Separation block

In this paper, in order to reduce the computational cost for estimating C_k,R(t) and D_k,R(t), we assumes that C_k,R(t) is a linear (R=1) polynomial ( dA_k(t)/dt=C_k,1(t)) and D_k,R(t) is zero ( $d\theta_{1k}(t)/dt=D_{k,0}=0$) in constraint (i). In this assumption, A_k(t) which can be allowed to undergo a temporal change in region, constrains the second-order polynomial ( $A_k(t)=\int C_{k,1}(t)dt+C_{k,0}$). Moreover, $\theta_{1k}(t)$, which is constrained (i.e. $\theta_{1k}(t)=D_{k,0}$), cannot be allowed to temporarily change. Here, if the number of channels K is very large, each frequency of the signal component that passed through the channel approximately coincides with the center frequency of each channel. Even if the above condition is false, its frequency difference can be represented by D_k,0.

In the segment T_h-T_h-1, C_k,1(t) and D_k,0 are determined by the following steps. First, let D_k,0 be any value within $-\pi/2 \leq D_{k,0} \leq \pi/2$. Next, using the Kalman filter, determine the estimated region, $\hat{C}_{k,0}(t)-P_k(t) \leq C_{k,1}(t) \leq \hat{C}_{k,0}(t)+P_k(t)$, where $\hat{C}_{k,0}(t)$ is the estimated value and P_k(t) is the estimated error. Then select candidates of C_k,1(t) using the spline interpolation in the estimated error region. Next, determine C_k,1(t) using

$$\hat{C}_{k,1}=\mathop{\arg\max}_{\hat{C}_{k,0}-P_k\leq C_{k,1... ...\vert\hat{A}_k\vert\vert \vert\vert\hat{\hat{A}}_k\vert\vert},$$ (18)

where $\hat{A}_k(t)$ is obtained by the spline interpolation and $\hat{\hat{A}}_k(t)$ is determined in across-channel which is satisfied constraint (iii). Finally, determine D_k,0 using

$$\hat{D}_{k,0}=\mathop{\arg\max}_{-\pi/2 \leq D_{k,0} \leq \pi... ...\vert\hat{A}_k\vert\vert \vert\vert\hat{\hat{A}}_k\vert\vert}.$$ (19)

Then, determining $\theta_{1k}(t)=\hat{D}_{k,0}$ and $\theta_{2k}(t)=\theta_k(t)+\theta_{1k}(t)$, we can determine A_k(t) and B_k(t) from Eqs. () and ().