A new generalized recursion for the fast computation of the Kalman gain to solve the covariance equations

A New Multichannel Recursive Least Squares Algorithm for Very Robust and Efficient Adaptive Filtering

“…Notice, that, although each one of the two formulae (3.28) and (3.29) is, formally, very similar to the one used for updating the alternative Kalman gain in the FAEST algorithm and its multichannel counterpart [3][4][5]11], these relations are in essence decisively different. This is so because, first, u m+1 n + 1 and v m+1 n + 1 are treated in this paper as a strongly interconnected pair, a fact that directly leads to parallelizable algorithms, and second, both A m n and B m n are updated here by means of (3.14) and (3.17), which are substantially different than the corresponding multichannel recursions used so far.…”

“…Order updating or rank displacement is a very powerful concept which was first introduced in [2] and subsequently used in [1,3,4,16] for the derivation of fast algorithms. In deriving order updating relations for the various useful quantities the key point is the observation that the shift invariance of the multichannel input signal permits the following partitioning…”

Recursive Fixed-Order Covariance Least-Squares Algorithms

“…Consider again the data vectors Xi and Y i defined by (23). Suppose we wish to compute the linear least-squares estimate of Y i given Xi in terms of a least-squares estimate that does not use the most distant or past values Yio and Xio' Clearly, this problem can be solved in exactly the same fashion as the time-update problem stated at the beginning of the last section.…”

“…23 The gain g(i) is the only variable needed at the next iteration of the fixed-order algorithm and can be computed by first using (96j) to calculate fJ(i) and then using (96m) to solve for g(i).…”