Chapter 12
Phase Reconstruction with Iterated
Hilbert Transforms
Erik Gengel and Arkady Pikovsky
Abstract We discuss theoretical and practical issues of data-driven phase reconstruction approaches for nonlinear oscillatory systems by means of the geometric
technique of embeddings and protophase-to-phase transformation. In this chapter,
we introduce a natural extension of the well-studied Hilbert transform by iteration.
The novel approach, termed iterated Hilbert transform embeddings, implements central assumptions underlying phase reconstruction and allows for exact demodulation
of purely phase modulated signals. Here, we examine the performance of the novel
method for the more challenging situation of generic phase-amplitude modulated
signals of a simple nonlinear oscillatory system. In particular we present the benefits
of the approach for secondary phase analysis steps illustrated by reconstruction of
the phase response curve. Limitations of the approach are disussed for a noise-driven
phase dynamics.
E. Gengel · A. Pikovsky (B)
Institute for Physics and Astronomy, Universit of Potsdam, Karl-Liebknecht-Str. 24/25,
14476 Potsdam, Germany
e-mail: pikovsky@uni-potsdam.de
E. Gengel
e-mail: egengel@uni-potsdam.de
E. Gengel
Friedrich-Ebert Stiftung, Bonn, Germany
A. Pikovsky
Higher School of Economics, Nizhny Novgorod, Russia
Department of Control Theory, Institute of Information Technologies, Mathematics
and Mechanics, Lobachevsky University Nizhny Novgorod, Nizhny Novgorod, Russia
© Springer Nature Switzerland AG 2021
A. Stefanovska and P. V. E. McClintock (eds.), Physics of Biological
Oscillators, Understanding Complex Systems,
https://doi.org/10.1007/978-3-030-59805-1_12
191
Phase Reconstruction with Iterated
Hilbert Transforms
Erik Gengel and Arkady Pikovsky
Abstract We discuss theoretical and practical issues of data-driven phase reconstruction approaches for nonlinear oscillatory systems by means of the geometric
technique of embeddings and protophase-to-phase transformation. In this chapter,
we introduce a natural extension of the well-studied Hilbert transform by iteration.
The novel approach, termed iterated Hilbert transform embeddings, implements central assumptions underlying phase reconstruction and allows for exact demodulation
of purely phase modulated signals. Here, we examine the performance of the novel
method for the more challenging situation of generic phase-amplitude modulated
signals of a simple nonlinear oscillatory system. In particular we present the benefits
of the approach for secondary phase analysis steps illustrated by reconstruction of
the phase response curve. Limitations of the approach are disussed for a noise-driven
phase dynamics.
E. Gengel · A. Pikovsky (B)
Institute for Physics and Astronomy, Universit of Potsdam, Karl-Liebknecht-Str. 24/25,
14476 Potsdam, Germany
e-mail: pikovsky@uni-potsdam.de
E. Gengel
e-mail: egengel@uni-potsdam.de
E. Gengel
Friedrich-Ebert Stiftung, Bonn, Germany
A. Pikovsky
Higher School of Economics, Nizhny Novgorod, Russia
Department of Control Theory, Institute of Information Technologies, Mathematics
and Mechanics, Lobachevsky University Nizhny Novgorod, Nizhny Novgorod, Russia
© Springer Nature Switzerland AG 2021
A. Stefanovska and P. V. E. McClintock (eds.), Physics of Biological
Oscillators, Understanding Complex Systems,
https://doi.org/10.1007/978-3-030-59805-1_12
191