Biomedical Engineering Reference
In-Depth Information
•
only harmonics which are not overlapping with the first five breathing harmonics
are used
•
equal amplitude
A
k
for all excited harmonics
•
the phase
φ
k
uniformly distributed between
[
0
,
2
π
]
•
one so-called
detection line
for each group of four excited odd harmonics is not
excited in order to check for odd non-linear distortion.
This algorithm has been broadly detailed elsewhere [
77
,
136
]. The flowchart for the
measurement procedure can be summarized as follows:
•
estimate the fundamental breathing frequency,
•
make the excitation signal taking into account the information from the previous
step and
•
filter the measured data and perform further signal processing (i.e. non-linear
distortions detection algorithm).
9.1.2 Non-linear Distortions
The standard procedure to obtain the impulse response
g(t)
of a linear system is
based on the correlation analysis:
R
yu
(t)
=
g(t)
∗
R
uu
(t)
(9.13)
with
u(t)
the input signal,
y(t)
the output signal and
denoting the convolution
product.
R
yu
(t)
and
R
uu
(t)
are the cross- and auto-correlations, respectively:
R
yu
(τ )
=
E
y(t)u(t
−
τ)
R
uu
(τ )
=
E
u(t)u(t
−
τ)
∗
(9.14)
with
τ
the shift interval. Applying Fourier transform to (
9.13
) results in
S
YU
(jω)
S
UU
(jω)
G(jω)
=
(9.15)
where the cross-spectrum
S
YU
(j ω)
, the auto-spectrum
S
UU
(j ω)
, and the frequency
response function (FRF)
G(jω)
are the Fourier transforms of
R
YU
(t)
,
R
UU
(t)
and
g(t)
, respectively.
The Best Linear Approximation (BLA) [
136
,
137
] of a non-linear system
g
BLA
(t)
minimizes the mean squared error (MSE) between the real output of a
non-linear system
y(t)
−
E
{
y(t)
}
and the output of a linear model approximation
g
BLA
(t)
∗
(u(t)
−
E
{
u(t)
}
)
:
E
y(t)
E
u(t)
E
y(t)
−
∗
u(t)
2
−
g
BLA
(t)
−
(9.16)
