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Computationally Efficient FIR Filtering of Polynomial Signals in DFT Domain

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Abstract

Fast unbiased finite impulse response (UFIR) filtering of polynomial signals can be provided in the discrete Fourier transform (DFT) domain employing fast Fourier transform (FFT). We show that the computation time can further be reduced by utilizing properties of UFIR filters in the DFT domain. The transforms have been found and investigated in detail for low-degree FIRs most widely used in practice. As a special result, we address an explicit unbiasedness condition uniquely featured to UFIR filters in DFT domain. The noise power gain and estimation error bound have also been discussed. An application is given for state estimation in a crystal clock employing the Global Positioning System based measurement of time errors provided each second. Based upon it, it is shown that filtering in the time domain takes about 1 second, which is unacceptable for real-time applications. The Kalman-like algorithm reduces the computation time by the factor of about 8, the FFT-based algorithm by about 18, and FFT with the UFIR filter DFT properties by about 20.

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Notes

  1. Here and in the following, \(\hat{x}_{n\mid v}\) means the estimate of x n at n via measurement from the past to v.

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Authors and Affiliations

  1. Department of Electronics, University of Guanajuato, Salamanca, 36855, Mexico

    Paula Castro-Tinttori, Oscar Ibarra-Manzano & Yuriy S. Shmaliy

Authors
  1. Paula Castro-Tinttori
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  2. Oscar Ibarra-Manzano
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  3. Yuriy S. Shmaliy
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Correspondence to Yuriy S. Shmaliy.

Appendix: Low-degree unbiased FIR filter impulse responses

Appendix: Low-degree unbiased FIR filter impulse responses

The following impulse responses [25] are unique for unbiased FIR filtering of linear, quadratic, and cubic signals, respectively, on an interval of N past-points from n−N+1 to n:

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Castro-Tinttori, P., Ibarra-Manzano, O. & Shmaliy, Y.S. Computationally Efficient FIR Filtering of Polynomial Signals in DFT Domain. Circuits Syst Signal Process 31, 2153–2166 (2012). https://doi.org/10.1007/s00034-012-9433-y

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