On the Cramer-Rao Lower Bound for Frequency Correlation Matrices of Doubly Selective Fading Channels for OFDM Systems

On the Cram ´ er -Rao Lo wer Bound for Frequenc y Correlati on Matrices of Doubly Selecti v e F ading Channels for OFDM Systems Xiaochuan Zhao, Ming Y ang, T ao Peng and W enbo W a ng W ireless Sign al Processing and Network Lab Ke y La boratory of Univ ersal W ireless Commun ication, Ministry of Education Beijing University of Posts and T elecomm unications, Beijing, China Email: zhao xiaochuan @gmail.com Abstract —In this paper , the Cram ´ er -Rao lower b ound (CRLB) of the sample frequency correlation matrices (S FCM) is d eriv ed based on a rigoro us model of the doubly selectiv e fadi ng chann el fo r orthogonal frequency division multipl exing (OFDM) systems with pilot-symbol-aided modulation. By assuming a fixed pilot sequence and ind ependent samples, SFCM is complex Wishart distributed. Then, th e maximum likelihood estimator (MLE) and the exact expression of CRLB are obtained. F rom CRLB, the lo wer bounds of total m ean squared error (TMSE) and a vera ge mean squared error (A vgMSE) indepen dent of the pi lot sequence are deduced, whi ch rev eal that th e amount of samples is the dominant factor affecting A vgMS E wh ile th e signal-to-noise ratio an d the maximum Doppler spread hav e negligible effect. Numerical simul ations demonstrate the analytic results. Index T erms —CRLB, Frequency correla tion matrix, Doubly selectiv e fading channels, OFDM. I . I N T RO D U C T I O N Playing a key role in the cha nnel estimatio n for orth ogonal frequen cy di vision multiplexing (OFDM) systems, the fre- quency correlation matrix (FCM) is utilized by many statistics- based chann el estimation algorithm s, e.g., the linear mini- mum mean-squ ared error ( LMMSE) estima tor an d its optimal low-rank appro ximations [1], the MMSE estimato r explo ring both time and f requency correlatio ns [ 2], the two-dimension al W iene r filterin g [3], and those algorithm s b ased o n parametric channel model [4] [5] [6]. In real app lications, th e sample FCM (SFCM) is u sed in stead of th e true one, and u sually obtained throu gh the least squa red (LS) channel estimation . For fixed or slo wly moving rad io chann els whose Doppler spreads ar e relatively small, the channels reveal a featu re of b lock-fading [7 ]. Hen ce, th e Do ppler spr ead affects the accumulatio n of SFCM n egligibly . Howe ver, for fast movin g radio c hannels, intra- symbol fading become s so pro minent that inter-carrier interf erence (ICI) takes effect b y not only degrading the perform ance o f OFDM systems [8], but also This work is sponsored in part by the National Natural Science Foundati on of China under grant No.60572120 and 6060205 8, and in part by the national high technolo gy researching and dev elopin g program of China (Natio nal 863 Program) under grant No.2006AA01Z257 and by the National Basic Research Program of China (Nationa l 973 Program) under grant No.2007CB310602 . affecting the mean and co variance of SFCM. In [9], the bounds of the ICI power is d eriv ed. The bias-prop erty of SFCM in dou bly selecti ve fading channels has been in vestigated in [1 0]. As a counterp art, in this paper, we find out the Cram ´ er-Rao lower bo und (CRLB) for FCM to ev aluate the p erforman ce of m aximum likelihood estimator (MLE) and uncover the factors influencing the estimation accuracy . This p aper is o rganized as fo llows. In Section II, the OFDM system and chan nel model are introduced. Then, in Section III, CRLB f or FCM is derived a nd fu rther discu ssed to uncover the essential factors. Numerical results appear in Section IV. Finally , Section V co ncludes the p aper . Notation : Lowercase an d uppercase bold face letters d enote column vectors and matrices, r espectiv ely . ( · ) ∗ , ( · ) H , and || · || F denote conju gate, conjugate transposition , and Frobenius norm, respec ti vely . ⊗ de notes the Kro necker pr oduct. E ( · ) represents expectation . [ A ] i,j and [ a ] i denotes the ( i , j ) -th element o f A and th e i -th element of a , respectively . diag ( a ) is a diagon al matrix by plac ing a on the diagonal. I I . S Y S T E M M O D E L Consider an OFDM system with a band width of B W = 1 /T Hz ( T is the sampling period). N den otes the total number of to nes, and a cyclic prefix ( CP) o f leng th L cp is in serted before eac h sy mbol to elimina te inter-block in terference . Thus the whole symbol dur ation is T s = ( N + L cp ) T . The complex baseb and mo del of a linear time-variant m o- bile channel with L path s can be described by [1 1] h ( t, τ ) = L X l =1 h l ( t ) δ ( τ − τ l T ) (1) where τ l ∈ R is th e normalized non-samp le-spaced delay of the l - th path, and h l ( t ) is the co rrespond ing complex amplitude. Acco rding to the wid e-sense stationary u ncorre- lated scattering (WSSUS) assumptio n, h l ( t ) ’ s are modeled as uncorr elated narrowband comp lex Gaussian processes. Furthermo re, by a ssuming the un iform scattering en viron- ment introd uced by Clarke [1 2], h l ( t ) ’ s have the id entical normalized time correlation fun ction ( TCF) for all l ’ s, thus the TCF of the l ’ s path is r t,l (∆ t ) = σ 2 l J 0 (2 π f d ∆ t ) (2) where σ 2 l is the p ower of the l -th path, f d is the maxim um Doppler spread, and J 0 ( · ) is the zeroth order Bessel function of the first k ind. Ad ditionally we assume th e p ower of chan nel is normalized , i.e., P L − 1 l =0 σ 2 l = 1 . Assuming a sufficient CP , i.e., L cp ≥ L , the discrete sign al model in the fre quency d omain is written as y f ( n ) = H f ( n ) x f ( n ) + n f ( n ) (3) where x f ( n ) , y f ( n ) , n f ( n ) ∈ C N × 1 are the n -th transmit- ted and r eceiv ed sign al and ad ditiv e white Gaussian n oise (A WGN) vectors, respectively , an d H f ( m ) ∈ C N × N is the channel transfer matrix with the ( k + ν , k ) -th ele ment as [ H f ( n )] k + υ ,k = 1 N N − 1 X m =0 L X l =1 h l ( n, m ) e − j 2 π ( υ m + kτ l ) / N (4) where h l ( n, m ) = h l ( nT s + ( L cp + m ) T ) is the sampled com - plex amplitude of the l -th path. k and υ de note freq uency an d Doppler spr ead, respectively . App arently , as H f ( n ) is non- diagona l, ICI is p resent. In fact, when the n ormalized max - imum Do ppler sp read f d T s ≤ 0 . 1 , the signal-to-in terference ratio (SIR) is over 17. 8 d B [13]. I I I . C R L B F O R F R E Q U E N C Y C O R R E L AT I O N M A T R I C E S Usually SFCM is o btained throu gh the L S chann el e sti- mation. W e co nsider OFDM systems adopting p ilot-symbo l- assisted modu lation (PSAM) [1], hence on ly pilo t symbo ls, denoted as y p ( n ) ∈ C N × 1 , a re extrac ted and used to per form LS channel estimation. I n ad dition, the pilot seque nce is assumed to be inv ar iant along the time. Th erefore, h p,ls ( n ) = X − 1 p y p ( n ) = X − 1 p H p ( n ) x p + X − 1 p n p ( n ) (5) where X p = diag ( x p ) is a diag onal m atrix consisting of pilot symbols, and the noise term is n p ( n ) ∼ C N (0 , σ 2 n I N ) . Denote the instantaneou s chan nel impu lse resp onse (CIR) vector as h t ( n, m ) = [ h 1 ( n, m ) , . . . , h L ( n, m )] T , m = 0 , . . . , N − 1 , accord ing to th e assumptions of WSSUS and unifor m scattering, h t ( n, m ) is co mplex nor mal, i.e., h t ( n, m ) ∼ C N L (0 , D ) where D = diag ( σ 2 l ) , l = 1 , . . . , L . Then fo rm th e CIR matrix as H t ( n ) = [ h t ( n, 0 ) , . . . , h t ( n, N − 1)] , so we ha ve vec ( H t ( n )) ∼ C N LN (0 , Ω ⊗ D ) where Ω ∈ C N × N is a T oeplitz time correlation matrix ( TCM), defined as [ Ω ] m 1 ,m 2 = J 0 (2 π f d ( m 1 − m 2 ) T ) (6) Then accordin g to (4), the channel transfer matrix H f ( n ) = F τ H t ( n ) , whe re F τ ∈ C N × L is the unbalance d Fourier transform matrix, defined as [ F τ ] k,l = e − j 2 π kτ l / N . T hus H f ( n ) ∼ C N N × N (0 , Ω ⊗ ( F τ DF H τ )) (7) Assuming CIR is independe nt of the therm al noise, with ( 5) and (7), we have h p,ls ( n ) ∼ C N N (0 , Σ ) (8) where the covariance m atrix Σ is defin ed as Σ = ω X − 1 p ( R p + σ 2 n ω I N ) X − H p (9) where ω = x H p Ωx p , and R p = F τ DF H τ is the true FCM. When the LS estimated CFR’ s, i.e., h p,ls ( n ) ’ s, are av ailable , SFCM is constru cted as ˆ R p,ls = 1 N t N t X n =1 h p,ls ( n ) h H p,ls ( n ) (10) where N t is the amoun t o f samples. T o derive the proba bility density function (PDF) of SFCM, we assum e that samples are indepen dent o f each oth er , which may be a strict constrain t. Howe ver, when the maximu m Doppler spre ad is large and the spacing between two contigu ous pilot sy mbols is com para- ti vely small, the correlatio n b etween th em is rath er low , wh ich alleviates the effect of model m ismatch. Th en, based o n the assumption of independ ence and (8), we know that SFCM has the complex central W ishart distribution with N t degrees of freedom and covariance matrix Σ ′ = Σ / N t [14], denoted as ˆ R p,ls ∼ C W N ( N t , Σ ′ ) (11) and its PDF is f ( ˆ R p,ls ) = etr ( − Σ ′− 1 ˆ R p,ls )(det( ˆ R p,ls )) N t − N C Γ N ( N t )(det( Σ ′ ) N t (12) where etr ( · ) = exp( tr ( · )) and C Γ N ( N t ) is th e complex multiv a riate gamm a fun ction, d efined as C Γ N ( N t ) = π N ( N − 1) / 2 N Y k =1 Γ( N t − k + 1) Then, from (12), the likelihood functio n is written a s L ( R p ) = tr ( − Σ ′− 1 ˆ R p,ls ) + ( N t − N ) ln(det( ˆ R p,ls )) − ln ( C Γ N ( N t )) − N t ln(det( Σ ′ )) Therefo re, th e score function with respect to the p arameter matrix R p is score ( R p ) = ∂ L ( R p ) ∂ vec ( R p ) = ∂ vec ( Σ ′ ) T ∂ vec ( R p ) × ∂ L ( R p ) ∂ vec ( Σ ′ ) (13) where the first term on th e rig ht-hand sid e of (13) is ∂ vec ( Σ ′ ) T ∂ vec ( R p ) = ω N t ( X − H p ⊗ X − 1 p ) (14) and the second term is ∂ L ( R p ) ∂ vec ( Σ ′ ) = vec [( Σ ′− 1 ˆ R p,ls Σ ′− 1 − N t Σ ′− 1 ) T ] (15) By letting the sco re fun ction equal zero an d with (9), the MLE of FCM is derived as MLE ( R p ) = X p ˆ R p,ls X H p − σ 2 n I N x H p Ωx p (16) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 200 400 600 800 1000 1200 Normalized maximum Doppler spread Maximum Eigenvalue of Γ N = 256 N = 128 N = 1024 N = 512 Fig. 1. Fitting the maximum eigen value s of Ω with (27) for dif ferent sizes of FFT ( N ) and normaliz ed Doppler’ s ( f d T s ). Note that (1 6) relies on th e pre- known TCM, i.e., Ω , an d noise power . For Rayleig h fading chann els, it m eans the max imum Doppler spread, f d , is known. Further, according to the score functio n, the Fisher Info r- mation matrix with respect to R p is J ( R p ) = E "  ∂ L ( R p ) ∂ vec ( R p )   ∂ L ( R p ) ∂ vec ( R p )  H # (17) W ith ( 13)(14)(15), we hav e ∂ L ( R p ) ∂ vec ( R p ) = ω N t ( X − H p ⊗ X − 1 p ) × vec [( Σ ′− 1 ˆ R p,ls Σ ′− 1 − N t Σ ′− 1 ) T ] so, (17) is r ewritten into ( 18), shown a t the bottom of the n ext page. Notice that Σ ′− 1 ˆ R p,ls Σ ′− 1 ∼ C W N ( N t , Σ ′− 1 ) and E [ Σ ′− 1 ˆ R p,ls Σ ′− 1 ] = N t Σ ′− 1 therefor e E { vec [( Σ ′− 1 ˆ R p,ls Σ ′− 1 − N t Σ ′− 1 ) T ] × vec [( Σ ′− 1 ˆ R p,ls Σ ′− 1 − N t Σ ′− 1 ) T ] H } = V ar { vec [( Σ ′− 1 ˆ R p,ls Σ ′− 1 ) T ] } (19) Giv en S ∼ C W N ( N t , Σ ′ ) , the entry of its secon d orig in moment is [15 ] E ([ S ] i,j [ S ] k,l ) = N 2 t [ Σ ′ ] i,j [ Σ ′ ] k,l + N t [ Σ ′ ] k,j [ Σ ′ ] i,l Therefo re, the entry of its seco nd central momen t is E [([ S ] i,j − E ([ S ] i,j ))([ S ] k,l − E ([ S ] k,l ))] = N t [ Σ ′ ] k,j [ Σ ′ ] i,l According ly , (1 9) is rewritten as V ar { vec [( Σ ′− 1 ˆ R p,ls Σ ′− 1 ) T ] } = N t ( Σ ′− H ⊗ Σ ′− T ) (20) Then, with (20), J ( R p ) is J ( R p ) = ω 2 N t ( X − H p ⊗ X − 1 p )( Σ ′− H ⊗ Σ ′− T )( X − 1 p ⊗ X − H p ) From the Fisher In formation matrix, the CRLB o f R p can be derived a s [16] [17] CRLB ( R p ) = J − 1 ( R p ) = N t ω 2 ( X p ⊗ X H p )( Σ ′ H ⊗ Σ ′ T )( X H p ⊗ X p ) = 1 N t ( 1 ω X p Σ H X H p ) ⊗ ( 1 ω X H p Σ T X p ) (21) W ith ( 9), ( 21) can be further written as CRLB ( R p ) = 1 N t ( R p + σ 2 n ω I N ) ⊗ ( R p + σ 2 n ω I N ) T (22) Based on (22), a lower boun d of the total me an squared error (TMSE) for MLE ( R p ) is TMSE LB ( R p ) = tr ( CRLB ( R p )) = 1 N t tr 2 ( R p + σ 2 n ω I N ) = N 2 N t (1 + 1 ω γ ) 2 (23) where γ = σ − 2 n is the signal- to-noise ratio (SNR). And, accordin gly , the lower b ound of the average mean squa red error (avgMSE) is A vg MSE LB ( R p ) = TMSE LB ( R p ) N 2 = 1 N t (1 + 1 ω γ ) 2 (24) (24) verifies the common sense th at the m ore samples col- lected, the more ac curate estimation acquired . And it also reveals that increasing SNR can r educe the estimation err or . Furthermo re, since ω = x H p Ωx p = k x p k 2 2 × x H p Ωx p x H p x p = k x p k 2 2 × R x p ( Ω ) (25) where R x p ( Ω ) is th e Rayleigh q uotient o f Ω a ssociated w ith the pilot seq uence x p , and R x p ( Ω ) ≤ λ max where λ max is the max imum eigenv alue of Ω . Besides, whe n the p ower of pilot symbo l is normalized , k x p k 2 2 = N . Hen ce (24) is f urther lower bou nded by A vg MSE LB ( R p ) = 1 N t (1 + 1 N λ max γ ) 2 (26) T o f urther look into the relationship b etween f d T s and λ max , we examine the extrem e eig en values of Ω f or different f d T s ’ s and N ’ s numerically , and th e re sults are plo tted in Fig.1. Moreover , we find a simple fu nction fitting the max imum eigenv alue s of all cases very well. The fu nction is λ max ( Ω ) = N J 0 (2 π cf d T s ) (27) 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 0 . 0 0 0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 0 0 8 0 . 0 1 0 0 . 0 1 2 N u m b e r o f sa m p l e s A vg M S E a n a l y t i c E V A E T U Fig. 2. Comparison of analytic results (24) and numerical results for EV A and E TU channels when γ = 20 dB and f d = 200 Hz. where c = 0 . 35 when f d T s ≤ 0 . 3 5 ∗ . T herefore , a m ore insightful lower bou nd can be achie ved by A vg MSE LB ( R p ) = 1 N t (1 + 1 N 2 J 0 (2 π cf d T s ) γ ) 2 (28) According to ( 28), we k now that the amoun t of samples, i.e., N t , ef fects the estimation accuracy do minantly but SNR and maximum Doppler spread do not, since N 2 is sufficiently large for most of cur rent systems. I V . N U M E R I C A L R E S U LT S The OFDM system in simulations is of B W = 1 . 25 MHz ( T = 1 / B W = 0 . 8 ms), N = 128 , an d L cp = 16 . T wo 3GPP E-UTRA chan nel mode ls a re adop ted: E xtended V ehicular A model (EV A) and E xtended T ypical U rban model (E TU) [18]. The excess tap delay of E V A is [ 0 , 30 , 1 50 , 310 , 370 , 7 10 , 1090 , 173 0 , 2 510 ] n s, and its r elativ e power is [ 0 . 0 , − 1 . 5 , − 1 . 4 , − 3 . 6 , − 0 . 6 , − 9 . 1 , − 7 . 0 , − 12 . 0 , − 16 . 9 ] dB. For E TU, they are [ 0 , 50 , 1 20 , 2 00 , 230 , 500 , 16 0 0 , 230 0 , 5000 ] ns and [ − 1 . 0 , − 1 . 0 , − 1 . 0 , 0 . 0 , 0 . 0 , 0 . 0 , − 3 . 0 , − 5 . 0 , − 7 . 0 ] dB, respectively . The classic Doppler spectru m, i.e., Jakes’ spectrum [11], is ap plied to generate the Rayleigh fading channel. In Fig.2, we co mpare th e an alytic r esults (24) and the numerical results over a range of N t ’ s for EV A and ET U chan- nels, respectively , when γ = 20 dB and f d = 20 0 Hz. The pilot sequences are QPSK mo dulated an d ra ndomly cho sen. And the collected samples ar e apart from each others far enough ∗ This condition ensures that J 0 (2 παf d T s ) is positi ve and monotonical ly decrea sing with respect to f d T s . In f act, this con dition is al ways satisfied since current applied OFDM systems hav e f d T s ≤ 0 . 1 to maintai n the power of ICI within a tolerable range [13]. 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 0 . 0 0 0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 0 0 8 0 . 0 1 0 0 . 0 1 2 A vg M S E N u m b e r o f sa m p l e s a n a l y t i c E V A E V B Fig. 3. Comparison of analytic results (28) and numerica l results for EV A and E TU channels when γ = 20 dB and f d = 200 Hz. to guaran tee the assump tion o f indep endence. App arently , the analytic results meet the nume rical o nes q uite well. In Fig.3, we compare the ana lytic results (28) and th e nu - merical r esults f or EV A a nd E TU cha nnels, r espectiv ely , when γ = 20 dB a nd f d = 200 Hz. The pilot sequences a re QPSK modulated . In order to examine the effect o f d ifferent pilot sequences on ω , one hu ndred different sequen ces random ly generated are tested and th eir MSE’ s ar e averaged and plotted. From the figure, we find that (28) is a tight b ound ev en for an arbitrary pilot sequence. The distributions of avgMSE for different SNR’ s and Doppler’ s ar e plotted in Fig.4 throug h ten thou sands estima- tions fo r EV A and ETU chan nels, respectively . The amount of sam ples of ea ch test is 200, and the pilot sequences are QPSK modulated and randomly generated. Clearly , avgMSE’ s are centere d arou nd ze ro and most o f th em are within the range o f zer o to CRLB, w hich f ollows that (1 6) is an u nbiased estimator . Moreover , it is also obvio us that th e distributions o f avgMSE fo r EV A and ETU channels a re negligibly in fluenced by γ a nd f d , which follows the ana lytic lower bound (28). V . C O N C L U S I O N In this p aper, the maximu m likeliho od estimator and CRLB of the frequ ency correlatio n matrix for OFDM systems in doubly selective fadin g chann els are de riv ed and analy zed. Throu gh the analyses, we obtain an insigh tful lower bo und o f av erage MSE, i.e., ( 28), an d acco rding to which, the amou nt of samples shows a domin ant impac t on the accu racy of estimation while SNR and maxim um Dopp ler spre ad have relativ ely sma ll effect whe n the nu mber of subcarrier s ar e sufficiently large, althou gh increasing SNR and decre asing maximum Dopple r spread can help to red uce MSE slig htly . J ( R p ) = ω 2 N 2 t ( X − H p ⊗ X − 1 p ) E { vec [( Σ ′− 1 ˆ R p,ls Σ ′− 1 − N t Σ ′− 1 ) T ] vec [( Σ ′− 1 ˆ R p,ls Σ ′− 1 − N t Σ ′− 1 ) T ] H } ( X − H p ⊗ X − 1 p ) H (18) 0 . 0 0 0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 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 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 0 . 1 2 0 . 1 4 P r o p o r t i o n A v g M S E S N R = 5 d B S N R = 1 5 d B S N R = 2 5 d B S N R = 3 5 d B (a) E V A, f d = 200 Hz. 0 . 0 0 0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 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 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 0 . 1 2 0 . 1 4 P r o p o r t i o n A v g M S E S N R = 5 d B S N R = 1 5 d B S N R = 2 5 d B S N R = 3 5 d B (b) ET U, f d = 200 Hz. 0 . 0 0 0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 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 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 0 . 1 2 0 . 1 4 P r o p o r t i o n A v g M S E f d = 2 0 0 H z f d = 4 0 0 H z f d = 6 0 0 H z (c) E V A, γ = 15 dB. 0 . 0 0 0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 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 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 0 . 1 2 0 . 1 4 P r o p o r t i o n A v g M S E f d = 2 0 0 H z f d = 4 0 0 H z f d = 6 0 0 H z (d) ETU, γ = 15 dB. Fig. 4. Distributi ons of av erage MSE for EV A and ETU channel s when N t = 200 . R E F E R E N C E S [1] O. Edfors, M. Sandell , J. v an de Beek, S. Wilson , and P . Borjesson, “OFDM Channel Estimation by Singular V alue Decompostion, ” IEEE T rans. Commun. , vol. 46, pp. 931–939, July 1998. [2] Y . Li, L. Cimini, and N. Sollenbe rger , “Robust Channe l Estimation for OFDM Systems with Rapid Dispersi ve Fading Channels, ” IEE E T rans. Commun. , vol. 46, pp. 902–915, July 1998. [3] P . Hoe her , S. Kaiser , and P . Robert son, “T wo-Dimensional Pilot-Symbol - Aided Channel Estimation by Wi ener Filte ring, ” in IEEE ICASSP 1997 , April 1997, pp. 1845–1848. [4] B. Y ang et al. , “Channel Estimation for OFDM Transmission in Mul- tipat h Fading Channels Based on Parametri c Channel Modeling, ” IEEE T rans. Commun. , vol. 49, pp. 467–479, March 2001. [5] M. Ragha vend ra et al. , “Parame tric Channel Estimation for Pseudo- Random Tile-Allo cation in Uplink OFDMA, ” IEEE T rans. Signal Pr o- cess. , vol. 55, pp. 5370–5381, Novembe r 2007. [6] X. Zhao and T . Peng and W . W ang, “Paramet ric Channel Estimation by Exploitin g Hopping Pilots in Uplink OFDMA, ” in IEEE PIMRC 2008 , Cannes, France, September 2008. [7] D. Shiu, G. Foschini, and M. Gans, “Fading Correl ation and Its E f fect on the Capacity of Multielement Antenna Systems, ” IEEE T rans. Commun. , vol. 48, pp. 502–513, March 2000. [8] T . W ang et al. , “Performance Degrad ation of OFDM Systems Due to Doppler Spreadi ng, ” IEEE T rans. W ireless Commun. , vol. 5, pp. 1422– 1432, June 2006. [9] Y . Li and L. Cimini, “Bounds on the Interchannel Interference of OFDM in Time-V arying Impairments, ” IEEE T rans. Commun. , vol. 49, pp. 401– 404, March 2001. [10] X. Zhao and M. Y ang and T . Peng and W . W ang, “An Anal ysis of the Bias-Property of the Sample A uto-Corre lati on Ma trices of Doubly Select i ve Fading Chann els for OFDM systems, ” Submitted to IEEE ICC 09 , Dresden, Germany, June 2009. [11] R. Steele, Mobile Radio Communications . IEEE Press, 1992. [12] R. Clarke, “A Statist ical Theory of Mobile Radio Reception, ” Bell Syst. T ech. J . , pp. 957–1000, July-Auguest 1968. [13] Y . Choi, P . V oltz, and F . Cassara, “On Channel Estimati on and De- tecti on for Multicarri er Signals in Fast and Select i ve Rayleig h Fading Channel s, ” IEEE T rans. Commun. , v ol. 49, pp. 1 375–1387, August 2001. [14] T . Ratnara jah, R. V aillancour t, and M. Alvo, “Complex Random Ma- trices and Rayleigh Channel Capacity , ” Commun. Inf. Syst. , vol. 3, pp. 119–138, October 2003. [15] D. Maiwal d and D. Kraus, “C alcula tion of Moments of Comple x W ishart and Complex In verse Wish art Distributed Matrices, ” IE E Pr oc.-Radar . Sonar Navig. , vol. 147, pp. 162–168, August 2000. [16] K. Mard ia, J. K ent, and J. Bibby , Mult ivariate A nalysis . Academic Press, 1979. [17] G. Golub and C. V . Loan, Matrix Computati ons , 3rd ed. Ne w Y ork: Johns Hopkins Univ ersity Press, 1996. [18] “3GPP T S 36.101 v8.2.0 – Evolv ed Univ ersal Terrestrial Radio Access (E-UTRA); User Equipment (UE) Radio Transmission and Reception (Relasa se 8), ” 3GPP , May 2008.