COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 16

as knowing z we already know its conjugate. This is an issue that needs special care in
evaluations such as integrals, which is needed for example, when using f ( z , z )to
denote probability density functions and calculating the probabilities with this form.
In the evaluation of integrals, when we consider f ( . ) as a function of real and ima-
ginary parts, the definition of an integral is well understood as the integral of function
f ( x , y ) in a region R defined in the ( x , y ) space as
ðð

f ( x , y ) dx dy:

R

However, the integral ÐÐ f ( z , z ) dz dz is not meaningful as we cannot vary the two
variables z and z independently, and cannot define the region corresponding to R
in the complex domain. However, this integral representation serves as an intermediate
step when writing the real-valued integral as a contour integral in the complex domain
using Green's theorem [1] or Stokes's theorem [44, 48] as noted in [87]. We can use
Green's theorem (or Stokes's theorem) along with the definitions for the complex
derivative given in (1.5) to write
ðð þ

j
2

F ( z , z ) dz

f ( x , y ) dx dy ¼ (1 : 9)

R C R

where

@ F ( z , z )
@z ¼ f ( z , z ) :

Here, we assume that f ( x , y ) is continuous through the simply connected region R
and C R describes its contour. Note that by transforming the integral defined in the
real domain to a contour integral when the function is written as f ( z , z ), the formula
takes into account the dependence of the two variables, z and z in a natural manner.
In [87], the application of the integral relationship in (1.9) is discussed in detail for
the evaluation of probability masses when f ( x , y ) defines a probability density func-
tion. Three cases are identified as important and a number of examples are studied
as application of the formula. The three specific cases to consider for evaluation of
the integral in (1.9) are when

† F ( z , z ) is an analytic function inside the given contour, that is, it is a function of
z only in which case the integral is zero by Cauchy's theorem;
† F ( z , z ) contains poles inside the contour, which in the case of probability
evaluations will correspond to probability masses inside the given region;
† F ( z , z ) is not analytic inside the given contour in which case the value of the
integral will relate to the size of the region R .

We demonstrate the use of the integral formula given in (1.9) in Section 1.6.4 in the
derivation of an efficient representation for the score function for complex maximum
likelihood based independent component analysis.