Calculation of the four parameters

The amplitude envelope S_k(t) and phase $\phi_k(t)$ of X_k(t) are determined by using the amplitude and the phase spectra defined by the complex wavelet transform. Since we assume $\theta_{1k}(t)=0$, $\theta_{k}(t)=\theta_{2k}(t)$, we must know the input phase $\theta_k(t).$The input-phase $\theta_k(t)$ is derived by applying three physical constraints related to regularities (ii) and (iv) as shown below[5].

1.

Gradualness of change

This constraint is dA_k(t)/dt=C_k,R(t), where C_k,R(t) is an Rth-order differentiable polynomial. By putting dA_k(t)/dt=C_k,R(t) into equation (7), and solving the resulting linear differential equation, we obtain

$$\theta_k(t)=\arctan\left(\frac{S_k(t)\sin\phi_k(t)}{S_k(t)\cos\phi_k(t)+C_k(t)}\right),$$ (9)

where unknown function C_k(t) is $-\int C_{k,R}(t)dt +C_{k,0}$. In order to determine C_k(t), we estimate C_k(t) using the Kalman filter.

2.

Smoothness

This constraint, the smoothness for A_k(t), is a function of the estimated C_k(t). By considering the relationship between A_k(t) and C_k(t) from Eqs. (7) and (9), we can interpret the smoothness for A_k(t) in order to determine the smoothest C_k(t). Therefore, by calculating the candidates of C_k(t) interpolated using the spline function within the estimated error, and then by calculating a correct solution from the candidates of C_k(t), the smoothest A_k(t) can be determined uniquely.

3.

Changes taken in an acoustic event

This constraint is

$$\frac{A_k(t)}{\Vert A_k(t)\Vert} \approx \frac{A_\ell(t)}{\Vert A_\ell(t)\Vert}, k\not=\ell.$$ (10)

With this constraint, $\theta_k(t)$ is determined when the correlation between A_k(t) and $A_{\ell}(t)$ becomes maximum at any C_k(t) within the estimated error-region.