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

Hussein Hijazi
email: hijazi.hussein@liu.edu.lb

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

✩✩ L Ros

Complex Amplitudes Tracking Loop for multipath channel estimation in OFDM systems over slow to moderate fading

Keywords: OFDM, Channel estimation, Rayleigh multipath channel with Jakes' Doppler spectrum, Phase-Locked Loop (PLL), Kalman filter

Introduction

Orthogonal Frequency-Division Multiplexing (OFDM) is an effective technique for alleviating frequency-selective channel effects in wireless communication systems. In this technique, a wideband frequency-selective channel is converted to a number of parallel narrow-band flat fading subchannels which are free of Inter-Symbol-Interference (ISI) and free of Inter-Carrier Interference (ICI) (for negligible channel time variation within one OFDM symbol period T ). For coherent detection of the information symbols, reliable estimation of the gain of each subchannel in the OFDM system is crucial.

Some approaches to channel estimation in OFDM

Most of the conventional methods work in a symbol-by-symbol scheme [START_REF] Zhao | A novel Channel estimation method for OFDM Mobile Communications Systems based on pilot signals and transform domain processing[END_REF][START_REF] Hsieh | Channel estimation for OFDM systems based on combtype pilot arrangement in frequency selective fading channels[END_REF][START_REF] Coleri | Channel estimation techniques based on pilot arrangement in OFDM systems[END_REF] by using the correlation of the channel only in the frequency-domain (FD), i.e. the correlation between subchannels. Generally, they consist in estimating the channel at the pilot subcarrier position and then interpolating it over the entire frequency grid [START_REF] Zhao | A novel Channel estimation method for OFDM Mobile Communications Systems based on pilot signals and transform domain processing[END_REF]. The channel estimation at the pilot frequencies can be based on the Least-Squares (LS) criterion, or, for better performance, on the Linear-Minimum-Mean-Square-Error (LMMSE) criterion [START_REF] Hsieh | Channel estimation for OFDM systems based on combtype pilot arrangement in frequency selective fading channels[END_REF]. In [START_REF] Coleri | Channel estimation techniques based on pilot arrangement in OFDM systems[END_REF], Low-Pass Interpolation (LPI) has been shown to perform better than all interpolation techniques used in channel estimation. This channel estimator will be called conventional LS(FD)-LPI in this paper.

Though the conventional methods can operate with time-varying channels, the information of the time-domain correlation is not exploited. However, the channel estimation process can be theoretically greatly improved by using the previous OFDM symbols, according to the on-line Bayesian Cramer-Rao Bound (BCRB) analysis in [START_REF] Hijazi | Bayesian Cramer-Rao bounds for complex gain parameters estimation of slowly varying Rayleigh channel in OFDM systems[END_REF]. Thus, Chen and Zhang proposed in [START_REF] Chen | Kalman filter Channel estimator for OFDM systems in time and frequency-selective fading environment[END_REF] a structure to track the complex gains of each subchannel by using one Kalman filter (KF) per sub-channel. In practice, only a subset of pilot-subcarriers is used to perform the per-subchannel KF, and the global frequency response of the channel is still obtained by LPI interpolation. This estimator will be named Kalman(FD)-LPI in the paper. Other works still exploit time and frequency correlation for OFDM channel estimation by using additional assumptions or different approaches. Assuming the availability of the power delay profile, a data-aided KF estimator (derived from the Expectation-Maximization algorithm framework) is employed in [START_REF] Al-Naffouri | An EM-based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM[END_REF] to track the discrete-time impulse response of the channel (i.e in Time-Domain (TD)). And a low-complexity parameter reduction approach based on the eigenvalue decomposition of the auto-correlation matrix of the channel (in FD) is proposed in [START_REF] Sohail | An EM-based frequency domain channel estimation algorithm for multi-access OFDM systems[END_REF]. It tracks the channel coefficients in the dominant eigenvectors subspace basis by KF, and then performs eigenvalues interpolation to compute the channel frequency response. This estimator will be denoted Kalman-EIG in this paper.

In the same context of reducing the signal subspace dimension, we now focus on the class of parametric channel estimators. Assuming a multipath channel structure, estimation can be reduced to the estimation of certain physical propagation parameters, such as multipath delays and multipath Complex Amplitudes (CAs) [START_REF] Yang | Channel estimation for OFDM transmission in multipath fading channels based on parametric channel modeling[END_REF][START_REF] Raghavendra | Exploiting hopping pilots for parametric channel estimation in OFDM systems[END_REF][START_REF] Chen | Improved Channel Estimation Based on Parametric Channel Approximation Modeling for OFDM Systems[END_REF][START_REF] Hijazi | Polynomial estimation of time-varying multi-path 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].

It is well known that in wireless radio channels, the delays are quasi-constant over a large number of symbols. Consequently, the number of paths and path delays can be very accurately estimated, for example by applying the MDL (Maximum Description Length) principle combined with the ESPRIT (Estimation of Signal Parameters by Rotational Invariance Techniques) method as proposed in [START_REF] Yang | Channel estimation for OFDM transmission in multipath fading channels based on parametric channel modeling[END_REF], and adopted many times [START_REF] Raghavendra | Exploiting hopping pilots for parametric channel estimation in OFDM systems[END_REF][START_REF] Hijazi | Polynomial estimation of time-varying multi-path 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]). Several papers on OFDM channel estimation focus their works on the crucial CAs tracking problem, assuming the delays are invariant and perfectly estimated. This approach will be adopted in this paper. In [START_REF] Hijazi | Polynomial estimation of time-varying multi-path 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] we have addressed this issue for the special case of fast time-varying channel (i.e. with normalized Doppler spread f d T ≥ 10 -2 ), by using polynomial modeling of the CA time-variation. We have also addressed it in [START_REF] Simon | Joint carrier frequency offset and channel estimation for OFDM systems via the EM algorithm in the presence of very high mobility[END_REF] for the joint carrier frequency offset and high speed channel estimation problem.

Motivation of the work and contributions

Second-order versus first-order algorithms. the use of KF for channel estimation has received great attention in recent years in the wireless communication literature. It is true for most systems, e.g. MIMO [START_REF] Komninakis | Multi-Input Multi-Output Fading Channel Tracking and Equalization Using Kalman Estimation[END_REF][START_REF] Liu | Space-time coding and Kalman filtering for time-selective fading channels[END_REF] or single-carrier systems [START_REF] Barbieri | On the ARMA Approximation for Frequency Channels Described by the Clarke Model with Applications to Kalman-based Receivers[END_REF][START_REF] Ghandour-Haidar | On the Use of first-order Autoregressive Modeling for Rayleigh Flat Fading Channel Estimation with Kalman filter[END_REF], as well as in OFDM systems, as mentioned before [START_REF] Chen | Kalman filter Channel estimator for OFDM systems in time and frequency-selective fading environment[END_REF][START_REF] Al-Naffouri | An EM-based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM[END_REF][START_REF] Sohail | An EM-based frequency domain channel estimation algorithm for multi-access OFDM systems[END_REF][START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF]. All the aforementioned works based their KF on the AR approximation of the widely accepted Rayleigh fading channel with the Jakes'Doppler spectrum [START_REF] Jakes | Microwave Mobile Communications[END_REF], called the "Rayleigh-Jakes" model in this paper, as developed in [START_REF] Baddour | Autoregressive modeling for fading channel simulation[END_REF]. The first-order Gauss-Markov assumption (AR1 model) is most often retained [START_REF] Al-Naffouri | An EM-based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM[END_REF][START_REF] Sohail | An EM-based frequency domain channel estimation algorithm for multi-access OFDM systems[END_REF][START_REF] Komninakis | Multi-Input Multi-Output Fading Channel Tracking and Equalization Using Kalman Estimation[END_REF][START_REF] Liu | Space-time coding and Kalman filtering for time-selective fading channels[END_REF][START_REF] Barbieri | On the ARMA Approximation for Frequency Channels Described by the Clarke Model with Applications to Kalman-based Receivers[END_REF][START_REF] Ghandour-Haidar | On the Use of first-order Autoregressive Modeling for Rayleigh Flat Fading Channel Estimation with Kalman filter[END_REF][START_REF] Wang | On verifying the first-order Markovian assumption for a Rayleigh fading channel model[END_REF][START_REF] Banelli | Data-Aided Kalman Tracking for Channel Estimation in Doppler-Affected OFDM Systems[END_REF]. The so-called AR1-Kalman estimators are convenient for the very high mobility case, leading to quasi-optimal performance, as seen, for example, in [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF][START_REF] Simon | Joint carrier frequency offset and channel estimation for OFDM systems via the EM algorithm in the presence of very high mobility[END_REF]. In these works an AR1-Kalman is actually used to track the polynomial Basis Expansion Model coefficients of the high speed channel. However, in the more common scenario of slow to moderate fading with negligible variation during one symbol (i.e. f d T ≤ 10 -2 , as in [START_REF] Chen | Kalman filter Channel estimator for OFDM systems in time and frequency-selective fading environment[END_REF][START_REF] Al-Naffouri | An EM-based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM[END_REF][START_REF] Komninakis | Multi-Input Multi-Output Fading Channel Tracking and Equalization Using Kalman Estimation[END_REF][START_REF] Liu | Space-time coding and Kalman filtering for time-selective fading channels[END_REF][START_REF] Barbieri | On the ARMA Approximation for Frequency Channels Described by the Clarke Model with Applications to Kalman-based Receivers[END_REF][START_REF] Ghandour-Haidar | On the Use of first-order Autoregressive Modeling for Rayleigh Flat Fading Channel Estimation with Kalman filter[END_REF]), the AR1-Kalman estimators of the literature seem to exhibit poor performance compared to BCRB lower bounds, as seen in [START_REF] Ros | Paths Complex Gain Tracking Algorithms for OFDM Receiver in Slowly-Varying Channels[END_REF]-Fig. 10. On the other hand, [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF] shows, in a single-carrier single-path context, that the MSE performance of a KF can be well improved by switching from the AR1 to a second-order model (of the integrated random walk (IRW) model type) for the approximation of the Jakes'process. Indeed, for low f d T , the exact channel CA continues in a given direction during several symbols, and a second-order approximation model can generally take into account this strong trend behaviour better than a first-order model [START_REF] Gazor | Prediction in LMS-type Additive Algorithms for Smoothly Time Varying Environments[END_REF][START_REF] Lindbom | Tracking of Time-Varying Mobile Radio Channels-Part II : A Case Study[END_REF].

Reduced complexity algorithms compared to Kalman. KF-based algorithms require the updating of the coefficients of the algorithm at each iteration (each new OFDM symbol), and are quite complex as a result. However, reduced complexity adaptive algorithms can be obtained, using constant coefficients. They can be designed, if an a priori model of the dynamic of time-varying parameters (i.e. hypermodel) is available, such as a Wiener LMS adaptation algorithm ( [START_REF] Lindbom | Tracking of Time-Varying Mobile Radio Channels-Part II : A Case Study[END_REF]), or as a steady-state version of the KF, since a time-varying KF becomes a time-invariant filter after convergence, see [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation Theory[END_REF], ch 13.5. Such algorithms are generally slower than the KF during the convergence, but can have the same asymptotic performance in tracking mode. In this family, the classical Least-Mean-Squared (LMS) algorithm can be regarded as a steady-state version of a KF based on a first-order random-walk (RW) model. Second-order channel tracking algorithms described as predictive LMS, or as a steady-state version of an IRW modelbased KF, have also been proposed in [START_REF] Gazor | Prediction in LMS-type Additive Algorithms for Smoothly Time Varying Environments[END_REF][START_REF] Lindbom | Tracking of Time-Varying Mobile Radio Channels-Part II : A Case Study[END_REF][START_REF] Nassir | Wireless Channel Tracking Based on Optimum Predictive LMS[END_REF]. However, they have been developed for the single-carrier transmission, and without simple closed-form formulas versus the channel state for the tuning and performance of the estimators.

Approach and contributions. in this paper, we propose and analyze a low-complexity

on-line recursive algorithm with constant coefficients for the multipath CAs estimation problem under the common slow to moderate channel variations scenario ( f d T ≤ 10 -2 ). It is developed for OFDM systems with comb-type pilots within the framework of parametric channel estimators, exploiting the availability of delay related information (assuming a primary acquisition as in [START_REF] Yang | Channel estimation for OFDM transmission in multipath fading channels based on parametric channel modeling[END_REF][START_REF] Hijazi | Polynomial estimation of time-varying multi-path 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][START_REF] Simon | Joint carrier frequency offset and channel estimation for OFDM systems via the EM algorithm in the presence of very high mobility[END_REF]) for tracking the CA variations. The proposed algorithm is based on a Complex Amplitudes Tracking Loop (CATL) structure. This structure is inspired by second-order digital Phase-Locked Loops (PLL) [START_REF] Mengali | Synchronization Techniques for Digital Receivers (Applications of Communications Theory)[END_REF][START_REF] Shayan | All digital phase-locked loop: concepts, design and application[END_REF], as well as by the "prediction-correction" principle of the KF (in the steady-state mode) given the close link between the two ( [START_REF] Patapoutian | On phase-locked loops and Kalman filters[END_REF][START_REF] Christiansen | Modeling of a PRML timing loop as a Kalman filter[END_REF]). The error signal that feeds the loop is based on the LS estimate of the paths CA, obtained for each current symbol from the pilot-subcarriers. The proposed LS-CATL algorithm can be seen as an extension for the multipath OFDM case of the second-order adaptive algorithms of [START_REF] Gazor | Prediction in LMS-type Additive Algorithms for Smoothly Time Varying Environments[END_REF][START_REF] Lindbom | Tracking of Time-Varying Mobile Radio Channels-Part II : A Case Study[END_REF](and also [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF]-ch 4.1), using the parametric estimation framework.

Our main contributions can be summarized below:

• proposition, interpretation, and analytical optimization of a simple on-line secondorder (multipath) CAs tracking algorithm with almost the same asymptotic MSE performance as a second-order KF derived with the same assumptions (parametric modeling and a priori knowledge), but with a reduced complexity,

• derivation of closed-form expressions usable to tune the coefficients of the CATL as well as to predict the MSE performance with respect to the channel state (Doppler spread, power-delay profile, SNR) under "Rayleigh-Jakes" assumption.

• evaluation of the benefit of the second-order proposed algorithm compared to first-order KF-based reference algorithms or other conventional (FD interpolation) methods of the literature, for the common slow to moderate fading channel.

The paper is organized as follows: Section II describes the system model and objectives. Section III derives the proposed algorithm and its analysis, and the different results are discussed in Section IV.

Notations: [x] k denotes the kth entry of vector x, and [X] k,m the [k, m]th entry of matrix X (indices begin from 1). I N is an N × N identity matrix. The notation diag{x} stands for a diagonal matrix with x on its diagonal, diag{X} is a vector whose elements are the elements of the diagonal of X, and blkdiag{X, Y} is a block diagonal matrix with the matrices X and Y on its diagonal. The superscripts (•) T , (•) H , | • |, and Tr(•) respectively stand for transpose and Hermitian operators, determinant and trace operations. J 0 (•) is the zeroth-order Bessel function of the first kind.

System Model

OFDM Transmission over multipath channel

Let us consider an OFDM system with N subcarriers, and a cyclic prefix length N g . The duration of an OFDM symbol is T = vT s , where T s is the sampling time

and v = N + N g . Let x (n) = x (n) [-N 2 ], x (n) [-N 2 + 1], ..., x (n) [ N 2 -1]
T be the vector containing the N QAM symbols for the nth OFDM symbol. After transmission over a multipath channel and FFT demodulation, the observation is given by [START_REF] Yang | Channel estimation for OFDM transmission in multipath fading channels based on parametric channel modeling[END_REF][START_REF] Hijazi | Polynomial estimation of time-varying multi-path gains with intercarrier interference mitigation in OFDM systems[END_REF]:

y (n) = H (n) x (n) + w (n) (1)
where w (n) is a N × 1 zero-mean complex circular Gaussian noise vector with covariance matrix σ 2 I N , and

H (n) is a N × N diagonal matrix with diagonal elements [H (n) ] k,k = 1 N L ∑ l=1 α l(n) × e -j2π( k-1 N -1 2 )τ l (2)
L is the total number of propagation paths, {α l(n) } is the lth CA at nth OFDM symbol with variance σ 2 α l (with ∑ L l=1 σ 2 α l = 1), and τ l × T s is the lth delay (τ l is not necessarily an integer, but τ L < N g ). The L individual elements of {α l(n) } are uncorrelated with respect to each other. Using (2), the observation model ( 1) can be re-written [START_REF] Yang | Channel estimation for OFDM transmission in multipath fading channels based on parametric channel modeling[END_REF] as

y (n) = diag{x (n) }F α (n) + w (n) (3)
where

α (n) = [α 1(n) , ..., α L(n) ]
T and F is an N × L Fourier matrix depending on the delay distribution, with elements given by

[F] k,l = e -j2π( k-1 N -1 2 )τ l .
We assume the "Rayleigh-Jakes" model [START_REF] Jakes | Microwave Mobile Communications[END_REF] for the channel, with Doppler frequency

f d .
It means the L CAs α l(n) are independent wide-sense stationary zero-mean complex circular Gaussian processes, with correlation coefficients for a time-lag k given by R (k)

α l = E[α l(n) α l(n-k) H ] = σ 2 α l .J 0 (2π f d T k) (4)

Pilot Pattern

The N p pilot subcarriers are evenly inserted into the N subcarriers at the positions

P = {p s | p s = (s -1)L f + 1, s = 1, ..., N p } with L f the distance between two adjacent
pilots. The received pilot subcarriers can be written as

y p(n) = K (n) α (n) + w p(n) (5)
where y p and w p are N p × 1 vectors. The N p × L matrix K (n) is defined by

K (n) = diag{x p(n) }F p (6)
Note that K (n) is computed for each OFDM symbol, using knowledge of the N p × 1 data pilot vector x p(n) and the delays {τ l } through the N p × L matrix F p with elements

[F p ] k,l = e -j2π( p k -1 N -1 2 )τ l (7)

Estimation objectives

We wish to estimate the CAs α (n) assuming the knowledge of pilots subcarriers

x p(n) and delays τ = [τ 1 , ..., τ L ] T . The estimation is based on the observation model (5) that can be reformulated as

y p(n) = K x p(n) , τ α (n) + w p(n) .
We restrict the problem to the on-line estimation, which means current and previous observations are available (i.e. for indices n, n -1, n -2, ...) to estimate α at time index n.

Complex Amplitudes Tracking algorithm

The proposed tracking algorithm, called the LS-CATL algorithm, is built from a general second-order recursive structure (CATL) presented below, and from a specific error signal (based on pilots and LS criterion) that will specify the error detector of the structure, presented subsequently.

Structure of the algorithm: CA Tracking Loop

The estimate of α (n) , denoted α(n|n) , is updated at a symbol rate by the computation of an error signal v ǫ(n) , next filtered by a second-order feedback loop. The recursive equations of the second-order CATL, using a PLL-type formulation ( [START_REF] Mengali | Synchronization Techniques for Digital Receivers (Applications of Communications Theory)[END_REF][START_REF] Shayan | All digital phase-locked loop: concepts, design and application[END_REF]), are:

Error detector : v ǫ(n) = function of { y p(n) ; α(n|n-1) } (8)
Final estimate :

α(n|n) = α(n|n-1) + µ 1 .v ǫ(n) (9)
Loop filter :

v Lag(n) = v Lag(n-1) + v ǫ(n) (10) v c(n) = µ 1 .v ǫ(n) + µ 2 .v Lag(n) (11)
NC Generator :

α(n+1|n) = α(n|n-1) + v c(n) (12)
Figure 1: Structure of the second-order complex amplitude tracking loop, inspired by second-order digital PLL where µ 1 , µ 2 are the (real positive) loop coefficients. Based on these equations, the structure of the CATL is shown in Fig. 1. As in a second-order digital PLL [START_REF] Shayan | All digital phase-locked loop: concepts, design and application[END_REF], we find in cascade an error detector that delivers error signal v ǫ(n) , a proportional-integral (PI) loop filter F PLL (z) = µ 1 + µ 2 1-z -1 (or lead-lag filter) parametrized by µ 1 and µ 2 , and a Numerically Controlled (NC) Generator delivering the predicted estimates α(n|n-1) .

However, the estimates are multiple complex amplitudes, instead of one real phase in a PLL (then delivered by a NC Oscillator). v ǫ(n) is a complex vector in the output of a "Complex Amplitude Error Detector" (CAED) (vs a real scalar in the output of a phase error detector in a PLL), to be defined in [START_REF] Al-Naffouri | An EM-based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM[END_REF] from the new measurement y p(n) and the prediction α(n|n-1) . Also, the final estimate α(n|n) is not directly the prediction α(n|n-1) as in conventional PLL, but is delivered after a correction step according to [START_REF] Sohail | An EM-based frequency domain channel estimation algorithm for multi-access OFDM systems[END_REF]. Thus, an additional branch is added as a dotted line in Fig. 1.

Using [START_REF] Sohail | An EM-based frequency domain channel estimation algorithm for multi-access OFDM systems[END_REF], we can compact the second equation of the loop filter [START_REF] Raghavendra | Exploiting hopping pilots for parametric channel estimation in OFDM systems[END_REF] and the NCG equation ( 12) by the unique equation [START_REF] Hijazi | Polynomial estimation of time-varying multi-path gains with intercarrier interference mitigation in OFDM systems[END_REF]: [START_REF] Hijazi | Polynomial estimation of time-varying multi-path gains with intercarrier interference mitigation in OFDM systems[END_REF] showing that the sum accumulator of the error signal weighted by µ 2 , i.e. µ 2 .v Lag(n) , is an estimate of the speed evolution (or slope) of α, useful to predict the CAs evolution.

α(n+1|n) = α(n|n) + µ 2 .v Lag(n)

Error signal specific to the LS-CATL algorithm

We now have to define an error signal vector in place of eq. ( 8). Inspired by PLL, a good candidate (among several possibilities [START_REF] Ros | Paths Complex Gain Tracking Algorithms for OFDM Receiver in Slowly-Varying Channels[END_REF]) is an error signal vector v ǫ(n) collinear (in absence of noise) with the prediction error vector ǫ Pred(n) = α (n)α(n|n-1) , in order to get a detector, which is perfectly linear and free from inter-path-interference. In this perspective, let us first consider the LS-estimator of α (n) that permits, among all estimators α(n) , us to minimize the squares error

(y p(n) -K (n) α(n) ) H .(y p(n) -K (n) α(n) )
for the current OFDM symbol:

α LS(n) = G (n) y p(n) (14)
with

G (n) = K H (n) K (n) -1 K H (n) (15)
We see from ( 14)&( 15)&( 5) that the LS estimator is unbiased, with α LS(n) = α (n) + N (n) where N (n) is a zero-mean complex Gaussian noise vector. So, we propose to use simply the difference between the LS estimator α LS(n) for the nth OFDM block and the prediction, α(n|n-1) , as an error signal vector in place of (8):

v ǫ(n) = G (n) y p(n) -α(n|n-1) (16)
Thus, this specific error signal vector ( 16) has a simple linear form versus the prediction error vector ǫ Pred(n) = α (n)α(n|n-1) , as seen while using ( 14)&( 15)&( 5):

v ǫ(n) = k d .{α (n) -α(n|n-1) } + N (n) (17)
The real coefficient k d is the gain factor of the CAED, reduced here to k d = 1. And

N (n) = [N 1(n) , ..., N L(n) ]
T is the (temporally uncorrelated) zero-mean disturbance due to the additive thermal noise w p(n) in the input of the CAED, and represents the so-called (input) loop noise (i.e. in the input of the loop but in the output of the CAED). We

have

N (n) = G (n) w p(n) , with a correlation matrix E{N (n) .N H (n) } = σ 2 . F H p F p -1
, and a mean variance (per path,

σ 2 N = 1 L . ∑ L l=1 σ 2 N l ): σ 2 N = σ 2 N p × λ N ( 18
) with λ N = 1 L .Tr{ 1 N p .F H p F p -1 } ≥ 1 ( 19
)
where N p is the number of pilot subcarriers. The (input) loop noise variance is mini-

mum (i.e. σ 2 N (min) = σ 2 N p and λ N = 1) if N (n)
is uncorrelated from one path to another, i.e when F H p F p is a diagonal matrix. This condition depends on the delays distribution.

Computational Complexity and comparison with reference KFs

The CATL can be interpreted as a reduced complexity approach compared to reference KFs designed for the complete multi-carrier and multi-path observation model [START_REF] Coleri | Channel estimation techniques based on pilot arrangement in OFDM systems[END_REF] such as presented in Appendix A (AR1-Kalman and Or2-Kalman). Indeed, the CATL acts in each branch l, i.e. for each multi-path component α l , as a "simplified" IRW model-based KF. Each KF is "simplified" in the sense it is designed for the simplified single-carrier and single-path scenario, as in [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF], and it operates only in steady-state mode. But we have chosen an error signal ( 16) able to cope with the more complicated multi-carrier and multi-path scenario. This interpretation of the CATL structure is more detailed (i.e. derived from the equations) in Appendix B.

Let us determine now the implementation complexity in terms of the number of the complex multiplications needed for each OFDM symbol for our LS-CATL algorithm.

The matrices

G (n) (size L × N p ) and K (n) (size N p × L)
p × L + 2L Or2-Kalman N 3 p + N 2 p × 3L + N p × (6L 2 + 3L) + 4L 2 + 2L 1st-order LS-CATL N p × L + L AR1-Kalman N 3 p + N 2 p × 2L + N p × (2L 2 + 2L) + L 2 + L
Table 1: Complexity of the LS-CATL proposed algorithm (first-order and second-order versions) compared to reference KFs (AR1-Kalman and Or2-Kalman, described in Appendix A)

General properties and theoretical MSE analysis

Second-order closed-loop transfer function

The estimation error of the tracking algorithm is defined as

ǫ (n) = α (n) -α(n|n) (20)
We want to obtain the transfer function between the true vector parameter and the estimate. Combining equations ( 13) and ( 9), we have:

α(n|n) = α(n-1|n-1) + µ 1 .v ǫ(n) + µ 2 .v Lag(n-1) (21)
By using [START_REF] Yang | Channel estimation for OFDM transmission in multipath fading channels based on parametric channel modeling[END_REF], the Z-domain transform of ( 21) leads to

α(z).[1 -z -1 ] = [µ 1 + µ 2 .z -1 1 -z -1 ].v ε (z) (22)
Combining the general loop equation [START_REF] Wang | On verifying the first-order Markovian assumption for a Rayleigh fading channel model[END_REF] with the specific (LS-based) error signal [START_REF] Liu | Space-time coding and Kalman filtering for time-selective fading channels[END_REF] rewritten versus the estimation error as

v ǫ(n) = k d 1 -k d µ 1 .{α (n) -α(n|n) } + 1 1 -k d µ 1 .N (n) (23)
we obtain in the Z-transform domain:

α(z) = L(z).α(z) + L(z) k d .N (z) (24)
where L(z) is the transfer function of the CATL defined by

L(z) = k d 1-k d µ 1 F(z) (1 -z -1 ) + k d 1-k d µ 1 F(z) ( 25
)
with respect to

F(z) = µ 1 + µ 2 .z -1 1-z -1 .
Hence, the CATL transfer function can be written versus the loop coefficients (µ 1 , µ 2 ) as1

L(z) = k d [(z -1) 2 .µ 1 + (z -1).(µ 1 + µ 2 ) + µ 2 ] (z -1) 2 + (z -1).k d (µ 1 + µ 2 ) + k d µ 2 (26)
or rewritten in a more interpretable form as a function of both the natural pulsation ω n (or natural frequency f n = ω n 2π ), and the damping factor ζ as

L(z) = 2ζ ω n .(1 -z -1 ) + ω 2 n (1 -z -1 ) 2 + 2ζ ω n .(1 -z -1 ) + ω 2 n ( 27
)
with:

(ω n T ) 2 = k d µ 2 1 -k d µ 1 (28) 2ζ ω n T = (µ 1 -µ 2 )k d 1 -k d µ 1 ( 29
)
And from ( 28) and ( 29), one given couple (ω n , ζ ) of the second-order low-pass transfer function can be obtained by tuning (µ 1 , µ 2 ) as

µ 1 = 1 k d . (ω n T ) 2 + 2ζ ω n T 1 + (ω n T ) 2 + 2ζ ω n T (30) µ 2 = 1 k d . (ω n T ) 2 1 + (ω n T ) 2 + 2ζ ω n T ( 31
)
The strict stability conditions of L(z) in ( 26) or ( 27) versus (µ 1 , µ 2 ) are given in [2],

but if we impose that 0 < ω n < +∞ and 0 < ζ < +∞ in order to preserve a physical meaning, we deduce from ( 30)&(31

) that 0 < µ 2 < µ 1 < 1/k d . We can rewrite L(z)
in the frequency-domain, by making z = e pT , with p = j2π f , and f is the frequency variable. Fig. 2 plots the modulus of the resulting function, L(e j2π f T ). Assuming slow reaction of the loop during one OFDM symbol T (i.e. f n .T ≪ 1) and for Low Frequency (LF) region (i.e. for f T ≪ 1, using the approximation z -1 ≈ 1p.T ), the transfer function of the digital loop in ( 27) is close to the second-order low-pass transfer function of the analog PLL (i.e. with an active analog lead-lag loop filter, see [START_REF] Gardner | Phaselock Techniques[END_REF], ch II):

L(e pT ) ≈ 2ζ ω n p + ω 2 n p 2 + 2ζ ω n p + ω 2 n (32)

Mean Squared Error analysis

The estimator is unbiased since the CA estimation error ǫ (n) defined in ( 20) is zeromean (see [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF]). Our aim is to compute the estimation error variance as

σ 2 ε def = 1 L .E{ǫ H (n) ǫ (n) } = σ 2 εα + σ 2 εN ( 33
)
where σ 2 εα is the dynamic error variance, due to the variation of the process α, and σ 2 εN is the static error variance, due to the additive thermal noise. According to [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF] and [START_REF] Jakes | Microwave Mobile Communications[END_REF], error ǫ (n) can be expressed in the Z-domain by ǫ(z) = (1-L(z)).α(z)k -1 d .L(z).N (z). Then, the two components of variance σ 2 ε can be easily expressed in the frequency-

σ 2 εα = + 1 2T -1 2T Γ α ( f ).|1 -L(e j2π f T )| 2 d f (34) with Γ α ( f ) = 1 L . ∑ L l=1 Γ α l ( f )
where Γ α l is the Power Spectrum Density (PSD) of α l . The component σ 2 εN results from the low-pass filtering (-k -1 d .L(z)) of the input loop noise N (n) :

σ 2 εN = + 1 2T -1 2T Γ N ( f ). 1 k 2 d .|L(e j2π f T )| 2 d f (35) with Γ N ( f ) = 1 L . ∑ L l=1 Γ N l ( f ) where Γ N l is the PSD of N l .
The couple ( f n , ζ ) has to be properly chosen for a good trade-off between the gain in tracking ability and the reduction in loop noise, for a given SNR and f d T scenario. Fig. where B L is the (double-sided normalized) noise equivalent bandwidth of the system:

B L = T × + 1 2T -1 2T |L(e j2π f T )| 2 d f (37)
An exact analytical expression of B L is derived for the exact second-order loop (( 26) or ( 27)) from the method presented by R. Winkelstein in [START_REF] Winkelstein | Closed Form Evaluation of Symmetric Two-Sided Complex Integrals[END_REF], resulting in

B L = [8ζ 2 + 2](ω n T ) + 6ζ (ω n T ) 2 + (ω n T ) 3 8ζ + [8ζ 2 + 4].(ω n T ) + 6ζ .(ω n T ) 2 + (ω n T ) 3 (38)
If f n .T ≪ 1, we can use the approximation (39) which coincides (see [START_REF] Gardner | Phaselock Techniques[END_REF], ch. III) with the noise equivalent bandwidth of the usual analog second-order PLL given in [START_REF] Christiansen | Modeling of a PRML timing loop as a Kalman filter[END_REF]:

B L ≈ 2π f n T.(ζ + 1 4ζ ) ( 39
)
Dynamic error variance σ 2 εα . the dynamic error variance depends on the Doppler spectrum Γ α ( f ) and on |1 -L(e j2π f T )| 2 via the integral form [START_REF] Winkelstein | Closed Form Evaluation of Symmetric Two-Sided Complex Integrals[END_REF]. According to [START_REF] Christiansen | Modeling of a PRML timing loop as a Kalman filter[END_REF], the squared modulus of the error transfer function of the second-order loop is

|1 -L(e j2π f T )| 2 ≈ f 4 f 4 n -f 2 f 2 n .(2 -4ζ 2 ) + f 4 for f ≪ 1/T (40)
On the other hand, the Doppler spectrum for the "Rayleigh-Jakes" model ( 4), Γ α ( f ) =

σ 2 α /L π f d 1-( f f d ) 2 for f ∈] -f d ; + f d [,
has a bounded support. Therefore, good tracking will require that the natural frequency of the second-order loop f n be greater than f d .

Then, we can deduce that only the LF part of the function |1 -L(e j2π f T )| 2 is used in the integral [START_REF] Winkelstein | Closed Form Evaluation of Symmetric Two-Sided Complex Integrals[END_REF], and we can use the LF approximation

|1 -L(e j2π f T )| 2 ≈ ( f f n ) 4 for f ≤ f d ≪ f n ≪ 1/T .
σ 2 εα ≈ + f d -f d Γ α ( f ). f f n 4 d f (41)
For the "Rayleigh-Jakes" model, a variable change cos θ = ( f / f d ) permits us to evaluate (41) analytically as

σ 2 εα (Jakes) ≈ ( 3 8
).

f d f n 4 . σ 2 α L (42)
Optimal natural frequency. the dynamic component σ 2 εα decreases proportionally to the 4th power of f n according to (42), whereas the static component σ [START_REF] Gardner | Phaselock Techniques[END_REF], by searching the zero of the derivative of σ 2 ε in ( 33) using ( 36)&( 39)&( 42):

f d < f n ≪ 1/T ) to minimize the global MSE σ 2 ε in (
( f n f d )(Jakes) = 3 4 . 1 π . 1 (ζ + 1 4ζ ) . 1 f d T . σ 2 α /L σ 2 N /k 2 d 1 5
(43)

The closed-form expression of the corresponding optimal MSE results in

σ 2 ε (Jakes) = λ • σ 2 α L 1 5 • σ 2 N k 2 d • f d T 4/5 (44) with λ = 15 8 • (ζ + 1 4ζ ) • 4π 3 4 5 (45)
It is noticeable that the asymptotic performance of the second-order CATL in (44)

coincides, for ζ = √ 2
2 and L = 1, with that of the second-order Kalman in [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF]-eq.( 39), derived for the simplified case of single-carrier and single-path channels.

Special case of the first-order CATL

In the special case where µ 2 = 0, the on-line estimation algorithm is reduced to a first-order low-pass filtering of the LS estimator (see [START_REF] Liu | Space-time coding and Kalman filtering for time-selective fading channels[END_REF] and ( 21)), such that α(n|n) = (1 -µ 1 ). α(n-1|n-1) + µ 1 .α LS(n) . The transfer function (25) of the system just depends on a cut-off pulsation ω c (or cut-off frequency f c = ω c 2π ), and is reduced to

L(z) = ω c T (1 -z -1 ) + ω c T with (ω c T ) = k d µ 1 1 -k d µ 1 ( 46
)
and then, approximately, (when f c .T ≪ 1) to an analog first-order low-pass transfer function L(e pT ) ≈ ω c p+ω c , as can be seen in Fig. 2. We have from (46) that µ 1 = 2π f c T 1+2π f c T . The noise-equivalent bandwidth (37) becomes B L1 = 2π f c T 2 + 2π f c T , and can be approxi-

mated by B L1 ≈ π f c T when f c T ≪ 1. As seen in (46) that |1 -L(e j2π f T )| 2 ≈ ( f f c ) 2 for f ≤ f d ≤ f c ≪ 1/T , the dynamic error variance (34) is σ 2 εα (Jakes) ≈ ( 1 2 ).( f d f c ) 2 . σ 2 α L . The minimum global MSE (33) is then reduced, for f d < f c ≪ 1/T , to σ 2 ε (Jakes) = 3 2 . σ 2 α L 1 3 . π. σ 2 N k 2 d . f d T 2/3 (47) obtained for ( f c f d )(Jakes) = 1 π . 1 f d T . σ 2 α /L σ 2 N /k 2 d 1 3 (48)
It is noticeable that formula (47) of the first-order CATL coincides for L = 1 with the approximate expression of the asymptotic estimation variance of the AR1-Kalman in [START_REF] Ghandour-Haidar | On the Use of first-order Autoregressive Modeling for Rayleigh Flat Fading Channel Estimation with Kalman filter[END_REF]-eq.( 25), derived for the simplified case of single-carrier and single-path channels.

Simulations

In this section, the performance of the LS-CATL algorithm is evaluated, first confronted with theoretical analysis and natural references (BCRB and KFs using the same a priori model), and, then, with other algorithms from the literature. By default, we have used a 4QAM-OFDM system, with an FFT size N = 128 subcarriers, N g = N 8 = 16 samples for the cyclic prefix, and 1 T s = 28 25 × 1.25 = 1.4MHz. These parameters are selected in order to be in concordance with one configuration of the standard Mobile WiMAX Scalable [START_REF] Andrews | Fundamentals of WiMAX: Understanding Broadband Wireless Networking[END_REF], with a subcarrier spacing of 10.94 kHz. The number of pilot subcarriers was N p = 6, 8, 16, or 32, corresponding to a distance between pilot subcarriers L f = 22, 16, 8, or 4 respectively. The channel model is the Jakes'spectrum Rayleigh channel model with L = 6 paths and a maximum delay τ max = 10 (expressed as a fraction of T s ) given in [START_REF] Hijazi | Polynomial estimation of time-varying multi-path gains with intercarrier interference mitigation in OFDM systems[END_REF][START_REF] Zhao | A novel Channel estimation method for OFDM Mobile Communications Systems based on pilot signals and transform domain processing[END_REF] and recalled in table 2. The performance is evaluated under a time-varying channel with f d T ≤ 10 -2 (corresponding to a vehicle speed

V m ≤ 52.5 km/h for f c = 2 GHz), with a default value f d T = 10 -3 . Path number l 1 2 3 4 5 6 σ 2 αl /σ 2 α (dB) -3 0 -2 -6 -8 -10 τ l 0 1 2 3 4 10
Table 2: Average powers and (normalized) delays of the channel First of all, we can observe that all the theoretical curves are very close to the simulated ones. Therefore, the abscissa to the minimum of the simulated MSE σ 2 ε corresponds very well to the theoretical optimal natural frequency (in (43), such that f n / f d (Jakes) = 3, 7.4 and 18.7 respectively for SNR = 0, 20, and 40 dB, with N p = 16). It is interesting to note that there is a large range around the optimal natural frequency for which the MSE remains very close to the minimum value (for any SNR). Hence, the tuning of the natural frequency of the loop coefficients does not need to be very accurate.

For the rest of the section, we will use the parameters that yield around the minimum possible MSE for the various algorithms. Tables 3(a (theory (43))

Comparison with KFs using the same a priori knowledge and parametric model

We now compare the asymptotic performances to those obtained with two KFs directly derived from our OFDM parametric channel modeling-based estimation problem defined in section 2.3, and using the same a priori knowledge as the proposed LS-CATL. The first one is the AR1-Kalman, which uses an AR1 model to approximate the CA dynamic, and that can be found in [START_REF] Ros | Paths Complex Gain Tracking Algorithms for OFDM Receiver in Slowly-Varying Channels[END_REF]-section IV for our specific OFDM model (or in [START_REF] Al-Naffouri | An EM-based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM[END_REF] 2 or [14] 3 after slight adaptations). But, we also consider as a reference the Or2-Kalman, a Kalman based on a second-order model to better approximate the trend 2 considering in [START_REF] Al-Naffouri | An EM-based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM[END_REF] the Kalman-(forward-only)-initial estimation based on pilots. More precisely [START_REF] Al-Naffouri | An EM-based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM[END_REF] is the Time-Domain channel estimator that estimates the discrete-time impulse response including both physical channel (CAs at known positions τ ) and receive filter.

3 adapted for a pilot-aided mode and one polynomial coefficient.

behaviour of the CAs, as in our second-order CATL. It is a kind of extension for multiple carriers and multipath channel of the steady-state version of the KF in [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF]. Details about the design of these KFs are given in Appendix A, and table 4 gives the values of the parameters used, yielding around the minimum possible MSE. discussed in the introduction, motivated by some works about phase estimation [START_REF] Patapoutian | On phase-locked loops and Kalman filters[END_REF][START_REF] Christiansen | Modeling of a PRML timing loop as a Kalman filter[END_REF] or about CA estimation in single-carrier flat fading channel [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF](ch. 4.1) [START_REF] Gazor | Prediction in LMS-type Additive Algorithms for Smoothly Time Varying Environments[END_REF][START_REF] Lindbom | Tracking of Time-Varying Mobile Radio Channels-Part II : A Case Study[END_REF].

f d T = 10 -3 f d T = 10 -2 AR1-Kalman Or2-Kalman AR1-
Indeed, the authors of these papers have previously proved that PLL, or time-invariant CATL-based filters, can be interpreted, if a satisfactory dynamic model is available, as forms of KF in steady-state mode, with equivalent MSE asymptotic performance.

Secondly, we observe that the performance of the (well-tuned) AR1-Kalman, despite its complexity, does not reach the BRCB in case of slow to moderate channel variation (more notable for f d T = 10 -3 than for f d T = 10 -2 ). We can also incidentally remark an additional degradation (in dashed line), if the AR1-parameter is tuned by the standard correlation matching criterion as in [START_REF] Al-Naffouri | An EM-based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM[END_REF][START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF] (i.e. with ε = 0 in (A.3), Appendix A) instead of a minimum variance criterion. This last point corroborates the recent results established in [START_REF] Barbieri | On the ARMA Approximation for Frequency Channels Described by the Clarke Model with Applications to Kalman-based Receivers[END_REF][START_REF] Ghandour-Haidar | On the Use of first-order Autoregressive Modeling for Rayleigh Flat Fading Channel Estimation with Kalman filter[END_REF].

On the other hand, with a second-order loop (or an Or2-Kalman), the MSE becomes closer to the BCRB (obtained from [START_REF] Hijazi | Bayesian Cramer-Rao bounds for complex gain parameters estimation of slowly varying Rayleigh channel in OFDM systems[END_REF], see also [START_REF] Ros | Complex Amplitudes Tracking Loop for multipath slow fading channel estimation in OFDM systems[END_REF]-ch IV). This point reveals the advantage of a second-order loop versus a first-order loop (i.e. with µ 2 = 0) in slow to moderate fading scenarios. It allows a higher decrease in the MSE that is proportional to the4 5 power of the SNR (in full agreement with the theory (44)), versus the 2 3 power for the first-order algorithms ((47) and [START_REF] Ghandour-Haidar | On the Use of first-order Autoregressive Modeling for Rayleigh Flat Fading Channel Estimation with Kalman filter[END_REF]). Fig. 6 shows the evolution of the Bit Error Rate (BER) in the case of 4-QAM, 16-QAM and 64-QAM modulations for the previous channel estimators completed by a Zero-Forcing (ZF) frequency-domain equalizer. The channel frequency response is previously estimated from the CA estimates α(n) by Ĥ(n) = F α(n) . The BER results agree with the previous MSE results, but with a lower difference between the curves due to the decision process. Hence, the performance with our 2nd-order LS-CATL algorithm is the same in terms of BER as with the Or2-Kalman, and is slightly better (for low SNR regions) than with the AR1-Kalman (and then the 1st-order LS-CATL).

The BER performances are close to those found with a ZF equalizer using perfect channel knowledge.

BER comparison with other literature algorithms

Fig. 7 shows for f d T = 10 -3 the BER performances of the proposed LS-CATL algorithm, using a ZF equalizer and a 4-QAM modulation. Also shown are the performances of the previous AR1-Kalman and Or2-Kalman references, together with three estimators that have been suggested in the literature and discussed in the introduction: the "conventional" LS(FD)-LPI [START_REF] Hsieh | Channel estimation for OFDM systems based on combtype pilot arrangement in frequency selective fading channels[END_REF], the Kalman(FD)-LPI [START_REF] Chen | Kalman filter Channel estimator for OFDM systems in time and frequency-selective fading environment[END_REF] 4 , and the Kalman-EIG [START_REF] Sohail | An EM-based frequency domain channel estimation algorithm for multi-access OFDM systems[END_REF] 5 . Note that the latter requires the availability of the power-delay profile to perform "eigenvalue interpolation" of the channel, unlike the two previous algorithms that performed a "bind" LPI interpolation. It is, first of all, highly noteworthy that a "first- mit us to measure the gain when exploiting time-domain correlation, frequency-domain correlation, as well as knowledge of the delays-related information ("first-category") versus only frequency correlation (conventional).

When the conventional symbol by symbol LS(FD)-LPI method is extended into Kalman(FD)-LPI algorithm to improve the estimation of the channel at pilot frequency positions (before performing the LPI interpolation in frequency-domain), we can measure the increase in performance due to the use of the past symbols. The benefit of the time-filtering is mainly observed in low SNR regions, and more notable for the lower f d T , because of a stronger channel time-correlation. But, the resulting performance still remains far from that of the "first category" of algorithms, unless if N p = 64 pilot subcarriers are used (i.e. a distance L f = 2 between two pilot subcarriers). Hence, the availability or the non-availability of the delay-related information is an assumption that influences strongly the channel estimator performance (as discussed more in [START_REF] Ros | Complex Amplitudes Tracking Loop for multipath slow fading channel estimation in OFDM systems[END_REF] as well as the effect of an imperfect delay knowledge). We may also note that the BER obtained with the Kalman-EIG is almost the same as with the AR1-Kalman, for a comparable complexity (both algorithms use KF based on the same AR1 state-space model with N c = 6 dominant eigenvalues tracked in the Kalman-EIG, versus L = 6 paths CAs in the AR1-Kalman). In conclusion, among the algorithms in the "first category" regarding the asymptotic performance, the proposed LS-CATL algorithm is the one with the lowest complexity, as seen in section 3.3 and table 1.

Conclusion

A complex amplitude (CA) estimator of the channel paths over slow to moderate fading channels has been proposed and analyzed. It can be directly useful for either Data Aided or Data Directed single-carrier systems over flat fading channels. Applied to OFDM systems with a comb-type pilot sub-carrier arrangement, it belongs to the class of algorithms that perform the tracking of the CAs of a multipath channel from the information related to path delays. Therefore, it is assumed that an acquisition procedure has already been put into place to calculate path delays. The proposed algorithm is based on a 2nd-order recursive loop, that integrates an error signal created from the pilot-based LS estimates of the CAs. It allows the time-domain correlation of the equations (9)(B.2)-(B.4) coincide for one given path l with the steady-state equations of the KF in [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF] derived under the assumption of a dynamic IRW model for α l , but for a simplified single-carrier and single-path channel scenario. The vector (µ 1 , µ 2 ) in

1 Figure 2 :

 12 Figure 2: Transfer function L(z = e j2π f T ) versus normalized frequency f T for the 2nd-order LS-CATL (continuous line) with various normalized natural frequencies f n T = 3.10 -4 to = 3.10 -1 and a damping factor ζ = 1 2 , and for a 1st-order LS-CATL (dashed line) with various normalized cut-off frequencies f c T = 3.10 -4 to = 3.10 -1 .

2 ε (numerical integration) for different ζ σ 2 ε 5 0 2 Figure 3 :

 22523 Figure 3: Network of curves of global σ 2 ε = σ 2 εα + σ 2 εN (continuous line) versus f n / f d (for a fixed f d T = 1.10 -3 ) for the second-order LS-CATL with various damping factors ζ = 0.2,0.3,0.5,0.7, 1, 1.5, 2 computed numerically from (34) assuming Rayleigh-Jakes model (with σ 2 α = 1,L = 6), and from (35) assuming k 2 d = 1 and σ 2 N = 6,25.10 -2 (top of the figure) or σ 2 N = 6,25.10 -4 (bottom of the figure). Theoretical reference (dashed line) given from closed-form expressions (36)&(38) for σ 2 εN , and from (42) for σ 2 εα

3

  gives results obtained by numerical integration for σ 2 ε assuming a "Rayleigh-Jakes" model for the CA dynamic, and a (temporally uncorrelated) input loop noise with two different variances σ 2 N . It is shown that fixing ζ = 1 2 and varying f n can be a strategy to obtain the best minimum of σ 2 ε . Our objective now is to give some approximate closed-form expressions for σ 2 εα and σ 2 εN , especially for ζ = 1 2 , approximately.Static error variance σ 2 εN . using the whiteness of N l(n) with the PSD Γ N ( f ) = σ 2

2 ε 2 ε = σ 2 α

 222 (simulation), SNR = 0,20, or 40 dB σ

Figure 4 : 4 . 1 .

 441 Figure 4: Comparison between simulated and theoretical MSEs vs f n T for f d T = 10 -3 , ζ = 0.5, SNR = 0, 20, or 40 dB, N p = 16 for the proposed 2nd-order LS-CATL algorithm. Theoretical values are given from (36)&(38) for σ 2 εN , and from (42) for σ 2 εα .

  ) and 3(b) give the CATL parameters used for f d T = 10 -3 and f d T = 10 -2 , with N p = 16 pilot subcarriers.

2 Table 3 :

 23 for f d T = 10 -Loop parameters f n / f d (2nd-order) and f c / f d (1st-order), for N p = 16 and f d T = 10 -3 , 10 -2 .

Fig. 5 (

 5 Fig. 5(a) and Fig. 5(b) show the evolution of MSE versus SNR, respectively for f d T = 10 -3 and f d T = 10 -2 . First of all, the MSE of the proposed 2nd-order LS-CATL algorithm is very close to that obtained by the Or2-Kalman algorithm. Likewise, the MSE of the proposed 1st-order LS-CATL algorithm is very close to that obtained by the AR1-Kalman algorithm. It is, therefore, gratifying to verify that the reduced complexity proposed algorithm exhibits almost the same asymptotic variance as the reference Kalman algorithm, for a same model order. It was our desired objective as

  f d T = 10 -2

Figure 5 :

 5 Figure 5: MSE vs SNR for f d T = 10 -3 (a) and f d T = 10 -2 (b), with N p = 16

Figure 6 :Figure 7 :

 67 Figure 6: BER comparison for 4-QAM, 16-QAM or 64-QAM modulations, f d T = 10 -3 and N p = 8

( 9 )

 9 &(B.2) is the steady-state Kalman gain for this simplified situation. In other words, the more complex Or2-Kalman described in Appendix A-(A.6)-(A.10) would be reduced in steady-state mode to the CATL if the observation model (5) were simplified in y p(n) = α (n) + w p(n) , leading to the simple error signal v ǫ(n) = y p(n)α(n|n-1) .

  are assumed to be precomputed and stored, if the delays are invariant for a great number of OFDM symbols. Then, we just have N p × L multiplications for the LS estimate[START_REF] Komninakis | Multi-Input Multi-Output Fading Channel Tracking and Equalization Using Kalman Estimation[END_REF] used in the error detector[START_REF] Al-Naffouri | An EM-based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM[END_REF], plus 2L multiplications (or just L if µ 2 = 0) in the loop filter[START_REF] Raghavendra | Exploiting hopping pilots for parametric channel estimation in OFDM systems[END_REF]. Table1compares this complexity to reference KFs presented in Appendix A. It is noteworthy that the LS-CATL algorithm is computationally less demanding technique than Kalman filters, since the latter require inversion of matrices of size N p × N p for the Kalman gain matrix computation (A.8), plus multiplications of matrix with size N p × 2L, 2L × 2L, or N p × N p for the update of the filters coefficients (see (A.6)-(A.10)). The LS-CATL approach presents finally a linear complexity in terms of the number of pilot subcarriers N p (O(N p )) versus a cubic complexity (O(N 3 p )) for the reference Kalman algorithms.

		Number of complex multiplications per OFDM symbol
	2nd-order LS-CATL	N

  This approximation is still accurate for f ≈ f n for the special

case ζ ≈ 1 2 (see (40)). It results that the CA dynamic error variance σ 2 εα in (34) can finally be approximated (for f d < f n ≪ 1/T , and ζ ≈ 1 2 ) by

Table 4 :

 4 Parameters (β , γ) used (and related ε) for the AR1-Kalman and Or2-Kalman, for f d T = 10 -3 ,10 -2

	Kalman Or2-Kalman

L(z) is the same in[START_REF] Ros | Paths Complex Gain Tracking Algorithms for OFDM Receiver in Slowly-Varying Channels[END_REF] withβ d = k d 1-k d .µ 1, but differs slightly from the closed-loop transfer function of a

2nd-order digital PLL[START_REF] Shayan | All digital phase-locked loop: concepts, design and application[END_REF][START_REF] Patapoutian | On phase-locked loops and Kalman filters[END_REF], due to the additional branch in dashed line in Fig.1.

the per-subchannel KF has been adapted to our pilot scheme since[START_REF] Chen | Kalman filter Channel estimator for OFDM systems in time and frequency-selective fading environment[END_REF] considered a full block of pilots

corresponds to the initial (pilot-based) channel estimator in[START_REF] Sohail | An EM-based frequency domain channel estimation algorithm for multi-access OFDM systems[END_REF]

✩ Part of this work was presented in the conference ISCCSP 2010

channel to be exploited more simply than the Kalman-based methods, which require matrix inversion at each iteration. Simulation results show that the MSE performance of our 2nd-order algorithm is very close to that of a Kalman estimator based on a 2nd-order approximation of the actual channel. Moreover, our 2nd-order algorithm outperforms the more complex Kalman estimator when the latter is based only on a 1st-order Auto-Regressive model. This emphasizes the advantage of 2nd-order versus 1st-order methods in the case of slow to moderate fading variation ( f d T ≤ 10 -2 ).

We have given closed-form expressions of the optimal natural frequency of the loop, and the corresponding minimum MSE (assuming Rayleigh-Jakes channel). We have demonstrated that the MSE of our 2nd-order algorithm decreases proportionally to the 4 5 power of the SNR, and increases proportionally to the 4 5 power of the normalized Doppler frequency f d T . Moreover, BER comparison through simulation has shown that the proposed algorithm outperforms the basic conventional method based on LPI interpolation in the frequency-domain.

Appendix A. AR1-Kalman (review) and Or2-Kalman filters

We present two KFs as obvious benchmarks for our specific (parametric channel modeling-based) estimation problem defined in section 2.3. Since exact linear state evolution equation for the Jakes' process is not available, the flat fading CA dynamic has to be approximated in the perspective to use KF (without guarantee of optimality).

Let us first introduce the general dynamic model, which will be next declined into AR1 and Or2 models, to approximate the variation of one Jakes'process α l(n) by αl(n) :

where γ and β are two positive scalars with values lower or equal to 1, and u l(n) is zero-mean Gaussian complex circular with a variance σ 2 u l .

Appendix A.1. AR1-Kalman

The special case of AR1-model corresponds to β = 0 (no drift) and γ < 1. The specific equations of the AR1-Kalman applied to our OFDM parametric model can be found in [START_REF] Ros | Paths Complex Gain Tracking Algorithms for OFDM Receiver in Slowly-Varying Channels[END_REF]-section IV. We just want to briefly report here recent results about the choice of γ for the "Rayleigh-Jakes" channel estimation. For the AR1 model, the autocorrelation function of the approximated process is R

αl . Assuming the same variance for the approximated process and the true process (i.e. R (0) αl = σ 2 α l ), the variance of the state noise is directly fixed by the choice of the AR1-coefficient γ as σ 2

αl of the approximate process perfectly matches the Bessel auto-correlation function

α l of the true Jakes' process in (4) for lag k ∈ {0, 1} (or for k ∈ {0, 1, ..., p} for a model with order p). This choice corresponds to a correlation matching (CM) criterion, and is the most often used in the literature ( [START_REF] Al-Naffouri | An EM-based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM[END_REF][START_REF] Sohail | An EM-based frequency domain channel estimation algorithm for multi-access OFDM systems[END_REF][START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF][START_REF] Komninakis | Multi-Input Multi-Output Fading Channel Tracking and Equalization Using Kalman Estimation[END_REF][START_REF] Wang | On verifying the first-order Markovian assumption for a Rayleigh fading channel model[END_REF][START_REF] Banelli | Data-Aided Kalman Tracking for Channel Estimation in Doppler-Affected OFDM Systems[END_REF]). However, imposing the matching of the first two taps (R

α l and R

(1)

α l ) for p = 1 does not ensure a short distance between the two auto-correlation functions, and, even less, the minimum estimation variance of the AR1-Kalman. This is especially true for low f d T ≪ 1 (see [START_REF] Komninakis | Multi-Input Multi-Output Fading Channel Tracking and Equalization Using Kalman Estimation[END_REF], Fig. 1) where the taps for lags 1 and 0 have very close values (since J 0 (2π

, and then the exponential decay of the AR1 autocorrelation function R (k) αl = σ 2 α l • γ |k| is imposed so as to be too slow compared to the Bessel function decay, σ 2 α l • J 0 (2π f d T × k). Thereby, we consider possible lower values for the AR coefficient γ as in [START_REF] Barbieri | On the ARMA Approximation for Frequency Channels Described by the Clarke Model with Applications to Kalman-based Receivers[END_REF][START_REF] Ghandour-Haidar | On the Use of first-order Autoregressive Modeling for Rayleigh Flat Fading Channel Estimation with Kalman filter[END_REF][START_REF] Baddour | Autoregressive modeling for fading channel simulation[END_REF]:

where ε is a very small positive amount (≪ 1). Such a slight change can decrease strongly the estimation variance of the AR1-Kalman as proved recently in [START_REF] Barbieri | On the ARMA Approximation for Frequency Channels Described by the Clarke Model with Applications to Kalman-based Receivers[END_REF][START_REF] Ghandour-Haidar | On the Use of first-order Autoregressive Modeling for Rayleigh Flat Fading Channel Estimation with Kalman filter[END_REF].

Appendix A.2. Or2-Kalman

For the second-order model, the coefficients γ and β are non zero. The special case γ = β = 1 corresponds to the IRW model [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF], with just one parameter (σ 2 u l ) to be adjusted. But we can keep a more general second-order model with two coefficients γ and β to be adjusted with values lower and close to one. In this case, the state-noise variance will be fixed versus the 2 coefficients (β , γ) by σ 2

, where

αl is the variance of the drift δ l . In order to more easily tune the couple of coefficients in section 4 empirically, we use eq. (A.4) (derived from (A.1)&(A.2)) that gives γ wrt to β and the two correlation coefficients R

(1)

αl , the latter being fixed in (A.5) wrt f d T by CM criterion (with possible adjustment ε ≪ 1):

To design the KF, the multi-path multi-carrier problem and the CAs evolution have to be re-formulated in a state-space model. The state vector is a Measurement Update Equations:

where K (n) is the Kalman gain matrix (with size 2L×N p ) and U = diag 0, σ 2 u 1 , ..., 0, σ 2 u L .

Appendix B. Interpretation and Kalman-type formulation of the CATL structure

We define δ(n+1|n) and δ(n|n) as the a priori and a posteriori slope estimates of the CAs, forced to be equal and related to v Lag(n) by:

Then, the recursive equations of the CATL ( 8)-( 12) can be rewritten by ( 8


Wilhem Leconet

Christel Larbouret

Thierry Chardès

Gaëlle Thomas

Madeline Neiveyans

Muriel Busson

Marta Jarlier

Nina Radosevic-Robin

Martine Pugnière

Preclinical validation of AXL receptor

est destinée au dépôt

SupplementaryInformation

Materials and Methods

Generation of monoclonal antibodies

Balb/c mice were subcutaneously immunized with 10 grhAXL-EDC at day 0 in complete Freund adjuvant, 14 and 28 in incomplete Freund adjuvant. Three days after a last intravenous boost, spleen cells were fused with non-secreting P3X63Ag8.653 murinemyeloma cells using polyethylene glycol, as previously described (21) . Antibodies in hybridoma supernatants were screened by ELISA.

Hybridoma cells that secreted anti-AXL antibodies were cloned by limiting dilution and cryopreserved. Antibodies were purified on protein G-agarose (Sigma-Aldrich, St Louis, MO) columns, as recommended by the manufacturer, sterilized through 0.2 m filters, quantified and stored at -20°C in phosphate buffered saline (PBS).

Fluorescence-Activated Cell sorting (FACS)

Cells were trypsin-detached and incubated with 20µg/ml of purified anti-AXL mAbs in PBS/0.1%BSA on ice for 1.5h. After washing, cells were incubated with FITCconjugated anti-mouse IgGs (Fc specific) (Sigma-Aldrich) on ice in the dark for 1h.

Antibody binding to the cells was then quantified (minimum of 10000 events) using a Beckman-Coulter FC500 MPL Flow Cytometer.

ELISA

Maxisorb 96-well plates (Nunc, Paisley, UK) were coated with 200 ng/ml rhAXL, rhMER, rhTYRO3 or rmAxl in PBS at 4°C overnight and then saturated with PBS/0.1% BSA. Purified anti-AXL mAbs were added at 37°C for 2h. After washes, horseradish peroxidase-conjugated anti-mouse antibodies (Sigma-Aldrich) were added for 1h and o-Phenylenediaminedihydrochloride(Sigma-Aldrich) was used as substrate as recommended by the manufacturer. Absorbance was measured at 450 nm with an ELISA plate reader.

Phosphorylated AXL measurement by ELISA 1 x 10 6 cells were grown in 6-well plates for 24h and then switched to serum-free medium. Cells were then incubated, or not, with 100 g/ml purified anti-AXL mAbs for 1.5h, followed by 200 ng/ml rhGAS6 for 30 minutes. Cells were then lysed and protein concentration determined with the BC assay Protein Quantitation Kit (Interchim, Montluçon, France). The PathScan® Phospho-Axl (PanTyr) Sandwich ELISA Kit (Cell Signaling Technology) was used for quantifying phosphorylated AXL according to the manufacturer's instructions. Absorbance was measured at 450 nm with an ELISA reader.

Surface Plasmon Resonance analysis by BIACORE

Analysis of the interaction of the anti-AXL mAbs with AXL ECD and of their competition with GAS6 was performed on a BIACORE 3000 instrument (BIACORE AB, Uppsala, Sweden) at 25°C with HBS-EP (10 mM HEPES, 150 mMNaCl, pH 7.4, 0.005% surfactant P20 buffer) as running buffer. Recombinant human AXL-Fc fusion protein was immobilized on a CM5 sensor chip using an amine coupling kit (Biacore AB). The control flow cell was treated only with chemical reagents without protein.

For competition studies, a saturating concentration (625 nM) of rhGAS6 was injected onto the AXL-Fc-coated chip before injection of 666 nM anti-AXL mAbs. To validate our results, the reverse experiment (first injection of anti-AXL mAbs and then of rhGAS6) was carried out.