This paper considers the problem of filter design with secrecy constraints, where two legitimate parties (Alice and Bob) communicate in the presence of an eavesdropper (Eve), over a Gaussian multiple-input multiple-output (MIMO) wiretap channel. This problem involves the design of transmit and receive filters which minimize the mean-square error (MSE) between the legitimate parties, whilst assuring that the eavesdropper MSE remains above a certain level. We characterize the form of the optimal transmit filter when both the legitimate receiver and the eavesdropper employ Zero-Forcing (ZF) filters. By capitalizing on the dual problem, we also show that the original matrix optimization problem can be reduced to a simple scalar optimization problem, whose solution can be readily computed by employing a simple bisection method. Numerical results illustrate the main conclusions.
