Pascal Bianchi

Cooperative spectrum sensing using random matrix theory

In this paper, using tools from asymptotic random matrix theory, a new cooperative scheme for frequency band sensing is introduced for both AWGN and fading channels. Unlike previous works in the field, the new scheme does not require the knowledge of the noise statistics or its variance and is related to the behavior of the largest and smallest eigenvalue of random matrices. Remarkably, simulations show that the asymptotic claims hold even for a small number of observations (which makes it convenient for time-varying topologies), outperforming classical energy detection techniques.

Performance of Statistical Tests for Single-Source Detection Using Random Matrix Theory

This paper introduces a unified framework for the detection of a single source with a sensor array in the context where the noise variance and the channel between the source and the sensors are unknown at the receiver. The Generalized Maximum Likelihood Test is studied and yields the analysis of the ratio between the maximum eigenvalue of the sampled covariance matrix and its normalized trace. Using recent results from random matrix theory, a practical way to evaluate the threshold and the p-value of the test is provided in the asymptotic regime where the number K of sensors and the number N of observations per sensor are large but have the same order of magnitude. The theoretical performance of the test is then analyzed in terms of Receiver Operating Characteristic (ROC) curve. It is, in particular, proved that both Type I and Type II error probabilities converge to zero exponentially as the dimensions increase at the same rate, and closed-form expressions are provided for the error exponents. These theoretical results rely on a precise description of the large deviations of the largest eigenvalue of spiked random matrix models, and establish that the presented test asymptotically outperforms the popular test based on the condition number of the sampled covariance matrix.

Convergence of a Multi-Agent Projected Stochastic Gradient Algorithm for Non-Convex Optimization

We introduce a new framework for the convergence analysis of a class of distributed constrained non-convex optimization algorithms in multi-agent systems. The aim is to search for local minimizers of a non-convex objective function which is supposed to be a sum of local utility functions of the agents. The algorithm under study consists of two steps: a local stochastic gradient descent at each agent and a gossip step that drives the network of agents to a consensus. Under the assumption of decreasing stepsize, it is proved that consensus is asymptotically achieved in the network and that the algorithm converges to the set of Karush-Kuhn-Tucker points. As an important feature, the algorithm does not require the double-stochasticity of the gossip matrices. It is in particular suitable for use in a natural broadcast scenario for which no feedback messages between agents are required. It is proved that our results also holds if the number of communications in the network per unit of time vanishes at moderate speed as time increases, allowing potential savings of the networks energy. Applications to power allocation in wireless ad-hoc networks are discussed. Finally, we provide numerical results which sustain our claims.

Asynchronous distributed optimization using a randomized alternating direction method of multipliers

Consider a set of networked agents endowed with private cost functions and seeking to find a consensus on the minimizer of the aggregate cost. A new class of random asynchronous distributed optimization methods is introduced. The methods generalize the standard Alternating Direction Method of Multipliers (ADMM) to an asynchronous setting where isolated components of the network are activated in an uncoordinated fashion. The algorithms rely on the introduction of randomized Gauss-Seidel iterations of Douglas-Rachford splitting leading to an asynchronous algorithm based on the ADMM. Convergence to the sought minimizers is provided under mild connectivity conditions.

Resource Allocation for Downlink Cellular OFDMA Systems—Part I: Optimal Allocation

In this pair of papers (Part I and Part II in this issue), we investigate the issue of power control and subcarrier assignment in a sectorized two-cell downlink OFDMA system impaired by multicell interference. As recommended for WiMAX, we assume that the first part of the available bandwidth is likely to be reused by different base stations (and is thus subject to multicell interference) and that the second part of the bandwidth is shared in an orthogonal way between the different base stations (and is thus protected from multicell interference). Although the problem of multicell resource allocation is nonconvex in this scenario, we provide in Part I the general form of the global solution. In particular, the optimal resource allocation turns out to be ¿binary¿ in the sense that, except for at most one pivot-user in each cell, any user receives data either in the reused bandwidth or in the protected bandwidth, but not in both. The determination of the optimal resource allocation essentially reduces to the determination of the latter pivot-position.

Explicit Convergence Rate of a Distributed Alternating Direction Method of Multipliers

Consider a set of N agents seeking to solve distributively the minimization problem inf_∞ Σ_n=1^N f_n(x) where the convex functions f_n are local to the agents. The popular Alternating Direction Method of Multipliers has the potential to handle distributed optimization problems of this kind. We provide a general reformulation of the problem and obtain a class of distributed algorithms which encompass various network architectures. The rate of convergence of our method is considered. It is assumed that the infimum of the problem is reached at a point x_*, the functions f_n are twice differentiable at this point and Σ ∇²f_n(x_*) > 0 in the positive definite ordering of symmetric matrices. With these assumptions, it is shown that the convergence to the consensus x_* is linear and the exact rate is provided. Application examples where this rate can be optimized with respect to the ADMM free parameter ρ are also given.

Performance of a Distributed Stochastic Approximation Algorithm

In this paper, a distributed stochastic approximation algorithm is studied. Applications of such algorithms include decentralized estimation, optimization, control or computing. The algorithm consists in two steps: a local step, where each node in a network updates a local estimate using a stochastic approximation algorithm with decreasing step size, and a gossip step, where a node computes a local weighted average between its estimates and those of its neighbors. Convergence of the estimates toward a consensus is established under weak assumptions. The approach relies on two main ingredients: the existence of a Lyapunov function for the mean field in the agreement subspace, and a contraction property of the random matrices of weights in the subspace orthogonal to the agreement subspace. A second-order analysis of the algorithm is also performed under the form of a central limit Theorem. The Polyak-averaged version of the algorithm is also considered.

Performance analysis of some eigen-based hypothesis tests for collaborative sensing

In this contribution, we provide a theoretical study of two hypothesis tests allowing to detect the presence of an unknown transmitter using several sensors. Both tests are based on the analysis of the eigenvalues of the sampled covariance matrix of the received signal. The Generalized Likelihood Ratio Test (GLRT) derived in [1] is analyzed under the assumption that both the number K of sensors and the length N of the observation window tend to infinity at the same rate: K/N → c ∈ (0, 1). The GLRT is compared with a test based on the condition number used which is used in cognitive radio applications. Using results of random matrix theory for spiked models and tools of Large Deviations, we provide the error exponent curve associated with both test and prove that the GLRT outperforms the test based on the condition number.

Asymptotically Efficient Reduced Complexity Frequency Offset and Channel Estimators for Uplink MIMO-OFDMA Systems

In this paper, we address the joint data-aided estimation of frequency offsets and channel coefficients in uplink multiple-input multiple-output orthogonal frequency-division multiple access (MIMO-OFDMA) systems. As the maximum-likelihood (ML) estimator is impractical in this context, we introduce a family of suboptimal estimators with the aim of exhibiting an attractive tradeoff between performance and complexity. The estimators do not rely on a particular subcarrier assignment scheme and are, thus, valid for a large number of OFDMA systems. As far as complexity is concerned, the computational cost of the proposed estimators is shown to be significantly reduced compared to existing estimators based on ML. As far as performance is concerned, the proposed suboptimal estimators are shown to be asymptotically efficient, i.e., the covariance matrix of the estimation error achieves the Cramer-Rao bound when the total number of subcarriers increases. Simulation results sustain our claims.

Outage Probability-Based Power and Time Optimization for Relay Networks

In the context of cooperative wireless networks that convey data on slow fading channels, outage probability P o is the relevant performance index from the point of view of information theory. Derivation and minimization of this probability with respect to the relaying protocol parameters is of central importance. However, it is often hard to derive its expression, let alone to find its exact minimum for all possible values of the signal-to-noise ratio (SNR). This problem can be simplified by studying the behavior of P o in the asymptotic regime where the SNR rho converges to infinity. In this regime, usually rho N+1 P o converges to a constant xi where N is the number of relays. In this paper, a simple and general method for deriving and minimizing xi with respect to the power distribution between the source and the relays, and with respect to the durations of the slots specified by the relaying protocol, is developed. While the proposed approach is designed for the high SNR regime, simulations show that outage probability is reduced in a similar proportion at moderate SNR. The method applies to a general class of radio channels that includes the Rayleigh and the Rice channels as particular cases. Convexity of xi with respect to the design parameters is shown. Decode-and-forward, as well as amplify-and-forward protocols are considered in the half duplex mode.