Acknowledgments

The authors wish to thank H. Kawahara of Wakayama University and T. Irino of ATR-HIP for helpful suggestions about this work, and Y. Ichinose and S. Katagiri of ATR-HIP for the arrangement. This work was supported by a Grant-in-Aid for science research from the Ministry of Education (Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists and No. 10680374) and by CREST (Core Research for Evolutional Science and Technology) of Japan Science and Technology Corporation (JST).

 

 

Table: Definitions of symbols for Kalman filtering

Symbol	Est. of Bk(t)	Est. of $\theta_{2k}(t)$	Est. of Ck,0(t)	Est. of Dk,0(t)
Observed signal ${\bf {y}}_m$	Xk(tm)	$\exp(j\phi_k(t_m)$	Xk(tm)	$\exp(j\phi_k(t_m))$
State variable ${\bf {x}}_m$	Bk(tm)	$\exp(j\theta_{2k}(t))$	Ck(tm)	$\exp(j D_{k}(t_m))$
Observed noise ${\bf {v}}_m$	X1,k(tm)	X1,k(tm)/Sk(tm)	X2,k(tm)	X2,k(tm)/Sk(tm)
System noise ${\bf {w}}_m$	wm	wm	wm	wm
State transition matrix ${\bf {F}}_m$	$1+\frac{B_k(t_m)-B_k(t_{m-1})}{B_k(t_m)}$	$1+\frac{\theta_{2k}(t_m)-\theta_{2k}(t_{m-1})}{\theta_{2k}(t_m)}$	$1+\frac{C_k(t_m)-C_k(t_{m-1})}{C_k(t_m)}$	$1+\frac{D_k(t_m)-D_k(t_{m-1})}{D_k(t_m)}$
Observation matrix ${\bf {H}}_m$	$\exp(j\omega_k t_m)$	Bk(tm)/Sk(tm)	$\exp(j\omega_k t_m)$	$\hat{C}_{k}(t_m)/S_k(t_m)$
Driving matrix ${\bf {G}}_m$	1	1	1	1
Initial value $\hat{\bf {x}}_{0\vert-1}$	${\rm {std}}(S_k(t))^2$	${\rm {cov}}(\exp(j\phi_k(t)))$	${\rm {std}}(\hat{A}_k(t))^2$	${\rm {cov}}(\exp(j\hat{\theta}_{1k}(t)))$
Initial value $\hat{\bf {\Sigma}}_{0\vert-1}$	Sk(t0)	$\exp(j\phi_k(t_m))$	$\hat{A}_k(t_0)$	$\exp(j\hat{\theta}_{1k}(t_0))$