Complex-Valued Adaptive Signal Processing using Wirtinger Calculus and its Application to Independent Component Analysis
Creators: name: Li, Hualiang
Publisher: name: Maryland Shared Open Access Repository
Year: 2008
Subjects: Complex signal processing
Independent Component Analysis
Optimization
Wirtinger calculus
Engineering, Electronics and Electrical (0544)

description: Complex-valued signals arise frequently in applications as diverse as communications, radar, and biomedicine. The complex domain is the natural home for the processing of these signals, however, it also poses a  number of challenges in the derivation and analysis of algorithms. As a result, most  algorithms developed for the complex domain take shortcuts that limit the usefulness of the algorithms. In this dissertation, we introduce a framework based on Wirtinger calculus that enables working completely in the complex domain for the derivation and analysis of signal processing algorithms. Using this framework, we can perform algorithm derivation and analysis completely in the complex domain in a straightforward and  efficient manner. Beyond offering simple convenience, this approach makes many signal processing tools developed for the real-valued domain readily available for complex-valued signal processing and eliminates the need to make simplifying assumptions in the derivations and analyses that have become common place for many signal processing algorithms.   Using Wirtinger calculus, we establish the fundamental relationships for optimization between the real and the complex domains, thus show how many real-valued optimization algorithms can be equivalently derived for the complex domain, such as the gradient-based and Newton update algorithms. We provide a number of key examples to demonstrate the application of the framework to complex-valued adaptive signal processing such that the true processing power of the complex domain can be realized. The first example is the design of a multilayer perceptron (MLP) filter and the derivation of the gradient update (back-propagation) rule. Compared to previous work on complex MLP, our framework allows the derivation of the complex back-propagation rule in a very straightforward manner.   Then, 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.