Biomedical Engineering Reference
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in which the subscripts '
sn
' and '
n
' refer to the signal-noise and noise subspaces,
respectively. Removing the noise part,
H
can be formed as follows:
H
U
sn
Σ
sn
V
sn
.
=
(10)
H
could further be factorized as
Using the balanced coordinate method [
28
],
H
=
ΩΓ
,
(11)
where
Σ
sn
1/2
V
sn
,
U
sn
Σ
sn
1
/
2
Ω
=
and
Γ
=
(12)
where
are the observability and controllability matrices, respectively.
A
could be derived from both the observability or the controllability matrices;
here
Ω
and
Γ
Ω
is used to derive
A
.
Ω
−
rl
Ω
−
rl
)
−
1
(
A
=
(
Ω
−
rl
Ω
−
rf
),
(13)
Ω
−
rl
and
Ω
−
rf
are obtained by removing the last and first rows of
Ω
where
,
respectively. Now
α
i
and
R
i
are related to the eigenvalues of
A
by
α
i
=
−
log
|
λ
i
|
i
4
πf
,
and
R
i
=−
c
(14)
f
where
R
i
is the range and
α
i
is the damping factor of the sinusoid related to the
i
th
scattering point.
i
is the phase of
λ
i
,
the
i
th
eigenvalue of
A
. To find the constant
coefficients
a
i
, we use the following equation:
(
Cm
i
)(
v
i
B
)
λ
f
1
/f
i
a
i
=
,
(15)
where
m
i
are eigenvectors of
A
and
v
i
are defined as
⎡
⎣
⎤
⎦
v
1
.
m
p
]
−
1
V
=
[
m
1
···
=
.
(16)
v
p
,
f
1
is the carrier frequency of the pulse and the
k
th
element of frequency vector is related to carrier frequency by
In Eq. 15,
C
is the first row of
Ω
f
k
=
f
1
+
(
k
−
1)
f .
(17)
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