Doppler Spread Estimation by Subspace Tracking for OFDM Systems

Doppler Spread Estimation b y Subsp ace T racking for OFDM Systems Xiaochuan Zhao, T ao Peng, Ming Y ang and W en bo W ang W ireless Signal Processing an d Network Lab Ke y Lab oratory of Un i versal W ireless Comm unication, Ministry o f Ed ucation Beijing Un i versity of Posts and T eleco mmunicatio ns, Beijing, China Email: zh aoxiaoch uan@gmail.co m Abstract —In this paper , a nove l maximum Doppler sprea d estimation algorithm is presented fo r OFDM systems with the comb-type pilot pattern. The least squared estimated channel frequency responses (CFR’ s) on pilot t ones are used to generate the auto-corr elation matrices with/without a known lag, fr om which the time correlation function can be measured. The maximum Doppler sp read is acqu ired b y in verting the time correla tion function. Since the noise term will bi as the estimator , the estimated CFR’s are projected onto the d elay su bspace of the chan nel to reduce the bi as term as well as the computation complexity . Furthermore, th e sub space tracking algor ithm is adopted to automatically u pdate t he d elay sub space. Simulation results demonstrate the proposed algorithm can q uickly con verge to t he true values for a wide range of SN R’ s and Doppler sprea ds in Rayleigh fading channels. Index T erms —Doppler sprea d, Estimation, Sub space tracking, OFDM, Time correlation, Comb-type pil ot. I . I N T R O D U C T I O N In ord er to co pe with various radio tra nsmission scenarios, adaptive strategies, for example, adaptive modu lation and coding and dynamic resource allocation, are widely employed by many orthogon al frequen cy division multiplexing (OFDM)- based wireless standa rds, e.g., T AB, TV B, IEEE 802 .11/16 and 3GPP L TE [1] . Adaptive schemes auto matically adjust the system c onfiguratio ns and transmission profiles according to some criter ia to accommodate the varying radio environments. The maximum Doppler sp read is one of the key p arameters of criteria for ada ptiv e strategies. It d etermines th e fading rate of the radio cha nnel and its r eciprocal is a metric of the coheren t time of the chann el. W ith th e knowledge of it, wir e- less systems can change the depths of inter leav ers to redu ce coding/d ecoding latencies, decr ease u nnecessary handoffs a nd adjust the rate of power c ontrol to red uce signallin g overhea d. More imp ortant, for O FDM systems, when the Dop pler spread is co mparab le to th e tone spacing, the orthog onality between tones would b e corru pted, which would arise the in ter-carrier interferen ce (I CI) and consequen tly deterior ate the system perfor mance. Howev er, if the Doppler sprea d can be estimated, This work is sponsored in part by the Nationa l Natural Science Foundation of China under grant No.60572120 and 606020 58, and in part by the national high technology researching and de velopi ng program of China (National 863 Program) under grant No.2006AA01Z257 and by the National Basic Research Program of China (National 973 Program) under grant No.2007CB310602. it will facilitate the adap ti ve schedule/con trol alg orithms to approp riately tun e systems to alleviate ICI. Most of existing m ethods of estimatin g the max imum Doppler spre ad are c ategorized into two classes [2] : lev el crossing r ate (L CR)-based and covariance (Cov)-b ased tech- niques. Since the algorithm s reviewed in [2] were no t specif- ically designed for OFDM systems, they did n ot exploit the spec ial signal structure o f OFDM systems. For OFDM systems, most liter atures are Co v-b ased. [ 3] determined the maximum Dopp ler spread through estimating the sma llest positive zer o crossing p oint. In [4], Cai prop osed to obtain the tim e auto -correlation function (T A CF) in time do main by exploiting the cyclic prefix (CP) and its counterpar t. Howe ver , Y ucek [5] pointed out that for scalable OFDM systems whose CP sizes wer e varying over time, [4] was difficult to offer a sufficient estimation of T AC F , which would de gra de its estima- tion accu racy significan tly . On the contrary , Y ucek prop osed to e stimate the Doppler spread throug h the chann el impulse responses (CIR’ s) which were estimated from the period ically inserted train ing symbols. Although the meth od in [5] seems to be more robust than in [4], its shortcomin gs are also evident. For example, in o rder to reduce system overheads, training sym bols are arrang ed to b e far from eac h other, and typically tra nsmitted as p reambles to facilitate the fram e timin g a nd carr ier f requen cy synch roniza- tion. Once the du ration of fr ame is longer than the co herent time o f the chan nel, the maximu m Dopp ler spread cannot be attained becau se T ACF turns to ir rev ersible. Moreover, for sparse training symbols, the co n verging speed of [ 5] would be very slow , which hin ders its em ployment. In this pap er , we propo se to estimate T AC F by exp loiting the comb -type pilot tones [6] which are widely adop ted in wireless standar ds. In order to reduc e noise pertu rbation, th e estimated chann el fr equency respon ses (CFR’ s) are pr ojected onto the delay subspac e [7] to obtain CIR’ s, and the sub space tracking algo rithm [8] is ad opted as we ll to track the drifting delay sub space. This p aper is o rganized as fo llows. In Section I I, the OFDM system and chan nel model are introduced . Then , the maximum Doppler spread estimation algo rithm is pr esented in Section III. Simulation results and analyses are provid ed in Section IV. Finally , Section V conc ludes the p aper . I I . S Y S T E M M O D E L Consider an OFDM system with a ba ndwidth of B W = 1 /T Hz ( T is the sampling period ). N denotes the to tal number o f tones, and a CP o f le ngth L cp is inserted before each symbol to elimin ate inter -b lock interference. Thus the whole symb ol d uration is T s = ( N + L cp ) T . In each OFDM symbol, P ( < N ) to nes ar e u sed as p ilots to assist chan nel estimation. In a ddition, optim al pilot pattern, i.e., e quipowered and equispaced [9], is assumed. Pilot indexes are collected in the set I P , i.e., I P = { φ + p × θ } , ( p = 0 , ..., P − 1) , wh ere φ an d θ a re the offset an d interval, r espectively . The discrete comp lex baseba nd rep resentation of a multi- path CIR of length L can be describ ed b y [10 ] h ( n, τ ) = L − 1 X l =0 γ l ( n ) δ ( τ − τ l ) where τ l is the delay o f the l -th path, normalized by the sampling period T , and γ l ( n ) is th e co rrespond ing com plex amplitude. Due to the mo tion of u sers, γ l ( n ) ’ s are wide-sense stationary (WSS) n arrowband c omplex Gaussian p rocesses, and unco rrelated with ea ch other based o n the assumption of uncorr elated scatterin g (US). In the sequel, P ≥ L and P × θ = N are assumed for determin ability and simplicity , respectively . Furthermo re, we assume the unif orm scattering en vironme nt introdu ced by Clarke [ 11], thus γ l ( n ) ’ s have the iden tical normalized T AC F J 0 (2 π f d t ) for all l ’ s, wh ere f d is the maximum Do ppler spread and J 0 ( · ) is the zeroth order Bessel function o f the first kind . He nce, the discrete T A CF is r t,l ( m ) = σ 2 l J 0 (2 π | m | f d T ) (1) where σ 2 l is the power of the l -th path. Additionally we assume the p ower of ch annel is normalize d, i.e., P L − 1 l =0 σ 2 l = 1 . Assuming a sufficient CP , i.e., L cp ≥ L , the signal model in th e time d omain can b e expressed as y m ( n ) = L − 1 X l =0 h m ( n, τ l ) x m ( n − τ l ) + w m ( n ) where x m ( n ) and y m ( n ) are the n -th samples of th e m -th transmitted an d received OFDM symbols, r espectiv ely , w m ( n ) is the sample of a dditive wh ite Gaussian noise ( A WGN), i.e., E [ w m ( n ) w ∗ m ( n + q )] = σ 2 n δ ( q ) , an d h m ( n, τ l ) is the correspo nding sample of the time-varying CIR. Throug h some simple man ipulations, the sign al mode l in th e frequ ency do - main is written a s Y m = H m X m + W m (2) where X m , Y m , W m ∈ C N × 1 are the m - th tran smitted and received si gn al and noise vectors in the freq uency d o- main, respectively , and H m ∈ C N × N is the channel trans- fer matrix whose ( ν + k , k ) -th elem ent, i.e., [ H m ] ν + k , k , is 1 N P N − 1 n =0 P L − 1 l =0 h m ( n, τ l ) e − j 2 π ( ν n + k τ l ) / N , where k deno tes subcarrier while ν denote s Doppler spread. Apparently , as H m is n on-diag onal, ICI is p resent. Howev er, wh en th e no rmalized maximum Do ppler sprea d, i.e., f d T s , is less than 0.1 , the signal-to-in terference ra tio (SIR) is over 1 7.8 dB [12], which enables us to discard non-diag onal elements of H m with a negligible perfor mance pena lty . As the comb- type pilot p attern is adopte d, only pilot tones, denoted as Y P ; m ∈ C P × 1 , ar e extracted from Y m . By approx imating H m to b e diagon al, ( 2) is m odified to Y P ; m = X P ; m H P ; m + W P ; m (3) where X P ; m ∈ C P × P is a pre-k nown d iagonal m a- trix, and H P,m ∈ C P × 1 consists of diag onal elements of H m . Hen ce, b y denoting the in stantaneous CFR as H m ( n, k ) = P L − 1 l =0 h m ( n, τ l ) e − j 2 π kτ l / N , we ha ve [ H P ; m ] p = 1 N P N − 1 n =0 H m ( n, φ + p × θ ) . Denote the Fourier tr ansform matrix on the pilo t to nes as F P ∈ C P × N , that is, [ F P ] p,n = 1 √ N e − j 2 π ( φ + p × θ ) n/ N , then W P ; m = F P w m , wh ere w m = [ w m (0) , . . . , w m ( N − 1 )] T , so, E [ W P ; m W H P ; m ] = σ 2 n I P . I I I . M A X I M U M D O P P L E R S P R E A D E S T I M A T I O N At the receiver , the least-squa red (LS) chan nel estimation on pilo t tones is carried ou t firstly , i.e., ˆ H P ; m = X − 1 P ; m Y P ; m = H P ; m + V P ; m (4) where ˆ H P ; m ∈ C P × 1 is the estimated CFR, an d V P ; m ∈ C P × 1 is the noise vector exp ressed as V P ; m = X − 1 P ; m W P ; m , hence, V P ; m ∼ C N (0 , σ 2 n I P ) when X H P ; m X P ; m = I P for PSK mod ulated pilot ton es with eq ual power . In the following, w e will intro duce a metho d of estimating the max imum Doppler sprea d based on T ACF measured fro m significant paths o f the cha nnel obtained by projectin g th e LS estimated CFR onto th e delay sub space. A. Measur ement o f the T ime Auto-Corr elation Func tion First, b y definin g the Fourier transfo rm matrix as F P,τ ∈ C P × L with [ F P,τ ] p,l = e − j 2 π ( φ + p × θ ) τ l / N , H P ; m can be expressed as H P ; m = 1 N N − 1 X n =0 H P ; m ( n ) = 1 N N − 1 X n =0 F P,τ h m ( n ) where H P ; m and h m ( n ) are CFR a nd instantan eous CIR vectors, respectively . Regardless of noise, the 0 -lag au to- correlation matrix of H P ; m is R H P (0) = E  H P ; m H H P ; m  = 1 N 2 N − 1 X n =0 N − 1 X q =0 F P,τ A m ( n, q ) F H P,τ (5) where A m ( n, q ) = E [ h m ( n ) h H m ( q )] , an d based on th e as- sumption of WSSUS an d Clarke mode l, its ( l 1 , l 2 ) -th elemen t is [ A m ( n, q )] l 1 ,l 2 = σ 2 l 1 r t ( n − q ) δ ( l 1 − l 2 ) , where r t ( n ) is the n ormalized T ACF , hence A m ( n, q ) is d iagonal. Den oting D = dia g ( σ 2 l ) , l = 0 , . . . , L − 1 , we have A m ( n, q ) = r t ( n − q ) D (6) Substitute (6) into (5) and with som e manipu lations R H P (0) = ξ (0) F P,τ DF H P,τ (7) ξ (0) = 1 N 2 N − 1 X n =0 N − 1 X q =0 r t ( n − q ) (8) Similarly , the β - lag auto-cor relation m atrix of H P ; m ( β ≥ 0 ), defined as R H P ( β ) = E [ H P ; m + β H H P ; m ] , can be written as R H P ( β ) = ξ ( β ) F P,τ DF H P,τ (9) ξ ( β ) = 1 N 2 N − 1 X n =0 N − 1 X q =0 r t ( n − q + ( N + L cp ) β ) (10 ) Then, with (7) an d (9), we have η = s || R H P ( β ) || 2 F || R H P (0) || 2 F = ξ ( β ) ξ (0) (11) where ||·|| F denotes the Frobenius norm. When the no rmalized Doppler spread f d T s ≤ 0 . 1 , re ferring to ( 1), we can make a n approx imation (which we will examine later) a s η ≈ J 0 (2 π β ( N + L cp ) f d T ) (12) Since when β ( N + L cp ) f d T = β f d T s ≤ 0 . 3 8 , J 0 ( · ) is po siti ve and reversible, me anwhile, in o rder to hold the ortho gonality between sub carriers, f d T s is usually smaller than 0.1, th us β ≤ 3 is the feasible rang e. Then f d can be estimated b y ˆ f d = J − 1 0 ( η ) 2 π β T s (13) Now we conside r the effect of no ise. When no ise is present, (7) an d (9) a re rewritten into ˆ R H P (0) = ξ (0) F P,τ DF H P,τ + σ 2 n I P (14) ˆ R H P ( β ) = ξ ( β ) F P,τ DF H P,τ (15) Correspon dingly , ( 11) cha nges into η = s || ˆ R H P ( β ) || 2 F || ˆ R H P (0) || 2 F = s ξ ( β ) 2 ξ (0) 2 + ρ 2 (16) where ρ is de fined as ρ = s P σ 4 n || F P,τ DF H P,τ || 2 F (17) As pilot tones ar e equispaced , F H P,τ F P,τ = P I L , the n || F P,τ DF H P,τ || 2 F = P 2 L − 1 X l =0 σ 4 l therefor e, (17) is ρ = s σ 4 n P P L − 1 l =0 σ 4 l (18) Note P L − 1 l =0 σ 4 l ≤ ( P L − 1 l =0 σ 2 l ) 2 , we h av e ρ ≥ 1 √ P × S N R (19) where S N R = σ − 2 n for the nor malized power of the c hannel and pilo t tones. B. Impr oving the E stimation Accuracy by th e Delay Space Although the maximum Doppler spre ad can be e valuated from (16) and (13), the effect of noise will b ias the result of estimation heavily when P is small and SNR is low . On the other hand, when P is large, the ef fect of noise is negligible, but the sizes of ˆ R H P (0) a nd ˆ R H P ( β ) tu rn to b e so large tha t evaluating their Frobenius nor ms wou ld requ ire a large amo unt o f c alculations, which h inders the employm ent of this metho d in real applicatio ns. Ther efore, w e introduc e the delay subspace on to wh ich the auto-co rrelation matrice s are projected to reduce the effect of no ise as we ll as th e computatio n com plexity . Firstly , the e igen value deco mposition (EVD) is perfor med ˆ R H P (0) = U τ  ξ (0) Λ + σ 2 n I P  U H τ (20) ˆ R H P ( β ) = U τ [ ξ ( β ) Λ ] U H τ (21) Since the numb er of channel taps is L , r ank ( Λ ) = L ≤ P , the last P − L eige n values o f ˆ R H P (0) and ˆ R H P ( β ) ar e σ 2 n and 0, r espectively . Onc e L is available, U τ can be divided into two p arts nam ed as th e ” signal” an d ”noise” su bspaces, respectively , i.e., U τ = [ U τ ,s , U τ ,n ] , where U τ ,s ∈ C P × L , and so d oes Λ , i.e., Λ = diag ( Λ s , 0 P − L ) , wher e Λ s ∈ C L × L . Hence, U H τ ,s ˆ R H P (0) U τ ,s = ξ (0) Λ s + σ 2 n I L (22) U H τ ,s ˆ R H P ( β ) U τ ,s = ξ ( β ) Λ s (23) Based on (22) and (23), ( 16) can be refined as η = v u u t || U H τ ,s ˆ R H P ( β ) U τ ,s || 2 F || U H τ ,s ˆ R H P (0) U τ ,s || 2 F = s ξ ( β ) 2 ξ (0) 2 + ρ 2 r (24) where ρ r is d efined as ρ r = s Lσ 4 n || Λ s || 2 F (25) From (1 4)(15)(20)(21), || F P,τ DF H P,τ || 2 F = || Λ s || 2 F Hence, compar ing ( 25) with (17), the b ias term is reduc ed because ρ r ρ = r L P ≤ 1 (26) Actually , in th e real circumstance, the number of significant taps of wireless ch annels is far less than of p ilot ton es, thereby projecting auto-cor relation matrices o nto th e delay su bspace, like ( 22) and (2 3), can effectiv ely r educe the b ias term and ease the calculation o f η . C. T rac king the Dela y Su bspace – the Pr oposed Alg orithm When th e user is moving , the tap d elays of the chan nel, i.e., τ l ’ s, are slowly drifting [13] [7], which causes F P ; τ to vary and so does U τ ,s . T o accomm odate this variation, the sub space tracking algorithm is ado pted to autom atically u pdate the delay subspace. In addition , if th e num ber of significan t taps of the ch annel is unknown, minimu m description length (MDL) [14] is employed to estimate it. The prop osed algorithm is summarized as follows. Initialize : ( n = 0 ) Q 0 (0) = Q β (0) = [ I L m , 0 T L m ,P − L m ] T A 0 (0) = A β (0) = 0 P ,L m C 0 (0) = C β (0) = I L m Run : ( n = n + 1 ) Input: ˆ H P ( n ) 1) Updating for the 0-lag auto-correl ation matrix: Z 0 ( n ) = Q 0 ( n − 1) ˆ H P ( n ) A 0 ( n ) = α A 0 ( n − 1) C 0 ( n − 1) + (1 − α ) ˆ H P ( n ) Z 0 ( n ) H A 0 ( n ) = Q 0 ( n ) R 0 ( n ) : QR- factorization C 0 ( n ) = Q 0 ( n − 1) H Q 0 ( n ) ˆ L ( n ) = M D L ( dia g ( R 0 ( n ))) 2) Updating for the β -la g auto-correlation matrix: Z β ( n ) = Q β ( n − 1) ˆ H P ( n − β ) A β ( n ) = α A β ( n − 1) C β ( n − 1) + (1 − α ) ˆ H P ( n ) Z β ( n ) H A β ( n ) = Q β ( n ) R β ( n ) : QR-factor ization C β ( n ) = Q β ( n − 1) H Q β ( n ) 3) Estimating η according to (24): ˆ η = v u u t P ˆ L ( n ) l =1 | [ R β ( n )] l,l | 2 P ˆ L ( n ) l =1 | [ R 0 ( n )] l,l | 2 4) Estimating f d according to (13). Remark : α is a posi tive exponential forgetting factor close to 1. L m is the maximum rank to be tested. M D L ( · ) denotes the MDL detector and diag ( R ) deno tes the diagonal elements of R . In the simulation, we set α = 0 . 995 a nd L m = 10 . D. Other Con siderations In this subsection, several fu rther discussions about the propo sed algor ithm are p resented. First, numerica l r esults are shown in T able I to examine (12) wh en N = 51 2 , L cp = 64 and β = 1 . From T able I, we find (12) is a goo d appro ximation when β f d T s is sma ll. It is worth noting that J 0 (2 π β f d T s ) ≤ η , hence f d ≥ J − 1 0 ( η ) 2 π β T s , in other words, (12) tends to over-estimate the maximum Doppler spread a bit. Then we compare th e computatio n co mplexity of the pro- posed alg orithm with (1 6). Th e com putation comp lexity of the subspace tr acking is O ( P × L 2 ) [8]. Since it takes a dominan t propo rtion of the total numb er of instructions requ ired by the pr oposed alg orithm, we use it instead. Mean while, the computatio n com plexity of (16), which dire ctly co mputes the Frobeniu s norm of P × P matric es, is O ( P 2 ) . Apparently , T ABLE I A T A B L E O F V A L U E S O F ( 1 2 ) f d T s 0.02 0.04 0.06 0.08 J 0 (2 πβ f d T s ) 0.9961 0.9843 0.9648 0.9378 η  = ξ ( β ) ξ (0)  0.9961 0.9843 0.9649 0.9381 J 0 (2 πβ f d T s ) η 1.0000 1.0000 0.9999 0.9997 5 1 0 1 5 2 0 2 5 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 0 2 0 0 0 E st i m a t e d M a x. D o p p l e r S p r e a d ( H z) S N R ( d B ) cp 1 ( f d = 2 0 0 H z ) cp 2 ( f d = 4 0 0 H z ) cp 3 ( f d = 6 0 0 H z ) f r o 1 ( f d = 2 0 0 H z ) f r o 2 ( f d = 4 0 0 H z ) f r o 3 ( f d = 6 0 0 H z ) ss1 ( f d = 2 0 0 H z ) ss2 ( f d = 4 0 0 H z ) ss3 ( f d = 6 0 0 H z ) cp Fig. 1. Performance comparison for the CP-based [4], Frobenius-norm- based (16) and subspace-b ased (24) algorit hms when a 20ms frame is used and β = 1 . when P > L 2 , wh ich is usually th e case for sparse multip ath channels, the pro posed alg orithm can redu ce th e compu tation complexity c onsiderably . I V . S I M U L AT I O N R E S U L T S The pe rforman ce of the proposed algo rithm is ev aluated for an OFDM system with B W = 5 MHz ( T = 1 /B W = 20 0 ns), N = 512 , L cp = 64 and P = 64 . ITU V ehicular A Chan- nels [1 5] is adop ted, which consists o f six ind ividually 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 classic Do ppler spec trum, i.e., Jakes’ spectru m [10] , is applied to gener ate the Rayleigh fading chann el. In Fig.1, a CP-based algorith m reported in [4] and (16), which is based on the Froben ius norm , ar e compared with the propo sed subspace-ba sed algor ithm (24) with β = 1 . A 20ms frame inclu ding 19 4 OFDM symb ols is used to generate the statistics. f d = 200 , 40 0 and 60 0 Hz are tested under a range of SNR’ s, respe cti vely . Ap parently , the CP-based algo rithm fails for all f d ’ s when th e SNR is below 20 d B, mean while (16) and (24) work very well for all f d ’ s and SNR’ s but with a moder ate positi ve bias for S N R = 5 dB. In fact, when S N R = 5 dB, reso rting to (19) and (2 6), the lower bou nd of th e bias terms ρ and ρ r are 0.0395 fo r (1 6) and 0.009 9 for (24), respectively . And wh en f d = 600 Hz, accord ing to (8) and (1 0), ξ (0) = 0 . 9938 and ξ ( β ) = 0 . 9476 . Th us, th e relativ e err ors of η are 0. 0007 and 0 .0000 for (16) an d (2 4), respectively , whic h alm ost hav e no effect on the estimation of 0 1 0 2 0 3 0 4 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 6 5 0 7 0 0 7 5 0 8 0 0 8 5 0 E st i m a t e d M a x. D o p p l e r S p r e a d ( H z) F r a m e D u r a t i o n ( m s) f r o 1 ( S N R = 0 d B ) f r o 2 ( S N R = 2 d B ) f r o 3 ( S N R = 4 d B ) ss1 ( S N R = 0 d B ) ss2 ( S N R = 2 d B ) ss3 ( S N R = 4 d B ) Fig. 2. Performanc e comparison between the Frobenius-norm-based (16) and subspace-b ased (24) algorithms when f d = 400 Hz and β = 1 . the maxim um Dop pler spr ead. Therefo re, (16) and (2 4) show almost the same perform ance when SNR’ s are a bove 5 dB. Fig.2 sh ows the perfor mance compar ison between the Frobenu s-norm-b ased (1 6) and subspace-based (24) in the low SNR r egime, specifically , below 5 dB, to emp hasize the capability of noise depr ession of the latter . Different frame duration s ar e used to obtain T A CF . From the figure we can find (24) o utperfo rms ( 16) fo r all the SNR’ s an d fr ame dur ations, although both of them over-estimate th e m aximum Do ppler spread due to th e non-n egligible noise bias term in the low SNR regime, wh ich is also the reaso n why in creasing the length o f ob servation record does n ot help to d ecrease th e bias in this regime. The conv ergence of the pro posed subspace -tracking- based algorithm is shown in Fig.3. Three different max imum Dop pler spreads are tested, i.e., f d = 2 00 , 40 0 and 600 Hz, when S N R = 15 dB. The curves are plotted fro m the thirtieth OFDM symbo l. It is observed from the p lot that all the three curves con verge to their true values after nu mbers of OFDM symb ols and, furth er , the higher the Dopp ler spr ead is, the faster the curve converges. T his is due to the subspace is up dating faster when the Dop pler spread is high er . In addition, after con verging, th e estimated maximum Doppler spread is fluctuating around its true value in a small r ange, hence a dditional time-averaging can be employed to smooth the cu rve. V . C O N C L U S I O N S In this pape r , we pr opose a sub space-trackin g-based max i- mum Dopp ler spr ead estima tion alg orithm which is applicable to O FDM systems with the com b-type pilot patter n. It enjoys three main advantages: i) alleviating the noise term; ii) re- ducing the co mputation comp lexity; iii) tracking the d rifting delay subspace. T hroug h simulations, the perfo rmance of the propo sed algorith m is d emonstrated to outp erform the CP- based a lgorithm [4]. Moreover , since the p roposed algo rithm is based o n the su bspace trackin g, it ca n be easily integrated 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 6 5 0 7 0 0 7 5 0 E st i m a t e d M a x. 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