Essential statistics and tools for complex random variables

Authors:
Jan Eriksson;Esa Ollila;Visa Koivunen
Affiliations:
SMARAD CoE, Signal Processing and Acoustics, Aalto University School of Science and Technology, Aalto, Finland;SMARAD CoE, Signal Processing and Acoustics, Aalto University School of Science and Technology, Aalto, Finland;SMARAD CoE, Signal Processing and Acoustics, Aalto University School of Science and Technology, Aalto, Finland
Venue:
IEEE Transactions on Signal Processing
Year:
2010

Citing 10
Cited 1

Statistics for complex variables and signals—Part I: variables

Signal Processing
Testing for spherical symmetry of a multivariate distribution

Journal of Multivariate Analysis
Deviation from normality in statistical signal processing: parameter estimation with alpha-stable distributions

A practical guide to heavy tails
Invariant tests for symmetry about an unspecified point based on the empirical characteristic function

Journal of Multivariate Analysis
Statistics for complex random variables revisited

ICASSP '09 Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing
On testing for impropriety of complex-valued Gaussian vectors

IEEE Transactions on Signal Processing
Widely linear estimation with complex data

IEEE Transactions on Signal Processing
Complex ICA Using Nonlinear Functions

IEEE Transactions on Signal Processing
Complex random vectors and ICA models: identifiability, uniqueness, and separability

IEEE Transactions on Information Theory
The multivariate complex normal distribution-a generalization

IEEE Transactions on Information Theory

On the asymptotic distribution of GLR for impropriety of complex signals

Signal Processing

Quantified Score

Hi-index	35.68

Visualization

Abstract

Complex random signals play an increasingly important role in array, communications, and biomedical signal processing and related fields. However, the fundamental properties of complex-valued signals and mathematical tools needed to process them are scattered in literature. We provide a concise, unified, and rigorous treatment of essential properties and tools of complex random variables, and apply these fundamentals to derive complex extensions of Leibniz rule, Faá di Bruno's formula, and Taylor's series. The extensions allow establishing relationships among complex moments and cumulants, and characterizing the circularity property. We propose measures for testing and quantifying circularity, and observe that non-circularity may be more common in practical applications than previously thought. All results are rigorously proved and supplemented with clarifying examples.