Information Technology Reference
In-Depth Information
CPV
+
∞
−
∞
CPV
+
∞
−
∞
f
(
t
)=
f
(
τ
)
g
(
t
−
τ
)
d
τ
=
g
(
τ
)
h
(
t
−
τ
)
d
τ
,
(18.25)
where
1
π
g
(
t
)
:
=
t
,
t
∈
T
,
and symbol CPV signifies that the integral is taken in the sense of
Cauchy principal
value
.
Notice that
f
(
t
)
can be viewed as a convolution
g
(
t
)
×
f
(
t
)
of the original func-
tion
f
. This means that the Hilbert transform can be per-
formed by applying an ideal filter, whose amplitude response equals to 1, and phase
response is a constant
(
t
)
with the function
g
(
t
)
π
/
2 lag at all frequencies.
Given an arbitrary continuous real-valued time series
X
(
)
, the corresponding
analytic signal
is defined as the following complex-valued function:
t
X
ξ
X
(
t
)=
X
(
t
)+
ı
·
(
t
)=
a
X
(
t
)
·
exp
{
ı
·
φ
X
(
t
)
},
(18.26)
where
t
denotes time,
ı
is a unit on the complex axis,
X
(
t
)
denotes the Hilbert trans-
form of the time series
X
(
t
)
,
a
X
(
t
)
is the corresponding instantaneous amplitude,
and
represents the instantaneous phase of the signal via Hilbert convolution.
It follows from (18.26) that the
instantaneous phase
φ
X
(
t
)
φ
X
(
t
)
of
X
(
t
)
can be com-
puted as
arctan
X
(
t
)
φ
X
(
t
)=
.
(18.27)
(
)
X
t
A key advantage of the analytic approach is that the phase can be easily computed
for an arbitrary broadband signal. On the other hand, instantaneous amplitude and
phase have a clear physical meaning only if
X
is a narrowband signal. Therefore,
filtration is required in order to separate the frequency band of interest from the
background brain activity.
Various measures of phase synchrony between two signals are proposed based on
the phases extracted via the Hilbert and the wavelet transforms, including standard
deviation, mutual information, and Shannon entropy [31, 14]. However, most of the
currently used measures of phase synchronization are based on
bivariate
indexes. In
the next section, we propose a novel
multivariate
approach to detecting phase syn-
chronization in the phases extracted from multiple time series, such as multichannel
EEG.
(
t
)
18.5 Multivariate Approach to Phase Synchrony via
Cointegrated VAR
We develop a new method for measuring the synchrony among the instantaneous
phases extracted from multivariate time series. Our technique is based on the coin-
tegrated VAR modeling of time series.
