A general approach of least squares estimation and optimal filtering

TL;DR

This paper presents a comprehensive framework for least squares estimation, including its relation to optimal filtering, with detailed analysis of estimators and variances in both time and frequency domains.

Contribution

It introduces a general approach to least squares estimation, formalizes its properties, and establishes its equivalence with optimal filtering techniques.

Findings

The estimator and variance are characterized in time and Fourier domains.

The equivalence between Generalized Least Squares and optimal filtering is proven.

Numerical considerations for implementation are discussed.

Abstract

The least squares method allows fitting parameters of a mathematical model from experimental data. This article proposes a general approach of this method. After introducing the method and giving a formal definition, the transitivity of the method as well as numerical considerations are discussed. Then two particular cases are considered: the usual least squares method and the Generalized Least Squares method. In both cases, the estimator and its variance are characterized in the time domain and in the Fourier domain. Finally, the equivalence of the Generalized Least Squares method and the optimal filtering technique using a matched filter is established.