arXiv:0808.0544v1 [cs.IT] 5 Aug 2008Generalized Cross-correlation Properties of ChuSequencesJae Won Kang, Younghoon Whang, Hyo Yol Park andKwang Soon Kim, Senior Member, IEEEDepartment of Electrical and Electronic Engineering,Yonsei University,134 Shinchon-dong, Seodaemun-gu, Seoul 120-749, KoreaTel: +82-2-2123-5861Fax: +82-2-313-2879E-mail: ks.kim@yonsei.ac.kr Corresponding AuthorAbstractIn this paper, we analyze the cross-correlation properties for Chu sequences, which provide information on thedistribution of the maximum magnitudes of the cross-correlation function. Furthermore, we can obtain the numberof available sequences for a given maximum magnitude of the cross-correlation function and the sequence length.Index TermsChu sequences, cross-correlation function.I. INTRODUCTIONIN general, it is desired to design a set of sequences with an impulsive autocorrelation function anda zero cross-correlation function for many practical applications. However, according to the Welchbound, theSarwatebound, theSidelnikovbound, theMasseyboundandotherbounds[1][4], it wasshown to be impossible to construct such an ideal set of sequences. Therefore, searching large families ofsequences with good auto-correlation function and cross-correlation function properties has been one ofthe most interesting topics in sequence design. For evaluating the correlation properties, one good choiceis to use the maximum sidelobe magnitude of the autocorrelation function and the maximum magnitudeofthecross-correlationfunction, whicharerespectivelydenotedasaandcinthispaper. Here, thefollowingquestionsarisenaturally: howmanypairsof sequencesareavailableforagivenmaximumvalues ofa andc and what is the distribution of the magnitude of the cross-correlation function?Among well known good sequences are Kasami [5], Gold [5], Chu [7][9] and complex four-phase [10]sequences. ForKasamiandGoldsequences, itwasshownthatthereare N+ 1sequencessatisfyinga=1 andc=1 +_2/N[5][6][11][12], whereNis the sequence length. For four-phase sequences,the number of sequences satisfyinga= 1 +Nandc= 1 +NisN+ 2 [10][13][14]. On the otherhand, the autocorrelation function of Chu sequences is known to be zero except at the lag of an integermultiple of the sequence length [7][9][15][17].A set of Chu sequences with lengthNis dened asC= {ar| 0 < r < N, gcd(N, r) = 1}, where thekth element ofar, ar(k), is dened asar(k) =___exp_jrk2N_, Neven,exp_jrk(k + 1)N_, Nodd.(1)The periodic autocorrelation function with lag, r(), of the sequencearis dened asr() =N1

k=0ar(k)ar(k + )+N1

k=Nar(k)ar(k + N). (2)In [8], it was shown that the periodic autocorrelation function of Chu sequences satisesr() =___N, modN= 0,0, modN = 0.(3)Let arandasbe any two Chu sequences with lengthN. Then, the cross-correlation functionr,s() ofarandas with lagis dened asr,s() =N1

k=0ar(k)as(k + ) +N1

k=Nar(k)as(k + N)=N1

k=0ar(k)as(k + ),(4)where the last equality comes from the fact thatar(k +d) = ar(k +d +N) for an arbitrary integer d [8].In [1] and [18], it was shown that the maximum magnitude of the cross-correlation functionc can belower-bounded as a function of the sequence length and the maximum magnitude of the autocorrelationfunction,a. By using this lower-bound, the optimum correlation properties of a set of sequences can bedened and it follows that the lower bound ofcis equal to Nwhenaequals zero. Certain pairs ofChu sequences, arandas, meet this lower-bound whengcd(r s, N) =1. However, in order to obtainmoreChusequenceswithrelativelylowcross-correlationvalues, weneedtoinvestigatemoregeneralcross-correlation properties.In this paper, we derive general properties for cross-correlation function of Chu sequences. Using thederived properties, we can obtain the magnitude distribution of the cross-correlation function. Here, themaximummagnitudedenotesthemaximummagnitudevalueof thecross-correlationfunctionof twogiven Chu sequences among all possible lags and its distribution is taken over all possible pairs of Chusequences. In addition, the number of available sequences can be obtained for a given value ofc and thegiven sequence length.Theremainingofthispaperisorganizedasfollows. InSectionII, themagnitudeofcross-correlationfunction of Chu sequences are described. In Section III, the distribution of the maximum magnitude ofthecross-correlationfunctionandthenumber of availableChusequencesfor givenmaximumcross-correlation value and the sequence length are investigated. Finally, Section IV concludes this paper.II. CHARACTERISTICOF THE CROSS-CORRELATIONFUNCTIONOF CHU SEQUENCESIn order to investigate the cross-correlation function of Chu sequences in detail, we need to nd whatare the possible values that the cross-correlation function of Chu sequences can take, which are given inthe following theorem.Denition 1: Letr ands be positive integers satisfying0 < r, s < N,gcd(N, r) = 1 andgcd(N, s) =1. Also dene gr,s= gcd(N, rs),ur,s= N/gr,s and vr,s= (rs)/gr,s. Then ur,s is relatively prime withvr,s. Also, for a given lag, we can rewrite it as= igr,s+d, wherei= /gr,s andd= igr,s.Theorem 1: The magnitude of the cross-correlation functionr,s(), |r,s()|, is given as|r,s()| =____Ngr,sK (d) , Nandur,svr,s even, orNodd,_Ngr,sK_d gr,s2_, Neven andur,svr,s odd,0, otherwise,whereK () is the Kroneker delta function.To prove Theorem 1, the following lemmas are useful.Lemma1[7][8]:Thehthprimitiveroot ofunityhcanbedenedas h=exp_j2uh_, whereuis any integer relatively prime toh. Then, for any integerv, 0 < v h 1,h1

k=0vkh= 0, h = 1.Lemma 2: The squared magnitude of the cross-correlation function is given as|r,s()|2=___ur,sgr,sgr,s1

m=0(1)ur,svr,sm2exp_j2smdgr,s_,N even,ur,sgr,sgr,s1

m=0(1)vrm(ur,s+1)exp_j2smdgr,s_,N odd.The proof of Lemma 2 is given in Appendix A.Now, the proof of Theorem 1 is given as follows.Proof of Theorem 1: First, consider the case whenNandur,svr,sare even. Then, from Lemma 2,we obtain|r,s()|2= ur,sgr,sgr,s1

m=0exp_j2smdgr,s_. (5)If d=0,|r,s(iur,s)|2=ur,sg2r,s=Ngr,s. When, d=0, since sis relativelyprimewith gr,s,|r,s(igr,s + d)|2= 0.Now, consider the case whenNis even andur,svr,s is odd. Whend= gr,s/2, we obtain from Lemma2 thatr,s_igr,s +gr,s2_2= ur,sgr,sgr,s1

m=0exp_j2_m(ur,svr,sm + s)2__. (6)We know that s is odd because s is relatively prime with N. If m is odd, ur,svr,sm is odd and ur,svr,sm+sis even. On the other hand, ifm is even, ur,svr,sm is also even. Thus, m(ur,svr,sm + s) is always evenanditshows that |r,s(igr,s + gr,s/2)|2=ur,sg2r,s=Ngr,s. Whend =gr,s/2, fromLemma 2, we canrewrite |r,s()|2asr,s(igr,s +gr,s2+ d)2= ur,sgr,sgr,s1

m=0exp_j2_m(ur,svr,sm+ s)2+smdgr,s__= ur,sgr,sgr,s1

m=0exp_j2smdgr,s_= 0,(7)whered= d gr,s/2 and the last equality comes from the fact thats is relatively prime withgr,s.Finally, consider the case whereNis odd. Thengr,s andur,s should be odd. Then, from Lemma 2, weobtain|r,s(igr,s + d)|2= ur,sgr,sgr,s1

m=0exp_j2smdgr,s_=_Ngr,sk (d) ,(8)which concludes the proof.III. DISTRIBUTIONOF THE MAXIMUMMAGNITUDESOF THE CROSS-CORRELATIONFUNCTIONA. The uniform propertyTheorem 1tells us that the characteristic of the cross-correlation function of twoChusequences, arandas, dependsonly ongr,s=gcd(r s, N). For example, whengr,s=1,cmeetsthe lower boundof N. On the other hand, whengr,s=N,cbecomes the largest value ofN. However, it has not yetbeen investigated how many sequences are available for a given values ofc and the sequence length. Toanswer the question, it is required to investigate the distribution of the maximum magnitude values of thecross-correlation function.Denition 2: Anygiveninteger Ncanberepresentedas N= ki=1pcii , where pidenotestheithsmallest prime factor ofN. Let us deneN= {n|0 < n < N,gcd(n, N) = 1} as the index set of Chusequences of lengthN. Also, for a given integerc, dene the following sets and function as follows. UN,c= {n c | 0 n < N} RN,c= {n c | n N} DN,c= {n c | 0 n < Nandn/ N} PmN,c= {npm c | 0 n < N/pm} GN,x(S) = {n| n Sandgcd(n, N) = x} for a given integer setS. |A| : The cardinality of a set |A|.FromTheorem1, wecanseethat themaximummagnitudeof thecross-correlationfunctionbetweenarandasisr,s=max|r,s()| =_gr,sN. Thus, forgivenN, s Nandx, it iseasilyseenthatGN,x(RN,s)= {r s | gr,s=gcd(r s, N)=xandr N}isthe setof differencesbetweensandall Chu sequence indices whose maximum squared magnitude of the cross-correlation function withasis equal tox. Then, |GN,x(RN,s)| is the number of available Chu sequences satisfying2r,s/N= x. Then,the main result of this subsection is given in the following theorem.Theorem 2: Let 1 s = s N be two different integers relatively prime with N. Then, |GN,x(RN,s)| =|GN,x(RN,s )|.Theorem2 indicates that the distribution of the maximum magnitudes of the cross-correlation functionfor agivenChusequenceset canbeobtainedbyxingonesequencearbitrarilyandexaminingthecross-correlation functions with the other sequences. The following Lemmas 35 are useful the proof ofTheorem 2.Lemma 3: For any two different integersc andc, GN,x(UN,c) = GN,x(UN,c ).Proof: Ithas beenproved that gcd(c + mN, N)=gcd(c, N)[19][20]. Then, it iseasily seenthat{ gcd(c + 1, N), gcd(c + 2, N), ,gcd(c + N, N) } = { gcd(1, N), gcd(2, N), ,gcd(N, N) }for any integerc. Therefore, GN,x(UN,c) =GN,x(UN,c ).Lemma4:Let aandbbepositive integerssatisfyinggcd(a, b) =1. Also, foranarbitrarypositiveintegerm, deneC= {na c | k n < k}, wherek is an arbitrary integer andk= k + mb. Then, Ccontains exactlym integer multiples ofb.Proof:Fortheithelement ci=kia cof C, wecanrepresent it asci=q(ki)b + e(ki), whereq(ki) = ci/b ande(bi) = ci modb. Note that such a pair ofq(k) ande(k) is unique for a given ci [20].Letdij= ci cj. Thendij= (ki kj)a = {q(ki) q(kj)}b + e(ki) e(kj). Thus, e(ki) = e(kj) impliesthat (kikj) is an integer multiple of b and vice versa because a is relatively prime with b. Now, considerthe partition {Cr, 0 r