Rémy Boyer

Oblique Projections for Direction-of-Arrival Estimation With Prior Knowledge

Estimation of directions-of-arrival (DOA) is an important problem in various applications and a priori knowledge on the source location is sometimes available. To exploit this information, standard methods are based on the orthogonal projection of the steering manifold onto the noise subspace associated with the a priori known DOA. In this paper, we derive and analyze the Cramer-Rao bound associated with this model and in particular we point out the limitations of this approach when the known and unknown DOA are closely spaced and the associated sources are uncorrelated (block-diagonal source covariance). To fill this need, we propose to integrate a priori known locations of several sources into the MUSIC algorithm based on oblique projection of the steering manifold. Finally, we show that the proposed approach is able to almost completely cancel the influence of the known DOA on the unknown ones for block-diagonal source covariance and for sufficient signal-to-noise ratio (SNR).

Performance Bounds and Angular Resolution Limit for the Moving Colocated MIMO Radar

To identify a target, the moving noncoherent colocated multiple-input multiple-output (MIMO) radar system takes advantage of multiple antennas in transmission and reception which are close in space. In this paper, we study the estimation performance and the resolution limit for this scheme in which each array geometry is described by the sample-variance of the sensor distribution. So, our analysis encompasses any sensor distributions, including varying intersensors distances or/and lacunar (missing sensors) configuration. As in the space-time MIMO model considered here the radar is moving, the target Doppler frequency cannot be assumed invariant to the target position/angle. The first part of this paper derives and analyzes closed form (nonmatrix) expressions of the deterministic Cramér-Rao lower bound (CRB) for the direction and the velocity of a moving target contaminated by a structured noise (clutter echoes) and a background noise, including the cases of the clutter-free environment and the high signal-to-noise ratio (SNR) regime. The analysis of the proposed expressions of the CRB allows to better understand the characterization of the target. In particular, we prove the coupling between the direction parameter and the velocity of the target is linear with the radar velocity. In the second part, we focus our study on the analytical (closed form) derivation and the analysis of the angular resolution limit (ARL). Based on the resolution of an equation involving the CRB, the ARL can be interpreted as the minimal separation to resolve two closely spaced targets. Consequently, the ARL is a key quantity to evaluate the performance of a radar system. We show that the ARL is in fact quasi-invariant to the movement of the MIMO radar.

Fast Multilinear Singular Value Decomposition for Structured Tensors

The higher-order singular value decomposition (HOSVD) is a generalization of the singular value decomposition (SVD) to higher-order tensors (i.e., arrays with more than two indices) and plays an important role in various domains. Unfortunately, this decomposition is computationally demanding. Indeed, the HOSVD of a third-order tensor involves the computation of the SVD of three matrices, which are referred to as “modes” or “matrix unfoldings.” In this paper, we present fast algorithms for computing the full and the rank-truncated HOSVD of third-order structured (symmetric, Toeplitz, and Hankel) tensors. These algorithms are derived by considering two specific ways to unfold a structured tensor, leading to structured matrix unfoldings whose SVD can be efficiently computed.

Statistical Resolution Limit of the Uniform Linear Cocentered Orthogonal Loop and Dipole Array

Among the family of polarization sensitive arrays, we can find the so-called cocentered orthogonal loop and dipole uniform linear array (COLD-ULA). The COLD-ULA exhibits some interesting properties, e.g., the insensibility of the polarization vector with respect to the source localization in the plan of the array. In this correspondence, we derive the statistical resolution limit (SRL) characterizing the minimal separation, in terms of direction-of-arrivals, to resolve two closely spaced known polarized sources impinging on a COLD-ULA. Toward this end, nonmatrix closed form expressions of the deterministic Cramér-Rao bound (CRB) are derived and thus, the SRL is deduced. A comparison between the SRL of the COLD-ULA and the classical ULA are given. Particularly, it is shown that, in the case of orthogonal known signal sources, the SRL of the COLD-ULA is equal to the SRL of the ULA, meaning that it is not a function of polarization parameters. Furthermore, due to the derived SRL, it is shown that, under some general conditions, the SRL of the COLD-ULA is smaller than the one of the ULA.

Conditional and Unconditional Cramér–Rao Bounds for Near-Field Source Localization

Near-field source localization problem by a passive antenna array makes the assumption that the time-varying sources are located near the antenna. In this context, the far-field assumption (i.e., planar wavefront) is, of course, no longer valid and one has to consider a more complicated model parameterized by the bearing (as in the far-field case) and by the distance, named range, between the source and a reference coordinate system. One can find a plethora of estimation schemes in the literature, but their ultimate performance in terms of mean square error (MSE) have not been fully investigated. To characterize these performance, the Cramer-Rao bound (CRB) is a popular mathematical tool in signal processing. The main cause for this is that the MSE of several high-resolution direction of arrival algorithms are known to achieve the CRB under quite general/weak conditions. In this correspondence, we derive and analyze the so-called conditional and unconditional CRBs for a single time-varying near-field source. In each case, we obtain non-matrix closed-form expressions. Our approach has two advantages: i) due to the fact that one has to inverse the Fisher information matrix, the computational cost for a large number of snapshots (in the case of the conditional CRB) and/or for a large number of sensors (in the case of the unconditional CRB), of a matrix-based CRB can be high while our approach is low and ii) some useful information can be deduced from the behavior of the bound. In particular, an explicit relationship between the conditional and the unconditional CRBs is provided and one shows that closer is the source from the array and/or higher is the signal carrier frequency, better is the range estimation.

Audio modeling based on delayed sinusoids

In this work, we present an evolution of the Damped and Delayed Sinusoidal (DDS) model introduced within the framework of the general signal modeling. This model, named the Partial Damped and Delayed Sinusoidal (PDDS) model, takes into account a single time delay parameter for a set (sum) of damped sinusoids. The proposed modification is more consistent with the transient audio modeling problem. The validity of the approach is shown by its comparison with the well-known Exponentially Damped Sinusoids (EDS) approach. Finally, the performances of the three models based high-resolution parameter estimation algorithms are compared on synthetic fast time-varying signals, and on two typical audio transients.

A Cramér Rao bounds based analysis of 3D antenna array geometries made from ULA branches

In the context of passive sources localization using antenna array, the estimation accuracy of elevation, and azimuth are related not only to the kind of estimator which is used, but also to the geometry of the considered antenna array. Although there are several available results on the linear array, and also for planar arrays, other geometries existing in the literature, such as 3D arrays, have been less studied. In this paper, we study the impact of the geometry of a family of 3D models of antenna array on the estimation performance of elevation, and azimuth. The Cramér-Rao Bound (CRB), which is widely spread in signal processing to characterize the estimation performance will be used here as a useful tool to find the optimal configuration. In particular, we give closed-form expressions of CRB for a 3D antenna array under both conditional, and unconditional observation models. Thanks to these explicit expressions, the impact of the third dimension to the estimation performance is analyzed. Particularly, we give criterions to design an isotropic 3D array depending on the considered observation model. Several 3D particular geometry antennas made from uniform linear array (ULA) are analyzed, and compared with 2D antenna arrays. The isotropy condition of such arrays is analyzed. The presented framework can be used for further studies of other types of arrays.

Statistical resolution limit for the multidimensional harmonic retrieval model: hypothesis test and Cramér-Rao Bound approaches

The statistical resolution limit (SRL), which is defined as the minimal separation between parameters to allow a correct resolvability, is an important statistical tool to quantify the ultimate performance for parametric estimation problems. In this article, we generalize the concept of the SRL to the multidimensional SRL (MSRL) applied to the multidimensional harmonic retrieval model. In this article, we derive the SRL for the so-called multidimensional harmonic retrieval model using a generalization of the previously introduced SRL concepts that we call multidimensional SRL (MSRL). We first derive the MSRL using an hypothesis test approach. This statistical test is shown to be asymptotically an uniformly most powerful test which is the strongest optimality statement that one could expect to obtain. Second, we link the proposed asymptotic MSRL based on the hypothesis test approach to a new extension of the SRL based on the Cramér-Rao Bound approach. Thus, a closed-form expression of the asymptotic MSRL is given and analyzed in the framework of the multidimensional harmonic retrieval model. Particularly, it is proved that the optimal MSRL is obtained for equi-powered sources and/or an equi-distributed number of sensors on each multi-way array.

Statistical Resolution Limit for Source Localization With Clutter Interference in a MIMO Radar Context

During the last decade, multiple-input multiple-ouput (MIMO) radar has received an increasing interest. One can find several estimation schemes in the literature related to the direction of arrivals and/or direction of departures, but their ultimate performance in terms of the statistical resolution limit (SRL) have not been fully investigated. In this correspondence, we fill this lack. Particulary, we derive the SRL to resolve two closely spaced targets in clutter interference using a MIMO radar with widely separated antennas. Toward this end, we use a hypothesis test formulation based on the generalized likelihood ratio test (GLRT). Furthermore, we investigate the link between the SRL and the minimum signal-to-noise ratio (SNR) required to resolve two closely spaced targets for a given probability of false alarm and for a given probability of detection. Finally, theoretical and numerical analysis of the SRL are given for several scenarios (with/without clutter interference, known/unknown parameters of interest and known/unknown noise variance).

Deterministic asymptotic Cramér-Rao bound for the multidimensional harmonic model

The harmonic model sampled on a P-dimensional grid contaminated by an additive white Gaussian noise has attracted considerable attention with a variety of applications. This model has a natural interpretation in a P-order tensorial framework and an important question is to evaluate the theoretical lowest variance on the model parameter (angular-frequency, real amplitude and initial phase) estimation. A standard Mathematical tool to tackle this question is the Cramer-Rao bound (CRB) which is a lower bound on the variance of an unbiased estimator, based on Fisher information. So, the aim of this work is to derive and analyze closed-form expressions of the deterministic asymptotic CRB associated with the M-order harmonic model of dimension P with P>1. In particular, we analyze this bound with respect to the variation of parameter P.