# Accession Number:

## AD0713098

# Title:

## AN EXACT SOLUTION TO AN ADAPTIVE LINEAR ESTIMATION PROBLEM,

# Descriptive Note:

# Corporate Author:

## FRANK J SEILER RESEARCH LAB UNITED STATES AIR FORCE ACADEMY COLO

# Personal Author(s):

# Report Date:

## 1970-09-01

# Pagination or Media Count:

## 37.0

# Abstract:

An exact analysis of a particular form of adaptive estimator is presented. The result is the explicit evolution of the mean square estimation errors in general time-varying environments, as well as necessary and sufficient conditions for convergence of the estimates and explicit corresponding solutions in the presence of step changes in signal statistics. The estimator in question involves a gradient-following algorithm for recursively computing the solution to a discrete aprroximation to the Wiener-Hopf integral equation. The term adaptive is used to indicate that the gradient-following is permitted to continue indefinitely resulting in the capability of adjusting to sudden or unexpected evolutionary changes in process statistics at any time. Convergence of the estimates is defined in terms of the fractional excess mean square error or misadjustment maintained by the random estimator weights. Adaptive estimator convergence is intentionally weaker than stochastic approximation convergence so that precise knowledge of the time-varying nature of f the statistics is not required in order to obtain satisfactory mean-square-error performance. The specific problem which is analyzed exactly involves zero-mean, mutually gaussian distributed, independent sample vectors. The assumption of intersample independence is in part justified by the detailed analytical results which provide for the first time a prototype for the synthesis of adaptive estimators suitable for truly nonstationary environments. Author

# Descriptors:

# Subject Categories:

- Cybernetics