Analysis of Fast Algorithm of Matrix-Vector Multiplication for the Bank of Digital Filters

Abstract

Algorithms of implementation of vector-matrix multiplication are presented, which are intended for application in banks (sets) of digital filters. These algorithms provide significant savings in computational costs over traditional algorithms. At the same time, reduction of computational complexity of algorithms is achieved without any performance loss of banks (sets) of digital filters. As the basis for the construction of algorithms proposed in the article, the previously known Winograd method of multiplication of real matrices and vectors and two versions of the method of type 3M for multiplication of complex matrices and vectors are used. Methods of combining these known methods of multiplying matrices and vectors for building digital filter banks (sets) are considered. The analysis of computing complexity of such ways which showed a possibility of reduction of computing complexity in comparison with a traditional algorithm of realization of bank (set) of digital filters approximately in 2.66 times - at realization on the processor without hardware multiplier is carried out; and by 1.33 times - at realization on the processor with the hardware multiplier. These indicators are markedly higher than those of known algorithms. Analysis of sensitivity of algorithms proposed in this article to rounding errors arising by digital signal processing was carried out. Based on this analysis, an algorithm is selected that has a computational complexity smaller than that of a traditional algorithm, but its sensitivity to rounding errors is the same as that of a traditional algorithm. Recommendations are given on its practical application in the development of a bank (set) of digital filters.