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3.3
Instantaneous Phase Complexity
In this section we propose a novel approach to measure phase complexity in cir-
cadian rhythms. It is based on a simple idea: the application of the conventional
complexity measures
ApEn
and
LZ
to the instantaneous phase of recorded sig-
nals. We consider that the presence of phase variability and phase shifts might
be associated to a lack of circadian robustness. Complexity measures of instanta-
neous phase might be an appropriate indicator of cycle robustness. In this paper,
instantaneous phase has been computed by means of the Hilbert transform and
by means of a complex continuous wavelet transform (CWT).
Givenanarrowbandsignal
x
(
t
), its Hilbert transform is computed as [17]:
x
H
(
t
)=
π
−
1
PV
∞
−∞
x
(
τ
)
t
τ
dτ
(4)
−
where
PV
is the Cauchy Principal Value method for evaluating improper inte-
grals. An analytic signal
χ
(
t
) can be obtained from
x
H
as
χ
(
t
)=
x
(
t
)+
ix
H
(
t
).
The instantaneous phase is then defined as:
ϕ
=arctan
x
H
(
t
)
x
(
t
)
(5)
In this way
ϕ
lies in the interval (
π, π
].
ϕ
is called the wrapped instantaneous
phase and it has discontinuities. In order to obtain meaningful results when
applying the
LZ
and
ApEn
complexity measures, the wrapped phase is required.
Examples of wrapped phases are shown in Figures 1(a) and 1(b).
ϕ
depends on
the magnitude of the DC continuous component, since the DC component is
lost in
x
H
(
t
), but not in
x
(
t
) itself. In order to normalize, a zero-mean signal
x
(
t
) is required (that is simply obtained by removing the average value from the
signal).
Additionally, we propose the application of the complex 1st order gaussian
CWT at scale 43 as an alternate way to compute the instantaneous phase:
−
ϕ
=arctan
Im
[
C
(
x
)]
Re
[
C
(
x
)]
(6)
Here
x
is an LP filtered signal and
C
(
x
) is the CWT of the signal at the given
scale. This method also provides values of
ϕ
in the interval (
π, π
].
We have studied three kind of phase complexity measures: the
LZ
complexity
index
γ
of the Hilbert instantaneous phase (
LZHP
), the approximate entropy
ApEn
of the Hilbert instantaneous phase (
ApEnHP
), and the approximate en-
tropy
ApEn
of the CWT instantaneous phase (
ApEnWP
).
On the one hand, for a pure sinusoidal function, unwrapped instantaneous
phase is a perfectly linear decaying line. A lack of regularity in the signal cycles
produces an unwrapped phase that is quasi-linear but exhibits irregularities and
occasional phase shifts. On the other hand, the wrapped instantaneous phase
of a pure sinusoidal signal has an approximate sawtooth shape perfectly regular
(Figure 1(a)). However, a lack of regularity in the signal cycles produces an irreg-
ularly spaced sawtooth shape that is strongly affected by occasional small phase
−
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