Assumptions and constraints

In this paper, it is assumed that the desired signal f₁(t) is a harmonic tone, consisting of the fundamental frequency F₀(t) and the harmonic components, which are multiplies of F₀(t). The proposed model segregates the desired signal from the mixed signal by constraining the temporal differentiation of the instantaneous amplitude A_k(t), the instantaneous phase $\theta_{1k}(t)$, and the fundamental frequency F₀(t). Constraints used in this model are shown in Table 1. Constraint (ii) for the above parameters gives dA_k(t)/dt=C_k,R(t), $d\theta_{1k(t)}/dt=D_{k,R}(t)$, and dF₀(t)/dt=E_0,R(t), where C_k,R(t), D_k,R(t), and E_0,R(t) are R-th-order differentiable piecewise polynomials (using Table 1 (ii)). Then, substituting dA_k(t)/dt=C_k,R(t) into Eq. (), we get the linear differential equation of the input phase difference $\theta_k(t)=\theta_{2k}(t)-\theta_{1k}(t)$. By solving this equation, a general solution is determined by

$$\theta_k(t)=\arctan\left(\frac{S_k(t)\sin(\phi_k(t)-\theta_{1k}(t))}{S_k(t)\cos(\phi_k(t)-\theta_{1k}(t))+C_k(t)}\right),$$ (4)

where $C_k(t)=-\int C_{k,R}(t)dt-C_{k,0}=-A_k(t)$.