Digital Signal Processing Reference
In-Depth Information
1
COMPLEX-VALUED
ADAPTIVE SIGNAL
PROCESSING
T¨ lay Adalı and Hualiang Li
University of Maryland Baltimore County, Baltimore, MD
1.1 INTRODUCTION
Complex-valued signals arise frequently in applications as diverse as communi-
cations, 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
