Parametric Channel Estimation by Exploiting Hopping Pilots in Uplink OFDMA

P aramet ric Channel Estimation by Exploiting Hopping Pilots in Uplink OFDMA Xiaochuan Zhao, T ao Peng and W enbo W ang W ireless Signal Processing an d Network Lab Ke y Laborato ry of Univ ersal W ireless Co mmunicatio n, Ministry of Edu cation Beijing University of Posts and T elecom municatio ns, Beijing, China Email: zhao xiaoch uan@gmail.co m Abstract —In this paper , a p arametric chann el estimation algorithm applicable to uplink s of orthogonal f requency division multiple access (OFDMA) systems whose sub carriers are pseud o- randomly allocated is proposed. By exploiting pilot hopping, estimation of signal parameters via rotational in variance tech- nique (ESPRIT) is employed to estimate the p ath delays of the sparse multipath fadin g chann el. From the delay infor mation, a channel interpolator utilizing global pilots, which c an be irr egular distributed, is d eriv ed to estimate the channel state information on the desired tones. Moreov er , a simple method of estimating the time correlation of th e channel taps is introduced and integrated in the proposed algorithm. Simulation results demonstrate that the proposed algorithm outperfor ms the l ocal linear ch annel interpolator with in a wi de range of SNR’s and Dopp ler’ s. Index T erms —Parame tric channel estimation, Uplink OFDMA, Pilot hopping, ESPRIT . I . I N T RO D U C T I O N Recently , the pseudo-r andom resource a llocation scheme is adopted by wireless standar ds suc h as IE EE 80 2.16e wireless MAN stand ard [1] to provide frequ ency diversity an d co- channel interference averaging in orthogo nal frequ ency di vi- sion multiple access (OFDMA) systems. However , the p rice paid f or th e flexibility of such complex allocatio n sche me is the incr eased difficulty in estimating the ch annel impulse response (CIR) or channel frequency response (CFR) of the uplink. Since the pilot tones allocated to a certain user are pseudo- random ly interlea ved with the oth ers, traditional channel in terpolator s [2] [3] [4] using global pilots are im - practicable, wh ile the local lin ear interpo lators suffer from significant performance degra dation at high SNR regime where the estimation e rror floor shows itself. In ord er to overcome the irregular pilot pattern, [ 5] has propo sed a par ametric ch annel estimation [6] scheme which can greatly redu ce the channel estimation err or for sparse multipath channels. [5] has applied estimation of signal param- eters via rotational in variance techn ique ( ESPRIT) [7] which exploits the shif t-in variance p roper ty of uplink tile structure of IEEE 802 .16e to estimate a nd track multip ath delay spread of the re ceiv ed sign al. W ith the multipath d elays, the ch annel This work is sponsored in part by t he Nat ional Natura l Scienc e Foundat ion of China under grant No.60572120 an d 60602058, and in part by the natio nal high techn ology researching and de velo ping progra m of China (Nationa l 863 Program) under grant No.2006AA01Z257. frequen cy correlation in formatio n can be deriv ed to enable th e sinc-functio n interpolator to estimate the CFR on the de sired tones. The downside of [5] is its dependence o n the special tile structure wh ich intro duces the shift- in variance pro perty — the base of ESPRIT , howe ver this symm etric stru cture may be corrupted. For example, when u plink virtu al multiple- input multiple- outpu t (MI MO) is activ e, pilots in each tile are divided into two exclusive su bsets an d allocated to two users, respectively . For any one, the pilots in the su bset lose the shif t- in variance prop erty . As a result, [5] fails un der uplink virtu al MIMO. In this paper , the strict restriction of the special symmetr y structure of p ilot pattern is relaxed, in stead, pilot hop ping is exploited to en able the par ametric channe l estimation un der arbitrarily irregular pilot distribution. By comb ining two vec- tors of pilots of two contiguous OFDMA symbols, the shift- in variance prop erty is recovered. Th e sample au to-corr elation matrix of the com bined p ilot vectors is formed , an d from which, time c orrelation of the mu ltipath fadin g ch annel is estimated to com pensate the time variance indu ced b y Doppler . Minimum description length (MDL) [8] is ad opted to identify the numbe r of significan t paths. W ith the delay inform ation, a global channel interpo lator is in troduc ed to estimate the CFR on the desired to nes. The rest of this paper is organiz ed as follows. Section I I introdu ces the channel model and the OFDMA system model. The pro posed ch annel estimation algo rithm is presen ted in Section III. Simulation results and analy sis are provided in Section IV . Finally , Section V concludes the pa per . Notation : Bold face letters den ote ma trices or column vec- tors. tr ( · ) , ( · ) ∗ , ( · ) H , ( · ) † , an d || · || F denote trace, conjugate, conjuga te transposition, Moore-Penro se pseud o in verse, and Frobeniu s no rm, respectively . E ( · ) represents the expectation of a stochastic process. [ · ] i,j denotes the ( i , j ) -th element of a matrix. span ( A ) d enotes the colu mn space spanned by A . I I . S Y S T E M M O D E L Consider the uplink of an OFDMA system with a bandwidth of B W = 1 / T Hz ( T is th e samp ling p eriod) . W e use N to denote the tota l numb er of tone s and N used the total nu mber of useful tones, including data an d pilot tones. A cyclic prefix (CP) of len gth L cp is inserted bef ore ea ch OFDMA symbol to eliminate inte r-block interf erence. Thus the whole sym bol duration is T s = ( N + L cp ) T . Now we conside r a certain u ser is scheduled to transmit ov er M ( M ≤ N used ) ton es per sym bol. The in dexes of these tones are co llected in a set, d enoted a s I 0 ( n ) , which can b e chan ged along the time. Regardless of CP , the transmitted sig nal o f th is user can be expressed in matrix fo rm as x ( n ) = F H s ( n ) , where s ( n ) ∈ C N × 1 has no n-zero elements at I 0 ( n ) , and F ∈ C N × N is th e b alanced Fourier tr ansform matr ix with the ( m, n ) -th entry 1 √ N e − j 2 π mn/ N . The discrete comp lex baseban d rep resentation of a mobile wireless CIR o f length L ca n be de scribed by [9] h ( n, τ ) = L − 1 X l =0 γ l ( n ) δ ( τ − τ l ) (1) where τ l ∈ R is the no n-sample- spaced d elay o f th e l - th pa th norm alized by the sampling period T , and γ l ( n ) is the cor respond ing comp lex amp litude. Due to the mo tion of mob ile stations (MS’ s), γ l ( n ) ’ s are wide-sense stationary (WSS) nar rowband co mplex Gaussian pro cesses, which a re uncorr elated amon g d ifferent paths based on the assum ption of uncorrelated scatter ing (US). Furthe rmore we assume delays of multip ath are static dur ing the process of estimation , as they are varying very slowly compa red with th e co mplex amplitude of paths [ 10]. W e co llect τ l ’ s into a set, den oted as I τ . In add ition, W e assume that γ l ( n ) ’ s have the same no rmal- ized correlatio n function r t ( m ) for all l ’ s, he nce r γ l ( m ) = E [ γ l ( n + m ) γ ∗ l ( n )] = σ 2 l r t ( m ) (2) where σ 2 l is the power of the l - th p ath. Assuming a sufficient CP , i.e., L cp ≥ τ L − 1 , the received signal of the base station (BS) fo r the gi ven user, d enoted as y ( n ) ∈ C N × 1 , can be described as y ( n ) = X ( n ) H ( n ) + v ( n ) (3) where X ( n ) ∈ C N × N is a d iagonal matrix with non-zero elements drawn from x ( n ) , H ( n ) ∈ C N × 1 is CFR experienced by the n - th OFDMA symb ol, and is expressed as H ( n ) = F τ h ( n ) (4) where h ( n ) = [ h ( n, τ 0 ) , ..., h ( n, τ L − 1 )] T , and F τ ∈ C N × L is the no n-balan ced Fourier transform matrix with the ( k , l ) - th entry e − j 2 π kτ l / N , and v ( n ) ∈ C N × 1 is the zero- mean complex Gau ssian noise vector with variance σ 2 n , i.e., v ( n ) ∼ C N (0 , σ 2 n I N ) . I I I . C H A N N E L E S T I M A T I O N Denote the numbe r of pilots for th e given user as P ≤ M , the indexes of pilots are collected into a set denoted as I p ( n ) ⊂ I 0 ( n ) . A t th e BS, pilots a re firstly extracted after Fourier transfor mation to per form least-squ ared (LS) channel estimation in f requen cy domain, i.e., ˆ H p ( n ) = X − 1 p ( n ) y p ( n ) = H p ( n ) + w p ( n ) (5) where, X p ( n ) ∈ C P × P is the pre-known pilot diagonal matrix, ˆ H p ( n ) ∈ C P × 1 is the estimated CFR, an d w p ( n ) ∈ C P × 1 is the noise vector expressed as w p ( n ) = X − 1 p ( n ) v p ( n ) (6) and w p ( n ) ∼ C N (0 , σ 2 n I P ) wh en X H p X p = I P for PSK modulated pilots. In the following section, we will introduce a simple pilot hoppin g scheme based o n which the par ametric chan nel esti- mation can be ca rried out. A. Pilot Hopp ing P attern Pilot hoppin g is made in two ph ases: the inner hop ping between two contigu ous OFDMA symb ols and the ou ter hoppin g supe rposed over the inn er one. First, the inner h oppin g is car ried ou t, wh ich shif ts all p ilots of odd OFDMA symbo ls by a pr e-defined o ffset deno ted as ν . Theref ore, after th at, I p,od − I p,ev = ν , where I p,od and I p,ev are ind exes of pilots o f odd and even OFDMA symbols, respecti vely . Seco nd, the ou ter ho pping is perform ed by shif ting the pilots of both even and od d symbols with the same offset. For th e proposed channel estimation algorithm, the inner hoppin g is indispensable, while th e outer h oppin g is option al — although it can acc elerate th e estimation of the au to- correlation matrix of CFR by simulatin g mo re channel fading at the r eceiver throug h includ ing ph ase rotation [1 1]. For simplicity , the outer hopp ing is n ot app lied in this paper . Later on, the shift-inv ariance structure is exploited from the inner pilot hopp ing sche me. B. Estimation of Multip ath Delays Stack two contiguou s estimated CFR vectors into one with the ev en on th e odd, i.e., ˆ H con,p ( n ) = " ˆ H p (2 n ) ˆ H p (2 n + 1) # = H con,p ( n ) + w con,p ( n ) (7) where H con,p ( n ) , ˆ H con,p ( n ) ∈ C 2 P × 1 are the conc atenated CFR vector and its estimation, r espectively , and w con,p ( n ) ∈ C 2 P × 1 is the co rrespon ding noise vector, and w con,p ∼ C N (0 , σ 2 n I 2 P ) . From (4), we h av e H p (2 n ) = F p,ev h (2 n ) (8) H p (2 n + 1) = F p,od h (2 n + 1) (9) where F p,ev , F p,od ∈ C P × L are subm atrices drawn fro m F τ with r ows indexes belong ed to I p,ev and I p,od , respectiv ely . P ≥ L is required so that F p,ev and F p,od are of fu ll co lumn rank. As I p,od − I p,ev = ν , it is straightfo rward that F p,od = F p,ev Φ (10) where Φ ∈ C L × L is a diag onal phase-twisted matrix with the l -th diagonal element [ Φ ] l,l = e − j 2 π ν τ l / N . Apparen tly , Φ contains the mu ltipath delay inform ation τ l ’ s as expected. According to (8 )-(10 ), (7) can be rewritten into ˆ H con,p ( n ) = " F p,ev 0 0 F p,ev Φ # " h (2 n ) h (2 n + 1) # + w con,p ( n ) (11) Consequently , the auto-cor relation matrix of the co ncate- nated estimation of CFR is g iv en in ( 12) shown at the bottom of the next page, w here R h ( m ) ∈ C L × L denotes the auto-co rrelation matr ix o f the CIR vector with the lag m . From the WSSUS assump tion and (2), the ( i, j ) -th elem ent [ R h ( m )] i,j = σ 2 i r t ( m ) δ ( i − j ) , where δ ( · ) is the Kronecker function . T herefor e, R h ( m ) can b e written as R h ( m ) = r t ( m ) R h (0) (13) where R h (0) is a diago nal matrix, an d the l -th diag onal element [ R h (0)] l,l = σ 2 l . Discarding the noise co mpon ent temporally , the useful com- ponen t on the right-h and side o f (12 ) is rewritten in to R con,p = " F p,ev R h (0) F H p,ev η F p,ev R h (0) Φ H F H p,ev η F p,ev ΦR h (0) F H p,ev F p,ev ΦR h (0) Φ H F H p,ev # ∆ = " A 11 A 12 A 21 A 22 # (14) where η = r t (1) = r t ( − 1) , since r t ( m ) = r t ( − m ) for the WSS Gaussian pr ocess h ( n, τ ) . If η is pre -known — fo r example, whe n the maximu m Doppler, denoted as f d , can be measured, according to the Jakes’ model, r t (1) = J 0 (2 π f d T s ) , where J 0 ( · ) is the zeroth o rder Bessel fun ction of the first k ind — we can e liminate the effect of Dop pler from R con,p by e R con,p = " A 11 A 12 /η A 21 /η A 22 # (15) Howe ver , wh en η can no t be ob tained in advance, we can approx imately evaluate η by ˆ η = s k A 12 k 2 F + k A 21 k 2 F k A 11 k 2 F + k A 22 k 2 F (16) Noting when the pilots are equ ispaced and P is a factor of N , which is an o ptimal case [12], ( 16) is accurate. In fact, at this circum stance, F H p,ev ( od ) F p,ev ( od ) = P I L , therefor e k A 11 k 2 F = k A 22 k 2 F = P 2 L − 1 X l =0 σ 4 (17) k A 12 k 2 F = k A 21 k 2 F = η 2 P 2 L − 1 X l =0 σ 4 (18) From (17 )(18), ( 16) is an accurate estimator of η . Denoting F p = h F T p,ev , ( F p,ev Φ ) T i T , (15) is re written into e R con,p = FR h (0) F H (19) and with the noise compone nt of (1 2), (19 ) is ˆ e R con,p ∆ = e R con,p + σ 2 n I 2 P = FR h (0) F H + σ 2 n I 2 P (20) Now (20 ) is of th e standard form to a pply ESPRIT algo- rithm, which we will briefly discuss in the following. Firstly , the eigenv alue dec omposition (EVD ) is app lied to ˆ e R con,p . Then L dominant e igenv alues a re distinguished an d the corre- sponding eigen vectors are collected in a matrix denoted as U . As span ( U ) = span ( F p ) , there exists a nonsing ular matrix T such that U = F p T . V ertically splitting U into two parts, i.e., U =  U T up , U T dw  T , it f ollows that U up = F p,ev T and U dw = F p,ev ΦT . Accord ingly we have U dw = U up T − 1 ΦT = U up Q (21) where Q = T − 1 ΦT is similar with Φ , in oth er words, they have the common eigenv alues. W e can solve Q fro m ( 21) by the linear least-squares (LS) criterion or the non-lin ear total least-squares (T LS) criterion for a b etter solutio n. When Q is obtained, its eigenv alues, denoted as λ l , are calculated thro ugh EVD. Then the tap delays are estimated as ˆ τ l = arg ( λ ∗ l ) N 2 π ν , l = 0 , ..., L − 1 (22) where arg ( λ ) denotes the phase angle of λ in the range [0 , 2 π ) . Finally , as the phase angle wrap s aroun d with a p eriod of 2 π , the multipath delay is un iquely identified only when τ < N /ν , where τ is no rmalized by the samplin g period T . C. Estimation of th e Number of Significan t P a ths In practical applications, th e auto-corr elation ma trix ˆ R con,p in (12) is un attainable. With a finite nu mber o f received symbols, the samp le auto-cor relation matrix is obtained as ˆ R ′ con,p = 2 N t N t / 2 X n =1 ˆ H con,p ( n ) ˆ H H con,p ( n ) (23) where N t is the nu mber of sample O FDMA symbols. ˆ R ′ con,p is an asym ptotic unbiased consistent estimator o f ˆ R con,p . Since ˆ R con,p is af fected by the Doppler, it is not used in the e stimation of the numb er of significant path s. Instead, the auto-cor relation matrix of ˆ H p in (5) is adopted . W ith (14), it can be easily o btained throu gh ˆ R con,p as ˆ R p = ( A 11 + A 22 ) / 2 (24) where we n ote ΦR h (0) Φ H = R h (0) . When ˆ R ′ con,p is available, the sample auto-co rrelation ma- trix ˆ R ′ p is obtained through (24) accordingly , with wh ich, then, MDL is perf ormed to estimate the nu mber of sign ificant paths, denoted as ˆ L . D. Channel Interpo lation After estimating the m ultipath delay s, further modification s are made to enhan ce the perf orman ce. Due to many p otential factors, such as the inaccurate e stimation of η , finite obser- vations o f CFR and add iti ve noise, ESPRIT is impair ed: the estimated multipath delay s fluctuate around th eir tr ue v alues within a certain range, which results in an incomplete path subspace spann ed. In addition , sinc e the multipath delays are non-samp le-spaced, all p aths lea k their power to the samples nearby . An d the closer a sample g ets to a p ath, th e stronger it is influenced by th e path. Considering the flu ctuating estimated d elays as well as the power leaka ge, we prefer to broaden th e ”observation wind ow” around eac h estimated path delay to capture most of its p ower , i.e., for each estimated path ˆ τ l , a captur ing win dow , denoted as W ˆ τ l = {⌊ ˆ τ l ⌋ − β , ..., ⌈ ˆ τ l ⌉ + β } , is inclu ded, where β is a prede fined param eter suc h that 2 β ˆ L ≤ P . Throu gh the simulation, we find β ≤ 5 is sufficient fo r a wide range of SNR’ s and Doppler’ s. Consequently , the expande d set of estimated multipath d elays I ˆ τ is I ˆ τ =   ˆ L − 1 [ l =0 W ˆ τ l   \ I cp (25) where I cp = { 0 , ..., L cp − 1 } denotes the range of CP , hence limits the estimated multipath delays within it. The param eter β requires car efully designin g, since ther e exists a tradeoff: the larger β is, the mo re ene rgy of paths is captured , meanwhile, the mor e noise is a lso in troduced , and vice versa. The channel interpolato rs, denoted as G ev and G od for even and odd sy mbols, respectively , are given as G ev ( od ) = F d,ev ( o d ) F † p,ev ( od ) (26) where F d,ev ( o d ) is a matrix with the ( k , l ) -th elemen t [ F d,ev ( o d ) ] k,l = e − j 2 π φ k ¯ τ l / N , wh ere φ k ∈ I d,ev ( o d ) and ¯ τ l ∈ I ˆ τ , respectively; similarly , F p,ev ( od ) is a matrix with the ( k , l ) - th e lement F p,ev ( od ) k,l = e − j 2 π θ k ¯ τ l / N , wh ere θ k ∈ I p,ev ( od ) and ¯ τ l ∈ I ˆ τ , resp ectiv ely; I d,ev ( o d ) denotes the indexes of da ta tones allocated to th e g iv en user in the ev en (odd ) OFDMA symbols. Finally , the ch annel estimation on the data tones is ˆ H d,ev ( o d ) = G ev ( od ) ˆ H p,ev ( od ) (27) where ˆ H p,ev ( od ) denotes the LS chan nel estimation on the pilot tones of even ( odd) OFDMA sym bols. I V . S I M U L AT I O N R E S U LT S The p erform ance of the prop osed channel estimation algo- rithm is ev aluated for a WiMAX system [1 ] with B W = 1 0 MHz, N = 10 24 , N used = 840 , L cp = 128 , f c = 3 . 5 GHz. For simplicity , data tones are QPSK modulated, an d no forward error codin g is applied. ITU V ehicular A channels 5 1 0 1 5 2 0 2 5 3 0 3 5 - 4 0 - 3 5 - 3 0 - 2 5 - 2 0 - 1 5 - 1 0 - 5 D P , 2 0 & 4 0 m s P H , 2 0 & 4 0 m s P H & D P , 1 0 m s N M S E ( d B ) S N R ( d B ) L L 1 0 m s L L 2 0 m s L L 4 0 m s P H 1 0 m s P H 2 0 m s P H 4 0 m s D P 1 0 m s D P 2 0 m s D P 4 0 m s L o ca l L i n e a r Fig. 1. NMSE comparison for L L, PH and DP under differe nt N t ’ s when f d = 200 Hz and N sch = 20 . [13] is adopted, which co nsists of six individually faded taps with relative d elays as [0, 310, 71 0, 1090 , 1730, 2510 ] ns and average power as [ 0, -1, -9, -1 0, -15 , -20] dB. The Jakes’ s spectru m [9] is ap plied to g enerate the Rayleigh fading channel. Besides, id eal synchronizatio n is assumed, thus the delay of the first tap is always zero ( τ 0 = 0 ). The perfo rmance of the pro posed pilo t ho pping based chan- nel estimation algorithm (PH) is compared with th e local linear interpolatio n algorithm (LL) when up link vir tual M IMO is activ e. The alg orithm in [5] which can be called the ”d oublet- pilots” algo rithm (DP) is also evaluated as a benchmark , and for it, MIMO is inacti ve. For LL, the CFR of a certain data tone is simp ly eq ual to the arithm etic average of the two pilo t ton es within the same tile, since the irregular distributed pilot pattern prohib its the glo bal interpolatio n. Since the second symbol in a tile contains no pilots, PH is adjusted to accommo date this situation: the secon d symbol in each tile is rem oved an d CFR of this symbol is linearly interp olated f rom th e first and third symbols within the same tile. T here are thr ee main factors influence the perfo rmance of PH an d DP do minantly: the number of o bserved OFDMA symbo ls ( N t ), Dopp ler ( f d ), and the numb er o f allocated sub channels 1 ( N sch ). First, different N t ’ s ar e evaluated to examine the con ver- gence o f PH. In Fig.1 , three values o f N t , i.e., 1 0ms, 20ms and 40ms, whic h are 96, 192 and 3 87 OFDMA symbols, cor- respond ingly , are plo tted when f d = 200 Hz and N sch = 20 . Obviously , the per forman ce of LL is ind epende nt of N t , an d it 1 Each subcha nnel con sists of six tile s random distrib uted in frequenc y domain. ˆ R con,p = " F p,ev 0 0 F p,ev Φ # " E  h (2 n ) h H (2 n )  E  h (2 n ) h H (2 n + 1)  E  h (2 n + 1) h H (2 n )  E  h (2 n + 1) h H (2 n + 1)  # " F H p,ev 0 0 Φ H F H p,ev # + σ 2 n I 2 P = " F p,ev 0 0 F p,ev Φ # " R h (0) R h ( − 1) R h (1) R h (0) # " F H p,ev 0 0 Φ H F H p,ev # + σ 2 n I 2 P (12) 5 1 0 1 5 2 0 2 5 3 0 3 5 - 4 0 - 3 5 - 3 0 - 2 5 - 2 0 - 1 5 - 1 0 - 5 P H 1 0 0 & 2 0 0 , D P 1 0 0 ~ 3 0 0 P H & D P N M S E ( d B ) S N R ( d B ) L L 1 0 0 H z L L 2 0 0 H z L L 3 0 0 H z L L 4 0 0 H z L L 5 0 0 H z L L 6 0 0 H z P H 1 0 0 H z P H 2 0 0 H z P H 3 0 0 H z P H 4 0 0 H z P H 5 0 0 H z P H 6 0 0 H z D P 1 0 0 H z D P 2 0 0 H z D P 3 0 0 H z D P 4 0 0 H z D P 5 0 0 H z D P 6 0 0 H z L L Fig. 2. NMSE comparison for LL, PH and DP under dif ferent f d ’ s when N t = 192 and N sch = 20 . lev els o ff at high SNR r egime. For PH and DP , no significan t error floor can be observed. W hen N t = 10 ms, PH at least has a 2 .5dB gain over LL at low SNR r egime, and far better than LL at high SNR regime. More over , PH is about 2dB worse than DP fo r a ll N t ’ s. However , it is worth noting that for the same N t , the number of pilots av ailable fo r DP is two times for PH, since MI MO is active f or PH but not fo r DP . Fig.2 plo ts the perf ormance s o f LL, PH a nd DP un der different Do ppler’ s, when N t = 192 and N sch = 20 . For LL, the perfo rmance gets worse alon g the incr easing of f d , and lev els off at high SNR regime for all f d ’ s. Ho wever , for PH and DP , no significant e rror floor can b e o bserved for lower f d ’ s. Althou gh PH also levels off when f d is h igh, it still outperf orms LL over 10dB at high SNR regime. Moreover , when f d is lower , the performan ce difference betwe en PH and DP is subtle, and when f d is hig her, the NMSE difference is about 5d B at high SNR regime. Finally , d ifferent N sch ’ s are ev aluated in Fig.3. From the figure, it is obvious that the perfor mance difference be tween PH and DP is subtle when N sch ≥ 20 , and both of them are far better than LL. When N sch = 10 , PH outper forms LL abo ut 2dB at low SNR regime an d over 8dB at high SNR regime, meanwhile, it is worse than DP f or abou t 1dB at low SNR regime a nd 6dB a t high SNR r egime. From simulatio ns, we find th at PH o utperfo rms L L over a wide ra nge of SNR’ s and Doppler’ s when th e num ber of observed OFDMA symbols and the number of allocated subchann els ar e not too sm all, i.e., N t ≥ 96 and N sch ≥ 10 . V . C O N C L U S I O N In th is pape r , we p ropose an ESPRIT - based chan nel esti- mation algo rithm ap plicable to up link OFDMA by exploiting pilot ho pping . T hroug h a very simple pilot hopping scheme, the shift-inv ariance pr operty based on which ESPRIT is capa- ble is acquired. Since this special property is derived from a pair of contiguo us OFDMA symbo ls, no special pilot pa ttern, e.g., the dou blet pilo ts in [5], is indispensable within one OFDMA symbol. Hence, the propo sed a lgorithm increases 5 1 0 1 5 2 0 2 5 3 0 3 5 - 4 0 - 3 5 - 3 0 - 2 5 - 2 0 - 1 5 - 1 0 - 5 P H 2 0 , D P 1 0 P H 3 0 , D P 2 0 & 3 0 N M S E ( d B ) S N R ( d B ) L L 1 0 su b ch a n L L 2 0 su b ch a n L L 3 0 su b ch a n P H1 0 su b ch a n P H2 0 su b ch a n P H3 0 su b ch a n DP 1 0 su b ch a n DP 2 0 su b ch a n DP 3 0 su b ch a n L L 1 0 ~ 3 0 Fig. 3. NMSE comparison for LL, PH and D P under dif ferent N sch ’ s when N t = 192 and f d = 200 Hz. the spec trum efficiency and eases the design of the p ilot pattern. Estimating the auto-correlatio n matrix over nu mbers of OFDM A symb ols, the p ropo sed algorith m outp erform s the linear local interp olator within a wid e range of SNR’ s and Doppler’ s. Besides, low rank adaptive filter [14 ] can be integrated to abate the estimatio n latency . R E F E R E N C E S [1] Draft Standar d for Local and Metr opolitan Area Networks Part 16 - Air Inte rface for Fixed B r oadband Wi rel ess Access Systems , IEEE Std., March 2007. [2] O. Edfors et al. , “OFDM Channel Es timatio n by Singular Value De- compostion, ” IEEE T rans. Commun. , vol. 46, pp. 931–939, July 1998. [3] S. Coleri et al. , “Channel Estimation T echnique s Ba sed on Pilot Ar- rangement in OFDM Systems, ” IEEE T rans. Broadca st. , vol. 48, pp. 223–229, September 2002. [4] X. Dong et al. , “Linear Interpolation in Pilot Symbol Assisted Channel Estimation for OFDM, ” IEEE T rans. W ire less Commun. , v ol. 6, pp. 1910–1920, May 2007. [5] M. Ragh av endra et al. , “Paramet ric Channel Estimati on for Pseudo- Random Tile-Allocati on in Uplink OFDMA, ” IEEE Tr ans. Signal Pr o- cess. , vol. 55, pp. 5370–5 381, Nov ember 2007. [6] B. Y ang et al. , “Channel E stimation for OFDM Transmission in Mul- tipat h Fa ding Channels Based on Parametri c Channe l Modeling, ” IEEE T rans. Commun. , vo l. 49, pp. 467–479, March 2001. [7] R. Roy and T . Kailath, “ESPRIT - Estimation of Signal Parameters via Rotati onal In va riance Techniqu es, ” IEE E T rans. Acoust., Sp eech, Signal Pr ocess. , vol. 37, pp. 984–995 , July 1989. [8] M. W ax and T . Kaila th, “Detecti on of Signals by Information Theore tic Criteri a, ” IEE E T rans. A coust., Speec h, Signal Pro cess. , vol. 33, pp. 387–392, April 1985. [9] R. Steele, Mobile Radio Communications . Ne w Y ork: IEEE Press, 1992. [10] D. Tse and P . V iswanath, Fundament als of Wir eless Communic ation . Ne w Y ork: Cambridge Unive rsity Press, 2005. [11] M. Raghave ndra et al. , “Exploiting Hopping Pilots for Para metric Channel Estimation in OFDM Systems, ” IEEE Signal Pr ocess. Lett. , vol. 12, pp. 737–740, N ov ember 2005. [12] S. Ohno and G. Giannakis, “Capacity Maximizing MMSE-Optimal Pilots for Wire less OFDM Over Freque ncy-Sel ecti ve Block Rayleig h- Fading Channel s, ” IEEE T rans. Inf. T heory , vol. 50, pp. 2138–2145, September 2004. [13] “Guideli nes for Eva luation of Radio Tra nsmission T echnolo gies for IMT-2000, ” Recommendations ITU-R M.1225, 1997. [14] P . St robach, “Low-Rank Adapti ve Fil ters, ” IEEE T rans. Signal P r ocess. , vol. 44, pp. 2932–2947, Decembe r 1996.