Figure 2

(a) Short epoch of noisy data \({\bar{s}}\) (blue) and causally obtained amplitudes \(a_N\) (red) and \(a_R\) (magenta); the latter provides the most smooth envelope (cf. \(a_H\) in Fig. 1a). When the signal’s amplitude is very small, the noise dominates and phase determination becomes complicated. The HT approach fails here (blue line in (b,c)), while both locked and resonant oscillator “devices” provide reasonable results. The performance of the non-resonant technique is poor: when the amplitude nearly vanishes, the phase estimated by this technique (not shown) is not better than the Hilbert phase.