Biomedical Engineering Reference
In-Depth Information
ʣ
zz
are covariance matrices of the complex Gaussian distribu-
tion. Using exactly the same derivation, we can obtain
ʣ
xx
,
ʣ
yy
, and
Here,
d
1
1
I(
,
)
=
xy
=
−
ʳ
j
,
x
y
log
log
(C.52)
1
−
ʣ
−
1
xx
ʣ
xy
ʣ
−
1
I
yy
ʣ
j
=
1
ʣ
−
1
xx
ʣ
xy
ʣ
−
1
xy
.
where
ʳ
j
is the
j
th eigenvalue of
yy
ʣ
C.3.3
Covariance of Residual Signal and Conditional Entropy
Let us next consider the regression problem:
y
=
Ax
+
e
,
(C.53)
where the vector
e
represents the residual of this regression. The
p
×
q
coefficient
matrix
A
can be estimated using the least-squares fit,
E
2
E
2
A
=
argmin
A
e
=
argmin
A
y
−
Ax
.
(C.54)
Here,
E
2
is expressed as
E
e
2
E
yy
T
x
T
A
T
Ax
x
T
A
T
y
y
T
Ax
e
=
−
−
+
.
Differentiating
E
2
with respect to
A
gives
e
A
E
2
∂
∂
yx
T
xx
T
e
=−
2
E
(
)
+
2
A
E
(
).
Setting this derivative to zero, we obtain
A
yx
T
xx
T
)
−
1
=
ʣ
yx
ʣ
−
1
=
E
(
)
E
(
xx
,
(C.55)
yx
T
xx
T
where
ʣ
yx
=
E
(
)
and
ʣ
xx
=
E
(
)
. The covariance matrix of the residual
e
is
defined as
