COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 35

Taking the transpose of the first equality in (1.27), we have

@ w R ¼ @ f

@f

U: (1 : 28)

@ w C

We regard the kth element of the two row vectors in (1.28) as two equal scalar-valued
functions defined on w R and w C , and take their derivatives to obtain

@f
@ w R

@f
@ w C U

@

@ w R ¼ U T @

k k

:

@ w C

We can then take the conjugate on each side and write the equality in vector form as

2 f
@ w R @ w R ¼ U H

2 f
@ w C @ w C

2 f
@ w C @ w C

@ @ @

U ¼ U T U

to obtain the second-order relationship given in (1.26).

The second-order differential relationship for vector parameters given in (1.26) is
first reported in [111] but is defined with respect to variables w R and w C using
element-wise transforms given in Table 1.2. Using the mapping w C as we have
shown here rather than the element-wise transform enables one to easily reduce the
dimension of problem from C 2 N to C N . The second-order Taylor series expansion
using the two forms ( w C and w C ) are the same, as expected, and we can write
using either w C or w C

Df D w C @f

2 f
@ w C @ w C

1
2 D w C

@

w C þ

D w C

(1 : 29)

@

as in (1.19), a form that demonstrates the fact that the C 2 N 2 N Hessian in (1.29) can be
decomposed into three C NN Hessians which are given in (1.17).
The two complex-to-real relationships given in (1.25) and (1.26) are particularly
useful for the derivation of update rules in the complex domain. Next, we show
their application in the derivation of the complex gradient and the complex Newton
updates, and note the connection to the corresponding update rules in the real domain.

Complex Gradient Updates Given a real-differentiable function f as defined
in Proposition 1, the well-known gradient update rule for f ( w R )is
D w R ¼m @ f
@ w R

which can be mapped to the complex domain using (1.25) as

D w C ¼ UD w R ¼mU @f

@ w R ¼ 2 m @ f @ w C