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Independent component analysis for separating complex-valued signals has found utility in many applications such as communications, analysis of functional magnetic resonance imaging and electroencephalography data, face recognition, and radar beamforming. In this dissertation, we show the importance of matching the cost function to the source density in the complex case by showing the connection between maximization of non-Gaussianity, maximum likelihood, and minimization of mutual information. This connection emphasizes that optimal source separation algorithms must consider the bivariate distribution especially for the case of noncircular sources, i.e., sources that are not rotation invariant. Based on this result, we develop four density-matching ICA algorithms that are well suited for a wide range of circular and noncircular distributions. Along with the derivations of these algorithms, we also present a rigorous local stability analysis of the cost functions that explicitly show the effects of noncircularity on performance. We test the effectiveness of the four density-matching algorithms using simulations and real-world radar data and wind data. To effectively test the algorithms, we extend the bivariate generalized Gaussian distribution to the fully-complex case, present a maximum likelihood estimate for its shape and covariance parameters, provide a method for generating complex random variables from its distribution, and derive a generalized likelihood ratio test for testing whether a signal is noncircular or Gaussian.
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