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

T. Adali, P. J. Schreier, L. L. Scharf
2011 IEEE Transactions on Signal Processing  
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,
more » ... re 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 complexvalued 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 T. for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. selected recent developments in the field of complex-valued signal processing, addressing the topics of model selection, filtering, and source separation.
doi:10.1109/tsp.2011.2162954 fatcat:geh5hio6hvad5nzh73tz2m6am4