Biomedical Engineering Reference
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with Welch's method (averaged periodogram) show significant reduction of variation
compared with modified periodogram. AR power spectrum is unbiased, free of win-
dowing effects, and it has better statistical properties than Fourier spectrum, since
the number of the degrees of freedom in case of AR is greater; namely it is equal
to
N
p
,where
N
is the number of points of the data window and
p
is a model or-
der. Correct estimation of the model order is important, but small deviations of
p
do
not drastically change the power spectrum. In case of AR model fitting, the signal
should not be oversampled. Sampling should be just slightly above the Nyquist fre-
quency. Oversampling does not bring any new information and it causes redundancy.
In case of the autoregressive model in order to account for all frequencies, especially
low frequencies, we have to increase the number of steps backward in autoregressive
process, which means that we have to increase the model order. Estimating more pa-
rameters (from the same amount of data) increases the uncertainty of the estimates.
/
2.3.2.2.4 Parametric description of the rhythms by AR model, FAD method
The transfer function
H
has maxima for
z
values corresponding to the zeroes of
the denominator of the expression (2.58). These values of
z
are the poles
z
j
lying
inside the unit circle
(
z
)
f
N
in polar coordinates (
Figure2.10)
corresponds to the counterclockwise traverse of
the upper half of this unit circle. The angular position of
z
j
lying closest to the unit
circle determines the frequency of the corresponding peak, and the radial distance
depends on the damping of the relevant frequency component. From the coefficients
of the AR model we can derive parameters which characterize oscillatory properties
of the underlying process. Namely the transfer function
H
|
z
|
=
1 in the complex plane. The frequency axis 0
<
f
<
(
z
)
equation (2.58) for the
corresponding continuous system takes a form:
p
j
=
1
C
j
z
H
(
z
)=
(2.61)
z
−
z
j
It can be interpreted as a system of parallel filters each of them corresponding to a
characteristic frequency (
Figure 2.11)
.
For
z
=
z
j
the resonance for filter
j
occurs. We
can introduce parameters characterizing these filters [Franaszczuk and Blinowska,
1985]:
1
Δ
t
log
z
j
|
(2.62)
where Δ
t
is the sampling interval. The imaginary part of α
j
corresponds to the res-
onance frequency ω
j
and the real part to the damping factor β
j
of the oscillator
generating this frequency. By rewriting
H
α
j
=
,
β
j
=
−
Re
(
α
j
)
,
ω
j
=
Im
(
α
j
)
,
φ
j
=
Arg
(
C
j
)
,
B
j
=
2
|
C
j
in terms of parameters from (2.62) and
performing inverse Laplace transform we find the impulse response function in the
form of damped sinusoids (Figure 2.11):
(
z
)
p
j
=
1
B
j
e
−
β
j
t
cos
(
ω
j
t
+
φ
j
)
h
(
t
)=
(2.63)
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