Biomedical Engineering Reference
In-Depth Information
(Z
m
−
Z
3
)Z
r
Z
m
Z
m
+
=
Z
r
)Z
1
·
U
g
+
Z
r
·
P
U
r
(3.1)
(Z
m
+
(Z
m
−
Z
3
)
1
Z
m
+
Q
=
Z
r
)Z
1
·
U
g
−
Z
r
·
U
r
(3.2)
(Z
m
+
Z
1
·
Z
2
with
Z
m
=
Z
2
.
This can be written as a system
P(s)
Q(s)
Z
3
+
Z
1
+
U
g
(s)
U
r
(s)
=
H(s)
·
(3.3)
with two
inputs U
g
and
U
r
,two
outputs P
and
Q
and transfer matrix:
(Z
m
−
Z
3
)Z
r
(Z
m
+
Z
m
Z
m
+
Z
r
)Z
1
Z
r
=
H
(3.4)
(Z
m
−
Z
3
)
(Z
m
+
Z
r
)Z
1
1
Z
m
+
Z
r
−
(all impedances in
Z
being also a function of
s
, the Laplace operator). Define now
the vectors:
S
PU
g
S
QU
g
S
U
g
U
g
S
U
r
U
g
S
YU
=
and
S
UU
=
(3.5)
containing cross-power spectra
S
YU
(j ω)
between two signals
y(t)
and
u(t)
and
auto-power spectra
S
UU
(ω)
of a signal
u(t)
. From well-known identification and
signal-processing theory it then follows that [
136
]
S
YU
(jω)
=
H(jω)S
UU
(j ω)
(3.6)
In the case of
absence of breathing (U
r
=
0
)
(
3.6
) reduces to
(Z
m
−
Z
3
)Z
r
(Z
m
+
S
PU
g
S
QU
g
Z
r
)Z
1
(Z
m
−
=
·
S
U
g
U
g
(3.7)
Z
3
)
(Z
m
+
Z
r
)Z
1
It follows that the respiratory impedance
Z
r
can be defined as their spectral (fre-
quency domain) ratio relationship [
24
,
67
]:
S
PU
g
(jω)
S
QU
g
(j ω)
Z
r
(jω)
=
(3.8)
=
√
−
where
ω
=
2
πf
is the angular frequency and
j
1, the result being a complex
variable.
However, it is supposed that the test is done under
normal breathing conditions
,
which may result in an interference between the (unknown) breathing signal
U
r
and
the test signal
U
g
, making the identification exercise more difficult. From the point
of view of the forced oscillatory experiment, the signal components of respiratory
origin (
U
r
) have to be regarded as pure noise for the identification task! Neverthe-
less, if the test signal
U
g
is designed to be uncorrelated with the normal respiratory
breathing signal
U
r
, then
S
U
r
U
g
=
0, and the approach (
3.8
) is still valid, based on
(
3.6
) with
S
U
r
U
g
=
0[
24
,
67
].
