Deterministic Cramer-Rao bound for strictly non-circular sources and analytical analysis of the achievable gains

TL;DR

This paper derives the deterministic Cramer-Rao bound for strictly non-circular sources, analyzes the potential estimation gains, and provides simplified expressions and conditions for maximum achievable NC gain in specific scenarios.

Contribution

It introduces a closed-form deterministic R-D NC CRB for multi-dimensional parameter estimation of non-circular sources and analyzes the achievable NC gain analytically.

Findings

In some cases, no NC gain is achievable.

Simplified CRB expressions for two closely-spaced sources.

Conditions for maximum NC gain depending on physical parameters.

Abstract

Recently, several high-resolution parameter estimation algorithms have been developed to exploit the structure of strictly second-order (SO) non-circular (NC) signals. They achieve a higher estimation accuracy and can resolve up to twice as many signal sources compared to the traditional methods for arbitrary signals. In this paper, as a benchmark for these NC methods, we derive the closed-form deterministic R-D NC Cramer-Rao bound (NC CRB) for the multi-dimensional parameter estimation of strictly non-circular (rectilinear) signal sources. Assuming a separable centro-symmetric R-D array, we show that in some special cases, the deterministic R-D NC CRB reduces to the existing deterministic R-D CRB for arbitrary signals. This suggests that no gain from strictly non-circular sources (NC gain) can be achieved in these cases. For more general scenarios, finding an analytical expression of the NC gain for an arbitrary number of sources is very challenging. Thus, in this paper, we simplify the derived NC CRB and the existing CRB for the special case of two closely-spaced strictly non-circular sources captured by a uniform linear array (ULA). Subsequently, we use these simplified CRB expressions to analytically compute the maximum achievable asymptotic NC gain for the considered two source case. The resulting expression only depends on the various physical parameters and we find the conditions that provide the largest NC gain for two sources. Our analysis is supported by extensive simulation results.