Modular and Numerically Stable Multichannel Fast Transversal Filters

Abstract

In this paper, we present scalar implementations of multichannel fast Recursive Least-Squares algorithms in transversal filter form (so-called FTF algorithms). The point is that by processing the different channels sequentially, ie one at a time, the processing of any channel reduces to that of the single-channel algorithm. This sequential processing decomposes the multichannel algorithm into a set of intertwined single-channel algorithms. Geometrically, this corresponds to a modified Gram-Schmidt orthogonalization of multichannel error vectors. Algebraically, this technique corresponds to matrix triangularization of multichannel error covarianre matrices and converts matrix operations into a regular set of scalar operations.

Algorithm structures that are amenable to VLSI implementation on arrays of parallel processors naturally follow from our approach. Numerically, the resulting algorithm benefits from the advantages of triangularization techniques in block-processing, which are a well-known part of Kalman filtering expertise. Furthermore, recently introduced stablization techniques for proper control of the propagation of numerical errors in the update recursions are also incorporated.

Indexing terms:

• Multichannel fast transversal filter
• Triangularization techniques