Biomedical Engineering Reference
In-Depth Information
2
y
k
y
k
1
ʓ
−
1
H
T
=
ʲ
I
−
ʲ
H
ʲ
2
y
k
y
k
1
H
T
H
)
−
1
H
T
=
ʲ
I
−
ʲ
H
(
ʦ
+
ʲ
ʲ
2
y
k
ʦ
−
1
H
T
−
1
y
k
.
1
ʲ
−
1
I
=
+
H
(4.99)
In the equation above, we use the matrix inversion formula in Eq. (C.91). Using the
model covariance matrix
ʣ
y
defined in Eq. (
4.25
), we get
K
y
k
ʦ
−
1
H
T
−
1
K
1
2
1
2
ʲ
−
1
I
y
k
ʣ
−
1
=
+
H
y
k
=
y
k
.
y
k
=
1
k
=
1
The above equation is equal to Eq. (
4.29
).
4.10.3 Proof of Eq.
(
4.50
)
The proof of Eq. (
4.50
) begins with
ʲ
x
k
ʥ
−
1
x
k
2
x
k
=
argmin
x
k
y
k
−
Hx
k
+
.
(4.100)
The solution of this minimization,
x
k
, is known as the weighted minimum-norm
solution. To derive it, we define the cost function
F
,
2
x
k
ʥ
−
1
x
k
.
F
=
ʲ
y
k
−
Hx
k
+
Let us differentiate
F
with respect to
x
k
, and set it to zero,
H
T
y
k
−
Hx
k
+
∂
ʥ
−
1
x
k
=
x
k
F
=−
2
ʲ
2
0
.
∂
Thus, the weighted minimum-norm solution is given by
H
T
H
−
1
ʥ
−
1
H
T
y
k
.
x
k
=
ʲ
+
ʲ
(4.101)
The above
x
k
in Eq. (
4.14
). Therefore,
according to the arguments in the preceding subsection, we have the relationship,
x
k
is exactly the same as the posterior mean
¯
ʲ
x
k
ʥ
−
1
x
k
x
k
x
k
ʥ
−
1
2
2
y
k
−
Hx
k
+
=
ʲ
y
k
−
x
k
¯
+ ¯
¯
min
x
k
H
y
k
ʣ
−
1
=
y
k
.
(4.102)
y
