Biomedical Engineering Reference
In-Depth Information
For a real signal
s
(
t
), the power spectrum is real and nonnegative, whereas
bispectra and the higher order spectra are, in general, complex. For a real, discrete,
zero-mean, stationary process
s
(
t
)
,
we can determine the third-order cumulant as we
did for (3.44). Subsequently, the bispectrum in (3.48) becomes:
∞
∞
{
}
(
)
( ) (
) (
)
(
)
∑
∑
S
ωω
,
=
E sn sn l
+
sn l
+
exp
−
j
ω
l
−
j
ω
l
(3.49)
3
1
2
1
2
1
1
2
2
l
=−∞
l
=−∞
1
2
ω
2
, and that it
does not depend on a linear time shift of
s
. In addition, the bispectrum quantifies the
presence of quadratic phase coupling between any two frequency components in the
signal. Two frequency components are said to be quadratically phase coupled
(QPC) when a third component, whose frequency and phase are the sum of the fre-
quencies and phases of the first two components, shows up in the signal's
bispectrum [see (3.50)]. Whereas the power spectrum gives the product of two iden-
tical Fourier components (one of them taken with complex conjugation) at one fre-
quency, the bispectrum represents the product of a tuple of three Fourier
components, in which one frequency equals the sum of the other two [60]. Hence, a
peak in the bispectrum indicates the presence of QPC. If there are no phase-coupled
harmonics in the data, the bispectrum (and, hence, the second-order cumulant) is
essentially zero. Interesting properties of the bispectrum, besides its ability to detect
phase couplings, are that the bispectrum is zero for Gaussian signals and that it is
constant for linearly related signals. These properties have been used as test statistics
to rule out the hypothesis that a signal is Gaussian or linear [59]. Under conditions
of symmetry, only a small part of the bispectral space would have to be further ana-
lyzed. Examples of such symmetries are [60]:
Equation (3.49) shows that the bispectrum is a function of
ω
1
and
(
)
(
)
(
)
(
)
(
)
*
S
ωω
,
=
S
ωω
,
=
S
−
ωω
,
−
=
S
−
ωωω
−
,
=
S
ωωω
,
−
−
3
1
2
3
2
1
3
2
1
3
2
1
1
3
2
2
1
For a detailed discussion on the properties of the bispectrum, we refer the reader
to [59].
3.2.1.4 Estimation of Bispectrum
The bispectrum can be estimated using either parametric or nonparametric estima-
tors. The nonparametric bispectrum estimation can be further divided into the indi-
rect method and the direct method. The direct and indirect methods discussed herein
have been shown to be more reliable than the parametric estimators for EEG signal
analysis. The bias and consistency of the different estimators for bispectrum are
addressed in [60].
Indirect Method
We first estimate the cumulants as described in Section 3.2.1.2. The first three steps
therein are followed to obtain the cumulant
ckl
3
(,)per segment
s
i
. Then, the
i
two-dimensional Fourier transform
S
i
3
(
ω
1
,
ω
2
) of the cumulant is obtained. The
average of
S
i
3
(
ω
1
,
ω
2
) over all segments
i
=
1, …,
R
, gives the bispectrum estimate
S
3
(
ω
1
,
ω
2
).
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