Minimax Mean-Squared Error Estimation of Multichannel Signals Amir Beck ∗ , Yonina C. Eldar † and Aharon Ben-Tal ‡ October 28, 2005 Abstract We consider the problem of multichannel estimation, in which we seek to estimate N deterministic input vectors x k that are observed through a set of linear transformations and corrupted by additive noise, where the linear transformations are subjected to uncertainty. To estimate the inputs x k we propose a minimax mean-squared error (MSE) approach in which we seek the linear estimator that minimizes the worst-case MSE over the uncertainty region, where we assume that the weighted norm of each of the inputs x k is bounded and that each of the linear transformations is perturbed by a bounded norm disturbance. For an arbitrary choice of weighting, we show that assuming a block circulant structure on the resulting model matrix, the minimax MSE estimator can be formulated as a solution to a semidefinite programming problem (SDP), which can be solved efficiently. For an Euclidean norm bound on x k , the SDP is reduced to a simple convex program with N + 1 unknowns. Finally, we demonstrate through examples, that the minimax MSE estimator can significantly increase the performance over conventional methods. 1 Introduction Estimation of multiple signals from multiple outputs is an important problem that appears in a variety of applications, such as blind multichannel estimation [7, 5, 8, 34], speech separation [32, 38, 40, 12] and image restoration [22, 14, 13]. In a multichannel estimation problem, we seek to estimate multiple input vectors {x k , 0 ≤ k ≤ N − 1}, that are observed through a set of linear transformations H k,j and corrupted by additive noise. Thus, the kth output vector y k is given by the superposition y k = ∑ N −1 j =0 H k,j x j + w k , where H k,j is the transfer function from the kth input x j to the j th output y k , and w k are the noise vectors. There is a vast body of literature that treats the multichannel estimation problem, under the assumption that the input vectors are random with known statistics. If the transfer matrices H k,j as well as the second order statistics of the input vectors and the noise vectors are known, then we can design an estimator to minimize the mean-squared error (MSE). The resulting estimator is the well-known Wiener estimator, or the minimum MSE (MMSE) estimator. However, if the inputs are deterministic or H k,j are unknown, then the Wiener estimator cannot be implemented. A significant amount of prior knowledge is required for MMSE multichannel estimation. Specifically, the statistics of all the inputs and noise vectors must be known. In many practical scenarios, this information may not be available. Although the MMSE estimator minimizes the MSE, its success depends on accurate statistical knowledge of the inputs and noise characteristics. Therefore, in many cases it is more reasonable to assume that the inputs are deterministic but unknown. * MINERVA Optimization Center, Department of Industrial Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel. E-mail: becka@tx.technion.ac.il. † Department of Electrical Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel. E-mail: yonina@ee.technion.ac.il. ‡ MINERVA Optimization Center, Department of Industrial Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel. E-mail: morbt@ie.technion.ac.il. 1