An Order-Recursive Lattice Algorithm for H^∞ Adaptive Filtering

Abstract

We present an order-recursive lattice algorithm for H ∞ adaptive filtering. The standard H ∞ filter algorithm is expressed in an alternative formula that is closed in a relaxed sense. On the basis of this formula, the algorithm is reformulated as an indefinite least-squares problem that can be solved by a two-dimensional block recursive least-squares (RLS) algorithm. Then an H ∞ lattice filter is derived by transforming the block RLS algorithm into its multivariable least-squares lattice form. The resulting computational complexity is still proportional to the filter order, although 2 × 2 matrix computations are involved. The equivalence between the standard H ∞ filter and its lattice version is verified by numerical experiments. Keywords: adaptive filtering, H ∞ filter, lattice filter This paper is organized as follows. Section 2 introduces a typical H ∞ adaptive filtering problem and its recursive solution. Section 3 derives a pseudo-closed-form expression for the standard solution, which is used to reformulate H ∞ adaptive filtering as an indefinite least-squares problem. In Sec. 4, the indefinite least-squares problem is solved by employing a two-dimensional block RLS algo-

A QRD-Based Array Lattice Form for H^∞ Adaptive Filtering

“…For such lters, there exits a standard algorithm whose complexity is on the order of M 2 [5]. Also, an orderrecursive lattice form is presented in [8].…”

“…Particularly in the case of leastsquares lattice (LSL) lters, although several array forms have been proposed, a QRD-based algorithm that employs QR decomposition (QRD) for triangularization would be the most popular one. For this reason, in this paper, we also attempt to derive such a QRD-based array form for the H ∞ lattice lter presented in [8].…”

“…This paper is organized as follows. In section 2, we formulate the H ∞ adaptive ltering problem and review the lattice form solution derived in [8]. In Section 3, we introduce a decomposition for a matrix variable related to the conversion factor and thereby transform the lattice algorithm into an array form.…”

“…Section 4 presents several numerical examples to demonstrate the numerical stability of the H ∞ array lattice lter. This research note is an addendum to [8] and readers are invited to refer to it. In particular, an error in [8] is corrected in Appendix 1.…”

“…This research note is an addendum to [8] and readers are invited to refer to it. In particular, an error in [8] is corrected in Appendix 1.…”