Efficient multichannel FIR filtering using a single step versatile order recursive algorithm

Abstract

Abstract In this paper two highly efficient, order recursive algorithms for least squares multichannel FIR filtering and multivariable system identification are developed. Multichannel FIR filters with different number of delay elements for each input channel are allowed. The first algorithm uses two terms Levinson type recursions. The later utilizes Schur type formulas for updating the pertinent parameters, thus being suitable for parallel implementation. Multichannel FIR filters are described by a multi-index [ m 1 , m 2 ,…, m κ ] where m i equals the number of delay elements associated with the i -input channel, i = 1, 2, …, k . The novel feature of the proposed algorithms is that they employ updates of the form [ m 1 , m 2 , …, m i , …, m k ] → [ m 1 , m 2 , …, m i + 1, …, m k ]. Therefore, and in contrast to existing methods, they offer the greatest possible maneuverability in the index space. This flexibility can be taken into advantage when the true index is not known, except from being an element of a set. Computationally efficient paths that search for the true index are described. If the true filter order [ p 1 , p 2 , …, p k ] is known, the filter coefficients are computed at P = ( p 1 + p 2 + … + p k ) steps, by a repetitive application of single step recursions. The computational complexity of the method is O( kP 2 ), while execution time could be reduced to O(1) or O( P ) if the Schur type algorithm is implemented in a parallel processing environment on a rectangular or on a linear array, respectively. The final filter can be approached by P !/( p 1 ! p 2 !… p k !) distinct order updating paths, each time passing through different lower dimension filters. This feature can be utilized for the efficient determination of the order of a multichannel process, accelerating the exhaustive searching procedure required by most of the order determination criteria. Finally, the mean squared error is considered with potential applications to the optimal two-dimensional (2-D) FIR filtering and 2-D system identification.