GENERATION OF GOLD-SEQUENCES WITH APPLICATIONS TO SPREAD SPECTRUM SYSTEMS

Transcription

1 GENERATION OF GOLD-SEQUENCES WITH APPLICATIONS TO SPREAD SPECTRUM SYSTEMS F. Rodríguez Henríquez (1), Member, IEEE, N. Cruz Cortés (1), Member, IEEE, J.M. Rocha-Pérez (2) Member, IEEE. F. Amaro Sánchez (3). Abstract--In ths paper we dscuss some of the most reevant theoretca and practca aspects reated to the Hardware mpementaton of pseudo-random God-sequences. The man contrbuton of ths work s the proposa of a genera methodoogy that aows us to generate God-sequences for any arbtrary order n an effcent way. A specfc desgn exampe that ustrates how to use our method to obtan a set of Godsequences of order n5 together wth ts appcaton n a smpfed spread spectrum communcatons system s aso gven. A the resuts presented here were tested and smuated va MatLab programs that were wrtten foowng our proposed methodoogy. Index terms God sequences, Pseudo-random sequences, Spread spectrum. I. INTRODUCTION The ever ncreasng mportance of appcaton areas such as cryptography and spread-spectrum communcatons has ed to a renovated nterest n perodc correaton parameters for pseudorandom sequences. Spread spectrum moduaton uses a transmsson bandwdth many tmes greater than the nformaton bandwdth. For ths knd of systems, the frst ssue that a desgner needs to sove s the generaton of a nose-ke sgna. Smary, most cryptographc areas need the mpementaton of pseudo-random perodc sequences wth hgh cross-correaton propertes. The study of pseudorandom and reated sequences spans for more than ffty years. Durng that tme, resuts have been obtaned on structura propertes, correaton functons, method of generaton, and appcatons to varous eectronc systems probems. So far, the sequences that have receved more attenton n the terature are the maxma-ength near feedback shft regster bnary sequences, whch we refer to as m-sequences. As the name suggests, these are precsey the sequences of maxmum possbe perod (whch s N 2 n 1) that can be 1. CINVESTAV-IPN, Computer Scence Secton. Av. IPN 258, 736 Méxco, D.F. 2. INAOE, Tonantznta, Pueba. 3. BUAP, Facutad de Eectrónca obtaned from an n-stage bnary shft regster wth near feedback. One of the key features of an m-sequence s ts autocorreaton functon θ x,x () N and θ x,x () -1 for 1 <N. It s ths dea perodc autocorreaton property that was expoted n most of the eary appcatons of the m- sequences. The mathematca study of maxma-ength sequences (msequences) seems to have started n the md-195s. Much of the eary research was concerned wth the autocorreaton propertes and the nose-ke aspects of m-sequences. However, some attenton was gven to the probem of seectng sets of m-sequences wth good cross-correaton propertes and by the ate 196 s severa theoretca and expermenta resuts were known. However, even at the present tme the key cross-correaton propertes of m- sequences are consderaby ess wdey known than the autocorreaton propertes, and yet the former s more mportant n many appcatons such as the ones mentoned above. The man desgn probem address n ths paper s how to generate n the cheapest way, Gaussan nose sgnas wth ow cross-correaton vaues. There are two qute mportant characterstcs that ths generaton technque shoud exhbt: Impementaton. Shoud be easy mpemented and reproducbe; because the same generatng process must be used at the transmtter (for encodng/spreadng); and at the recever (for decodng/undo-spreadng). The sgnas generated at the recever must be perfecty synchronzed wth the tmng of the receved transmsson. In ths research work we descrbe a practca methodoogy to generate a speca cass of perodc m-sequences caed God sequences. God sequences yed the theoretcay mnmum cross-correaton vaues that one can possby expect from perodc m-sequences. Impementaton detas and the appcaton of God sequences to spread spectrum systems are aso outned. The remanng part of ths paper s organzed as foows. Secton II ntroduces the most mportant mathematca concepts and defntons reated to m-sequences n genera and ther reated cross-correaton propertes. Aso n secton II a three-step strategy that aows us to generate God

2 F. Rodríguez-Henríquez, J.M. Rocha-Pérez, F. Amaro-Sánchez y F. Sandova-Ibarra 2 sequences n a practca manner s outned. That strategy was coded n MatLab and n secton III the correspondng mpementaton detas are fuy dscussed. Then, n secton IV we present a desgn exampe of a spread spectrum system usng a God sequence of order fve. Fna concusons and remarks are gven n secton V. II. MATHEMATICAL BACKGROUND We start our dscusson wth the foowng mathematca n n 1 defntons. Let h ( x) h x h1 x... hn 1x hn denote a bnary monc poynoma of degree n where hh n 1 and the other h s can take vaues of or 1. It s customary to represent such poynoma as a bnary vector h h h,..., h ), and to express that vector n octa (, 1 n notaton. For exampe, the poynomas x 1 and 5 2 x x 1 are represented by the bnary vectors 111 and 111, respectvey, and the octa notaton for those poynomas s 23 and 45, respectvey. A perodc bnary sequence u s sad to be a sequence generated by the bnary poynoma h(x) defned above f for a ntegers j: h u h u h u... h u (1) j 1 j 1 2 j 2 n j n A gven poynoma h(x) s caed prmtve, f ts assocated perodc sequence has the maxmum theoretca perod of repetton gven by N 2 n -1. From equaton (1) t foows that the perodc sequence u can be generated by an n-stage bnary near feedback shft regster whch has a feedback tap connected to the -th ce f h 1, < < n as s shown n Fg. 1 [Gssc95, Vterb95]. Notce that snce h n 1, there s aways a connecton for the nth ce. Lnea Feedback Shft Regster LFSR n n-1 n FF FF FF FF FF FF Code Mask n n-1 n x 4 Sada Fg. 1 Reazaton of a maxmum ength sequence A shft regster can generate severa dfferent perodc sequences, one of whch s the a-zeroes sequence. Of course, ony the nonzero sequences are of nterest. The foowng propertes of shft regster perodc sequences are we known and are stated here for convenence: If u s a perodc sequence generated by h(x), then for a ntegers, T u s aso a sequence generated by h(x); where the operator T shfts the perodc sequence u by paces. Hence, any shft regster can generate dfferent phases of the same perodc sequence. If u and v are generated by h(x), then so s u v. The perod of u s at most N 2 n -1, where n s the number of ces n the shft regster, or equvaenty, the degree of h(x). There are exacty N nonzero perodc sequences generated by h(x), and they are just N dfferent phases of u; namey u, Tu, T 2 u,, T N-1 u. A sequence of perod N s a maxmum ength sequence f and ony f t has the shft-and-add property, whch can be formuated as foows. Gven dstnct ntegers and j,, j < N, there s a unque nteger k, dstnct from j k and j, such that k<n and: T u T u T u. In most practca appcatons, a bnary sequence s actuay transmtted as a sequence of postve and negatve puses of unt amptude. By conventon, the waveform to be transmtted s obtaned by repacng each 1 of the orgna bnary sequence by a 1 and each by a 1. We can map any arbtrary {, 1}-vaued perodc sequence u, by usng the functon χ such that: χ(α) (-1) α for α {, 1}. (2) Notce that T ( u) ( ) χ( T u) χ and ( u) χ( u ) χ( u ) χ( u ) χ 1... N 1 N 2hw( u) (3) Where hw(u) denotes the Hammng weght of u, and N s the perod of the sequence u. Havng defned bnary perodc sequences and ther assocated generatng poynoma, et us defne now the mportant concept of perodc cross-correaton between sequences. Defnton 1 For sequences x and y of perod N, we defne the perodc cross-correaton functon by: x, y N 1 () n θ x y (4) n n It s straghtforward to verfy that for each Ζ, ( ) ( N ) θ (5) x, y θ x, y Aso, we defne the perodc auto-correaton functon θ x () for the sequence x as θ x,x (). Wthn the context of appcatons for spread-spectrum communcatons we are nterested n the probem of how to fnd sets of perodc pseudo-random sequences wth the foowng two propertes: For each sequence x x n n the set, θ x,x () s as sma as possbe for 1 N-1; For each par of sequences x x n and y y n, θ x,y () s as sma as possbe for a. Sequences that exhbt the above two propertes and the desgn probem of how to mpement them effcenty n hardware are the man subject of ths paper. In the remanng part of ths secton, we w derve the theoretcay best

3 3 F. Rodríguez-Henríquez, J.M. Rocha-Pérez, F. Amaro-Sánchez and F. Sandova-Ibarra resuts that one can possby expect from such a set of perodc sequences. Appyng the defnton gven n (2) to the crosscorreaton formua of equaton (4) we obtan: θ u, v () θ χ ( u ), χ ( v)() χ( u ) χ( v ) u ( 1) ( 1) ( 1) χ u v ( u v ) v Puggng the resut of equaton (3) nto the ast equaton, we obtan: u, v () N 2hw( u T v θ ) (7) Let us now, defne t(n) as n t ( n), (8) Where α denotes the nteger part of the rea number α. Then wth n mod 4, there aways exst pars of sequences wth three vaued cross-correaton functons, where the three vaues are: -1, -t(n), and t(n) 2 [Prasad96, Zemer92, Smon94]. A cross-correaton functon takng these and ony these vaues s caed preferred three-vaued cross-correaton functon and the correspondng par of poynomas assocated wth t, s caed a preferred par of poynomas. In the next subsecton we w show how to use a preferred par of poynomas to generate a set of pseudo-random sequences wth theoretca-mnmum cross-correaton vaues. A. God Sequences God sequences form an mportant cass of perodc sequences, whch provde arger sets of sequences wth good perodc cross-correaton. A set of God sequences of perod N 2 n 1, conssts of N2 sequences for whch we have exceent cross-correaton propertes. A set of God sequences can be constructed from appropratey seected maxmum ength sequences as descrbed beow. Suppose a shft regster poynoma f(x) factors nto h(x)g(x) where h(x) and g(x) have no factors n common. Then the set of a sequences generated by f(x) s just the 8set of a sequences of the form a b where a s some sequence generated by h(x), b s some sequence generated by g(x), and where both, a and b can be ether nonzero or zero sequences. Assumng now that h(x) and g(x) are two preferred par of prmtve poynomas of degree n and that each one of them generates the m-sequences u and v of maxmum perod N 2 n -1, respectvey. If y denotes a nonzero sequence generated (6) by f(x) h(x)g(x) then, from the above propertes of m- sequences, we get that ether: or or y T u; j y T v; j y T u T v. Where, j N-1. From equaton (9) t foows that y must be some phase of one of the sequences n the set G(u, v) defned as, N 1 ( v) { u v u v u Tv u T v u T v} 2,,,,,,..., G u (9) (1) Note that G(u, v) contans a tota of N 2 2 n 1 sequences of perod N. Every snge par of sequences that beongs to the set G(u, v), has the property that ts cross-correaton functon can ony take one of the three dfferent vaues defned for the preferred par of sequences: -1, -t(n), and t(n) 2. Therefore, n order to construct a God-sequence set for a gven order n, we can use the foowng methodoogy: 1) Fnd a preferred par of prmtve poynomas h(x) and g(x) of order n. 2) By usng the shft-regster archtecture, mpement the sequences u and v correspondng to the poynomas h(x) and g(x), respectvey. 3) Use the N dfferent phases of ether u or v, n order to fnd each of the N2 God sequences as they are gven n equaton (1). In the next secton we expan how we can obtan God sequences based on the above three-step strategy. Input: Two monc poynomas of degree n P1, P2 Output: Cross-correaton pot. functon o crossc(poy1, poy2) % Intazaton fgure(2); crossc []; % 1. Generatng sequences u, v from the gen. poynomas. u pn(poy1); v pn(poy2); % 2. Fndng the order&ength of the generator poynomas. ord ength(oct2bn(poy1)); N (2^(ord-1) - 1)*2; % 3. Obtanng the perodc Cross-correaton spectrum % f poy1 and poy2 form a preferred par of m sequences % then, ts cross-correaton can ony take three vaues: 1, -% t(n) and t(n) - 2, where t(n) 1 2^ (n2)/2 for : N-1 % 4 Crosscorraatng wth shft. crossc [crossc sum(summ(u, [v(1: N) v(1:)]))]; end % 5 Pottng. Fgure; stem(crossc); grd; tte('crosscorreaton functon'); xabe('shft (from to N-1)'); yabe('teta');

4 F. Rodríguez-Henríquez, J.M. Rocha-Pérez, F. Amaro-Sánchez y F. Sandova-Ibarra 4 Fgure 2. Fndng the cross-correaton of two prmtve poynomas. end Fg. 3 Generaton of a God Sequence set. III. GENERATION OF GOLD SEQUENCES Accordng to the methodoogy outned n the ast secton, the frst probem that a desgner needs to face n order to generate a set of god sequences s to fnd egbe coupes of poynomas (.e. coupes of prmtve poynomas) canddates to conform a preferred par of poynomas. The probem to cacuate a possbe prmtve poynomas s unfeasbe because ther grown s exponenta wth the order n. For exampe, for n2, the number of prmtve poynoma s 24,. Fortunatey there are severa heurstc and probabstc tests that effcenty determne f a gven poynoma s prmtve or not. In the rest of ths secton we w assume that the user possesses a st of prmtve poynomas of the desred order n. In order to test f two prmtve poynomas have or not the preferred cross-correaton characterstc, we can use the MatLab-ke pseudo-code shown n Fg. 2. For n 7, the agorthm n Fg. 2 fnds (among others) the foowng canddates as a preferred par of poynomas (a of them n octa representaton): 221 and 345; 211and 217; 211 and 247; 211 and 235; 31and 313, etc. On the other hand, one can check that the par of prmtve poynomas: 211 and 221; 313 and 323, do not form a preferred par of poynomas. Those resuts competey agree wth the dagram of maxmum connected sets descrbed n [Sarwate8]. Fg. 3 shows a code that stores a the God-sequences generated by a preferred par of poynomas. The output matrx has a sze of N2 rows and N-1 coumns, and t contans a the N2 god sequences assocated wth the preferred par, each of them havng a tota ength of N-1 (where N 2 n -1, and where n s the gven order of the prmtve poynomas). Input: Two preferred monc poys of degree n P1, P2 Output: Assocated God Sequences functon seq_m god(poy1, poy2) % Each row of the seq_m matrx, contans one of the 2^n 1 % God sequences generated by the gven preferred par of % poynomas poy1 and poy2. % Intazaton fgure; % 1. Generatng sequences u, v from the gen. poynomas. u pn(poy1); v pn(poy2); % 2. Fndng the order&ength of the generator poynomas. ord ength(oct2bn(poy1)); N (2^(ord-1) - 1)*2; % 3. Generaton of God Sequences. seq_m(1,:) u; seq_m(2,:) v; for 1: N seq_m(2,:) u.*[v(1: N) v(1:)]; IV. IMPLEMENTATION EXAMPLE In ths secton we present an expct exampe of how the generaton of a set of God sequences of order n5 can be used n practce on a spread spectrum communcatons system. Let us dscuss frst the smpfed spread spectrum system shown n Fgure 4, where t s assumed that a the users subscrbed to the system may gan smutaneous access to the channe n a Code-Dvson Mutpe Access (CDMA) fashon. Random Source Informaton m Transmtter X c Spreadng Code t Nose AWGN w r t w Σ Channe X Despreadng Code (Loca) Fg. 4 Spread Spectrum concept c Recever Decoded Informaton In the transmtter sde of the system shown n Fg. 4, a random bnary message m, wth bandwdth B s spread nto a bandwdth PG tmes greater through the use of a pseudorandom PN sequence c. PG s caed the processng gan. We aso assume that the dstorton ntroduced by the channe can be modeed as an addtve whte Gaussan nose, AWGN composed of two man components: The whte nose ntrnsc to the channe and, the nose contrbuton of a the other users sharng the channe. In the recever sde, a process of de-spreadng s carred out by mutpyng the receved sgna wth an exact copy of the PN sequence used durng transmsson. The receved sgna (before de-spreadng) ooks as shown n Fgure 5. However, after the process of de-spreadng s performed, the system s abe to recover the orgna nformaton, as s shown n Fg. 6, where both, the orgna and the recovered sgnas can be compared. It s apparent that for ths knd of systems the generaton of hgh quaty pseudorandom m-sequences s one of the cruca desgn ssues n order to obtan a correct performance of the communcatons system

5 5 F. Rodríguez-Henríquez, J.M. Rocha-Pérez, F. Amaro-Sánchez and F. Sandova-Ibarra Fg. 5 Typca receved sgna (nformaton nose). 2. a4 a3 a2 a1 a God Sequence g(d)45 Seña después de Despread 1. a'4 a'3 a'2 a'1 a'. g'(d)75 Fg. 7 God sequence generator Cross Correaton 8 6 Cross-correaton Funton Fg. 6 Comparson between the orgna (contnuous ne) and the recovered (dashed ne) nformaton 4 2 As t was expaned n secton II, God sequences are a speca cass of m-sequences that happen to be partcuary sutabe for spread spectrum appcatons. An attractve characterstc of ths knd of sequences s that the generated sequences are very easy and cheap to mpement, and hence, they represent a good opton when compared wth other reazatons. Aso n secton II, t was ponted out that t s aways possbe to obtan speca pars of God m-sequences havng a three-vaued cross-correaton functon among them [Prasad96, Zemer92, and Smon94]. For exampe, for n1, usng equaton (8), we have t(1) and the ony three vaues for the cross-correaton functon of the preferred God sequences w be: {-1, -65, 63}. For spread spectrum appcatons, we can choose the ength of the sequence accordng to the desred capacty of the channe (n terms of the number of users). Snce the spread spectrum processng gan PG s drecty proportona to the ength of the m-sequences, a more common crteron used frequenty n practce s to seect frst the gan (n db) n order to compy wth some gven specfcaton n the recepton sde of the communcaton channe [Jen-Sh97]. Once that the ength of the sequences has been fxed we need to seect a par of preferred sequences. A dfferent nta phase of the sequence can be assgned to each user n the system. In ths way, each user has a code that dentfes hm/her unquey. As an ustratve exampe, et n5 be the requred order for a set of God sequences. In order to fnd a preferred par of m-sequences we fed the prmtves poynomas n octa notaton to the Matab program shown n Fg. 2. The program gves nformaton about whch poynomas form preferred pars. For the case n hand, the program found the par 45 and 75 as one of the preferred pars. Wth ths nformaton we can construct the crcut of Fgure 7. It s very mportant to verfy that the crcut ndeed accompshes the cross-correaton property defned by equaton (8). The pot of the correspondng cross-correaton functon s obtaned wth the Matab code of Fg. 2. Notce that for ths case n5, hence t(n) {-1, -9, 7}. As we can see n Fg. 8, the cross-correaton functon vaues obtaned correspond to the predcted ones Shft ( to N-1) Fg. 8 Cross-correaton functon for God sequences of order 5. V. CONCLUSIONS The most mportant propertes of m-sequences, God sequences and ther appcaton to spread spectrum have been presented. God-sequence mpementaton requres fndng a preferred par of m-sequences whch s n genera a dffcut task. In ths paper an effcent and practca methodoogy has been ntroduced together wth ts correspondng Matab mpementaton. By foowng our methodoogy and through the use of our Matab-coded agorthms we can fnd and generate preferred m-sequences and therefore we can construct God sequences of any order. For ustraton purposes a specfc desgn exampe for Goden sequences of order n5 was presented, athough the program s abe to fnd preferred pars of any order. V. REFERENCES [Fkkema97] Fkkema, Pau G., Spread Spectrum Technques for Wreess Communcaton IEEE Sgna Processng Magazne, May [Zemer92] Zemer Rodger E., Peterson Roger L., Introducton to Dgta Communcaton, MacMan Pubshng Company, [Gsc95] Gsc Savo G., Leppanen Pentt A., Code Dvson Mutpe Access, Kuwer Academc Pubshers, 1995 [Pckhotz82] Pckhotz, Schng, Laurence, Msten, Theory of Spread Spectrum Communcatons - A Tutora, IEEE Trans. on Comm, May [Vterb95] Vterb Andrew J., CDMA Prncpes of Spread Spectrum Communcaton, Addson-Wesey Pubshng Company, 1995

6 F. Rodríguez-Henríquez, J.M. Rocha-Pérez, F. Amaro-Sánchez y F. Sandova-Ibarra 6 [Jen-Sh97] Jen-Sh Wu, et a, A 2.6-V, 44-Mhz A- Dgta OPSK Drect-Sequence Spread-Spectrum Transcever IC, IEEE JSSC, October [Prasad96] Prasad Ramjee, CDMA for Wreess Communcatons, Artech House, [Smon94] Smon, Marvn K. Et a, Spread Spectrum Communcatons Handbook, Mc Graw-H, 1994.