COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 21

region, the Taylor series expression assumes the same form as in the real case given by
f ( z ) ¼ 1
k¼ 0

f ( k ) ( z 0 )
k!

( zz 0 ) k : (1 : 15)

If f ( z ) is analytic for jzjR , then the Taylor series given in (1.15) converges uniformly
in jzjR 1 , R . The notation f ( k ) ( z 0 ) refers to the k th order derivative evaluated
at z 0 and when the power series expansion is written for z 0 ¼ 0, it is called the
Maclaurin series.
In the development of signal processing algorithms (parameter update rules) and in
stability analyses, the first- and second-order expansions prove to be the most useful.
For an analytic function f ( z ): C

N

7! C , we define Df ¼ f ( z )- f ( z 0 ) and Dz ¼ z - z 0 to
write the second-order approximation to the function in the neighborhood of z 0 as

1
2 Dz T H ( z ) Dz

Df Dz T

r z f þ

þ

2 H ( z ) Dz , Dz

1

¼r z f , Dz

(1 : 16)

where

z 0

r z f ¼ @ f ( z )
@z

is the gradient evaluated at z 0 and

@z@z T z 0
is the Hessian matrix evaluated at z 0 . As in the real-valued case, the Hessian matrix in
this case is symmetric and constant if the function is quadratic.
Second-order Taylor series expansions as given in (1.16) help summarize main
results for optimization and local stability analysis. In particular, we can state the
following three important observations for the real-valued case, that is, when the
argument z and the function are real valued, by directly studying the expansion
given in (1.16).

2 f ( z )

z f W H ( z ) ¼ @

2

r

† Point z 0 is a local minimum of f ( z ) when r z f ¼ 0 and H ( z ) is positive semi-
definite, that is, these are the necessary conditions for a local minimum.
† When H ( z ) is positive definite and r z f ¼ 0, z 0 is guaranteed to be a local
minimum, that is, positive-definiteness and zero gradient, together, define the
sufficient condition.
† Finally, z 0 is a locally stable point if, and only if, H ( z ) is positive definite
and r z f ¼ 0, that is, in this case, the two properties define the sufficient and
necessary conditions.

When deriving complex-valued signal processing algorithms, however, the func-
tions of interest are real valued and have complex arguments z , hence are not analytic