Biomedical Engineering Reference
In-Depth Information
FIGURE 2.10: Left: poles of illustrative AR transfer function in the complex
plane. For real signals the transfer function has pairs of complex conjugate poles
(here: a and a , b and
b , c and c , d and
d ). These poles z j lie inside the unit circle
|
| =
(
<
<
)
cor-
responds to the angular position of the pole. Right: traces a-d are impulse response
functions corresponding to the poles a-d; each of them is in the form of an exponen-
tially damped sinusoid. Trace e—total impulse response of the AR model is the sum
of the components a, b, c, and d.
z
1 in the complex plane. The frequency f related to the pole
0
f
f N
In this way we can find directly (not from the spectral peak positions) the fre-
quencies of the rhythms present in the signal, which can be quite useful, especially
for weaker components. We can also find the amplitudes and damping factors of
the oscillations. The method of description of time series by means of parameters:
frequency (ω j ), amplitude ( B j ), damping (β j ) was named FAD. The number of iden-
tified oscillatory components depends on the model order; for even model order it
is p
2. Model of odd order contains the non-oscillatory component e β , which ac-
counts for the usually observed form of the power spectrum decaying with frequency.
The correct estimation of the model order is important for description of signals in
terms of FAD parameters. For a too low model order, not all components will be
accounted for. For a too high model order, too many poles in the complex plane will
appear. They will lie close to the main frequency components and will give the effect
of “sharpening” the spectral peaks.
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