Biomedical Engineering Reference
In-Depth Information
The ARMA linear system model is depicted in Figure 3.6 in its discrete form.
The frequency-domain transfer function
H
can be obtained using the
Z
-transform
as follows:
p
q
()
∑
()
∑
()
Yz
=−
az Yz
−
k
+
bz Xz
−
k
(3.21)
k
k
k
=
1
k
=
0
q
∑
−
k
bz
()
()
Yz
Xz
k
()
k
=
0
Hz
=
=
(3.22)
p
∑
1
+
az
−
k
k
k
=
1
The absolute squared value of
H
(
z
) evaluated at
ze
j
ω
is:
=
2
q
∑
−
k
bz
k
()
2
=
0
j
ω
k
He
=
(3.23)
2
p
∑
−
k
1
+
az
k
k
=
1
Several algorithms are used to estimate the model's coefficients. The most popu-
lar are the Yule-Walker, the Burg, and the covariance and modified covariance
methods. All of these methods are available in MATLAB.
The following MATLAB functions are used to estimate the AR model parame-
ters (
a
) using the above methods where
x
is the sequence that contains the
time-series data and
p
is the order of the AR model [41]:
a = arburg(x,p)- Burg's method
a = aryule(x,p)- Yule-Walker
a = arcov(x,p)- covariance
a = armcov(x,p)- modified covariance
The following MATLAB functions are used to estimate the PSD of the modeled
signal using the above methods where
x
is the sequence that contains the time-series
data and
p
is the order of the AR model [41]:
Pxx = pburg(x,p) - Burg's method
Pxx = pyulear(x,p) - Yule-Walker
Pxx = pcov(x,p) - covariance
Pxx = pmcov(x,p) - modified covariance
The following example illustrates the difference between the classical versus
modern spectral estimation techniques. Figure 3.7(a) shows a 2,048-point EEG
X
()
Yz
()
Xz Hz
(). ()
=
H
()
Figure 3.6
The ARMA linear system model in the discrete form.
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