Classical Widely Linear Estimation of Real Valued Parameter Vectors in Complex Valued Environments

TL;DR

This paper introduces widely linear estimators for accurately estimating real-valued parameters from complex measurements, outperforming standard methods and requiring fewer measurements in practical applications.

Contribution

It proposes the widely linear least squares and the best widely linear unbiased estimators for real-valued parameters in complex environments, improving accuracy and efficiency.

Findings

Widely linear estimators outperform standard estimators in accuracy.

Proposed methods require only half as many measurements.

Application to impulse response estimation shows superior performance.

Abstract

This work investigates the task of estimating a real valued parameter vector based on complex valued measurements in a classical set-up. The application of standard estimators in general results in complex valued estimates of the real valued parameter vector. To avoid this systematic error, widely linear classical estimators that produce real valued estimates are investigated. One of these estimators is the widely linear least squares (WLLS) estimator proposed in this work, which does not utilize any noise statistics. Further, we introduce the best widely linear unbiased estimator (BWLUE) for real valued parameter vectors. The proposed estimators in general outperform their standard counterparts LS estimator and BWLUE, respectively, and they only require half as many complex valued measurements. We compare the novel approaches to standard classical estimators in two application scenarios. One of these applications considers the estimation of a real valued impulse response based on noisy measurements of the system's magnitude and phase response. For this problem, we propose a novel two-step approach based on the introduced widely linear concepts that outperforms standard estimators.