Biomedical Engineering Reference
In-Depth Information
appropriate model to this pattern helps to reconstruct the signal beyond the bandwidth
of the system and thus leads to a higher image resolution without additional hardware
cost. Cumo et al. [
26
] have used this bandwidth extrapolation method to estimate
the frequency response of the backscattered signal in the gap between two different
incoherent sub-bands.
The number of harmonics in frequency domain is the same as the number of
the scattering points in the view of the antenna and multiple scattering effect [
27
].
Mathematical model of the frequency-domain signal is [
28
]
N
A
i
(
f
)
e
(
j
4
c
R
i
)
f
,
=
y
(
f
)
(2)
i
=
1
where f is the frequency, c the speed of light, N the number of scattering points and
R
i
is the range of the
i
th
scattering point.
A
i
(
f
) is the frequency dependence function
corresponding to the
i
th
scattering point. This frequency dependence function is of the
form
f
α
,
where the exponent
α
is known for some common scattering mechanisms.
For example, a flat plate has
α
0[
27
].
As stated by Cumo et al. [
26
],
f
α
scattering behavior can accurately be estimated by
exponential functions over a finite bandwidth interval. Hence, the following discrete
model can represent the frequency behavior of the reflected signal given in Eq. 2
above:
=
1 while a spherical object will have
α
=
N
a
i
e
−
(
α
i
+
j
4
c
R
i
)
kf
,
y
(
k
)
=
(3)
i
=
1
where
a
i
are the constant coefficients of the sinusoids,
α
i
and
R
i
refer to the frequency
decay/growth factor and the range of the
i
th
scatterer, respectively and
f
is the
sampling frequency. In the rest of this section, we provide the formulation to estimate
model parameters in Eq. 3. More details and derivations are available in [
28
].
From system theory, we know that the following state-space equations hold for
input-output relation in a linear system:
x
(
k
+
1)
=
Ax
(
k
)
+
Bw
(
k
)
(4)
y
(
k
)
=
Cx
(
k
)
+
w
(
k
),
where
x
(
k
) is the state vector,
w
(
k
) the input vector and
y
(
k
) is the output of the
system.
A
,
B
and
C
are matrices characterizing the system and define its state-space
behavior. The transfer function of the system described in Eq. 4 is given in Eq. 5:
Y
(
z
)
X
(
z
)
=
A
)
−
1
B
T
(
z
)
=
C
(
z
I
−
+
1
.
(5)
The impulse response of such a system in general comprises a number of complex
sinusoids or poles of the system which are the roots of the denominator or as seen
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