Nacerredine Lassami
email: lassami.nacerredine@gmail.com

Abdeldjalil Aïssa-El-Bey
email: abdeldjalil.aissaelbey@imt-atlantique.fr

Karim Abed-Meraim
email: karim.abed-meraim@univ-orleans.fr

Blind Joint MIMO Channel and Data Estimation based on Regularized ML

Keywords: Blind MIMO identification, deterministic ML, regularized estimation, joint channel and data estimation, sparsity, simplicity

The problem of blind joint FIR-MIMO channel and data estimation is addressed in this paper. Based on a regularized DML (Deterministic Maximum Likelihood) formulation of the problem, a bilinear approach is used in order to estimate jointly the channel impulse responses and the input data. This regularization is introduced as a penalty function added to the classical DML criterion representing the a priori information about the problem in order to enhance the accuracy of the estimation. Two types of priors information are considered for the transmitted data: the finite alphabet simplicity or the sparsity. The sparsity prior was also considered for channel impulse responses. The key advantage of the proposed criteria is their convexity when optimized alternatively over the channel and the input data. The proposed approach allows to improve further the estimation accuracy of such a blind estimation problem but suffers from a relatively high computational cost. Hence, a reduced complexity implementation of the latter has been proposed at the end of the paper, in an adaptive scheme for high dimensional or streaming data situations.

Introduction

Blind system identification (BSI) problems are receiving considerable interest from both signal processing and communications communities since accurate channel estimates are likely to become more challenging in future generation wireless systems. Furthermore, blind techniques are necessary in some communication systems where the synchronization between the receiver and the transmitter is not possible and training sequences are not exploitable. This can be due to security problems such is the case in military communication systems or to the impossibility to influence the input such as the problem of extracting the fetal electrocardiogram (fECG) or in speech separation. Blind techniques present a reduced need for overhead information which increases the bandwidth efficiency. Although, even if some training sequence exists, combining them with blind techniques, in a semi-blind scheme, often leads to improved performance [START_REF] Abuthinien | Joint maximum likelihood channel estimation and data detection for MIMO systems[END_REF][START_REF] Abuthinien | Semi-blind joint maximum likelihood channel estimation and data detection for MIMO systems[END_REF][START_REF] Palally | Particle swarm optimisation aided IEEE International Conference on Communications (ICC)[END_REF][4][5][START_REF] Prasad | Joint approximately sparse channel estimation and data detection in OFDM systems using sparse bayesian learning[END_REF][START_REF] Ishihara | Iterative frequency-domain joint channel estimation and data detection of faster-than-nyquist signaling[END_REF][START_REF] Hong | Semi-blind joint channel estimation and data detection on sphere manifold for MIMO with high-order QAM signaling[END_REF].

Recently, the problem of joint channel estimation and data detection has been the subject of much research and different solutions have been proposed depending on the considered application. For instance, in [5] the authors analyzed the performance of the data-aided channel estimation scheme in a multicell large antenna system and they can effectively suppress the contamination effect and achieve improved performance in large antenna systems. A solution to enhance the reliability of a single-input single-output (SISO) underwater acoustic communication system was proposed in [START_REF] Torrico | Joint channel estimation and data detection on commercially available underwater acoustic modems[END_REF] based on a maximum likelihood (ML) method for both channel estimation and data detection followed by a dataprojection step. In the case of SISO orthogonal frequency-division multiplexing (SISO-OFDM) model and under the assumption of sparse channel, a recursive algorithm was proposed in [START_REF] Prasad | Joint approximately sparse channel estimation and data detection in OFDM systems using sparse bayesian learning[END_REF] based on sparse Bayesian learning and Kalman filtering for the joint channel estimation and data detection. In [START_REF] Ishihara | Iterative frequency-domain joint channel estimation and data detection of faster-than-nyquist signaling[END_REF], an iterative frequency-domain joint channel estimation and data detection solution was proposed for the problem of faster-than-Nyquist signaling in a SISO system. A Bayes-optimal inference approach based on the bilinear generalized approximate message passing (BiG-AMP) algorithm was used in [4] to address the problem of joint channel and data estimation in a multiple-input multiple-output (MIMO) system. Recently, a low-complexity semi-blind scheme is proposed in [START_REF] Hong | Semi-blind joint channel estimation and data detection on sphere manifold for MIMO with high-order QAM signaling[END_REF] for joint channel estimation and data detection on sphere manifold MIMO systems with high-order quadrature amplitude modulation (QAM) signaling. It uses training symbols with zero-forcing detector to get an initial channel and data estimation, then, a Riemannian conjugate gradient method is used followed by another zero-forcing detector. The MIMO-OFDM system is considered in [START_REF] Yang | Joint adaptive blind channel estimation and data detection for MIMO-OFDM systems[END_REF],

where the authors proposed an adaptive blind scheme for joint channel and data estimation based on parallel factor decomposition (PARAFAC).

The joint channel estimation and data detection (data-aided) problem is generally formulated as a deterministic maximum likelihood (DML) estimation where the input signals are considered as part of the unknown parameters with the channel coefficients. The ML methods have the advantage of being efficient for high signal-to-noise ratio (SNR) [START_REF] Tong | Multichannel blind identification: from subspace to maximum likelihood methods[END_REF]. The ML methods usually cannot be obtained in closed form and require an optimization in the presence of local minima. In addition, the dimension of the problem increases with the sample size, which makes this approach not practical for large data size applications. However, ML approaches can be made very effective by including initialization procedure such as least squares channel estimate based on training symbols, the blind channel subspace method or other sub-optimal approaches. As major contributions to DML methods, we can cite the two-step maximum likelihood (TSML) [START_REF] Hua | Fast maximum likelihood for blind identification of multiple FIR channels[END_REF], the iterative quadratic maximum likelihood (IQML) [START_REF] Slock | Further results on blind identification and equalization of multiple FIR channels[END_REF] and its dual algorithm proposed in [START_REF] Feder | Algorithms for joint channel estimation and data recovery-application to equalization in underwater communications[END_REF] which considered single-input multiple-output (SIMO) finite impulse response (FIR) channels. Another DML method, the maximum likelihood block algorithm (MLBA), has been proposed in [START_REF] Gesbert | Blind least-squares approaches for joint data/channel estimation[END_REF] where for both the channels and the symbols, least squares estimation is performed in an alternating manner. Following the same formulation, the Maximum Likelihood Adaptive Algorithm (MLAA) is derived in [START_REF] Alberge | Adaptive solution for blind identification/equalization using deterministic maximum likelihood[END_REF] which presents reducedcomplexity in term of computation cost.

Another important feature of the considered DML formulation is the facility to introduce any further information about the channel impulse response or the input signal to the cost function. This priors depends on the considered system and adding them to the DML formulation as regularization will likely enhance the estimation accuracy. For instance, most of the digital communications are based on transmitted signals that belong to a finite-alphabet set. The finite alphabet property was first considered with DML formulation in [START_REF] Seshadri | Joint data and channel estimation using fast blind trellis search techniques[END_REF][START_REF] Ghosh | Maximum-likelihood blind equalization[END_REF][START_REF] Talwar | Blind estimation of multiple co-channel digital signals using an antenna array[END_REF]. Unfortunately, the convergence of such methods is often not guaranteed due to the increased number of the local minima caused by the incorporation of the finite alphabet property. The solution to such a problem is computationally expensive and can be achieved using the Viterbi-like algorithms. In order to reduce the computational complexity, Bayesian maximum a posteriori approaches were proposed such as in [4,[START_REF] Alberge | Blind identification/equalization using deterministic maximum likelihood and a partial prior on the input[END_REF]. Recently, the simplicity property was considered for the recovery of finite-alphabet signals as is the case in [START_REF] Hajji | Simplicity-based recovery of finite-alphabet signals for large-scale MIMO systems[END_REF] for large-scale MIMO systems. The sparsity prior of the transmitted signals was also considered in different applications such as in [START_REF] Zibulevsky | Blind source separation by sparse decomposition in a signal dictionary[END_REF][START_REF] Gribonval | Sparse component analysis[END_REF] for blind source separation when the sources are known to be sparse or can be sparsely represented. The sparsity of the channel impulse response can also be used with the DML formulation. In fact, it seems likely that in case of very long impulse response and sparse channel, most of the state-of-the-art methods perform poorly. Such sparse channels can be encountered in many communication applications including High-Definition television (HDTV) channels and underwater acoustic channels [START_REF] Aïssa-El-Bey | Blind and semi-blind sparse channel identification in MIMO OFDM systems[END_REF].

Paper contributions

In this paper, we aim to estimate blindly both the channel state information and the transmitted data in the case of a MIMO FIR channel (convolutive system). We propose a blind scheme based on a regularized DML formulation by considering priors on either or both the channel impulse response and the transmitted data. First, a blind second-order subspace method [START_REF] Abed-Meraim | A subspace algorithm for certain blind identification problems[END_REF] is used to estimate the channel followed by a resolving ambiguity step which is necessary in second-order method. We will use for the ambiguity removal, either the sparsity assumption in case of sparse channel prior or the independence assumption of the transmitted signals in case of non-sparse channel. The key advantage of the proposed regularized DML formulation is its convexity when optimized alternatively over the channel and the input data. Two regularized quadratic convex problems will be iteratively optimized to estimate the data in the first one and the channel in the second. Different priors are considered in our work: the finite alphabet simplicity, the sparsity of the transmitted data, or the sparsity of the channels finite responses. The proposed constrained ML-like approach allows to improve further the estimation accuracy of such a blind estimation problem but suffers from a relatively high computational cost. Hence, an adaptive version is also proposed to reduce the numerical complexity and to handle the case of time varying channel. The specific contributions of this paper include:

• Providing a unified convex regularized DML formulation for different types of priors: sparsity or simplicity of input signals and sparsity of channel impulse responses while considering the MIMO convolutive model. To the best of our knowledge, this is the first work that solves the joint channel and input data estimation in a fully blind context using simultaneously channel sparsity and input signal simplicity priors.

• Resolving the ambiguity of blind second-order subspace method in case of sparse channel impulse responses.

• A reduced complexity implementation of the proposed constrained MLlike approach, proposed in an adaptive scheme for high dimensional or streaming data situations.

Data model

Our focus is on the MIMO channels case in a blind identification context.

We aim to estimate jointly the channel state information and the transmitted data using only the observation data and some additional prior information.

Considering a mathematical model where the input and the output are both discrete but we have access only to the output. The system is driven by N t input sequences s 1 (t), ..., s Nt (t), and yields N r output sequences x 1 (t), ..., x Nr (t) at time t. We assume that the discrete channel between the N t transmit antennas and the N r receive antennas are modeled as a N r × N t FIR filter with L as the upper bound on the orders of these channels i.e. H = [H T (0), ..., H T (L)] T with

H(l) =      h 11 (l) . . . h 1Nt (l) . . . . . . h Nr1 (l) . . . h NrNt (l)     
The system can be described by:

x(t) = L k=0 H(k)s(t -k) + n(t) = [H(z)]s(t) + n(t) (1)
where

[H(z)] = L k=0 H(k)z -k (2)
and n(t) is an additive N r -dimensional white noise, independent from the symbol sequences with E[n(t)n T (t)] = σ 2 I Nr , where σ 2 is the unknown noise power.

We denote stacked observations over a window of length N by the vector

x N (t) = [x T (t), x T (t -1), ..., x T (t -N + 1)] T . The following linear model holds:

x N (t) = T (H)s N (t) + n N (t) (3)
with s N (t) = [s T (t), s T (t -1), ..., s T (t -N -L)] T and n N (t) = [n T (t), ..., n T (t -

N + 1)] T . The (N N r ) × ((N + L + 1)N t ) block-Toeplitz matrix T (H) associated
with the filter H is given by:

T (H) =      H(0) . . . H(L) 0 . . . . . . 0 H(0) . . . H(L)      (4)
From now on, we make the following important assumptions:

• H(z) is irreducible (Rank(H(z)) = N t ; ∀z) and H(L) is of full column rank

• The channels have a known maximum order L.

• Number of sensors is strictly greater than the number of sources N r > N t

• Large enough observation Window N > (L + 1)N t Under this assumptions, T (H) is a full rank column matrix and the channel matrix H is identifiable using second order statistics up to a N t × N t constant full rank matrix [START_REF] Abed-Meraim | A subspace algorithm for certain blind identification problems[END_REF].

Blind deterministic maximum likelihood DML estimation approach

The DML approach assumes no statistical model for the input sequences.

In other words, both the channel matrix H and the input source vectors s i (t) i = 1, ..., N t are parameters to be estimated. Considering the MIMO FIR model expressed in above, the DML problem can be stated as: given the observation

x N (t), we want to estimate:

Ĥ, ŝN (t) = arg max f x N ; H, s N (5)
where f x N ; H, s N is the density function of the observation vectors parameterized by both the channel matrix H and the input sources vector s N . In the case of zero-mean Gaussian noise with covariance σ 2 I, the above DML estimator lead to the nonlinear least squares optimization:

Ĥ, ŝN (t) = arg min

T (H),s N (t) x N (t) -T (H)s N (t) 2 2 (6)
The above function can also be written as :

Ĥ, ŝN (t) = arg min T (H),s N (t) x N (t) -F s N (t) h 2 2 (7)
where h = vec(H) is the vectorized version of the matrix H. The operator F . transforms a vector s N (t) into an N N r × (L + 1)N t N r matrix, in such a way that:

F s N (t) h = T (H)s N (t). ( 8
)
Under the considered assumptions, T (H) is a full rank column matrix and the variables H, s N (t) are identifiable up to a N t × N t constant full rank matrix [START_REF] Abed-Meraim | A subspace algorithm for certain blind identification problems[END_REF] which we will denote Q. The identifiability condition for the DML approach is the same as that for the deterministic second-order subspace methods. The reason is that, when the noise is Gaussian (assumed true in DML), all information about the channel in the likelihood function resides in the secondorder moments of the observations. Readers are referred to [START_REF] Abed-Meraim | A subspace algorithm for certain blind identification problems[END_REF] for a more detailed discussion about the identifiability of the MIMO FIR channel with the noise subspace method.

Resolving the ambiguity caused by the matrix Q in second-order based blind channel estimation of MIMO-FIR channels is equivalent to instantaneous Blind Source Separation BSS problem [START_REF] Visuri | Resolving ambiguities in subspace-based blind receiver for MIMO channels[END_REF]. Hence, the indeterminacy can be reduced to a complex-valued diagonal matrix and a real-valued permutation matrix.

The blind DML criterion as stated in Eq. ( 6) and Eq. ( 7) is a complicated and a non-convex optimization problem when estimating jointly both parameters H, s N (t) . However, it is convex for each unknown parameter. Hence, Eq. ( 6)

and Eq. ( 7) can be resolved by alternatively optimizing over H and s N (t). This idea was proposed in many previous papers [START_REF] Slock | Further results on blind identification and equalization of multiple FIR channels[END_REF][START_REF] Gesbert | Blind least-squares approaches for joint data/channel estimation[END_REF][START_REF] Alberge | Adaptive solution for blind identification/equalization using deterministic maximum likelihood[END_REF][START_REF] Alberge | Blind identification/equalization using deterministic maximum likelihood and a partial prior on the input[END_REF] but only in the SIMO case.

Regularized blind deterministic maximum likelihood DML estimation approach

In the case where the system is not time-varying and if the data sequence is long enough so that a reliable statistical model can be built, then, a statistical maximum likelihood SML method should be used, since in that case, the statistical method outperforms the deterministic one in terms of estimation accuracy. However, in a fast fading environment, data related to a given channel are not numerous which makes a reliable statistical estimation less effective.

In such a situation, the symbols are assumed arbitrary and a DML method is used instead. The regularized DML approach is a trade-off between DML and SML approaches, where we take advantage of some prior information about input data or/and transmission channels to enhance the DML results. Next, we present different types of priors and how to integer them into the DML criterion.

Simplicity and the finite alphabet property

The finite alphabet property in digital communication signals was considered in [START_REF] Seshadri | Joint data and channel estimation using fast blind trellis search techniques[END_REF][START_REF] Ghosh | Maximum-likelihood blind equalization[END_REF][START_REF] Talwar | Blind estimation of multiple co-channel digital signals using an antenna array[END_REF] with the DML approach. In this case, an alternating optimization is used to minimize the criterion given by:

ŝN (t) = arg min s N (t)∈S x N (t) -T (H)s N (t) 2 2 (9)
Ĥ = arg min h x N (t) -F s N (t) h 2 2 (10)
where s N (t) belongs to a complex-valued finite alphabet S. It can be decomposed from its real and imaginary parts as

s N (t) = R(s N (t)) + jI(s N (t)) where (R(s N (t)), I(s N (t))) ∈ A Nt(N +L) × A Nt(N +L) and A = {α 1 , ..., α m } is a real- valued finite alphabet of cardinal m.
The optimization in Eq. ( 10) is a linearleast squares problem whereas the optimization in Eq. ( 9) is computationally expensive and can be achieved by using the Viterbi algorithm. The convergence of such approaches is not guaranteed due to the numerous local minima induced by the discrete set constraint. We propose to replace this constraint by the signal simplicity. Simplicity property was first introduced in [START_REF] Donoho | Counting faces of randomly projected polytopes when the projection radically lowers dimension[END_REF] where we say that a signal is simple if most of its elements are equal to the extremes of the finite alphabet. Recently, the authors in [START_REF] Hajji | Simplicity-based recovery of finite-alphabet signals for large-scale MIMO systems[END_REF] proposed a simplicity-based detector for finite alphabet source separation in both determined and underdetermined large-scale MIMO systems. They relaxed the problem of the finite alphabet constraint as stated in Eq. ( 9) to convex box constraint depending only on the constellation extremes {α 1 , α m } (e.g. {-1, +1} for BPSK and QPSK modulations, or {-3, +3} for 16-QAM modulation). The new optimization problem is given by: r = arg min

r x N (t) -T (H)B α r 2 2 subject to B 1 r = 1 2N , r ≥ 0 (11)
where s N (t) = B α r. The matrices B α and B 1 are defined by:

B α = I 2Nt(N +L) ⊗ [α 1 , α m ] and B 1 = I 2Nt(N +L) ⊗ [1, 1]
where (. ⊗ .) stands for the Kronecker product. The underlined notation is reserved to the complex-real transformation where T (H), x N (t) and s N (t) are given by

T (H) =   R(T (H)) -I(T (H)) I(T (H)) R(T (H))   x N (t) =   R(x N (t)) I(x N (t))   s N (t) =   R(s N (t)) I(s N (t))  
In addition to the convexity of this formulation, the computation cost of the resulting detector does not depend on the constellation size [START_REF] Hajji | Simplicity-based recovery of finite-alphabet signals for large-scale MIMO systems[END_REF]. The simplicity is used generally in case of modulations where the constellation is shaped as a square box such in the case of BPSK, QPSK and M-QAM modulations where M is a square number (e.g., [START_REF] Alberge | Adaptive solution for blind identification/equalization using deterministic maximum likelihood[END_REF]-QAM, 64-QAM). Note that the performance of the simplicity approach depends on the number of the points in the square box defined by the boundary of the constellation.

Sparsity of the input signals

The sparsity property of the input signals was introduced earlier as a contrast function for blind source separation [START_REF] Zibulevsky | Blind source separation by sparse decomposition in a signal dictionary[END_REF]. In this paper, the sparsity is introduced as a regularization constraint to Eq. ( 6) in order to enhance the estimation and improve the robustness to outliers. If the sources are sparse, the regularization can be carried out directly in the time domain by considering the least absolute shrinkage and selection (LASSO) problem [START_REF] Hastie | Statistical Learning with Sparsity: The LASSO and Generalizations[END_REF] (or basis pursuit equivalent problem [START_REF] Hastie | Statistical Learning with Sparsity: The LASSO and Generalizations[END_REF]). In this case, an alternating optimization is used to minimize the criterion given by:

ŝN (t) = arg min s N (t) x N (t) -T (H)s N (t) 2 2 + λ s s N (t) 1 (12) Ĥ = arg min h x N (t) -F s N (t) h 2 2 ( 13
)
where λ s is a weighting parameter that controls the trade-off between approximation error and sparsity level of the input signals.

This formulation can also be interpreted as a Maximum a Posteriori (MAP) approach where the input signals s N are estimated using its conditional probability distribution as follows:

ŝN =arg max s N f x Nr |s N f s N f x Nr |s N f s N ds N (14)
=arg max

s N f x Nr |s N f s N (15)
Generally speaking, the MAP allows us to exploit prior information about the desired parameter. Hence, one needs to a priori know the probability distribution function f (s N ) of the input signals vector. This a priori depends on the application context and its physical environment. In our case, it is the transmitted vector sparsity that we model by considering the input signals components i.i.d. as the Generalized Gaussian Distribution (GGD) given by:

f (s N ) = p 2βΓ( 1 p ) -Nt(N +L+1) exp - s N p p β p (16)
where β > 0 is a scale parameter, 0 < p ≤ 1 and Γ(z) = ∞ 0 t z-1 e -t dt, z > 0 is the Gamma function. Using this pdf, one increases the chances to get signal coefficients close to zero. Next, we will consider the special case of p = 1 which is equivalent to the Laplace prior distribution because of the convexity of the

1 -norm.
The problem in Eq. ( 12) is convex but has no closed-form solution. The Least Angle Regression (LARS) [START_REF] Efron | Least angle regression[END_REF]

with s N (t) = Φr.

Sparsity of the channel impulse response

The sparsity of the channel impulse response was first studied in the case of SIMO systems, then extended to the MIMO, e.g. [START_REF] Aïssa-El-Bey | Blind and semi-blind sparse channel identification in MIMO OFDM systems[END_REF]. In order to exploit the sparsity a priori information of the channel impulse response, we introduce an additional cost function based on the GGD model of channel coefficients in the same manner as we did previously in case of sparse input signals. Under the assumption that all the component of H are i.i.d, the GGD model is expressed in the same way as in Eq. ( 16). This model enhances the sparsity of the channel H. Taking the logarithm of the a posteriori estimator leads to the objective function:

Ĥ = arg min h x N (t) -F s N (t) h 2 2 + λ h h q q ( 18
)
where λ h is a weighting parameter that controls the trade-off between approximation error and sparsity represented by the q -norm of the channel impulse response. This function has a LASSO-like formulation for the case q = 1 and can be solved using the LARS method.

Actually, representing the sparsity of the source signals or the channel coefficients with the 1 -norm minimization is suitable due to the convexity property.

However, another heuristic penalization function (generally non-convex) was used in the literature to enhance the sparsity such as the Reweighted-1 [START_REF] Candès | Enhancing sparsity by reweighted 1 minimization[END_REF] criterion, which generally out performs the 1 -based criterion. The cost function, in this case, is given by:

Ĥ = arg min x N (t) -F s N (t) h 2 2 + λ h NrNt(L+1) i=1 log |h(i)| + ( 19
)
where > 0 is relatively small positive constant. Both cost functions in Eq. [START_REF] Ghosh | Maximum-likelihood blind equalization[END_REF] and Eq. ( 19) are optimized under the constraint H(1, j) = 1 for j = 1, ..., N t to avoid the trivial null solutions.

Initialization and resolving ambiguities in subspace-based blind identification of MIMO channels

As we stated before, ML methods solutions usually cannot be obtained in closed-form and require optimization in the presence of local minima. The dimension of the problem to optimize depends on the sample size which is problematic in case of large data size applications. As a solution, generally ML approaches can be very effective if they are initialized by some suboptimal procedure. In our case, we have chosen the MIMO second-order based subspace method [START_REF] Abed-Meraim | A subspace algorithm for certain blind identification problems[END_REF] because of its nice convergence properties.

The choice of the subspace method as an initialization procedure will lead to another problem in our formulation which concerns the ambiguity of the blind identification MIMO problem itself. In the MIMO case (N t > 1), the channel matrix H is estimated up to a N t × N t full rank matrix Q i.e.

H = ĤSS Q ( 20
)
where ĤSS is the estimated channel with the second order subspace method [START_REF] Abed-Meraim | A subspace algorithm for certain blind identification problems[END_REF].

Hence, the input signal is also estimated up to full rank matrix Q-1 expressed as:

s N (t) = Q-1 ŝN (t) SS (21)
where

T (H) = T ( ĤSS ) Q (22) and Q
=         Q 0 . . . 0 0 Q . . . 0 . . . . . . 0 0 0 . . . Q         (23)
Hence, we can not start the iterative optimization of Eq. ( 6) or any other regularized without resolving this ambiguity. This ambiguity can not be resolved without using-higher order methods or using additional information about the system. In a total blind scheme, the best we can aim for, is to estimate the channel matrix H up to a scaling and permutation matrices i.e. Ĥ is a solution if:

Ĥ = HPΛ ( 24
)
where P is a permutation matrix and Λ is a non-singular complex diagonal matrix.

The authors in [START_REF] Visuri | Resolving ambiguities in subspace-based blind receiver for MIMO channels[END_REF] show that resolving this ambiguity can be treated as an instantaneous Blind Source Separation BSS problem and they solve it using independent component analysis (ICA) under the assumption that transmitted sequences are statistically independent and non-Gaussian (which is generally true for communication sources). Hence, the indeterminacy can be reduced to a complex-valued diagonal scaling matrix and a permutation matrix.

In the case of sparse MIMO channel H, the problem is similar to the sparse principal subspace estimation problem discussed in our previous work [START_REF] Lassami | Low cost subspace tracking algorithms for sparse systems[END_REF]. In this case, the subspace is represented by the estimation ĤSS and we search for the rotation matrix Q that leads to the sparse channel H (i.e. H = ĤSS Q).

In a more recent result [START_REF] Lassami | Low cost sparse subspace tracking algorithms[END_REF], we have discussed the conditions to have a unique solution that corresponds to the sparse MIMO channel H. Without going deep into details, our results stated that if we have high level of sparsity of the MIMO channel H and a large number of receivers ((L + 1)N r > CN t log(N t )

with C is a positive constant related to the sparsity of the channel), then we can estimate the sparse channel H up to complex diagonal scaling matrix and permutation matrix. If these conditions are satisfied, we can use iteratively the second step of the algorithm SS-FAPI [START_REF] Lassami | Low cost sparse subspace tracking algorithms[END_REF] (System matrix Sparsity based on Fast Approximated Power Iterations) in order to estimate the sparse channel matrix H from the subspace channel method estimation ĤSS . The SS-FAPI algorithm is an adaptive algorithm based on a two step approach, where the first one uses the FAPI algorithm [START_REF] Badeau | Fast approximated power iteration subspace tracking[END_REF] for the adaptive extraction of an orthonormal basis of the principal subspace equivalent to Ĥt SS (changes over the time t). Then, an estimation of the desired sparse weight matrix H t is done in the second step using the natural gradient scheme. Taking iteratively the second Algorithm 1 Regularized blind DML joint estimation Requires: x N (t), regularization parameters and number of iterations N iter .

Ensures: Ĥ and ŝN (t).

1: Find ĤSS using MIMO subspace method [START_REF] Aïssa-El-Bey | Blind and semi-blind sparse channel identification in MIMO OFDM systems[END_REF] 2: Resolve the ambiguity of the subspace method by using either:

Sparsity of H with the second step of SS-FAPI.

Independence of the transmitted signals with ICA methods. Resolve Eq. ( 12) or Eq. ( 17) depending on the sparse domain if Channel sparsity prior then Resolve Eq. [START_REF] Yang | Joint adaptive blind channel estimation and data detection for MIMO-OFDM systems[END_REF] 13: end if 14: end for step of SS-FAPI is equivalent to fixing the Ĥt SS over t and looking for the best Q to get the sparset channel matrix H.

Finally, the resulting scaling and permutation indeterminacy is inherent to the blind identification problem and requires additional information (semi-blind) in order to be resolved. In our case, we generalize the same assumptions used in the SIMO case such as having the first row of H equal to ones and sort its columns depending on their 2 -norm. The full regularized blind DML approach is summarized in Algorithm 1.

Complexity analysis

Note that we consider here the dominant costs and hence, the computational complexity of the subspace method and the ambiguity resolving step (steps 1-2 of Algorithm 1) are disregarded. Steps 3-14 of algorithm 1 are based on convex optimization problems. In order to solve these convex problems, we have used the CVX MATLAB Toolbox which is based on the interior point method whose complexity is a function of the length of the unknown vector and the number of constraints [START_REF] Grant | CVX: Matlab software for disciplined convex programming[END_REF]. It is shown in [START_REF] Boyd | Convex Optimization[END_REF] that interior-point methods can solve a convex optimization problem defined over R n1 and subject to n 2 constraints in a number of steps or iterations that is almost always in the range between 10 and 100. Each step requires max(n 3 1 , n 2 1 n 2 , F ) operations, where F is the cost of evaluating the first and second derivatives of the objective and constraint functions. In our case, this is reduced to O(N 3 ) for all the cases depending on the considered prior in Algorithm 1 (recall that N is bigger than L, N r and N t ). This complexity can be reduced further, according to [START_REF] Boyd | Convex Optimization[END_REF], if the problem has some structure which is the case of sparse matrices T (H) and F s N (t) .

Adaptive regularized blind DML estimation approach

In real applications, channels are usually time-varying, which means that we need to develop an adaptive solution where processing is faster compared to acquisition. We provide here an adaptive solution based on the proposed regularized DML approach. First, the subspace MIMO method is used to estimate the channel response of the first observed data block. Then, an iterative estimation of the transmitted data and the channel response is done using the appropriate regularized DML approach depending on the considered case (simplicity, signal sparsity, or channel sparsity). The processing is done block by block. These blocks of size Ñ need to respect identifiability conditions and can be chosen with some overlapping. To enhance the processing speed, we propose to use the alternating direction method of multipliers (ADMM) algorithm [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF] for the optimization of the convex regularized objective functions of the DML approach. The ADMM algorithm can solve convex optimization problems by breaking them into smaller problems, each of which is then easier to handle, especially when we use proximal operators. We have chosen to use ADMM algorithm because it is less time-consuming compared to interior point solutions.

Algorithm 2 Adaptive regularized blind DML joint estimation

Requires: x Ñ (t) the t th block of size Ñ , regularization parameters and number of iterations N iter .

Ensures: Ĥ(t) and ŝ Ñ (t).

1: if t = 0 then Find ĤSS (0) using MIMO subspace method [START_REF] Aïssa-El-Bey | Blind and semi-blind sparse channel identification in MIMO OFDM systems[END_REF] Resolve the ambiguity of the subspace method by using either:

Sparsity of H with the second step of SS-FAPI.

Independence of the transmitted signals with ICA methods. 2: end if 3: Apply steps 3 → 14 of Algorithm 1 on x Ñ (t) while taking as initialization Ĥ(t -1) to get Ĥ(t) and ŝ Ñ (t)

More details about the rate of convergence and how to use ADMM for solving convex problems such as LASSO can be found in [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF]. The proposed adaptive algorithm is summarized in Algorithm 2 Table.

Simulation results

To assess the performance of the proposed solutions, we consider the MIMO problem with N t transmitters and N r receivers and the channels are represented by a polynomial transfer function of maximum degree L. The channels' impulse responses are generated randomly according to a Gaussian distribution for the non-sparse case. Sparse channels impulse responses are generated according to the Bernoulli-Gaussian distribution. In our simulation, we have used a sparsity level of 30%, which means that 30% of vector h entries are non-zero.

The input signals are a random binary sequences and the additive white Gaussian noise has a variance σ 2 chosen according to the target signal to noise ratio SN R = 10 log(

h 2 2 σ 2 ).
The used performance factor is the normalized mean-square error (NMSE) criterion (generalized from the SIMO case to the MIMO) which was used in many previous work such as in [START_REF] Aïssa-El-Bey | Blind and semi-blind sparse channel identification in MIMO OFDM systems[END_REF]. In our case, due to the inherent scaling indeterminacy of the full blind scheme, we have normalized the entries of the first row of H to be equal to one. Hence, we can use directly the standard NMSE (i.e. without scale ambiguity removal) defined as:

N M SE = 1 K K k=1 ĥk -h 2 h 2 (25)
where ĥk = vec( Ĥk ) is the estimated channel vector corresponding to the k th Monte-Carlo simulation and K = 100 is the number of Monte-Carlo runs. All the proposed criteria are convex with respect to the channel or the input data separately. Hence, for alternative optimization, we have used the Matlab CVX toolbox [START_REF] Grant | CVX: Matlab software for disciplined convex programming[END_REF].

In the first simulation, we consider a real-valued case with a Gaussian noise, a real non-sparse channel with N t = 3, N r = 8 and L = 10. We consider a BPSK constellation which will be used for the simplicity prior with T = 1000 received symbols. We have used as performance factors the NMSE of the estimated channels and the Bit Error Rate (BER) of the estimated input signals. The proposed regularized DML method which in this case considers the simplicity prior (BPSK constellation) is compared to the initialization subspace method, the Least-Squares LS solution (no regularization) and the oracle solution. In the latter, the exact input signal is used to estimate the channel matrix and the exact channel matrix is used to estimate the transmitted signal. We have considered one iteration of Algorithm 1 for the regularized DML (N iter = 1) and two iterations for the LS solutions.

It is clear in Fig. 1 that using the simplicity prior in the DML formulation enhances the performance in terms of NMSE and BER. Furthermore, we can reach the oracle performance for a certain SNR level (above 13dB). Same observations can be made in Fig. 2 where we have considered QPSK modulation, circularly symmetric complex Gaussian noise and a complex non-sparse channel with (N t = 3, N r = 8, L = 10) and T = 1000 symbols. However, compared to the BPSK case, we require a much higher SNR level to see the impact of the proposed approach.

In both previous simulations, we have considered only one iteration in our regularized DML approach. However, we can use multiple iterations to have better performance. Fig. 3 shows the results of using 1, 2, 5 and 10 iterations of the DML approach based on the simplicity prior for the case of BPSK constellation, (N t = 3, N r = 8, L = 10) and T = 1000 symbols. It is clear that increasing the number of iterations improves the performance, especially for the first ones.

In terms of NMSE, we can observe that the amount of SNR needed to reach the oracle performance is going lower with the increase of the iterations number.

In terms of BER, the same can be noted, in addition to outperforming the LS channel matrix H over the total number of elements) on the performance of the proposed solution, we consider the parameters N r = 5; N t = 2, L = 2, T = 400, QPSK modulation and SNR = 12 dB. Figure 6 shows the NMSE and BER versus the sparsity level ratio changing from 0.1 to 0.9. We have kept the LASSO parameter fixed at λ h = 0.5 for all the sparsity levels which explains that the performance is better for certain sparsity levels (mainly between 0.3 and 0.6) and severely deteriorates for other levels. Note that the important choice of the LASSO parameter λ h depends on the sparsity level and the SNR and its optimal tuning remains an open problem. Most of the works in the literature consider the semi-blind case and the instantaneous channel model due to its relation with the OFDM formulation.

Even though the contexts are different (i.e. semi-blind instead of blind and instantaneous channel instead of convolutive channel), we compare our solution with the recent work proposed in [START_REF] Hong | Semi-blind joint channel estimation and data detection on sphere manifold for MIMO with high-order QAM signaling[END_REF] in the particular case of instantaneous (no memory) channel. We consider a non-sparse channel with N t = 4, N r = 10 and L = 0. We consider a QPSK constellation which will be used for the simplicity prior with T = 250 received symbols. The solution in [START_REF] Hong | Semi-blind joint channel estimation and data detection on sphere manifold for MIMO with high-order QAM signaling[END_REF] is semi-blind and needs some initial training data. Hence, we have chosen, for each user, the first N t + 2 = 6 symbols as training sequence. Figure 7 shows that the semiblind method labeled as ref. [START_REF] Hong | Semi-blind joint channel estimation and data detection on sphere manifold for MIMO with high-order QAM signaling[END_REF] is slightly better than our solution. However, this small performance difference is admissible since our approach is fully blind compared to ref. [START_REF] Hong | Semi-blind joint channel estimation and data detection on sphere manifold for MIMO with high-order QAM signaling[END_REF]. Next, we consider the sparse channel case with parameters N t = 4, N r = 15, L = 0, T = 250 and QPSK constellation. Figure 8 shows how the performance of the semi-blind method in ref. [START_REF] Hong | Semi-blind joint channel estimation and data detection on sphere manifold for MIMO with high-order QAM signaling[END_REF] deteriorates in the case of sparse channel as compared to the proposed one, especially when we are combining the data simplicity and the channel sparsity priors. In the case of sparse signals, we will use the mean squared error (MSE)

M SE = 1 T E s N -ŝN 2
2 between the original signal and the estimated one as a performance factor. Fig. 9 shows the improved performance of the proposed approach in case of sparse a priori on the input signals compared to the leastsquares method with (N t = 3, N r = 8, L = 10) and T = 1000.

In the next simulation, in addition to the sparsity of the input signals, we consider longer L = 20 and sparse channels with 30% of non-zeros elements.

Hence, we will use the sparse regularization DML approach on both the channels and the signals. proposed procedure to the oracle cases, where the channel matrix H or the transmitted data S are known with or without regularization. We compare also to the case where we use the subspace method as initialization to each block followed by iterative least-squares without regularization. In the previous simulations, we have chosen ad-hoc parameters λ s and λ h .

This choice has an important influence on the performance of the proposed formulation. Choosing the "best" LASSO regularization parameter which depends on the sparsity level and the SNR, is an open problem. Some solutions can be adopted from the literature such as those presented in [START_REF] Hastie | Statistical Learning with Sparsity: The LASSO and Generalizations[END_REF].

Conclusion

In this paper, we have considered the problem of blind identification of MIMO FIR systems. We have presented a bilinear approach based on the regularized DML formation of the problem. This formulation has the advantage of alternatively estimate the channel impulse responses and the transmitted data while adding the a priori information about the problem as a regularization penalty. Different a priori are considered: the finite alphabet simplicity or the sparsity of the transmitted data, the sparsity of the channels finite responses. As an initialization of our blind framework, we have proposed to use the subspace blind identification method followed by a step to resolve its full rank matrix ambiguity for both sparse and non-sparse channels. Then, an iterative convex optimization is applied over the channel and the transmitted data. An adaptive version is also proposed to handle the case of time varying channel. Simulations results showed that the proposed method has improved the estimation accuracy which reaches the same level as the oracle solutions above a certain SNR. [5] J. Ma, L. Ping, Data-aided channel estimation in large antenna systems, in: 2014

2 + λ r r 1

 21 is a less greedy version of traditional forward selection methods for model selection problems and solves the LASSO problem efficiently. In the case where the input signals are not sparse in the time domain, we can use sparse representations methods in order to transform the signals into a dictionary where they are more sparse. For instance, speech signals have a more sparse representations in the time-frequency domain than in the time domain, therefore, the Short-time Fourier transform (STFT) transform is used in this case. If we denote by Φ the considered transformation (dictionary) matrix, then the problem in Eq. (12) can be rewritten as: r = arg min r x N (t) -T (H)Φr2

3 :

 3 for k = 1 → N iter do 4: if Signal simplicity prior then 5:Resolve Eq. (11)6:else if Signal sparsity prior then 7:

Figure 1 :

 1 Figure 1: NMSE in dB and BER performance of the regularized-DML versus SNR for BPSK modulation and (Nt = 3, Nr = 8, L = 10) and T = 1000.

Figure 2 :

 2 Figure 2: NMSE in dB and BER performance of the regularized-DML versus SNR for QPSK modulation and (Nt = 3, Nr = 8, L = 10) and T = 1000.

Figure 4

 4 Figure4shows the evolution of the NMSE and BER for different values of SNR (5, 10 and 15 dB) versus the number of receivers N r . Increasing the number of receivers N r will result in a significant performance gain especially at low SNR levels. However, if the number of receivers is very large, one has to use a wider data block (greater T ) to maintain the estimation accuracy of such large dimensional systems.

Figure 3 : 1 BERFigure 4 :

 314 Figure 3: NMSE in dB and BER performance of the regularized-DML with different number of iterations versus SNR for BPSK modulation and (Nt = 3, Nr = 8, L = 10) and T = 1000.

Figure 5 :

 5 Figure 5: NMSE in dB and BER performance of the regularized-DML versus SNR for BPSK modulation and sparse channel with (Nt = 3, Nr = 8, L = 20) and T = 1000.

Figure 6 :

 6 Figure 6: NMSE in dB and BER performance of the regularized-DML versus sparsity level ratio for QPSK modulation and sparse channel with (Nt = 2, Nr = 5, L = 2) and T = 400.

Figure 7 :

 7 Figure 7: NMSE in dB BER performance of the regularized-DML versus SNR for QPSK modulation with (Nt = 4, Nr = 10, L = 0) and T = 250.

Figure 8 :

 8 Figure 8: NMSE in dB BER performance of the regularized-DML versus SNR for QPSK modulation and sparse channel with (Nt = 4, Nr = 20, L = 0) and T = 250.

Fig. 10 illustrates 1 MSEFigure 9 :

 1019 Figure 9: NMSE in dB and MSE performance of the regularized-DML versus SNR for sparse signals with (Nt = 3, Nr = 8, L = 10) and T = 1000.

Fig. 11 confirms

 11 the superiority of the proposed approach and how its performance approaches the oracle performance for both SN R = 8dB and SN R = 12dB. It shows also how the sudden change in the channel impacts the performance. One can reduce the time reaction of the proposed adaptive approach by combining the previous channel estimation with a subspace-based new estimation at the start of each block. This will enhance the tracking capability but increase the complexity (apply subspace method at each block). The use of ADMM optimization technique reduce considerably the calculation time compared to using CVX toolbox. For instance, CVX toolbox-based solution needs on average 10s to solve the problem compared to 0.2s for the ADMM-based solution. One should keep in mind that the solution proposed by CVX toolbox is based on interior point methods which should give more precise results (the error is not important compared to the calculation time difference).

Figure 10 :Figure 11 :

 1011 Figure 10: NMSE in dB and MSE performance of the regularized-DML versus SNR for sparse signals and sparse channels with (Nt = 3, Nr = 8, L = 20) and T = 1000.


Predicting Nurses' Occupational Commitment and Turnover Intention: The Role of Autonomous Motivation and Supervisor and Coworker Behaviors

Keywords: Nurse turnover and retention, autonomous motivation, occupational commitment, transformational leadership, autonomy support

Aim. To examine whether supportive supervisor (transformational leadership) and coworker (autonomy-supportive) behaviors predict occupational commitment and turnover intention over time through autonomous motivation.

Background. Nurse turnover is a serious issue in several countries, straining the efficiency of the healthcare system and compromising both the quality and accessibility of healthcare.

Method. Longitudinal data were collected over 12 months from 387 French-Canadian registered nurses. Structural equation modelling was used to test the hypothesized model.

Results:

The relationships between predictors at Time 1 (supervisor and coworker behaviors) and occupational commitment and turnover intention at Time 2 are mediated by autonomous motivation at Time 1.

Conclusion:

In times of global scarcity, the present findings provide insights into how the healthcare work environment acts on nurses' occupational turnover and commitment.

Implications for Nursing Management: Healthcare organizations are advised to foster supportive work environments and promote autonomous motivation to sustain the nursing workforce.

Introduction

Although research has suggested for some time that nurse turnover is a serious concern in several countries [START_REF] Perreira | The employee retention triad in health care: Exploring relationships amongst organisational justice, affective commitment and turnover intention[END_REF], most empirical studies have focused on inter-role transitions, such as organizational commitment, with less attention directed toward occupational turnover indicators including affective commitment and intention to leave the profession [START_REF] Van Der Heijden | Job characteristics and experience as predictors of occupational turnover intention and occupational turnover in the European nursing sector[END_REF]).

Yet, if staff turnover incurs significant direct costs (e.g., recruiting, replacement) as well as indirect costs (e.g., loss of productivity) for healthcare organizations, quitting the profession substantially adds to this burden, further straining the efficiency of the healthcare system and compromising both access to and quality of care and services [START_REF] Hayes | Nurse turnover: A literature review-an update[END_REF]. This situation is all the more alarming given the global shortage of nurses and midwives which is estimated to reach nine million by 2030 (World Health Organization, 2020).

Several psychosocial factors have been associated with nurses' retention in the occupation. These factors generally fall into two categories, namely (1) work environment (e.g., supervisor and coworker support, job overload, lack of professional autonomy), and (2) individual (e.g., job attitudes, demographic characteristics such as age and job experience) factors (e.g., [START_REF] Pursio | Professional autonomy in nursing-An integrative review[END_REF][START_REF] Van Der Heijden | Job characteristics and experience as predictors of occupational turnover intention and occupational turnover in the European nursing sector[END_REF]. However, among the individual factors assessed, the role of autonomous motivation at work has not been clearly established. Yet, recent work (e.g., [START_REF] Fernet | Self-determination trajectories at work: A growth mixture analysis[END_REF] suggests that nurses who emotionally disengage from their occupation previously experience a gradual decline in autonomous motivation, defined by self-determination theory (SDT; [START_REF] Ryan | Self-Determination theory: Basic psychological needs in motivation, development, and wellness[END_REF] as the sense of being able to act according to their own choices, values, and interests. Despite this finding, research is currently limited as to the temporal relationship between nurses' autonomous motivation and occupational commitment and turnover intention. It remains unclear whether, and how, social work environment factors may contribute to nurses' occupational commitment and turnover intention over time. Turnover intention refers to a deliberate and conscious willingness to quit the occupation or the organization [START_REF] Tett | Job satisfaction, organizational commitment, turnover intention, and turnover: Path analyses based on meta-analytic findings[END_REF]. As for commitment, it reflects nurses' emotional attachment, involvement and identification with the occupation [START_REF] Meyer | Commitment to organizations and occupations: Extension and test of a three-component conceptualization[END_REF]. In addition to being associated with significant organizational and societal costs (Douada et al., 2021;[START_REF] Hayes | Nurse turnover: A literature review-an update[END_REF], these two variables are associated with actual turnover [START_REF] Van Der Heijden | Job characteristics and experience as predictors of occupational turnover intention and occupational turnover in the European nursing sector[END_REF].

To address this issue, we tested a model which posits that nurses' perceptions of supportive behavior from their immediate supervisor and coworkers relate to their autonomous motivation at work, which simultaneously predicts greater affective commitment to the occupation and lower intention to leave it over time. This study offers three main research contributions. First, we address the need for research to improve our understanding of why nurses intend to leave the occupation [START_REF] Baljoon | Factors affecting nurses' work motivation level at a governmental hospital: A cross-sectional study[END_REF]. To date, research has primarily focused on motivational correlates (e.g., job satisfaction), rather than examining the underlying reasons for being fully engaged in the job [START_REF] Fernet | Motivational pathways of occupational and organizational turnover intention among newly registered nurses in Canada[END_REF]. The present longitudinal study allows us to address this gap by examining the temporal relationship between autonomous motivation and two well-established occupational turnover indicators (commitment and turnover intention). As the occupational turnover literature is largely based on cross-sectional evidence, we aimed to offer a more comprehensive understanding of the dynamic nature of nurses' autonomous motivation. Despite the relatively stable nature of occupational commitment and turnover intention [START_REF] Fernet | Self-determination trajectories at work: A growth mixture analysis[END_REF], this would suggest that autonomous motivation can produce a motivational drive that stimulates nurses to become more committed and less inclined to leave the profession over time. Second, we assess the contribution of the perceived behaviors of two key agents of the organization (i.e., immediate supervisor and coworkers) in the prediction of nurses' motivation. By drawing attention to the complementary and relative role of these agents' behaviors on work motivation [START_REF] Jungert | Whose autonomy support is more effective? Managers' or co-workers'? An experimental comparison of source and occupational context on intrinsic motivation[END_REF], this research suggests different avenues for intervention with nurses. Third, this study allows us to better understand the psychological mechanisms involved in occupational commitment and turnover intention. By proposing autonomous motivation as an explanatory mechanism, we advance research by identifying psychological pathways by which the prevailing social context in health care organizations is liable to act on nurses' occupational turnover over time.

Theoretical Background

Autonomous Motivation

SDT [START_REF] Ryan | Self-Determination theory: Basic psychological needs in motivation, development, and wellness[END_REF] provides a useful framework for understanding nurses' motivation.

Focusing on both the quantitative and qualitative aspects of motivation, SDT distinguishes two main forms of motivation: Autonomous and controlled. In autonomous motivation, an effort is made for the pleasure and satisfaction of performing the task (intrinsic motivation) or because it allows for the attainment of objectives aligned with one's personal values (identified regulation). In contrast, controlled motivation refers to efforts made in response to internal (introjected regulation; avoiding anxiety or guilt or achieving a sense of self-worth) or external (external regulation; avoiding constraints or obtaining material or social rewards) pressures. In the present study, we focus on autonomous motivation, as it is more sensitive than controlled motivation to work environment factors [START_REF] Fernet | Transformational leadership and optimal functioning at work: On the mediating role of employees' perceived job characteristics and motivation[END_REF] and is a better predictor of change in attitudes at work (e.g., commitment, turnover intention; [START_REF] Austin | Fatigue in new registered nurses: A 12-month cross-lagged analysis of its association with work motivation, engagement, sickness absence and turnover intention[END_REF].

Social Work Environment Predictors of Autonomous Motivation

According to SDT, employee motivation is largely influenced by environmental conditions in the workplace including the interpersonal work climate [START_REF] Gagné | Self-determination theory and work motivation[END_REF]). An environment that provides conditions supporting employees' growth and development contributes to their autonomous motivation [START_REF] Ryan | Self-Determination theory: Basic psychological needs in motivation, development, and wellness[END_REF]. Research has shown that supportive behaviors, both from the immediate supervisor and coworkers in the form of leadership or autonomy support, can facilitate employees' autonomous motivation [START_REF] Jungert | Whose autonomy support is more effective? Managers' or co-workers'? An experimental comparison of source and occupational context on intrinsic motivation[END_REF]. Supportive supervisor behaviors. Transformational leadership theory [START_REF] Bass | Leadership and performance beyond expectations[END_REF] arguably provides one of the best-established conceptualizations of supportive supervisor behaviors in the nursing context [START_REF] Cummings | Leadership styles and outcome patterns for the nursing workforce and work environment: A systematic review[END_REF]. This framework is based on four types of behavior: Idealized influence (serving as role model), inspirational motivation (inspiring others to become fully committed to organizational goals), intellectual stimulation (encouraging others to be creative and innovative), and individualized consideration (providing a supportive climate, paying attention to each nurse's needs). Several studies support the effect of an immediate supervisor's transformational leadership behaviors on nurses' autonomous motivation (e.g., [START_REF] Fernet | Transformational leadership and optimal functioning at work: On the mediating role of employees' perceived job characteristics and motivation[END_REF]. These behaviors promote autonomous motivation as they create the conditions required for the satisfaction of basic psychological needs [START_REF] Ryan | Self-Determination theory: Basic psychological needs in motivation, development, and wellness[END_REF] and for the appropriation of work resources conducive to performance (e.g., [START_REF] Fernet | Transformational leadership and optimal functioning at work: On the mediating role of employees' perceived job characteristics and motivation[END_REF]. The theory and available findings lead us to propose the following hypothesis: Hypothesis 1. Supportive supervisor behaviors at Time 1 are positively associated with nurses' autonomous motivation at Time 1.

Supportive coworker behaviors.

Although research in the nursing context has focused more on the leadership behaviors of the immediate supervisors, the behaviors of coworkers likely play an important role in motivating nurses (e.g., [START_REF] Jungert | Whose autonomy support is more effective? Managers' or co-workers'? An experimental comparison of source and occupational context on intrinsic motivation[END_REF] for two main reasons. First, the reality of many healthcare organizations, particularly in a context of a shortage, leads them to orient their management strategies towards the achievement of specific results, such as accessibility and quality of services, thus promoting task-focused leadership [START_REF] Cummings | Leadership styles and outcome patterns for the nursing workforce and work environment: A systematic review[END_REF]. This approach leaves immediate supervisors with fewer opportunities and time to support, coach, and motivate their team members [START_REF] Shao | Creativity under workload pressure and integrative complexity: The double-edged sword of paradoxical leadership[END_REF]. Second, the departments and units in which nurses operate generally rely on small teams to perform well, which makes collaboration and supportive behaviors essential for motivation and commitment [START_REF] Galletta | A cross-lagged analysis of the relationships among workgroup commitment, motivation and proactive work behaviour in nurses[END_REF].

According to SDT [START_REF] Ryan | Self-Determination theory: Basic psychological needs in motivation, development, and wellness[END_REF], supportive coworker behaviors foster autonomous motivation by creating an environment conducive to communication, information sharing, and recognition. A nurse can help another by giving constructive comments that allow, for example, the colleague to grasp the meaningful rationale or the merits of a task [START_REF] Reeve | Teachers become more autonomy supportive after they believe it is easy to do[END_REF]. A nurse can also respect a colleague's perspective, by being available to give information, clarify any ambiguities related to the position or task, answer questions, and offer guidance [START_REF] Fernet | Transformational leadership and optimal functioning at work: On the mediating role of employees' perceived job characteristics and motivation[END_REF]. [START_REF] Jungert | Whose autonomy support is more effective? Managers' or co-workers'? An experimental comparison of source and occupational context on intrinsic motivation[END_REF] have shown that such coworkers' autonomy-supportive behaviors enhance autonomous motivation (i.e. intrinsic motivation) of employees in a social occupational context.

Although few studies have examined the relationship between autonomy-supportive behaviors from coworkers and nurses' autonomous motivation, the simultaneous consideration of supportive behaviors of the immediate supervisor and coworkers will provide insight into their relative contribution. We thus propose the following hypothesis: Hypothesis 2. Supportive coworker behaviors at Time 1 are positively associated with nurses' autonomous motivation at Time 1.

Autonomous Motivation and Occupational Commitment and Turnover Intention

Growing empirical evidence suggests that autonomous motivation at work promotes employee adaptation and functioning [START_REF] Ryan | Self-Determination theory: Basic psychological needs in motivation, development, and wellness[END_REF]. In the nursing context, nurses' autonomous motivation has been correlated with a variety of job attitudes related to retention, including occupational commitment, and negatively correlated with the intention to leave the organization and the nursing profession [START_REF] Fernet | Motivational pathways of occupational and organizational turnover intention among newly registered nurses in Canada[END_REF][START_REF] Fernet | Self-determination trajectories at work: A growth mixture analysis[END_REF]Galleta et al., 2016). However, as past research has mainly utilized cross-sectional designs, little is currently known about the temporal relationship between motivation and occupational commitment and turnover intention.

Occupational affective commitment. Occupational affective commitment is a unique construct that captures the psychological relationship that nurses develop with their occupation [START_REF] Van Der Heijden | Job characteristics and experience as predictors of occupational turnover intention and occupational turnover in the European nursing sector[END_REF]. The emotional attachment that nurses experience towards their profession allows them to maintain positive feelings towards their position and sustains their occupational commitment [START_REF] Meyer | Commitment to organizations and occupations: Extension and test of a three-component conceptualization[END_REF]. According to SDT, this level of emotional attachment is expected to depend on the reasons that led the nurses to invest in their occupation. When nurses' efforts are fully aligned with their choices and values (i.e., high autonomous motivation), this should strengthen their emotional attachment and commitment to their role. We thus propose the following hypothesis: Hypothesis 3. Nurses' autonomous motivation at Time 1 positively predicts occupational affective commitment at Time 2.

Occupational turnover intention.

As the intent to leave refers to the conscious and deliberate desire to quit the profession [START_REF] Tett | Job satisfaction, organizational commitment, turnover intention, and turnover: Path analyses based on meta-analytic findings[END_REF], it could stem from underlying reasons which motivate nurses to invest effort into their work [START_REF] Fernet | Motivational pathways of occupational and organizational turnover intention among newly registered nurses in Canada[END_REF]. Whereas autonomous motivation is characterized by volition and self-endorsement of choices and actions, nurses who do not find pleasure, satisfaction, or a genuine interest in performing tasks aligned with their personal values could be inclined to leave their occupation. According to [START_REF] Sheldon | Goal striving, need satisfaction, and longitudinal well-being: The self-concordance model[END_REF], the self-consistent nature of autonomously motivated activities is likely to create a sense of genuineness, which should allow autonomously motivated nurses to feel alive and authentic. Because they appreciate having the opportunity to express their true selves, autonomously motivated nurses would be more inclined to remain in the profession. Accordingly, we propose the following hypothesis: Hypothesis 4. Autonomous motivation at Time 1 negatively predicts occupational turnover intention at Time 2.

The Mediating Role of Autonomous Motivation

Implicit to all four hypotheses and in SDT [START_REF] Ryan | Self-Determination theory: Basic psychological needs in motivation, development, and wellness[END_REF] is the notion that supportive behaviors have an effect on occupational commitment and turnover intention through autonomous motivation. Despite the lack of empirical evidence in a nursing context, results from several organizational studies support this motivational sequence (see [START_REF] Ryan | Self-Determination theory: Basic psychological needs in motivation, development, and wellness[END_REF]. Thus, we explicitly predict a mediated model.

Hypothesis 5. Autonomous motivation at Time 1 mediates the relationships between perceived supportive supervisor and coworker behaviors at Time 1, and occupational commitment and turnover intention at Time 2.

Method Study Design and Sample

The participants of this 12-month longitudinal study were French-Canadian registered nurses working in the public healthcare sector. Our sample was drawn from a random mailing list of 665 nurses, members of the Quebec Nursing Association who agreed to be contacted for academic research purposes. Eligibility criteria was limited to the ability to read and understand French. All participants were asked to provide informed consent, before completing each online questionnaire.

They were informed about the objectives of the research, told that participation was voluntary and confidential, and notified that they could freely withdraw from the project at any time. Approval for this study was obtained from the Ethic Committee of the principal researcher's institution.

Measures

All variables were administered in French at both time points with the exception of the predictors (supportive supervisor and coworker behaviors), which were assessed at Time 1 only (means, standard deviations, latent correlations are presented in Table 1). Validity and reliability of the French-Canadian version of these measures have been supported in prior studies [START_REF] Fernet | Motivational pathways of occupational and organizational turnover intention among newly registered nurses in Canada[END_REF][START_REF] Fernet | Self-determination trajectories at work: A growth mixture analysis[END_REF].

For all variables, higher scores reflect greater levels of each corresponding construct.

Supportive supervisor behaviors. The seven-item Global Transformational Leadership scale (GTL; [START_REF] Carless | A short measure of transformational leadership[END_REF] was used to assess nurses' perceptions of supportive leadership behaviors of their immediate supervisor (α = .94; "He/she fosters trust, involvement, and cooperation among team members"). Items were scored on a five-point scale ranging from "never" to "almost always". [START_REF] Blais | L'inventaire des perceptions du style de supervision au travail[END_REF] scale was used to assess autonomy-supportive behaviors from coworkers (α = .75). Participants were asked to rate on a seven-point scale from "do not agree at all" to "very strongly agree" the extent to which their colleagues interacted with them in a given way (e.g., "My colleagues provide me with constructive feedback regarding my work and help me improve it").

Supportive coworker behaviors. A three-item adapted version of

Autonomous motivation. Two dimensions of the Revised Motivation at Work Scale (R-MAWS; [START_REF] Gagné | The Multidimensional Work Motivation Scale: Validation evidence in seven languages and nine countries[END_REF] were used to assess autonomous motivation (T1 α = .82; T2 α = .87): Intrinsic motivation (three items; e.g., "Because my work is stimulating") and identified regulation (three items; e.g., "Because this job has a personal significance for me"). Respondents scored on a seven-point scale from "not at all for this reason" to "exactly for this reason" the primary reasons for performing their job.

Occupational commitment. Three items adapted from [START_REF] Meyer | Commitment to organizations and occupations: Extension and test of a three-component conceptualization[END_REF] were used to assess affective occupational commitment (T1 α = .85; T2 α = .78). A sample item is "The nursing profession means a lot to me". Each item was scored on a five-point scale from "strongly disagree" to "strongly agree".

Occupational turnover intention. Three items adapted from O'Driscoll and Beehr (1994) were used to assess occupational turnover intention (T1 α = .88; T2 α = .93). A sample item is "I'm thinking about leaving the nursing profession". Each item was scored on a seven-point scale from "strongly disagree" to "strongly agree".

Statistical Analysis

A series of models were tested through structural equation modeling using a robust maximum likelihood estimator (MLR) in Mplus 8.4 [START_REF] Muthén | Mplus user's guide[END_REF]. Full Information Maximum Likelihood (FIML) was used to handle missing data (e.g., [START_REF] Enders | Applied missing data analysis[END_REF]. To determine the goodnessof-fit of these models, various fit indices were used: The Comparative Fit Index (CFI), the Tucker-Lewis Index (TLI), and the Root Mean Square Error of Approximation (RMSEA). Values greater than .90 for the CFI and TLI and smaller than .08 for the RMSEA indicate acceptable fit [START_REF] Marsh | Goodness of fit evaluation in structural equation modeling[END_REF].

Results

Preliminary Analysis

Three measurement models were tested. The first model ( M1a Following previous recommendations (Little et al., 2007), each item at Time 1 was also allowed to covary with its corresponding item at Time 2. Results indicated that this model presented a satisfactory fit to the data: χ² (df) = 851.693 (487), CFI = .926, TLI = .915, and RMSEA = .044 [.039;

.049]. Parameter estimates are reported in Table 1 (latent correlations) and Table 2 (loadings and uniquenesses).

Main Analysis

The proposed model (see Figure 1) presented satisfactory fit to the data: χ² (df) = 895.721 (499), CFI = .919, TLI = .909, and RMSEA = .045 [.041; .050]. Supportive supervisor (β = .162, p < .05) and coworker (β = .465, p < .001) behaviors at Time 1 positively predicted autonomous motivation at Time 1. In addition, each of the following variables at Time 1 (autonomous motivation, and occupational commitment and turnover intention) were significantly and positively related to its corresponding variable at Time 2 (βs ranging from .445 to .760, p < .001). Finally, results show that controlling for baseline effects, autonomous motivation at Time 1 positively predicted occupational commitment at Time 2 (β = .232, p < .01) and negatively predicted occupational turnover intention at Time 2 (β = -.200, p < .05).

Bootstrapping analyses [START_REF] Cheung | Testing mediation and suppression effects of latent variables: Bootstrapping with structural equation models[END_REF] were then conducted to analyze the mediating role of autonomous motivation at Time 1 in the relationships, on the one hand, between supportive supervisor and coworker behaviors at Time 1, and on the other hand, occupational commitment and turnover intention at Time 2. The indirect effects were tested with 95% confidence intervals computed from 5000 bootstrap samples. First, the relationship between supportive supervisor behaviors at Time 1 and both outcomes at Time 2 was mediated by autonomous motivation at Time 1 (occupational commitment [indirect effect = .069; s.e. = .043; 95% CI = .007 to .183] and turnover intention [indirect effect = -.038; s.e. = .027; 95% CI = -.122 to -.004]). Second, the relationship between supportive coworker behaviors at Time 1 and both outcomes at Time 2 was mediated by autonomous motivation at Time 1 (occupational commitment [indirect effect = .199; s.e. = .089; 95% CI = .055 to

.403] and turnover intention [direct effect = -.109; s.e. = .067; 95% CI = -.293 to -.012]).

Discussion

Theoretical Implications

The main contribution of this study is the identification of autonomous motivation as a dynamic individual factor that simultaneously acts on occupational commitment and turnover intention. The greater autonomous motivation the nurses showinvesting in their work for pleasure and satisfaction or to achieve professional goals that they valuethe more they will become affectively committed to their occupation and the less inclined they will be to leave it over time. While these results align with those of recent cross-sectional studies carried out within the nursing context [START_REF] Fernet | Motivational pathways of occupational and organizational turnover intention among newly registered nurses in Canada[END_REF]Galleta et al., 2016), they could constitute the first empirical evidence that nurses' autonomous motivation acts on occupational commitment and turnover intention over time.

The present study also sheds new light on the predictors and occupational outcomes of nurses' autonomous motivation. First, our results show that supportive behaviors by both the immediate supervisor and coworkers relate to nurses' work autonomous motivation. By revealing the distinct and complementary contribution of these key agents in their environment, our results underline the need to consider the role of social resources separately (rather than as an aggregate variable, irrespective of the source; e.g., supervisor, coworkers). These findings lend additional support to recent Job Demands-Resources model based-studies having shown the relative contribution of various job resources [START_REF] Hakanen | The relative importance of various job resources for work engagement: A concurrent and follow-up dominance analysis[END_REF], including social support from the supervisor and coworkers in nursing studies (e.g., Van der Heijden et al., 2019). Whereas SDT-based research has heavily focused on supervisor behaviors, our results also invite to examine the contribution, not only of the immediate supervisor, but of other key agents in the work environment. Interestingly, the present findings are aligned with a recent experimental study [START_REF] Jungert | Whose autonomy support is more effective? Managers' or co-workers'? An experimental comparison of source and occupational context on intrinsic motivation[END_REF] showing that autonomy-supportive behaviors from coworkers increased autonomous motivation more than from a supervisor in a social occupational context.

Limitations

First, the study relies exclusively on self-report data that may be subject to self-presentation report bias. Future research integrating multi-source (e.g., coworker leadership assessment) and objective (e.g., actual turnover) data would increase the range of the results. Second, the study is based on a partial longitudinal design, which does not allow us to establish causal relationships between all the variables examined. Third, although we have relied on proven theoretical perspectives to determine predictors of autonomous motivation, our analysis relies on a limited set of predictors.

Further research should enhance our understanding of predictors, especially the larger work environment, at various levels (e.g., organizational culture and climate, human resource practices).

Finally, our study should be replicated in other countries with more heterogeneous samples (e.g., wider age range, racial and ethnic groups).

Implications for Nursing Management

From a social perspective, keeping nurses in the profession is a major concern given the current global shortage, but also in terms of associated societal issues [START_REF] Duffield | A comparative review of nurse turnover rates and costs across countries[END_REF]. The COVID-19 pandemic context reminds us on the need to count on an adequate influx of healthcare personnel in order to maintain the efficiency of our healthcare systems. What emerges from our results is that nurses' occupational affective commitment (and the desire not to abandon it) can be cultivated within an establishment by certain key agents, for example through the supportive nature of relationships with the immediate supervisor and coworkers. Organizational efforts to strengthen nurses' autonomous motivation, through supportive supervisor and coworker behaviors, should prove to be a beneficial strategy for contributing to a well-rooted workforce in the profession.

Among the possible means to facilitate such efforts, healthcare managers, including supervisors in departments or units, could promote a culture of support, for example by encouraging equity in the application of policies and relying on shared leadership [START_REF] Pursio | Professional autonomy in nursing-An integrative review[END_REF]. The finding that supportive coworker behaviors would be particularly important to facilitate nurses' autonomous motivation adds to the relevance of bringing particular attention to shared leadership in which team members with particular knowledge, skills or abilities assume greater leadership in some areas of their work. Coaching activities, as well as formal and informal mentoring, may also help to build a more supportive workplace climate for nurses [START_REF] Newman | The impact of employee perceptions of training on organizational commitment and turnover: A case of multinationals in the Chinese service sector[END_REF]. SDT has great potential in this regard, particularly for understanding the need-supportive conditions (e.g., autonomy support, appropriate structure and interpersonal involvement strategies) that facilitate optimal functioning and might therefore serve as an impetus to increase the scope of such activities. In parallel, organizational

  Out of this list, 387 nurses took part in our survey either at T1 (May 2014; n = 302) or T2 (May 2015; n = 206) and represented the final sample (overall response rate: 58%). Most participants were women (88.9%; n = 344) and held a permanent position (78.2%; n = 280) and 54.9% (n = 196) worked full time. The mean sample age was 29.15 years (SD = 9.25) and average length of experience in nursing was 4.19 years (SD = 9.41).

  ) consisted of all latent variables and indicators at Time 1 (supervisor and coworker behaviors, autonomous motivation, and occupational commitment and turnover intention), while the second model (M1b) included all latent variables and indicators at Time 2 (autonomous motivation, and occupational commitment and turnover intention). Both models presented satisfactory fit indices: χ² (df) = 427.735 (199), CFI = .923, TLI = .911, and RMSEA = .062 [.054; .070] for Model 1a; and χ² (df) = 109.530 (51), CFI = .946, TLI = .930, and RMSEA = .066 [.049; .084] for Model 1b. A third measurement model was tested (M1c) by combining all latent variables and indicators at both measurement times, and consisted of eight latent variables (i.e., autonomous motivation, and occupational commitment and turnover intention at Times 1 and 2, as well as supervisor and coworker behaviors at Time 1 only) and 34 indicators.

  interventions could be developed to raise awareness of nurses' autonomous motivation. Van Dorssen-Boog et al. (2021) recently provided evidence that the implementation of a self-leadership training program (based on motivation principles such as mastery, proactive problem-solving, and strengthbased coaching) facilitates healthcare employees' autonomous motivation and fosters their work engagement and performance. Future interventions efforts along these lines are encouraged as they offer promising avenues to foster high quality work motivation and subsequently occupational commitment and retention in nurses. workers' selfdetermination: The impact of a self-leadership intervention on work engagement, health, and performance. Journal of Occupational and Organizational Psychology. Advance online publication. World Health Organization (2020). Nursing and midwifery. Retrieved March 10 2021, from https://www.who.int/fr/news-room/fact-sheets/detail/nursing-and-midwifery.

Figure 1

 1 Figure 1

Table 1

 1 Latent Correlations Between the Variables

	Variable	1	2	3	4	5	6	7	8
	1. Autonomous motivation (T1)	─
	2. Occupational commitment (T1)	.588	─
	3. Occupational turnover intention (T1)	-.330 -.733	─
	4. Supportive supervisor behaviors (T1)	.346	.279 -.202	─
	5. Supportive coworker behaviors (T1)	.571	.418 -.366 .352	─
	6. Autonomous motivation (T2)	.778	.511 -.438 .269	.540	─
	7. Occupational commitment (T2)	.647	.944 -.729 .340	.471	.682	─
	8. Occupational turnover intention (T2)	-.392 -.590 .684 -.224 -.439 -.511 -.667	─

Note. All associations are significant at p < .001.

Table 2

 2 Standardized Factor Loadings (λ) and Uniquenesses (δ) for the Measurement Model Factor loading; δ: Item uniqueness; ω: Omega coefficient of model-based composite reliability; non-significant parameters (p > .05) are marked in italics.

	Time 1	Time 2