Complex-valued signal processing — essential models, tools and statistics

Complex-valued signals arise in many diverse fields such as communications, radar, biomedical sciences, physical sciences, and related fields. This paper briefly reviews some important tools, statistics, models and estimators that are useful for handling complex-valued random signals. Over the past four decades, circularity (i.e. invariance of the distribution under multiplication by a unit complex number) or second-order circularity (i.e. uncorrelatedness of the random vector with its complex conjugate) has been a common implicit assumption. Hence in this paper a special emphasis is put on this circularity property, as optimal signal processing methods for circular and non-circular signals are often different and choosing the right type of processing can provide significant performance gains. Topics reviewed in this paper include different types of circularity measures and detectors of circularity, complex elliptical symmetry of random variables, Cramér-Rao lower bounds on the estimation of complex-valued parameters, optimization of a real-valued cost function with respect to complex-valued parameters using CR-calculus, and complex-valued independent component analysis.