Annual Report 2000 - WIT
Annual Report 2000 - WIT
Annual Report 2000 - WIT
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g . The minimization of ª ¾ Í“Î <br />
$<br />
¿<br />
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–<br />
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À<br />
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265<br />
have the same importance as that of the primary field has along the trace (see Figure 1<br />
and 2).<br />
The WHL filter governing equations is a<br />
¾ —–<br />
natural stationary model, and the filter<br />
is the unknown time-invariant operator that is constrained —–<br />
to satisfy a desired output<br />
through the commonly referred to as the Wiener-Hopf ¿ integral equation. We<br />
continue with the equations in the convenient canonic forms, with a uniform discretization<br />
as already established in the Goupillaud model. The criteria<br />
<br />
for<br />
g<br />
the<br />
<br />
filter<br />
g<br />
is<br />
<br />
the<br />
g g<br />
minimization of the variance error, (desired signal)<br />
ç<br />
(real output) in time domain:<br />
<br />
and À<br />
g<br />
, between ¿<br />
ýÁÀ<br />
¯ÇÆ<br />
Ä Å<br />
ª ¾<br />
g<br />
g<br />
è (5)<br />
± ùœÊ Ë Ì<br />
¥ÃÂ<br />
The filter real output is simply:<br />
results in the general WHL equation:<br />
g<br />
¾ g<br />
¾ g<br />
QÐ<br />
QÐ<br />
è (6)<br />
¢3ÏqÏ<br />
ýÑ¢ÒšÏ<br />
The solution defines the coefficients g which depends on what kind of operation<br />
is to be intended for, and accomplished by a priori conditions.<br />
¾ è represents the<br />
theoretical autocorrelation of the input, and<br />
¢:ÏqÏ<br />
è is the theoretical stochastic unilat-<br />
¢ÒšÏ<br />
eral crosscorrelation between the desired and the observed signals. The filter quality<br />
is here also measured by the formula of the normalized minimum error given by the<br />
summation:<br />
è (7)<br />
ªXÓ<br />
Þ<br />
g<br />
¾ g<br />
š§<br />
ý©´<br />
¢ÒšÏ<br />
¤ ¢3ÏqÏ<br />
6 -<br />
The KBC filter governing equations in a natural non-stationary model, and the data<br />
window does not satisfy the principles underlined by the convolution integral. For this<br />
reason, the equation is rewritten in the form of a moving average according to the<br />
commonly referred to as the Wiener-Kolmolgorov problem, and it is expressed by the<br />
matrix integral equation:<br />
Ô–<br />
Ô–<br />
Ô–<br />
—–<br />
—–<br />
¢ ÒšÏ<br />
ý×Ö<br />
¢ ÏqÏ<br />
ý’Ö<br />
q Ú<br />
ÚÛ<br />
(8)<br />
,¿<br />
a¿<br />
#Õ<br />
bÚ<br />
Ú€#Õ<br />
Ú€<br />
Ú€#Õ<br />
bÚ<br />
Ô–<br />
Ô–<br />
where is the actual output, and ¾ is the corresponding desired optimum<br />
À<br />
time-variant operator. The criterion used is the minimization of the residue covariance<br />
° bÚ<br />
expressed as:<br />
è (9)<br />
ÙØ<br />
ÙØ<br />
ª ¾<br />
—–<br />
Ô–<br />
jß<br />
¥ÝÜQÞ À<br />
ùaà
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