@article{GengelPikovskij2022,
  author    = {Gengel, Erik and Pikovskij, Arkadij},
  title     = {Phase reconstruction from oscillatory data with iterated Hilbert transform embeddings},
  series = {Physica : D, Nonlinear phenomena},
  volume    = {429},
  journal   = {Physica : D, Nonlinear phenomena},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0167-2789},
  doi       = {10.1016/j.physd.2021.133070},
  pages     = {9},
  year      = {2022},
  abstract  = {In the data analysis of oscillatory systems, methods based on phase reconstruction are widely used to characterize phase-locking properties and inferring the phase dynamics. The main component in these studies is an extraction of the phase from a time series of an oscillating scalar observable. We discuss a practical procedure of phase reconstruction by virtue of a recently proposed method termed iterated Hilbert transform embeddings. We exemplify the potential benefits and limitations of the approach by applying it to a generic observable of a forced Stuart-Landau oscillator. Although in many cases, unavoidable amplitude modulation of the observed signal does not allow for perfect phase reconstruction, in cases of strong stability of oscillations and a high frequency of the forcing, iterated Hilbert transform embeddings significantly improve the quality of the reconstructed phase. We also demonstrate that for significant amplitude modulation, iterated embeddings do not provide any improvement.},
  language  = {en}
}
@phdthesis{Gengel2021,
  author    = {Gengel, Erik},
  title     = {Direct and inverse problems of network analysis},
  doi       = {10.25932/publishup-51236},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-512367},
  school      = {Universit{\"a}t Potsdam},
  pages     = {VIII, 102},
  year      = {2021},
  abstract  = {Selfsustained oscillations are some of the most commonly observed phenomena in biological systems. They emanate from non-linear systems in a heterogeneous environment and can be described by the theory of dynamical systems. Part of this theory considers reduced models of the oscillator dynamics by means of amplitudes and a phase variable. Such variables are highly attractive for theoretical and experimental studies. Theoretically these variables correspond to an integrable linearization of the generally non-linear system. Experimentally, there exist well established approaches to extract phases from oscillator signals. Notably, one can define phase models also for networks of oscillators. One highly active field examines effects of non-local coupling among oscillators, which is thought to play a key role in networks with strong coupling. The dissertation introduces and expands the knowledge about high-order phase coupling in networks of oscillators. Mathematical calculations consider the Stuart-Landau oscillator. A novel phase estimation scheme for direct observations of an oscillator dynamics is introduced based on numerics. A numerical study of high-order phase coupling applies a Fourier fit for the Stuart-Landau and for the van-der-Pol oscillator. The numerical approach is finally tested on observation-based phase estimates of the Morris-Lecar neuron. A popular approach for the construction of phases from signals is based on phase demodulation by means of the Hilbert transform. Generally, observations of oscillations contain a small and generic variation of their amplitude. The work presents a way to quantify how much the variations of signal amplitude spoil a phase demodulation procedure. For the ideal case of phase modulated signals, amplitude modulations vanish. However, the Hilbert transform produces artificial variations of the reconstructed amplitude even in this case. The work proposes a novel procedure called Iterative Hilbert Transform Embedding to obtain an optimal demodulation of signals. The text presents numerous examples and tests of application for the method, covering multicomponent signals, observables of highly stable limit cycle oscillations and noisy phase dynamics. The numerical results are supported by a spectral theory of convergence for weak phase modulations.},
  language  = {en}
}
@article{GengelPikovskij2019,
  author    = {Gengel, Erik and Pikovskij, Arkadij},
  title     = {Phase demodulation with iterative Hilbert transform embeddings},
  series = {Signal processing},
  volume    = {165},
  journal   = {Signal processing},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0165-1684},
  doi       = {10.1016/j.sigpro.2019.07.005},
  pages     = {115 -- 127},
  year      = {2019},
  abstract  = {We propose an efficient method for demodulation of phase modulated signals via iterated Hilbert transform embeddings. We show that while a usual approach based on one application of the Hilbert transform provides only an approximation to a proper phase, with iterations the accuracy is essentially improved, up to precision limited mainly by discretization effects. We demonstrate that the method is applicable to arbitrarily complex waveforms, and to modulations fast compared to the basic frequency. Furthermore, we develop a perturbative theory applicable to a simple cosine waveform, showing convergence of the technique.},
  language  = {en}
}