Biomedical Engineering Reference
In-Depth Information
9.5.2 Mixed Algorithms: Bispectrum
The effort to glean useful information from the EEG has led from first-order (mean
and variance of the amplitude of the signal waveform) to second-order (power spec-
trum, or its time-domain analog, autocorrelation) statistics, and now to higher
order statistics (HOS). HOS include the bispectrum and trispectrum (third- and
fourth-order statistics, respectively). Little work has been published to date on
trispectral applications in biology, but there are currently well over 1,000
peer-reviewed papers to date related to bispectral analysis of the EEG. Whereas the
phase spectrum produced by Fourier analysis measures the phase of component fre-
quencies relative to the start of the epoch, the bispectrum measures the correlation
of phase between different frequency components as described later. What exactly
these phase relationships mean physiologically is uncertain. One simple teleological
model holds that strong phase relationships relate inversely to the number of
independent EEG generator elements.
Bispectral analysis has several additional characteristics that may be advanta-
geous for processing EEG signals: Gaussian sources of noise are suppressed, thus
enhancing the SNR for the non-Gaussian EEG, and bispectral analysis can identify
nonlinearities that may be important in the signal generation process. A complete
treatment of higher order spectra may be found in the text by Proakis et al. [60]. As
described later in this chapter, the commercial exemplar of bispectral EEG process-
ing, the Aspect BIS monitor, actually mixes parameters from both the time, fre-
quency, and HOS domains to produce its output.
As noted earlier, the bispectrum quantifies the relationship among the underly-
ing sinusoidal components of the EEG. Specifically, bispectral analysis examines the
relationship between the sinusoids at two primary frequencies,
f
1
and
f
2
, and a mod-
ulation component at the frequency
f
1
+
f
2
. This set of three frequency components
is known as a triplet (
f
1
,
f
2
, and
f
1
+
f
2
). For each triplet, the bispectrum,
B
(
f
1
,
f
2
), a
quantity incorporating both phase and power information, can be calculated as
described next. The bispectrum can be decomposed to separate out the phase infor-
mation as the bicoherence,
BIC
(
f
1
,
f
2
), and the joint magnitude of the members of
the triplet, as the real triple product,
RTP
(
f
1
,
f
2
). The defining equations for
bispectral analysis are described in detail next.
A high bicoherence value at (
f
1
,
f
2
) indicates that there is a phase coupling
within the triplet of frequencies
f
1
,
f
2
and
f
1
+
f
2
. Strong phase coupling implies that
the sinusoidal components at
f
1
and
f
2
may have a common generator, or that the
neural circuitry they drive may, through some nonlinear interaction, synthesize a
new, dependent component at the modulation frequency
f
1
+
f
2
. An example of such
phase relationships and the bispectrum is illustrated in Figure 9.7.
Calculation of the bispectrum,
B
(
f
1
,
f
2
), of a digitized epoch,
x
(
i
), begins with an
FFT to generate complex spectral values,
X
(
f
). For each possible triplet, the complex
conjugate of the spectral value at the modulation frequency
X
*(
f
1
+
f
2
) is multiplied
against the spectral values of the primary frequencies of the triplet:
(
)
( ) ( )
(
)
Bf f
,
=
Xf
⋅
Xf
⋅
X f
*
+
f
12
1
2
1
2
This multiplication is the heart of the bispectral determination: If, at each fre-
quency in the triplet, there is a large spectral amplitude (i.e., a sinusoid exists for
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