Complex-Valued Signal Processing: The Proper Way to Deal With Impropriety

Complex-valued signals occur in many areas of science and engineering and are thus of fundamental interest. In the past, it has often been assumed, usually implicitly, that complex random signals are proper or circular. A proper complex random variable is uncorrelated with its complex conjugate, and...

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Bibliographic Details
Published in:IEEE transactions on signal processing 2011-11, Vol.59 (11), p.5101-5125
Main Authors: Adali, T., Schreier, P. J., Scharf, L. L.
Format: Article
Language:English
Subjects:
Applied sciences
Calculus
Cost function
CR calculus
Detection, estimation, filtering, equalization, prediction
estimation
Exact sciences and technology
Filtering
Filtration
improper
independent component analysis
Information, signal and communications theory
Invariants
Materials
Mathematical models
model selection
noncircular
Random signals
Random variables
Signal and communications theory
Signal processing
Signal, noise
Studies
Telecommunications and information theory
Vectors
widely linear
Wirtinger calculus
Zirconium
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Summary:Complex-valued signals occur in many areas of science and engineering and are thus of fundamental interest. In the past, it has often been assumed, usually implicitly, that complex random signals are proper or circular. A proper complex random variable is uncorrelated with its complex conjugate, and a circular complex random variable has a probability distribution that is invariant under rotation in the complex plane. While these assumptions are convenient because they simplify computations, there are many cases where proper and circular random signals are very poor models of the underlying physics. When taking impropriety and noncircularity into account, the right type of processing can provide significant performance gains. There are two key ingredients in the statistical signal processing of complex-valued data: 1) utilizing the complete statistical characterization of complex-valued random signals; and 2) the optimization of real-valued cost functions with respect to complex parameters. In this overview article, we review the necessary tools, among which are widely linear transformations, augmented statistical descriptions, and Wirtinger calculus. We also present some selected recent developments in the field of complex-valued signal processing, addressing the topics of model selection, filtering, and source separation.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2011.2162954