Digital Signal Processing Reference
In-Depth Information
2
and the superscript in (
.
)
H
denotes the transpose of the
complex conjugate. The norm we consider in this chapter is the Frobenius—also
called the Euclidean—norm. For vectors, the definition simplifies to
hw
,
vi¼v
H
w
.
The definition of an inner product introduces a well-defined notion of orthogonality
as well as of norm, and provides both computational and conceptual convenience.
Inner product satisfies certain properties.
such that
hW
,
Wi¼kWk
Properties of inner product:
positivity: hV
,
Vi
.
0
for all
V
[
V
;
definiteness: hV
,
Vi¼0
if and only if
V ¼ 0
;
linearity (additivity and homogeneity): ha
(
UþW
),
Vi¼ahU
,
ViþahW
,
Vi
for all
W
,
U
,
V
[
V
;
conjugate symmetry: hW
,
Vi
¼hV
,
Wi
for all
V
,
W
[
V
.
In the definition of the inner product, we assumed linearity in the first argument, which
is more commonly used in engineering texts, though the alternate definition is also
possible. Since our focus in this chapter is the finite-dimensional case, the inner pro-
duct space also defines the Hilbert space.
A complex matrix
W
[ C
NN
is called symmetric if
W
T
¼W
and Hermitian
if
W
H
¼W
. Also,
W
is orthogonal if
W
T
W ¼ I
and unitary if
W
H
W ¼ I
where
I
is the identity matrix [49].
1.2.2 Efficient Computation of Derivatives in the
Complex Domain
Differentiability and Analyticity
Given a complex-valued function
f
(
z
)
¼ u
(
x
,
y
)
þ j
v
(
x
,
y
)
where
z ¼ xþ jy
, the derivative of
f
(
z
) at a point
z
0
is written similar to the real case as
f
(
z
0
þ
Dz
)
f
(
z
0
)
Dz
f
0
(
z
0
)
¼
lim
Dz!
0
:
However, different from the real case, due to additional dimensionality in the complex
case, there is the added requirement that the limit should be independent of the
direction of approach. Hence, if we first let
Dy ¼
0 and evaluate
f
0
(
z
) by letting
Dx !
0, we have
f
0
(
z
)
¼ u
x
þ j
v
x
(1
:
1)
and, similarly, if we first let
Dx ¼
0, and then
Dy !
0, we obtain
f
0
(
z
)
¼
v
y
ju
y
(1
:
2)
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