Filter Design with Secrecy Constraints: The Degraded Parallel Gaussian Wiretap Channel

We introduce the problem of filter design with secrecy constraints in the classical wiretap scenario, where two legitimate parties, Alice and Bob, communicate in the presence of an eavesdropper, Eve. In particular, we consider the design of transmit and receive filters that minimize the mean-squared error (MSE) between the legitimate parties whilst guaranteeing a certain eavesdropper MSE level subject to a total average power constraint, in the situation where the main channel and the wiretap channel consist of a bank of parallel independent degraded Gaussian channels (a scenario representative of orthogonal frequency division multiplexing (OFDM) communications systems). We derive the form of the optimal receive filter and the optimal transmit filter, which are diagonal. In the regime of low available power, we demonstrate that the diagonal elements of the optimal transmit filter satisfy a waterfilling type of expression. In contrast, in the regime of high available power the diagonal elements of the optimal transmit filter satisfy a simple expression involving the linear minimum mean-squared error (LMMSE) that obeys a simple operational interpretation. A range of numerical results corroborate the conclusions.