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
> ( 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 1 1 1 1 ) 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 1 1 1 {1} n N n tt + < = + p 1 1 1 1 ( 1 1 1 1 1 1 1 1 1 1 1 1 ) 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 1 1 + ζ 1 + ζ + ζ n+ ζn 1 + λ 1 1 + 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
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. 1 1 1 1 1 1 1 1 1 1. (14) 1 1 1 1 1 1 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 1 1 1+ 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 1 1 1+ 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
1+ j 1+ j 1 j 1+ j 1 j 1 1 j 1 j 1 j 1 j + + + + 1+ 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 1 1 1+ 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 1 1 1+ 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. 7 6 5 Autocoelaton 4 3 1 5 tau1-5 -1-5 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 687164 6134 61713 and 61471336 Chna Postdoctoal Scence Foundaton Gant 14M55318 Natual Scence Foundaton Poject of CQ Gant cstc14jcyj A45 and 46
Chongqng Postdoctoal Scence Specal Foundaton Gant Xm1431. Refeences [1] Pngzh Fan Mchael Danell. Sequence desgn fo communcatons applcatons. John Wley & Sons INC. 1996 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. 1 13-134. [3] Fanxn Zeng et al. Pefect 8-QAM+ sequences. IEEE Weless Commun. Lett. 1 1(4) 388-391. [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) 45-48. [5] Fanxn Zeng et al. 8-QAM+ peodc complementay sequence sets IEEE Commun. Lett. 1 16(1) 83-85. [6] Fanxn Zeng et al. A note on 8-QAM+ sequences IEICE Tans. Fundamentals 14 E97-A(3) 888-893. [7] Yubo L et al. Zeo coelaton zone sequence sets ove the 8-QAM+ constellaton. IEEE Commun. Lett. 1 16(11) 1844-1847. [8] Fanxn Zeng et al. Pefect aays ove the 8-QAM+ constellaton. IEICE Tans. Fundamentals. 15 E98-A(4) 138-143. [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 14 1185-119. 47