Candidate selection of C_k,1(t) using the spline interpolation

In order to determined whether A_k(t) satisfied Eq. (15), consider the selection of candidates for A_k(t). Suppose that A_k^(R+1)(t) is the instantaneous amplitude of f₁(t) given by any C_k,R(t), and $\tau_1, \tau_2, \cdots, \tau_i$ are within the opened-duration (t_a,t_b), where $t_a<\tau_1< \cdots < \tau_i < t_b$. In addition, suppose that A_k,i:=A_k^(R+1)(t_i) is the value of the instantaneous amplitude at time $\tau_i$.

To estimate C_k,1(t), where R=1, that satisfies Eq. (15) we interpolate A_k^(R+1)(t), where R=1 and, A_k^(R+1)(t)=A_k,i, $i=1,2,\cdots, I$ in [t_a, t_b]. According to constraint 4, the smoothest interpolation function is the (2R+1)th-order spline function. This spline function is unique [de Boor1978].

As shown in Fig. 7, first we determine candidates of C_k,1(t) using the spline function within the estimated error region: $\hat{C}_{k,0}(t)-P_k(t) \leq C_{k,1}(t) \leq \hat{C}_{k,0}(t)+P_k(t)$. Then, select a correct solution from the candidates of C_k,1(t), we can uniquely determine the smoothest A_k(t) from C_k,1(t).

In this paper, we use the cubic spline function (2R+1), where R=1. The interpolated region is from t_a=T_h-1 to t_b=T_h. The interpolated interval is $\Delta \tau=15\times (2\pi/\omega_k)\Delta t$. Therefore, $I=\lceil (t_b-t_a)/\Delta \tau \rceil$.