Biomedical Engineering Reference
In-Depth Information
(
)
The first term on the right-hand side is expressed as
E
x
, which indicates the
expectation of
x
. This term represents how the solution deviates from its true value
even in the noiseless cases . The second term indicates the influence of the noise
ʵ
.
The first term is rewritten as
F
T
FF
T
)
−
1
Fx
E
(
x
)
=
(
=
Qx
,
(2.28)
where
F
T
FF
T
)
−
1
F
Q
=
(
.
(2.29)
Apparently, the first term is not equal to the true value
x
, and the matrix
Q
in
Eq. (
2.29
) expresses the relationship between the true value
x
and the estimated
value
E
.
Denoting the
(
x
)
(
i
,
j
)
th element of
Q
as
Q
i
,
j
,the
j
th element of
E
(
x
)
,
E
(
x
j
)
,is
expressed as
N
E
(
x
j
)
=
Q
j
,
k
x
k
.
(2.30)
k
=
1
The above equation shows how each element of the true vector
x
affects the value of
E
(
x
j
)
. That is,
Q
j
,
k
expresses the amount of leakage of
x
k
into
x
j
when
j
=
k
.If
the weight
Q
j
,
1
,...,
x
j
may be close to the true value
x
j
. If the weight has no clear peak or if the weight has a peak at
j
Q
j
,
N
has a sharp peak at
j
,
that is different
from
j
,
x
j
may be very different from
x
j
. Because of such properties, the matrix
Q
is called the resolution matrix.
We next examine the second term, which expresses the noise influence. The noise
influence is related to the singular values of
F
. The singular value decomposition of
F
is defined as
M
T
F
=
1
ʳ
j
u
j
v
j
,
(2.31)
j
=
where we assume that
M
<
N
, and the singular values are numbered in decreasing
order. Using
N
1
ʳ
j
v
j
u
j
,
F
T
FF
T
)
−
1
(
=
(2.32)
j
=
1
we can express the second term in Eq. (
2.27
)as
N
u
j
ʵ
)
ʳ
j
(
v
j
.
(2.33)
j
=
1
The equation above shows that the denominator contains the singular values. Thus,
if higher order singular values are very small and close to zero, the terms containing
such small singular values amplify the noise influence, resulting in a situation where
