Adaptive Filters

Summary and Notes

The chapters in this part developed three array variants for recursive least-squares solutions: inverse QR, QR, and extended QR (also known as square-root RLS, square-root information RLS, and extended square-root information RLS, respectively).

SUMMARY OF MAIN RESULTS

1. Array methods are based on transforming a pre-array of numbers into a post-array of num bers by means of unitary transformations. Such transformations are easy to implement as a sequence of elementary rotations or reflections (e.g., as Givens rotations or Householder reflections).
2. Array methods are self-contained in that quantities that are needed to form the pre-array are propagated in the post-array.
3. The variables involved in array methods are usually square-root factors whose entries assume values within smaller dynamic ranges. As a result, array methods are more reliable and have better numerical properties in finite-precision arithmetic than a direct RLS implementation — see, e.g., the computer project at the end of this part.
4. The array variants developed in this part have the same computational complexity as RLS, namely, O(M²) operations per iteration.

BIBLIOGRAPHIC NOTES

QR methods. It is generally accepted that the QR method for solving and updating least-squares solutions is among the most reliable procedures for finite word-length implementations. The origin of the method can be traced back to the works of Householder (1953, pp. 72–73), Golub (1965), and Businger and Golub (1965) ...