F₀(t) estimation block

In the proposed model, the fundamental frequency F₀(t) is estimated using Comb filtering on the auditory-motivated filterbank. This Comb filter is defined by

$${\rm{Comb}}(k,\ell)=\left\{\begin{array}{ll} 2, & \omega_k=n\... ...ll, 4\leq n \leq N\\ 0, & \mbox{otherwise} \end{array}\right.$$ (5)

where k and $\ell$ are indices, $\omega_k$ and $\omega_\ell$ are the center frequencies in channels, and N is the number of harmonics of the highest order. Then, $\hat{\ell}$, which corresponds to the channel containing the fundamental wave, is determined by

$$\hat{\ell} = \mathop{\arg \max}_{\ell\leq L}\sum_{k=1}^K {\rm{Comb}}(k,\ell)S_k(t),$$ (6)

where L is the upper-limited search region of $\ell$. The estimated F₀(t) is determined by $\omega_{\hat{\ell}}/2\pi$.

Since the number of channels in the auditory-motivated filterbank is finite, the estimated fundamental frequency F₀(t) takes a discrete value. In addition, the fluctuation of the estimated F₀(t) behaves like a stair shape and the temporal differentiation of F₀(t) is zero at any segment. Therefore, this paper assumes that E_0,R(t)=0 in constraint (ii) for a segment. Here, the above segment is determined using the following equation, as the duration for which the temporal variation of F₀(t) has variance of zero as F₀(t).

$$\frac{1}{T_{h}-T_{h-1}}\int_{T_{h-1}}^{T_h} \left\vert F_0(t)-\overline{F_0(t)}\right\vert^2 dt = 0,$$ (7)

where the length of the segment is T_h-T_h-1. In this paper, let the parameters be N=10 and L=K/4.