Ami Wiesel

Linear precoding via conic optimization for fixed MIMO receivers

In this paper, the problem of designing linear precoders for fixed multiple-input-multiple-output (MIMO) receivers is considered. Two different design criteria are considered. In the first, the transmitted power is minimized subject to signal-to-interference-plus-noise-ratio (SINR) constraints. In the second, the worst case SINR is maximized subject to a power constraint. It is shown that both problems can be solved using standard conic optimization packages. In addition, conditions are developed for the optimal precoder for both of these problems, and two simple fixed-point iterations are proposed to find the solutions that satisfy these conditions. The relation to the well-known uplink-downlink duality in the context of joint transmit beamforming and power control is also explored. The proposed precoder design is general, and as a special case, it solves the transmit rank-one beamforming problem. Simulation results in a multiuser system show that the resulting precoders can significantly outperform existing linear precoders.

Zero-Forcing Precoding and Generalized Inverses

We consider the problem of linear zero-forcing precoding design and discuss its relation to the theory of generalized inverses in linear algebra. Special attention is given to a specific generalized inverse known as the pseudo-inverse. We begin with the standard design under the assumption of a total power constraint and prove that precoders based on the pseudo-inverse are optimal among the generalized inverses in this setting. Then, we proceed to examine individual per-antenna power constraints. In this case, the pseudo-inverse is not necessarily the optimal inverse. In fact, finding the optimal matrix is nontrivial and depends on the specific performance measure. We address two common criteria, fairness and throughput, and show that the optimal generalized inverses may be found using standard convex optimization methods. We demonstrate the improved performance offered by our approach using computer simulations.

Shrinkage Algorithms for MMSE Covariance Estimation

We address covariance estimation in the sense of minimum mean-squared error (MMSE) when the samples are Gaussian distributed. Specifically, we consider shrinkage methods which are suitable for high dimensional problems with a small number of samples (large p small n). First, we improve on the Ledoit-Wolf (LW) method by conditioning on a sufficient statistic. By the Rao-Blackwell theorem, this yields a new estimator called RBLW, whose mean-squared error dominates that of LW for Gaussian variables. Second, to further reduce the estimation error, we propose an iterative approach which approximates the clairvoyant shrinkage estimator. Convergence of this iterative method is established and a closed form expression for the limit is determined, which is referred to as the oracle approximating shrinkage (OAS) estimator. Both RBLW and OAS estimators have simple expressions and are easily implemented. Although the two methods are developed from different perspectives, their structure is identical up to specified constants. The RBLW estimator provably dominates the LW method for Gaussian samples. Numerical simulations demonstrate that the OAS approach can perform even better than RBLW, especially when n is much less than p . We also demonstrate the performance of these techniques in the context of adaptive beamforming.

Robust Shrinkage Estimation of High-Dimensional Covariance Matrices

We address high dimensional covariance estimation for elliptical distributed samples, which are also known as spherically invariant random vectors (SIRV) or compound-Gaussian processes. Specifically we consider shrinkage methods that are suitable for high dimensional problems with a small number of samples (large p small n). We start from a classical robust covariance estimator [Tyler (1987)], which is distribution-free within the family of elliptical distribution but inapplicable when n <; p. Using a shrinkage coefficient, we regularize Tylers fixed-point iterations. We prove that, for all n and p , the proposed fixed-point iterations converge to a unique limit regardless of the initial condition. Next, we propose a simple, closed-form and data dependent choice for the shrinkage coefficient, which is based on a minimum mean squared error framework. Simulations demonstrate that the proposed method achieves low estimation error and is robust to heavy-tailed samples. Finally, as a real-world application we demonstrate the performance of the proposed technique in the context of activity/intrusion detection using a wireless sensor network.

Semidefinite relaxation for detection of 16-QAM signaling in MIMO channels

We develop a computationally efficient approximation of the maximum likelihood (ML) detector for 16 quadrature amplitude modulation (16-QAM) in multiple-input multiple-output (MIMO) systems. The detector is based on a convex relaxation of the ML problem. The resulting optimization is a semidefinite program that can be solved in polynomial time with respect to the number of inputs in the system. Simulation results in a random MIMO system show that the proposed algorithm outperforms the conventional decorrelator detector by about 2.5 dB at high signal-to-noise ratios.

Non-data-aided signal-to-noise-ratio estimation

Non-data-aided (NDA) signal-to-noise-ratio (SNR) estimation is considered for binary phase shift keying systems where the data samples are governed by a normal mixture distribution. Inherent estimation accuracy limitations are examined via a simple, closed-form approximation to the associated Cramer-Rao bound which eliminates the need for numerical integration. The expectation-maximization algorithm is proposed to iteratively maximize the NDA likelihood function. Simulation results show that the resulting estimator offers statistical efficiency over a wider range of scenarios than previously published methods.

SNR estimation in time-varying fading channels

Signal-to-noise ratio (SNR) estimation is considered for phase-shift keying communication systems in time-varying fading channels. Both data-aided (DA) estimation and nondata-aided (NDA) estimation are addressed. The time-varying fading channel is modeled as a polynomial-in-time. Inherent estimation accuracy limitations are examined via the Cramer-Rao lower bound, where it is shown that the effect of the channels time variation on SNR estimation is negligible. A novel maximum-likelihood (ML) SNR estimator is derived for the time-varying channel model. In DA scenarios, where the estimator has a simple closed-form solution, the exact performance is evaluated both with correct and incorrect (i.e., mismatched) polynomial order. In NDA estimation, the unknown data symbols are modeled as random, and the marginal likelihood is used. The expectation-maximization algorithm is proposed to iteratively maximize this likelihood function. Simulation results show that the resulting estimator offers statistical efficiency over a wider range of scenarios than previously published methods.

Unified Framework to Regularized Covariance Estimation in Scaled Gaussian Models

We consider regularized covariance estimation in scaled Gaussian settings, e.g., elliptical distributions, compound-Gaussian processes and spherically invariant random vectors. Asymptotically in the number of samples, the classical maximum likelihood (ML) estimate is optimal under different criteria and can be efficiently computed even though the optimization is nonconvex. We propose a unified framework for regularizing this estimate in order to improve its finite sample performance. Our approach is based on the discovery of hidden convexity within the ML objective. We begin by restricting the attention to diagonal covariance matrices. Using a simple change of variables, we transform the problem into a convex optimization that can be efficiently solved. We then extend this idea to nondiagonal matrices using convexity on the manifold of positive definite matrices. We regularize the problem using appropriately convex penalties. These allow for shrinkage towards the identity matrix, shrinkage towards a diagonal matrix, shrinkage towards a given positive definite matrix, and regularization of the condition number. We demonstrate the advantages of these estimators using numerical simulations.

Geodesic Convexity and Covariance Estimation

Geodesic convexity is a generalization of classical convexity which guarantees that all local minima of g-convex functions are globally optimal. We consider g-convex functions with positive definite matrix variables, and prove that Kronecker products, and logarithms of determinants are g-convex. We apply these results to two modern covariance estimation problems: robust estimation in scaled Gaussian distributions, and Kronecker structured models. Maximum likelihood estimation in these settings involves non-convex minimizations. We show that these problems are in fact g-convex. This leads to straight forward analysis, allows the use of standard optimization methods and paves the road to various extensions via additional g-convex regularization.

Power System State Estimation Using PMUs With Imperfect Synchronization

Phasor measurement units (PMUs) are time synchronized sensors primarily used for power system state estimation. Despite their increasing incorporation and the ongoing research on state estimation using measurements from these sensors, estimation with imperfect phase synchronization has not been sufficiently investigated. Inaccurate synchronization is an inevitable problem that large scale deployment of PMUs has to face. In this paper, we introduce a model for power system state estimation using PMUs with phase mismatch. We propose alternating minimization and parallel Kalman filtering for state estimation using static and dynamic models, respectively, under different assumptions. Numerical examples demonstrate the improved accuracy of our algorithms compared with traditional algorithms when imperfect synchronization is present. We conclude that when a sufficient number of PMUs with small delays are employed, the imperfect synchronization can be largely compensated in the estimation stage.