Calculation of $\theta_k(t)$

In this paper, we assume $\theta_{1k}(t)=0$ and $\theta_{k}(t)=\theta_{2k}(t)$. Therefore, we must know the input phase $\theta_k(t)$. The input phase $\theta_k(t)$ can be determined by applying three physical constraints derived from regularities(ii) and (iv) as follows.

Firstly, we use regularity (ii). This regularity means that ``a single sound tends to change its properties smoothly and slowly (gradualness of change)''. We consider this regularity as the following physical constraint, to apply it to the amplitude envelope A_k(t).

Physical constraint 1  

Temporal differentiation of the amplitude envelope A_k(t) must be represented by Rth-order differentiable polynomial C_k,R(t) as follows:

$$\frac{dA_k(t)}{dt}=C_{k,R}(t).$$ (18)

$\Box$

A general solution of the input phase $\theta_k(t)$ is determined by solving the linear differential equation obtained by applying Physical constraint 1 to Eq. ().

Lemma 2   A general solution of the input phase $\theta_k(t)$ is determined by

$$\theta_k(t)=\arctan\left(\frac{S_k(t)\sin\phi_k(t)}{S_k(t)\cos\phi_k(t)+C_k(t)}\right),$$ (19)

where $C_k(t)=-\int C_{k,R}(t)dt+C_{k,0}$. C_k(t) is called the ``unknown function''. $\Box$

Therefore, if C_k(t) is determined, then $\theta_k(t)$ is uniquely determined by Eq. (). In this paper, we estimate C_k(t) using the Kalman filter.