Observation sampling and quantisation for continuous-time estimators

A Bayesian estimation problem is considered, in which the observation is a vector-valued, continuous-time stochastic process of the ‘signal-plus-white-noise’ variety and approximations based on sampling and quantisation of this process are developed. The problem includes continuous-time nonlinear filters, interpolators and extrapolators as special cases. The effect of quantisation is characterised by means of a central limit theorem, which shows that the resulting loss of information is asymptotically equivalent to a modest reduction in signal-to-noise ratio – even when the quantisation is quite coarse. Optimal thresholds are derived that minimise this loss for a given degree of quantisation. The approximations based on quantised samples are shown to be significantly simpler than those based on raw samples, and this, in particular, allows the use of higher sampling rates, reducing approximation errors and increasing estimator currency.