Phase demodulation with iterative Hilbert transform embeddings

TL;DR

The paper tackles demodulating phase-modulated signals $x(t)=S(\varphi(t))$ with arbitrary waveform $S$ and monotone phase $\varphi(t)$. It introduces iterative Hilbert transform embeddings that generate a sequence of proxi-phases via $\theta_{n+1}=\hat{P}[x(\theta_n)]$, starting from $\theta_0=t$, and demonstrates convergence toward the true protophase, supported by a perturbative theory for the cosine case showing exponential damping of high-frequency modulations. Numerical experiments on simple and complex waveforms—including fast modulations—confirm near machine-precision demodulation (errors around $10^{-7}$ to $10^{-8}$) and reveal the superior robustness of the length-based proxi-phase in fast-modulation regimes. The study also discusses transforming a protophase to a true phase with a uniform average rotation via a $C(\cdot)$ mapping, highlighting practical limits from discretization and data length. Overall, the approach offers a filter-free, broadly applicable framework for phase demodulation in driven oscillatory systems and signals with arbitrary waveform modulations.

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 the 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 simple cosine waveforms, showing convergence of the technique.

Figures (7)

• Figure 1: Iterated HT embeddings for the phase-modulated signals $x_{1,2}(t)=S_{1,2}(\varphi(t))$ (see expressions (\ref{['eq:swf']},\ref{['eq:cwf']})) with modulation $\varphi(t)=t+1.2(\sin(0.25\sqrt{2}t) + \cos(0.25\sqrt{3}t))$. Panels (a)-(d) show the simple waveform case, here the analytic proxi-phase was adopted. Panels (e)-(h) show the complex waveform, here the length proxi-phase was used. The iteration steps: (a),(e): $n=0$; (b),(f): $n=1$; (c),(g): $n=2$, and (d),(h): $n=10$.
• Figure 2: Depicted is the damping of Fourier modes in the range $2K<\omega<2K+2$ for the simple waveform $S(\varphi) = \cos(\varphi)$. In each iteration step, the method damps $\mathcal{F}_n(\omega)$ (dashed vertical arrows) and additionally generates a new Fourier mode at a smaller frequency (dashed diagonal arrows). At step $K$, a Fourier mode with frequency less than 1 is generated in the low-frequency region (bold green-white arrow), where it disappears before the next iteration. The damping factors are given by \ref{['eq:damp']}.
• Figure 3: We show the most essential spectral components of the phase demodulation error $\theta^{(a)}_n-\varphi$ for the first 4 iterations (squares, circles, up and bottom triangles, respectively). The observed spectral components are marked with arrows on top of panels. In all cases the original signal is $x(t)=\cos(\varphi(t))$, where the modulation is given by Eq. \ref{['eq:tfm']}. The vertical scale is logarithmic, with the tics marking factor 2, to allow for a visual comparison with the theory, where the damping factors are $1/2$.
• Figure 4: Errors of demodulation for simple waveform $S_1$ and both proxi-phases calculations in dependence on the iteration index $n$ and $R$. For $R=0.25,\;0.5,\;1$, results for the analytic and length proxiphases are depicted in the same color, but with open and filled markers, respectively.
• Figure 5: Errors of demodulation in dependence on the iteration index $n$ and $R$ for the complex waveform $S_2$.