Digital Signal Processing Reference
In-Depth Information
the function as we used the Cauchy-Riemann conditions, that is, a strict relationship
between the real and imaginary parts of the function.
The same approach of treating the variable and its complex conjugate as indepen-
dent variables, can be used when taking derivatives of functions of matrix variables as
well so that expressions given for real-valued matrix derivatives can be directly used as
shown in the next example.
B
EXAMPLE 1.4
Let
g
(
Z
,
Z
)
¼
Trace(
ZZ
H
). We can calculate the derivatives of
g
with respect to
Z
and
Z
by simply treating one variable as a constant and directly using the results
from real-valued matrix differentiation as
@Z
¼
@
Trace[
Z
(
Z
)
T
]
@g
¼ Z
@Z
and
@g
@Z
¼ Z
A good reference for real-valued matrix derivatives is [88] and a number of
complex-valued matrix derivatives are discussed in detail in [46].
For computing matrix derivatives, a convenient tool is the use of differentials. In
this procedure, first the matrix differential is computed and then it is written in the
canonical form by identifying the term of interest. The differential of a function is
defined as the part of a function
f
(
ZþDZ
)
f
(
Z
) that is linear in
Z
. For example
when computing the differential of the function
f
(
Z
,
Z
)
¼ ZZ
, we can first write
the product of the two differentials
(
ZþdZ
)(
Z
þdZ
)
¼ ZZ
þ
(
dZ
)
Z
þZ dZ
þdZ dZ
and take the first-order term (part of the expansion linear in
Z
and
Z
) to evaluate the
differential of the function as
d
(
ZZ
)
¼
(
dZ
)
Z
þZdZ
as discussed in [74, 78]. The approach can significantly simplify certain derivations.
We provide an example for the application of the approach in Section 1.6.1.
Integrals of the Function f ðz
,
z
Þ
Though the three representations of a func-
tion we have discussed so far:
f
(
z
),
f
(
x
,
y
), and
f
(
z
,
z
) are all equivalent, certain care
needs to be taken when using each form, especially when using the form
f
(
z
,
z
). This
is the form that enables us to treat
z
and
z
as independent variables when taking
derivatives and hence provides a very convenient representation (mapping) of a com-
plex function in most evaluations. Obviously, the two variables are not independent
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