An Improved Scheme for Initial Ranging in OFDMA-based Networks

An Impro v ed Scheme for Initial Ranging in OFDMA-based Netw orks Luca Sanguine tti* and Michele Morelli Departmen t of I nform ation Engineer ing University of Pisa Pisa, I taly luca.sangu inetti, michele. morelli@iet.unip i.it H. V incent Poor Departmen t of Electrical Engine ering Princeton Un iv ersity Princeton, N J USA poor@pr inceton.ed u Abstract — An efficient scheme fo r in itial ranging has recently been proposed by X. Fu et al. in the context of orthogonal frequency-division multiple-access ( OFDMA) networks based on the IEEE 802 .16e-2005 stand ard. The proposed solution aims at estimating the power le vels and timing o ffsets of the ranging subscriber stations (RSSs) with out taking into account the effect of possible carrier fre quency o ffsets (CFOs) between th e recei ved signals a nd the base station local reference. Motiva ted by the abov e problem, in the present work we d esign a novel ranging scheme for OFDMA in which the ranging signals are assumed to be misaligned both in t ime and frequency . Our goal is to estimate the timing errors and CFOs of each active RSS . Specif- ically , CF O estimation is accomplished by resorting to su bspace- based methods wh ile a least-squares app roach is employed fo r timing reco very . Computer simulations are used to assess the effectiv eness of the proposed solut ion and to make comparisons with existing alternatives. I . I N T R O D U C T I O N The main im pairmen t o f an or thogon al freq uency-division multiple-acce ss ( OFDMA) n etwork is r epresented by its re - markable sen siti vity to tim ing er rors an d car rier frequ ency offsets (CFOs) be tween th e uplink signals and the ba se statio n (BS) lo cal refer ences. For this reason, the IEEE 802 .16e-2 005 standard for OFDMA-based wireless metrop olitan area n et- works (WMANs) spec ifies a synchronization pr ocedur e called Initial Ranging (IR) where subscriber stations that in tend to establish a link with the BS can use some dedicated subcarriers to tran smit th eir specific r anging codes [1]. On ce the BS has revealed the presence o f ra nging subscrib er stations ( RSSs), it has to estimate some fund amental parameters includ ing timing errors, CFOs an d p ower lev els. T w o promin ent schemes for initial syn chron ization and power con trol in OFDMA wer e prop osed in [2] an d [3]. In these works, a long pseudo-n oise sequence is transmitted by each RSS ov er the a vailable ranging subcarriers. T iming recov- ery is then acco mplished o n the b asis o f suitab le cor relations computed in the freq uency- and time -domain , respectively . The main d rawback o f these m ethods is their sensitivity to *This work was completed while the author was with Princet on Uni versit y and it was supported by the U.S. National Science Foundatio n under Grants ANI-03-38807 and CNS-06-25637 . multipath distortion , which destroys orth ogon ality among the employed codes and gi ves rise to m ultiple access interferen ce (MAI). Better results ar e o btained in [4] by using a set of generalized chirp -like (GCL) sequen ces, which echibits increased ro bustness against the channel selectivity . A differ - ent appro ach to man aging the IR p rocess h as recen tly b een propo sed in [5]. Her e, the pilot streams transm itted by RSSs are spread in the time-d omain over ad jacent OFDM blocks using orthogon al codes. In this way , signals of different RSSs can be easily separated at the BS as they r emain orthogo nal after p ropag ating thro ugh the ch annel. Timing info rmation is ev entually acquired in an itera tiv e fashion by explo iting the autocorr elation p roper ties of the received samples induced by the use of the cyclic p refix (CP). Unfo rtunately , this sche me is derived under th e assumptio n of perfect frequ ency align ment between the receiv ed signals and the BS local reference . Actually , the occ urrence o f residual CFOs r esults into a loss of orthogon ality am ong ranging codes and may lead to se vere degradations of the system performan ce in ter ms of mis- detection p robability and estimation a ccuracy . In the present w ork we prop ose a n ovel rang ing scheme for OFDMA systems that is robust to time and fr equency misalignments. The go al is to estimate timing error s and CFOs o f all active RSSs. The number of acti ve co des is found by resorting to the minimum description length (MDL) principle [6] while the multiple signal classification (MUSIC) algorithm [7] is employed to detect which cod es are actua lly activ e and to deter mine their co rrespond ing CFOs. Timing estimation is e ventu ally ach iev ed throug h least-squares (LS) methods. Although the prop osed solu tion allows one to esti- mate the timing errors o f ea ch RSS in a de coupled fashion , it may in volve hu ge computational burden in applications characterized by large pr opagatio n delays. For this reason, we also p resent an alternative sch eme derived from ad hoc - reasoning wh ich results into substantial computational saving. It is worth noting that timing synch ronization in OFDMA uplink transmissions h as received little attention so f ar . A well-established way to hand le timing e rrors is to design the CP length large enoug h to in clude bo th the chann el delay spread and the two-way prop agation d elay between the BS and the user station [8]. This leads to a qu asi-synchro nous system in which timing errors can be viewed as part of the channel impulse respo nse (CIR) and are compensated fo r by the channel eq ualizer . Unfo rtunately , this approach p oses an upper limit to th e max imum tolerable p ropag ation delay or , equiv alently , to the maxim um distance between the BS and the subscriber stations [9]. For this reason, its app lication to a scen ario with large cells (as en visioned in n ext broad band wireless networks) is ha rdly viable. In the latter case, accurate knowledge of the timing erro rs is req uired in orde r to align the uplink signals to the BS time scale. I I . S Y S T E M D E S C R I P T I O N A N D S I G N A L M O D E L A. System description The in vestigated OFDMA n etwork employs N subcar riers with freq uency s pacing ∆ f and indices in the set J = { 0 , 1 , . . . , N − 1 } . Following [5], we deno te R the number of sub channe ls re served fo r the IR pr ocess. Each subchann el is di vided in to Q sub bands un iformly spa ced over th e signal bandwidth at a distance ( N /Q )∆ f fr om each o ther . A gi ven subband is co mposed o f a set o f V ad jacent sub carriers. The subcarrier indices within the q th su bband ( q = 0 , 1 , . . . , Q − 1) of the r th ran ging subc hannel ( r = 0 , 1 , . . . , R − 1) ar e collected into a set J ( r ) q = { i ( r ) q,ν ; ν = 0 , 1 , . . . , V − 1 } with entries i ( r ) q,ν = q N Q + rN QR + ν. (1) The r th subcha nnel is thus composed of subcarriers with indices tak en from J ( r ) = ∪ Q − 1 q =0 J ( r ) q . Hence, a total of N R = QV R r anging subcar riers are av ailable in th e sy stem with indices in the set J R = ∪ R − 1 r =0 J ( r ) . The remain ing N − N R subcarriers are used f or data transmission and are assigned to data subscrib er stations (DSSs) which ha ve already completed their IR proc ess and are assum ed to be p erfectly synchro nized to th e BS time an d f requen cy scales [5]. W e denote by M th e number of con secutive OFDM block s reserved for IR and assume th at each ran ging subchan nel can be accessed at most by M − 1 RSSs. The latter are separated by means o f spec ific rang ing codes selected in a pseud o-rand om fashion fr om a pre defined set { c 1 , c 2 , . . . , c M − 1 } , with c k = [ c k (1) , c k (2) , . . . , c k ( M )] T (the super script T denotes the transpose operation ). As in [5], we assume th at different RSSs employ dif feren t code s. W ithou t loss o f generality , in what follows we concentr ate on the r th rang ing subch annel and denote by K ( r ) ≤ M − 1 th e nu mber of simultaneously activ e RSSs. Also, to simplify the no tation, the subchannel index ( r ) is d ropped in all sub sequent de riv a tions. The wa veform tra nsmitted by the k th RSS ( 1 ≤ k ≤ K ) propag ates th rough a multipath chan nel charac terized by a n impulse r esponse h k = [ h k (0) , h k (1) , . . . , h k ( L − 1)] T of length L (in samplin g periods). At the BS, the re ceiv ed samples are not synch ronized with the local ref erences. W e denote by θ k the timing error expressed in sampling period s while ε k is the frequ ency offset normalized to the sub carrier spacing. As discussed in [8], subscriber station s th at in tend to start the rangin g pr ocess com pute in itial frequ ency and timing estimates on th e ba sis of a downlink control signal broadc ast by the BS. The estimated pa rameters are then employed by each RSS as synch ronization refer ences for the uplink ran ging tran smission. Th is m eans that du ring IR the CFOs are only due to Do ppler shifts and/o r estimation errors and , in consequen ce, they are assumed to lie within a small fraction of th e subcarrier spacing. Furtherm ore, in order to eliminate interblock interf erence (IBI), we assume that d uring the rang ing p rocess th e CP leng th com prises N G ≥ θ max + L sampling periods, where θ max is the maximum expected timing error [9]. Th is assumption is n ot restrictive, since in many standardized OFDM systems the initializatio n blocks are u sually p receded by lo ng CPs. B. Signa l model W e denote by Y m the QV -dimensional vector that co llects the DFT outputs cor respond ing to the consider ed su bchann el during the m th OFDM block. Sin ce the DSSs are assumed to be perfec tly synchron ized to the BS r eferences, their sig- nals will not contribute to Y m . In contrast, the presenc e o f uncomp ensated CFOs destroys orthog onality among ranging signals, th ereby leading to some interchan nel interf erence (ICI). Howe ver, as the subchan nels are well separated in the frequen cy d omain, we can reaso nably neglect in terferen ce on Y m arising f rom ranging signals o f sub channe ls other th an the co nsidered o ne. Under this assump tion, we may write Y m = K X k =1 c k ( m ) e j mω k N T A ( ω k ) S k ( θ k ) + n m (2) where ω k = 2 π ε k / N , N T = N + N G is the duration of the cyclically extended block and n m is a Gau ssian vector with zero mean an d cov ariance matrix σ 2 I QV (we denote by I N the ide ntity m atrix o f o rder N ). Also, w e have defined A ( ω k ) = FV ( ω k ) F H (3) where V ( ω k ) ac counts for the CFOs an d is given by V ( ω k ) = dia g  e j nω k ; n = 0 , 1 , . . . , N − 1  (4) while F =  F H 0 , F H 1 , . . . , F H Q − 1  H (the superscript H denotes the Hermitian transpo sition) with F q ( q = 0 , 1 , . . . , Q − 1) denoting a V × N matrix with entries [ F q ] ν,n = 1 √ N e − j 2 π ni q,ν / N 0 ≤ ν ≤ V − 1 , 0 ≤ n ≤ N − 1 . (5) V ector S k ( θ k ) in ( 2) can be partitioned as S k ( θ k ) =  S T k ( θ k , i 1 ) , S T k ( θ k , i 2 ) , . . . , S T k ( θ k , i Q − 1 )  T , where S k ( θ k , i q ) is a V -d imensional vector with eleme nts S k ( θ k , i q,ν ) = e − j 2 π θ k i q,ν / N H k ( i q,ν ) , 0 ≤ ν ≤ V − 1 (6) while H k ( i q,ν ) denotes the cha nnel frequ ency respon se over the i q,ν th subcarrier and is g iv en by H k ( i q,ν ) = L − 1 X ℓ =0 h k ( ℓ ) e − j 2 π ℓi q,ν / N . (7) From (6) w e see th at θ k simply a ppears as a phase shift across the DFT outputs. The reason is that the CP length is larger than th e maxim al expected prop agation d elay , thereb y making the rangin g signals quasi-synch ronou s. In the following sectio ns we show how vecto rs { Y m } M − 1 m =0 can be exploited to comp ute frequency an d timing estimates for all a ctiv e ra nging c odes. I I I . E S T I M AT I O N O F T H E C F O S T o simp lify th e der iv atio n, we assume that the CFOs are adequately smaller than the subcarrier spacing, i. e., | ω k | ≪ 1 . In su ch a case, ma trices A ( ω k ) in (3) can reaso nably be replaced b y I N to ob tain [ 8] Y m ≈ K X k =1 c k ( m ) e j mω k N T S k ( θ k ) + n m . (8) This equation indicates that each CFO results only in a phase shift between contig uous OFDM blocks. Collecting the i q,ν th DFT output o f all M rang ing blocks into a vector Y ( i q,ν ) = [ Y 0 ( i q,ν ) , Y 1 ( i q,ν ) , . . . , Y M − 1 ( i q,ν )] T , we may write Y ( i q,ν ) = K X k =1 S k ( θ k , i q,ν ) Γ ( ω k ) c k + n ( i q,ν ) (9) where n ( i q,ν ) is Gau ssian distributed with zero-mean an d covariance matr ix σ 2 I M , while Γ ( ω k ) = d iag { e j mω k N T ; m = 0 , 1 , . . . , M − 1 } is a diagonal matrix that accounts fo r the phase sh ifts in duced by ω k . Inspection o f (9) reveals that, apart f rom thermal noise, vector Y ( i q,ν ) is a linear com bination of the frequ ency-rotated codes { Γ ( ω k ) c k } . T his means th at the signal space is spanned by the K vectors { Γ ( ω k ) c k } that corr espond to the active RSSs [1 0]. Then, if we tempo rarily assume that the n umber K of acti ve co des is known at the receiv er, an estimate of ω k ( k = 1 , 2 , . . . , K ) can be obtain ed b y resorting to the MUSIC alg orithm [ 7]. T o see how this comes abo ut, we use the ob servations { Y ( i q,ν ) } to obtain the following sample correlation matrix ˆ R Y = 1 QV V − 1 X v =0 Q − 1 X q =0 Y ( i q,ν ) Y H ( i q,ν ) . (10) Next, ba sed o n the forward-backward (FB) approach [1 0], we compute e R Y = 1 2 ( ˆ R Y + J ˆ R T Y J ) (11) where J is the exchange matrix with 1’ s on th e anti- diagon al and 0 ’ s elsewhere. W e denote by λ 1 ≥ λ 2 ≥ · · · ≥ λ M the eigenv alu es of e R Y arrange d in no n-increa sing or der, an d by { s 1 , s 2 , . . . , s M } the correspondin g eigenv ectors. The MUSIC algorithm r elies o n the fact that the eigenv ectors associated with the M − K smallest eig en values are an estimated basis of the noise subspace a nd, accordin gly , they ar e ap proxim ately orthog onal to all vecto rs in the signal space [7]. Hen ce, an estimate of ω k is obtained b y minimizing the projection of Γ ( e ω ) c k onto the n oise sub space, i. e., ˆ ω k = arg ma x e ω { Ψ k ( e ω ) } , (12) with Ψ k ( e ω ) = 1 P M m = K +1   c H k Γ H ( e ω ) s m   2 . (13) It is w orth observing that C FO recovery must be accom- plished for any acti ve RSS. Howe ver, since the BS has no prior knowledge as to which codes have been transmitted in the c onsidered subch annel, it must evaluate th e quan tities { ˆ ω 1 , ˆ ω 2 , . . . , ˆ ω M − 1 } for the com plete set { c 1 , c 2 , . . . , c M − 1 } . At this stage the pro blem arises of identifyin g whic h codes are actually acti ve. The iden tification algorithm look s for the K largest v alues in the set { Ψ k ( ˆ ω k ) } M − 1 k =1 , say { Ψ u k ( ˆ ω u k ) } K k =1 , and d eclare a s active th e correspon ding codes { c u k } K k =1 . The CFO estimates are eventually fou nd as ˆ ω u = [ ˆ ω u 1 , ˆ ω u 2 , . . . , ˆ ω u K ] T . At this stage we are left with the p roblem of estimating the parameter K to be used in ( 13). For this purp ose, we adopt the MDL ap proach an d o btain [ 6] ˆ K = a rg min ˜ K n F ( ˜ K ) o (14) where F ( ˜ K ) is the fo llowing metric F ( ˜ K ) = 1 2 ˜ K (2 M − ˜ K ) ln( QV ) − QV ( M − ˜ K ) ln ρ ( ˜ K ) ( 15) with ρ ( ˜ K ) denoting the ratio between th e geom etric and arithmetic mean of { λ ˜ K +1 , λ ˜ K +2 , . . . , λ M } . Finally , rep lacing K by ˆ K in (13) leads to the proposed MUSIC-based frequency estimator (MFE) while the de scribed identification alg orithm is called the MUSIC-based code detector (MCD) I V . E S T I M AT I O N O F T H E T I M I N G D E L AY S After code detection and CFO recovery , the BS must acquire informa tion about the tim ing d elays o f all ran ging sign als. This pro blem is now addressed b y resorting to LS meth ods. In doing so we still assume that the num ber o f activ e codes has been correctly e stimated so that ˆ K = K . Also, to simplify the no tation, the in dices { u k } K k =1 of the detected codes are relabeled f ollowing the map u k − → k for k = 1 , 2 , . . . , K . W e begin b y reformu lating (9) in a more c ompact form. For this purpose, we collect the CFOs and timing error s in two K -dimensional vectors ω = [ ω 1 , ω 2 , . . . , ω K ] T and θ = [ θ 1 , θ 2 , . . . , θ K ] T . Then , af ter de fining the matrix C ( ω ) = [ Γ ( ω 1 ) c 1 Γ ( ω 2 ) c 2 · · · Γ ( ω K ) c K ] and the vector S ( θ , i q,ν ) = [ S 1 ( θ 1 , i q,ν ) , S 2 ( θ 2 , i q,ν ) , . . . , S K ( θ K , i q,ν )] T , we may rewrite (9) in the eq uiv alent form Y ( i q,ν ) = C ( ω ) S ( θ , i q,ν ) + n ( i q,ν ) . (16) Omitting for simplicity the functional depende nce o f S ( θ , i q,ν ) on θ and assuming ˆ ω ≈ ω , from (16) the m aximum likelihood estimate of S ( i q,ν ) is found to be ˆ S ( i q,ν ) = [ C H ( ˆ ω ) C ( ˆ ω )] − 1 C H ( ˆ ω ) Y ( i q,ν ) . (17) Substituting (1 6) into ( 17) yie lds ˆ S ( i q,ν ) = S ( i q,ν ) + ξ ( i q,ν ) (18) where ξ ( i q,ν ) is a zero-mea n disturba nce term. From (6) and (7) it f ollows that ˆ S k ( i q,ν ) = e − j 2 πθ k N i q,ν L − 1 X ℓ =0 h k ( ℓ ) e − j 2 πn N i q,ν + ξ k ( i q,ν ) . (19) On deno ting ˆ S k ( ν ) = h ˆ S k ( i 0 ,ν ) , ˆ S k ( i 1 ,ν ) , . . . , ˆ S k ( i Q − 1 ,ν ) i T , and Φ ( θ k , ν ) = dia g { e − j 2 πθ k N i q,ν ; q = 0 , 1 , . . . , Q − 1 } , we may r ewrite (19) as follows ˆ S k ( ν ) = Φ ( θ k , ν ) F ( ν ) h k + ξ k ( ν ) (20) where ξ k ( ν ) = [ ξ k ( i 0 ,ν ) , ξ k ( i 1 ,ν ) , . . . , ξ k ( i Q − 1 ,ν )] T while F ( ν ) is a matrix of dimension Q × L with entries [ F ( ν )] q,ℓ = e − j 2 πℓ N i q,ν for 0 ≤ q ≤ Q − 1 and 0 ≤ ℓ ≤ L − 1 Equation (20) indicates that, apart from the disturban ce term ξ k ( ν ) , ˆ S k ( ν ) is only con tributed by the k th RSS, meaning that ranging signals h av e been successfu lly decoup led at the BS. W e may thus exploit vectors { ˆ S k ( ν ) ; ν = 0 , 1 , . . . , V − 1 } to get LS estimates of ( θ k , h k ) separately fo r each RSS. This amounts to minimizing the following o bjective function with respect to ( ˜ θ k , ˜ h k ) Λ k ( ˜ θ k , ˜ h k ) = V − 1 X ν =0    ˆ S k ( ν ) − Φ ( ˜ θ k , ν ) F ( ν ) ˜ h k    2 . (21) For a fixed ˜ θ k , the min imum o f Λ k ( ˜ θ k , ˜ h k ) is achieved at ˆ h k = 1 QV V − 1 X ν =0 F H ( ν ) Φ H ( ˜ θ k , ν ) ˆ S k ( ν ) (22) where we have used the iden tity F H ( ν ) F ( ν ) = Q · I L . The n, substituting (22) in to (21) and m inimizing with respect to ˜ θ k yields th e timing estimate in the f orm ˆ θ k = a r g max 0 ≤ ˜ θ k ≤ θ max n Υ( ˜ θ k ) o (23) where Υ( ˜ θ k ) is given by Υ( ˜ θ k ) = ˜ θ k + L − 1 X ℓ = ˜ θ k      V − 1 X ν =0 ˆ s k ( ν, ℓ ) e j 2 π ℓν / N      2 (24) and we have denoted by ˆ s k ( ν, ℓ ) the Q -point IDFT of the sequence { ˆ S k ( i q,ν ); 0 ≤ q ≤ Q − 1 } . In the sequel ˆ θ k is termed the L S-based timing estimator (LS-T E). Once ˆ θ k has been computed f rom (23), it is used in (22) to estimate the CIR of the k th RSS as ˆ h k = 1 QV V − 1 X ν =0 F H ( ν ) Φ H ( ˆ θ k , ν ) ˆ S k ( ν ) . (25) It is w orth noting t hat for V = 1 the timing metric (24) reduces to Υ( ˜ θ k )    V =1 = ˜ θ k + L − 1 X ℓ = ˜ θ k | ˆ s k (0 , ℓ ) | 2 (26) and beco mes p eriodic in ˜ θ k with per iod Q . In such a case, the estimate ˆ θ k is af fected by an ambiguity of multiples o f Q . This amb iguity do es n ot rep resent a serious pro blem as long as Q can be chosen to be greater th an θ max . Un fortun ately , in some applications this may not be the case. For example, in [5] we have Q = 8 while θ max = 1 0 2 . A. Reduced -complexity timing estimation Although separating the RSS sig nals at the BS con siderably reduces the system complexity , ev alu ating Υ( ˜ θ k ) for ˜ θ k = 0 , 1 , . . . , θ max may still be comp utationally demand ing, espe- cially in applications whe re θ max is large. For this r eason, we now develop an ad-h oc r educed complexity timin g estimator (RC-TE). W e begin by d ecompo sing the timing error θ k into a fractional part β k , less than Q , plus an integer part which is mu ltiple o f Q , i.e., θ k = β k + p k Q (27) where β k ∈ { 0 , 1 , . . . , Q − 1 } wh ile p k is an in teger p arameter taken fro m { 0 , 1 , . . . , P − 1 } with P = ⌊ θ max /Q ⌋ . Omitting the de tails, it is possible to rewrite Υ( ˜ θ k ) as Υ( ˜ θ k ) = Υ 1 ( ˜ β k ) + Υ 2 ( ˜ β k , ˜ p k ) (28) where Υ 1 ( ˜ β k ) = ˜ β k + L − 1 X ℓ = ˜ β k V − 1 X ν =0 | ˆ s k ( ν, ℓ ) | 2 (29) while Υ 2 ( ˜ β k , ˜ p k ) is shown in ( 30) at the top of the next pag e. The RC-TE is a suboptimal schem e which , starting from (28), estimates β k and p k in a deco upled fashion. More precisely , an estimate of β k is first obtain ed lookin g for th e maximum of Υ 1 ( ˜ β k ) , i. e., ˆ β k = arg max 0 ≤ ˜ β k ≤ Q − 1 n Υ 1 ( ˜ β k ) o . (31) Next, replacing β k with ˆ β k in the r ight-ha nd-side of (28) and maximizing with resp ect to ˜ p k , pr ovides an e stimate of p k in the fo rm Υ 2 ( ˜ β k , ˜ p k ) = 2 ℜ e    ˜ β k + L − 1 X ℓ = ˜ β k V − 2 X ν =0 V − ν − 1 X n =1 ˆ s k ( ν, ℓ ) ˆ s ∗ k ( ν + n, ℓ ) e − j 2 π n ( ℓ + ˜ p k Q ) / N    (30) ˆ p k = arg max 0 ≤ ˜ p k ≤ P − 1 n Υ 2 ( ˆ β k , ˜ p k ) o . (32) A fu rther redu ction o f com plexity is possible when V = 2 . Actu ally , in th is case it can be shown that ma ximizing Υ 2 ( ˆ β k , ˜ p k ) is equ iv alen t to m aximizing cos( ϕ k − 2 π ˜ p k Q/ N ) , where ϕ k = arg    ˆ β k + L − 1 X ℓ = ˆ β k ˆ s k (0 , ℓ ) ˆ s ∗ k (1 , ℓ ) e − j 2 π ℓ/ N    . (33) The estima te of p k is thu s obtained in closed -form as ˆ p k = N ϕ k 2 π Q . (34) V . S I M U L A T I O N R E S U LT S A. System parameters The simulated system is inspired by [5]. The total num ber of sub carriers is N = 1024 while the numb er o f ran ging subchann els is R = 8 . E ach subchannel is composed by Q = 16 subband s unif ormly spaced at a distan ce N / Q = 6 4 . Any subb and com prises V = 2 ad jacent su bcarriers while the rangin g time-slot includes M = 4 OFDM blocks. The ranging cod es are taken fr om a Fourier set of len gth 4 an d are ra ndomly selected by the RSSs at eac h simu lation run (expect for the sequence [1 , 1 , 1 , 1 ] T ). The discrete-time CIRs have L = 12 channel coefficients. The latter are mod eled as circu larly sym metric and in depend ent Gau ssian ra ndom variables with zero mean s (Ray leigh fading) and expo nential power delay profiles, i.e., E {| h k ( ℓ ) | 2 } = σ 2 h · exp( − ℓ/ 12) with ℓ = 0 , 1 , . . . , 1 1 and σ 2 h chosen such that E {k h k k 2 } = 1 . Channels of different u sers ar e a ssumed to be statistically indepen dent. They are gen erated at each new simu lation run and kept fixed over a n entire time-slot. The normalized CFOs are uniformly distributed within the interval [ − Ω , Ω] and vary at each sim ulation run . W e consider a c ell radius of 10 km , correspo nding to a maximum transmission dela y θ max = 204 . A CP of le ngth N G = 256 is chosen to av oid IBI. B. P erformance evaluation W e begin by in vestigating the performanc e of MCD i n terms of prob ability of m aking an incorrect d etection, say P f . This parameter is illustrated in Fig. 1 as a f unction of SNR = 1 /σ 2 under different oper ating condition s. The num ber of ac tiv e RSSs varies from 2 to 3 while the max imal fr equency offset is either Ω = 0 . 0 5 or 0 . 075 . Co mparisons are made with the ranging scheme discussed b y Fu, Li an d Minn (FLM) in [5], where the k th rangin g cod e is declared acti ve p rovided that the quantity 10 -4 10 -3 10 -2 10 -1 10 0 P f 22 20 18 16 14 12 10 8 6 4 2 0 -2 SNR, dB FLM MCD Ω = 0.05 Ω = 0.075 K = 2 K = 3 Fig. 1. P f vs. SNR for K = 2 or 3 when Ω is 0.05 or 0.075. Z k = 1 M 2 Q − 1 X q =0 V − 1 X ν =0   c H k Y ( i q,ν )   2 (35) exceeds a suitable thresh old η wh ich is proportion al to the estimated noise p ower ˆ σ 2 . The results of Fig. 1 indica tes that the pr oposed scheme performs remarka bly better th an FLM because o f its intrinsic robustness against CFOs. As e xpected , the system per forman ce deterio rates fo r large values of K a nd Ω . The rea son is tha t increasing K r educes the dime nsionality of the noise su bspace, whic h degrades the accuracy of the MUSIC estima tor . Fu rthermo re, large CFO values r esult in to significant ICI which is n ot accounted for in the signal mod el (8), whe re A ( ω k ) has b een replaced b y I N . Fig. 2 illustrates the root m ean-squar e-error (RMSE) of the frequen cy estimates obtained with MFE vs. SNR. Again, we see th at th e system p erform ance deteriorates whe n K and Ω are relatively la rge. N ev ertheless, the accur acy o f MFE is satisfactory un der all inv estigated cond itions. The p erform ance o f the timing estimators is measured in terms of pro bability of mak ing a timing error , say P ( ǫ ) , a s defined in [ 8]. More pre cisely , an error e vent is declar ed to occur wh enever the estimate ˆ θ k giv es rise to IBI durin g the data section of the fr ame. In such a c ase, the q uantity ˆ θ k − θ k + ( − N G,D + L ) / 2 is larger than zero or smaller than − N G,D + L − 1 , where N G,D is the CP length d uring the data transmission phase. Fig. 3 illu strates P ( ǫ ) vs. SNR as obtained with RC-TE and FLM wh en N G,D = 64 . The operating conditions are the same of the previous figu res. S ince the performance of LS-TE is virtu ally iden tical to that of RC- TE, it is no t repo rted in o rder no t to overcro wd the figure. W e 10 -4 10 -3 10 -2 10 -1 10 0 Frequency RMSE 22 20 18 16 14 12 10 8 6 4 2 0 -2 SNR, dB Ω = 0.05 Ω = 0.075 K = 2 K = 3 MFE Fig. 2. Accurac y of the frequenc y estimates vs. SNR for K = 2 or 3 when Ω is 0.05 or 0.075. 10 -4 10 -3 10 -2 10 -1 10 0 P ( ε ) 22 20 18 16 14 12 10 8 6 4 2 0 -2 SNR, dB FLM RC-TE Ω = 0.05 Ω = 0.075 K = 2 K = 3 Fig. 3. P ( ǫ ) vs. SNR for K = 2 or 3 when Ω is 0.05 or 0.075. see that f or SNR values larger th an 6 dB the proposed sch eme provides muc h better results than FLM. V I . C O N C L U S I O N S W e have derived a n ovel timing a nd frequen cy synchroniza- tion scheme f or in itial ranging in OFDMA-based networks. The p roposed solution aims at d etecting which codes are ac- tually b eing em ployed and p rovides timing and CFO estimates for all activ e RSSs. CFO estimation is accomplish ed by resort- ing to the MUSIC algorithm while a LS app roach is emp loyed for tim ing recovery . Compar ed to the timing synch ronization algorithm discussed in [5], the propo sed scheme is more robust to freque ncy m isalignments and exhibits improved accuracy . R E F E R E N C E S [1] “IEEE standard for local and metropolitan area networks: Air interface for fixed and mobile broadband wireless access systems amendment 2 : Physi cal and medium access control la yers for combined fixed and mobile operation in license d bands and corrigendum 1, ” IEEE Std 802.16e-2005 and IE EE Std. 802.16-2004/Cor 1-2005 Std. 2006, T ech. Rep., 2006. [2] J. Krinock , M. Singh, M. 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