Adaptive Filters

CHAPTER 33 Norm and Angle Preservation

Array methods are powerful algorithmic variants that are theoretically equivalent to the recursive least-squares algorithm, but they nevertheless perform the required computations in a more reliable manner. In these array forms, an algorithm is described not as an ex plicit set of equations, but as a sequence of elementary operations on arrays of numbers (or matrices). Usually, a pre-array of numbers has to be triangularized by a sequence of elementary rotations in order to yield a post-array of numbers. The quantities needed to form the next pre-array are read off from the entries of the post-array, and the procedure can be repeated. The explicit forms of the rotation matrices are not needed in most cases, and they can be implemented in a variety of well-known ways, e.g., as a sequence of ele mentary circular or hyperbolic rotations. The purpose of this chapter is to develop several array-based methods for RLS filtering. In order to motivate such array algorithms, we shall first consider a simple (yet contrived) example that helps highlight some important issues.

33.1 SOME DIFFICULTIES

Thus, consider the update equation (30.12) for the variable P_N in the RLS algorithm, and assume that all variables are real and scalar-valued (and, hence, we shall write {u(N), P(N)} instead of {U_N, P_N})’ Assume further that at some iteration n_o, especially during the ini tial stages of adaptation where P(N) is more likely to assume large values, u(n