Full-length non-linear binary sequences with Zero Correlation Zone for multiuser communications

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1 94 Iteratioal Joural of Commuicatio Networks ad Imatio Security (IJCNIS) Vol. 7, No. 2, August 2015 Full-legth o-liear biary sequeces with Zero Correlatio Zoe multiuser commuicatios Mahdiyar Sarayloo 1, Eio Gambi 1 ad Susaa Spisate 1 1 Dipartimeto di Igegeria dell Imazioe, Uiversita olitecica delle Marche, Italy S @pm.uivpm.it, e.gambi@uivpm.it, s.spisate@uivpm.it Abstract: The research o ew sets of sequeces that ca be applied as spreadig codes i multiple user commuicatios is still a active area, eve if this topic has bee extesively ivestigated sice log time. I fact, ew commuicatio paradigms like dese ad decetralized wireless etworks, where there is o cetral cotroller to assig the resources to the odes, are revampig the iterest o fidig large sets of sequeces providig adequate correlatio properties to support a big umber of odes, i potetially hostile chaels. This paper focuses o the Zero Correlatio Zoe (ZCZ) property exhibited by a family of oliear biary sequeces featurig a great cardiality of their set, ad good security-related features, ad provides evidece of their suitability to multiuser commuicatios, i chaels affected by multipath. Keywords: Zero Correlatio Zoe, o-liear biary sequeces, CDMA, Gold codes, boud. 1. Itroductio Despite the idea of Spread Spectrum (SS) commuicatios dates back more tha oe cetury, this techology is still applied i a wide set of differet fields, ragig from satellite positioig, to multiuser mobile commuicatios, to wireless local ad persoal area etworks. SS commuicatios have bee recetly revisited, i the framework of decetralized wireless etworks [1] comprisig several separate trasmitter-receiver pairs, where users treat each other as oise, ad there is o cetral cotroller, as it happes i traditioal uplik/dowlik multiuser systems, to assig the resources to them. New Medium Access Cotrol (MAC) protocols are also beig ivestigated [2], to support emergig commuicatio techologies ad paradigms, such as the Iteret of Thigs (IoT). The differet SS techiques available, such as Direct Sequece (DS), Frequecy Hoppig (FH), Time Hoppig (TH), ad Hybrid (H) [3] [5] feature differet properties, limitatios, ad advatages, that are strogly related to the iheret ature of the sequeces selected ad used as spreadig codes [6]. Moder commuicatio systems also require the capability to support icreasig umber of users, which relies o the selectio of sets of spreadig codes featurig large cardiality, i.e. a large amout of sequeces belogig to the same family [7]. To reduce the Multiple Access Iterferece (MAI) ad cochael iterferece i SS systems ad multiuser applicatios, such as radars, positio detectio, ad ultrasoic imagig, Zero Correlatio Zoe (ZCZ) sequeces may be applied, i.e. sequeces whereby the out-of-phase auto-correlatio fuctio is equal to zero, i a specified zoe of phase shift [8]. I SS commuicatios, the ZCZ property ca make spreadig codes more robust to the effects of multipath propagatio. I fact, a delay spread τ max withi the ZCZ iterval may be tolerated, assumig that each i th user s sigal, affected by a relative path delay τ i, satisfies the coditio: τ = Z (1) i τ max 0 T C where T C is the chip period, ad Z 0 is the ZCZ width. Whe the ZCZ property holds, geeralized orthogoal (GO) sequeces may be defied, alterative to ormal orthogoal (NO) oes. I fact, a set of M sequeces of legth N is said to be ormal orthogoal if the periodic correlatio fuctio amog two sequeces a i ad a j, Ca i, j { 1, K, M } satisfies the coditios: N, k = 0, i = j Ca a = i j 0, k = 0, i j, 0 k N-1, The sequece set is said to be geeralized orthogoal, if the periodic correlatio fuctio Ca i, j { 1, K, M } satisfies the coditio: (2), 0 k N-1, N, k = 0, i = j Ca = 0, k = 0, i j (3) 0, 0 < k Z0 beig Z 0 the width of the ZCZ. The defiitio of GO sequeces implies that the greater the ZCZ width Z 0, the better the sequece set, i.e. the system will be robust agaist a greater delay spread. For a shift k value outside the rage ( 0 0 a Z, Z ), C ca take ay value: the periodic crosscorrelatio caot be cotrolled outside the ZCZ. It would be desirable to get both a wide ZCZ ad oliearity, withi the same set of sequeces used as spreadig codes assiged to multiple users, i order to get more robust ad secure commuicatios. Based o the above premises, this paper aalyses the permace obtaiable i a Direct Sequece Code Divisio Multiple Access (DS-CDMA) system, o a multipath chael, where o-liear biary sequeces exhibitig the ZCZ property are chose as spreadig codes. The family of o-liear biary sequeces selected evaluatio is the set of De Bruij (DB) sequeces [9]. With respect to the work preseted i [10], the sequeces assiged to users are selected accordig to their ZCZ properties. As a matter of ( 1) 2 fact, the great cardiality of the set, give by M = 2, where is the spa of the sequeces, allows to apply suitable

2 95 Iteratioal Joural of Commuicatio Networks ad Imatio Security (IJCNIS) Vol. 7, No. 2, August 2015 strategies to select the spreadig codes providig the best permace i terms of miimal error probability at the receiver. The results herei preseted also itegrate the prelimiary aalysis developed i [11], where a Additive White Gaussia Noise (AWGN) chael was assumed. The paper is orgaized as follows: Sectio 2 discusses related work, whereas Sectio 3 presets the mai properties of the biary DB sequeces ivestigated by the paper. The sequece selectio strategy is detailed i Sectio 4. The results obtaied from simulatios are preseted i Sectio 5, whereas Sectio 6 draws the mai coclusio of the paper ad highlights future research directios. 2. Related Work All SS systems make use of code geerators, usually desiged as pseudoradom, or pseudooise (N) geerators. Each code i a give family is used to set the frequecy spectrum that the output sigal will occupy. It also determies ad cotrols the spreadig patter of the system: by usig a carefully desiged ad selected set of spreadig sequeces, which beefits from good correlatio properties, it is possible to implemet multiple access techologies, ad multi user trasmissio at the same time, i the same frequecy bad. Several families of spreadig codes have bee proposed i the literature, either biary (two-valued), ad polyphase oes (i.e. multi-valued). Codes may be eve specifically desiged to satisfy oe or more requiremets related to SS commuicatios. Utuately, some requiremets may be coflictig. As a example, sets of spreadig codes that are optimal to reduce the MAI term, ofte result i a few spreadig codes that may ot suffice to support as may users as requested. Comparig differet sets of spreadig codes may be ot a straightward process, due to the amout of differet codes-related parameters that shall be take ito accout, ad to the iheret differeces i the code geeratio process. Asychroous systems, or frequecy selective propagatio chaels, may destroy or severely impact the orthogoality of the NO spreadig codes used, thus causig a icreasig iterferece, beig both Iter Symbol Iterferece (ISI) ad MAI geerated by radom time offsets amog the sigals, that make ufeasible the code wavems to be completely orthogoal. Well-kow families of liear spreadig sequeces, such as m-sequeces, Kasami, Gold, ad Walsh codes, traditioally employed as chaelizatio codes, exhibit o-zero ad sometime o egligible auto- ad cross-correlatio out-of-phase values, which limit the achievable permace, i asychroous or i quasisychroous scearios [11]. O the other had, full legth sequeces obtaied through o-liear geeratio, such as biary De Bruij (DB) sequeces [12], [13], provide attractive features as well, such as log periods ad large complexities, great cardiality of the set [14], ad good radomess properties, that ca be fruitfully exploited i commuicatio systems, codig theory, ad cryptography [15]. 3. No-liear Biary De Bruij Sequeces No-liear feedback shift registers (NLFSRs) are a geeralizatio of liear feedback shift registers, i which a curret state is a o-liear fuctio of the previous state. While the structure of a -bit LFSR ca be easily deduced from 2 cosecutive bit of its sequeces, by applyig the Berlekamp-Massey algorithm, up to O(2) bits would be eeded to determie the structure of a -bit NLFSR geeratig a give sequece [12], thus deotig a stroger resistace to attacks aimed at breakig the sequece geeratio algorithm. No-liearity brigs better security to spreadig codes, as they become more difficult to guess by uauthorized users. It is kow that a -stage LFSR has a maximum period of 2 1, if ad oly if its characteristic polyomial is primitive. Sequeces geerated by full-period NLFSRs have a legth equal to 2, ad they are also kow as De Bruij sequeces. I a DB sequece of order (or spa), all the possible 2 differet biary -tuples appear exactly oce. The periodic auto-correlatio fuctio C aa of ay DB sequece a of spa, a give shift k, exhibits the followig properties: C aa 2, = 0, 0, k = 0 1 k 1 k = which meas that ay biary De Bruij sequece of spa exhibits a ZCZ of legth (-1) chips i its periodic autocorrelatio fuctio. For a arbitrary pair of De Bruij sequeces a i ad a j, i j, of spa ad period N, the followig boud o the periodic cross-correlatio fuctio C a a i j holds: Ca 0 k N 1 (5) By adoptig suitable strategies, ad exploitig the great cardiality of the set, it is possible to select specific subsets of DB sequeces featurig the desired periodic auto- ad cross-correlatio properties. About the aperiodic autocorrelatio (AAC) of the sequeces, Table 1 compares the maximum ad average AAC values of differet families, amely DB sequeces of legth 32 bits, Gold, ad m- sequeces of legth 31 bits, computed over the whole sets. Table 1. Compariso of maximum ad average AAC values of differet spreadig codes families DB (32) Gold (31) M (31) Maximum Average Despite the apparetly worse behavior of DB sequeces i terms of maximum AAC value, their greater cardiality allows the miimum average AAC value, with respect to the other sets cosidered. The metioed maximum value of AAC is ot frequetly occurred the DB sequeces; further, DB sequeces havig the maximum AAC may be removed from the selected subset, to improve the permace attaiable. (4)

3 96 Iteratioal Joural of Commuicatio Networks ad Imatio Security (IJCNIS) Vol. 7, No. 2, August 2015 Aother iterestig parameter is the umber of zero AAC values at each shift idex, that ca impact the permace of the codes i sychroous systems, ad cotrol the selectio of the proper codes to use. Figure 1 illustrates the percet umber of sequeces with ull AAC values at each rotatio DB, Gold ad m-sequeces. Cosiderig the umber of distict sequeces of each sequece family, it appears that DB sequeces may be chose as the best set. Figure 1. ercet amout of ull AAC values each shift idex, the differet sets of spreadig codes cosidered (DB, Gold ad m-sequeces) Cosiderig the umber of sequeces featurig the same amout of ull AAC values, Figure 2 shows how DB sequeces icredibly outperm the other sets. sequeces to be used as spreadig codes. O the cotrary, bigger values of the spa, the exhaustive geeratio of the sequeces becomes extremely resource-cosumig, ad oly partial subsets could be cosidered, thus gettig a suboptimal selectio oly. The algorithm desiged to extract the spa 5 sequeces from their full set looks clusters of at least 5 sequeces, which have to show a ZCZ i their cross-correlatio, or a maximum out-of-phase value of the cross-correlatio equal to 4 or 8. The full set of 2048 De Bruij sequeces provides up to periodic cross-correlatio fuctios that exhibit a ZCZ of variable legth. Accordig to the costraits set above, up to clusters of at least 5 sequeces may be foud from the origial family of 2048 biary DB sequeces; amog them, 708 clusters iclude up to 6 differet sequeces. 5. Simulatio Results I order to gather sigificat results that ca be assumed as represetative of a geeral behavior, due to the big amout of differet clusters possible, the simulatios have bee permed by radomly selectig the clusters of DB sequeces to test, ad comparig them agaist the best clusters of m-sequeces ad Gold codes possible. I fact, a spa = 5, oly 6 m-sequeces exist, ad oly 33 Gold codes, so it is easy to fid the clusters of 5 sequeces that provide the best permace these families, accordig to the selectio rules stated above. Figure 2. Amout of sequeces with the same umber of zero AAC values (DB, Gold ad m-sequeces) 4. Sequeces Selectio Strategy The evaluatio of full-legth o-liear biary sequeces with ZCZ property, to be used as spreadig codes i a DS- CDMA system, requires to desig a proper sequece selectio strategy, i order to optimize the expected permace. Due to the great cardiality featured by families of DB sequeces, the aalysis herei preseted cosiders oly the full set of spa 5 sequeces, i.e. a total amout of 2048 sequeces of legth 32. This way, it is possible to apply a optimal selectio strategy to select so called clusters of Figure 3. robability of Symbol Error provided by a radomly selected cluster of 5 De Bruij sequeces of legth 32, compared to ideal curve (dashed lie) Figure shows the permace of a DS-CDMA system employig a radomly chose cluster of 5 DBs sequeces as spreadig codes (i.e. 5 users), assumig a propagatio chael affected by multipath, with a multipath delay profile give by [0,3,4] T C, ad a Ricea factor K = 0.2. The Ricea factor may take differet values: K = 100 a chael with o multipath; K 1 a chael affected by Rice fadig; ad K < 1 a chael affected by Rayleigh fadig. Each curve correspods to a differet user; the dashed lie gives the ideal behavior. Compariso amog differet sets of sequeces is give i Table 2, that provides the miimum ad maximum differece of the robability of Symbol Error deoted as, betwee the spreadig codes i the cluster SE

4 97 Iteratioal Joural of Commuicatio Networks ad Imatio Security (IJCNIS) Vol. 7, No. 2, August 2015 ad the ideal curve, at E b / N 0 = 4 db, DB, m-sequeces ad Gold codes, i the same chael coditios stated above. DB sequeces provide the less scattered behavior with respect to the ideal oe, better tha the optimal clusters of Gold ad m-sequeces. Table 2. Spreadig codes permace, with respect to the ideal behavior, i terms of miimum ad maximum differece of the robability of Symbol Error ( ) E b / N 0 = 4 Cluster Code Legth mi SE max SE db SE De Bruij M-sequece Gold Radomly chose clusters of DB sequeces remai better tha m-sequeces ad Gold codes, compared to the ideal curve, eve whe 6 users are cosidered. Table 3 refers to a multipath delay profile of [0,3,6] T C, ad a Ricea factor K = 0.7. Agai, DB sequeces outperm other clusters, eve i a propagatio chael affected by a stroger multipath. Table 3. Spreadig codes permace, with respect to the ideal behavior, i terms of miimum ad maximum differece of the robability of Symbol Error ( ) E b / N 0 = 4 db SE Cluster Code Legth mi SE max SE De Bruij M-sequece Gold As a fial remark, the permace improvemets of raw bit error probability (BE) attaiable with DB sequeces ca be coverted ito dramatic permace improvemets after deiterleavig ad decodig [16]. Reed-Solomo (RS) codig has bee selected simulatio, ad the evaluatio of RScoded BE (i.e.: BE after RS decodig) has bee obtaied by usig the well-kow aalytic approximatio that maps the raw BE ito the RS-coded BE. Figure 4 compares the lower ad upper bouds o RS-coded BE DB sequeces ad Gold codes, assumig 4 users i the DS-CDMA system, ad RS(31,23) codig. The better permace of DB sequeces are clearly visible. 6. Coclusios ad Future Work The great cardiality of the De Bruij sequeces, exhibited eve small values of the spa, allows to fid a big amout of subsets providig the ZCZ property amog the sequeces, joit security-related features. This paper preseted simulatio results related to radomly chose subsets of spa 5 De Bruij sequeces, composed by 5 ad 6 sequeces, to be used as spreadig codes i multiuser commuicatios over multipath chaels. The De Bruij sequeces belogig to these subsets provide better permace tha m-sequeces or Gold codes, typically adopted their good autocorrelatio properties. I chaels affected by multipath, the ZCZ property esures to keep the probability of symbol error limited to values ot far from the ideally expected oes. The adoptio of chael codig further emphasizes the improvemets attaiable. Future research o this topic will be focused o ivestigatig the behavior of loger De Bruij sequeces, i.e. sequeces featurig a spa > 5, o chaels affected by possibly severe multipath, as i vehicular commuicatios. Figure 4. Improved RS-coded BE provided by DB sequeces with respect to Gold codes, i a 4 users DS- CDMA system with RS(31,23) codig 7. Ackowledgemet The authors ackowledge the support provided by Luca Di Cesare i desigig part of the simulatios eeded to fid the sequece subsets. Refereces [1] K. Moshksar, A. K. Khadai, "Decetralized Wireless Networks: Spread Spectrum Commuicatios Revisited," IEEE Trasactios o Imatio Theory, Vol. 60, No. 5, pp , [2] M. Guerroumi, A.-S. Kha atha, N. Badache, S. Moussaoui, "O the Medium Access Cotrol rotocols Suitable Wireless Sesor Networks - A Survey," Iteratioal Joural of Commuicatio Networks ad Imatio Security, Vol. 6, No. 2, pp , [3] M. B. ursley, Direct-sequece spread-spectrum commuicatios multipath chaels, IEEE Trasactios o Microwave Theory ad Techology, Vol. 50, No. 3, pp , [4] R. Muammar, Frequecy-hopped multilevel frequecy shift keyig spread spectrum mobile radio commuicatio systems, IEEE Trasactios o Imatio Theory, Vol. 28, No. 6, p. 979, [5] A. R. Forouza, M. Nasiri-keari, ad J. A. Salehi, ermace aalysis of time-hoppig spread-spectrum multiple-access systems: ucoded ad coded schemes, IEEE Trasactios o Wireless Commuicatios, Vol. 1, No. 4, pp , [6] F. D. Garber ad M. B. ursley, Optimal phases of maximal legth sequeces asychroous spread-spectrum multiplexig, Electroics Letters, Vol. 16, No. 19, pp , [7] W. A. Imtiaz, N. Ahmad, "Cardiality Ehacemet of SAC- OCDMA Systems Usig ew Diagoal Double Weight Code," Iteratioal Joural of Commuicatio Networks ad Imatio Security, Vol. 6, No. 3, pp , [8] T. Hayashi, A Class of Zero-Correlatio Zoe Sequece Set Usig a erfect Sequece, IEEE Sigal rocessig Letters IEEE, Vol. 16, No. 4, pp , [9] N. De Bruij, A combiatorial problem, roceedigs Koiklijke Nederlads Akademie va Weteschappe, [10] S. Spisate, S. Adreacci, ad E. Gambi, Biary De Bruij sequeces DS-CDMA systems: aalysis ad results, EURASI Joural of Wireless Commuicatios ad Networkig, Vol. 4, [11] M. Sarayloo, E. Gambi, S. Spisate, De Bruij Sequeces as Zero Correlatio Zoe Codes Satellite Navigatio

5 98 Iteratioal Joural of Commuicatio Networks ad Imatio Security (IJCNIS) Vol. 7, No. 2, August 2015 Systems, 21st Iteratioal Coferece o Telecommuicatios, Lisbo (T), [12]. Fa, Spreadig Sequece Desig ad Theoretical Limits Quasisychroous CDMA Systems, EURASI Joural of Wireless Commuicatios ad Networkig, Vol. 2004, No. 1, p , [13] T. Etzio ad A. Lempel, Algorithms the geeratio of full-legth shift- register sequeces, IEEE Trasactios o Imatio Theory, Vol. 30, No. 3, pp , [14] M. Sarayloo, E. Gambi, ad S. Spisate, A Large Set of Orthogoal Codes the V2V Sceario, Iteratioal Coferece o Coected Vehicles ad Expo, Viea (AT), pp , [15] S. Spisate, C. Warty, E. Gambi, "DS-SS with de Bruij sequeces secure Iter Satellite Liks," IEEE Aerospace Coferece, Big Sky (MT, USA), pp.1-8, 2013 [16] S. Spisate, M. Sarayloo, E. Gambi, C. Warty, C. Sacchi, De Bruij Sequeces DS/CDMA Trasmissio: Efficiet Geeratio, Statistical Aalysis ad ermace Evaluatio, IEEE Aerospace Coferece, Big Sky (MT, USA), 2015.

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