Noisy data-rate limited estimation: renewal codes

We consider the problem of estimating the state of a system whose observations are transmitted from the sensor to the estimator over a binary symmetric channel (BSC). A coder at the sensor encodes its data into channel inputs. The estimator receives the channel output and estimates the state. There is a perfect ´reverse´ channel from the estimator back to the coder, so the coder has full knowledge of the decoder. Such a system model is appropriate when the coder is power limited, but the decoder is not. The existence of a stable estimator (i.e. one with bounded estimation error) depends on two parameters: the degree of instability of the system, and the channel error probability. We present estimators for a subset of these parameters. It is difficult to design of a coder-decoder pair for a stable estimator. Classical codes for the BSC cannot be used, because they introduce large coding delays, hence large errors in the state estimate. We need codes that simultaneously impose very small delays and reduce decoding errors. These considerations point to a rich class of design problems that lie in the intersection of real-time control and communication. Previous work addressed this problem using a non-constructive random coding argument. The estimator presented here is constructive and has very low complexity. Surprisingly, too, our estimator works for parameter values not covered by the random coding argument.