COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 8

derivatives is [88]. As we show through a number of examples of interest for adaptive
signal processing in Sections 1.4-1.6, these formulas can be used without much
alteration for the complex case.
In the development, we use various representations for a given function f ( . ), that
is, write it in terms of different arguments. When doing so, we keep the function vari-
able, which is f ( . ) in this case, the same. It is important to note, however, that even
though these representations are all equivalent, different arguments may result in
quite different forms for the function. A simple example is given below.

B EXAMPLE 1.1
For a given function f ( z ) ¼jzj

2 , where z ¼ x þ jy , we can write

f ( z , z ) ¼ zz

or

f ( x , y ) ¼ x 2 þy 2

:

It is also important to note that in some cases, explicitly writing the function in
one of the two forms given above—as f ( z , z )or f ( x , y )—is not possible. A simple
example is the magnitude square of a nonlinear function, for example, f ( z ) ¼

2 . In such cases, the advantage of the approach we emphasize in this chapter,
that is, directly working in the complex domain, becomes even more evident.
Depending on the application, one might have to work with functions defined to
satisfy certain properties such as boundedness. When referring to such functions,
that is, those that are defined to satisfy a given property, as well as traditional
functions such as trigonometric functions, we use the terminology introduced in
[61] to be able to differentiate among those as given in the next definition.

j tanh( z ) j

Definition 1 (Split-complex and fully-complex functions) Functions that are
defined in such a way that the real and imaginary—or the magnitude and the
phase—are processed separately using real-valued functions are referred to as
split-complex functions. An example is

f ( z ) ¼ tanh xþ j tanh y:
Obviously, the form f ( x, y ) follows naturally for the given example but the form f ( z , z )
does not.
Complex functions that are naturally defined as f : C 7! C , on the other hand,
are referred to as fully-complex functions. Examples include trigonometric func-
tions and their hyperbolic counterparts such as f ( z ) ¼ tanh( z ) . These functions
typically provide better approximation ability and are more efficient in the charac-
terization of the underlying nonlinear problem structure than the split-complex
functions [62].

We define the scalar inner product between two matrices W , V [ V as
hW , Vi¼ Trace( V H W )