Tülay Adalı and Hualiang Li
University of Maryland Baltimore County, Baltimore, MD
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 (MRI) 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. 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 through, such as the phase and magnitude distortion a communications signal experiences. In all these cases, the processing also needs to be carried out in the complex domain in such a way that the complete information—represented by the interrelationship of the real and imaginary parts or the magnitude and phase of the signal—can be fully exploited.
In this chapter, 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, and in such a way that all of the computations can be performed in a straightforward manner, very similarly to the real-valued case. In the derivation of adaptive algorithms, we need to evaluate the derivative ...