Estimation of C_k(t) using the Kalman filter

We formulate the problem of estimating C_k(t) by using the Kalman filter.

A complex representation of the output of the kth channel X_k(t) represented by Eq. () is the wavelet transform given by Eq. () as follows.

 

$$S_k(t)e^{j(\omega_k t+\phi_k(t))}$$ X_k(t) =

$$\tilde{f}(a,b),\quad a=\alpha^{k-\frac{K}{2}},b=t_m,$$ := (20)

where $t_m=m/f_s, m=0,1,\cdots,M$. From Eq. (), this is expressed as the sum of the wavelet transforms of f₁(t) and f₂(t). Hence,

$$\tilde{f}(\alpha^{k-\frac{K}{2}},t_m)=\tilde{f}_1(\alpha^{k-\frac{K}{2}},t_m)+ \tilde{f}_2(\alpha^{k-\frac{K}{2}},t_m)$$ (21)

where

$$\tilde{f}_1(\alpha^{k-\frac{K}{2}},t_m)=A_k(t_m)e^{(j\omega_k t_m+\theta_{1k}(t))}$$ (22)

and

$$\tilde{f}_2(\alpha^{k-\frac{K}{2}},t_m)=B_k(t_m)e^{(j\omega_k t_m+\theta_{2k}(t))}.$$ (23)

On the other hand, from Eqs. () and (), we obtain the following relation.

C_k(t)=-A_k(t). (24)

Suppose that a displacement of C_k(t) in discrete time t_m is represented by

$$C_k(t_{m+1})=C_k(t_m)\Delta C_k+w_m,$$ (25)

where

$$\Delta C_k=1+\frac{C_k(t_m)-C_k(t_{m-1})}{C_k(t_m)\cdot f_s}.$$ (26)

That is, C_k(t_m+1) is represented by C_k(t_m) times $\Delta C_k$, and represented-error w_m follows a white Gaussian probability process with average 0 and variance $\sigma_w$.

   

Figure: Basic system of the Kalman filter.
Figure: Segregation algorithm

file=FIGURE/basicmodel.eps,width=0.47

decompose f(t) into its frequency components using the


wavelet filterbank (wavelet transform) as Eq. ();

for k:=1 to K do

  $\theta_{1k}(t)=0$
and $\theta_{k}(t)=\theta_{2k}(t)$;


determine Sk(t) and $\phi_k(t)$
from Lemma 1;


determine onset $T_{k,{\rm{on}}}$
and offset $T_{k,{\rm{off}}}$;


the segregated duration is $T_{k,{\rm{on}}}\leq t \leq T_{k,{\rm{off}}}$;

  if Physical constraint 4 or 5 is satisfied ${\bf{then}}$


estimate Ck(t) using the Kalman filter;


determine the interpolated duration;


let I be the number of the interpolated samples;

  for i=1 to I do 

determine the candidates for Ck(t), which 

interpolated by the spline function within 
  $\hat{C}_k(t_i)-P_k(t_i) \leq C_k(t_i) \leq \hat{C}_k(t_i)+P_k(t_i)$;


determine $\hat\theta_k(t)$
from Eq. ();


determine $\hat{A}_k(t)$
from Eq. ();


determine $\hat{\hat{A}}_k(t)$
from Eq. ();


determine ${\rm{Corr}}(\hat{A}_k(t),\hat{\hat{A}}_k(t))$
from Eq. ();

  end


determine Ck(t) when ${\rm{Corr}}(\hat{A}_k(t),\hat{\hat{A}}_k(t))$


becomes a maximum within the estimated 
  -error;


determine $\theta_k(t)$
from Eq. ();

  else


set Ak(t)=0, Bk(t)=Sk(t) and $\theta_k(t)=\phi_k(t)$;

  end


determine Ak(t) and Bk(t) from Eqs. () and (); 

determine each frequency components of f1(t) 

and f2(t) from Eqs. () and ();

end


reconstruct $\hat{f}_1(t)$
and $\hat{f}_2(t)$
using the wavelet filterbank 
  (inverse wavelet transform) from Eqs. () and ();