On the distributions of some statistics related to adaptive filters trained with $t$-distributed samples

TL;DR

This paper investigates how adaptive filters behave when trained with t-distributed samples instead of Gaussian, analyzing the impact on key statistics and deriving properties of related matrices to quantify performance degradation.

Contribution

It introduces new statistical representations for adaptive filter metrics trained with t-distributions, extending existing Gaussian-based models.

Findings

t-distribution training causes measurable performance degradation

Derived properties of complex F-distributed matrices facilitate analysis

Numerical simulations confirm theoretical predictions

Abstract

In this paper we analyse the behaviour of adaptive filters or detectors when they are trained with $t$-distributed samples rather than Gaussian distributed samples. More precisely we investigate the impact on the distribution of some relevant statistics including the signal to noise ratio loss and the Gaussian generalized likelihood ratio test. Some properties of partitioned complex $F$ distributed matrices are derived which enable to obtain statistical representations in terms of independent chi-square distributed random variables. These representations are compared with their Gaussian counterparts and numerical simulations illustrate and quantify the induced degradation.