## Complex-valued Adaptive Signal Processing Using Wirtinger Calculus and Its Application to Independent Component AnalysisThen, we study complex independent component analysis (ICA) using our framework. ICA has emerged as a powerful and attractive statistical tool for revealing hidden factors for many types of signals. Two of the most important guiding principles for performing ICA are maximum likelihood and maximization of non-Gaussianity. Following the principle of maximization of non-Gaussianity, we derive a class of effective complex ICA algorithms that provide reliable performance for a wide range of input source distributions. Stability analysis is provided to show its superior convergence rate. We also derive a class of complex ICA algorithms based on maximum likelihood estimation. We perform local stability analysis of maximum likelihood ICA algorithms and show that the complex ICA problem is more difficult to solve with non-circular sources. We also show that the stability conditions are easier to be satisfied when the mixtures are whitened and unitary constraints are imposed. Simulation results further demonstrate these observations with generalized Gaussian distributed sources. |

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### Contents

PRELIMINARIES | 8 |

OPTIMIZATIONINTHE COMPLEX DOMAINUSINGWIRTINGER | 30 |

COMPLEX INDEPENDENT COMPONENT ANALYSIS | 49 |

COMPLEX INDEPENDENT COMPONENT ANALYSIS | 70 |

CONCLUSIONS AND FUTURE WORK | 89 |

### Common terms and phrases

activation functions Adalı adaptive signal processing analytic approach Average ISI back-propagation c-FastICA Chapter circular sources complex conjugate complex domain complex gradient complex ICA algorithms complex ICA model complex ML ICA conjugate gradient conjugate gradient algorithm convergence cost function deﬁned df df differential dissertation efﬁcient estimation theory evaluate example FastICA algorithm ﬁrst ﬁrst-order ﬁxed-point framework function f(z GGD sources gradient update rule Hessian matrix ICA problem ICA update rule independent component analysis JADE KM algorithms KM-F KM-G kurtosis maximization linear maximum likelihood ML ICA algorithm ML ICA update ML-unitary ICA algorithm Newton algorithm Newton rule Newton update rule non-circular sources optimization performance random variable real and imaginary real domain real-differentiable satisﬁed second-order circular shape parameter shown signal processing algorithms split-complex stability analysis stability conditions sufﬁcient Taylor series Taylor series expansion theorem transformations update rule given vector whitened Wirtinger calculus Wirtinger derivatives Wolfe condition write