COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 75

The definition of analyticity and commonly invoked assumptions such as circularity
have been primarily motivated by this desire, such that the computations in the com-
plex domain parallel those in the real domain. Wirtinger calculus, on the other hand,
generalizes the definition of analyticity and enables development of a very convenient
framework for complex-valued signal processing, which again, as desired, allows
computations to be performed similar to the real case. Another important fact to
note is that the framework based on Wirtinger calculus is a complete one, in the
sense that the analytic case is included as a special case. Another attractive feature
of the framework is that promising nonlinear structures such as fully complex (ana-
lytic) functions can be easily incorporated in algorithm development both for use
within nonlinear filter structures such as MLPs and for the development of effective
algorithms for performing independent component analysis. Commonly invoked
assumptions such as circularity can be also easily avoided in the process, making
the resulting algorithms applicable to a general class of signals, thus not limiting
their usefulness. The only other reference besides this chapter—to the best of our
knowledge—that fully develops the optimization framework including second-order
relationships is [64], where the term CR calculus is used instead of Wirtinger calculus.
Though very limited in number, various textbooks have acknowledged the impor-
tance of complex-valued signals. In the most widely used topic on adaptive filtering,
[43], the complete development is given for complex signals starting with the 1991
edition of the topic. In [43, 60, 97, 105], special sections are dedicated to complex sig-
nals, and optimization in the complex domain is introduced using the forms of the
derivatives given in (1.5), also defined by Brandwood [15]. The simple trick that
allows regarding the complex function as a function of two variables, z and z ,
which significantly simplifies all computations, however, has not been noted in gen-
eral. Even in the specific instance where it has been noted—a recent topic [75] follow-
ing [4, 6, 70, 71]—Wirtinger calculus is relegated to an afterthought as derivations are
still given using the unnecessarily long and tedious split approach as in the previous
work by the authors, for example, as in [38, 39, 41]. The important point to note is
that besides simplifying derivations, Wirtinger calculus eliminates the need for
many restrictive assumptions and extends the power of many convenient tools in
analysis introduced for the real-valued case to the complex one. A simple example
is the work in [72] where the second-order analysis of maximum likelihood indepen-
dent component analysis is performed using a transformation introduced in [7] for the
real-valued case while bypassing the need for any circularity assumption.
It is also interesting to note that the two forms for the derivatives given in [15],
which are the correct forms and include the analytic case as well, have not been
widely adopted. In a recent literature search, we noted that a significant portion of
the papers published in the IEEE Transactions on Signal Processing and IEEE
Transactions on Neural Networks within the past five years define the complex deriva-
tive differently than the one given in [15], which was published in 1983. The situation
regarding contradictory statements and conflicting definitions in the complex domain
unfortunately becomes more discouraging when we look at second-order expansions
and algorithms. Even though the algorithms developed with derivative definitions
other than those in (1.5) still provide reasonable—and in certain cases—equivalent