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By Randall L. Eubank

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2 Recursions for L For the developments in this and subsequent sections it will be convenient to introduce a final piece of notation for the so-called Kalman gain matrices. These matrices arise naturally in formulae for both L and L−1 and, not surprisingly, appear in various signal and state vector prediction formulae that we will encounter in the next chapter. They are defined by K(t) = F (t)S(t|t − 1)H T (t)R −1 (t), t = 1, . , n. 17) via the relation M (t) = F (t) − K(t)H(t). With this notational preliminary, we can now give a detailed description of the form of L.

F (n−2)  F (n − 2) · · · F (2)S(2|1)H T (2) ×F (n−1)  F (n − 1) · · · F (2)S(2|1)H T (2) By extrapolating from what we have observed in these special cases we can determine that the diagonal and below diagonal blocks of ΣXε can be computed on a row-by-row basis by simply “updating” entries from previous rows through pre-multiplication by an appropriate F (·) matrix. That is, we can use the elements of the tth block row of ΣXε to evaluate all the (strictly) below diagonal blocks of the (t + 1)st row block through premultiplication by F (t).

T. 3. This two-stage approach can be perfectly satisfactory and we will see this reflected in some of the forward and backward recursions for computing signal and state vector estimators in Chapters 4 and 5. However, there are also cases where it would be more convenient to have the entire matrix ΣXε in hand after a single forward pass. One way to evaluate the entirety of ΣXε in a single recursion is to work forward from the upper left hand corner of the matrix in an L-shaped pattern. 2. Then the above diagonal blocks for the (t + 1)st column block are evaluated.

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