Optimal design of real-time causal differential encoding systems with Lloyd-Max quantization

This paper studies the design of a real-time causal encoder and decoder for discrete-time linear systems with noiseless communication channels. The encoder consists of a Kalman filter and a Lloyd-Max quantization law. The performance of the encoder is measured by the mean-square quantization distortion. An optimal design scheme is presented to the real-time causal differential encoder. To synchronize the encoder and decoder, the protocol on the updating rules of the encoder and decoder based on quantized measurements is proposed. For the decoder, given the differential encoder above, two recursive algorithms for the minimum mean-square error (MMSE) estimator of the state of the system are presented. Then an upper bound for the ensemble average performance is presented, which is very tight and approaches the average performance in the high-resolution case.