12 Phase Reconstruction with Iterated Hilbert Transforms
201
STD
q, ˙
q
n
tend to zero only, if the reconstructed protophases and transformed protophases q n = {θ n (t), ψ n (t)} are close to the true phase ϕ(t) of the system (see
Eq.(12.4)). In the integration, we skip the outer ten percent at the beginning and at
the end of the time series, to avoid boundary effects. Estimations of the instantaneous
frequency ϕ(t) and ˙
q n (t) are performed by a 12th order polynomial filter (SavitzkyGolay filter) with a window of 25 points and four times repetition [33] denoted as
SG(12,25,4). Throughout the chapter we use a sampling rate of dt = 0.01, such that
the smoothing window has a width of dt = 0.25, corresponding to roughly 11% of
the fastest forcing period (r = 14.3). The estimated average growth rate ˜
ω is obtained
by linear regression. Note that the normalization integral ˆ
N 1 is suitable for all phases
where the average growth is linear.
12.4 Numerical Experiments
12.4.1 Deterministic Oscillations
Here we consider the SL system (12.3) with μ = 8, α = 0.1, ν = 1. As the observables we explore functions X 1,2,3 [a(t)] defined above. The system is forced harmonically by εP(η) = ε cos(η(t)) with amplitude ε = 0.1. The external force phase is
η(t) = r ωt, for the explored range of driving frequencies r ω the SL operated in the
asynchronous regime. We observe 100 periods with a time step of dt = 0.01.
In Fig. 12.6 the phase and the frequency errors according to Eq. (12.17) for the first
20 iteration steps are shown. While for slow modulations (r < 1), the reconstruction
is already accurate in the first step, for fast forcing frequencies (r > 1) indeed several
iterations are needed for precise reconstruction. The reason for this is that for highfrequency modulations iterative HT embeddings first shift high-frequency Fourier
components of the phase modulation to lower frequencies, where they eventually
disappear. This mechanism is closely related to the Bedrosian identities [40] and is
explained in detail in [12]. For the reconstruction of phases in case of X 2,3 [a(t)], we
have to calculate the transformed phase ψ(t), because here the protophases deviate
from uniform growth. The results show, that IHTE combined with the protophaseto-phase transformation provides proper phase reconstructions for the fairly stable
limit cycle oscillator under study.
Figure 12.5 presents comparisons of the inferred modulation u n (t) := ψ n (t) −
˜
ωt with the true one q(t) = ϕ − ωt, and of the inferred instantaneous frequencies
˙
θ n (t)/ ˙
ψ n (t) with ˙
ϕ, for a quite fast external force r = 5.6 (black bold dots in Fig.
12.6). While the first iterate is by far not accurate, iterations provide the reconstructed
estimation of the phases ψ 20 (t) which is very close to ϕ(t).
201
STD
q, ˙
q
n
tend to zero only, if the reconstructed protophases and transformed protophases q n = {θ n (t), ψ n (t)} are close to the true phase ϕ(t) of the system (see
Eq.(12.4)). In the integration, we skip the outer ten percent at the beginning and at
the end of the time series, to avoid boundary effects. Estimations of the instantaneous
frequency ϕ(t) and ˙
q n (t) are performed by a 12th order polynomial filter (SavitzkyGolay filter) with a window of 25 points and four times repetition [33] denoted as
SG(12,25,4). Throughout the chapter we use a sampling rate of dt = 0.01, such that
the smoothing window has a width of dt = 0.25, corresponding to roughly 11% of
the fastest forcing period (r = 14.3). The estimated average growth rate ˜
ω is obtained
by linear regression. Note that the normalization integral ˆ
N 1 is suitable for all phases
where the average growth is linear.
12.4 Numerical Experiments
12.4.1 Deterministic Oscillations
Here we consider the SL system (12.3) with μ = 8, α = 0.1, ν = 1. As the observables we explore functions X 1,2,3 [a(t)] defined above. The system is forced harmonically by εP(η) = ε cos(η(t)) with amplitude ε = 0.1. The external force phase is
η(t) = r ωt, for the explored range of driving frequencies r ω the SL operated in the
asynchronous regime. We observe 100 periods with a time step of dt = 0.01.
In Fig. 12.6 the phase and the frequency errors according to Eq. (12.17) for the first
20 iteration steps are shown. While for slow modulations (r < 1), the reconstruction
is already accurate in the first step, for fast forcing frequencies (r > 1) indeed several
iterations are needed for precise reconstruction. The reason for this is that for highfrequency modulations iterative HT embeddings first shift high-frequency Fourier
components of the phase modulation to lower frequencies, where they eventually
disappear. This mechanism is closely related to the Bedrosian identities [40] and is
explained in detail in [12]. For the reconstruction of phases in case of X 2,3 [a(t)], we
have to calculate the transformed phase ψ(t), because here the protophases deviate
from uniform growth. The results show, that IHTE combined with the protophaseto-phase transformation provides proper phase reconstructions for the fairly stable
limit cycle oscillator under study.
Figure 12.5 presents comparisons of the inferred modulation u n (t) := ψ n (t) −
˜
ωt with the true one q(t) = ϕ − ωt, and of the inferred instantaneous frequencies
˙
θ n (t)/ ˙
ψ n (t) with ˙
ϕ, for a quite fast external force r = 5.6 (black bold dots in Fig.
12.6). While the first iterate is by far not accurate, iterations provide the reconstructed
estimation of the phases ψ 20 (t) which is very close to ϕ(t).