Grouping constraints

For the fundamental frequency F₀(t) in each small segment, the constraint of harmonicity, and the constraint of common onset and offset are implemented as follows.

First, from constraint 2, the channel number $\ell$ of $X_\ell(t)$, in which the harmonic components exist in the output of the $\ell$-th channel, is determined by

$$\ell=\frac{K}{2}-\left\lceil \frac{\log(n\cdot F_0(t)/f_0)}{\log\alpha} \right\rceil,\quad n=1,2,\cdots, N_{F_0},$$ (25)

where $\alpha$ is the scale parameter shown in Table II and $\lceil\cdot\rceil$ is the ceil symbol, meaning the approximation of the closest integer value toward positive infinity.

Next, from constraint 3, let the onset and offset of the fundamental component, $T_{\rm {S}}$ and $T_{\rm {E}}$, be $T_{\rm {S}}=T_{h-1}$ and $T_{\rm {E}}=T_h$, respectively. Moreover, we assume that $\Delta T_{\rm {S}}=50$ msec and $\Delta T_{\rm {E}}=100$ msec. Here $\Delta T_{\rm {S}}$ is taken from the result of a psychoacoustical experiment on the synchronism of onset [Kashino and Tanaka1994].

In this paper, onset $T_{k,\rm {on}}$ and offset $T_{k,\rm {off}}$ in X_k(t) are determined as follows.

1.

Onset $T_{k,\rm {on}}$ is determined by the nearest maximum point of $\vert\frac{d\phi_k(t)}{dt}\vert$ (within 25 ms) to the maximum point of $\vert\frac{dS_k(t)}{dt}\vert$.

2.

Offset $T_{k,\rm {off}}$ is determined by the nearest maximum point of $\vert\frac{d\phi_k(t)}{dt}\vert$ (within 25 ms) to the minimum point of $\vert\frac{dS_k(t)}{dt}\vert$.

Here, constraint 2 acts on the signal component of f₁(t) in the log-frequency domain, and constraint 3 acts the signal component of f₁(t) in the time domain.