Biomedical Engineering Reference
In-Depth Information
Notice that if the breathing frequency
F
0
is perfectly known, the estimation is
straightforward. However, if the baseline frequency is not well known, then also
the harmonics are erroneously estimated [
41
,
111
]. Therefore, the baseline fre-
quency needs to be estimated as well, resulting in a non-linear least squares prob-
lem [
124
,
138
]. In addition, a higher polynomial degree
β
can lead to numerical
instability of the convergence matrix from (
9.3
). The numerical conditioning can be
improved by making the values dimensionless, i.e. introducing the variable:
t
N
·
T
s
t
=
(9.8)
Additionally, one can introduce a diagonal matrix
S
:
b(n)
=
·
Θ
=
Φ
·
S
·
S
−
1
Φ
·
Θ
=
Φ
·
Θ
(9.9)
with
Φ
=
S
and
Θ
=
S
−
1
Θ
. The matrix
S
is a diagonal matrix, with the
diagonal elements the Root Mean Squared values calculated per each column, which
can be depicted as
Φ
·
·
⎡
⎣
⎤
⎦
s
11
0
.
.
.
Φ
=
(9.10)
0
s
pk
To determine
F
0
, we make use of the FFT-interpolation method. The method is
simplified for the case of a frequency
F
0
surrounded by two integer multiples of the
spectrum
S
xx
for the resolution frequency (
f
0
)[
52
]:
f
1
=
max
(S
xx
)
2
f
1
+
f
0
f
1
−
1
δ
=
(9.11)
f
1
+
f
0
f
1
+
1
F
0
=
(f
1
−
+
·
f
0
)
δ
(f
0
)
The optimized excitation signal is an odd random phase multisine defined as:
109
A
k
sin
2
π(
2
k
φ
k
U
FOT
=
+
1
)f
0
t
+
(9.12)
k
=
0
with
•
frequency interval from 0.1 to 21.9 Hz
•
frequency resolution
f
0
of 0.1 Hz
•
only odd harmonics
