Estimating Random Integrals from Noisy Observations: Sampling Designs and Their Performance.
James Antonio Bucklew, Stamatis Cambanis · 1985
Ahtract -The problem of estimating a weighted average of a random process from noisy observations at a finite number of sampling points is considered.The performance of sampling designs with optimal or suboptimal, but easily computable, estimator coefficients is studied.Several examples and special cases are studied including additive independent noise, nonlinear distortion with noise, and quantization noise. I. INTRODUCTIONHE PROBLEM of estimating a weighted integral of a T random process from observations of the process at a finite number of sampling points has been studied by several authors (see the survey [2]).It is an important problem of interest in several areas of communications, information theory, statistics, and signal processing.The usual questions of interest are to find the optimal sampling design of size n, or sampling designs which are asymptotically optimal as the sample size tends to infinity.Coupled with these is the problem of estimator design and the study of how the mean square estimation error tends to zero as the sample size tends to infinity.In this paper we consider these problems for the case where the observations are corrupted by noise.We allow the noise to be possibly dependent upon the random process whose integral we are trying to estimate, henceforth called the signal process.In t h s case, as the number of sampling points increases to infinity, the mean square approximation error no longer tends to zero but instead to some positive least possible value.We consider estimators which use optimal coefficients as well as suboptimal (but simple) coefficients.As far as the authors are aware, the only case of noisy observations considered in the literature is in [ 5 ] , [6], where the observation noise is assumed white and the signal Gauss-Markov.The optimal sampling designs are determined in [ 5 ] , and the rates of convergence of the mean square estimation error are found in [6] to be l / n with noise and l / n 2 with no noise.