Calculation of the four physical parameters

The amplitude envelope S_k(t) and phase $\phi_k(t)$ of X_k(t) are determined using the amplitude and phase spectra. Since $\theta_{1k}(t)=0$, we must find the input phase $\theta_{2k}(t)$. It can be determined by applying three physical constraints, derived from regularities (ii) and (iv), as shown below[7].

1.

Gradualness of change (slowness)

Regularity (ii) means that ``a single sound tends to change its properties smoothly and slowly (gradualness of change)'' [6]. First constraint, considered as ``slowness'', is dA_k(t)/dt=C_k,R(t), where C_k,R(t) is an R-th-order differentiable polynomial. By applying it to Eq. (), and solving the resulting linear differential equation, we obtain

$$\theta_{2k}(t)=\arctan\left(\frac{S_k(t)\sin\phi_k(t)}{S_k(t)\cos\phi_k(t)+C_k(t)}\right),$$ (7)

where $C_k(t)=-\int C_{k,R}(t)dt + C_{k,0}$. Here, we assume that, in small segment $\Delta t$, C_k,R(t)=C_k,0.

2.

Gradualness of change (smoothness)

Second constraint, considered as ``smoothness'', is that, in the bound (t=T_r) between pre-segment ( $T_r-\Delta t \leq t < T_r$) and post-segment ( $T_r \leq t < T_r+\Delta t$),

 

$$\left\vert A_k(T_r+0) - A_k(T_r-0)\right\vert \leq \Delta A$$ (8)

$$\left\vert B_k(T_r+0) - B_k(T_r-0)\right\vert \leq \Delta B$$ (9)

$$\left\vert \theta_{2k}(T_r+0) - \theta_{2k}(T_r-0)\right\vert \leq \Delta \theta.$$ (10)

From the above relationships, we can interpret this constraint in order to determine C_k,0 , which is restricted within $C_{k,\alpha} \leq C_{k,0} \leq C_{k,\beta}$, where $C_{k,\alpha}$ and $C_{k,\beta}$ are the upper-limited and lower-limited C_k,0 in the bound between the two segments.

3.

Changes taken in an acoustic event

Regularity (iv) means 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'' [6]. Third constraint, considered as this regularity, is

$$\frac{B_k(t)}{\Vert B_k(t)\Vert}\approx \frac{B_{k\pm \ell}(t)}{\Vert B_{k \pm\ell}(t)\Vert},\qquad \ell=1,2,\cdots ,L.$$ (11)

Here, a masker envelope B_k(t) is a function of C_k,0 from Eqs. () and (). We consider this constraint to select an optimal coefficient C_k,0 using

$$\mathop{\max}_{C_{k,\alpha} \leq C_{k,0} \leq C_{k,\beta}} \... ...\vert\vert \vert\vert\hat{\hat{B}}_k\vert\vert}, \nonumber \\$$

where $\hat{B}_k(t)$ is the masker envelope given by any C_k,0, and

$$\hat{\hat{B}}_k(t)=\frac{1}{2L}\sum_{\ell=-L,\ell \not= 0}^L \frac{\hat{B}_{k+\ell}(t)}{\Vert\hat{B}_{k+\ell}(t)\Vert}.$$ (12)

Hence, the above computational process can be summarized as follows: (a) a general solution of $\theta_{2k}(t)$ is determined using physical constraint 1; (b) candidates of C_k,0 that can uniquely determine $\theta_{2k}(t)$, is determined using physical constraint 2; (c) an optimal C_k,0 is determined using physical constraint 3; and (d) $\theta_{2k}(t)$ can be uniquely determined by the optimal C_k,0.