Figure 1
From: An extended Hilbert transform method for reconstructing the phase from an oscillatory signal
Reconstruction of the instantaneous phase from an observed signal. (a) Observed signal x(t) (Eq. 6) with \(u(t) = 0.2 \left( \sin \sqrt{2} \hat{\omega } t + \cos \sqrt{3} \hat{\omega } t \right) \). (b) Instantaneous phase \(\phi (t)\) (Eq. 6). (c) Phase-modulation u(t). (d) Power spectrum of the phase-modulation. The dashed line in (b–d) represents the true phase, the phase-modulation, and its power spectrum, respectively. The red and blue lines represent the reconstructions by the conventional HT method and the proposed method, respectively. Dotted vertical lines in (d) represent the dominant frequencies of the true phase-modulation: \(\sqrt{2} \hat{\omega }\) and \(\sqrt{3} \hat{\omega }\), where \(\hat{\omega }= 2\pi \) is the effective frequency. Note that we plotted a part of the signal and the start time of the plot is redefined as 0.
