The KALMAN filter is derived in the way outlined in ref. /2/. 
Adopting the statement derived there from the Fokker-Planck 
equation for the development of the first two cumulants of the 
state distribution, we can generally state for linear systems: 
rw lt) (7) 
il 
il 
Rb). zi ok). dedil SEGA, m (8) 
in which z(t) is the first and K(t) the second cumulant and 
A the system matrix. Substitution of our model in these evo- 
lution equation with 
2(t) = A(t) and their solution supply the equations for the 
evolution of the first two cumulants with an unobserved pro- 
cess. If the process is sampled, as it is the case here, the 
integrals are computed only between two sampling points. The 
solution is obtained in terms of the transition from the 
so-called a-posteriori to the a priori value of the cunulants 
with sampling period I . For the first cumulant we obtain, 
with À (0) = A 4p and A(T) = Aa» 
and for the second cumulant, with K(0) = Ky 1p and K(T) = Kips 
; 2 
Kyp= Ky_qp+ D7 T (10) 
The observation of the process via the linear observation 
equation (4) permits us to state the function 
5 (44/ 1n.) which in case of Gaussian observation noise 
p j , 
is a Gaussian distribution. It is termed a Likelihood function 
and makes it possible that the a-priori values computed in the 
unobserved system motion can be improved in accordance with 
the information obtained by observation. Ref. /2/ contains 
a derivation of the transition from a-priori to a-posteriori 
values. Here it may suffice to state the results for our case: 
  
Bip = Dy + mee (y,~2,,) (11) 
jP JA w J JA 
N, + Kia 
K^ 
’ JA 
T P 
Nt LST 
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