Nobutaka Ito

Emmanuel Vincent

Nobutaka Ono

Shigeki Sagayama

† Nobutaka

Robust Estimation of Directions-of-Arrival in Diffuse Noise Based on Matrix-Space Sparsity

Keywords: diffuse noise, DOA estimation, microphone arrays, MUSIC, matrix completion bruit diffus, estimation de direction d'arrivée, antenne de microphones, MUSIC, complétion de matrice

We consider the estimation of the Directions-Of-Arrival (DOA) of target signals in diffuse noise. The state-of-the-art MUltiple SIgnal Classification (MUSIC) algorithm necessitates accurate identification of the signal subspace. In diffuse noise, however, it is difficult to identify it directly from the observed spatial covariance matrix. In our approach, we estimate the target spatial covariance matrix, so that we can identify the orthogonal complement of the signal subspace as its null space. We present a unified framework for modeling noise covariance in a matrix space, which generalizes four state-of-the-art diffuse noise models. We propose two alternative algorithms for estimating the target spatial covariance matrix, namely Low-rank Matrix Completion (LMC) and Trace Norm Minimization (TNM). These rely on denoising of the observed spatial covariance matrix via orthogonal projection onto the orthogonal complement of the noise matrix subspace. The missing component lying in the noise matrix subspace is then completed by exploiting the low-rankness of the target spatial covariance matrix. Large-scale experiments with real-world noise show that TNM with a certain noise model outperforms conventional MUSIC based on Generalized EigenValue Decomposition (GEVD) by 5% in terms of the precision averaged over the dataset.

Estimation Robuste de Directions d'Arrivée dans du Bruit Diffus Basée sur la Parcimonie dans un Espace Matriciel

Résumé : Nous considérons l'estimation des directions d'arrivée de sources sonores dans du bruit diffus. L'algorithme de l'état de l'art MUSIC (MUltiple SIgnal Classification) nécessite l'identification précise du sous-espace signal. En présence de bruit diffus, cependant, il est difficile de l'estimer directement à partir de la matrice de covariance spatiale observée. Dans notre approche, nous estimons la matrice de covariance spatiale de la source cible, de sorte à pouvoir identifier le complément orthogonal du sous-espace signal comme son espace nul. Nous présentons un cadre unifié pour la modélisation de la matrice de covariance du bruit dans un espace matriciel, qui généralise quatre modèles de bruit diffus de l'état de l'art. Nous proposons deux algorithmes pour estimer la matrice de covariance spatial de la cible, basés soit sur la complétion de matrice de rang faible soit sur la minimisation de la norme trace. Ces algorithmes reposent sur le débruitage de la matrice de covariance spatiale observée par projection orthogonale sur le complément du sous-espace matriciel correspondant au bruit. La composante manquante dans le sous-espace matriciel correspondant au bruit est alors complétée en utilisant le faible rang de la matrice de covariance spatiale de la cible. Des expériences à grande échelle montrent que, pour l'un des modèles de bruit, la minimisation de la norme trace dépasse l'approche classique par MUSIC avec décomposition en valeurs propres généralisée de 5% en terme de précision en moyenne.

Introduction

We address the DOA estimation of multiple target signals in diffuse noise. This has many applications epitomized by automatic camera steering [START_REF] Wang | Voice source localization for automatic camera pointing system in videoconferencing[END_REF]. For instance, we encounter diffuse noise when many people are speaking at the same time in the street or at a party. Another example is noise in a car or on a train that is caused by the vibration of the body and the windows, which constitute surface noise sources instead of point noise sources.

While single-source DOA estimation techniques are now established [START_REF]Microphone Arrays: Signal Processing Techniques and Applications[END_REF][START_REF] Foy | Position-location solutions by Taylor-series estimation[END_REF][START_REF] Smith | Closed-form least-squares source location estimation from range-difference measurements[END_REF][START_REF] Stoica | Source localization from range-difference measurements[END_REF][START_REF] Knapp | The generalized correlation method for estimation of time delay[END_REF], multi-source DOA estimation has remained challenging to date. There are three main approaches, namely clustering methods [START_REF] Izumi | Sparseness-based 2ch BSS using the EM algorithm in reverberant environment[END_REF][START_REF] Sawada | Grouping separated frequency components by estimating propagation model parameters in frequency-domain blind source separation[END_REF][START_REF] Mandel | An EM algorithm for localizing multiple sound sources in reverberant environments[END_REF], SRP-Phat [START_REF] Dibiase | Robust localization in reverberant rooms[END_REF] and its variants, and MUSIC [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF][START_REF] Nakadai | Real-time tracking of multiple sound sources by integration of in-room and robot-embedded microphone arrays[END_REF].

MUSIC is based on the assumption that the target signals reside in a lowdimensional target signal subspace of the signal space C M spanned by the multichannel observed signals. It is based on null steering into the target DOAs by exploiting vectors orthogonal to the target signal subspace. Therefore, accurate identification of the target signal subspace is crucial. MUSIC can estimate DOAs even in diffuse noise in principle, because its presence does not violates the assumption that the target signals reside in a low-dimensional target signal subspace. However, the identification of the target signal subspace in diffuse noise has been difficult because of its unknown spatial correlation.

In this paper, we propose methods for estimating the target spatial covariance matrix from the observed signals, so that the orthogonal complement of the target signal subspace can be identified as its null space. We present a unified framework for modeling noise covariance in a matrix space, which generalizes four state-of-the-art diffuse noise models. We propose two algorithms for estimating the target spatial covariance matrix, namely LMC and TNM. These rely on denoising of the observed spatial covariance matrix via orthogonal projection onto the orthogonal complement of the noise matrix subspace. The missing component lying in the noise matrix subspace is then completed by exploiting the low-rankness of the target spatial covariance matrix.

We extend our preliminary paper [START_REF] Ito | Crystal-MUSIC: Accurate localization of multiple sources in diffuse noise environments using crystal-shaped microphone arrays[END_REF] by introducing a general noise model and a new matrix completion algorithm and by performing a large-scale experiment.

The rest of this paper is structured as follows. Section 2 formulates the task considered, and reviews the state-of-the-art MUSIC. Section 3 presents the proposed noise modeling framework, and compares the fit of noise models to real-world noise. Section 4 presents the proposed algorithms for estimating the target spatial covariance matrix. Section 5 describes the experiment, and Section 6 concludes this paper.

Definition of DOA Estimation and Review of MUSIC 2.1 Definition of DOA estimation

We use the following notation throughout. The superscripts * and H denote complex conjugation and Hermitian transposition, respectively. Signals are represented in the Short-Time Fourier Transform (STFT) domain as, e.g. α(τ, ω), with τ and ω denoting the frame index and the angular frequency. The covariance matrix of a zero-mean vector signal α(τ, ω) is denoted by

Φ αα (τ, ω) E [α(τ, ω)α H (τ, ω)], (1)
where E [•] is expectation. DOA estimation is the task of estimating the DOAs of the target signals given the observed multichannel signals. When the target sources are in the far field of the array, their location is specified by two parameters, namely the azimuth and the zenith angle. We assume that the target signals are at the same height as the microphone array, and focus on the estimation of the azimuth for simplicity. However, the proposed techniques can easily be extended to the estimation of both.

Formally, we assume that an array of M microphones receives L target signals from unknown azimuths in the presence of diffuse noise. Let s(τ, ω) ∈ C L be the vector of the target signals observed at a reference point, and x(τ, ω) ∈ C M and v(τ, ω) ∈ C M be the vector of the observed signals and diffuse noise at the microphones, respectively. We denote the steering vector of a planewave impinging the array from a horizontal direction with azimuth ξ by h(ω; ξ) e -jωδ1(ξ) e -jωδ2(ξ) . . . e -jωδM (ξ)

T , (2)
where δ m (ξ) is the time the planewave takes to propagate from the reference point to the m-th microphone. Therefore, denoting the target azimuths by Ξ {ξ l } L l=1 and defining

H(ω; Ξ) = h(ω; ξ 1 ) h(ω; ξ 2 ) • • • h(ω; ξ L ) , (3)
we can model the observed multichannel signal by [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF]:

x(τ, ω) = H(ω; Ξ)s(τ, ω) + v(τ, ω). (4)
Assuming that s(τ, ω) and v(τ, ω) are mutually uncorrelated, we have the following relationship among covariance matrices:

Φ xx (τ, ω) = Φ cc (τ, ω) + Φ vv (τ, ω), (5)
where Φ xx (τ, ω), Φ cc (τ, ω) H(ω; Ξ)Φ ss (τ, ω)H H (ω; Ξ), and Φ vv (τ, ω) are the observed, target, and noise covariance matrices, respectively.

Consequently, our task is now formally defined as that of estimating Ξ given x(τ, ω), where the number L of target signals is assumed to be known [START_REF] Paulraj | A subspace rotation approach to signal parameter estimation[END_REF].

Review of MUSIC

The observation model (4) implies that, if there are less target sources than the microphones (L < M ), the target component H(ω; Ξ)s(τ, ω) resides in the low-dimensional target signal subspace defined by

S (ω) span{h(ω; ξ l )} L l=1 . (6)
Let us denote by {e i (ω)} M -L i=1 some basis vectors of S ⊥ (ω) (the orthogonal complement of S (ω)) . Each of them forms a directivity pattern with nulls at ξ ∈ Ξ:

|e H i (ω)h(ω; ξ)| 2 = 0, ∀ξ ∈ Ξ. ( 7
)

Inria

Equivalently, the inverses of these directivity patterns have peaks at ξ ∈ Ξ, and so does the narrowband MUSIC spectrum defined by [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF]:

f N (ω; ξ) M -L i=1 |e H i (ω)h(ω; ξ)| 2 -1 . (8)
In order to integrate the information at different frequencies, the narrowband spectrum is averaged over the frequency to obtain the wideband MUSIC spectrum. We employ geometric averaging as in [START_REF] Wax | Spatio-temporal spectral analysis by eignestructure methods[END_REF]:

f W (ξ) ωmax ωmin f N (ω; ξ) 1 K , (9)
Here [ω min , ω max ] denotes the frequency range of averaging, and K the corresponding number of frequency bins. The azimuth estimates { ξl } L l=1 are obtained by picking the L largest peaks in f W (ξ) up to the minimum angular distance of 15 • .

Target signal subspace identification

It is essential in MUSIC to accurately identify S ⊥ (ω) or its basis vectors {e i (ω)} M -L i=1 in [START_REF] Sawada | Grouping separated frequency components by estimating propagation model parameters in frequency-domain blind source separation[END_REF]. In the noiseless case, S ⊥ (ω) is obtained as the null space of Φ xx = Φ cc . Also, for spatially white noise, which is spatially uncorrelated and has the same power at any microphones, S ⊥ (ω) is obtained via EigenValue Decomposition (EVD) of Φ xx (τ, ω) as the eigenspace of Φ xx (τ, ω) corresponding to M -L smallest eigenvalues. Even when noise is not spatially white, if the noise covariance matrix is known up to a scalar, S ⊥ (ω) can still be obtained via Generalized EigenValue Decomposition (GEVD) as the generalized eigenspace corresponding to the M -L smallest generalized eigenvalues of the matrix pencil (Φ xx (τ, ω), Γ(ω)), where Γ(ω)) is the scaled noise spatial covariance matrix [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF]. The noise covariance matrix is known a priori up to a scale, for some ideal noise fields such as the spherically isotropic noise field [START_REF] Cook | Measurement of correlation coefficients in reverberant sound fields[END_REF], which is composed of an infinite number of noise planewaves with the identical power spectrum from all three-dimensional directions in the free field. However, real-world diffuse noise can deviate from this ideal model due to the geometry of the noise sources, the room shape, and diffraction effects. Thus, the identification of S ⊥ (ω) by this approach can be unreliable.

In the following, we focus on the estimation of Φ cc (τ, ω) from Φ xx (τ, ω) in the presence of real-world diffuse noise, so that S ⊥ (ω) can be identified as the null space of Φ cc (τ, ω).

Unified Noise Covariance Modeling in a Matrix Space

In order to address this estimation problem, appropriate models of Φ cc and Φ vv are needed. While we can exploit the low-rankness of Φ cc assuming L < M , we propose a unified matrix-space model of Φ vv (τ, ω).

Proposed noise modeling framework

Array signal processing techniques are typically formulated in the signal space C M . Directional noise spans a noise signal subspace

N (ω) span{g l (ω)} L ′ l=1 (10)
of C M , where g l (ω) are the noise steering vectors and L ′ the number of noise sources. Thus, if L ′ < M , noise can be eliminated by orthogonally projecting x(τ, ω) onto the orthogonal complement N ⊥ (ω) of N (ω). However, diffuse noise spans whole C M , so that it cannot be eliminated via orthogonal projection in C M . By contrast, we model Φ vv (τ, ω) as belonging to a noise matrix subspace V (ω) of the vector space of Hermitian matrices

H {A ∈ C M ×M |A H = A} (11)
over R. H is endowed with the Euclidian inner product

A, B tr(AB H ) = tr(AB) (12)
and the Frobenius norm

A F A, A . (13)
This model enables discrimination between the target signals and diffuse noise by orthogonal projection of Φ xx (τ, ω) onto V ⊥ (ω). This is exploited in the DOA estimation algorithms proposed in Section 4.

Although Φ vv (τ, ω) belongs more specifically to the set of Hermitian positive semidefinite matrices, this set is not a linear space. Linear-space modeling leads to efficient algorithms by using the orthogonal projections P ω and P ⊥ ω onto V (ω) and its orthogonal complement V ⊥ (ω) as will be shown in Section 4:

P ω [A] P i=1 A, Q i (ω) Q i (ω), (14)
P ⊥ ω [A] = A -P ω [A], (15)
where {Q i (ω)} P i=1 denotes an orthonormal basis of V (ω) with P dimV (ω). Explicit forms of these projections are given in Section 3.2.

New interpretation of state-of-the-art diffuse noise models

The proposed matrix space model includes four state-of-the-art noise models.

Spatially uncorrelated noise model

Zelinski [START_REF] Zelinski | A microphone array with adaptive post-filtering for noise reduction in reverberant rooms[END_REF] proposed a method for diffuse noise suppression based on the assumption that diffuse noise is spatially uncorrelated. The assumption implies that Φ vv (τ, ω) is diagonal, so that the corresponding noise matrix subspace is the following M -dimensional matrix subspace

V (ω) = {A ∈ H |A : diagonal}. ( 16
)
Inria P ω and P ⊥ ω are given by

P ω [A] = D(A), (17)
P ⊥ ω [A] = O(A), (18)
where D(•) is the operation of replacing the off-diagonal entries by zeros, and O(•) that of replacing the diagonal entries by zeros. While diffuse noise can be reasonably modeled as uncorrelated when the distances between microphones are large enough compared to the wavelength, it is highly correlated for small arrays or at low frequencies [START_REF] Mccowan | Microphone array post-filter based on noise field coherence[END_REF].

Fixed noise coherence model

To take noise correlation into account, McCowan et al. [START_REF] Mccowan | Microphone array post-filter based on noise field coherence[END_REF] assumed in their method for diffuse noise suppression that Φ vv (τ, ω) is fixed up to an unknown scale factor. This corresponds to the one-dimensional noise matrix subspace

V (ω) {kΓ(ω)|k ∈ R}, (19)
where Γ(ω) denotes the so-called noise coherence matrix. The orthogonal projection operators are given by

P ω [A] = (A, Γ(ω)) Γ(ω) 2 F Γ(ω).P ⊥ ω [A] = A -P ω [A]. (20)
While the noise coherence matrix for some ideal noise fields is known [START_REF] Cook | Measurement of correlation coefficients in reverberant sound fields[END_REF][START_REF] Mccowan | Microphone array post-filter based on noise field coherence[END_REF], real-world noise deviates from this ideal model as pointed out in Section 2.3.

Blind Noise Decorrelation (BND) model

Instead of assuming specific noise coherences, Shimizu et al. considered isotropic noise satisfying the following assumptions [START_REF] Ito | Diffuse noise suppression using crystal-shaped microphone arrays[END_REF][START_REF] Shimizu | Isotropic noise suppression in the power spectrum domain by symmetric microphone arrays[END_REF]:

equal power spectrogram φ vmvm (τ, ω) at all microphones m, equal cross-spectrogram φ vmvn (τ, ω) for microphone pairs (m, n) spaced by the same distance.

Under these assumptions, the unknown spatial covariance matrix of isotropic noise is diagonalized by a known unitary matrix for certain classes of symmetrical arrays or crystal arrays [START_REF] Ito | Diffuse noise suppression using crystal-shaped microphone arrays[END_REF][START_REF] Shimizu | Isotropic noise suppression in the power spectrum domain by symmetric microphone arrays[END_REF]. This BND technique has been applied to the estimation of the target power spectrogram in diffuse noise environments [START_REF] Shimizu | Isotropic noise suppression in the power spectrum domain by symmetric microphone arrays[END_REF] and to diffuse noise suppression in the signal domain [START_REF] Ito | Diffuse noise suppression using crystal-shaped microphone arrays[END_REF]. BND implies that Φ vv (τ, ω) is expressed as Φ vv (τ, ω) = P Λ(τ, ω)P H for some unknown diagonal matrix Λ(τ, ω) ∈ R M ×M and some known unitary matrix P ∈ C M ×M . (??) can be rewritten as

Φ vv (τ, ω) = M m=1 λ m (τ, ω)p m p H m
with λ m (τ, ω) denoting m-th diagonal entry of Λ(τ, ω) and p m the m-th column of P . This implies that Φ vv (τ, ω) belongs to the M -dimensional subspace

V (ω) = span{p m p H m } M m=1 . (21)
The projectors P ω and P ⊥ ω are then given by

P ω [A] = P D(P H AP )P H , (22)
P ⊥ ω [A] = P O(P H AP )P H . ( 23
)
While BND is valid for a wide range of noise fields, the array geometry is restricted to certain classes.

Real-valued noise covariance model

We introduced a flexible model of diffuse noise applicable to arbitrary array geometries in our diffuse noise suppression method [START_REF] Ito | Designing the Wiener post-filter for diffuse noise suppression using imaginary parts of inter-channel cross-spectra[END_REF]. The isotropy assumptions of BND imply that φ vmvn (τ, ω) = φ vnvm (τ, ω), and, by definition of the cross-spectrum, we have φ vnvm (τ, ω) = φ * vmvn (τ, ω). Therefore, φ vmvn ∈ R, and Φ vv belongs to the M (M + 1)/2-dimensional noise matrix subspace

V (ω) {A ∈ R M ×M |A T = A}. ( 24
)
The projectors are given by

P ω [A] = ℜ[A], (25)
P ⊥ ω [A] = jℑ[A], (26)
where ℜ[•] and ℑ[•] denotes the operations of taking the real part and the imaginary part, respectively. This model is more flexible than the spatially uncorrelated noise model and the fixed noise coherence model for spherically or cylindrically isotropic noise. Indeed, these models are real-valued [START_REF] Cook | Measurement of correlation coefficients in reverberant sound fields[END_REF][START_REF] Elko | Spatial coherence functions for differential microphones in isotropic noise fields[END_REF], and thus noise matrix subspaces of the real-valued noise covariance model. However, it has many parameters, which can cause overfitting.

Assessment of noise models with real-world noise

Before evaluating these four noise models for DOA estimation in Section 5, we assess their potential independently of the application as follows. We investigate two different aspects which are important to predict the performance of a certain noise model: the number of parameters of the model dimV (ω) compared to the number of observations dimH , the fit to real-world noise spatial covariance matrices.

Ideally, for e.g. twice as many parameters, we expect the fit to increase a lot. If the fit is only marginally better, the increased number of parameters is likely to result in a poorer performance due to overfitting. These two pieces of information together hence enable to predict the outcomes of subsequent experiments to a certain degree.

A 1 minute-long real-world noise signal was taken from each noise environment in the dataset described in Section 5. Note that a square array geometry was used, to which the BND model is applicable. The empirical noise covariance matrix Φ vv (ω) was computed by temporal averaging of v(τ, ω)v H (τ, ω) over the whole duration of each signal. We define the discrepancy index between the noise data and V (ω) as the average over frequency and noise environments of

P ⊥ ω [Φ vv ](ω) F Φ vv (ω) F . ( 27
)
Inria Table 1: The dimensions of H and each noise matrix subspace V (ω) as a function of the number of microphones.

# microphones H uncor coh BND real

M M 2 M 1 M M (M + 1)/2
This quantity is the distance between Φ vv (ω) and V (ω) normalized by Φ vv (ω) F , where the normalization is aimed to remove the dependency on the scale of Φ vv (ω).

The dimensions of V (ω) and H as functions of M are shown in Table 1. The fixed noise coherence model (denoted by coh) has one dimension independently of M . The dimensions of spatially uncorrelated noise model (denoted by uncor) and the BND model grow linearly w.r.t. M , whereas that of the real-valued noise covariance model (denoted by real) grow quadratically.

Fig. 1 plots the discrepancy index of each model versus its dimensionality. A model closer to the origin is a good model that is able to fit real-world noise better with a smaller number of parameters. The real-valued noise covariance model gave the smallest discrepancy index of 0.16, but its largest dimension can lead to overfitting. In comparison, the BND model has only 0.4 time as high a dimension with an slight increase of 0.06 in the discrepancy index. Furthermore, its discrepancy index of 0.22 was lower than those of the remaining two models. Therefore, this model is expected to work quite well, provided that a crystal array geometry is available. The spatially uncorrelated noise model gave the highest discrepancy index of 0.57. This poor fit is due to the high spatial correlation of real-world noise. The fixed noise coherence model has only dimension 1, but nevertheless it fitted noise reasonably well with a discrepancy index of 0.36.

Target Covariance Estimation Based on Matrix-Space Sparsity

As pointed out in Section 2, it is essential in MUSIC to accurately estimate the orthogonal complement of the target signal subspace S ⊥ (ω), and this boils down to the estimation of the target spatial covariance matrix Φ cc (τ, ω). In this section, we propose methods for estimating Φ cc (τ, ω) from the observed spatial covariance matrix Φ xx (τ, ω). As a direct consequence of the proposed unified noise model, the component of Φ cc (τ, ω) lying in the orthogonal complement V ⊥ (ω) of the noise matrix subspace V (ω) is easily obtained as

P ⊥ ω [Φ cc ](τ, ω) = P ⊥ ω [Φ xx ](τ, ω), (28)
Therefore, the problem becomes that of estimating the remaining component

P ω [Φ cc (τ, ω)] in V (ω).
In the machine learning literature, matrix completion techniques [START_REF] Srebro | Weighted low-rank approximations[END_REF][START_REF] Toh | An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems[END_REF][START_REF] Candès | Exact matrix completion via convex optimization[END_REF][START_REF] Ji | An accelerated gradient method for trace norm minimization[END_REF] have been proposed to recover a low-rank matrix from part of its entries. We extend these techniques to recovery of the low-rank matrix Φ cc (τ, ω) from its projection P ⊥ ω [Φ cc ](τ, ω) under a positive semidefiniteness constraint. Here, the positive semidefiniteness constraint is important, so that we can identify S ⊥ (ω) as the null space of the estimated matrix.

We present two methods based on LMC (Section 4.1) and TNM (Section 4.2), which are applied in each time-frequency bin. The former is based on the main assumption that an upper bound R of rank cc (τ, ω) is given, while the latter does not require that information. the rest of this section, the time-frequency indices (τ, ω) are omitted for simplicity.

Target covariance estimation based on Low-rank Matrix Completion (LMC)

Instead of regarding P ⊥ [Φ xx ] as exactly noise-free, we leave some room for possible errors due to the misestimation of Φ xx or to the possible inaccuracy of the noise model V (ω). Specifically, we consider the following constrained minimization problem:

min Φcc Ψ comp (Φ cc ) P ⊥ [Φ cc ] -P ⊥ [Φ xx ] 2 F (29) s.t. Φ cc ∈ H positive semidefinite, rank(Φ cc ) ≤ R.
We rely on the following lemma, which can be proved in line with [START_REF] Eckart | The approximation of one matrix by another of lower rank[END_REF]:

Lemma 1 Consider the following optimization problem min X X -Y 2 F , (30)
s.t. X ∈ H positive semidefinite, rank(X) ≤ R,

Inria

where Y ∈ H . Denote the eigenvalue decomposition of Y by

Y = U ΣU H , ( 31
)
where U ∈ C M ×M is unitary and Σ ∈ R M ×M is diagonal and composed of

σ 1 ≥ • • • ≥ σ M .
Then the optimal solution to ( 30) is given by

X = U max{Σ R , 0}U H , ( 32
)
where

Σ R diag(σ 1 , . . . , σ R , 0, . . . , 0), ( 33
)
with diag(α 1 , . . . , α M ) denoting the M × M diagonal matrix composed of α 1 , . . . , α M and max{•, 0} the operation of replacing the negative entries of a matrix with zeros.

Compared to ( 30), ( 29) includes an orthogonal projection P ⊥ ω . Therefore, from the principle of Majorization-Minimization (MM) [START_REF] Hunter | A tutorial on MM algorithms[END_REF], we design the auxiliary function

Ψ + comp (Φ cc , Z) Ψ comp (Φ cc ) + P[Φ cc ] -Z 2 F , (34)
with an auxiliary variable Z ∈ V , so that

Ψ comp (Φ cc ) = arg min Z Ψ + comp (Φ cc , Z). ( 35
)
Indeed, (34) can be rewritten as

Ψ + comp (Φ cc , Z) = Φ cc -Y 2 F , (36)
which has the same form as [START_REF] Ma | Fixed point and Bregman iterative methods for matrix rank minimization[END_REF], where

Y Z + P ⊥ [Φ xx ]. ( 37
)
The MM algorithm amounts to iteratively applying the following update rules:

Z ← arg min Z Ψ + comp (Φ cc , Z) s.t. Z ∈ V , (38)
Φ cc ← arg min Φcc Ψ + comp (Φ cc , Z) (39)
s.t. Φ cc : Hermitian positive semidefinite, rank(Φ cc ) ≤ R.

The solution to (39) is given by

Φ cc ← U max{Σ R , 0}U H , ( 40
)
where U and Σ are defined in the same manner as in Lemma 1 using the eigenvalue decomposition of Y . On the other hand, (38) amounts to

Z ← P[Φ cc ], (41)
because only the second term in (34) depends on Z.

RR n 8120

We iterate the algorithm until a preset maximum number of iterations K is reached or

Φ (k+1) cc -Φ (k) cc F Φ (k) cc F < ǫ, ( 42
)
where ǫ is a preset small constant, which means Φ The algorithm is summarized in the following:

Algorithm 1 Set Φ (0) cc = Φ xx , K ≥ 1, and 1 ≤ R < M . Set k ← 0. repeat Y (k) = P[Φ (k) cc ] + P ⊥ [Φ xx ]. Calculate the eigenvalue decomposition of Y (k) : Y (k) = U (k) Σ (k) U (k)H
, where U (k) is unitary and Σ (k) is real-valued and diagonal with the diagonal entries σ

(k) 1 , . . . , σ (k) M arranged in nonincreasing order. Φ (k+1) cc = U (k) max{Σ (k) R , 0}U (k)H . k ← k + 1. until k = K or (42).
As a general property of the MM algorithm, we have the following theorem:

Theorem 1 The sequence Ψ comp (Φ (k) cc ), k = 1, 2, . . . generated by Algorithm 1 is nonincreasing. Indeed, defining Z (k) by Z (k) arg min Z Ψ + (Φ (k) cc , Z), (43)
we have

Ψ(Φ (k) cc ) = Ψ + (Φ (k) cc , Z (k) ). (44)
Furthermore, defining

Φ (k+1) cc by Φ (k+1) cc arg min Φcc Ψ + (Φ cc , Z (k) ), (45)
we have

Ψ + (Φ (k+1) cc , Z (k) ) ≤ Ψ + (Φ (k) cc , Z (k) ). (46)
Therefore, from (44) and ( 46) and from

Ψ(Φ (k+1) cc ) ≤ Ψ + (Φ (k+1) cc , Z (k) ), (47)
we have

Ψ(Φ (k+1) cc ) ≤ Ψ(Φ (k) cc ). (48)
From Theorem 1, the convergence of Ψ comp (Φ cc ) to a local minimum is guaranteed, because Ψ comp (Φ cc ) > 0.

This algorithm can be regarded as an extension of Srebro's algorithm [START_REF] Srebro | Weighted low-rank approximations[END_REF]. The extension is twofold. First, we consider the completion of a missing noise matrix subspace instead of missing entries. Second, we consider the completion of a complex-valued matrix with an Hermitian positive semidefiniteness constraint instead of the completion of a real-valued matrix without such a constraint. We propose an alternative algorithm that does not require the upper bound R on the rank. This is advantageous because the the upper bound is not always given in practice due to an unknown number of sources and/or to reverberation. We utilize the trace norm Φ cc * , i.e. the sum of the singular values, to construct a cost function that favors a low-rank solution. The trace norm is known to be a convex relaxation of the rank function [START_REF] Toh | An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems[END_REF], and can be regarded as the matrix version of the popular l 1 -norm for vector. Specifically, we consider the following optimization problem:

min Φcc Ψ trace (Φ cc ) 1 2 P ⊥ [Φ cc ] -P ⊥ [Φ xx ] 2 F + µ Φ cc * , s.t. Φ cc ∈ H positive semidefinite, ( 49
)
where µ is a positive weight.

(49) can be solved efficiently by generalizing Toh's algorithm [START_REF] Toh | An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems[END_REF] to the completion of a subspace of a complex-valued matrix subject to a Hermitian positive semidefiniteness constraint. The value of µ is decreased at each iteration as proposed in [START_REF] Toh | An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems[END_REF]:

µ (k) = max{0.7 k , 10 -4 } × P ⊥ [Φ xx ] F . (50)
The stopping condition is defined in the same manner as in Algorithm 1.

Algorithm 2 Set Φ (0) cc = Φ (-1) cc = Φ xx ; t (-1) = t (0) = 1. k ← 0. repeat µ ← max{0.7 k , 10 -4 } × P ⊥ [Φ xx ] F . Z (k) = Φ (k) cc + t (k-1) -1 t (k) (Φ (k) cc -Φ (k-1)

cc

).

Y (k) = P[Z (k) ] + P ⊥ [Φ xx ]. Calculate the eigenvalue decomposition of Y (k) : Y (k) = U (k) Σ (k) U (k)H
, where U (k) is unitary and Σ (k) is real-valued and diagonal. Φ

(k+1) cc = U (k) max{Σ (k) -µI, 0}U (k)H . t (k+1) = 1 + √ 1 + 4t (k)2 2 . k ← k + 1. until k = K or (42)
The following theorem guarantees the convergence of Ψ trace (Φ cc ) in Algorithm 2 to a global minimum of (49).

|Ψ trace (Φ (k) cc ) -Ψ trace (Φ opt cc )| ≤ 2 Φ (0) cc -Φ opt cc 2 F (k + 1) 2 , ( 51
)
where Φ opt cc is a solution to (49). This can be proven in line with [START_REF] Toh | An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems[END_REF], because Lemmas 2.1-2.3 in [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF] and Theorem 3 in [START_REF] Ma | Fixed point and Bregman iterative methods for matrix rank minimization[END_REF] for real-valued vectors can be extended to complex-valued matrices.

Signal subspace identification

The target signal subspace S (ω) and its orthogonal complement S ⊥ (ω) are calculated as follows

S = span{u m } L m=1 , (52)
S ⊥ = span{u m } M m=L+1 , (53)
where u m denotes the m-th column of U .

5 Large-scale evaluation with real-world noise

Created dataset

We created a dataset of multichannel reverberant speech mixtures with realworld noise to evaluate the proposed algorithms. Noise was recorded in a station square in Japan with a 4-channel square array with a diameter of 5 cm [START_REF] Ito | Designing the Wiener post-filter for diffuse noise suppression using imaginary parts of inter-channel cross-spectra[END_REF]. To better control experimental conditions, the target components were simulated by the image method [START_REF] Allen | Image method for efficiently simulating small-room acoustics[END_REF] implemented in Roomsimove1 ) and added to noise.

In the simulation, we assumed the room dimensions to be 3.3 × 7.8 × 2.4 m, the array to be at the room center, and the target sources to be at 1 m from the array and at the height of 1.2 m. The dry speech sources were taken from the ATR Japanese dataset [START_REF] Kurematsu | ATR Japanese speech database as a tool of speech recognition and synthesis[END_REF]. The mixtures were 10 s long sampled at 16 kHz. The dataset includes (3 × 3 -1) × 3 × 3 = 72 mixtures with various values of the following four parameters:

the number L of target sources: 2, 4, or 6, the azimuth separation between successive sources: 30 • , 60 • , or 90 • (90 • was not considered for L = 6.), the absorption coefficient of the walls: 0.4, 0.7, or 1.0 (i.e. reverberation time RT 60 : 186, 79, or 0 ms), the input SNR: 10, 0, or -10 dB.

The input SNR refers to the energy ratio between a target signal and noise at the first microphone, where all sources were assumed to have the same energy.

Methods compared and evaluation metric

We compared the following ten methods differing in the way the basis vectors e i (ω) of S ⊥ (ω) were identified.

Conventional MUSIC with the spatially white noise model (denoted by conv-white). e i (ω) were identified by EVD of Φ xx .

Conventional MUSIC with the fixed noise coherence model in Section 3.2.2 (denoted by conv-coh). e i (ω) were identified by GEVD of the matrix pencil (Φ xx , Γ). Proposed MUSIC based on TNM with the four noise models in Section 3 (denoted by trace-uncor, trace-coh, trace-BND, or trace-real). e i (ω) were identified by EVD of Φ cc (τ, ω) estimated by TNM.

We estimated the target azimuths by picking the L largest peaks in the MUSIC spectrum, and assessed their precisions [START_REF] Blandin | Multi-source TDOA estimation in reverberant audio using angular spectra and clustering[END_REF], which is the ratio of correct azimuth estimates and L (assumed to be known). Here, the correct estimates are defined as those within 5 • from a true azimuth.

We used the cylindrically isotropic noise model [START_REF] Elko | Spatial coherence functions for differential microphones in isotropic noise fields[END_REF] to calculate Γ in convcoh, comp-coh, and trace-coh. We set ω min = 94 Hz and ω max = 8 kHz in [START_REF] Mandel | An EM algorithm for localizing multiple sound sources in reverberant environments[END_REF], so as to remove the low frequency bins with extremely low SNRs. We divided the data duration into B segments, calculated the wideband MUSIC spectrum in each segment, and geometrically averaged them to obtain a single MUSIC spectrum.

Experimental results

Table 2 shows the precision for each method averaged over all mixtures. We set R = 2 and B = 16, because R ≥ 2 resulted in a reasonable performance, and B had only a little impact in our preliminary experiments. The trace-BND algorithm gave the highest precision of 0.63 higher than that of state-of-the-art conv-coh by 0.05. The precisions of comp-coh and trace-coh were comparable to that of conv-coh based on the same noise model. The precisions of the other proposed algorithms were lower than that of conv-coh. The poor performance with the uncorrelated noise model and the real-valued noise covariance model is accounted for by high spatial correlation of real-world noise and the high model dimension, respectively. For this reason, the following evaluation is focused on the proposed trace-BND, comp-coh, and trace-coh and on the conventional conv-white and conv-coh as baselines.

Figure 2 shows examples of MUSIC spectra for two sources under a highly reverberant and noisy condition: RT 60 = 186 ms; SNR: -10 dB. The true azimuths are depicted by the vertical lines in the figure. The proposed three algorithms resulted in smaller estimation error than the conventional methods, and trace-BND resulted in the most accurate estimation.

Figs. 3 and4 show the impact of the input SNR and reverberation. We set R = 2 and B = 16. As we see in Fig. 3, the precisions decreased when the input SNR decreased. While the precision of trace-BND was comparable to those of the methods based on the fixed noise coherence model at 10 dB, it was higher by 0.06-0.09 at 0 dB and -10 dB, showing that it is more robust against noise. As seen from Fig. 4, the precision of trace-BND was comparable to those of the methods based on the fixed noise coherence model for the absorption coefficients of 1 and 0.7, whereas it was higher by 0.12-0.18 for that of 0.4. This robustness of trace-BND against reverberation is likely because late reverberation is uncorrelated to the direct path and can be regarded as diffuse noise, so that it is well explained by the BND model.

Conclusion

This paper proposed a framework for robust DOA estimation of multiple target sources in diffuse noise. Our approach is based on estimating the target spatial covariance matrix from the observed spatial covariance matrix. This enables the identification of the orthogonal complement of the target signal subspace as the null space of the estimated matrix, whereby enabling accurate DOA estimation via MUSIC. We presented a unified framework for modeling noise covariance in a matrix space. This enables us to obtain a noise-free component of the observed spatial covariance matrix through projecting it onto the orthogonal complement of the noise matrix subspace. This noise model was shown to include four state-of-the-art diffuse noise models as special cases, namely the spatially uncorrelated noise model, the fixed noise coherence model, the blind noise decorrelation model, and the real-valued noise covariance model. We express the target spatial covariance matrix as the sum of two components: one belonging to the noise matrix subspace and one orthogonal to it. The latter is obtained by the projection, whereas the former is reconstructed exploiting the low-rankness of the target spatial covariance matrix. We proposed two algorithms for this matrix completion, namely the low-rank matrix completion and the trace norm minimization algorithms. We evaluated the performance of the proposed methods using a large dataset, and showed that the proposed trace norm minimization algorithm with the blind noise decorrelation model outperformed the conventional MUSIC methods essentially in terms of the precision.

Figure 1 :

 1 Figure 1: Discrepancy index vs. model dimension for each noise model on the noise dataset of Section 5 (M = 4).

  cc has become almost constant. As for the initial value Φ (0) cc , we propose to simply use Φ xx .

Inria

4. 2

 2 Target covariance estimation based on Trace Norm Minimization (TNM)

Theorem 2

 2 Let Φ (k) cc (k = 1, 2, . . . ) be the sequence generated by Algorithm 2. Then,

Figure 2 :

 2 Figure 2: Examples of MUSIC spectra. L = 2; angle between adjacent sources: 60 • ; absorption coefficient: 0.4; SNR: -10 dB; R = 2; B = 1.

Figure 3 :

 3 Figure 3: Average precision as a function of the SNR for the conventional and proposed methods for R = 2 and B = 16.

Inria

Figure 4 :

 4 Figure 4: Precision as a function of the absorption coefficient of the walls for the conventional and proposed methods for R = 2 and B = 16.

Table 2 :

 2 Precision averaged over all mixtures for R = 2 and B = 16.Proposed MUSIC based on LMC with the four noise models in Section 3 (denoted by comp-uncor, comp-coh, comp-BND, or comp-real). e i (ω) were identified by EVD of Φ cc (τ, ω) estimated by LMC.

	method	conventional		LMC			TNM
	noise model white coh uncor coh BND real uncor coh BND real
	precision	0.51	0.58	0.41	0.59	0.42	0.48	0.51	0.58 0.63 0.35

RR n 8120

E. Vincent and D. R. Campbell, "Roomsimove," (2010, Nov.

[START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF].[Online]. Available: http://www.irisa.fr/metiss/members/evincent/software. Inria

Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399

Acknowledgments

This work was supported by Grant-in-Aid for JSPS Fellows 22-6927 from MEXT, Japan, and by INRIA under the Associate Team Program VERSAMUS


Vinh Le Quang

Olivier Tran

Olivier Berder
email: oberder@irisa.fr

Sentieys
email: sentieys@irisa.fr

Non-Regenerative Full Distributed Space-Time Codes in Cooperative Relaying Networks

Distributed space-time codes (DSTC) are often used in cooperative relaying networks whose relays can support a single antenna due to the limited physical size. In this paper, full DSTC protocol in which there is a data exchange between relays before forwarding signals to destination is proposed to improve the performance of a cooperative relaying system. A lower bound for the average symbol error probability (ASEP) of full DSTC cooperative relaying system in a Rayleigh fading environment is provided. In the case when the Signal to Noise Ratio (SNR) of the relay-relay link is much greater than that of the source-relay link, the upper bound on ASEP of this system is also derived. From the simulations, we show that the average SNR gain of full DSTC system over DSTC system is 3.8dB and the maximum SNR gain is 5dB when the relay-relay distance is small and the relays are in the middle of the source and the destination. The effect of the distance between the relays shows that the performance does not degrade so much as the distance between relays is lower than a half of the source-destination distance. Moreover, we also show that when the error synchronization range is lower than 0.5, the impact of the transmission synchronization error of the relay-destination link on the performance is not considerable.

I. INTRODUCTION

Recently, relay transmission has been identified as one of the core technologies that could enable robust and highreliable information transfer over challenging wireless environment. By positioning relays between source and destination, the relaying networks attract significant interest in wireless communication as it achieves the performance of Multi-Input Multi-Output (MIMO) systems through multiple relays instead of using multiple antennas at terminals. In [START_REF] Laneman | Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks[END_REF], via the outage probability analysis of distributed space-time codes (DSTC) and repetition-based cooperative diversity protocol, the DSTC is shown to have full spatial diversity. In [START_REF] Barbarossa | Distributed space-time coding for multihop networks[END_REF], a DSTC multihop system in which source and relay share their antennas to create a virtual transmit array is considered in terms of average Bit Error Rate (BER) and achievable rate. Beside, in [START_REF] Anghel | On the performance of distributed spacetime coding systems with one and two non-regenerative relays[END_REF], the performance of DSTC systems with one and two nonregenerative relays is given, while the performance of DSTC systems with regenerative relays is done in [START_REF] Anghel | Distributed space-time cooperative systems with regenerative relays[END_REF]. The DSTC protocol for wireless multi-hop networks is lately designed and the performance for communication over more than two hops is analyzed in [START_REF] Vajapeyjan | Performance analysis of distributed spacetime coded protocols for wireless multi-hop communications[END_REF].

In the present paper, a cooperative relaying system having a source, a destination and two relays all equipped with single antenna is considered. The full DSTC protocol for cooperative relaying networks in which the relays exchange data with each other and then forward the signals to destination using non-regenerative DSTC is proposed to improve the performance of the system in terms of BER. To the best of our knowledge, this is the first time the data exchange between relays in cooperative relaying networks is presented and analyzed. The lower bound expression on average symbol error probability (ASEP) of full DSTC cooperative relaying system is derived. Beside, the upper bound expression on ASEP is given in the case that the distance between relays, drr is small enough in comparison with the source-destination distance d sd to be able to neglect the transmission errors in the data exchange of two relays. Simulation results will show that this condition is satisfied when drr = 0.1d sd . In addition, the effects of some practical problems, such as the relative distance of relays, the distance between two relays and the transmission synchronization errors of the relay-destination link, are also considered and simulated in comparison with DSTC cooperative relaying model in which there is no data exchange between the relays.

The remainder of this paper is organized as follows. In Section II, the protocol of full DSTC cooperative relaying system is thoroughly described. Then the average error probability analysis is considered in Section III. Section IV presents the simulation results, while Section V proposes a discussion and conclusion.

II. SYSTEM MODEL

The full distributed space-time coded cooperative relaying network considered in this paper includes one source, one destination and two relays which all have only one antenna. After receiving the signals from the source, the two relays exchange data with each other, then combine all the received signals using Maximum Ratio Combining (MRC) technique and finally forward them to destination using DSTC. We assume that all communications are performed over flat Rayleigh fading channel. The channel coefficients remain the same for two consecutive time intervals. Statistically, we model the channel coefficient h ij with i ∈ {s, r 1 , r 2 }, j ∈ {r 1 , r 2 , d} and i = j as zero mean, independent, complex Gaussian random variables with variances Ω ij . Similarly, we model the AWGN noise z ij [n] as zero mean mutually independent complex Gaussian random variable with variance N ij . Without loss of generality we assume that N ij = 1, ∀i, j. The transmission protocol of this model can be described as below. Firstly, the source transmits signals to the relays and the destination at the same time. The received signals at relays and destination can be represented as y sr1 , y sr2 and y sd respectively

y sj (n) = √ ǫsh sj x[n] + z sj [n], j ∈ {r 1 , r 2 , d} (1)
where x[n] is the source transmitted signal and ǫs is the transmitted source power. Secondly, the two relays exchange their data with each other. The received signals at relay R 1 from relay R 2 and vice versa, respectively symbolized as yr 2 r 1 and yr 1 r 2 , are given by

y ij (n) = h ij G si y si [n] + z ij [n], i, j ∈ {r 1 , r 2 }, i = j (2)
where G si is the maximum gain [START_REF] Laneman | Energy-efficient antenna sharing and relaying for wireless networks[END_REF] which satisfies the output power constraint of relay i and can be chosen as

G si = ǫ i ǫs|h si | 2 + 1 , i ∈ {r 1 , r 2 } (3)
with ǫ i the transmitted power of relay i.

Thirdly, each relay uses a Maximum Ratio Combiner to combine the signals received from source and from the other relay

u j (n) = h * sj y sj [n] + h * ij G * si h * si |h ij | 2 |G si | 2 + 1 y ij [n] (4)
where i, j ∈ {r 1 , r 2 } and i = j.

Finally, the two relays use non-regenerative DSTC protocol to transmit simultaneously the Alamouti re-encoded signals

U U U = G r 1 d ur 1 [2k] G r 2 d ur 2 [2k + 1] -G * r 1 d u * r 1 [2k + 1] G * r 2 d u * r 2 [2k]
to the destination. The signals received from the relays at destination, y rd

y rd y rd = y rd [2k] y rd [2k + 1] T , are y rd y rd y rd = U U Uh rd h rd h rd + z rd z rd z rd , (5)
where h rd

h rd h rd = h rd [2k] h rd [2k + 1]
T is the Rayleigh channel coefficient vector of the relay-destination link and z rd

z rd z rd = z rd [2k] z rd [2k + 1]
T is the AWGN noise vector of the relaydestination link.

Similarly, to prevent the saturation, the maximum gain G jd at relay j is chosen as

G jd = ǫ j ǫs|a j | 2 + a j , j ∈ {r 1 , r 2 } ( 6
)
where

a j = |h sj | 2 + |h ij G si h si | 2 |h ij G si | 2 +1
i, j ∈ {r 1 , r 2 }, i = j and ǫ j is the transmitted power of relay j.

At the destination, the Alamouti receiver is used to process the signals received from R 1 and R 2 . Then, the output of the Alamouti receiver is combined with y sd [2k] y sd [2k + 1] T using Maximum Ratio Combining technique.

III. AVERAGE SYMBOL ERROR PROBABILITY ANALYSIS

With the fading realizations h ij , i ∈ {s, r 1 , r 2 } , j ∈ {r 1 , r 2 , d} and i = j, the Signal to Noise Ratio at terminal j after receiving signals from terminal i is found as

γ ij = ǫ i |h ij | 2 /N ij which has exponential distribution with the mean γ ij = ǫ i Ω ij N ij .
The density probability function of γ ij is represented by

fγ ij (γ) = 1 γ ij e -γ γ ij (7)
With the choice of G si , i ∈ {r 1 , r 2 } in (3), the post-detection SNR at relay i can be referred as

γ i = γ si + γ sj γ ji γ sj + γ ji + 1 i, j ∈ {r 1 , r 2 }, i = j (8)
Similarly, the post-detection SNR at destination is

γ d = γ sd + ǫs j∈{r 1 ,r 2 } |h jd G jd a j | 2 j∈{r 1 ,r 2 } |h jd G jd | 2 a j + 1 (9)
Because the distribution of γ d is not easy to find, we would like to find and use the distribution of γ L and γ U to upper bound and lower bound the ASEP of full DSTC cooperative relaying system. We find that

γ L < γ d < γ U (10)
where γ L is the lower bound of γ d only when

γ ij ≫ γ sj , i, j ∈ {r 1 , r 2 }, i = j γ L = γ sd + γsr 1 γ r 1 d γsr 1 + γ r 1 d + 1 + γsr 2 γ r 2 d γsr 2 + γ r 2 d + 1 = γ sd + γ L sr 1 d + γ L sr 2 d (11)
and γ U is the upper bound of γ d

γ U = γ sd + (γsr 1 + γsr 2 )γ r 1 d γsr 1 + γ r 1 d + γsr 2 + 1 + (γsr 2 + γsr 1 )γ r 2 d γsr 1 + γsr 2 + γ r 2 d + 1 = γ sd + γ U sr 1 d + γ U sr 2 d (12)
From [START_REF] Emanian | Multi-user spatial diversity in a shadow-fading environment[END_REF], we find that the lower bound on ASEP of full DSTC cooperative relaying system is P L γ d = P γ U and the upper bound on ASEP of full DSTC cooperative relaying system is

P U γ d = P γ L . We have P L γ d = P γ U < Pγ d < P γ L = P U γ d (13)
In order to find the lower bound for the performance of full DSTC cooperative relaying system, we first derive the moment generating functions (MGFs) of γ U sid , i ∈ {r 1 , r 2 } which is proved in Appendix A.

M γ U sid (s) =e α i /2 2 p i √ p i p i γ id -σ i + σ i 2γ id δJ 0 δβ + σ i γ id + 2 δ 2 J 0 δαδβ + 2 p 2 i 2p i γ id + σ i δ 2 J 0 δα 2 + p i γ id δJ 0 δα - σ i 4 J 0 ( 14
)
where t 0 = γ id , σ = σ i = γ id + γ si , p = p i = γ id γ si , and α i = (σ i -p i s)/p i . The formula of J 0 , δJ 0 δα , δJ 0 δβ , δ 2 J 0 δα 2 and δ 2 J 0 δαδβ can be found in Appendix A. Note that with the assumption γsr = γsr 1 = γsr 2 , the random variable γsr = γsr 1 + γsr 2 will have the probability density function fγ sr (γ) = 1 γsr 2 γe -γ γsr . Then, the lower bound ASEP of full DSTC cooperative relaying system, P L γ d , for BPSK modulation can be derived based on MGF-based approach of [7, Eq. 9.12]

P L γ d = 1 π π/2 0 i∈{r 1 ,r 2 } M γ U sid (-g BP SK sin 2 θ ) 1 + g BP SK γ sd sin 2 θ dθ (15)
where g BP SK = 1 [START_REF] Simon | Digital Communication over Fading Channels[END_REF] and

Mγ sd = 1 + g BP SK γ sd sin 2 θ -1 [7, Tab 9.1].
Similarly, the upper bound ASEP of full DSTC cooperative relaying system, P U γ d , can be written

P U γ d = 1 π π/2 0 i∈{r 1 ,r 2 } M γ L sid (-g BP SK sin 2 θ ) 1 + g BP SK γ sd sin 2 θ dθ (16)
Fortunately, the moment generating functions of γ L sid , i ∈ {r 1 , r 2 } are already found in [8, Eq. 13]

M γ L sid (s) = 2 p i e α i /2 -2 δJ 0 δα - σ i p i δJ 0 δβ ( 17
)
where σ = γ id + γ si , p = γ id γ si and α i = (σ i -p i s)/p i . The formula of δJ 0 δα and δJ 0 δβ is given in Appendix A. On the other hand, for DSTC cooperative relaying model, where there is no data exchange between the two relays, the post-detection SNR at destination is found as

γ ′ d = γ sd + ǫs j∈{r 1 ,r 2 } |h jd G ′ jd a ′ j | 2 j∈{r 1 ,r 2 } |h jd G ′ jd | 2 + 1 (18)
where a

′ j = h sj and G ′ jd = h * sj |h sj | ǫ j ǫs|h sj | 2 +1
. We find that γ ′ d < γ L so P U γ d = P γ L is also the lower bound on ASEP of DSTC cooperative relaying system.

IV. SIMULATION RESULTS

For the convenience of the symbolization, we denote r the relative distance of relays which is defined by the ratio of the source-relay distance dsr and the source-destination distance d sd and let rr be the relative inter-distance of relays which is defined by the ratio of the relay-relay distance drr and the source-destination distance d sd . Besides, the power is equally allocated among transmitters, i.e. ǫ ij = ǫ, and the BPSK modulation is selected.

A. Bounds for full DSTC cooperative relaying system

For the comparison between the analytic results and simulation results, we first consider the special case of symmetric network in which the fading variances are identical, e.g, Ω ij = 1 ∀i, j. In this simulation, the condition that γ ij ≫ γ sj , i, j ∈ {r 1 , r 2 }, i = j is made sure. We can see from Fig. 2 that the performance simulation of full DSTC cooperative relaying system is exactly in the middle of the upper bound on ASEP, P U γ d and lower bound on ASEP, P L γ d . Moreover, from the simulation results, P U γ d is shown to be the lower bound on ASEP of DSTC cooperative relaying system.

B. Effect of the distance between two relays

For the following subsections, the effect of geometry is taken into account. Using a common model for path-loss, we set

Ω ij ∝ d -α
ij where d ij is the distance between terminal i and terminal j, and α is the path-loss exponent. In our work, α = 2 is used. Fig. 3 shows the effects on BER of full cooperative relaying system in the case that r = 0.5 versus the change of rr. We see that when rr = 0.1, the performance of the system is the same as the ideal case, rr = 0, where there is no errors in the inter-transmission between two relays. In fact, this case represents the most popular one in reality, for example: as dsr = 100m, we can choose without difficulty two relays so that drr = 10m because normally, the distance between two relays is small. Moreover, the performance of the system is not much degraded until rr = 0.5. Otherwise, when rr → ∞, i.e. rr = 1000 in Fig. 3, the performance of full DSTC cooperative relaying system logically goes to that of DSTC cooperative relaying system. Because, if the SNR of the relay-relay link is much worse than that of the source-relay link, the data exchange between two relays does not bring any gain in comparison with DSTC cooperative relaying system. So, the results here can illustrate the accuracy of the simulations.

C. Effect of the relative distance of relays

In this subsection, we take the effect of the relative distance of relay into account, as illustrated by Fig. 4, in the case of rr = 0.1. When the relative distance of relay is small, i.e. r = 0.1, the performance gain of full DSTC cooperative relaying system over DSTC cooperative relaying system is not considerable. However, when the relative distance of relay increases, that performance gain is much more significant. Fig. 5 shows the SNR gain of full DSTC cooperative relaying system over DSTC cooperative relaying system at BER = 10 -5 and rr = 0.1 as a function of the relative distance of relay, r. The reference performance is the performance of DSTC system in the case that r = 0.9. The average SNR gain of full DSTC cooperative relaying system over DSTC cooperative relaying system is 3.8dB. The maximum SNR gain of full DSTC cooperative relaying system obtained is 5dB in comparison with DSTC cooperative relaying system when r = 0.5.

D. Effect of transmission synchronization error

As we know, a cooperative system suffers a large impact from the lack of synchronization between distributed nodes. It is no exception for full DSTC cooperative relaying system and DSTC cooperative relaying system. Here, we face with the transmission synchronization errors of the relaydestination link. Using the same synchronization error model Fig. 6. Effect of transmission synchronization error on full DSTC cooperative relaying system and DSTC cooperative relaying system at the case of r = 0.5 and rr = 0.1.

as [START_REF] Nguyen | Impact of transmission synchronization error and cooperative reception techniques on the performance of cooperative mimo systems[END_REF] in which the interference inter-symbol (ISI) is just created by the four nearest neighbor symbols, the effect of transmission synchronization error on full DSTC cooperative relaying system and DSTC cooperative relaying system at the case of r = 0.5 and rr = 0.1 is considered in Fig. 6. In this simulation, the synchronization error δ k , [START_REF] Nguyen | Impact of transmission synchronization error and cooperative reception techniques on the performance of cooperative mimo systems[END_REF], is considered to have a uniform distribution in [-∆T /2, ∆T /2] with ∆T the error synchronization range. We see that the performance of two models is not much degraded until ∆T = 0.5. For full DSTC cooperative relaying model, the performance becomes saturated at ∆T = 0.7 while the performance of DSTC cooperative relaying model goes to saturation at ∆T = 0.8.

V. DISCUSSION AND CONCLUSION

In this paper, a full distributed space-time coded protocol for cooperative relaying model is proposed. High SNR gain of full DSTC cooperative relaying system over DSTC cooperative relaying system, as we illustrated in the above section, comes at a price of decreasing bandwidth efficiency because of the data exchange between two relays. Full DSTC protocol is, therefore, suitable for applications of wireless sensor networks (WSNs) where the most important constraint is not spectral efficiency, but energy efficiency. On the other hand, as compared to DSTC cooperative relaying system, full DSTC one requires more complicated signal processing. However, it is easier to synchronize the DSTC transmission from relays to destination in full DSTC cooperative relaying system than in DSTC cooperative relaying system by taking advantage of the data exchange between two relays. Due to the data exchange between two relays, we can add some information on the synchronization at the Medium Access Control (MAC) layer to synchronize the data transfer of two relays to destination.

In conclusion, this paper presents the advantages of full DSTC protocol in comparison with conventional DSTC protocol in terms of BER performance. The lower bound and upper bound on ASEP of full DSTC cooperative relaying system is also derived. We showed that the average SNR gain of full DSTC cooperative relaying system over DSTC cooperative relaying system is 3.8dB and the maximum SNR gain is 5dB when the relay-relay distance is small and the relays are in the middle of the source and the destination. In addition, the performance of full DSTC cooperative relaying system is not degraded so much when the relative inter-distance of relays, rr, is below 0.5. Finally, the impact of transmission synchronization errors of the relay-destination link is shown to be not considerable as the error synchronization range is below 50% of the symbol duration.

APPENDIX A PROOF OF (14)

To obtain the lower bound ASEP using MGF-based approach, we need to find the MGF of Γ(v, t) = vt v+t+1 , where v and t are two independent random variables with the probability density function (PDF)

fv(v) = 1 v 2 0 ve -v v 0 (19) ft(t) = 1 t 0 e -t t 0 (20)
The cumulative distribution function (CDF) of Γ(v, t) is found for γ > 0 based on [START_REF] Emanian | Multi-user spatial diversity in a shadow-fading environment[END_REF]

F Γ (γ) = 1 - 2 γ 2 + γ √ p 1 + γ t 0 e -σ p γ K 1 2 √ p γ 2 + γ - 2 p (γ 2 + γ)e -σ p γ K 0 2 √ p γ 2 + γ (21)
where σ = t 0 + v 0 , p = t 0 v 0 , and K l (.) denotes the modified Bessel function of the second kind with order l. The PDF of Γ(v, t) is found by taking the derivative of F Γ (γ) with respect to γ and using [11, Eq. 9. The moment generating function (MGF) is derived by using its definition and using the change of variable γ → γ 1 -1/2

where α = (σ -ps)/p, β = 2/ √ p and

J 0 = ∞ 1/2 e -αγ 1 K 0 β γ 2 1 -(1/2) 2 = 1 2c e -c 2 E 1 α -c 2 -e c 2 E 1 α + c 2 (24)
with c = α 2 -β 2 , Re{s} < σ/p + 2/ (p) and E 1 (.) the exponential integral function defined as [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]Eq. 5.1.1]. Note that J 0 is derived by using the integrand in [START_REF] Rappaport | Table of Integrals, Series and Products[END_REF]Eq. 646].

Taking the derivative of J 0 with respect to α or/and β with the respective order, we can get

δJ 0 δα = - α 4c 3 (2 + c)e -c

Fig. 1 .

 1 Fig. 1. Full Distributed Space Time Code Cooperative Relaying Model

Fig. 2 .

 2 Fig. 2. Upper bound and lower bound of full DSTC cooperative relaying system

Fig. 3 .

 3 Fig. 3. Effect of the distance between two relays with the relative distance of relay, r = 0.5

Fig. 4 .

 4 Fig. 4. Effect of the relative distance of relays on the performance of full DSTC and DSTC cooperative relaying system in the case of rr = 0.1

Fig. 5 .

 5 Fig. 5. The SNR gain of full DSTC cooperative relaying system over DSTC cooperative relaying system at BER = 10 -5 and rr = 0.1 with respect to the relative distance of relay, r.

  6.26]

	f Γ (γ) =	p	2 √ p	-	p t 0	-σ -1 +	σ t 0	+ 2 γ e	-σ p γ K 1	2 √ p	γ 2 + γ
		+	2 p 2		2p t 0	+ σ γ 2 +	p t 0	+ σ γ e	-σ p γ K 0	2 √ p	γ 2 + γ
												(22)


Quoc-Tuong Ngo

Olivier Berder

Pascal Scalart

Reducing the number of neighbors in the received constellation of d min precoded MIMO systems

Not only the minimum Euclidean distance but also the number of neighbors providing the dmin in the received constellation has an important role in reducing the error probability when Maximum Likelihood detection is considered. Assuming that the channel-state-information is available at the transmitter, a new precoder in which the rotation parameter has no influence is proposed for two independent data-streams. The expression of the new precoding strategy is less complex and the space of solution is, therefore, smaller. Simulation results over Rayleigh fading channel confirm a bit-error-rate improvement of the proposed solution in comparison with other precoders. The improvement depends on the channel characteristics and is more significant if the channel is dispersive.

I. INTRODUCTION

In recent years, many communication techniques have been developed to adapt the demand for high data rates in wireless transmission. One of the most popular technique, known as multiple-input multiple-output (MIMO), provides a large gain in capacity and achieves a higher spectral efficiency compared to single antenna links [START_REF] Telatar | Capacity of multi-antenna Gaussian channels[END_REF]. Spatial multiplexing is a simple MIMO technique over which multiple independent datastreams are transmitted. At the receiver of spatial multiplexing system, a low error rate Maximum Likelihood (ML) detection is usually utilized. However its performance depends on the behavior of the matrix channel. If the channel state information (CSI) is available at the transmitter, the linear precoding technique, where the transmitted symbols are premultiplied by a precoding matrix, is proposed to significantly improve the performance of the ML detection [START_REF] Vu | MIMO wireless linear precoding[END_REF].

Linear precoders can be used for a transmitter with full CSI [START_REF] Scaglione | Optimal designs for space-time linear precoders and decoders[END_REF], or limited feedback CSI [START_REF] Love | Limited feedback unitary precoding for spatial multiplexing systems[END_REF] from the receiver. They are designed to optimize various criteria such as maximizing the minimum singular value [START_REF] Scaglione | Optimal designs for space-time linear precoders and decoders[END_REF], maximizing the post-processing SNR [START_REF] Stoica | Maximum-SNR spatial-temporal formatting designs for MIMO channels[END_REF], and minimizing the mean square error (MSE) [START_REF] Sampath | Generalized linear precoder and decoder design for MIMO channels using the weighted MMSE criterion[END_REF]. These precoding matrices are all diagonal and lead to power allocation strategies. The authors in [START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF], [START_REF] Ngo | Minimum distance based precoder for mimo-ofdm systems using 16-qam modulation[END_REF], [START_REF] Ngo | 3-d minimum euclidean distance based sub-optimal precoder for mimo spatial multiplexing systems[END_REF] proposed a non-diagonal precoder which is based on the maximization of the minimum Euclidean distance (d min ) between two received symbols. The max-d min precoder achieves a significant biterror-rate (BER) improvement in comparisons with diagonal precoders, especially when a ML detection is considered at the receiver.

In this paper, we propose a new version of maximum d min based precoding, the N-d min precoder. This precoding strategy considers not only the minimum Euclidean distance but also the number of neighbors providing it. Simulation results for BPSK and QPSK modulations confirm the improvement of the new precoder on the BER performance in comparison with different traditional precoders.

The paper is organized as follows. Section II introduces the MIMO channel representation when a precoder is considered at the transmitter. The impact of the minimum Euclidean distance in the received constellation on the BER performance of the system is described in Section III. In Section IV, the optimization of the d min criterion which reduces the number of neighbors is detailed. Finally, we present in Section V the BER simulation results over a Rayleigh fading channel and the conclusions are given in Section VI.

II. SYSTEM MODEL

We consider a MIMO system with n T transmit and n R receive antennas. The system concerns the transmission of b independent datastreams over Rayleigh fading channel. The received signal is then given by

y = GHFs + Gν ( 1
)
where H is the n R × n T channel matrix, F is the n T × b precoding matrix, G is the b × n R decoding matrix, s is the b × 1 transmitted vector symbol, and ν is the n R × 1 additive Gaussian noise vector. If perfect channel state information (CSI) is considered at both the transmitter and receiver, it was shown that the channel matrix can be diagonalized by using a virtual transformation [START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF]. The precoder and decoder matrices are then decomposed as

F = F v F d and G = G d G v .
The received signal now can be re-expressed as

y = G d H v F d s + G d ν v (2)
where

H v = G v HF v = diag(σ 1 , ..., σ b ) is the b × b virtual channel matrix, ν v = G v ν is the b × 1 transformed additive Gaussian noise vector.
In the paper, an ML detection is considered at the receiver so the decoder matrix G d has no effect on the performance and is consequently assumed to be an identity matrix of size b. Therefore, the received signal can be simplified as

y = H v F d s + ν v . (3)
Let us note E s as the average transmit power. The power constraint is

trace{F d F * d } = E s . (4)
The precoding matrix F d is designed to adapt various forms of channel information. In the next section, we consider the impact of the minimum Euclidean distance on the error probability of the precoded MIMO system.

III. MINIMUM EUCLIDEAN DISTANCE ON PRECODER

Let us define a vector x = H v F d s, and denote by A ij the event that yx j < yx i when the symbol s i was sent at the transmitter. If the event A ij happens, there is an error detection. The received constellation is decoded correctly if yx i < yx j with ∀j = i when s i was sent. Then the average error probability can be defined by

P e = 1 M M i=1 P ei {s i sent} = 1 M M i=1 Prob{ M j=1 j =i A ij } ( 5
)
where M is the number of all possible transmitted vectors s.

The average error probability can be approximated by

P e 1 M M i=1 M j=1 j =i Prob{A ij } (6)
where

Prob{A ij } = Prob{ y -x j < y -x i | s i sent } = Prob{ x i + ν v -x j < x i + ν v -x i } = Prob{ ν v -(x j -x i ) < ν v }.
Let us define d ij = x j -x i and n v the projection of vector ν v onto the vector (x j -x i ), we have

Prob{ ν v -(x j -x i ) < ν v } = Prob{n v > d ij 2 } = Q d ij 2 √ N 0 = Q dij 2 √ N 0 × E s
where N 0 is the variance of the white Gaussian noise ν v , and dij is the normalized distance of vector (x j -x i ).

Therefore, the error probability can be simplified as

P e 1 M M i=1 M j=1 j =i Q dij 2 √ N 0 × E s . (7)
According to [START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF], we can appreciate the impact of the Euclidean distances on the BER performance of a MIMO system. Let us first consider the simplest case: there are only two Euclidean distances.

Lemma 1: For every d α < d β < d χ < d δ , we can find the value of R high enough satisfying the condition

Q(d α .R) + Q(d δ .R) > Q(d β .R) + Q(d χ .R) (8)
Proof : see Appendix A. It is obvious that we can improve the BER performance by increasing the minimum Euclidean distance of the received constellation. One should note that Q

(d β .R) > Q(d χ .R), so ∀d χ such that d α < d χ < d δ , we can obtain Q(d α .R) + Q(d δ .R) > 2 Q(d χ .R). (9)
This is an actual evidence that the optimized detection, in reality, is obtained when the minimum distance is reached by many Euclidean distances.

Lemma 2: With two arrays d αi and d βi which are sorted by increasing order, if d α1 < d β1 and k ≥ 2, we can find the value of R high enough such that

k i=1 Q(d αi .R) > k i=1 Q(d βi .R) (10)
Proof : see Appendix B. From the form of error probability in [START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF] and the remark in the Lemma 2, it can be concluded that the minimum Euclidean distance has a very important role in the BER improvement of the precoding strategies system. We can predict that the optimized precoder can be obtained when the minimum Euclidean distance on the received constellation is provided by many difference vectors.

IV. N-d min PRECODER

Let us note N i is the number of distances dij such that dij = d min where the minimum Euclidean distance d min is defined by

d 2 min = min s k ,s l ∈S,s k =s l H v F d (s k -s l ) 2 . ( 11
)
A numerical search over F d , which maximizes the minimum Euclidean distance obtained by many difference vectors, shows that the values of other Euclidean distances are much higher than the minimum distance when d min is optimized. In that case, the other distances have no much impact on the biterror-rate performance. The error probability in (7) can be then simplified as

P e ≈ 1 M M i=1 N i .Q dmin 2 √ N 0 × E s ≈ N dmin .Q dmin 2 √ N 0 × E s (12)
where

N dmin = 1 M M i=1 N i .
It is observed that to improve the BER performance of the precoding strategies system, we have to not only maximize the minimum Euclidean distance but also minimize the number of neighbors providing it. The new precoding strategy is, therefore, called as N-d min precoder.

To illustrate the method of optimization, let us consider a simple case: b = 2. By using a singular value decomposition, the authors in [START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF] and [START_REF] Ngo | Minimum distance based precoder for mimo-ofdm systems using 16-qam modulation[END_REF] simplified the virtual channel and precoding matrices as

H v = σ 1 0 0 σ 2 = ρ cos γ 0 0 sin γ (13)
where ρ = σ 2 1 + σ 2 2 and γ = arctan σ2 σ1 are the channel gain and channel angle, respectively.

F d = E s cos ψ 0 0 sin ψ cos θ sin θ -sin θ cos θ 1 0 0 e iϕ (14)
with 0 ≤ θ ≤ π/4 and 0 ≤ ψ, ϕ ≤ π/2. The parameter ψ controls the power allocation on the virtual subchannels, θ and ϕ correspond to scaling and rotation of the received constellation, respectively. When θ and ϕ are equal to 0, the precoding matrix is diagonal and equivalent to the power allocation strategies. We present, herein, an original idea not only considering d min and N dmin , simultaneously, but also reducing the complexity of the solution. It is realized that if the coefficients of the precoding matrix F D do not depend on the rotation parameter (ϕ = 0 or ϕ = π/2), the received constellation will have less distances which can reach the minimum Euclidean distance. The property could be explained by the non-rotated received constellation (or perpendicular rotated constellation) when a Quadrature Amplitude Modulation (QAM) is used at the transmitter.

For this reason, we propose a new precoding strategy in which we assume that the rotation parameter has no employ (ϕ = 0 or ϕ = π/2). By using the parameterized form of the precoder in (14), we are now looking for the angles ψ and θ which optimize the d min criterion for each channel angle γ. A numerical approach for MIMO system using BPSK and QPSK modulation, which is considered in the following of this section, confirms a bit-error-rate improvement of our new precoder.

A. For BPSK modulation

It is observed that the difference vector as given by the difference between the two transmitted vectors (s = s k -s l with s k = s l ) is a vector created by the elements of the set {0, 2, -2}. A numerical search over ψ and θ which optimize the minimum Euclidean distance for two independent datastreams shows that the N-d min precoder has the same form as the max-d min precoder presented in [START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF]

F D = E s 2 1 i 0 0 . ( 15
)
One should note that the N-d min solution pours power only on the strongest virtual sub-channels. The minimum Euclidean distance is then defined by

d 2 BP SK = 4E s ρ 2 cos 2 γ. (16)

B. For QPSK modulation

The transmitted symbols belong to the following set S =

1 √ 2 {1 + i, 1 -i, -1 + i, -1 -i}.
For QPSK modulation with two datastreams, the set of all difference vectors denoted as SQP SK contains 16 × 15 = 240 elements. By eliminating the collinear vectors, we can reduce the size of SQP SK to 14 elements.

A numerical search over ψ and θ which optimize the minimum Euclidean distance for each channel angle, shows that our precoder has two different expressions. The first one, denoted as F snr pours power only on the strongest virtual subchannel. The other, obviously, uses all two virtual subchannels to transmit symbols, and is denoted as F rec .

1) For the first expression: the power is concentrated only on the first virtual subchannel and the rotation parameter ϕ is not considered at the precoder. The form of the precoder F snr can be expressed as

F snr = E s 5 2 1 0 0 . ( 17
)
The optimized d min is provided by the difference vector

1 √ 2 [0 2
] T , and defined by

d 2 snr = 2 5 E s ρ 2 cos 2 γ. ( 18
)
The received constellation obtained by F snr looks like the 16-QAM constellation. Hence, the average number of neighbors providing d min is given by N dmin = 1 16 (4 × 2 + 8 × 3 + 4 × 4) = 3. This value is less than the number of the minimum Euclidean distances obtained by the precoder F r1 presented in [START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF] (N dmin = 1 16 (4×2+4×3+4×4+4×5) = 3.5). However, the distances d min provided by two precoders remain very close. This explains why the new precoder has a slight improvement of BER in comparison with the max-d min precoder (see section V).

2) For the second expression: the difference between two virtual subchannels is smaller than the case of F snr . A numerical search shows that the optimized solution is found when the angle θ = π/4 is fixed and ψ depends on the channel angle γ. Indeed, the optimization is obtained by three difference vectors

s1 = 1 √ 2 [0 2] T , s2 = 1 √ 2 [2 0] T and s3 = 1
By substituting ( 19) into ( 14), the precoder F rec is given by

F rec = E s 2 cos ψ 0 0 sin ψ 1 1 -1 1 . ( 20
)
The minimum Euclidean distance provided by F rec is then

d 2 rec = E s ρ 2 4 sin 2 γ 3 tan 2 γ + 1 (21)
Fig. 1 illustrates the received constellation of the precoder for QPSK modulation. It is observed that the average number of d min is defined by N dmin = 1 16 (4 × 4 + 8 × 5 + 4 × 6) = 5. In comparison with the precoder in [START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF] where N dmin = 1 16 (8 × 5 + 8 × 9) = 7, our new precoder has a good improvement. 3) The threshold γ 0 : To choose between F snr and F rec , we have to compare the error probabilities in [START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF] which are obtained by both precoders. It is observed that when the channel angle γ varies from 0 to π/2, the ratios of other distances to the distance d min are fixed. Furthermore, we realize that the minimum Euclidean distance d snr and d rec in (18, 21) is proportional to E s . For this reason, the threshold γ 0 is not constant and depends on the signal-to-noise ratio E s /N 0 . The angle γ 0 increases to γ c if the average transmit power E s augments. The critical angle γ c is given by

d 2 snr = d 2 rec ⇔ 2 5 cos 2 γ c = 4 sin 2 γ c 3 tan 2 γ c + 1 ⇔ γ c = atan 1/7. (22)
V. SIMULATION RESULTS Fig. 2 illustrates the normalized minimum Euclidean distance d min of the new precoder N-d min and other precoders in the case of QPSK modulation. The average transmit power E s for diagonal precoders is choosen large enough such that the power is always allocated on both virtual subchannels. It is observed that the N-d min solution is better than WaterFiling [START_REF] Telatar | Capacity of multi-antenna Gaussian channels[END_REF], max-λ min [START_REF] Scaglione | Optimal designs for space-time linear precoders and decoders[END_REF] and MMSE [START_REF] Sampath | Generalized linear precoder and decoder design for MIMO channels using the weighted MMSE criterion[END_REF] precoders in terms of d min .

The new precoder has a small difference of d min in comparison with max-d min precoder [START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF]. Furthermore, the difference remains constant for a small channel angle (γ < 17.28 o ). According to the improvement of not only the minimum Euclidean distance (except for max-d min , of course) but also the average number of d min , an increase of BER performance for N-d min precoder is expected for QPSK modulation. We consider a MIMO-OFDM system with n T = 3 transmit antennas and n R = 2 receive antennas, 128 subcarriers which occupy a bandwidth of 19 MHz. The channel matrix H is complex Gaussian and the noise element are additive white Gaussian.

Firstly, we compare the BER performances obtained by the new precoder N-d min and the max-d min solution. Fig. 3 shows a BER improvement of the precoder F snr and F rec in comparison with F r1 and F octa for small and large channel angle γ, respectively. We observe a large BER improvement of the precoder F rec compared to F octa , and a slight superiority of F snr in comparison with F r1 , although both new precoders are inferior in terms of d min . This result clearly demonstrates that the number of minimum Euclidean distances has an important role in reducing the error probability when an ML detection is considered at the receiver. The BER performance with respect to SNR for QPSK modulation is plotted in Fig. 4. One should note that the expression of N-d min precoder is associated to the max-d min solution with F snr and F rec are available in large and small dispersion of virtual channel, respectively. As expected, the N-d min precoder provides a significant improvement in term of BER compared to diagonal precoders. Furthermore, it has a slight improvement in comparison with max-d min precoder. This can be explained by the distribution of the channel angles γ: the MIMO system (3,2) uses more often the precoder F snr than F rec .

VI. CONCLUSION

In the first part of this paper, we investigated the impact of the minimum Euclidean distance on the performance of biterror-rate when a ML detection is considered at receive side. It is realized that the number neighbors providing d min has an important role in reducing the error probability. Therefore, a new precoder for MIMO transmission, which is based on the In the new precoding strategy, the rotation parameter ϕ is not considered. Hence, the degree of freedom in precoding matrix F d is decreased and the space of the solution is smaller. Not only reducing the complexity, the N-d min precoder presents also a significant improvement of BER compared to diagonal precoders such as MMSE, Waterfiling and max-λ min . In comparison with max-d min precoder, the new precoder provides a slight improvement. The BER enhancement depends on channel characteristics and is more significant if the virtual are far from dispersive. APPENDIX A PROOF OF LEMMA 1 For all d β < d χ , it is obvious that we can find a high value of R such that

d 2 χ -d 2 β + 2 R 2 log d β -d α d δ -d χ > 0 (
log d β -d α d δ -d χ > -(d 2 χ -d 2 β ).R 2 /2 ⇔ d β -d α d δ -d χ > e -(d 2 χ -d 2 β ).R 2 /2 ⇔ R(d β -d α ).e -d 2 β .R 2 /2 > R(d δ -d χ ).e -d 2 χ .R 2 /2 (24)
Using the monotonic decreasing property of the function e -x 2 /2 , we obtain

APPENDIX B PROOF OF LEMMA 2

The mathematical induction can be used to prove this lemma. One should note that Q(x) is a monotonic decreasing function. First we show that our statement holds for k = 2. Indeed, there are two cases: i) d α2 ≤ d β2 : it is obvious that Q(d α1 .R) > Q(d β1 .R) and Q(d α2 .R) ≥ Q(d β2 .R) with ∀R > 0, so we have

Q(d α1 .R) + Q(d α2 .R) > Q(d β1 .R) + Q(d β2 .R)
ii) d α2 > d β2 : obviously, this is the case of the Lemma 1.

Thus it has been shown that the lemma holds for k = 2. We assume that our statement is true for k. It must then be shown that our statement is true for k+1. Let us define d γ1 =

√ 2 [ 2 - 2 ]

 222 T . The three corresponding normalized distances can be expressed as   d2 x1 = A sin 2 θ + B cos 2 θ d2 x2 = A cos 2 θ + B sin 2 θ d2 x3 = A (cos θ -sin θ) 2 + B (cos θ + sin θ) 2where A = 4 cos 2 γ cos 2 ψ and B = 4 sin 2 γ sin 2 ψ. By considering d2 x1 = d2 x2 = d2 x3 in the interval value of θ and ψ,

Fig. 1 :

 1 Fig. 1: Received constellation of the precoder F rec for QPSK.

Fig. 2 :

 2 Fig. 2: Normalized minimum Euclidean distance for QPSK.

  γ < 15 o F snr , γ < 15 o F octa , γ > 35 o F rec , γ > 35 o

Fig. 3 :

 3 Fig. 3: BER comparison of max-d min and N-d min precoders.

Fig. 4 :

 4 Fig. 4: Uncoded BER performance for QPSK modulation. maximization of d min associated with the minimization of the neighbors providing it, has been introduced.In the new precoding strategy, the rotation parameter ϕ is not considered. Hence, the degree of freedom in precoding matrix F d is decreased and the space of the solution is smaller. Not only reducing the complexity, the N-d min precoder presents also a significant improvement of BER compared to diagonal precoders such as MMSE, Waterfiling and max-λ min . In comparison with max-d min precoder, the new precoder provides a slight improvement. The BER enhancement depends on channel characteristics and is more significant if the virtual are far from dispersive.

  23) with d β > d α and d δ > d χ . The inequality (23) can be rewritten as

e

  -x 2 /2 dx > (d β .R -d α .R).e -(d β .R) 2 /2 (25) (d δ .R -d χ .R).e -(dχ.R) 2 /2 > d δ .R dχ.R e -x 2 /2 dx (26)From (24), (25), and (26) we haved β .R dα.R e -x 2 /2 dx > d δ .R dχ.R e -x 2 /2 dx ⇔ Q(d α .R) -Q(d β .R) > Q(d χ .R) -Q(d δ .R) ⇔ Q(d α .R) + Q(d δ .R) > Q(d β .R) + Q(d χ .R).

1 3 (

 3 d α1 + d β1 ), and d γ2 = 2 3 (d α1 +d β1 ). It is clear that d α1 < d γ1 < d γ2 , so we can find value R 1 such that ∀R ≥ R 1 Q(d α1 .R) + Q(d α2 .R) > Q(d γ1 .R) + Q(d γ2 .R) (27) Since d γ2 < d β1 ≤ d β2 , so we get Q(d γ2 .R) > Q(d β2 .R).(28)Furthermore, we have d γ1 < d β1 . According to the statement in the case of k, there exist values R 2 which satisfies∀R ≥ R 2 Q(d γ1 .R)+ k+1 i=3 Q(d αi .R) > Q(d β1 .R)+ k+1 i=3 Q(d βi .R). (29)From (27), (28) and (29), it can be concluded that ∀R ≥ max(R 1 , R 2 ), we have k+1 i=1 Q(d αi .R) > k+1 i=1 Q(d βi .R).