Encoder and decoder design for signal estimation

In this paper, we study the joint design of optimal linear encoders and decoders for filtering and transmission of a signal over an additive Gaussian noise channel subject to a real-time constraint. The objective is to minimize the variance of the estimation error at the receiving end. The design problem is nonconvex, but it is shown that a global optimum can be found by solving a related two-stage problem. The first stage consists of a mixed H₂ and H₁ norm minimization problem, where the H₂ norm corresponds to the error variance in a corresponding Wiener-Kolmogorov filtering problem and the H₁ norm is induced by the channel noise. The second stage consists of a spectral factorization. The results are illustrated by a numerical example.