Xavier Mestre

Finite sample size effect on minimum variance beamformers: optimum diagonal loading factor for large arrays

Minimum variance beamformers are usually complemented with diagonal loading techniques in order to provide robustness against several impairments such as imprecise knowledge of the steering vector or finite sample size effects. This paper concentrates on this last application of diagonal loading techniques, i.e., it is assumed that the steering vector is perfectly known and that diagonal loading is used to alleviate the finite sample size impairments. The analysis herein is asymptotic in the sense that it is assumed that both the number of antennas and the number of samples are high but have the same order of magnitude. Borrowing some results of random matrix theory, the authors first derive a deterministic expression that describes the asymptotic signal-to-noise-plus-interference ratio (SINR) at the output of the diagonally loaded beamformer. Then, making use of the statistical theory of large observations (also known as general statistical analysis or G-analysis), the authors derive an estimator of the optimum loading factor that is consistent when both the number of antennas and the sample size increase without bound at the same rate. Because of that, the estimator has an excellent performance even in situations where the quotient between the number of observations is low relative to the number of elements of the array.

Modified Subspace Algorithms for DoA Estimation With Large Arrays

This paper proposes the use of a new generalized asymptotic paradigm in order to analyze the performance of subspace-based direction-of-arrival (DoA) estimation in array signal processing applications. Instead of assuming that the number of samples is high whereas the number of sensors/antennas remains fixed, the asymptotic situation analyzed herein assumes that both quantities tend to infinity at the same rate. This asymptotic situation provides a more accurate description of a potential situation where these two quantities are finite and hence comparable in magnitude. It is first shown that both MUSIC and SSMUSIC are inconsistent when the number of antennas/sensors increases without bound at the same rate as the sample size. This is done by analyzing and deriving closed-form expressions for the two corresponding asymptotic cost functions. By examining these asymptotic cost functions, one can establish the minimum number of samples per antenna needed to resolve closely spaced sources in this asymptotic regime. Next, two alternative estimators are constructed, that are strongly consistent in the new asymptotic situation, i.e., they provide consistent DoA estimates, not only when the number of snapshots goes to infinity, but also when the number of sensors/antennas increases without bound at the same rate. These estimators are inspired by the theory of G-estimation and are therefore referred to as G-MUSIC and G-SSMUSIC, respectively. Simulations show that the proposed algorithms outperform their traditional counterparts in finite sample-size situations, although they still present certain limitations.

Capacity of MIMO channels: asymptotic evaluation under correlated fading

This paper investigates the asymptotic uniform power allocation capacity of frequency nonselective multiple-input multiple-output channels with fading correlation at either the transmitter or the receiver. We consider the asymptotic situation, where the number of inputs and outputs increase without bound at the same rate. A simple uniparametric model for the fading correlation function is proposed and the asymptotic capacity per antenna is derived in closed form. Although the proposed correlation model is introduced only for mathematical convenience, it is shown that its shape is very close to an exponentially decaying correlation function. The asymptotic expression obtained provides a simple and yet useful way of relating the actual fading correlation to the asymptotic capacity per antenna from a purely analytical point of view. For example, the asymptotic expressions indicate that fading correlation is more harmful when arising at the side with less antennas. Moreover, fading correlation does not influence the rate of growth of the asymptotic capacity per receive antenna with high E/sub b//N/sub 0/.

Improved Subspace Estimation for Multivariate Observations of High Dimension: The Deterministic Signals Case

We consider the problem of subspace estimation in situations where the number of available snapshots and the observation dimension are comparable in magnitude. In this context, traditional subspace methods tend to fail because the eigenvectors of the sample correlation matrix are heavily biased with respect to the true ones. It has recently been suggested that this situation (where the sample size is small compared to the observation dimension) can be very accurately modeled by considering the asymptotic regime where the observation dimension M and the number of snapshots N converge to +∞ at the same rate. Using large random matrix theory results, it can be shown that traditional subspace estimates are not consistent in this asymptotic regime. Furthermore, new consistent subspace estimate can be proposed, which outperform the standard subspace methods for realistic values of M and N . The work carried out so far in this area has always been based on the assumption that the observations are random, independent and identically distributed in the time domain. The goal of this paper is to propose new consistent subspace estimators for the case where the source signals are modelled as unknown deterministic signals. In practice, this allows to use the proposed approach regardless of the statistical properties of the source signals. In order to construct the proposed estimators, new technical results concerning the almost sure location of the eigenvalues of sample covariance matrices of Information plus Noise complex Gaussian models are established. These results are believed to be of independent interest.

On the Asymptotic Behavior of the Sample Estimates of Eigenvalues and Eigenvectors of Covariance Matrices

This paper analyzes the asymptotic behavior of the sample estimators of the eigenvalues and eigenvectors of covariance matrices. Rather than considering traditional large sample-size asymptotics, in this paper the focus is on limited sample size situations, whereby the number of available observations is comparable in magnitude to the observation dimension. Using tools from random matrix theory, the asymptotic behavior of the traditional sample estimates is investigated under the assumption that both the sample size and the observation dimension tend to infinity, while their quotient converges to a positive quantity. Assuming that an asymptotic eigenvalue splitting condition is fulfilled, closed form asymptotic expressions of these estimators are derived, proving inconsistency of the traditional sample estimators in these asymptotic conditions. The derived expressions are shown to provide a valuable insight into the behavior of the sample estimators in the small sample size regime.

MUSIC, G-MUSIC, and Maximum-Likelihood Performance Breakdown

Direction-of-arrival estimation performance of MUSIC and maximum-likelihood estimation in the so-called ldquothresholdrdquo area is analyzed by means of general statistical analysis (GSA) (also known as random matrix theory). Both analytic predictions and direct Monte Carlo simulations demonstrate that the well-known MUSIC-specific ldquoperformance breakdownrdquo is associated with the loss of resolution capability in the MUSIC pseudo-spectrum, while the sample signal subspace is still reliably separated from the actual noise subspace. Significant distinctions between (MUSIC/G-MUSIC)-specific and MLE-intrinsic causes of ldquoperformance breakdown,rdquo as well as the role of ldquosubspace swaprdquo phenomena, are specified analytically and supported by simulation.

Performance Analysis and Optimal Selection of Large Minimum Variance Portfolios Under Estimation Risk

We study the realized variance of sample minimum variance portfolios of arbitrarily high dimension. We consider the use of covariance matrix estimators based on shrinkage and weighted sampling. For such improved portfolio implementations, the otherwise intractable problem of characterizing the realized variance is tackled here by analyzing the asymptotic convergence of the risk measure. Rather than relying on less insightful classical asymptotics, we manage to deliver results in a practically more meaningful limiting regime, where the number of assets remains comparable in magnitude to the sample size. Under this framework, we provide accurate estimates of the portfolio realized risk in terms of the model parameters and the underlying investment scenario, i.e., the unknown asset return covariance structure. In-sample approximations in terms of only the available data observations are known to considerably underestimate the realized portfolio risk. If not corrected, these deviations might lead in practice to inaccurate and overly optimistic investment decisions. Therefore, along with the asymptotic analysis, we also provide a generalized consistent estimator of the out-of-sample portfolio variance that only depends on the set of observed returns. Based on this estimator, the model free parameters, i.e., the sample weighting coefficients and the shrinkage intensity defining the minimum variance portfolio implementation, can be optimized so as to minimize the realized variance while taken into account the effect of estimation risk. Our results are based on recent contributions in the field of random matrix theory. Numerical simulations based on both synthetic and real market data validate our theoretical findings under a non-asymptotic, finite-dimensional setting. Finally, our proposed portfolio estimator is shown to consistently outperform a widely applied benchmark implementation.

An Asymptotic Approach to Parallel Equalization of Filter Bank Based Multicarrier Signals

A novel equalization structure for filter bank based multicarrier (FBMC) based modulations is proposed in this paper. The equalizer architecture is derived by assuming that the number of carriers is asymptotically large, and it consists of multiple parallel stages that are linearly combined on a per-subcarrier basis. The traditional single-tap per-subcarrier equalizer is obtained as a special case of the proposed architecture when the number of stages is fixed to one. An analytical characterization of the output signal to noise plus distortion ratio is provided, which can be used in practice to fix the number of parallel stages of the equalizer in order to obtain a certain performance level at the minimum complexity. Both simulations and analytical predictions indicate that high performance gains can be obtained with respect to more conventional FBMC equalization approaches. Furthermore, the proposed architecture is also shown to present important advantages in terms of computational complexity when compared with classical FBMC equalization architectures.

A subspace estimator for fixed rank perturbations of large random matrices

This paper deals with the problem of parameter estimation based on certain eigenspaces of the empirical covariance matrix of an observed multidimensional time series, in the case where the time series dimension and the observation window grow to infinity at the same pace. In the area of large random matrix theory, recent contributions studied the behavior of the extreme eigenvalues of a random matrix and their associated eigenspaces when this matrix is subject to a fixed-rank perturbation. The present work is concerned with the situation where the parameters to be estimated determine the eigenspace structure of a certain fixed-rank perturbation of the empirical covariance matrix. An estimation algorithm in the spirit of the well-known MUSIC algorithm for parameter estimation is developed. It relies on an approach recently developed by Benaych-Georges and Nadakuditi (2011) [8,9], relating the eigenspaces of extreme eigenvalues of the empirical covariance matrix with eigenspaces of the perturbation matrix. First and second order analyses of the new algorithm are performed.

MIMO Signal Processing in Offset-QAM Based Filter Bank Multicarrier Systems

Next-generation communication systems have to comply with very strict requirements for increased flexibility in heterogeneous environments, high spectral efficiency, and agility of carrier aggregation. This fact motivates research in advanced multicarrier modulation (MCM) schemes, such as filter bank-based multicarrier (FBMC) modulation. This paper focuses on the offset quadrature amplitude modulation (OQAM)-based FBMC variant, known as FBMC/OQAM, which presents outstanding spectral efficiency and confinement in a number of channels and applications. Its special nature, however, generates a number of new signal processing challenges that are not present in other MCM schemes, notably, in orthogonal-frequency-division multiplexing (OFDM). In multiple-input multiple-output (MIMO) architectures, which are expected to play a primary role in future communication systems, these challenges are intensified, creating new interesting research problems and calling for new ideas and methods that are adapted to the particularities of the MIMO-FBMC/OQAM system. The goal of this paper is to focus on these signal processing problems and provide a concise yet comprehensive overview of the recent advances in this area. Open problems and associated directions for future research are also discussed.