Distinct 8-QAM+ Perfect Arrays Fanxin Zeng 1, a, Zhenyu Zhang 2,1, b, Linjie Qian 1, c

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1 nd Intenatonal Confeence on Electcal Compute Engneeng and Electoncs (ICECEE 15) Dstnct 8-QAM+ Pefect Aays Fanxn Zeng 1 a Zhenyu Zhang 1 b Lnje Qan 1 c 1 Chongqng Key Laboatoy of Emegency Communcaton Chongqng Communcaton Insttute Chongqng 435 Chna College of Communcaton Engneeng Chongqng Unvesty Chongqng 444 Chna emal: a fzengx@163.com b zhenyu.zhang@cqu.edu.cn c chalge@16.com Keywods: pefect aays 8-QAM+ constellaton famly sze Abstact. Ths pape nvestgates the constucton of 8-QAM+ pefect aays wth at least an odd ntege n the szes and gves the suffcent condton poducng dstnct 8-QAM+ pefect aays. Fo dffeent choces of odd nteges n a gven sze of at least an odd ntege the esultant dstnct 8-QAM+pefect aays have the own szes. Fo each choce the numbe of the esultant dstnct 8-QAM+pefect aays s exactly equvalent to ths chosen odd ntege. Ths pape can povde moe canddates fo the applcatons of 8-QAM+ pefect aays n communcatons ada and so foth. Intoducton Pefect aays have the popety of mpulse-le autocoelaton whch esults n the fact that pefect aays ae wdely appled to communcatons ada tme-fequency-codng map matchng synchonzaton bult-n tests of VLSI-ccuts and so on [1]. Thee ae a lage numbe of pefect bnay tenay and ployphase aay [1]. Howeve 8-QAM+ pefect aays ae ae whose man eason les at the late ntoducton of 8-QAM+ constellaton []. In 1 Boztas and Paampall ntoduced the defnton of 8-QAM+ sgnal set []. Fom that tme on the sequences ove 8-QAM+ constellaton ae apdly developed. Pefect and odd pefect 8-QAM+ sequences ae gven n 1 and 14 espectvely [3] [4]. 8-QAM+ complementay sequences ae pesented n 1 and 14 espectvely [5] [6]. ZCZ 8-QAM+ sequences appeaed n 1 [7]. Recently 8-QAM+ pefect aays ae poposed n 14 and 15 [8] [9]. The am of ths pape s to nvestgate 8-QAM+ pefect aays to ncease the famly sze and to eflect the state-of-the-at of ths aea. A suffcent condton poducng dstnct 8-QAM+ pefect aays wll be ntoduced. The numbe of the esultant 8-QAM+ pefect aays wll be mpoved. The est of ths pape wll oganzed as follows. In Secton we wll evew some necessay concepts and the nown esults on the constuctons of 8-QAM+ pefect aays. In Secton 3 a suffcent condton poducng dstnct 8-QAM+ pefect aays wll be gven. The examples wll appea n Secton 4. Fnally we conclude ths pape n Secton 5. Necessay Concepts and Known Constuctons of 8-QAM+ Pefect Aays In a mathematcal tem an aay A of n - dmenson wth the sze N1 N Nn s A= a 1 1 n < N n (1) whose peodc autocoelaton s defned by N1 1N 1 Nn 1 * RAA ( τ1 τ τn) = a 1 a n 1 τ 1 τ n τ n () 1= = n = whee the symbol x * denotes the complex-conjugate of x and the subscpt addton + τ (1 s pefomed modulo N. If the autocoelaton of an aay A of n - dmenson satsfes 15. The authos - Publshed by Atlants Pess 43

2 > ( τ1 τ τn) = ( ) RAA ( τ1 τ τn) = ( τ1 τ τn) ( ) (3) we efe to ths aay as pefect aay. Let the symbol T denote the n -dmensonal cyclcal shft opeato n othe wods fo a gven aay say A n (1) and gven n nteges µ s (1 unde ths opeato a new aay T( µ µ µ ) A can be gven by 1 n T( µ 1 µ µ n) A= a 1+ µ 1 + µ 1 n+ µ n < N n (4) whee the subscpt addton + µ (1 s counted modulo N. Let A and B be two aays wth the same sze. If thee exst n nteges µ s (1 so that T( µ 1 µ µ n) A= B (5) we say these two aays to be equvalent o else to be dstnct. The 8-QAM+ constellaton means the sgnal set []: + Ω 8 = { 11 j j 1 j1 + j1 j 1 + j}. (6) Based on the dect poduct { 1 1} { 1 1} the 8-QAM+ constellaton can be poduced by the followng mappngs: [3] + π : { 1 1} { 1 1} Ω8 (7) ( ab ) π ( ab ) whee π1( a b) = aj + b and π( a b) = aj b. (8) By the afoementoned mappngs a tenay aay can be conveted an 8-QAM+aay [8] [9]. The constucton methods efeed to n ths pape ae gven by the followng two Theoems. Theoem 1 [8] [9]: Let A be the pefect tenay aay (PTA) wth the sze N1 N Nn and η and δ (1 ae the nteges satsfyng N η δ N 1. If all the nteges N s (1 ae even and we have η d (mod N / ) (1 (9) the 8-QAM+ aay Q= q 1 1 n < N n wth the sze N1 N Nn s pefect whee q 1 = π 1 ( a ) n + η + η a n+ ηn + δ + δ n+ δ. (1) n Theoem [8] [9]: Let A and B be two PTAs wth the same sze N1 N Nn whee at least an ntege n the set { N 1 n} s odd say N. Set the nteges η s δ s ζ s and λ s (1 satsfy N η δ ζ λ N 1. Constuct the 8-QAM+ aay P below. P= p {1} n N n tt + < = + p ( ) f p a n η η η η b n ηn δ δ δ δ n δn + = =.(11) p = p ( a b ) f = 1 1 n 1 + ζ ζ 1 + ζ + ζ n+ ζn 1 + λ λ 1 + λ + λ+ 1 n+ λn If we have d η λ ζ mod N (1 n) ζ d η λ mod N (1 n ) (1) ζ d η λ mod N the 8-QAM+ aay P wth the sze N1 N 1 N N+ 1 Nn s pefect. Theoem 3 [8] [9]: The equaton system (11) has the soluton: 44

3 ζ η (mod N) η (1 n ) λ δ (mod N) δ (1 n ) N. (13) ζ η + (mod N) η N λ δ + (mod N) δ Poducton Pocedue of New Aays In ths secton we nvestgate the constucton method poducng dstnct 8-QAM+ pefect aays based on Theoem. Assume we need such aays wth the sze N1 N 1 N N+ 1 Nn whee N s odd numbe. The dstnct 8-QAM+ pefect aays can be educed by followng the gven steps below. Step 1: Choose two PTAs A and B wth the sze N1 N 1 N N+ 1 Nn Step : Intalzaton. Abtaly choose the nteges η s δ s ζ s and λ s (1 n ) Step 3: Intalzaton. Abtaly choose the nteges δ s (1 Step 4: Intalzaton. Detemne the ntege λ by Step 5: Iteaton tmes: t 1 Step 6: Set η = t 1 N d (mod N) λ + N 1 Step 7: Detemne the ntege ζ by + ζ η + (mod N) Step 8: Poduce 8-QAM+ aay P denoted by P t by Eq. (11) n Theoem Step 9: If t N 1 tun to Step 6 wth t t o else tun to Step 1 Step 1: End. Summazng the above we can obtan N dstnct 8-QAM+ pefect aays. Incdentally fo the eade who wants to now the detaled devaton of ths pocedue please efe to [8]. Examples Consde the poducton of 8-QAM+ pefect aays wth the sze 3 7. Choose the PTA wth the sze 3 7 below (14) Note that = 1. In accodance wth ou steps we ntalze that ( δ1 δ ) = (1 ) ( η ζ ) = () and ( λ1 λ ) = (). Now we pefom ou pocedue as follows. (1) When the teaton tmes t = 1 we have ( η1 ζ ) = ( ). The esultant 8-QAM+ pefect aay s j j 1 1+ j 1+ 1 j j 1 j j 1 j 1 j 1+ j 1+ j 1 j 1+ j 1 j 1 j P =. (15) j j 1 1+ j 1+ j 1 j j 1 j j 1 j 1 j 1+ j 1+ j 1 j 1+ j 1 j 1 () When the teaton tmes t = we have ( η1 ζ ) = (1 ). The esultant 8-QAM+ pefect aay s gven by 45

4 1+ j 1+ j 1 j 1+ j 1 j 1 1 j 1 j 1 j 1 j j 1+ j 1 j 1+ j 1 j 1 j P1 =. (16) 1+ j 1+ j 1 j 1+ j 1 j 1 j 1+ j 1+ j 1+ j 1+ j 1+ j 1+ j 1 j 1+ j 1 j 1 (3) When the teaton tmes t = 3 we have ( η1 ζ ) = (1). The esultant 8-QAM+ pefect aay s equvalent to 1+ j 1+ j 1 j 1+ j 1 j 1 1 j j 1 j j 1 j 1 j j j 1 1+ j 1+ j P =. (17) 1+ j 1+ j 1 j 1+ j 1 j 1 j 1 j j 1 j j 1 j 1 j j j 1 1+ j 1+ The autocoelaton functons of all the aays P - P ae depcted n Fg.1 below. Appaently t s an mpulse-le functon as pomsed n ou pocedue Autocoelaton tau tau 5 1 Concluson Fg.1 Autocoelaton of the esultant 8-QAM+ pefect aays. Ths pape gves the poducton pocedue of 8-QAM+ pefect aays and some examples show ts valdaton. Howeve the flaw of ths pocedue les at the expanson of the sze of the esultant aays n compason wth the sze of the aays employed. It s ou futue wo to exploe new method so as to constuct the 8-QAM+ pefect aays wth the sze unalteed. Acnowledgement Ths wo was suppoted by Chongqng Scence and Technology Commttee (cstc14pt-sy43) and Natonal Natual Scence Foundaton of Chna (NSFC) unde Gants and Chna Postdoctoal Scence Foundaton Gant 14M55318 Natual Scence Foundaton Poject of CQ Gant cstc14jcyj A45 and 46

5 Chongqng Postdoctoal Scence Specal Foundaton Gant Xm1431. Refeences [1] Pngzh Fan Mchael Danell. Sequence desgn fo communcatons applcatons. John Wley & Sons INC Ch. 1. [] Seda Boztas Udaya Paampall. Nonbnay sequences wth pefect and nealy pefect autocoelatons. Poceedngs Intenatonal Symposum on Infomaton Theoy Austn Texas U.S.A [3] Fanxn Zeng et al. Pefect 8-QAM+ sequences. IEEE Weless Commun. Lett. 1 1(4) [4] Yubo L et al. Odd pefect sequences and sequence sets wth zeo odd coelaton zone ove the 8-QAM+ constellaton. IEICE Tans. Fundamentals 14 E97-A(1) [5] Fanxn Zeng et al. 8-QAM+ peodc complementay sequence sets IEEE Commun. Lett. 1 16(1) [6] Fanxn Zeng et al. A note on 8-QAM+ sequences IEICE Tans. Fundamentals 14 E97-A(3) [7] Yubo L et al. Zeo coelaton zone sequence sets ove the 8-QAM+ constellaton. IEEE Commun. Lett. 1 16(11) [8] Fanxn Zeng et al. Pefect aays ove the 8-QAM+ constellaton. IEICE Tans. Fundamentals. 15 E98-A(4) [9] Fanxn Zeng et al. Desgn of 8-QAM+ pefect aays fom pefect tenay aays. Poceedngs 14 7th Intenatonal Congess on Image and Sgnal Pocessng Dalan Chna