Optimal causal coding of general alphabet Markov Sources with entropy and channel input constraints

For Markov sources, the structure of optimal causal encoders minimizing the total communication rate subject to a mean-square distortion constraint is studied. Results on the nature of non-stationarity of the optimal quantizers are presented. The class of sources considered lives in a continuous alphabet, and the encoder is allowed to be variable-rate. Both the finitehorizon and the infinite-horizon problems are considered. In the finite-horizon case, the problem is non-convex, whereas in the infinite-horizon case the problem can be convexified under certain assumptions. For a finite horizon problem, a Lagrangian framework is adopted. Upon showing the existence of a Lagrange multiplier, it is shown that the optimal deterministic causal encoder for a kth-order Markov source uses only the most recent k source symbols and the information available at the receiver, whereas the optimal causal coder for a memoryless source is memoryless. For the infinite-horizon problem, a convex analytical approach is adopted. Randomized stationary quantizers are suboptimal in the absence of side information between the encoder and the decoder with regard to the randomization information, which will be referred to as common randomness. If there is common randomness, the optimal quantizer requires the randomization of at most two deterministic quantizers. In the absence of common randomness, the optimal quantizer is non-stationary and a recurrence-based time sharing of two deterministic quantizers is optimal. A linear source driven by Brownian disturbance is considered. If the process is stable, innovation coding is almost optimal at high-rates, whereas if the source is unstable, then even a high-rate time-invariant innovation coding scheme leads to an almost surely unstable estimation process.