Biomedical Engineering Reference
In-Depth Information
Direct Method
The direct method estimates the bispectrum directly from the frequency domain. It
involves the following steps:
1. Divide the EEG data of length
N
into
R
segments, each of length
M
, such
that
R·M N
. Let each segment be denoted by
s
i
.
2. In each segment
s
i
subtract the mean.
3. Compute the one-dimensional FFT for each of these segments to obtain
Y
i
(
).
4. The bispectrum estimate for the segment
s
i
is obtained by
ω
(
)
( ) ( ) (
)
i
i
i
i
*
S
ωω
,
=
Y
ω
Y
ω
Y
ω
+
ω
(3.50)
3
1
2
1
2
1
2
for all combinations of
ω
1
and
ω
2
, with the asterisk denoting complex
conjugation.
5. As in a periodogram, the bispectrum estimate of the entire data is obtained
by averaging the bispectrum estimate of individual segments:
1
R
(
)
(
)
=
∑
S
i
(3.51)
S
ωω
,
=
ωω
,
3
1
2
3
1
2
R
i
1
It is clear from (3.50) that the bispectrum can be used to study the interaction
between the frequency components
ω
1
+ ω
2
. A drawback in the use of
polyspectra is that they need long datasets to reduce the variance associated with
estimation of higher order statistics.
The bispectrum is also influenced by the power of the signal at its components;
therefore, it is not only a measure of quadratic phase coupling. The bispectrum
could be normalized in order to make it sensitive only to changes in phase coupling
(as we do for spectrum in order to generate coherence). This normalized bispectrum
is then known as bicoherence [60]. To compute the bicoherence (BIC) of a signal,
we define the real triple product RTP(
ω
1
,
ω
2
, and
ω
1
,
ω
2
) of the signal as follows:
(
) ( ) ( ) (
)
RTP
ωω
,
=
PP P
ω
ω
ω
+
ω
(3.52)
1
2
1
2
1
2
where
P
(
. The
bicoherence is then defined as the ratio of the bispectrum of the signal to the square
root of its RTP:
ω
) is the power spectrum of the signal at angular frequency
ω
(
)
S
ωω
,
(
)
3
1
2
BIC
ωω
,
=
(3.53)
1
2
(
)
RTP
ωω
,
1
2
If all freq
uencies are co
mpletely phase coupled to each other (identical phases),
S
3
(
ω
1
,
ω
2
)
=
RTP(
ωω
1
,
2
, and BIC(
)
ω
1
,
ω
2
)
=
1. If there is no QPC coupling at all,
ω
1
,
ω
2
), the signal is a nonlinear process. The variance of the bicoherence estimate is
directly proportional to the amount of statistical averaging performed during the
the bispectrum will be zero in the (
ω
1
,
ω
2
) domain. If |BIC(
ω
1
,
ω
2
)|
≠
1 for some (
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