Biomedical Engineering Reference
In-Depth Information
Chapter 8
Time Domain: Breathing Dynamics and Fractal
Dimension
8.1 From Frequency Response to Time Response
8.1.1 Calculating the Impulse Response of the Lungs
In the previous section, we have seen that fractional order models of the impedance
can be fitted on the frequency response (
3.8
) of the respiratory system in a given
frequency band. However, these models cannot be used directly to simulate the time
response of the respiratory system (e.g. impulse response). A feasible solution is to
use finite-dimensional transfer functions of integer order. A good overview of such
feasible implementations is given in [
105
]. From these methods, we shall adopt in
this chapter the classical method of pole-zero interpolation introduced by Oustaloup
in early 1990s [
118
]. In the remainder of this paper, we shall refer to this method as
the
Oustaloup filter
.
Oustaloup filter approximation to a fractional order differentiator is a widely
used method in fractional calculus. A generalized Oustaloup filter defined in the
frequency band
[
ω
b
,ω
h
]
can be represented as
N
pz
s
+
p
G(s)
=
K
(8.1)
s
+
z
k
=
1
with
p
poles,
z
zeros,
K
a gain, and
N
pz
the number of pole-zero pairs (i.e. a design
parameter). The poles, zeros, and gain can be calculated from:
ω
2
k
−
1
−
n/N
pz
p
=
ω
b
·
(8.2)
u
z
=
ω
b
·
ω
2
k
−
1
+
n/N
pz
(8.3)
u
K
=
ω
h
(8.4)
wi
th
n
t
he fractional order of the derivative
s
n
to be approximated and
ω
u
=
√
ω
h
/ω
b
. The result will be a
N
pz
th integer order transfer function.
