Optimum linear causal coding schemes for Gaussian stochastic processes in the presence of correlated jamming

The complete solution is obtained to the following problem. A Gaussian stochastic process {θ_t, t ε[0, t_f]} satisfying a certain stochastic differential equation is to be transmitted through a stochastic channel to a receiver under minimum mean-squared error distortion measure. The channel is to be used for exactly t_f seconds, and, in addition to white Gaussian noise with a given energy level, the channel is corrupted by another source whose output may be correlated with the input to the channel and which satisfies a given power constraint. There is an input power constraint to the channel, and noiseless feedback is allowed between the receiver (decoder) and the transmitter (encoder). The authors determine the linear causal encoder and decoder structures that function optimally under the worst admissible noise inputs to the channel. The least favorable probability distribution for this unknown noise is found to be Gaussian and is correlated with the transmitted signal. Also included is a comparative study of these results with earlier ones that addressed a similar problem without a causality restriction imposed on the transmitter.