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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

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Keywords Infinite unimodular sequences · Zero autocorrelation ·. Hadamard Long, M.L.: Radar Reflectivity of Land and Sea. Artech House
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