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In this paper, we use complex analytic functions to achieve independent component analysis (ICA) by maximization of non-Gaussianity and introduce the complex maximization of non-Gaussianity (CMN) algorithm. We derive both a gradient-descent and a quasi-Newton algorithm that use the full second-order statistics providing superior performance with circular and noncircular sources as compared to existing methods. We show the connection among ICA methods through maximization of non-Gaussianity, mutual information, and maximum likelihood (ML) for the complex case, and emphasize the importance of density matching for all three cases. Local stability conditions are derived for the CMN cost function that explicitly show the effects of noncircularity on convergence and demonstrated through simulation examples.

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Knowing the statistical properties of a complex-valued signal is important in many signal processing applications by providing the necessary information for choosing the appropriate algorithm. In this paper, we provide generalized likelihood ratio tests (GLRT), based on the complex generalized Gaussian distribution (CGGD), for detecting two important signal properties: 1) the circularity of a complex random variable, not constrained to the Gaussian case and 2) whether a complex random variable is complex Gaussian. These tests can be combined to statistically determine if a complex random variable is, the often assumed, circular Gaussian. Simulations are used to quantify the performance of the detectors followed by application to communication signals and actual radar data.