Fisher information as a performance metric for locally optimum processing

TL;DR

This paper demonstrates that Fisher information quantifies the asymptotic performance of locally optimum processors in weak signal detection and transmission, establishing it as an upper bound for performance.

Contribution

It links Fisher information to the performance limits of locally optimum processing for weak signals in noise, providing a theoretical performance bound.

Findings

Fisher information determines maximum SNR gain, ARE, and CG in LOPs.

Gaussian noise has minimal Fisher information, equal to one.

Dichotomous noise can have infinite Fisher information, indicating perfect processing.

Abstract

For a known weak signal in additive white noise, the asymptotic performance of a locally optimum processor (LOP) is shown to be given by the Fisher information (FI) of a standardized even probability density function (PDF) of noise in three cases: (i) the maximum signal-to-noise ratio (SNR) gain for a periodic signal; (ii) the optimal asymptotic relative efficiency (ARE) for signal detection; (iii) the best cross-correlation gain (CG) for signal transmission. The minimal FI is unity, corresponding to a Gaussian PDF, whereas the FI is certainly larger than unity for any non-Gaussian PDFs. In the sense of a realizable LOP, it is found that the dichotomous noise PDF possesses an infinite FI for known weak signals perfectly processed by the corresponding LOP. The significance of FI lies in that it provides a upper bound for the performance of locally optimum processing.