsofx 1
lim p = . (3.3)
N→∞ N np
iii. Finally, we prove that S1+Q →0 as N →∞. Note that S = S (M) goes
S2 2 2
toinfinityas Ngoestoinfinityand,S beingfinite, S1 goesto0asS goes
1 S2 2
to infinity. Thus, we only need to show that Q goes to 0 as S goes to
S2 2
infinity.Wecalculate
Q (M+1)nM+1 (M+1)nM+1
0< < (cid:15) < (cid:15)
S2 Mj=p jn2j−1+ P(M+1)nM+1 n1 Mj=p jn2j+ P(M+1)nM+1
(M+1)nM+1 (M+1)nM+1
< (cid:15) = (cid:15)
1 M n2j+ P(M+1)nM+1 1n2p M−pn2j+P(M+1)nM+1
n j=p n j=0
(M+1)nM+1
= → 0as N →∞,
n2p−1n2(M−p+1)−1 + P(M+1)nM+1
n−1
since Mgoesto∞alongwith N. (cid:9)(cid:10)
Theorem3.4 Let {w : j=0,...,n−1} be the n roots of unity: w =1,w =
j 0 1
e2nπi,...,wn−1 =e2πi(nn−1).Then,
(cid:5)
1ifk=0
A [k]=
x 0ifk(cid:8)=0.
Proof
i. Ifk=0,then
(cid:2)N
1
A [0]= lim |x[m]|2 =1.
x N→∞2N+1
m=−N
Unimodularsequenceswithzeroautocorrelation 199
ii. Note th(cid:15)at Ax[k]= Ax[−k]. Further, if k(cid:8)=0 and if
limN→∞ N1 mN=1x[m+k]x[m]=0,then Ax[k]=0.Thus,itisenoughto
showthat
(cid:2)N
1
∀k>0, lim x[m+k]x[m]=0.
N→∞ N
m=1
Forthesakeofsimplicityletusprovethecasewhenk=1andwhenwe
are dealing with n=3 roots of unity. One should note that if k= p the
value of x[m+ p]x[m] is the product of the first (actually, its conjugate)
and last elements of some sequence w ...w , i ∈{1,2,3}. There are
i1 ip+1 n
3p+1suchsequences.Whenk=1,x[m+1]x[m]cancomefromanyoneof
thefollowing32 =9-tuples:w1w1 =1, w1w2 =e23πi, w1w3 =e43πi, w2w1 =
e43πi, w2w2 =1, w2w3 =e23πi, w3w1 =e23πi, w3w2 =e43πi, w3w3 =1.
Therefore,fromLemma3.3,wehave
1 (cid:2)N Repetitionsofw w
lim x[m+1]x[m]= lim 1 1 ·1
N→∞ N N→∞ N
m=1
lim Repetitionsofw1w2 ·e23πi + lim Repetitionsofw1w3 ·e43πi
N→∞ N N→∞ N
+ lim Repetitionsofw2w1 ·e43πi + lim Repetitionsofw2w2 ·1
N→∞ N N→∞ N
+ lim Repetitionsofw2w3 ·e23πi + lim Repetitionsofw3w1 ·e23πi
N→∞ N N→∞ N
+ lim Repetitionsofw3w2 ·e43πi + lim Repetitionsofw3w3 ·1.
N→∞ N N→∞ N
(3.4)
Using(3.1)with p=2andn=3,(3.4)reducesto
(cid:2)N
lim 1 x[m+1]x[m]= 1 ·1+ 1 ·e23πi + 1 ·e43πi + 1 ·e43πi + 1 ·1
N→∞ N 32 32 32 32 32
m=1
+ 1 ·e23πi + 1 ·e23πi + 1 ·e43πi + 1 ·1
32 32 32 32
(cid:16) (cid:17)
= 1 · 1+e23πi +e43πi ·3=0.
32
Ingeneral,fork= pandn=3,wehave
(cid:2)N (cid:16) (cid:17)
Nl→im∞ N1 x[m+ p]x[m]= 3p1+1 1+e23πi +e43πi ·3p =0.
m=1
200 J.J.Benedetto,S.Datta
iii. Inasimilarmannerwehave
(cid:2)N (cid:16) (cid:17)
Nl→im∞ N1 x[m+ p]x[m]= np1+1 1+e2nπi +···+e2πi(nn−1) ·np =0
m=1
forarbitraryn. (cid:9)(cid:10)
4ZeroautocorrelationsequencesconstructedfromHadamardmatrices
Definition4.1 A real Hadamard matrix is a square matrix whose entries are
either+1or−1andwhoserowsaremutuallyorthogonal.
The examples of Hadamard matrices below, of order 2n,n=1,..., were
first constructed by Sylvester in 1867. Hadamard constructed Hadamard ma-
trices of order 12 and 20. He also proved the following. IfU is a unimodular
matrix of order n, then |det(U)|(cid:2)nn/2, with equality in the case U is real if
and only if U is Hadamard. The Hadamard conjecture (due to Paley) is that
Hadamardmatricesoforder4kexistforeachk.
Let HbeaHadamardmatrixofordern.Then,thematrix
(cid:18) (cid:14)
H H
H −H
is a Hadamard matrix of order 2n. This observation can be applied repeat-
edly (as Kronecker products) to obtain the following sequence of Hadamard
matrices.
(cid:19) (cid:20)
H = 1 ,
1
(cid:18) (cid:14) (cid:18) (cid:14)
H H 1 1
H = 1 1 = ,
2 H −H 1−1
1 1
⎡ ⎤
(cid:18) (cid:14) 1 1 1 1
H4 = HH2 −HH2 =⎢⎢⎣11−11−11−−11⎥⎥⎦,··· .
2 2
1−1−1 1
Thus,
⎡ ⎤
(cid:18) (cid:14) H2k−2 H2k−2 H2k−2 H2k−2
H2k = HH22kk−−11 −HH2k2−k1−1 =⎢⎢⎣HH22kk−−22 −HH2k2−k2−2 −HH2k2−k2−2 −−HH22kk−−22⎥⎥⎦. (4.1)
H2k−2 H2k−2 −H2k−2 H2k−2
Example4.2 Toconstructaunimodularwaveformx,let H berepeatedonce
1
(20 =1), H berepeatedtwice(21), H berepeated22 times, H berepeated
2 4 8