The amplitude of the high-energy regions vary with the environmental context while the positions in frequency is constant. Consequently, it is pos-sible to use a simpli.ed model which explicitly de.nes ωk for each component and optimizes only b0 and rk. An
altered
version
(Eq.
5.7)
of
the
original
AR prototype is de.ned, forcing every pole to have a complex conjugate. This simpli.cation can be made without loss of generality as no acquisitions have meshing tone harmonics or modulation sidebands at dc or π frequency, meaning that all poles representing meshing tone harmonics or sidebands must have a complex conjugate. The adjustable parameters b0 and rk are estimated using Trust Region (Sec. A.4.1).
This estimation can also be performed using an evolutionary algorithm or other gradient-based methods.
H(pe)(ω)= b0 (5.7)
(pe) .(pe)
e.jωze.jω
1+ z
k∈Kk k
(x)(x) jω(x)
k
zk = rk e(5.8)
As an alternative, it is still possible to use the original AR de.nition and a
textbook
estimator
like
LPC
or
Burg
[36].
This
will
however
require
an
algorithm for keeping track of the pole angles relative to their indexes, as these might change order from signal to signal using a textbook optimizer.
The number of complex conjugate poles is chosen to match the number of high-energy regions, and the pole angles ωk are set to match the frequency of these regions. By estimating each signal X(pe)(ω) in the dataset, the
(pe)(pe)
corresponding approximate H(pe)(ω), given by b0 and rk , are obtained (Eq.
5.9).
The
variable
E(pe)(ω) is approximation error.
H(pe)(ω)= X(pe)(ω) . E(pe)(ω) (5.9)
Figure
5.6
shows
the
magnitude
PSD
of
the
same
set
of
signals
as
in
.gure
5.5,
but
with
each
signal
X(pe)(ω) replaced by its AR approximate
(pe)(pe)
H(pe)(ω). The parameters making up each AR model, b0 and rk , can themselves be modeled as a function of the contextual parameters pe using a parametric model. This example used a third order polynomial model, although this might not necessarily be the optimal choice for the pole radius, as this parameter is always between zero and one. A better model for this parameter might be a sigmoid, or some other function with output con.ned between zero and one.
5.4. SIGNAL CORRECTION
The meshing tone amplitude is signi.cantly smaller for H(pe)(ω) than (pe)(pe)
the non-parametric PSD. This is because b0 and rk are estimated in the least-square-error sense, and the meshing tone represents only a single point in the PSD. If need be, this problem can be amended by giving the meshing tone frequency higher weight than the rest when optimizing, though at the cost of less precision for the other frequencies. The di.erence in amplitude is however of less importance, as it is proportional for all values of pe and it is the ratio between di.erent values of pe that is of interest.
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