Statistical Performance Analysis of MDL Source Enumeration in Array Processing

In this correspondence, we focus on the performance analysis of the widely-used minimum description length (MDL) source enumeration technique in array processing. Unfortunately, available theoretical analysis exhibit deviation from the simulation results. We present an accurate and insightful performance analysis for the probability of missed detection. We also show that the statistical performance of the MDL is approximately the same under both deterministic and stochastic signal models. Simulation results show the superiority of the proposed analysis over available results.Comment: Accepted for publication in IEEE Transactions on Signal Processing, April 200

Source enumeration in non-white noise and small sample size via subspace averaging

This paper addresses the problem of source enumeration by an array of sensors in the challenging conditions of: i) large uniform arrays with few snapshots, and ii) non-white or spatially correlated noises with arbitrary correlation. To solve this problem, we combine a subspace averaging (SA) technique, recently proposed for the case of independent and identically distributed (i.i.d.) noises, with a majority vote approach. The number of sources is detected for increasing dimensions of the SA technique and then a majority vote is applied to determine the final estimate. As illustrated by some simulation examples, this simple modification makes SA a very robust method of enumerating sources in these challenging scenarios.This work was supported by the Ministerio de Economía y Competitividad (MINECO) of Spain, and AEI/FEDER funds of the E.U., under grants TEC2016-75067-C4-4-R (CARMEN) and BES-2017-080542

Robust Multiple Signal Classification via Probability Measure Transformation

In this paper, we introduce a new framework for robust multiple signal classification (MUSIC). The proposed framework, called robust measure-transformed (MT) MUSIC, is based on applying a transform to the probability distribution of the received signals, i.e., transformation of the probability measure defined on the observation space. In robust MT-MUSIC, the sample covariance is replaced by the empirical MT-covariance. By judicious choice of the transform we show that: 1) the resulting empirical MT-covariance is B-robust, with bounded influence function that takes negligible values for large norm outliers, and 2) under the assumption of spherically contoured noise distribution, the noise subspace can be determined from the eigendecomposition of the MT-covariance. Furthermore, we derive a new robust measure-transformed minimum description length (MDL) criterion for estimating the number of signals, and extend the MT-MUSIC framework to the case of coherent signals. The proposed approach is illustrated in simulation examples that show its advantages as compared to other robust MUSIC and MDL generalizations

Subspace averaging for source enumeration in large arrays

Subspace averaging is proposed and examined as a method of enumerating sources in large linear arrays, under conditions of low sample support. The key idea is to exploit shift invariance as a way of extracting many subspaces, which may then be approximated by a single extrinsic average. An automatic order determination rule for this extrinsic average is then the rule for determining the number of sources. Experimental results are presented for cases where the number of array snapshots is roughly half the number of array elements, and sources are well separated with respect to the Rayleigh limit.The work of I. Santamaría has been partially supported by the Ministerio de Economía y Competitividad (MINECO) of Spain, and AEI/FEDER funds of the E.U., under grant TEC2016-75067-C4-4-R (CARMEN). The work of D. Ramírez has been partly supported by Ministerio de Economía of Spain under projects: OTOSIS (TEC2013-41718-R) and the COMONSENS Network (TEC2015-69648-REDC), by the Ministerio de Economía of Spain jointly with the European Commission (ERDF) under projects ADVENTURE (TEC2015-69868-C2-1-R) and CAIMAN (TEC2017-86921- C2-2-R), and by the Comunidad de Madrid under project CASI-CAM-CM (S2013/ICE-2845). The work of L. L. Scharf was supported by the National Science Foundation (NSF) under grant CCF-1018472

Subspace-based order estimation techniques in massive MIMO

Order estimation, also known as source enumeration, is a classical problem in array signal processing which consists in estimating the number of signals received by an array of sensors. In the last decades, numerous approaches to this problem have been proposed. However, the need of working with large-scale arrays (like in massive MIMO systems), low signal-to-noise- ratios, and poor sample regime scenarios, introduce new challenges to order estimation problems. For instance, most of the classical approaches are based on information theoretic criteria, which usually require a large sample size, typically several times larger than the number of sensors. Obtaining a number of samples several times larger than the number of sensors is not always possible with large-scale arrays. In addition, most of the methods found in literature assume that the noise is spatially white, which is very restrictive for many practical scenarios. This dissertation deals with the problem of source enumeration for large-scale arrays, and proposes solutions that work robustly in the small sample regime under various noise models. The first part of the dissertation solves the problem by applying the idea of subspace averaging. The input data are modelled as subspaces, and an average or central subspace is computed. The source enumeration problem can be seen as an estimation of the dimension of the central subspace. A key element of the proposed method is to construct a bootstrap procedure, based on a newly proposed discrete distribution on the manifold of projection matrices, for stochastically generating subspaces from a function of experimentally determined eigenvalues. In this way, the proposed subspace averaging (SA) technique determines the order based on the eigenvalues of an average projection matrix, rather than on the likelihood of a covariance model, penalized by functions of the model order. The proposed SA criterion is especially effective in high-dimensional scenarios with low sample support for uniform linear arrays in the presence of white noise. Further, the proposed SA method is extended for: i) non-white noises, and ii) non-uniform linear arrays. The SA criterion is sensitive with the chosen dimension of extracted subspaces. To solve this problem, we combine the SA technique with a majority vote approach. The number of sources is detected for increasing dimensions of the SA technique and then a majority vote is applied to determine the final estimate. Further, to extend SA for arrays with arbitrary geometries, the SA is combined with a sparse reconstruction (SR) step. In the first step, each received snapshot is approximated by a sparse linear combination of the rest of snapshots. The SR problem is regularized by the logarithm-based surrogate of the l-0 norm and solved using a majorization-minimization approach. Based on the SR solution, a sampling mechanism is proposed in the second step to generate a collection of subspaces, all of which approximately span the same signal subspace. Finally, the dimension of the average of this collection of subspaces provides a robust estimate for the number of sources. The second half of the dissertation introduces a completely different approach to the order estimation for uniform linear arrays, which is based on matrix completion algorithms. This part first discusses the problem of order estimation in the presence of noise whose spatial covariance structure is a diagonal matrix with possibly different variances. The diagonal terms of the sample covariance matrix are removed and, after applying Toeplitz rectification as a denoising step, the signal covariance matrix is reconstructed by using a low-rank matrix completion method adapted to enforce the Toeplitz structure of the sought solution. The proposed source enumeration criterion is based on the Frobenius norm of the reconstructed signal covariance matrix obtained for increasing rank values. The proposed method performs robustly for both small and large-scale arrays with few snapshots. Finally, an approach to work with a reduced number of radio–frequency (RF) chains is proposed. The receiving array relies on antenna switching so that at every time instant only the signals received by a randomly selected subset of antennas are downconverted to baseband and sampled. Low-rank matrix completion (MC) techniques are then used to reconstruct the missing entries of the signal data matrix to keep the angular resolution of the original large-scale array. The proposed MC algorithm exploits not only the low- rank structure of the signal subspace, but also the shift-invariance property of uniform linear arrays, which results in a better estimation of the signal subspace. In addition, the effect of MC on DOA estimation is discussed under the perturbation theory framework. Further, this approach is extended to devise a novel order estimation criterion for missing data scenario. The proposed source enumeration criterion is based on the chordal subspace distance between two sub-matrices extracted from the reconstructed matrix after using MC for increasing rank values. We show that the proposed order estimation criterion performs consistently with a very few available entries in the data matrix.This work was supported by the Ministerio de Ciencia e Innovación (MICINN) of Spain, under grants TEC2016-75067-C4-4-R (CARMEN) and BES-2017-080542