Beyond Thresholding: Analysis and Improvements for Deterministic Parameter Estimation

TL;DR

This paper analyzes hard-threshold estimators in signal processing, showing that optimized piecewise-linear estimators outperform traditional hard thresholding in deterministic signal estimation under Gaussian noise.

Contribution

It introduces and compares piecewise-linear estimators as an improvement over hard thresholding, with performance benefits demonstrated through theoretical analysis.

Findings

Piecewise-linear estimators outperform hard thresholding when optimized.

Performance improvements are validated against Cramér-Rao bounds.

Optimized estimators achieve better decay rate performance.

Abstract

Hard-threshold estimators are popular in signal processing applications. We provide a detailed study of using hard-threshold estimators for estimating an unknown deterministic signal when additive white Gaussian noise corrupts observations. The analysis, depending heavily on Cram{\'e}r-Rao bounds, motivates piecewise-linear estimation as a simple improvement to hard thresholding. We compare the performance of two piecewise-linear estimators to a hard-threshold estimator. When either piecewise-linear estimator is optimized for the decay rate of the basis coefficients, its performance is better than the best possible with hard thresholding.