Survey and some new results on performance analysis of complex-valued parameter estimators

Recently, there has been an increased awareness that simplistic adaptation of performance analysis developed for random real-valued signals and parameters to the complex case may be inadequate or may lead to intractable calculations. Unfortunately, many fundamental statistical tools for handling complex-valued parameter estimators are missing or scattered in the open literature. In this paper, we survey some known results and provide a rigorous and unified framework to study the statistical performance of complex-valued parameter estimators with a particular attention paid to properness (i.e., second order circularity), specifically referring to the second-order statistical properties. In particular, some new properties relative to the properness of the estimates, asymptotically minimum variance bound and Whittle formulas are presented. A new look at the role of nuisance parameters is given, proving and illustrating that the noncircular Gaussian distributions do not necessarily improve the Cramer–Rao bound (CRB) with respect to the circular case. Efficiency of subspace-based complex-valued parameter estimators that are presented with a special emphasis is put on noisy linear mixture.