Convergent and roundoff error properties of reflection coefficients in adaptive spatial recursive least squares lattice algorithm

The spatial recursive least-squares lattice (RLSL) algorithm is considered and the convergent properties as well as the finite-precision roundoff effects of the reflection coefficients are studied in detail. It is shown that when the forgetting factor λ is set equal to one, the reflection coefficients can converge to a constant with probability one with the rate of convergence being O(1/T ^3/2) where T is the number of terms involved. When O<λ<1, the reflection coefficients converge weakly. Furthermore, the variable of the reflection coefficient is decreased by increasing λ. Thus, under stationary conditions λ should be taken as large as possible, while under nonstationary conditions λ should be taken such that the depth of memory in the system matches the rate of change of nonstationarity. Under a finite-wordlength constraint, λ should be taken to be less than some threshold, which is a function of the number of significant digits, such that the system can continuously adapt. In addition, it was found that the recursive form of computation for the reflection coefficient is better than direct form of computation