Biomedical Engineering Reference
In-Depth Information
signals independent of their amplitudes. The basic idea is to generate an analytic sig-
nal from which a phase, and a phase difference between two signals, can be defined.
Suppose we have a continuous signal
x
(
t
), from which we can define an analytic
signal
~
() () ()
()
( )
j
φ
t
Zt
=
xt
+
jxt
=
Ate
(4.8)
x
x
x
~
()
where
xt
is the Hilbert transform of
x
(
t
):
()
xt
tt
dt
′
−′
+∞
∫
1
π
~
() ( )()
xt
≡
Hx t
=
PV
..
′
(4.9)
−∞
φ
y
from a second signal
y
(
t
). Then, we define the (
n
,
m
) phase difference of the analytic
signals as
where P.V. refers to the Cauchy principal value. Similarly, we can define
A
y
and
()
()
()
φ
t
≡
nt
φ
−
mt
φ
(4.10)
xy
x
y
with
n, m
integers. We say that
x
and
y
are
m
:
n
synchronized if the (
n
,
m
) phase dif-
ference of (4.10) remains bounded for all
t
. In most cases, only the (1:1) phase syn-
chronization is considered. The phase synchronization index is defined as follows
[16-18]:
2
2
( )
()
()
j
φ
t
γ
≡
e
=
cos
φ
t
+
sin
φ
t
(4.11)
xy
xy
xy
t
t
t
where the angle brackets denote average over time. The phase synchronization
index will be zero if the phases are not synchronized and will be one for a constant
phase difference. Note that for perfect phase synchronization the phase difference is
not necessarily zero, because one of the signals could be leading or lagging in phase
with respect to the other. Alternatively, a phase synchronization measure can be
defined from the Shannon entropy of the distribution of phase differences
φ
xy
(
t
)or
from the conditional probabilities of
φ
y
(
t
) [19].
An interesting feature of phase synchronization is that it is parameter free.
However, it relies on an accurate estimation of the phase. In particular, to avoid
misleading results, broadband signals (as it is usually the case of EEGs) should be
first bandpass filtered in the frequency band of interest before calculating phase
synchronization.
It is also possible to define a phase synchronization index from the wavelet
transform of the signals [20]. In this case the phases are calculated by convolving
each signal with a Morlet wavelet function. The main difference with the estimation
using the Hilbert transform is that a central frequency
0
and a width of the wavelet
function should be chosen and, consequently, this measure is sensitive to phase syn-
chronization in a particular frequency band. It is of particular interest to mention
that both approaches—either defining the phases with the Hilbert or with the wave-
let transform—are intrinsically related (for details, see [8]).
φ
x
(
t
) and
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