12 Phase Reconstruction with Iterated Hilbert Transforms
205
The results in this case have to be interpreted relative to the smoothing parameters
which are chosen in such a way that they preserve essential local features of the
dynamics. Indeed, we observe negative instantaneous frequencies ϕ(t) pointing to
the need of a high polynomial order of smoothing (see Fig. 12.9c, d, e). Also, the
noise causes diffusion of phase (see Fig. 12.9).
From the viewpoint of phase extraction via embeddings, the white noise forcing
represents a “worst case”. On the contrary, in all situations where coloured noise with
a bounded spectrum is present, we expect IHTE to be the more easily applicable.
Depending on the spectral composition of noise, small-scale loops in the embedding
may be not present at all, or may be eliminated with minimal filtering. If the noise has
only relatively low-frequency component, the embedding will be relatively smooth,
and no additional processing is needed.
Our method is restricted to the conditions (12.11). Since all of these conditions are
not fulfilled in this example, the actual phase dynamics is only partly reconstructed,
as can be seen also from Fig. 12.9a where the reconstruction error decay is much
less pronounced than in Fig. 12.6. In view of this, the presented example can be
considered as a proof of concept for IHTE of noisy signals. The method improves
the estimation of the phase, as the examples of Fig. 12.9 show, by factor up to 2.
We add the following preprocessing to IHTE:
1. Given X (t), apply a high-order SG-filter making the signal smooth with a large
number of inflection points.
2. Next, smooth ϕ(t) by the same SG-filter.
3. Proceed signal X (t) with IHTE as described in Sect. 12.3.3.
12.5 Conclusion and Open Problems
In summary, the IHTE approach solves the problem of phase demodulation for purely
phase modulated signals. Here, we present results for a dynamical system, where the
amplitude dynamics is also present and linked to the dynamics of ϕ(t). We have
demonstrated that IHTE indeed provides a good reconstruction of the phase dynamics, if the amplitude variations are relatively small (see Fig. 12.4, 12.6, 12.5). We
show that iterations drastically improve the reconstruction of the phase, in comparison to the previously employed approach based on a single Hilbert transform (see
Fig. 12.5) and Z (ϕ) (see Fig. 12.8). However, the analysis of the performance of
IHTE in the case of larger amplitude variations is a question to be discussed in the
future.
An important issue in the phase reconstruction is the protophase-to-phase transformation. It is particularly relevant for generic observables like X 3 [.], with complex
waveforms. While handling such observables in the framework of IHTE does not
state a problem, influence of amplitude variations may depend drastically on the
complexity of the waveform. It should be stressed here, that while construction of
the protophase via IHTE is almost exact, the protophase-to-phase transformation is
205
The results in this case have to be interpreted relative to the smoothing parameters
which are chosen in such a way that they preserve essential local features of the
dynamics. Indeed, we observe negative instantaneous frequencies ϕ(t) pointing to
the need of a high polynomial order of smoothing (see Fig. 12.9c, d, e). Also, the
noise causes diffusion of phase (see Fig. 12.9).
From the viewpoint of phase extraction via embeddings, the white noise forcing
represents a “worst case”. On the contrary, in all situations where coloured noise with
a bounded spectrum is present, we expect IHTE to be the more easily applicable.
Depending on the spectral composition of noise, small-scale loops in the embedding
may be not present at all, or may be eliminated with minimal filtering. If the noise has
only relatively low-frequency component, the embedding will be relatively smooth,
and no additional processing is needed.
Our method is restricted to the conditions (12.11). Since all of these conditions are
not fulfilled in this example, the actual phase dynamics is only partly reconstructed,
as can be seen also from Fig. 12.9a where the reconstruction error decay is much
less pronounced than in Fig. 12.6. In view of this, the presented example can be
considered as a proof of concept for IHTE of noisy signals. The method improves
the estimation of the phase, as the examples of Fig. 12.9 show, by factor up to 2.
We add the following preprocessing to IHTE:
1. Given X (t), apply a high-order SG-filter making the signal smooth with a large
number of inflection points.
2. Next, smooth ϕ(t) by the same SG-filter.
3. Proceed signal X (t) with IHTE as described in Sect. 12.3.3.
12.5 Conclusion and Open Problems
In summary, the IHTE approach solves the problem of phase demodulation for purely
phase modulated signals. Here, we present results for a dynamical system, where the
amplitude dynamics is also present and linked to the dynamics of ϕ(t). We have
demonstrated that IHTE indeed provides a good reconstruction of the phase dynamics, if the amplitude variations are relatively small (see Fig. 12.4, 12.6, 12.5). We
show that iterations drastically improve the reconstruction of the phase, in comparison to the previously employed approach based on a single Hilbert transform (see
Fig. 12.5) and Z (ϕ) (see Fig. 12.8). However, the analysis of the performance of
IHTE in the case of larger amplitude variations is a question to be discussed in the
future.
An important issue in the phase reconstruction is the protophase-to-phase transformation. It is particularly relevant for generic observables like X 3 [.], with complex
waveforms. While handling such observables in the framework of IHTE does not
state a problem, influence of amplitude variations may depend drastically on the
complexity of the waveform. It should be stressed here, that while construction of
the protophase via IHTE is almost exact, the protophase-to-phase transformation is