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Constructing Fast Algorithms by Expanding a Set of Matrices into Rank-1 Matrices

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Abstract

This paper introduces the notion of numerical basis for a numerical space and uses it to establish a relation between a fast algorithm for computing a discrete linear transform and the problem of expanding a given finite set of matrices as a linear combination of rank-1 matrices. It is shown that the number of multiplications of the algorithm is given by the number of rank-1 matrices in the expansion. Applying this approach, an algorithm for computing three components of the nine-point discrete Fourier transform (DFT) and an algorithm to compute the seven-point DFT with the least possible number of multiplications are shown.

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Notes

  1. The ROSME procedure is directly related to the problem of obtaining the tensor rank of a set of matrices [15].

  2. The Goertzel and the optimized single component algorithms [6, 8] are applied to each component individually.

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Acknowledgements

The authors are grateful to Drs. J. B. Lima and H. M. de Oliveira and Professor V. C. da Rocha Jr. for their careful review of this work and for their helpful comments.

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  1. Department of Electronics and Systems, Federal University of Pernambuco - UFPE, Avenida da Arquitetura, s/n Cidade Universitária, Bloco B, 4o Andar, Recife, PE, 50740-550, Brazil

    G. Jerônimo da Silva Jr. & R. M. Campello de Souza

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  1. G. Jerônimo da Silva Jr.
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  2. R. M. Campello de Souza
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Correspondence to G. Jerônimo da Silva Jr..

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Jerônimo da Silva, G., Campello de Souza, R.M. Constructing Fast Algorithms by Expanding a Set of Matrices into Rank-1 Matrices. Circuits Syst Signal Process 39, 1630–1648 (2020). https://doi.org/10.1007/s00034-019-01228-5

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