Optimum coding and decoding schemes for the transmission of a stochastic process over a continuous-time stochastic channel with partially unknown statisticst†

Abstract

In this paper, we obtain the complete solution of the following information transmission problem: A second-order Gaussian stochastic process [u t ,t∊[0,t ƒ]} is to be transmitted through a stochastic channel to a receiver under minimum mean square error distortion measure. The channel is to be used for exactly T seconds (where T may be smaller or larger than t ƒ) and, in addition to the White-Gaussian noise with a given energy level, the channel is also 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 a noiseless feedback is allowed between the receiver (decoder) and the transmitter (encoder). The paper provides expressions for the best encoder and decoder structures that function optimally under 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.