Eric Pierre Simon
email: eric.simon@univ-lille1.fr

Hussein Hijazi
email: hussein.hijazi@hotmail.fr

Laurent Ros
email: laurent.ros@gipsa-lab.inpg.fr

Marion Berbineau
email: marion.berbineau@inrets.fr

Joint Estimation of Carrier Frequency Offset and Channel Complex Gains for OFDM Systems in Fast Time-varying Vehicular Environments

In this paper, the physical nature of the radiochannel is considered by using an L-path channel model to developing an algorithm for OFDM systems operating in fast timevarying vehicular environment. Assuming the Time of Arrival (TOA) is known, a novel iterative pilot-aided algorithm for joint estimation of multi-path Rayleigh channel complex gains and Carrier Frequency Offset (CFO) is introduced. Each complex gain time-variation, within one OFDM symbol, is approximated by a Basis Expansion Model (BEM) representation. An autoregressive (AR) model is built for the parameters to be estimated (the AR model for the BEM coefficients is based on the Jakes process). The algorithm performs recursive estimation using Extended Kalman Filtering. Hence, the channel matrix is easily computed, and the data symbol is estimated with free intersub-carrier-interference (ICI) when the channel matrix is QRdecomposed. It is shown that only one iteration is sufficient to achieve the performance of the ideal case where knowledge of channel response and CFO is available.

I. INTRODUCTION

O RTHOGONAL frequency division multiplexing (OFDM) has become a standard technique for many wireless networks. However, it is well known that small carrier frequency offsets (CFOs) yield severe degradation in OFDM modulation since it produces inter-carrier interference (ICI) and attenuates the desired signal. These effects reduce the effective signal-to-noise ratio (SNR) in OFDM reception resulting in degraded system performance [START_REF] Pollet | BER Sensitivity of OFDM Systems to Carrier Frequency Offset and Wiener Phase Noise[END_REF] [START_REF] Steendam | Sensitivity of Orthogonal Frequency-Division Multiplexed Systems to Carrier and Clock Synchronisation Errors[END_REF]. CFOs are mainly caused by two different factors: Doppler effects and oscillator inaccuracies in the transmitter and receiver. Frequently, these two quantities are grouped together and modeled as a single frequency offset as in [START_REF] Moose | A technique for orthogonal frequency division multiplexing frequency offset correction[END_REF] [START_REF] Roman | Joint Time-Domain Tracking of Channel and Frequency Offsets for MIMO OFDM Systems[END_REF]. However, this model is not sufficiently accurate since separate offset parameter is needed for each propagation path given that the Doppler shift depends on the angle of arrival, which is peculiar to each path. Recently, it has been proposed to track directly the channel paths, which permits to take into account separate Doppler for each path [START_REF] Hijazi | Time-varying channel complex gains estimation and ICI suppression in OFDM systems[END_REF] [START_REF] Tang | Pilot-assisted timevarying channel estimation for OFDM systems[END_REF]. Those works estimate the equivalent discrete-time channel taps ( [START_REF] Tang | Pilot-assisted timevarying channel estimation for OFDM systems[END_REF]) or the real path complex gains ( [START_REF] Hijazi | Time-varying channel complex gains estimation and ICI suppression in OFDM systems[END_REF]), which are both modeled by a basis expansion model (BEM). The BEM methods are Karhunen-Loeve BEM (KL-BEM), prolate spheroidal BEM (PS-BEM), complex exponential BEM (CE-BEM) and polynomial BEM (P-BEM).

However the CFO due to the mismatch between transmitter and receiver oscillators is not taken into account in those algorithms. In this paper, we propose a complete algorithm capable of estimating this CFO together with the time-variation of each channel path. To further improve the estimation accuracy, the algorithm uses decision feedback. Hence, the accuracy of the channel estimation, frequency offset estimation and symbol detection are enhanced simultaneously. Note also that since the pilots are used for both channel and frequency offset estimation, the pilot usage efficiency is greatly improved.

Generally, it is preferable to directly estimate the physical channel parameters [START_REF] Yang | Channel estimation for OFDM transmisson in mutipath fading channels based on parametric channel modeling[END_REF] [5], instead of the equivalent discretetime channel taps [START_REF] Tang | Pilot-assisted timevarying channel estimation for OFDM systems[END_REF]. Indeed, as the channel delay spread increases, the number of channel taps also increases, thus leading to a large number of BEM coefficients, and consequently more pilot symbols are needed. Estimating the physical propagation parameters means estimating multi-path TOAs and multi-path complex gains. Note that the TOAs can be safely assumed invariant (at least during a block of several OFDM symbols). In this work, they are assumed perfectly estimated. It should be noted that an initial, and generally accurate estimation of the number of paths and TOAs can be obtained by using the MDL (minimum description length) and ESPRIT (estimation of signal parameters by rotational invariance techniques) methods [START_REF] Yang | Channel estimation for OFDM transmisson in mutipath fading channels based on parametric channel modeling[END_REF] [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF].

Generally, the CFO due to the oscillator mismatch is considered constant during the transmission. One reason for this is that oscillators drift with temperature, supply voltage, load, and other slowly changing environmental parameters. However, depending on communication duration, time-varying CFO can be significant in a real physical environment. Our algorithm is able to track the CFO in case of variation. Our algorithm is a recursive algorithm based on Extended Kalman Filtering (EKF) combined with QR-equalization for data detection.

This paper is organized as follows: Section II introduces the OFDM system and the BEM modeling. Section III describes the AR model for the BEM coefficients and the Extended Kalman Filter. Section IV covers the algorithm for joint channel estimation and data recovery. Section V presents the simulations results which validate our technique. Finally, our conclusions are presented in Section VI.

The notations adopted are as follows: Upper (lower) bold face letters denote matrices (column vectors). [x] k denotes the kth element of the vector x, and [X] k,m denotes the [k, m]th element of the matrix X (note that the indices begin from 0). I N is a N × N identity matrix and 0 N is a N × N matrix of zeros. diag{x} is a diagonal matrix with x on its main diagonal and blkdiag{X, Y} is a block diagonal matrix with the matrices X and Y on its main diagonal. The superscripts (•) T , (•) * and (•) H stand respectively for transpose, conjugate and Hermitian operators. E[•] is the expectation operation. J 0 (•) is the zerothorder Bessel function of the first kind. ∇ x represents the firstorder partial derivative operator i.e., ∇ x = [ ∂ ∂x1 , ..., ∂ ∂xN ] T .

II. SYSTEM MODEL

A. OFDM System Model

Consider an OFDM system with N sub-carriers, and a cyclic prefix length N g . The duration of an OFDM block is T = N s T s , where T s is the sampling time and

N s = N + N g . Let x n = x n [-N 2 ], x n [-N 2 + 1], ..., x n [ N 2 -1] T be the nth transmitted OFDM symbol and {x n [b]} are normalized symbols (i.e., E x n [b]x * n [b] = 1)
. The frequency mismatch between the oscillators used in the radio transmitters and receivers causes a CFO ∆F . The normalized CFO is denoted ν = ∆F N T s . After transmission over a multi-path Rayleigh channel, the nth received OFDM symbol y n , where

y n = y n [-N 2 ], y n [-N 2 + 1], ..., y n [ N 2 -1]
T is given by [START_REF]Polynomial estimation of time-varying multi-path gains with intercarrier interference mitigation in OFDM systems[END_REF] [6]:

y n = H n x n + w n (1)
where

w n = w n [-N 2 ], w n [-N 2 + 1], ..., w n [ N 2 -1]
T is a white complex Gaussian noise vector of covariance matrix σ 2 I N and H n is the N × N channel matrix. The elements of H n can be written in terms of equivalent channel taps [START_REF] Kim | MAP-Based Channel Estimation for MIMO-OFDM Over Fast Rayleigh Fading Channels[END_REF] g

(n) l (qT s ) = g l (nT + qT s ) : [H n ] k,m = 1 N L -1 l=0 e -j2π( m N -1 2 )•l N -1 q=0 e j2π νq N g (n) l (qT s )e j2π m-k N q , (2)
or in terms of physical channel parameters [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF] (delays τ l and complex gains α

(n) l (qT s ) = α l (nT + qT s ) ), yielding: [H n ] k,m = 1 N L-1 l=0 e -j2π( m N -1 2 )τ l N -1 q=0 e j2π νq N α (n) l (qT s )e j2π m-k N q . (3)
L < N g and L are respectively the number of channel taps and the number of paths. The delays are normalized by T s and not necessarily integers (τ l < N g ). The L elements of α

(n) l (qT s ) are uncorrelated. However, the L elements of g (n) l (qT s ) are correlated, unless that the delays are multiple of T s as mostly assumed in the literature. They are wide-sense stationary (WSS), narrow-band zero-mean complex Gaussian processes of variances σ g l 2 and σ α l 2 , with the so-called Jakes' power spectrum of maximum Doppler frequency f d [START_REF] Jakes | Microwave Mobile Communications[END_REF]. The average energy of the channel approach is normalized to one, i.e.,

L -1 l=0 σ g l 2 = 1 and L-1 l=0 σ α l 2 = 1.
In the sequel, we will make the calculus based on the second approach (physical channel) and we can deduce the results of the first approach (channel taps) by replacing L by L and the set of delays τ l by l, l = 0 : L -1 .

B. BEM Channel Model

Since the number of samples to be estimated LN s is greater than the number of observation equations N , it is not efficient to estimate the time-variation of the complex gains, using directly the observation model in [START_REF] Pollet | BER Sensitivity of OFDM Systems to Carrier Frequency Offset and Wiener Phase Noise[END_REF]. Thus, we need to reduce the number of parameters to be estimated. In this section, our aim is to accurately model the timevariation of α (n) l (qT s ) by using a BEM. Collecting the samples of the lth path within the nth OFDM symbol in a N s × 1 vector α

(n) l = α (n) l (-N g T s ), ..., α (n) l (N -1)T s T , we can express α (n) l as: α (n) l = α (n) BEM l + ξ (n) l = Q c (n) l + ξ (n) l (4)
where

Q = [q 0 , ..., q Nc-1 ] is a N s × N c matrix that col- lects N c orthonormal basis function q d as columns, c (n) l = c (n) (0,l) , ..., c (n) (Nc-1,l)
T represents the N c BEM coefficients for the lth complex gain of the nth OFDM symbol, and ξ (n) l represents the corresponding BEM modeling error, which is assumed to be minimized in the MSE sense [START_REF] Leus | On the Estimation of Rapidly Time-Varying Channels[END_REF] [START_REF] Hijazi | Analytical Analysis of Bayesian Cramer-Rao Bound for Dynamical Rayleigh Channel Complex Gains Estimation in OFDM System[END_REF]. Under this criterion, the optimal BEM coefficients and the corresponding model error are given by:

c (n) l = Q H Q -1 Q H α (n) l (5) ξ (n) l = (I Ns -S)α (n) l (6)
where

S = Q Q H Q -1 Q H is a N s × N s matrix.
It provides the MMSE approximation for all BEM containing N c coefficients, given by:

MMSE l = 1 N s E ξ (n) l ξ (n) l H (7) = 1 N s Tr I Ns -S R (0) α l I Ns -S H (8)
where R (s)

α l = E α (n) l α (n-s) l H is the N s × N s correlation matrix of α (n) l
with elements given by:

[R (s) α l ] k,m = σ α l 2 J 0 2πf d T s (k -m + sN s ) (9)
Various traditional BEM designs have been reported to model the channels time-variations, e.g., the CE-BEM [START_REF] Hijazi | Analytical Analysis of Bayesian Cramer-Rao Bound for Dynamical Rayleigh Channel Complex Gains Estimation in OFDM System[END_REF] and the DKL-BEM which employs basis sequences that corresponds to the most significant eigenvectors of the autocorrelation matrix R (0) α l [START_REF] Kannu | MSE-optimal Training for Linear Timevarying Channels[END_REF]. From now on, we can describe the OFDM system model derived previously in terms of the BEM. Substituting (4) in [START_REF] Pollet | BER Sensitivity of OFDM Systems to Carrier Frequency Offset and Wiener Phase Noise[END_REF] and neglecting the BEM model error, we obtain after some algebra:

[Q] k,m = e j2π( k-N g Ns )(m-Nc-1 2 ) [14], the GCE-BEM [Q] k,m = e j2π( k-N g av )(m-Nc-1 2 ) with 1 < a ≤ Nc-1 2f d T [12], the P-BEM [Q] k,m = (k -N g) m
y n = K n (ν) • c n + w n ( 10
)
where the LN c × 1 vector c n and the N × LN c matrix K n are given by:

c n = c (n) T 0 , ..., c (n) T L-1 T (11)
K n (ν) = 1 N Z (n) 0 (ν), ..., Z (n) L-1 (ν) (12)
Z (n) l (ν) = [M 0 (ν) diag{x n } f l , ..., M Nc-1 (ν) diag{x n } f l ] (13)
where vector f l is the lth column of the N × L Fourier matrix F and M d (ν) is a N × N matrix given by:

[F] k,l = e -j2π( k N -1 2 )τ l (14)
[M d (ν)] k,m = N -1 q=0 e j2π νq N [Q] q+Ng ,d e j2π m-k N q (15)
Moreover, the channel matrix can be easily computed by using the BEM coefficients [START_REF]Polynomial estimation of time-varying multi-path gains with intercarrier interference mitigation in OFDM systems[END_REF]:

H n = Nc-1 d=0 M d (ν)diag{Fχ (n) d } (16)
where χ

(n) d = c (n) (d,0) , ..., c (n) (d,L-1)
T .

III. AR MODEL AND EXTENDED KALMAN FILTER

A. The AR Model for c n

The optimal BEM coefficients c

(n) l are correlated complex Gaussian variables with zero-means and correlation matrix given by: R (s)

c l = E[c (n) l c (n-s) l H ] = Q H Q -1 Q H R (s) α l Q Q H Q -1 (17)
Hence, the dynamics of c

(n) l can be well modeled by an autoregressive (AR) process [START_REF] Baddour | Autoregressive modeling for fading channel simulation[END_REF] [17] [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF] . A complex AR process of order p can be generated as:

c (n) l = p i=1 A (i) c (n-i) l + u (n) l (18)
where A (1) , ..., A (p) are N c ×N c matrices and u (n) l is a N c ×1 complex Gaussian vector with covariance matrix U l . From [START_REF]Polynomial estimation of time-varying multi-path gains with intercarrier interference mitigation in OFDM systems[END_REF], it is sufficient to choose p = 1 to correctly model the coefficients. The matrices A (1) = A and U l are the AR model parameters obtained by solving the set of Yule-Walker equations defined as:

A = R (1) c l R (0) c l -1 (19) U l = R (0) c l + AR (-1) c l (20)
Using [START_REF] Hijazi | OFDM High Speed Channel Complex Gains Estimation Using Kalman Filter and QR-Detector[END_REF], we obtain the AR model of order 1 for c n :

c n = A c • c n-1 + u cn (21)
where

A c = blkdiag {A, ..., A} is a LN c × LN c matrix and u cn = u (n) T 0 , ..., u (n) T L-1
T is a LN c × 1 zero-mean complex Gaussian vector with covariance matrix U c = blkdiag {U 0 , ..., U L-1 }.

B. The AR Model for ν n

Let us write the order 1 AR model for ν n as follows:

ν n = a • ν n-1 + u νn (22)
Since the CFO can be assumed as constant during the observation interval, a is considered to be close to 1, a = 0.99.

The state noise parameter u νn is assumed to be zero-mean complex Gaussian with variance σ 2 ν .

C. State equation

Now, let us write the state-variable model. The state vector at time instance n consists of the BEM coefficients c n and the vector of CFO ν n :

µ n = c T n , ν T n T (23)
There are LN c BEM coefficients and 1 CFO values in the state vector of dimension LN c + 1 × 1. Then the linear state equation may be written as follows:

µ n = A • µ n-1 + u n ( 24
)
where the state transition matrix is defined as follows:

A = blkdiag {A c , a} (25)
The LN c + 1 × 1 noise vector is such that u n = u T cn , u νn T with covariance matrix U = blkdiag U c , σ 2 ν .

D. Extended Kalman Filter (EKF)

The measurement equation ( 10) can be reformulated as:

y n = g (µ n ) + w n ( 26
)
where the nonlinear function g of the state vector µ n is defined as

g (µ n ) = K n (ν) • c n .
Nonlinearity of the measurement equation ( 26) is caused by CFO. The BEM coefficients are still linearly related to observations. Since the measurement equation is nonlinear, we use the Extended Kalman filter to adaptively track µ n . Let μ(n|n-1) be our a priori state estimate at step n given knowledge of the process prior to step n, μ(n|n) be our a posteriori state estimate at step n given measurement y n and, P (n|n-1) and P (n|n) are the a priori and the a posteriori error estimate covariance matrix of size LN c + 1 × LN c + 1, respectively. We initialize the EKF with μ(0|0) = 0 LNc+1,1 and P (0|0) given by:

P (0|0) = blkdiag R (0) c , b (27)
R (s) c = blkdiag R (s) c0 , ..., R (s) cL-1
where R (s) c l is the correlation matrix of c

(n) l defined in [START_REF] Anderson | Optimal filtering[END_REF]. To derive the EKF equations, we need to compute the Jacobian matrix G n of g (µ n ) with respect to µ n and evaluated at μ(n|n-1) :

G n = ∇ T µn g (µ n ) µn= μ(n|n-1) = ∇ T cn g (µ n ) µn= μ(n|n-1) , ∇ T νn g (µ n ) µn= μ(n|n-1) (28)
After computation, we find:

G n = K n (ν (n|n-1) ), K n (ν (n|n-1) )ĉ (n|n-1) (29)
where

K n (ν) = 1 N Z (n) 0 (ν), ..., Z (n)
L-1 (ν) (30)
Z (n) l (ν) = M 0 (ν) diag{x n } f l , ..., M Nc-1 (ν) diag{x n } f l (31)
The N × N matrix M d (ν) is given by:

M d (ν) k,m = N -1 q=0 j2π q N e j2π νq N [Q] q+Ng ,d e j2π m-k N q (32)
The EKF is a recursive algorithm composed of two stages: Time Update Equations and Measurement Update Equations. These two stages are defined as: Time Update Equations:

μ(n|n-1) = A μ(n-1|n-1) P (n|n-1) = AP (n-1|n-1) A H + U (33)
Measurement Update Equations:

K n = P (n|n-1) G H n G n P (n|n-1) G H n + σ 2 I N -1 μ(n|n) = μ(n|n-1) + K n y n -g μ(n|n-1) P (n|n) = P (n|n-1) -K n G n P (n|n-1) (34)
where K n is the Kalman gain. The Time Update Equations are responsible for projecting forward (in time) the current state and error covariance estimates to obtain the a priori estimates for the next time step. The Measurement Update Equations are responsible for the feedback, i.e., for incorporating a new measurement into the a priori estimate to obtain an improved a posteriori estimate. The Time Update Equations can also be thought of a predictor equations, while the Measurement Update Equations can be thought of a corrector equations.

IV. JOINT DATA DETECTION AND EKF

In the iterative algorithm for joint data detection, channel and CFO extended Kalman estimation, the N p pilots subcarriers are evently inserted into the N subcarriers at the positions P = {p r | p r = (r -1)L f + 1, r = 1, ..., N p }, where L f is the distance between two adjacent pilots. We use the the QR-equalizer [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF] [START_REF] Hijazi | OFDM High Speed Channel Complex Gains Estimation Using Kalman Filter and QR-Detector[END_REF] for the data detection. The QRequalizer allows us to estimate the data symbol with free ICI by performing a so-called QR-decomposition. The algorithm proceeds as follows:

initialization: • μ(0|0) = 0 LNc+N R N T ,1 • compute P (0|0) as (27) n ← n + 1 :
• execute the Time Update Equations of EKF (33)

• compute the channel matrix by substituting µn with the prediction parameters μ(n|n-1) in (16) • recursion:i ← 1 remove the pilot ICI from the received data subcarriers -Detection of data symbols execute the Measurement Update Equations of EKF (34) compute the channel matrix using [START_REF] Baddour | Autoregressive modeling for fading channel simulation[END_REF] with the updated parameters

-i ← i + 1
where i represents the iteration number.

V. SIMULATION

In this section, the performance of our recursive algorithm is evaluated in terms of MSE for the channel and CFO estimation and BER for data detection. The normalized channel model is GSM Rayleigh model with L = 6 paths and maximum delay τ max = 10T s (see table I) [START_REF] Zhao | A novel Channel estimation method for OFDM Mobile Communications Systems based on pilot signals and transform domain processing[END_REF][20] [START_REF]Polynomial estimation of time-varying multi-path gains with intercarrier interference mitigation in OFDM systems[END_REF]. A 4QAM-OFDM system with normalized symbols, N = 128 subcarriers, N g = N 8 subcarriers, N p = 32 pilots (i.e., L f = 4) and

1 Ts = 2M Hz is used (note that (SN R)dB = ( E b N0 )dB+3dB
). These parameters are selected in order to be in concordance with the standard Wimax IEEE802.16e. The MSE and the BER are evaluated under a rapid time-varying channel such as f d T = 0.1 corresponding to a vehicle speed V m = 600km/h for f c = 2.5GHz. We choose a GCE-BEM [START_REF] Leus | On the Estimation of Rapidly Time-Varying Channels[END_REF] and N c = 3 coefficients to model the path complex gain of the channel. Most advanced technologies have oscillator frequency tolerance less than 1 ppm (i.e. ν = 0.16 in normalized units). For the simulation, we choose ν = 0.1.

Fig. 1 shows the MSE as a function of SNR. The MSE is simulated for three iterations and is shown for the CFO on the one hand and for the multi-path complex gain on the other hand. For reference, the MSEs obtained in Data Aided (DA) mode have been plotted. As expected, the MSEs obtained in Data Aided mode are lower than the MSEs obtained with just the pilots, especialy at low SNR where the detection errors are the most important. At high SNR, the MSEs tend to the MSEs in DA mode. It is also observed that after 20 dB, there is no improvement any more. This is due to the fact that beyond 20 dB, the matrix to be inversed to compute the Kalman Gain in Eq. (34) becomes badly scaled. This issue can be attenuated by adding some noise on the main diagonal of this matrix.

Fig. 2 gives the BER performance of our proposed iterative algorithm. For reference, we also plotted performance obtained with perfect knowledge of channel response and CFO. It is shown that after just one iteration, our joint estimator can achieve the same Bit Error Rate (BER) as that of an ideal reference receiver with perfect knowledge of channel response and carrier frequency offset. This shows that our proposed iterative joint channel and frequency offset estimator is effective. Again, it is observed a degradation in the performance after 20 dB. However, most of the typical modern systems operate with a SNR less than 20 dB.

VI. CONCLUSION

A new iterative algorithm which jointly estimates multipath complex gain and CFO has been presented. The algorithm is based on a parametric channel model. Extended Kalman filtering is used for parameter estimation and the data recovery is carried out by means of a QR-equalizer. Simulation results show that by estimating and removing the ICI at each iteration, the BER is greatly improved, especially after the first iteration. For a SNR < 20 dB (typical modern systems), our algorithm needs only one iteration to achieve the same performance than the ideal case where knowledge of the channel response and CFO is available. However, for a SNR > 20 dB, performance is degraded compared to the ideal case.

Fig. 1 .

 1 Fig. 1. Mean Square Error (MSE) as a function of SNR for f d T = 0.1

  Fig. 2. Bit Error Rate (BER) as a function of SNR for f d T = 0.1

ACKOWLEDGEMENT This work has been carried out in the framework of the CISIT (Campus International sur la Sécurité et l Intermodalité des Transports) project and funded by the French Ministry of Research, the Region Nord Pas de Calais and the European Commission (FEDER funds)


Hussein Hijazi
email: hussein.hijazi@hotmail.fr

Eric Pierre Simon
email: eric.simon@univ-lille1.fr

Martine Liénard
email: martine.lienard@univ-lille1.fr

Laurent Ros
email: laurent.ros@gipsa-lab.inpg.fr

Channel Estimation for MIMO-OFDM Systems in Fast Time-Varying Environments

Keywords: MIMO systems, OFDM, time-varying Rayleigh channels, channel estimation, Kalman filter

A channel estimation algorithm for MIMO-OFDM systems in Fast Time-Varying Environments is proposed. The channel estimation function is based on the equivalent discretetime channel taps or on the physical propagation channel parameters. To handle rapid variations of channels within a transmission block, we approximate the channel by a basis expansion model (BEM). Based on the Jakes process, an auto-regressive (AR) model of the BEM coefficients dynamics is built, making it possible to estimate and track the BEM coefficients using Kalman filter . Hence, the channel matrix is easily computed, and the data symbol is detected with free ICI . Our claims are supported by theoretical analysis and simulation results, which are obtained considering Jakes' channels with high Doppler spreads.

I. INTRODUCTION

Multiple-Input-Multiple-Output (MIMO) antennas with Orthogonal Frequency Division Multiplexing (OFDM) provide high data rates and are robust to multi-path delay in wireless communications. Channel parameters are required for diversity combining, coherent detection and decoding. Therefore, channel estimation is essential in MIMO-OFDM system design. Various OFDM channel estimation schemes have been proposed in the literature, mostly for single antenna systems [START_REF] Hijazi | Polynomial Estimation of Time-varying Multipath Gains with Intercarrier Interference Mitigation in OFDM Systems[END_REF] [START_REF] Rugini | Block DFE and windowing for Doppler-affected OFDM systems[END_REF] [14] [START_REF] Hijazi | Rayleigh Time-varying Channel Complex Gains Estimation and ICI Cancellation in OFDM Systems[END_REF] [START_REF] Simon | Synchronization over Rapidly Timevarying Multipath Channel for CDMA Downlink RAKE Receivers in Time-Division Mode[END_REF]. For MIMO-OFDM systems, most of the channel estimation schemes have focused on pilot-assisted approaches [START_REF] Wang | A MIMO-OFDM Channel Estimation Approach Using Time of Arrivals[END_REF] [3] [START_REF] Wang | MIMO-OFDM Channel Estimation via Probabilistic Data Association Based TOAs[END_REF], based on a quasi-static fading model that allows the channel to be invariant within a MIMO-OFDM block. However, in fast fading channels, the time-variation of the channel within a MIMO-OFDM block results in a loss of subcarrrier orthogonality which leads to intercarrier interference (ICI) [START_REF] Hijazi | Polynomial Estimation of Time-varying Multipath Gains with Intercarrier Interference Mitigation in OFDM Systems[END_REF] [START_REF] Kim | MAP-Based Channel Estimation for MIMO-OFDM Over Fast Rayleigh Fading Channels[END_REF]. To support high speed mobile channels, the channel time-variation within a block must be considered. Channel estimation can be summarized by two approaches. The first one is to estimate the equivalent discretetime channel taps [START_REF] Kim | MAP-Based Channel Estimation for MIMO-OFDM Over Fast Rayleigh Fading Channels[END_REF] [START_REF] Rugini | Block DFE and windowing for Doppler-affected OFDM systems[END_REF] whereas the second approach is to directly estimate the physical propagation channel parameters such as multi-path delays and multi-path complex gains [14] [8]. In Radio-Frequencies transmission, the delays are quasi invariant over several MIMO-OFDM blocks [START_REF] Tang | Pilot-assisted Timevarying Channel Estimation for OFDM Systems[END_REF] [START_REF] Hijazi | Rayleigh Time-varying Channel Complex Gains Estimation and ICI Cancellation in OFDM Systems[END_REF] whereas the complex gains may change significantly, even within one MIMO-OFDM block. Exploiting the channel nature and assuming the availability of delay information, a lot of methods estimate the multi-path complex gains [START_REF] Hijazi | Polynomial Estimation of Time-varying Multipath Gains with Intercarrier Interference Mitigation in OFDM Systems[END_REF] [10] [START_REF] Hijazi | Rayleigh Time-varying Channel Complex Gains Estimation and ICI Cancellation in OFDM Systems[END_REF]. There are more channel parameters in fast fading channels than in quasi-static fading channels. Therefore, in single antenna systems, many existing works resort to the basis expansion model (BEM) in order to estimate the time-variation of the channel taps or the multi-path complex gains within a block [14] [8]. The BEM methods are discrete Karhunen-Love BEM (DKL-BEM), discrete prolate spheroidal BEM (PS-BEM), complexexponential BEM (CE-BEM), generalized CE-BEM (GCE-BEM) and polynomial BEM (P-BEM).

In this paper, we present a new iterative algorithm for joint Rayleigh channel (taps or complex gains) estimation and data recovery in fast time-varying environments. After approximated the time-variation of the channel by a BEM, an autoregressive (AR) model of the BEM coefficients dynamics is built based on the Jakes process. Hence, the BEM coefficients are estimated and tracked by using the Kalman filter. In order to perform BEM coefficients estimation, we use the estimate along with the channel matrix output to recover the transmitted data. One can, in turn, use the detected data along with pilots to enhance the BEM coefficients estimate giving rise to an iterative technique for channel estimation and data recovery. The data detection is performed by using the MMSE-equalizer [START_REF] Yang | Channel Estimation for OFDM Transmisson in Mutipath Fading Channels based on Parametric Channel Modeling[END_REF] or the QR-equalizer [8] [7]. This paper is organized as follows: Section II introduces the MIMO-OFDM system and the BEM modeling. Section III describes the AR model for the BEM coefficients and the Kalman filter. Section IV covers the algorithm for joint channel estimation and data recovery. Section V presents the simulations results which validate our technique. Finally, our conclusions are presented in Section VI.

The notations adopted are as follows: Upper (lower) bold face letters denote matrices (column vectors). [x] k denotes the kth element of the vector x, and [X] k,m denotes the [k, m]th element of the matrix X. We will use the matlab notation X[k1:k2,m1:m2] to extract a submatrix within X from row k 1 to row k 2 and from column m 1 to column m 2 . I N is a N × N identity matrix and 0 N is a N ×N matrix of zeros. diag{x} is a diagonal matrix with x on its main diagonal and blkdiag{X, Y} is a block diagonal matrix with the matrices X and Y on its main diagonal. The superscripts (•) T , (•) * and (•) H stand respectively for transpose, conjugate and Hermitian operators. Tr(•) and E[•] are respectively the determinant and expectation operations. J 0 (•) is the zeroth-order Bessel function of the first kind.

II. MIMO-OFDM SYSTEM AND CHANNEL MODELS

A. MIMO-OFDM System Model

Consider a MIMO-OFDM system with N T transmit antennas, N R receive antennas, N sub-carriers, and a cyclic prefix length N g . The duration of a MIMO-OFDM block is T = vT s , where T s is the sampling time and

v = N + N g . Let x (n) = x T (1,n) , x T (2,n) , ..., x T (NT ,n) ]
T be the nth transmitted MIMO-OFDM block, where

x (t,n) = x (t,n) [-N 2 ], x (t,n) [-N 2 + 1], ..., x (t,n) [ N 2 -1] T is
(n) = y T (1,n) , y T (2,n) , ..., y T (NR,n) ]
T , where

y (r,n) = y (r,n) [-N 2 ], y (r,n) [-N 2 + 1], ..., y (r,n) [ N 2 -1]
T is the nth received OFDM symbol by the rth receive antenna, is given by [9] [12]:

y (n) = H (n) x (n) + w (n) (1)
where

w (n) = w T (1,n) , w T (2,n) , ..., w T (NR,n) ] T , with w (r,n) = w (n) [-N 2 ], w (n) [-N 2 + 1], ..., w (n) [ N 2 -1]
T is a white complex Gaussian noise vector of covariance matrix N T .σ 2 I N and

H (n) is a N R N × N T N MIMO channel matrix
given by:

H (n) =    H (1,1,n) • • • H (1,NT ,n) . . . . . . . . . H (NR,1,n) • • • H (NR,NT ,n)    (2)
where H (r,t,n) is the channel matrix between the tth transmit antenna and the rth receive antenna. The elements of channel matrix H (r,t,n) can be written in terms of equivalent channel taps [START_REF] Kim | MAP-Based Channel Estimation for MIMO-OFDM Over Fast Rayleigh Fading Channels[END_REF] g

(l,r,t) (qT s ) = g (l,r,t) (nT + qT s ) or in terms of physical channel parameters [START_REF] Hijazi | Polynomial Estimation of Time-varying Multipath Gains with Intercarrier Interference Mitigation in OFDM Systems[END_REF]: delays τ (l,r,t) and complex gains α

(n) (l,r,t) (qT s ) = α (l,r,t) (nT + qT s ) , as: [H (r,t,n) ] k,m = 1 N L ′ (r,t) l=1 e -j2π( m-1 N -1 2 )(l-1) N -1 q=0 g (n) (l,r,t) (qTs)e j2π m-k N q (3) = 1 N L (r,t) l=1 e -j2π( m-1 N -1 2 )τ (l,r,t) N -1 q=0 α (n) (l,r,t) (qTs)e j2π m-k N q (4)
where L ′ (r,t) < N g and L (r,t) are respectively the number of channel taps and the number of paths for the branch between the tth transmit antenna and the rth receive antenna. The delays are normalized by T s and not necessarily integers (τ (l,r,t) < N g ). The L (r,t) elements of α

(n) (l,r,t) (qT s ) are uncorrellated. Howerver, the L ′ (r,t) elements of g (n)
(l,r,t) (qT s ) are correlated, unless that the delays are multiple of T s as mostly assumed in the litterature. They are wide-sense stationary (WSS), narrow-band zero-mean complex Gaussian processes of variances σ 2 g (l,r,t) and σ 2 α (l,r,t) , with the socalled Jakes' power spectrum of maximum Doppler frequency f d [START_REF] Baddour | Autoregressive modeling for fading channel simulation[END_REF]. The average energy of each channel approach is normalized to one, i.e.,

L ′ (r,t) l=1 σ 2 g (l,r,t) = 1 and L (r,t) l=1 σ 2 α (l,r,t) = 1.
In the sequel, we will make the calculus based on the second approach (physical channel) and we can deduce the results of the first approach (channel taps) by replacing L (r,t) by L ′ (r,t) and the set of delays τ (l,r,t) by l -1, l = 1 : L ′ (r,t) .

B. BEM Channel Model

Let L = NR r=1 L (r) = NR r=1 NT t=1
L (r,t) be the total number of complex gains for the MIMO channel and designate by (t, r) the branch between the tth transmit antenna and the rth receive antenna. Since the number of samples to estimate Lv is greater than the number of observation equations N R N , it is not efficient to estimate the time-variation of the complex gains, using directly the observation model in [START_REF] Wang | A MIMO-OFDM Channel Estimation Approach Using Time of Arrivals[END_REF]. Thus, we need to reduce the number of parameters to estimate. In this section, our aim is to accurately model the time-variation of α

(n) (l,r,t) (qT s ) by using a BEM. Collecting the samples of the lth path of the branch (r, t) within the nth MIMO-OFDM block in a v × 1 vector α

(n) (l,r,t) = α (n) (l,r,t) (-N g T s ), ..., α (n) (l,r,t) (N -1)T s T ,
we can express α

(l,r,t) as:

α (n) (l,r,t) = α (n) BEM (l,r,t) + ξ (n) (l,r,t) = Q c (n) (l,r,t) + ξ (n) (l,r,t) (5) where Q = [q 1 , ..., q Nc ] is a v × N c matrix that collects N c orthonormal basis function q d as columns, c (n) (l,r,t) = c (n) (1,l,r,t) , ..., c (n) (Nc,l,r,t)
T represent the N c BEM coefficients for the lth complex gain of the branch (r, t) of the nth MIMO-OFDM block, and ξ (n) (l,r,t) represents the corresponding BEM modeling error, which is assumed to be minimized in the MSE sense [START_REF] Kannu | MSE-optimal Training for Linear Timevarying Channels[END_REF] [START_REF] Hijazi | Analytical Analysis of Bayesian Cramer-Rao Bound for Dynamical Rayleigh Channel Complex Gains Estimation in OFDM System[END_REF]. Under this criterion, the optimal BEM coefficients and the corresponding model error are given by:

c (n) (l,r,t) = Q H Q -1 Q H α (n) (l,r,t) (6)
ξ (n) (l,r,t) = (I v -S)α (n) (l,r,t) (7)
where

S = Q Q H Q -1 Q H is a v × v matrix.
It provides the MMSE approximation for all BEM containing N c coefficients, given by:

MMSE (l,r,t) = 1 v E ξ (n) (l,r,t) ξ (n) (l,r,t) H (8) = 1 v Tr I v -S R (0) α (l,r,t) I v -S H (9)
where

R (s) α (l,r,t) = E α (n) (l,r,t) α (n-s) (l,r,t) H is the v × v correlation matrix of α (n)
(l,r,t) with elements given by:

[R (s) α (l,r,t) ] k,m = σ 2 α (l,r,t) J 0 2πf d T s (k -m + sv) (10)
Various traditional BEM designs have been reported to model the channels time-variations, e.g., the CE-BEM 1) [5] and the DKL-BEM which employs basis sequences that corresponds to the most significant eigenvectors of the autocorrelation matrix R (0) α (l,r,t) [START_REF] Jakes | Microwave Mobile Communications[END_REF]. From now on, we can describe the MIMO-OFDM system model derived previously in terms of the BEM. Substituting [START_REF] Hijazi | Analytical Analysis of Bayesian Cramer-Rao Bound for Dynamical Rayleigh Channel Complex Gains Estimation in OFDM System[END_REF] in [START_REF] Wang | A MIMO-OFDM Channel Estimation Approach Using Time of Arrivals[END_REF] and neglecting the BEM model error, we obtain after some algebra:

[Q] k,m = e j2π( k-N g-1 v )(m-1-Nc -1 2 ) [15], the GCE-BEM [Q] k,m = e j2π( k-N g-1 av )(m-1-Nc -1 2 ) with 1 < a ≤ Nc-1 2f d T [16], the P- BEM [Q] k,m = (k -N g -1) (m-
y (n) = K (n) c (n) + w (n) ( 11
)
where the LN c × 1 vector c (n) and the N R N × LN c matrix K (n) are given by:

c (n) = c T (1,1,n) , ..., c T (1,NT ,n) , ..., c T (NR,NT ,n) T c (r,t,n) = c (n) T (1,r,t) , ..., c (n) T (L (r,t) ,r,t) T K (n) = blkdiag K (1,n) , ..., K (NR,n) K (r,n) = K (r,1,n) , ..., K (r,NT ,n) K (r,t,n) = 1 N Z (n) (1,r,t) , ..., Z (n) (L r,t) ,r,t) Z (n) (l,r,t) = M (1) diag{x (t,n) }f (l,r,t) , ..., M (Nc) diag{x (t,n) }f (l,r,t)
where f (l,r,t) is the lth column of the N ×L (r,t) Fourier matrix F (r,t) and M (d) is a N × N matrix given by:

[F (r,t) ] k,l = e -j2π( k-1 N -1 2 )τ (l,r,t) (12)
M (d) k,m = N -1 q=0 [Q] q+Ng+1,d e j2π m-k N q (13)
Moreover, the channel matrix of the branch (r, t) can be easily computed by using the BEM coefficients [START_REF] Hijazi | Rayleigh Time-varying Channel Complex Gains Estimation and ICI Cancellation in OFDM Systems[END_REF]:

H (r,t,n) = Nc d=1 M (d) diag{F (r,t) χ (n) (d,r,t) } (14)
where

χ (n) (d,r,t) = c (n) (d,1,r,t) , ..., c (n) (d,L (r,t,) , r,t)
T . The matrices M (d) can be computed and stored.

III. AR MODEL AND KALMAN FILTER

A. The AR Model for c (n) The optimal BEM coefficients c

(n) (l,r,t) are correlated complex Gaussian variables with zero-means and correlation matrix given by:

R (s) c (l,r,t) = E[c (n) (l,r,t) c (n-s) (l,r,t) H ] = Q H Q -1 Q H R (s) α (l,r,t) Q Q H Q -1 (15)
Hence, the dynamics of c

(n) (l,r,t) can be well modeled by an auto-regressive (AR) process [START_REF] Anderson | Optimal filtering[END_REF] [20] [START_REF] Hijazi | Polynomial Estimation of Time-varying Multipath Gains with Intercarrier Interference Mitigation in OFDM Systems[END_REF] . A complex AR process of order p can be generated as:

c (n) (l,r,t) = - p i=1 A (i) c (n-i) (l,r,t) + u (n) (l,r,t) (16)
where A (1) , ..., A (p) are N c × N c matrices and u (n) (l,r,t) is a N c × 1 complex Gaussian vector with covariance matrix U (l,r,t) . The matrices A (1) , ..., A (p) and U (l,r,t) are the AR model parameters obtained by solving the set of Yule-Walker equations defined as:

TA = -V and U (l,r,t) = R (0) c (l,r,t) + p i=1 A (i) R (-i) c (l,r,t) (17)
where A = A (1) T , ..., A (p) T T is pN c × N c matrix, V and T are pN c × N c and pN c × pN c matrices defined by:

V = 1 σ 2 α (l,r,t) R (1) T c (l,r,t) , ..., R (p) T c (l,r,t) T (18)
T = 1 σ 2 α (l,r,t)     R (0) c (l,r,t) • • • R (-p+1) c (l,r,t) . . . . . . . . . R (p-1) c (l,r,t) • • • R (0) c (l,r,t)     (19)
Using ( 16), we obtain the AR model of order p for c (n) :

c (n) = - p i=1 A (i) c (n-i) + u (n) (20)
where

A (i) = blkdiag A (i) , ..., A (i) is a LN c × LN c matrix and u (n) = u (n) T (1,1,1) , ..., u (n) T (L (N R ,N T ) ,NR,NT ) T is a LN c × 1 complex Gaussian vector with covariance matrix U = blkdiag U (1,1,1) , ..., U (L (N R ,N T ) ,NR,NT ) .

B. The Kalman Filter

Based on the AR model of c (n) in (20), we define the state space model for the MIMO-OFDM system as g (n) = [c T (n) , ..., c T (n-p+1) ] T . Thus, using (20) and [START_REF] Tang | Pilot-assisted Timevarying Channel Estimation for OFDM Systems[END_REF], we obtain:

g (n) = S 1 g (n-1) + S 2 u (n) (21) y (n) = S 3 g (n) + w (n) (22)
where

S 2 = [I LNc , 0 LNc,(p-1)LNc ] T is a pLN c × LN c matrix, S 3 = [K (n) , 0 N,(p-1)
LNc ] is a N × pLN c matrix and S 1 is a pLN c × pLN c matrix defined as:

S 1 =        -A (1) -A (2) -A (3) • • • -A (p) I LNc 0 LNc 0 LNc • • • 0 LNc 0 LNc I LNc 0 LNc • • • 0 LNc . . . . . . . . . . . . . . . 0 LNc • • • 0 LNc I LNc 0 LNc        (23)
The state and the observation models (21) (22) allow us to use Kalman filter to adaptively track the BEM coefficients c (n) . Let ĝ(n) be our a priori state estimate at step n given knowledge of the process prior to step n, ĝ(n|n) be our a posteriori state estimate at step n given measurement y (n) and, P (n) and P (n|n) are the a priori and the a posteriori error estimate covariance matrix of size pLN c × pLN c , respectively. We initialize the Kalman filter with g (0|0) = 0 pLNc,1 and P (0|0) given by:

P (0|0) [u(s),u(s ′ )] = R (s-s ′ ) c (24) R (s) c = blkdiag R (s) c (1,1) , ..., R (s) c (N R ,N T ) R (s) c (r,t) = blkdiag R (s) c (1,r,t) , ..., R (s) c (L (r,t) ,r,t)
where u(s) = 1 + LN c : (s + 1)LN c with s ∈ [0, p -1] and R (s) c (l,r,t) is the correlation matrix of c

(n) (l,r,t) defined in [START_REF] Leus | On the Estimation of Rapidly Time-Varying Channels[END_REF]. The Kalman filter is a recursive algorithm composed of two stages: Time Update Equations and Measurement Update Equations. These two stages are defined as:

Time Update Equations: ĝ(n) = S 1 ĝ(n-1|n-1) P (n) = S 1 P (n-1|n-1) S H 1 + S 2 US H 2 ( 25
)
Measurement Update Equations:

K (n) = P (n) S H 3 S 3 P (n) S H 3 + N T .σ 2 I NRN -1 ĝ(n|n) = ĝ(n) + K (n) y (n) -S 3 ĝ(n) P (n|n) = P (n) -K (n) S 3 P (n) ( 26
)
where K (n) is the Kalman gain. The Time Update Equations are responsible for projecting forward (in time) the current state and error covariance estimates to obtain the a priori estimates for the next time step. The Measurement Update Equations are responsible for the feedback, i.e., for incorporating a new measurement into the a priori estimate to obtain an improved a posteriori estimate. The Time Update Equations can also be thought of a predictor equations, while the Measurement Update Equations can be thought of a corrector equations.

IV. JOINT DATA DETECTION AND KALMAN ESTIMATION

A. Iterative Algorithm

In the iterative algorithm for joint data detection and channel Kalman estimation, the N p pilots subcarriers are evently inserted into the N subcarriers at the positions P = {p r | p r = (r -1)L f + 1, r = 1, ..., N p }, where L f is the distance between two adjacent pilots. We test two channel equalizers for the data detection: the MMSE equalizer [START_REF] Yang | Channel Estimation for OFDM Transmisson in Mutipath Fading Channels based on Parametric Channel Modeling[END_REF] and the QRequalizer [START_REF] Hijazi | Polynomial Estimation of Time-varying Multipath Gains with Intercarrier Interference Mitigation in OFDM Systems[END_REF] [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF]. The first one is based on the noise variance N T .σ 2 whereas the second equalizer allows us to estimate the data symbol with free ICI by performing a so-called QRdecomposition. The algorithm proceeds as follows: initialization:

• g (0|0) = 0 pLNc,1

• compute P (0|0) as (24)

• n ← n + 1

• execute the Time Update Equations of Kalman filter (25)

• compute the channel matrix using [START_REF] Teo | Optimal MMSE Finite Parameter Model for Doubly-selective Channels[END_REF] • i ← 1 recursion: 1) remove the pilot ICI from the received data subcarriers 2) Detection of data symbols 3) execute the Measurement Update Equations of Kalman filter (26) 4) compute the channel matrix using (14)

5) i ← i + 1
where i represents the iteration number.

B. Mean Square Error (MSE) Analysis

The error between the lth exact complex gain and the lth estimated polynomial α(n) BEM (l,r,t) is given by:

e (n) (l,r,t) = α (n) (l,r,t) - α(n) BEM (l,r,t) = ξ (n) (l,r,t) + Qe (n) c (l,r,t) (27)
where e

(n)

c (l,r,t) = c (n) (l,r,t) - ĉ(n) (l,r ,t) and ξ (n)
(l,r,t) is the BEM model error defined in [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF]. Neglecting the cross-covariance terms between ξ (n) (l,r,t) and e (n) c (l,r,t) , the mean square error (MSE) between α (n) (l,r,t) and α (n) BEM (l,r,t) is given by:

MSE (l,r,t) = 1 v E e (n) H (l,r,t) e (n) (l,r,t) = MMSE (l,r,t) + 1 v Tr Q MSE c (l,r,t) Q H (28)
where

MSE c (l,r,t) = E e (n) c (l,r,t) e (n) c (l,r,t)

H

. Notice that, at the convergence of the Kalman filter, we have:

MSE c (l,r,t) = P (n|n) [f (l,r,t),f (l,r,t)] (29)
provided that the data symbols are perfectly estimated (i.e., data-aided context), where the sequence of indices

f (l, r, t) = 1 + (l -1 + a (r,t) )N c : (l + a (r,t) )N c with a (r,t) = r-1 r ′ =1 t-1 t ′ =1 L (r ′ ,t ′ ) .
The on-line Bayesian Cramer-Rao Bound (BCRB) is an important criterion for evaluting the quality of the complex gains Kalman estimation. The on-line BCRB for the estimation of α

(n) (l,r,t) , in data-aided (DA) context, is given by: BCRB(α

(∞) (l,r,t) ) = MMSE (l,r,t) + 1 v Tr QBCRB(c (∞) (l,r,t) )Q H (30) where BCRB(c (K) (l,r,t)
) is the on-line BCRB associated to the estimation of c (K) (l,r,t) which is given by: BCRB(c

(K) (l,r,t) ) = BCRB(c) [f (l,r,t),f (l,r,t)] (31
T in DA context which is given by:

BCRB(c) = blkdiag J (K) , ..., J (2) , J (1) + R -1 c -1 (32)
where R c is calculated in the same way as P (0|0) with s, s ′ ∈ [0, K -1], and J (n) is a LN c × LN c matrix given by [START_REF] Hijazi | Analytical Analysis of Bayesian Cramer-Rao Bound for Dynamical Rayleigh Channel Complex Gains Estimation in OFDM System[END_REF]: where M and F (r,t,n) are a N N c × N N c and a N N c × L (r,t) N c matrices, respectively, defined as:

J (n) = blkdiag J (1,n) , ..., J (NR,n) (33)
J (r,n) =    J (r,1,1,n) • • • J (r,1,NT ,n) . . . . . . . . . J (r,NT ,1,n) • • • J (r,NT ,NT ,n)    (34) J (r,t,t ′ ,n) = 1 N 2 N T σ 2 F H (r,t,n) MF (r,t ′ ,n) (35)
M =    M H (1) M (1) • • • M H (1) M (Nc) . . . . . . . . . M H (Nc) M (1) • • • M H (Nc) M (Nc)    (36) F (r,t,n) = F (1,r,t,n) • • • F (L (r,t) ,r,t,n) (37)
where the N N c × N c matrix F (l,r,t,n) is given by:

F (l,r,t,n) = (38)
blkdiag diag{x (t,n) }f (l,r,t) , ..., diag{x (t,n) }f (l,r,t)

V. SIMULATION

In this section, we verify the theory by simulation and we test the performance of the iterative algorithm based on the second approach (i.e., estimation of α assuming the availability of delay information). We assume that the channels from different transmitters to the different receivers have the same delay and fading property (i.e., same number of paths, σ 2 α (l,r,t) and τ (l,r,t) ) and we use the Rayleigh channel model given in [START_REF] Hijazi | Polynomial Estimation of Time-varying Multipath Gains with Intercarrier Interference Mitigation in OFDM Systems[END_REF] [START_REF] Hijazi | Analytical Analysis of Bayesian Cramer-Rao Bound for Dynamical Rayleigh Channel Complex Gains Estimation in OFDM System[END_REF]. It is reasonable because the transmitters and the receivers are very close to each other in practice. A normalized 4QAM MIMO-OFDM system, with two transmit and two receive antennas, N = 256 subcarriers, N g = N 8 , N p = N 4 pilots (i.e., L f = 4) and 1 Ts = 2M Hz is used (note that SNR = 1 σ 2 ). The MSE and the BER are evaluated under a rapid time-varying channel such as f d T = 0.5 corresponding to a vehicle speed V m = 400km/h for f c = 10GHz. In order to decrease the complexity of the Kalman filter, we choose an AR model of order p = 1.

Fig. 1 and Fig. 2 show respectively the evolution of MSE and BER versus SNR, with the iterations, for f d T = 0.5 and GCE-BEM. We select N c = 4 in order to reduce the impact of the BEM modeling error. The equalizers used for data detection are: the MMSE equalizer [START_REF] Yang | Channel Estimation for OFDM Transmisson in Mutipath Fading Channels based on Parametric Channel Modeling[END_REF] and the QRequalizer [START_REF] Hijazi | Polynomial Estimation of Time-varying Multipath Gains with Intercarrier Interference Mitigation in OFDM Systems[END_REF] [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF]. From Fig. 1, it is shown that, with DA, the MSE obtained by simulation agrees with the theoretical value of MSE given by (29). We also observe that MSE with DA is very close to the on-line BCRB. This means that the Kalman filter works very well. After three and nine iterations, a great improvement is realized and the MSE is close to the MSE with DA, particularly in regions of moderate and high SNR. In Fig. 2, we plotted, as reference, the performance obtained with perfect channel knowledge. After three and nine iterations, a significant improvement occurs; the performance of our algorithm and the performance obtained with perfect channel knowledge are very close, particularly in regions of low and moderate SNR. At very high SNR, it is normal to not reach the reference because we have a small error floor due to the data symbol detection error. It sould be noted that the error floor with QR-equalizer is larger than that of the MMSE equalizer.

VI. CONCLUSION

In this paper, we have presented a new iterative algorithm for joint multi-path Rayleigh channel estimation and data recovery in fast time-varying environments. The rapid timevariation of the channel within one MIMO-OFDM block are approximated by a BEM model. The BEM coefficients are tracked and estimated using the Kalman filter. The data symbols are estimated by performing a QR-decomposition of the channel matrix or by using a MMSE equalizer. Theoretical analysis and simulation results show that our algorithm has a good performance for high Doppler spread.

)

  BCRB(c) is the on-line BCRB for the estimation ofc = c T (K) , ..., c T

Fig. 1 .

 1 Fig. 1. MSE vs SNR for f d T = 0.5 and GCE-BEM with Nc = 4

Fig. 2 .

 2 Fig. 2. BER vs SNR for f d T = 0.5 and GCE-BEM with Nc = 4

  VII. ACKOWLEDGEMENT This work has been carried out in the framework of the CISIT (Campus International sur la Sécurité et Intermodalité des Transports) project and funded by the French Ministry of Research, the Region Nord Pas de Calais and the European Commission (FEDER funds)


Eric Pierre Simon
email: eric.simon@univ-lille1.fr

Hussein Hijazi
email: hussein.hijazi@hotmail.fr

Laurent Ros
email: laurent.ros@gipsa-lab.grenoble-inp.fr

Joint Carrier Frequency Offset and Fast Time-varying Channel Estimation for MIMO-OFDM Systems

In this paper, a novel pilot-aided iterative algorithm is developed for MIMO-OFDM systems operating in fast timevarying environment. An L-path channel model with known path delays is considered to jointly estimate the multi-path Rayleigh channel complex gains and Carrier Frequency Offset (CFO). Each complex gain time-variation within one OFDM symbol is approximated by a Basis Expansion Model (BEM) representation. An auto-regressive (AR) model is built for the parameters to be estimated. The algorithm performs recursive estimation using Extended Kalman Filtering. Hence, the channel matrix is easily computed and the data symbol is estimated with free intersub-carrier-interference (ICI) when the channel matrix is QRdecomposed. It is shown that only one iteration is sufficient to approach the performance of the ideal case for which the knowledge of the channel response and CFO is available.

I. INTRODUCTION

Multiple-Input-Multiple-Output (MIMO) antennas with Orthogonal Frequency Division Multiplexing (OFDM) provide high data rates and are robust to multi-path delay in wireless communications. Channel parameters are required for diversity combining, coherent detection and decoding. Therefore, channel estimation is critical to design MIMO-OFDM systems. For MIMO-OFDM systems, most of the channel estimation schemes have focused on pilot-assisted approaches [START_REF] Li | Simplified Channel Estimation for OFDM Systems with Multiple Transmit Antennas[END_REF][2] [START_REF] Wang | MIMO-OFDM Channel Estimation via Probabilistic Data Association Based TOAs[END_REF], based on a quasi-static fading model that allows the channel to be invariant within a MIMO-OFDM block. However, in fast-fading channels, the time-variation of the channel within a MIMO-OFDM block results in the loss of subcarrrier orthogonality, and consequently intercarrier interference (ICI) occurs [START_REF] Hijazi | Polynomial estimation of time-varying multi-path gains with intercarrier interference mitigation in OFDM systems[END_REF] [START_REF] Kim | MAP-Based Channel Estimation for MIMO-OFDM Over Fast Rayleigh Fading Channels[END_REF]. Therefore, the channel time-variation within a block must be considered to support high-speed mobile channels.

On the other hand, similarly to the single-input single-output (SISO) OFDM, one of the disadvantages of MIMO-OFDM lies in its sensitivity to carrier frequency offset (CFO) due to carrier frequency mismatches between transmitter and receiver oscillators. As for the Doppler shift, the CFO produces ICI and attenuates the desired signal. These effects reduce the effective signal-to-noise ratio (SNR) in OFDM reception such that the system performance is degraded [START_REF] Pollet | BER Sensitivity of OFDM Systems to Carrier Frequency Offset and Wiener Phase Noise[END_REF] [START_REF] Steendam | Sensitivity of Orthogonal Frequency-Division Multiplexed Systems to Carrier and Clock Synchronisation Errors[END_REF]. Most of the reported work consider that all the paths exhibit the same Doppler shift. Hence, they group together the Doppler shift and CFO due to oscillator mismatchs in order to obtain just one offset parameter [START_REF] Moose | A technique for orthogonal frequency division multiplexing frequency offset correction[END_REF] [START_REF] Roman | Joint Time-Domain Tracking of Channel and Frequency Offsets for MIMO OFDM Systems[END_REF] for each channel branch. However, this model is not sufficiently accurate since separate offset parameters are needed for each propagation path given that the Doppler shift depends on the angle of arrival, which is peculiar to each path. Recently, it has been proposed to directly track the channel paths, which permits to take into account separate Doppler shifts for each path ( [10][11] for SISO and [START_REF] Hijazi | Channel Estimation for MIMO-OFDM Systems in Fast Time-Varying Environments[END_REF] for MIMO). Those works estimate the equivalent discrete-time channel taps ( [START_REF] Tang | Pilot-assisted timevarying channel estimation for OFDM systems[END_REF]) or the real path complex gains ([10] [START_REF] Hijazi | Channel Estimation for MIMO-OFDM Systems in Fast Time-Varying Environments[END_REF]) which are both modeled by a basis expansion model (BEM). The BEM methods are Karhunen-Loeve BEM (KL-BEM), prolate spheroidal BEM (PS-BEM), complex exponential BEM (CE-BEM) and polynomial BEM (P-BEM).

However the CFO due to the mismatch between transmitter and receiver oscillators is not taken into account in those algorithms. In this paper, we propose a complete algorithm capable of estimating this CFO jointly with the time-variation of each channel path in MIMO environment.

Generally, it is preferable to directly estimate the physical channel parameters [START_REF] Yang | Channel estimation for OFDM transmisson in mutipath fading channels based on parametric channel modeling[END_REF] [10] [START_REF] Hijazi | Channel Estimation for MIMO-OFDM Systems in Fast Time-Varying Environments[END_REF] instead of the equivalent discrete-time channel taps [START_REF] Tang | Pilot-assisted timevarying channel estimation for OFDM systems[END_REF]. Indeed, as the channel delay spread increases, the number of channel taps also increases and a large number of BEM coefficients have to be estimated. This requires more pilot symbols. Additionally, estimating the physical propagation parameters means estimating multi-path delays and multi-path complex gains. Note that in Radio-Frequency transmissions, the delays are quasi-invariant over several MIMO-OFDM blocks [START_REF] Simon | Synchronization over Rapidly Timevarying Multipath Channel for CDMA Downlink RAKE Receivers in Time-Division Mode[END_REF] [4] (whereas the complex gains may change significantly, even within one MIMO-OFDM block). In this work, the delays are assumed perfectly estimated and quasi-invariant. It should be noted that an initial, and generally accurate estimation of the number of paths and delays can be obtained by using the MDL (minimum description length) and ESPRIT (estimation of signal parameters by rotational invariance techniques) methods [START_REF] Yang | Channel estimation for OFDM transmisson in mutipath fading channels based on parametric channel modeling[END_REF][10]. To further improve the estimation accuracy, our algorithm uses decision feedback. Hence, the accuracy of the channel estimation, frequency offset estimation and symbol detection are simultaneously enhanced. Note also that, since the pilots are used for both channel and frequency offset estimation, the pilot usage efficiency is greatly improved. Our algorithm is a recursive algorithm based on Extended Kalman Filtering (EKF) combined with QR-equalization for data detection.

This paper is organized as follows: Section II introduces the MIMO-OFDM system and the BEM modeling. Section III describes the state model and the Extended Kalman Filter. Section IV covers the algorithm for joint channel and CFO estimation together with data recovery. Section V presents the simulations results which validate our technique. Finally, our conclusions are presented in Section VI.

The notations adopted are as follows: Upper (lower) bold face letters denote matrices (column vectors). [x] k denotes the kth element of the vector x, and [X] k,m denotes the [k, m]th element of the matrix X. We will use the matlab notation X[k1:k2,m1:m2] to extract a submatrix within X from row k 1 to row k 2 and from column m 1 to column m 2 . I N is a N × N identity matrix and 0 N is a N ×N matrix of zeros. diag{x} is a diagonal matrix with x on its main diagonal and blkdiag{X, Y} is a block diagonal matrix with the matrices X and Y on its main diagonal. The superscripts (•) T , (•) * and (•) H stand respectively for transpose, conjugate and Hermitian operators. Tr(•) and E[•] are respectively the determinant and expectation operations. J 0 (•) is the zeroth-order Bessel function of the first kind. ∇ x represents the first-order partial derivative operator i.e.,

∇ x = [ ∂ ∂x1 , ..., ∂ ∂xN ] T .

II. MIMO-OFDM SYSTEM AND CHANNEL MODELS

A. MIMO-OFDM System Model

Consider a MIMO-OFDM system with N T transmitter antennas, N R receiver antennas, N sub-carriers, and a cyclic prefix length N g . The duration of a MIMO-OFDM block is T = N s T s , where T s is the sampling time and

N s = N + N g . Let x n = x (1) T n , x (2) T n , ..., x (NT ) T n T be the nth transmitted MIMO-OFDM block, where x (t) n = x (t) n [-N 2 ], x (t) n [-N 2 + 1], ..., x (t) n [ N 2 -1]
T is the nth transmitted OFDM symbol by the tth transmit antenna and {x

(t) n [b]} are normalized symbols (i.e., E x (t) n [b]x (t) * n [b] = 1)
. The frequency mimatch between the oscillators used in the radio transmitters and receivers causes a CFO. In multi-antenna systems, each transmitter and receiver typically requires its own Radio Frequency -Intermediate Frequency (RF-IF) chain. Consequently, each transmitter-receiver pair has its own mismatch parameter, yielding separate CFOs. In a N T × N R MIMO system this leads to N T N R different CFOs. However, if transmitter or receiver antennas share RF-IF chains, fewer different CFOs occured. The system model describes the general case where it is necessary to compensate for N T N R CFOs. Assume that the MIMO channel branch between the tth transmit antenna and the rth receive antenna (called (r, t) branch from now on) experiences a normalized frequency shift ν (r,t) = ∆F (r,t) N T s , where ∆F (r,t) is the absolute CFO. All the normalized CFOs can be stacked in vector form as: ν = ν (1,1) , . . . , ν (1,NT ) , . . . , ν (r,1) , . . . , ν (r,NT ) , . . . , ν (NR,NT ) T

(1)

After transmission over a multi-path Rayleigh channel, the nth received MIMO-OFDM block

y n = y (1) T n , y (2)
T n , ..., y

(NR) T n T , where y (r) n = y (r) n [-N 2 ], y (r) n [-N 2 + 1], ..., y (r) n [ N 2 -1]
T is the nth received OFDM symbol by the rth receive antenna, is given by [4] [11]:

y n = H n x n + w n (2)
where

w n = w (1) T n , w (2) T n , ..., w (NR) T n ] T with w (r) n = w (r) n [-N 2 ], w (r) n [-N 2 + 1], ..., w (r) n [ N 2 -1]
T a white complex Gaussian noise vector of covariance matrix N T σ 2 I N . The matrix H n is a N R N × N T N MIMO channel matrix given by:

H n =    H (1,1) n • • • H (1,NT ) n . . . . . . . . . H (NR,1) n • • • H (NR,NT ) n    (3)
where H (r,t) n is the (r, t) branch channel matrix. The elements of channel matrix H (r,t) n can be written in terms of equivalent channel taps [START_REF] Kim | MAP-Based Channel Estimation for MIMO-OFDM Over Fast Rayleigh Fading Channels[END_REF] g

(n,r,t) l (qT s ) = g (r,t) l
(nT + qT s ) or in terms of physical channel parameters [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF] (i.e. delays τ (r,t) l and complex gains α

(n,r,t) l (qT s ) = α (r,t) l
(nT + qT s ) ), yielding Eq. ( 4) and ( 5), respectively.

L (r,t) < N g and L (r,t) are respectively the number of channel taps and the number of paths for the (r, t) branch. The delays are normalized by T s and not necessarily integers (τ (r,t) l < N g ). The L (r,t) elements of α (n,r,t) l (qT s ) are uncorrelated. However, the L (r,t) elements of g (n,r,t) l (qT s ) are correlated, unless that the delays are multiple of T s as mostly assumed in the literature. They are wide-sense stationary (WSS), narrow-band zero-mean complex Gaussian processes of variances σ (r,t) g l 2 and σ (r,t) α l

2

, with the so-called Jakes' power spectrum of maximum Doppler frequency f d [START_REF] Jakes | Microwave Mobile Communications[END_REF]. The average energy of each (r, t) branch is normalized to one, i.e.,

L (r,t) -1 l=0 σ (r,t) g l 2 = 1 and L (r,t) -1 l=0 σ (r,t) α l 2 = 1.
In the next sections, we present the derivations for the second approach (physical channel). The results of the first approach (channel taps) can be deduced by replacing L (r,t) by L (r,t) and the set of delays τ (r,t) l by l, l = 0 : L (r,t) -1 .

B. BEM Channel Model

Let

L = NR r=1 L (r) = NR r=1
NT t=1 L (r,t) be the total number of complex gains for the MIMO channel. Since the number of samples to be estimated LN s is greater than the number of observation equations N R N , it is not efficient to estimate the time-variation of the complex gains, using directly the observation model in [START_REF] Wang | A MIMO-OFDM Channel Estimation Approach Using Time of Arrivals[END_REF]. Thus, we need to reduce the number of parameters to be estimated. In this section, our aim is to accurately model the time-variation of α (n,r,t) l (qT s ) from q = -N g to N -1 by using a BEM. Suppose α

(n,r,t) l represents an N s × 1 vector that collects the time-variation of the lth path of the (r, t) branch within the nth MIMO-OFDM block:

α (n,r,t) l = α (n,r,t) l (-N g T s ), ..., α (n,r,t) l (N -1)T s T (6) [H (r,t) n ] k,m = 1 N L (r,t) -1 l=0 e -j2π( m N -1 2 )•l N -1 q=0 e j2π ν (r,t) q N g (n,r,t) l (qT s )e j2π m-k N q (4) = 1 N L (r,t) -1 l=0 e -j2π( m N -1 2 )τ (r,t) l N -1 q=0 e j2π ν (r,t) q N α (n,r,t) l (qT s )e j2π m-k N q (5)
Then, each α (n,r,t) l can be expressed in terms of a BEM as: represents the corresponding BEM modeling error, which is assumed to be minimized in the MSE sense [START_REF] Leus | On the Estimation of Rapidly Time-Varying Channels[END_REF]. Under this criterion, the optimal BEM coefficients and the corresponding model error are given by:

α (n,r,t) l = α (n,r,t) BEM l + ξ (n,r,t) l = B c (n,r,t) l + ξ (n,r,t) l (7
c (n,r,t) l = B H B -1 B H α (n,r,t) l (8) ξ (n,r,t) l = (I Ns -S)α (n,r,t) l (9)
where

S = B B H B -1 B H is a N s × N s matrix.
Then, the MMSE approximation for all BEM with N c coefficients is given by:

MMSE (r,t) l = 1 N s E ξ (n,r,t) l ξ (n,r,t) l H (10) = 1 N s Tr I Ns -S R (0,r,t) α l I Ns -S H (11)
where R (s,r,t)

α l = E α (n,r,t) l α (n-s,r,t) l H is the N s × N s correlation matrix of α (n,r,t) l
with elements given by: [R (s,r,t)

α l ] k,m = σ (r,t) α l 2 J 0 2πf d T s (k -m + sN s ) (12)
Various traditional BEM designs have been reported to model the channel time-variations, e.g., the Complex Exponential BEM (CE-BEM)

[B] k,m = e j2π( k-N g Ns )(m-Nc-1 2
) which leads to a strictly banded frequency-domain matrix [START_REF] Teo | Optimal MMSE Finite Parameter Model for Doubly-selective Channels[END_REF], the Generalized CE-BEM (GCE-BEM)

[B] k,m = e j2π( k-N g av )(m-Nc-1 2 ) with 1 < a ≤ Nc-1
2f d T which is a set of oversampled complex exponentials [START_REF] Leus | On the Estimation of Rapidly Time-Varying Channels[END_REF], the Polynomial BEM (P-BEM) [B] k,m = (k -N g) m [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF] and the Discrete Karhuen-Loeve BEM (DKL-BEM) which employs basis sequences that corresponds to the most significant eigenvectors of the autocorrelation matrix R (0,r,t) α l [START_REF] Kannu | MSE-optimal Training for Linear Timevarying Channels[END_REF]. From now on, we can describe the MIMO-OFDM system model derived previously in terms of BEM. Substituting [START_REF] Steendam | Sensitivity of Orthogonal Frequency-Division Multiplexed Systems to Carrier and Clock Synchronisation Errors[END_REF] in [START_REF] Wang | A MIMO-OFDM Channel Estimation Approach Using Time of Arrivals[END_REF] and neglecting the BEM model error, we obtain after some algebra:

y n = K n (ν) • c n + w n ( 13
)
where the LN c × 1 vector c n and the N R N × LN c matrix K n (ν) are given by:

c n = c (1,1) T n , ..., c (1,NT ) T n , ..., c (NR,NT ) T n T (14) c (r,t) n = c (n,r,t) T 0 , ..., c (n,r,t) T L (r,t) -1 T ( 15
)
K n (ν) = blkdiag K (1) n (ν (1) ), ..., K (NR) n (ν (NR) ) ( 16)

K (r) n (ν (r) ) = K (r,1) n (ν (r,1) ), ..., K (r,NT ) n (ν (r,NT ) ) (17)
K (r,t) n (ν (r,t) ) = 1 N Z (n,r,t) 0 (ν (r,t) ), ..., Z (n,r,t) L (r,t) -1 (ν (r,t) ) (18)
Z (n,r,t) l (ν (r,t) ) = M (r,t) 0 (ν (r,t) ) diag{x (t) n } f (r,t) l , ..., M (r,t) Nc-1 (ν (r,t) ) diag{x (t) n } f (r,t) l ( 19
)
where ν (r) = ν (r,1) , . . . , ν (r,NT ) T . Vector f (r,t) l

is the lth column of the N ×L (r,t) Fourier matrix F (r,t) whose elements are given by:

[F (r,t) ] k,l = e -j2π( k-1 N -1 2 )τ (r,t) l , (20)
and M

(r,t) d is a N × N matrix whose elements are given by:

M (r,t) d (ν (r,t) ) k,m = N -1 q=0 e j2π ν (r,t) q N [B] q+Ng ,d e j2π m-k N q .
(21) Moreover, the channel matrix of the (r, t) branch can be easily computed by using the BEM coefficients [START_REF] Hijazi | Polynomial estimation of time-varying multi-path gains with intercarrier interference mitigation in OFDM systems[END_REF]:

H (r,t) n = Nc-1 d=0 M (r,t) d (ν (r,t) )diag{F (r,t) χ (n,r,t) d } (22) where χ (n,r,t) d = c (n,r,t) (d,0) , ..., c (n,r,t) (d,L (r,t) -1)
T .

III. AR MODEL AND EXTENDED KALMAN FILTER

A. The AR Model for c n

The optimal BEM coefficients c (n,r,t) l are correlated complex Gaussian variables with zero-means and correlation matrix given by: R (s,r,t)

c l = E[c (n,r,t) l c (n-s,r,t) l H ] = B H B -1 B H R (s,r,t) α l B B H B -1 (23)
Hence, the dynamics of c

(n,r,t) l can be well modeled by an auto-regressive (AR) process [START_REF] Baddour | Autoregressive modeling for fading channel simulation[END_REF] [20] [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF] . A complex AR process of order p can be generated as:

c (n,r,t) l = p i=1 A (i) c (n-i,r,t) l + u (n,r,t) l (24)
Time Update Equations:

μ(n|n-1) = A μ(n-1|n-1) P (n|n-1) = AP (n-1|n-1) A H + U (37)
Measurement Update Equations:

K n = P (n|n-1) G H n G n P (n|n-1) G H n + N T .σ 2 I NRN -1 μ(n|n) = μ(n|n-1) + K n y n -g μ(n|n-1) P (n|n) = P (n|n-1) -K n G n P (n|n-1) (38)
where K n is the Kalman gain. The Time Update Equations are responsible for projecting forward (in time) the current state and error covariance estimates to obtain the a priori estimates for the next time step. The Measurement Update Equations are responsible for the feedback, i.e., for incorporating a new measurement into the a priori estimate to obtain an improved a posteriori estimate. The Time Update Equations can also be thought of a predictor equations, while the Measurement Update Equations can be thought of a corrector equations.

IV. JOINT DATA DETECTION AND EKF

In the iterative algorithm for joint data detection, channel and CFO extended Kalman estimation, the N p pilots subcarriers are evently inserted into the N subcarriers at the positions P = {p r | p r = (r -1)L f + 1, r = 1, ..., N p }, where L f is the distance between two adjacent pilots. We use the the QR-equalizer [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF] for the data detection. The QRequalizer allows us to estimate the data symbol with free ICI by performing a so-called QR-decomposition. The algorithm proceeds as follows: initialization: where i represents the iteration number.

• μ(0|0) = 0 LNc+N R N T ,

V. SIMULATION

In this section, the performance of our recursive algorithm is evaluated in terms of Mean Square Error (MSE) for joint channel and CFO estimation and in terms of Bit Error Rate (BER) for data detection.

We assume that all the (r, t) channel branches, r = 1, . . . , N R , t = 1, . . . , N T have the same path delays and fading properties (i.e., the same number of paths, of σ ). This is understood when the antennas are very close to each other, which is typical in practice. The Rayleigh channel model given in [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF] [12](L (r,t) = 6 paths and maximum delay τ max = 10T s ) was chosen. A normalized 4QAM MIMO-OFDM system, with N T = N R = 2, N = 128 subcarriers, N g = N 8 , N p = N 4 pilots (i.e., L f = 4) and 1 Ts = 2M Hz was used. The MSE and the BER were evaluated under a rapid time-varying channel with f d T = 0.1 (corresponding to a vehicle speed of 600km/h at f c = 2.5GHz).

A GCE-BEM with N c = 3 was chosen to model the path complex gains of the channel. Most advanced technologies have an oscillator frequency tolerance less than 1 ppm (i.e. ν = 0.16 in normalized units with the given parameters).

For the simulation, we chose the configuration where each transmitter and receiver requires its own RF-IF chain, which is the configuration discussed in this article. For this scenario, the number of CFO parameters to be estimated (N T N R = 4) is the largest. Therefore, this is the most pessimistic configuration. The CFO values were arbitrarily chosen as ν (0,0) = 0.1, ν (0,1) = -0.1, ν (1,0) = 0.05 and ν (1,1) = -0.07. Fig. 1 shows the MSE of the channel complex gain and the MSE of the normalized CFO as a function of E b /N 0 . Seven iterations have been carried out. For reference, the MSEs obtained in Data Aided (DA) mode (knowledge of the data symbols) have been plotted. As expected, the MSEs obtained in Data Aided mode are lower than the MSEs obtained with just the pilots, especialy at low E b /N 0 where the detection errors are the most important. However, for E b /N 0 ≥ 20 dB, the MSE has a floor (especially the channel complex gain MSEs). This is due to the fact that beyond 20 dB, the matrix to be inversed in Eq. (38) becomes badly scaled.

Fig. 2 gives the BER performance of our proposed iterative algorithm. For reference, we also plotted BERs obtained with perfect knowledge of channel response and CFO. It is shown that one iteration is sufficient to approach the reference curve with perfect knowledge of channel response and carrier frequency offset.

VI. CONCLUSION

A new iterative algorithm which jointly estimates multipath complex gain and CFO in MIMO environment has been presented. The algorithm is based on a parametric channel model. Extended Kalman filtering is used for parameter estimation and the data recovery is carried out by means of a QR-equalizer. Simulation results show that by estimating and removing the ICI at each iteration, the BER is greatly improved, especially after the first iteration. Our algorithm needs only one iteration to approach the performance of the ideal case for which the knowledge of the channel response and CFO is available.

ACKOWLEDGEMENT

T

  ) where the N s ×N c matrix B is defined as: B = [b 0 , ..., b Nc-1 ]. The N s × 1 vector b d is termed as the dth expansion basis. c represents the N c BEM coefficients and ξ (n,r,t) l

Fig. 1 .

 1 Fig. 1. Mean Square Error (MSE) as a function of E b /N 0 for f d T = 0.1

0 Eb/No (dB) BER BER for iteration 0 BER for iteration 1 BER for iteration 2 BER for iteration 3 BER for iteration 4 BER for iteration 5 BER for iteration 6 BER with perfect knowledge of channel and CFO

Fig. 2. Bit Error Rate (BER) as a function of E b /N 0 for f d T = 0.1

where A (1) , ..., A (p) are N c × N c matrices and u (n,r,t) l is a N c ×1 complex Gaussian vector with covariance matrix U (r,t) l . From [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF], it is sufficient to choose p = 1 to correctly model the path complex gains. The matrices A (1) = A and U (r,t) l are the AR model parameters obtained by solving the set of Yule-Walker equations defined as:

Using (24), we obtain the AR model of order 1 for c n :

where

B. The AR Model for ν n

Let us write the AR model for ν n as follows:

where the state transition matrix is of size

Since the CFOs can be assumed as constant during the observation interval, A ν is considered to be close to the identity matrix A ν = aI NRNT , a = 0.99. The N R N T × 1 state noise vector u νn is assumed to be zero-mean complex Gaussian. The state noise covariance matrix is U ν = σ 2 ν I NRNT where σ 2 ν is the variance of the state noise associated with CFOs.

C. State equation

Now, let us write the state-variable model. The state vector at time instance n consists of the BEM coefficients c n and the vector of CFOs ν n :

There are LN c BEM coefficients and N T N R CFO values in the state vector of dimension LN c + N T N R × 1. Then, the linear state equation may be written as follows:

where the state transition matrix is defined as follows:

The

D. Extended Kalman Filter (EKF)

The measurement equation ( 13) can be reformulated as:

where the nonlinear function g of the state vector µ n is defined as g (µ n ) = K n (ν) • c n . Nonlinearity of the measurement equation ( 32) is caused by CFOs. The BEM coefficients are still linearly related to observations. Since the measurement equation is nonlinear, we use the Extended Kalman filter to adaptively track µ n . Let μ(n|n-1) be our a priori state estimate at step n given knowledge of the process prior to step n, μ(n|n) be our a posteriori state estimate at step n given measurement y n and, P (n|n-1) and P (n|n) are the a priori and the a posteriori error estimate covariance matrix of size

We initialize the EKF with μ(0|0) = 0 LNc+NRNT ,1 and P (0|0) given by:

where R (s,r,t) c l is the correlation matrix of c

(n,r,t) l defined in (23). To derive the EKF equations, we need to compute the Jacobian matrix G n of g (µ n ) with respect to µ n and evaluated at μ(n|n-1) :

Let us define µ . After computation, we find:

where

The elements of the N × N matrix M d (ν) are given by:

q N e j2π ν (r,t) n q N

[B] q+Ng ,d e j2π m-k N q (36) The EKF is a recursive algorithm composed of two stages: Time Update Equations and Measurement Update Equations. These two stages are defined as: