Calculation of S_k(t) and $\phi_k(t)$

The amplitude envelope S_k(t) and the output phase $\phi_k(t)$ can be calculated using the following lemma.

Lemma 1   The amplitude envelope S_k(t) is calculated by

$$S_k(t)=\vert\tilde{f}(\alpha^{k-\frac{K}{2}},t)\vert,$$ (16)

where $\vert\tilde{f}(a,b)\vert$ is the amplitude spectrum defined by the complex wavelet transform. The output phase $\phi_k(t)$ is calculated by

$$\phi_k(t)=\int \left(\frac{d}{dt}\arg\left(\tilde{f}(\alpha^{k-\frac{K}{2}},t)\right)-\omega_k\right)dt,$$ (17)

where $\arg(\tilde{f}(a,b))$ is the phase spectrum defined by the complex wavelet transform.

Proof. See appendix in [Unoki et al.1997]. $\Box$