Biomedical Engineering Reference
In-Depth Information
angles are not aligned, the product will be small. Finally, the complex bispectrum is
converted to a real number by computing the magnitude of the complex product.
If one starts by sampling EEGs at 128 Hz into 4-second epochs, then the result-
ing Fourier spectrum will extend from 0 to 64 Hz at 0.25-Hz resolution, or a total of
256 spectral points. If all triplets were to be calculated, there would be 65,536 (256
×
256) points. Fortunately, it is unnecessary to calculate the bispectrum for all possi-
ble frequency combinations. The minimal set of frequency combinations to calcu-
late a bispectrum can be visualized as a wedge of frequency versus frequency space
(Figure 9.7). The combinations outside this wedge need not be calculated because of
symmetry [i.e.,
B
(
f
1
,
f
2
)
=
B
(
f
2
,
f
1
)] and because the range of allowable modulation
f
2
, is limited to frequencies less than or equal to half of the sampling
rate. Still, because this calculation must be performed, using complex number arith-
metic, for at least several thousand triplets, it is easy to see that it is a major compu-
tational burden.
As noted earlier, computation of the bispectrum itself is only the beginning for
complete higher order spectral analysis. If one is interested in isolating and examin-
ing solely the phase relationships, as noted earlier, the bispectrum must have the
existing variations in signal amplitude normalized. Recall that the amplitude of a
particular Fourier spectral element
X
(
f
) is determined by the magnitude or the
length of its complex number vector. The RTP is formed from the multiplied prod-
uct of the squared magnitudes of the three spectral values in the triplet:
frequencies,
f
1
+
(
)
( )
2
() (
2
)
2
RTP f
,
f
=
X f
*
X f
*
X f
+
f
12
1
2
1
2
The square root of the RTP yields the joint amplitude of the triplet, the factor
that is used to normalize the bispectrum into the bicoherence. The bicoherence,
BIC
(
f
1
,
f
2
) is a number that varies from 0 to 1 in proportion to the degree of phase
coupling in the frequency triplet:
(
)
BIC f
,
f
(
)
12
BIC f
,
f
=
12
(
)
RTP f
,
f
12
Figure 9.8 illustrates some representative data during bispectral analysis. Com-
puting the bispectrum of a stochastic biological signal such as the EEG generally
requires that the signal be divided into relatively short epochs for calculation of the
bispectrum and bicoherence, which are then averaged over a number of epochs to
provide a relatively stable estimate of the true bispectral values. Figure 9.8(a) is a
two-dimensional plot of bispectrum
B
(
f
1
,
f
2
); Figure 9.8(b) is a three-dimensional
perspective illustration of the same data. Figure 9.8(c) is a three-dimensional illus-
tration of the bicoherence
BIC
(
f
1
,
f
2
).
9.5.3 Bispectral Index: Implementation
The BIS is a complex qEEG parameter, composed of a combination of time-domain,
frequency-domain, and higher order spectral subparameters. It was the first among
the qEEG parameters reviewed here to integrate several disparate descriptors of the
EEG into a single variable based on the post hoc analysis of a large volume of clini-
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