COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 34

1.3.2 Vector Optimization in C N
Given a real-differentiable cost function f ( w ): C 7! R , we can write f ( w ) ¼ f ( w , w )

and take advantage of Wirtinger calculus as discussed in Section 1.2.2. The first-order
Taylor series expansion of f ( w , w ) is given by (1.18), and as discussed in Section
1.2.4, it is the gradient with respect to the conjugate of the variable that results
in the maximum change for the complex case. Hence, the updates for gradient
optimization of f is written as

N

Dw ¼ w ( nþ 1) w ( n ) ¼mr w ( n ) f : (1 : 24)

2 ,
while the update that uses Dw ¼mr w ( n ) f , leads to changes of the form
D f ¼ 2 m Re{ hr w ( n ) f , r w ( n ) fi }, which are not guaranteed to be nonpositive.
Here, we consider only first-order corrections since m is typically very small.
The complex gradient update rule given in (1.24) can be also derived through
the relationship given in the following proposition, which provides the connection
between the real-valued and the complex-valued gradients. Using the mappings
defined in Table 1.2 (Section 1.2.3) and the linear transformations among them, we
can extend Wirtinger derivatives to the vector case both for the first- and second-
order derivatives as stated in the following proposition.

The update given in (1.24) leads to a nonpositive increment, D f ¼ 2 mkr w ( n ) fk

N N

Proposition 1 Given a function f ( w , w ): C 7! R that is real differentiable

up to the second-order. If we write the function as f ( w R ): R

C

2 N

7! R using the defi-

nitions for w C and w R given in Table 1.2 we have
@f
@ w R ¼ U H

@f
@ w C

(1 : 25)

2 f 2 f

@ @

w R ¼ U H

U (1 : 26)

w C @ w C @ w R @ @

I jI
I jI

where U ¼ :

2 U H w C . We can thus
write the two Wirtinger derivatives given in (1.5) in vector form as

Since we have UU H

1

Proof 1 ¼ 2 I , w C ¼ U w R and w R ¼

@f
@ w C ¼

1

2 U @ f

@ w R
in a single equation. Rewriting the above equality as

@f
@ w R ¼ U T

@f
@ w C ¼ U H

@f
@ w C

(1 : 27)

we obtain the first-order connection between the real and the complex gradient.