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. The complex domain not only provides a convenient representation for these signals but also a natural way to preserve the physical characteristics of the signals and the transformations they go though. The complex domain, however, also presents a number of challenges in the derivation and analysis of signal processing algorithms, and as a result, the vast majority of algorithms developed for the complex domain have taken shortcuts limiting their usefulness.
This research establishes a framework for complex-valued signal processing such that the full potential of complex-valued signal processing can be realized. It allows for all computations to be carried out in the complex domain eliminating the need for many simplifying assumptions, such as the circularity of signal, both in the derivation and the analysis of the algorithms. It also allows for the use of fully complex functions rather than the more commonly utilized bounded but non-analytic functions. These functions provide attractive alternatives for performing independent component analysis (ICA) by efficiently generating higher-order statistical information. Using this framework, a new class of efficient algorithms are derived for performing ICA in the complex domain, in particular, for studying brain function using the medical imaging data in its native, complex form.
