COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 5

involved, in order to transform the solution back to the complex domain, we
usually have to make additional assumptions such as analyticity of
the function. We give a simple example (Example 1.3) to highlight this
point in Section 1.2.2.
(3) When working in the real-dimensional space with the double dimension, many
quantities assume special forms. Matrices in this space usually have special
block structures which can make further analysis and manipulations more com-
plicated. In fact, these structures have been the primary motivation for invoking
certain simplifying assumptions in the analysis, such as the circularity of
signals. For example, this assumption is made in [13] in the derivation of an
independent component analysis algorithm when computing the Hessian pri-
marily for this reason. Circularity, which implies that the phase of the signal
is uniformly distributed and hence is noninformative, is in most cases an unrea-
listic assumption limiting the usefulness of algorithms. The communications
signals shown in Figure 1.1 as well as a number of other real-world signals
can be shown not to satisfy this property, and are discussed in more detail in
Section 1.2.5.

N

2 N , which is isomorphic,
we have to remember that mathematical equivalence does not imply that the optimiz-
ation, analysis, and numerical and computational properties of the algorithms will be
similar in these two domains. We argue that C

Thus, even though we can define a transformation C 7! R

N defines a much more desirable domain
for adaptive signal processing in general and give examples to support our point.
Using Wirtinger calculus, most of the processing and analysis in the complex
domain can be performed in a manner very similar to the real-valued case as we
describe in this chapter, thus eliminating the need to consider such transformations
in the first place.
The theory and algorithms using the widely linear and the fully complex filter can
be easily developed using Wirtinger calculus. Both of these filters are powerful tools
for complex-valued signal processing that allow taking advantage of the full proces-
sing power of the complex domain and without having to make limiting assumptions
on the nature of the signal, such as circularity.

1.1.2 Outline of the Chapter

To present the development, we first present preliminaries including a review of basic
results for derivatives and Taylor series expansions, and introduce the main idea
behind Wirtinger calculus that describes an effective approach for complex-valued
signal processing. We define first- and second-order Taylor series expansions in the
complex domain, establish the key relationships that enable efficient derivation of
first- and second-order adaptive algorithms as well as performing analyses such as
local stability using a quadratic approximation within a neighborhood of a local opti-
mum. We also provide a review of complex-valued statistics, again a topic that has
been, for the most part, treated in a limited form in the literature for complex signals.
We carefully define circularity of a signal, the associated properties and complete