Tracking Ill-Conditioning for the RLS-Lattice Algorithms

The numerical performance of lattice-based adaptive signal processing algorithms is shown to involve the conditioning of a 2 × 2 matrix whose off-diagonal elements contain reflection coefficients. Degraded algorithmic performance for the a posteriori Recursive Least Squares Lattice (RLSL) is shown to be attributed to the ill-conditioning of this matrix. Theoretical results are given which may be used to separate the conditioning of the underlying problem from issues concerning algorithmic stability. Although our results are not restricted to the all-pole case, for simplicity we make use of this well-understood example since the condition number of the autocorrelation matrix will become arbitrarily close to singularity as the poles of an all-pole filter approach the unit circle. For a second-order prediction problem, four case studies of varying conditioning are provided which demonstrate the appropriateness of the theoretical bounds which analytically describe the sensitivity to perturbations in the residual update recursions. In doing so, this paper illustrates the use of numerical linear algebra analysis techniques to better understand the numerical performance of algorithms in signal processing as well as emphasizing that numerical performance is a function of the problem’s conditioning as well as algorithmic stability.