MULTICARRIER CHIP PULSE SHAPE DESIGN WITH LOW PAPR

Transcription

1 MULTICARRIER CHIP PULSE SHAPE DESIGN WITH LOW PAPR Mariano Vergara, Felix Anreich German Aerospace Cener (DLR) Insiue for Communicaions and Navigaion Wessling, Germany {mariano.vergara, Gonzalo Seco-Granados SPCOMNAV Deparmen of Telecommunicaions and Sysems Engineering Universia Auónoma de Barcelona (UAB) Bellaera (Barcelona), Spain ABSTRACT In his paper we presen a mehodology o design pulse shapes for a direc sequence code division muliple access (DS-CDMA) ranging signal, using a mulicarrier (MC) modulaion. The advanage of his signal design mehodology is ha i allows us o perform specral shaping wih very low Peak-o-Average-Power Raio (PAPR). This feaure makes his approach very ineresing for ranging sysems for which flexible resource allocaion and power efficiency are major concerns, e.g. GNSS (Global Navigaion Saellie Sysems). Index Terms Mulicarrier ranging signal, low PAPR, chip pulse shape design. 1. INTRODUCTION Chip pulse shape design for direc sequence code division muliple access (DS-CDMA) sysems has a major impac on many aspecs of he ranging performance of a DS-CDMA sysem [1]. Moreover, a good chip pulse shape design can incorporae new services ensuring full backward compaibiliy wih legacy users [2]. More in general, a modern signal design can reallocae he available resources, i.e. power and bandwidh, o fully mach he demands of quickly evolving marke scenarios, wih he minimal impac on he ransmier and receiver hardware. Mulicarrier (MC) modulaions offer flexible allocaion of he bandwidh, where he conrollable radio resources are he sub-bands which he available bandwidh is divided ino. I is well known however ha MC modulaions can have higher Peak-o-Average Power Raios (PAPR) han single carrier modulaions. High PAPR values creae problems when he ranging signal passes hrough mixers and nonlinear componens such as a High Power Amplifier (HPA), causing power inefficiencies. In some applicaions where power efficiency plays a major role, e.g. Global Navigaion Saellie Sysems (GNSS), MC signals wih a very low PAPR can be of grea pracical ineres. Several mehods have been proposed o reduced he PAPR of MC signals [3] for daa ransmission. Neverheless, reducing he PAPR of unmodulaed mulione signals, which can be used for ranging purposes, is a problem wih differen challenges [4] [5], which has no been as widely invesigaed as he problem of PAPR minimizaion for mulione signals aimed a daa ransmission. Since he PAPR minimizaion for MC signals used for ranging need no be done in real-ime, more sophisicaed, ime-consuming algorihms can be implemened. In his paper we presen a new mehodology o design he pulse shape of a DS-CDMA signal, employing a MC modulaion wih very low PAPR. This approach is based on he applicaion of some codes developed more han half a cenury ago for radar pulse compression [6], and i allows o shape he Power Specral Densiy (PSD) of he ranging signal, wih he side consrain ha he PAPR can never exceed 3 db. This holds for any number of subcarriers. As a case sudy, in his paper we show how a MC chip pulse shape can have almos he same power specrum of a pulse shape used for GNSS, namely a filered BOCsin(1,1) signal [7], and ye have a PAPR which is roughly 1.5 db lower. This suggess ha he proposed chip-pulse shape design, besides offering high specral flexibiliy and power efficiency, i offers a cerain degree of backward compaibiliy. 2. SIGNAL MODEL The chip pulse shape of a DS-CDMA signal can be parameerized as windowed mulione signal: ( where rec p() = 1 ) n=0 c n e j2πn f } {{ } m() ( ) rec, (1) indicaes he recangular funcion cenered a 0 and of widh equal o, wih being he chip pulse duraion in seconds. The chip pulse shape (1) is a complex

2 mulione pulse creaed by windowing of he complex mulione signal m(). The frequency separaion among he ones consiuing he chip pulsep() is given by f. The complex vecor c = [c 0,c 1,...,c ] T C N 1, (2) is called he frequency code of he mulione signal and i deermines he ampliudes and he phases of N complex exponenials consiuing he signal. We say ha he mulione signal (1) is generaed by a frequency codec. A frequency code uniquely deermines a pulse shape (1). In he following, we assume ha he frequency code has uniary norm: c 2 2 = 1, (3) which implies ha he pulse (1) has uniary energy, if f = k,k N. The frequency code can be wrien as c = ρ θ, (4) where is he Hadamard-Schur produc and ρ = [ c 0, c 1,..., c ] T R N 1, (5) θ = [e jarg{c0},e jarg{c1},...,e jarg{c} ] T R N 1, (6) are respecively he code envelope and he phase vecor of he frequency code c. The vecor ρ mus fulfill he condiion: ρ 0 ρ = c 0 c = 0. (7) If his condiion is no fulfilled, eiher he firs or he las elemen of he frequency code, or boh, are zero. This means ha one or wo subcarriers a he side of he specrum conain no power, and hus hey do no exis. Consequenly, hese subcarriers wihou power can be eliminaed and hus insead of N, we will have N 1 or N 2 subcarriers. If he new subcarriers a he edges of he frequency code sill have zero power, he frequency code can be furher shorened. We say ha a mulione signal possesses N subcarriers, when he frequency code canno be furher shorened, i.e. when (7) holds. PSD of he signal (1) is mosly deermined by he vecor ρ, ha indicaes how he power is divided among he subcarriers. 3. PROBLEM STATEMENT Our objecive is o shape he specral conen ofp() in a cerain way, and a he same ime we wan p() o have a small PAPR. Tha is, we inend o deermine he vecor ρ so ha he PSD of he pulse shape fis a cerain specral mask and we wan o calculae he vecor θ ha for his ρ minimizes he PAPR. Since p() is a sricly ime-limied pulse, a DS- CDMA signal buil wih he pulsep() and spread wih a consan envelope spreading code has he same PAPR as p(). Deermining θ for a given ρ such ha he PAPR is minimal is already a very challenging problem for which only empirical or numerical soluions (e.g. [8]) have been proposed. In his work, we wan o solve his problem joinly wih he opimizaion of he vecorρ. The PAPR of he pulse p() generaed by he generic frequency code c, which for simpliciy will also be called he PAPR of he frequency code c, is given by { } p() 2 PAPR c = max 1 p() 2 d, (8) The relaionship beween he frequency code and he PAPR of he corresponding pulse can be described by means of he aperiodic auo-correlaion funcion of he frequency code, as we show. Le he aperiodic auo-correlaion of he frequency code c be r c [d] = N d 1 k=0 c k+d c k, d < 0 N+d 1 k=0 c k c k d, d 0 (9) The insananeous power of he pulse (1) can be expressed as { } p() 2 = 1+2 R r c [d]e j2πd f, (10) Since he frequency code has uniary energy we can sae ha: PAPR c = 1+2 I is possible o prove [9, Appendix] ha: rc [d] ) cos (2πd f +arg{r c [d]} (11) r c [d] 2 0 m.s. PAPR c 0 db (12) wih m.s. indicaing a convergence in he mean square sense. Minimizing he meric: r c [d] 2 (13) minimizes he maximum possible PAPR [10], ye his does no allow a full conrol on he PAPR of he MC signal, because he convergence (12) holds only in he mean square sense. Moreover, since he convergence (12) holds only in he mean square sense, a frequency code ha generaes a MC signal wih a small PAPR does no necessarily have a small value of he meric (13) [11]. On op of ha, he meric (13) depends boh on he vecor ρ and he vecor θ; while he vecor θ in principle can be devoed enirely o PAPR minimizaion, he

3 vecor ρ mus joinly opimise boh he specral shape and he PAPR and his consiues a very challenging ask. This non-rivial problem can be significanly simplified if all he coefficiens of he aperiodic auocorrelaion of he frequency code, excep he firs and he las ones, are se o zero: rc [d] = 0, d = 1,2,...,N 2. (14) If condiion (14) is fulfilled by a frequency code c, hen he respecive MC signal (and hus pulse) achieves he Friese s bound on he PAPR of a mulione signal [9]. A corollary o (14) is [6]: r c [N 1] 0.5. (15) Le any frequency code ha fulfills condiion (14) be denoed by c = ρ θ and is aperiodic auocorrelaion by r c [n]. A code c has he following properies: 1. The PAPR depends on a single parameerr c [N 1] and hus i can be easily seered. 2. The PAPR can never exceed 3 db, independenly from he number of subcarriers. 3. For a given value ofr c [N 1] a limied number of codes c exiss [6]. The PAPR of codes fulfilling (14) is given by: ) γ = 10log 10 (1+2r c [N 1], (16) which in conjuncion wih (15) explains why he PAPR is limied o 3 db. Moreover, i is worh noing ha he PAPR minimizaion problem (e.g. [8]): Given a vecor ρ, find he vecor θ ha minimizes he objecive funcion (8) is no a problem ha always possesses a global minimum. I is possible o prove ha condiion (14) resrics he se of all code envelopes o hose vecors ρ for which here exiss a vecor θ ha provides a global minimum for he PAPR minimizaion problem. In he following we explain how o generae codes fulfilling condiion (14) and how consrained specral shaping can be performed. 4. HUFFMAN CODES Condiion (14) idenifies a se of codes known as Huffman codes [6]. Huffman codes [6] are codes developed for radar pulse compression and are no o be confused wih eponymous codes used for source daa compression. A Huffman code of lenghn is uniquely idenified by wo parameers a vecor of binary symbols, which we call binary generaor and indicae by b k, wih he subscripk being he idenifier of he binary generaor, he sidelobe of he aperiodic correlaion, ha deermines PAPR of he corresponding MC signal hrough (16). The sidelobe of he aperiodic correlaionr c [N 1] deermines he wo radii of he circles on which hen 1 roos of he associaed polynomial are locaed [6]. The binary vecor b k deermines, for each of he angular locaions [6], on which of he wo radii he roos of he k-h Huffman code are locaed. A cyclic permuaion of he binary generaor corresponds o a roaion of he roos. This operaion corresponds o a muliplicaion for a complex exponenial in he Huffman code space and i is hus irrelevan [12]. Two binary generaors, which have he propery ha one canno be wrien as a cyclic permuaion of he oher, individuae wo differen Huffman codes wih disinc envelopes (5). Two Huffman codes wih his propery are said o be disinc, and such are heir binary generaors. Two disinc Huffman codes ha have he same value of aperiodic correlaion sidelober c [N 1] generae wo MC pulses (1) wih exacly he same PAPR. The se of all disinc Huffman codes ha have he same aperiodic auocorrelaion sidelobe represens he se of all possible codes saisfying (14) ha yield he same PAPR. A Huffman code family is idenified by he code lengh and he aperiodic auocorrelaion sidelobe (or equivalenly he PAPR γ). The generic Huffman code is indicaed by c Huff (b k,γ), where b k is he binary generaor idenifying he code. The number of codes for each family is given by he number of he disinc binary generaors. Specral shaping is hus performed by means of a line search in all Huffman code families, in order o find he code c Huff (b k,γ) wih he desired code envelope. In paricular: 1. All disinc binary vecors of lengh N 1 are generaed. 2. For each value of he aperiodic auocorrelaion sidelobe (beween 0 and 1 2 ), each binary vecor generaes a disinc Huffman code (wih a disinc PSD). 3. The PSD of each disinc Huffman code is inspeced and is fiing o he required specral shaping crieria is assessed. The line search is performed along wo dimensions: he binary generaors vecors b k, ha form a finie counable se, and he aperiodic auocorrelaion sidelobe, which forms an uncounable se. Making he search grid of he values of he aperiodic auocorrelaion sidelobe dense enough, one can be sure of having inspeced all chip pulse shapes (1) wih he minimal heoreical PAPR (Friese s bound (16) [9]). 5. PROOF OF CONCEPT: GNSS SIGNALS Using he mehod described above, a low-papr mulione pulse (1) can be adaped o mach any specral mask. In his

4 secion we show how such a signal can be used o mach a BOC (Binary Offse Carrier) pulse, used in GNSS. This is echnically relevan because i shows ha signals developed wih his approach can be backward compaible. As a secondary poin, in his secion we wan also o highligh how BOC signals are a paricular case of he signal (1). GNSS signals are based on BOC modulaion [13] o shape he pulse of he DS-CDMA signals used for ranging. BOC signals have been chosen for GNSS because of heir capabiliy o achieve adequae specral separaion wih oher GNSS signals in he same frequency band and for heir low PAPR. BOC signals modulae a recangular pulse wih a square wave subcarriers, insead of a sinusoidal subcarrier, so ha he signal envelope is consan, provided he number of he harmonics of he square wave is infinie. In his secion we show ha a MC pulse of he kind (1) can be shaped so ha is PSD is very similar o he PSD of a BOCsin(1,1) [7] and ye have a smaller PAPR. A BOCsin(1,1) can be seen as a Mancheser pulse [14, p.55]. In his example we consider an alernaive version of a BOCsin(1,1), obained by represening a BOCsin(1,1) wih he model (1), and using a finie number of harmonics (i.e. subcarriers). According o his represenaion, a BOCsin(1,1) can be seen as a signal of he kind (1) whose frequency code is c BOC n = j 2 πn, n odd, 0, n even (17) This represenaion suggess also a flexible and power efficien approach o generae represenaions of BOC signals on he saellie payload. If he lengh of he frequency code is infinie, we obain exacly he BOCsin(1,1) as defined in [7]. In his example we consider only 40 harmonics (20 wihou he null harmonics) from n = 19 ill n = 19. The chip duraion is = 0.977µsec. The frequency separaion among he non-zero harmonics is chosen equal o f = 2 Tc = MHz. The frequency code idenifying he BOCsin(1,1) pulse conains N = 20 non-zero elemens. Nex, we consider Huffman codes of lengh N = 20 and we choose a frequency separaion equal o f = 2. In order o perform specral shaping of he MC pulse (1), we have o define a meric. Le ρ BOC indicae he code envelope of he code (17) andρ Huff (b k,γ) he envelope of he Huffman code of he same lengh, generaed by he generaor b k and wih PAPR equal o γ db. Since he PSD of a MC pulse (1) depends predominanly on he code envelope, we look for he Huffman code whose envelope has he smalles Euclidean disance from he envelope of he BOCsin(1,1) code (17). The bes maching Huffman code is hus he one whose envelope is { } ρ Huff (b k,γ) ρ BOC 2 ρ Huff op = argmin ρ Huff (b k,γ) 2 (18) For each value ofγ, 4862 Huffman codes exis. The opimum Huffman code has a PAPR γ = db. A BOCsin(1,1) wih 20 non-zero harmonics (corresponding o roughly a 10 MHz one-sided bandwidh) has a PAPR equal o 1.52 db. The PSD of he BOCsin(1,1) and ha of he opimized pulse are shown in Fig.1. As i can be observed, he PSD of he opimized MC signal is almos idenical o he one of he BOCsin(1,1), and hus i could fulfill he curren frequency regulaions. The PAPR of he opimized MC pulse is one and a half db lower. Also he magniudes of he corresponding auocorrelaion funcions (Fig.2) are very similar oo, wih he opimized MC pulse having even a higher seepness around he main peak. The pulses can be observed in Fig.3. The pulse opimised according he meric (18) has a correlaion loss of roughly 1 db wih he BOC signal as defined in (17). This correlaion loss comes, however, wih he advanage of a smaller PAPR and hus of a higher ransmi power efficiency. Wih some oher meric his correlaion loss may be made even smaller. As a closing remark, we would like o poin ou ha he meric of his example has been chosen because i highlighs he similariy beween exising BOC pulse shapes and a more general way of designing low PAPR chip pulse shapes for DS-CDMA signals. PSD [dbw-hz] f (normalised frequency) BOCsin(1,1) Op. MC pulse Fig. 1. Power Specral Densiies (PSD). 6. CONCLUSIONS In his paper we have presened a mehodology o design a chip pulse shape for a DS-CDMA ranging signal. This chip pulse pulse design approach is based on a MC modulaion and uses some codes developed for radar pulse compression in order o have a deerminisic conrol on he PAPR of he signal. This approach keeps he PAPR of he MC pulse very low (i can be a mos 3 db) for any number of subcarriers and hus i is of grea ineres for he design of ranging signals

5 1 0.8 BOCsin(1,1) Op. MC pulse [2] F. Anreich, J.-L. Issler J.-J. Floch, and J. A. Nossek, On backward compaibiliy in gnss signal design, in in Proceedings of 6h European Workshop on GNSS Signals and Signal Processing, GNSS SIGNALS, Toulouse, France, December R(τ) [3] S.H. Han and J. H. Lee, An overview of peak-oaverage power raio reducion echniques for mulicarrier ransmission, IEEE Wireless Communicaion Magazine, pp. pp , April [4] S. Boyd, Mulione signals wih low cres facor, IEEE Trans. Circuis Sys., vol. CAS-33, pp , Ocober τ Fig. 2. Magniudes of he pulse auocorrelaion funcions Op. MC pulse, real par Op. MC pulse, imag. par Op. MC pulse, magniude BOCsin(1,1) pulse Fig. 3. Chip pulse shapes in ime domain. for sysems ha mus be highly power efficien. As an example, he proposed approach was applied for GNSS ranging signal design and he opimized signal showed an exremely low PAPR (0.015 db). Wih he proposed approach for ranging signal design high flexibiliy and high power efficiency, which are imporan fuure drivers for evoluion of GNSS, can be achieved. Moreover he new chip pulse design approach can also be backward compaible o curren GNSS signals. 7. REFERENCES [1] F.Anreich and J.A. Nossek, Opimum chip pulse shape design for iming synchronizaion, in in Proceedings of he IEEE Inernaional Conference on Acousics, Speech, and Signal Processing, ICASSP 2011, Prague, Czech Republic, May [5] D. R. Gimlin, On Minimizing he Peak-o-Average Power Raio for he Sum of N Sinusoids, IEEE Trans. on Communiaions, vol. 41, no. 4, April [6] D. A. Huffman, The generaion of impulse equivalen pulse rains, IRE. Trans. Inform. Theory, vol. ITS, no. 5, pp , [7] European GNSS (Galileo) Open Service, OS ICD, Issue 1, Tech. Rep., February [8] M. R. Schroeder, Synhesis of low-peak-facor signals and binary sequences wih low auocorrelaion, IEEE Trans. Inform. Theory (Corresp.), vol. IT-16, pp , June [9] M.Friese, Mulione signals wih low cres facorsfriese, IEEE Trans. on communicaions, vol. 45, no. 10, Ocober [10] C. Tellambura, Upper bound on peak facor of n- muliple carriers, Ele. Leers, vol. 33, pp , Sep [11] N.Y. Ermolova and P. Vainikainen, On he relaionship beween peak facor of a mulicarrier signal and aperiodic auocorrelaion of he generaing sequence, IEEE Comm. leers, vol. 7, no. 3, pp , [12] L. Bomer and M. Anweiler, Long energy efficien Huffman sequences, in Inernaional Conference on Acousics, Speech, and Signal Processing ICASSP-91, April 1991, pp [13] J.W. Bez, Binary offse carrier modulaions for radionavigaion, NAVIGATION: Journal of The Insiue of Navigaion, vol. 48, no. 4, [14] M.K. Simon, S.M. Hinedi, and W.C. Lindsey, Digial Communicaion Techniques, Signal Design and Deecion, Pearson Educaion, Inc, 1995.