Romain Couillet

Large System Analysis of Linear Precoding in Correlated MISO Broadcast Channels Under Limited Feedback

In this paper, we study the sum rate performance of zero-forcing (ZF) and regularized ZF (RZF) precoding in large MISO broadcast systems under the assumptions of imperfect channel state information at the transmitter and per-user channel transmit correlation. Our analysis assumes that the number of transmit antennas M and the number of single-antenna users K are large while their ratio remains bounded. We derive deterministic approximations of the empirical signal-to-interference plus noise ratio (SINR) at the receivers, which are tight as M, K → ∞. In the course of this derivation, the per-user channel correlation model requires the development of a novel deterministic equivalent of the empirical Stieltjes transform of large dimensional random matrices with generalized variance profile. The deterministic SINR approximations enable us to solve various practical optimization problems. Under sum rate maximization, we derive 1) for RZF the optimal regularization parameter; 2) for ZF the optimal number of users; 3) for ZF and RZF the optimal power allocation scheme; and 4) the optimal amount of feedback in large FDD/TDD multiuser systems. Numerical simulations suggest that the deterministic approximations are accurate even for small M, K.

A Deterministic Equivalent for the Analysis of Correlated MIMO Multiple Access Channels

In this article, novel deterministic equivalents for the Stieltjes transform and the Shannon transform of a class of large dimensional random matrices are provided. These results are used to characterize the ergodic rate region of multiple antenna multiple access channels, when each point-to-point propagation channel is modelled according to the Kronecker model. Specifically, an approximation of all rates achieved within the ergodic rate region is derived and an approximation of the linear precoders that achieve the boundary of the rate region as well as an iterative water-filling algorithm to obtain these precoders are provided. An original feature of this work is that the proposed deterministic equivalents are proved valid even for strong correlation patterns at both communication sides. The above results are validated by Monte Carlo simulations.

Large System Analysis of Linear Precoding in MISO Broadcast Channels with Confidential Messages

In this paper, we study the performance of regularized channel inversion (RCI) precoding in large MISO broadcast channels with confidential messages (BCC). We obtain a deterministic approximation for the achievable secrecy sum-rate which is almost surely exact as the number of transmit antennas M and the number of users K grow to infinity in a fixed ratio β=K/M. We derive the optimal regularization parameter ξ and the optimal network load β that maximize the per-antenna secrecy sum-rate. We then propose a linear precoder based on RCI and power reduction (RCI-PR) that significantly increases the high-SNR secrecy sum-rate for 1<;β<;2. Our proposed precoder achieves a per-user secrecy rate which has the same high-SNR scaling factor as both the following upper bounds: (i) the rate of the optimum RCI precoder without secrecy requirements, and (ii) the secrecy capacity of a single-user system without interference. Furthermore, we obtain a deterministic approximation for the secrecy sum-rate achievable by RCI precoding in the presence of channel state information (CSI) error. We also analyze the performance of our proposed RCI-PR precoder with CSI error, and we determine how the error must scale with the SNR in order to maintain a given rate gap to the case with perfect CSI.

Electrical Vehicles in the Smart Grid: A Mean Field Game Analysis

In this article, we investigate the competitive interaction between electrical vehicles or hybrid oil-electricity vehicles in a Cournot market consisting of electricity transactions to or from an underlying electricity distribution network. We provide a mean field game formulation for this competition, and introduce the set of fundamental differential equations ruling the behavior of the vehicles at the feedback Nash equilibrium, referred here to as the mean field equilibrium. This framework allows for a consistent analysis of the evolution of the price of electricity as well as of the instantaneous electricity demand in the power grid. Simulations precisely quantify those parameters and suggest that significant reduction of the daily electricity peak demand can be achieved by appropriate electricity pricing.

Large Dimensional Analysis and Optimization of Robust Shrinkage Covariance Matrix Estimators

This article studies two regularized robust estimators of scatter matrices proposed (and proved to be well defined) in parallel in Chen et al. (2011) and Pascal et al. (2013), based on Tyler’s robust M-estimator (Tyler, 1987) and on Ledoit and Wolf’s shrinkage covariance matrix estimator (Ledoit and Wolf, 2004). These hybrid estimators have the advantage of conveying (i) robustness to outliers or impulsive samples and (ii) small sample size adequacy to the classical sample covariance matrix estimator. We consider here the case of i.i.d. elliptical zero mean samples in the regime where both sample and population sizes are large. We demonstrate that, under this setting, the estimators under study asymptotically behave similar to well-understood random matrix models. This characterization allows us to derive optimal shrinkage strategies to estimate the population scatter matrix, improving significantly upon the empirical shrinkage method proposed in Chen et al. (2011).

Eigen-Inference for Energy Estimation of Multiple Sources

In this paper, a new method is introduced to blindly estimate the transmit power of multiple signal sources in multiantenna fading channels, when the number of sensing devices and the number of available samples are sufficiently large compared to the number of sources. Recent advances in the field of large dimensional random matrix theory are used that result in a simple and computationally efficient consistent estimator of the power of each source. A criterion to determine the minimum number of sensors and the minimum number of samples required to achieve source separation is then introduced. Simulations are performed that corroborate the theoretical claims and show that the proposed power estimator largely outperforms alternative power inference techniques.

Signal Processing in Large Systems: A New Paradigm

For a long time, detection and parameter estimation methods for signal processing have relied on asymptotic statistics as the number n of observations of a population grows large comparatively to the population size N, i.e., n/N → ∞. Modern technological and societal advances now demand the study of sometimes extremely large populations and simultaneously require fast signal processing due to accelerated system dynamics. This results in not-so-large practical ratios n/N, sometimes even smaller than one. A disruptive change in classical signal processing methods has therefore been initiated in the past ten years, mostly spurred by the field of large-dimensional random matrix theory. The early works in random matrix theory for signal processing applications are, however, scarce and highly technical. This tutorial provides an accessible methodological introduction to the modern tools of random matrix theory and to the signal processing methods derived from them, with an emphasis on simple illustrative examples.

Robust Estimates of Covariance Matrices in the Large Dimensional Regime

This paper studies the limiting behavior of a class of robust population covariance matrix estimators, originally due to Maronna in 1976, in the regime where both the number of available samples and the population size grow large. Using tools from random matrix theory, we prove that, for sample vectors made of independent entries having some moment conditions, the difference between the sample covariance matrix and (a scaled version of) such robust estimator tends to zero in spectral norm, almost surely. This result can be applied to various statistical methods arising from random matrix theory that can be made robust without altering their first order behavior.

Fluctuations of Spiked Random Matrix Models and Failure Diagnosis in Sensor Networks

In this paper, the joint fluctuations of the extreme eigenvalues and eigenvectors of a large dimensional sample covariance matrix are analyzed when the associated population covariance matrix is a finite-rank perturbation of the identity matrix, corresponding to the so-called spiked model in random matrix theory. The asymptotic fluctuations, as the matrix size grows large, are shown to be intimately linked with matrices from the Gaussian unitary ensemble. When the spiked population eigenvalues have unit multiplicity, the fluctuations follow a central limit theorem. This result is used to develop an original framework for the detection and diagnosis of local failures in large sensor networks, for known or unknown failure magnitude.

A mean field game analysis of electric vehicles in the smart grid

In this article, we present an economical analysis of the integration of purely electrical vehicles (EV) in the smart grid energy market using tools from mean field game theory. Our main contribution consist in a formal description of the mean field equilibrium of the resulting competitive interaction when EV owners buy and sell electricity from their cars, selfishly but rationally, based on collective price incentives. We present a comprehensive set of numerical results, which allows a consistent analysis of the evolution of the price of electricity, of the timely demand, and possibly of the energy reserves in the grid.