COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 22

on the complex plane. In this case, we can use Wirtinger calculus and write the
expansions by treating the function f ( z ) as a function of two arguments, z and z .In
this approach, the main idea is treating the two arguments as independent from each
other, when they are obviously dependent on each other as we discussed. When
writing the Taylor series expansion, the idea is the same. We write the series expansion
for a real-differentiable function f ( z ) ¼ f ( z , z )asif z and z were independent
variables, that is, as

@f
@z@z T Dz , Dz

1
2

Df ( z , z ) hr z f , Dz iþhr z f , Dziþ

@f
@z@z H Dz , Dz

1
2

@f
@z @z H Dz , Dz

þ þ : (1 : 17)

In other words, the series expansion has the same form as a real-valued function of two
variables which happen to be replaced by z and z as the two independent variables.
Note that when f ( z , z ) is real valued, we have

hr z f , Dz iþhr z f , Dzi¼ 2Re{ hr z f , Dzi } (1 : 18)

since in this case we have rf ( z ) ¼ [ rf ( z )] . Using the Cauchy-Bunyakovskii -
Schwarz inequality [77], we have

jDz H rf ( z ) jkDzkkrf ( z ) k

which holds with equality when Dz is in the same direction as rf ( z ). Hence, it is the
gradient with respect to the complex conjugate of the variable rf ( z ) that yields the
maximum change in function D f ( z , z ).
It is also important to note that when f ( z , z ) ¼ f ( z ), that is, the function is
analytic (complex differentiable), all derivatives with respect to z in (1.17) vanish
and (1.17) coincides with (1.16). As noted earlier, the Wirtinger formulation
for real-differentiable functions includes analytic functions, and when the function
is analytic, all the expressions used in the formulations reduce to the traditional
ones for analytic functions.
For the transformati ons that map the function to the real domain as those given in
Table 1.2, the ( ) R and ( ) R mappings, the expansion is straightforward since in this
case, the expansion is written in the real domain as in

1
2 hH ( z R ) Dz R , Dz R i:
By using the complex domain transformation defined by van den Bos (1.14), a very
similar form for the expansion can be obtained in the complex domain as well, and
it is given by [110]

D f ( z R ) hr z R f ( z R ), Dz R iþ

1
2 hH ( z C ) Dz C , Dz C i

D f ( z C ) hr z C f ( z C ), Dz C iþ (1 : 19)