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The backward filter defined the measurement and time updates eqs

12 1


issue of state estimation pertaining to the future data, we use backward filtering, which starts at the final time N and runs backwards. Let ^xf kand ^xb kdenote the state estimates obtained from the forward and backward recursions, respectively. Given these two estimates, the next issue to be considered is how to combine them into an overall smoothed estimate ^xk, which accounts for data over the entire time interval. Note that the symbol ^xk used for the smoothed estimate in this section is not to be confused with the filtered (i.e., a posteriori) estimate used in Section 1.3.

Sk ¼ ½Pb k��1;

S�k¼ ½Pb�k��1;


^zk ¼ ½Pb k��1^xb k¼ Sk^xb k;

^z�k¼ ½Pb�k��1^xb�¼ S�k^xb�k:

^zk ¼ ^z�kþ HT kR�1 kyk;


^z�k¼ FT kþ1;kðI � Gb kÞ^zkþ1;

where Gb kis the Kalman gain for backward filtering and Q�1

� The forward a posteriori estimate ^xf kby operating the Kalman filter on data yj for 0 < j � k.

� The backward a priori estimate ^xb� by operating the information filter on data yj for k < j � N.

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the backward filter defined the measurement and ti
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