Minimax Mean-Squared Error Estimation of Multichannel Signals

We consider the problem of multichannel estimation, in which we seek to estimate N deterministic input vectors xk 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 xk 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 xk 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 xk, 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.