Complex-valued signals arise frequently in applications as diverse as communications, radar, and biomedicine, as most practical modulation formats are of complex type and applications such as radar and magnetic resonance imaging lead to data that are inherently complex valued. When the processing has to be done in a transform domain such as Fourier or complex wavelet, again the data are complex valued. In order to perform independent component analysis of complex-valued data two main methods have been proposed: one method uses the signal’s modulus in the cost functions as a measure of independence, and the second path use a split-type cost function where the real and imaginary parts are processed separately. In both cases, the cost function does not use the phase information in the signal, and hence, do not perform in an optimal fashion. 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 mutual information. This result highlights the importance of matching the cost function to the source density in two ways: in terms of maximum likelihood, an asymptotically efficient estimator is realized, and second, mutual information is a natural measure of independence. Thus, density matching provides the optimal option for ICA, able to handle a wide range of circular and noncircular distributions from both the sub-Gaussian and super-Gaussian families.

Based on this result, we develop four flexible ICA algorithms that provide density matching through: transcendental functions as in the T-CMN algorithm, adaptive on-line density estimation using the A-CMN algorithm, a Gaussian mixture model approach to match BPSK and QAM signals with the C-QAM algorithm, and extending the c-FastlCA algorithm to noncircular sources as in the nc-FastlCA algorithm. 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, or more specifically, defines the space of noncircular distributions that are unstable using the algorithm.

We test the effectiveness of the four density-matching algorithms using simulations and real-world radar data and wind data. The results show that density-matching provides improved performance as compared with current complex-valued independent component analysis methods.

To effectively test the algorithms, a method for generating these varied distributions is necessary. To this end, 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.