Calculation of instantaneous amplitude S_k(t) and instantaneous output phase $\phi _k(t)$

The instantaneous amplitude S_k(t) of Eq. (5) and the instantaneous output phase $\phi _k(t)$ of Eq. (6) can be calculated using the following lemma.

Lemma 2   The instantaneous amplitude S_k(t) is calculated by

$$S_k(t)=\vert\tilde{f}(\alpha^{k-\frac{K}{2}},t)\vert,\qquad a=\alpha^{k-\frac{K}{2}},b=t,$$ (22)

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

$$\phi_k(t)=\int_0^t \left(\frac{d}{d\tau}\arg\left(\tilde{f}(\... ...au)\right)-\omega_k\right)d\tau, a=\alpha^{k-\frac{K}{2}},b=t,$$ (23)

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

Proof. See appendix C. $\Box$