I. COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING : RECENT ADVANCES

Complex-valued signals arise frequently in applications as diverse as commu- nications, 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. Complex-valued signal processing, however, has been long regarded as a simple extension of the real domain processing, and as a result, until recently most adaptive signal processing algorithms developed for the complex domain have failed to take full advantage of complex domain processing. Two key issues that need to be addressed in this regard pertain to optimization in the complex domain and the statistical characterization of complex signals. In this tutorial, we address both issues and first introduce Wirtinger calculus that allows development of an efficient framework for optimization in the complex domain. We introduce the fundamental relationships between the real and the complex domains, and show how many real domain optimization algorithms can be equivalently derived for the complex domain, such as the gradient-based and Newton update algorithms in a very straightforward manner. We then emphasize the importance of incorporating complete statistical information into estimation through the covariance and the so-called pseudo- covariance matrices, and the need to minimize assumptions on the statistics of complex signals such as circularity. We demonstrate how Wirtinger calculus enables one to minimize assumptions on the statistics of the complex signal, and allows one to easily incorporate the complete statistical information into the estimation. We provide a number of examples of practical significance that demonstrate how using these tools, one can easily derive algorithms and perform their analyses, and thus truly take advantage of complex domain processing. In particular, we discuss applications in widely linear filtering, training of complex-valued multilayer perceptron networks, and independent component analysis.