# Cram\'er-Rao Bounds for Complex-Valued Independent Component Extraction: Determined and Piecewise Determined Mixing Models

## TL;DR

This paper derives Cramér-Rao lower bounds for complex-valued blind source extraction under determined and piecewise determined mixing models, providing new theoretical performance limits and extending previous real-domain results to complex signals.

## Contribution

It introduces CRLB analysis for complex-valued BSE with determined and piecewise determined models, including new bounds for varying mixing scenarios.

## Key findings

- CRLB for ICE matches ICA when background non-Gaussianity is considered
- New bounds for piecewise determined mixing models
- Extension of real-domain results to complex signals

## Abstract

This paper presents Cram\'er-Rao Lower Bound (CRLB) for the complex-valued Blind Source Extraction (BSE) problem based on the assumption that the target signal is independent of the other signals. Two instantaneous mixing models are considered. First, we consider the standard determined mixing model used in Independent Component Analysis (ICA) where the mixing matrix is square and non-singular and the number of the latent sources is the same as that of the observed signals. The CRLB for Independent Component Extraction (ICE) where the mixing matrix is re-parameterized in order to extract only one independent target source is computed. The target source is assumed to be non-Gaussian or non-circular Gaussian while the other signals (background) are circular Gaussian or non-Gaussian. The results confirm some previous observations known for the real domain and bring new results for the complex domain. Also, the CRLB for ICE is shown to coincide with that for ICA when the non-Gaussianity of background is taken into account. %unless the assumed sources' distributions are misspecified. Second, we extend the CRLB analysis to piecewise determined mixing models. Here, the observed signals are assumed to obey the determined mixing model within short blocks where the mixing matrices can be varying from block to block. However, either the mixing vector or the separating vector corresponding to the target source is assumed to be constant across the blocks. The CRLBs for the parameters of these models bring new performance bounds for the BSE problem.