# Slepian-Bangs formula and Cramer Rao bound for circular and non-circular complex elliptical symmetric distributions

## TL;DR

This paper extends the Slepian-Bangs formula and analyzes the Cramer-Rao bound for non-circular complex elliptical symmetric distributions, revealing new insights into their statistical properties and bounds.

## Contribution

It introduces a new stochastic representation theorem for NC-CES distributions and extends the Slepian-Bangs formula to these distributions, including non-circular complex Gaussian cases.

## Key findings

- Gaussian distribution does not always maximize the stochastic CRB
- Derived closed-form SCRBs for noisy mixture models
- Established relations between CES and Gaussian CRBs

## Abstract

This paper is mainly dedicated to an extension of the Slepian-Bangs formula to non-circular complex elliptical symmetric (NC-CES) distributions, which is derived from a new stochastic representation theorem. This formula includes the non-circular complex Gaussian and the circular CES (CCES) distributions. Some general relations between the Cramer Rao bound (CRB) under CES and Gaussian distributions are deduced. It is proved in particular that the Gaussian distribution does not always lead to the largest stochastic CRB (SCRB) as many authors tend to believe it. Finally a particular attention is paid to the noisy mixture where closedform expressions for the SCRBs of the parameters of interest are derived.