Biomedical Engineering Reference
In-Depth Information
In this study, we have applied the approximation given by (
8.1
)forthe
s
α
r
and
the
s
β
r
coefficients from:
1
C
r
·
G(s)
=
L
r
·
g
1
+
(8.5)
g
2
with
g
1
and
g
2
the integer order approximations for
s
α
r
and
s
β
r
, respectively.
The Fourier transform of a signal
x(t)
in time is used to obtain the frequency
representation of that signal [
136
]:
∞
x(t)e
−
jωt
dt
FT
{
x
}=
X(jω)
=
(8.6)
−∞
where
ω
=
2
πf
(rad/s) with
f
the frequency (Hz) and
t
is time (s). If we assume
that
x
k
is a discrete sample of
x(t)
, we have the discrete Fourier transform (DFT):
N
k
−
1
x
k
e
−
jωkT
s
DFT
{
x
}=
(8.7)
k
=
0
with
k
the sample number,
T
s
the sampling period and
N
k
the number of samples.
Hence, the Fourier transform of a sequence of impulse functions, each of which is
an element of the sample vector
x
, is equal to the DFT of
x
at each frequency where
DFT is measured.
The inverse DFT is calculated as
∞
1
2
π
X(jω)e
jωt
dω
x(t)
=
(8.8)
−∞
The result of (
8.8
) is the impulse response of the system whose frequency response
is given by
X(jω)
.
It has been shown in Chap.
5
that the respiratory system can be successfully
modeled by recurrent ladder networks which preserve the morphology and the
anatomy. It was also shown that these ladder networks converge to a transfer func-
tion with fractional order operators. Since the fractal dynamics can be well mod-
eled by power-law models (decay function), one may expect this property also from
the respiratory system. We propose therefore to model the impulse response by the
power-law model:
t
B
(8.9)
with
A
and
B
identified constants using a similar nonlinear least-squares algorithm
as presented in Chap.
3
[
21
]. A Student
t
-test was used to derive the 95 % confidence
intervals and analysis of variance (i.e. ANOVA test) was used to compare model
parameters among the groups.
x(t)
=
A
·
8.1.2 Implications in Pathology
The same groups of adults and children have been employed here as in the previous
chapter. The estimated model parameter for (
8.5
) are given in Table
8.1
for all sub-
jects by means of mean and standard deviation values, along with 95 % confidence
