Calculation of input phase $\theta_k(t)$

Input phase $\theta_k(t)$ can be determined by applying three physical constraints: (i) gradualness of change, (ii) continuity (temporal proximity), and (iii) changes in an acoustic event. In particular, constraints (i) and (iii) are regularity (2) and (4) proposed by Bregman.

We will first apply regularity (i). This regularity states that ``a single sound tends to change its properties smoothly and slowly (gradualness of change)''[2]. We will consider this in the following physical constraint.

Constraint 1 (gradualness change)  

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)

$\Box$

Applying Physical constraint 1 into Eq. (), a linear differential equation is obtained as follows:

$$y'(t)+\frac{P'(t)}{P(t)}y(t)=\frac{Q'(t)-C_{k,R}(t)}{P(t)},$$ (19)

where $P(t)=S_k(t)\sin\phi_k(t)$, $Q(t)=S_k(t)\cos\phi_k(t)$, $y(t)=\cot\theta_k(t)$. $\theta_k(t)$ can be determined by solving the linear differential equation ().

Lemma 2   By solving the linear differential equation, a general solution of the input phase $\theta_k(t)$ is determined by

$$\theta_k(t)=\arctan\left(\frac{S_k(t)\sin\phi_k(t)}{S_k(t)\cos\phi_k(t)+C_k(t)}\right),$$ (20)

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

If C_k(t) is determined, then $\theta_k(t)$ is uniquely determined by the above equation. Although it is possible to estimate the coefficients $C_{k,r},r=0, 1, \cdots, R$ by considering as an optimization problem, we will assume, in order to reduce the computational costs that in small segment $\Delta t$, C_k,R(t)=C_k,0. As this point, Eq. () is equivalent to dA_k(t)/dt=0, and amplitude envelope A_k(t) does not fluctuate in small segment $\Delta t$.

Next, we will use regularity (ii) to segregate each small segment $\Delta t$. Regularity (ii) means that ``each physical parameter must retain temporal proximity in the bound (t=T_r) between pre-segment ( $T_r-\Delta t \leq t < T_r$) and post-segment ( $T_r \leq t < T_r+\Delta t$).'' In order to apply this regularities to physical parameters, it is considered in the following physical constraint.

Constraint 2 (proximity)   In the bound (t=T_r) between pre-segment and post-segment, each physical parameter A_k(t), B_k(t), and $\theta_k(t)$ must be connected within $\Delta A$, $\Delta B$, $\Delta \theta$, respectively. That is,

$$\left\{ \begin{array}{l} \left\vert A_k(T_r+0) - A_k(T_r-0)\r... ...ta_k(T_r-0)\right\vert \leq \Delta \theta . \end{array}\right.$$ (21)

$\Box$

From Eqs. (), () and (), amplitude envelopes A_k(t) and B_k(t), and input phase $\theta_k(t)$ are functions of the unknown coefficient. Therefore, by considering the above relationships, we can interpret physical constraint 2 in order to determine C_k,0, which is restricted within

$$C_{k,\alpha} \leq C_{k,0} \leq C_{k,\beta},$$ (22)

where $C_{k,\alpha}$ and $C_{k,\beta}$ are the upper limit and the lower limit C_k,0 in the bound between the two segments.

Finally, we will apply regularity (iii). This regularity states that ``many changes take place in an acoustic event that affect all the components of the resulting sound in the same way and at the same time'' [2]. This regularity is considered the following physical constraint.

Constraint 3 (Changes in an acoustic event)  

The amplitude envelope B_k(t) must be highly correlated with the amplitude envelope $B_{k\pm 1}(t)$ obtained by the output of adjacent channel:

$$B_k(t)\approx B_{k\pm 1}(t)$$ (23)

$\Box$

Since an amplitude envelope B_k(t) is a function of C_k,0 from Eqs. () and (), and let $\hat{B}_k(t)$ be an amplitude envelope B_k(t) determined by any C_k,0. In addition, a correlation between amplitude envelopes is defined by

$${\rm {Corr}}(\hat{B}_k,\hat{\hat{B}}_k)= \frac{\langle \hat{B... ...\vert\hat{B}_k\vert\vert \vert\vert\hat{\hat{B}}_k\vert\vert},$$ (24)

where $\hat{\hat{B}}_k(t)=(\hat{B}_{k+1}(t)+\hat{B}_{k-1}(t))/2$. Let T_B be an arbitrary past time, where integral region is $T_B\leq t < T_r+\Delta t$. $\hat{B}_{k\pm 1}(t)$ can be determined using $\hat{A}_k(t)$ and $\hat{B}_k(t)$ obtained by C_k,0, from Fig. . Here, physical constraint 3 can be considered as selecting C_k,0 when correlation () is a maximum. That is, by selecting C_k,0 as

$$\mathop{\max}_{C_{k,\alpha} \leq C_{k,0} \leq C_{k,\beta}} {\rm {Corr}}(\hat{B}_k,\hat{\hat{B}}_k),$$ (25)

the input phase $\theta_k(t)$ can be uniquely determined from Eq. ().