Biomedical Engineering Reference
In-Depth Information
as the “parametric” spectral estimation techniques. Although these methods are
based on time-domain analyses, they are used to characterize and estimate the spec-
trum of the signal. These methods are very useful when dealing with short segment
data sequences [39, 41]. The most popular of the parametric methods is the
autoregressive linear model. The input to the model is white noise, which contains
all frequencies, whereas the output is compared with the signal being modeled. The
model parameters are then adjusted to match the model output to the signal being
modeled. The resultant model parameters are then used to estimate the spectrum of
the signal under consideration. These models include [39] the autoregressive (AR),
moving average (MA), and autoregressive moving average (ARMA) models. The
AR method is usually used when the signal being modeled has spectral peaks,
whereas the MA method is useful for modeling signals with spectral valleys and
without spectral peaks.
The AR model of a single-channel EEG signal is defined as follows [41]:
p
()
∑
1
( ) ()
yn
=−
a yn
− +
k
x n
(3.16)
k
k
where
a
k
,
k
1, 2,
…
,
p
, are the linear model parameters,
p
is the model order,
n
denotes the discrete sample time, and
x
(
n
) is white noise input with zero mean and
unity variance. The output current value depends on the input signal and previous
output samples.
In the ARMA model, the signal is defined as
p
q
( )
∑
(
)
∑
(
)
yn
=−
aynk
− +
bxnk
−
(3.17)
k
k
k
=
1
k
=
0
where
b
k
,
k
1, 2,
…
,
q
, are the additional linear model parameters. The parameters
p
and
q
are the model orders. Although
p
and
q
are usually determined through trial
and error, Akaike criterion (AIC) can be used [45, 46] to determine the optimum
model order. For an
N
-length data sequence, the optimum AR model order
p
is
obtained by minimizing the AIC criterion:
(
)
()
AIC
pN
=
ln
ρ
+
2
p
(3.18)
p
where
is the model error variance defined as
N
−
∑
1
2
()
ρ
p
=
en
(3.19)
p
n
=
0
which can be determined using
p
()
=
∑
0
()( )
e n
=
a kxn k
−
(3.20)
p
p
k
In any case, the model order should be such that the model estimated spectrum
fits the signal spectrum.
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