Biomedical Engineering Reference
In-Depth Information
Bispectrum is a Fourier transform of third order cumulant
R
3
(
τ
1
,
τ
2
)
. For discrete
case (sampled signal) correlation is expressed by:
∞
∑
(
)=
(
)
(
+
)
R
2
τ
x
t
x
t
τ
(2.41)
t
=
−
∞
and third order cumulant by:
∞
∑
R
3
(
τ
1
,
τ
2
)=
x
(
t
)
x
(
t
+
τ
1
)
x
(
t
+
τ
2
)
(2.42)
t
=
−
∞
Bispectrum (BS) is defined as:
X
∗
(
BS
(
f
1
,
f
2
)=
X
(
f
1
)
X
(
f
2
)
f
1
+
f
2
)
(2.43)
where
X
denotes Fourier transform of signal
x
. Bispectrum quantifies the relation-
ship between the sinusoids at two primary frequencies
f
1
and
f
2
and the modulation
component at frequency
f
1
+
f
2
. Bispectrum is a function of the triplet of frequen-
cies
incorporating both power and phase information. Bicoherence,
which takes values from the range
(
f
1
,
f
2
,
f
1
+
f
2
)
[
0
−
1
]
,isdefined as a squared normalized version
of the bispectrum:
|
(
,
)
|
BS
f
1
f
2
B
2
(
f
1
,
f
2
)=
(2.44)
S
(
f
1
)
S
(
f
2
)
S
(
f
1
+
f
2
)
Bicoherence is a function which gives the information on non-linear interactions. It is
a measure for quantifying the extent of phase coupling between different frequencies
in the signal. Namely bicoherence measures the proportion of the signal energy at
any bifrequency that is quadratically phase coupled.
In practical applications, the computation of bicoherence is limited by the quality
of the data. The statistical error of spectral estimate is high as was mentioned already
in
Sect. 2.3.2.1.2.
For bicoherence these errors cumulate due to the multiplication
of spectral terms. Therefore reliable estimation of bicoherence is possible only for
a high signal to noise ratio and requires long enough data allowing to compute an
average estimate based on several equivalent segments of the signal.
2.3.2.2
Parametric models: AR, ARMA
A time series model that approximates many discrete stochastic and determinis-
tic processes encountered in practice is represented by the ARMA filter difference
equation:
p
i
=
1
a
i
x
t
−
i
−
r
k
=
0
b
k
y
n
−
k
x
t
=
(2.45)
Where
x
t
is the output sequence of a causal filter and
y
t
is an input driving se-
quence. In the ARMA model it is assumed that the driving sequence is white noise
process of zero mean.
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