Biomedical Engineering Reference
In-Depth Information
in Eq. 5 are the eigenvalues of
A
, the open-loop matrix of the system. Hence, the
output signal of the system
y
(
k
) can be written as
M
a
i
e
−
(
α
i
+
jβ
i
)
kt
.
y
(
k
)
=
(6)
i
=
1
In Eq. 6, M is the number of the poles of the system or the eigenvalues of
A
,
a
i
are the
constant coefficients of each complex sinusoid and
α
i
and
β
i
are the damping factor
and frequency of the
i
th
harmonic, respectively.
t
is the sampling time interval.
Comparing Eqs. 6 and 3 reveals that the frequency response of the backscattered
signal and the impulse response of a linear system have a similar mathematical
structure. Thus, we can use the mathematics of linear system identification to estimate
the parameters of the frequency model for the backscattered data.
Suppose that the frequency response of the backscattered signal is the impulse
response of a hypothetical linear system. Here we try to extract the system matrices
and consequently, a model for the impulse response based on the eigenvalues or
poles of this hypothetical system. As mentioned before, we can derive the desired
frequency model parameters from this impulse response model.
The process of finding the hypothetical system matrices involves forming forward
prediction or a Hankel matrix from the sample data of the frequency response of the
backscattered signal and deriving
A
through singular value decomposition of
H
, the
Hankel matrix which is defined as follows:
⎛
⎝
⎞
⎠
y
(1)
···
y
(
L
)
.
.
.
.
.
H
=
,
(7)
y
(
N
−
L
+
1)
···
y
(
N
)
where
y
(
i
) are the samples of frequency-domain response of the backscattered data,
N
is the number of data samples and
L
is chosen as
N
/3 [
25
]. By singular value
decomposition,
H
is decomposed into three matrices,
V
∗
,
H
=
U
Σ
(8)
where
U
is the left unitary matrix,
V
∗
is the right unitary matrix and
is a diagonal
matrix containing singular values of
H
in descending order.
∗
denotes the complex
conjugate and transpose.
Singular values of
H
could be separated into two subspaces, the signal-plus-noise
subspace and the noise-only subspace. If the signal-to-noise ratio (SNR) value is
high enough there would be a sharp transition between singular values of the signal
and those of the noise. The criterion for separating the two parts is described in [
29
].
Hence,
U
,
Σ
and
V
∗
could be divided into two different subspaces as follows:
Σ
Σ
sn
V
sn
V
n
,
0
=
U
sn
U
n
H
(9)
0
Σ
n
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