Digital Signal Processing Reference
In-Depth Information
N
Table 1.2 Four primary mappings defined for z
z
r
1
j z
i
[C
5
Complex-to-Real:
C
Complex-to-Complex:
C
N
2
N
N
2
N
7!
R
7!
C
Vector-concatenation type
mappings
z
r
z
i
z
z
z
R
¼
z
C
¼
2
4
3
5
2
4
3
5
Element-wise mappings
z
r
,1
z
i
,1
.
z
r
,
N
z
i
,
N
z
1
z
1
.
z
N
z
N
z
R
¼
z
C
¼
2
is a simple linear invertible mapping, one
can work in either space, depending on the convenience offered by each. In [110], it is
shown that such a transformation allows the definition of a Hessian, hence of a Taylor
series expansion very similar to the one in the real-case, and the Hessian matrix
H
defined in this manner is naturally linked to the complex
C
2
to
C
Since the transformation from
R
NN
Hessian matrix.
In the next section, we establish the connections of the results of [110] to
C
N
for
first- and second-order derivatives such that efficient second-order optimization
algorithms can be derived by directly working in the original
C
N
space where the
problems are typically defined.
Relationship Among Mappings
It can be easily observed that all four map-
pings defined in Table 1.2 are related to each other through simple linear transform-
ations, thus making it possible to work in one domain and then transfer the solution
to another. Two key transformations are given by
z
C
¼ Uz
R
and
z
C
¼ Uz
R
where
I jI
I jI
U ¼
. It is easy to observe that for the
transformation matrices
U
defined above, we have
UU
H
1
j
1
j
U ¼
diag{
U
2
,
...
,
U
2
} where
U
2
¼
and
¼ U
H
U ¼
2
I
making it
easy to obtain inverse transformations
as
we demonstrate in Section 1.3. For trans-
formations between the two mappings, (
) and (
), we can use permutation matrices
that are orthogonal, thus allowing simple manipulations.
1.2.4 Series Expansions
Series expansions are a valuable tool in the study of nonlinear functions, and for
analytic functions,
that
is, functions that are complex differentiable in a given
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