Weak convergence and asymptotic properties of adaptive filters with constant gains

Author

Kushner, Harold J. ; Shwartz, Adam

Volume

Issue

fYear

1984

fDate

3/1/1984 12:00:00 AM

Firstpage

177

Lastpage

182

Abstract

The basic adaptive filtering algorithm $X_{n+1}^{\\epsilon} = X_{n}^{\\epsilon} - \\epsilon Y_{n}(Y_{n}^{\´}X_{n}^{\\epsilon} - psi_{n})$ is analyzed using the theory of weak convergence. Apart from some very special cases, the analysis is hard when done for each fixed $\\epsilon > 0$ . But the weak convergence techniques are set up to provide much information for small $\\epsilon$ . The relevant facts from the theory are given. Define $x^{\\epsilon}(\\cdot)$ by $x^{\\epsilon}(t) = X_{n}^{\\epsilon}$ on $[n\\epsilon, n\\epsilon + \\epsilon)$ . Then weak (distributional) convergence of ${x^{\\epsilon}(\\cdot)}$ and of ${x^{\\epsilon}(\\cdot + t_{\\epsilon})}$ is proved under very weak assumptions, where $t_{\\epsilon} \\rightarrow \\infty$ as $\\epsilon \\rightarrow 0$ . The normalized errors ${(X_{n}^{\\epsilon} - \\theta ) / \\sqrt {\\epsilon} }$ are analyzed, where $\\theta$ is a "stable" point for the "mean" algorithm. The asymptotic properties of a projection algorithm are developed, where the $X_{n}^{\\epsilon}$ are truncated at each iteration, if they fall outside of a given set.

Keywords

Adaptive filters; Adaptive filters; Algorithm design and analysis; Convergence; Differential equations; Filtering algorithms; Filtering theory; Information analysis; Mathematics; Projection algorithms; Random variables;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/TIT.1984.1056897

Filename

1056897

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=938760