Adaptive Filters

Summary and Notes

The chapters in this part describe several efficient implementations of RLS.

SUMMARY OF MAIN RESULTS

1. By exploiting the fact that regressors have shift structure, the computational cost of RLS can be reduced from O(M²) to O(M) operations per iteration.
2. The key insight is to realize that the gain vector gi, and the conversion factor γ(i), can be time-updated without requiring explicit evaluation of the matrix P_i. Instead, only the low- rank difference
   

   needs to be time-updated.

3. Interestingly enough, the regularization matrix Π can be chosen so as to enforce a rank-two difference right at the first iteration. Once this is done, the rank-two property preserves itself over time.
4. Four efficient implementations were described in this part: (a) fast array filter, (b) fast transver sal filter (FTF), (c) fast a posteriori error sequential technique (FAEST), and (d) fast Kalman filter. The array form is the simplest to derive and to describe; it also shows reasonable ro bustness in finite-precision implementation, as illustrated in the computer project at the end of this part. The FTF and FAEST filters are the most efficient but tend to suffer from numerical difficulties. This is because many of the equalities used to derive them tend to break down in finite precision. The fast Kalman filter is the oldest among them.
5. The derivation of FTF, FAEST, and fast Kalman is based ...