Wiener Filter Approximations Without Covariance Matrix Inversion

Abstract

In this article, we address the problem of ill-conditioning of the Wiener filter, the optimal linear minimum mean square error estimator. Computing the Wiener filter involves the inverse of the observation covariance matrix. In practice, the observation covariance matrix has a large condition number, resulting in unreliable numerical computation of the Wiener filter. To address this issue, we develop four approximate Wiener filter formulas using a truncation technique based on the principal components of a composite covariance matrix. Our approximate filter formulas do not directly involve the inverse of the observation covariance matrix. As a result, our filters are well-conditioned and numerically reliable to compute. We also present an asymptotic analysis of our approximate filter formulas and show that they converge to the Wiener filter as certain approximating terms vanish. Using real data, we evaluate the performance of our filters in terms of accuracy and computation time against the Wiener filter. Our performance-computation tradeoff results show that, unlike the Wiener filter, our filters have stable performance without significantly more computation, even when the covariance matrix is ill-conditioned.

Keywords

• Estimation
• optimization
• random matrix theory
• Wiener filters