Mean-squared error estimation for linear systems with block circulant uncertainty

We consider the problem of estimating a vector x in the linear model Ax ≈ y, where A is a block circulant (BC) matrix with N blocks and x is assumed to have a weighted norm bound. In the case where both A and y are subjected to noise, we propose a minimax mean-squared error (MSE) approach in which we seek the linear estimator that minimizes the worst-case MSE over a BC structured uncertainty region. For an arbitrary choice of weighting, we show that the minimax MSE estimator can be formulated as a solution to a semidefinite programming problem (SDP), which can be solved efficiently. For a Euclidean norm bound on x, the SDP is reduced to a simple convex program with N + 1 unknowns. Finally, we demonstrate through an image deblurring example the potential of the minimax MSE approach in comparison with other conventional methods.