Biomedical Engineering Reference
In-Depth Information
Another approximation that has often been used is given by (15.26).
N
−
1
N
x
)
2
x
)
2
(
x
t
−
=
(
x
t
−
(15.26)
t
=
1
t
=
1
α
k
=
r
k
, where
r
k
is the
k
th
correlation coefficient.
Then the sample correlation coefficients are substituted into the Yule-Walker equations
(15.27), and the matrix equations are solved for
If (15.26) is used then
α
, the AR coefficients.
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
1
r
1
r
p−1
r
2
1
r
1
r
p−2
−−−−−
−−−−−
−−−−
α
r
1
r
2
−
−
r
p
1
α
2
−
−
−
α
x
=
(15.27)
r
1
−−−
1
r
p
−1
p
Since the Fourier transform is given by (15.28), the autospectral density function
is estimated by using an AR model of order
p
as given in (15.29).
+∞
k
e
−
j
ω
k
F
(
ω
)
=
γ
(15.28)
k
=−∞
1
k
e
j
ω
k
p
p
2
=
σ
k
e
−
j
ω
k
X
(
ω
)
+
1
α
+
1
α
(15.29)
π
k
=
k
=
2
where the autocovariance function is
γ
(
k
)
=
σ
α
k
.
It should be noted that the frequency
ω
varies within the interval (0
,π
), where
π
represents the Nyquist Frequency.
15.6.2 Summary
The cross-spectrum shows the relationship between two waveforms (input and output) in
the “Frequency Domain.” The cross-spectrum indicates what frequencies in the output
are related to frequencies in the input. The cross-spectrum consists of complex terms,
which are termed the
coincident spectral density function
[
Co-spectrum, Cxy( f )
] and the
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