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Realisations of Finite-Sample Frequency-Selective Filters

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  • D.S.G. Pollock

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Abstract

A filtered data sequence can be obtained by multiplying the Fourier ordinates of the data by the ordinates of the frequency response of the filter and by applying the inverse Fourier transform to carry the product back to the time domain. Using this technique, it is possible, within the constraints of a finite sample, to design an ideal frequency-selective filter that will preserve all elements within a specified range of frequencies and that will remove all elements outside it. Approximations to ideal filters that are implemented in the time domain are commonly based on truncated versions of the infinite sequences of coefficients derived from the Fourier transforms of rectangular frequency response functions. An alternative to truncating an infinite sequence of coefficients is to wrap it around a circle of a circumference equal in length to the data sequence and to add the overlying coefficients. The coefficients of the wrapped filter can also be obtained by applying a discrete Fourier transform to a set of ordinates sampled from the frequency response function. Applying the coefficients to the data via circular convolution produces results that are identical to those obtained by a multiplication in the frequency domain, which constitutes a more efficient approach.

Suggested Citation

  • D.S.G. Pollock, 2008. "Realisations of Finite-Sample Frequency-Selective Filters," Discussion Papers in Economics 08/32, Department of Economics, University of Leicester.
  • Handle: RePEc:lec:leecon:08/32
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    File URL: http://www.le.ac.uk/economics/research/RePEc/lec/leecon/dp08-32.pdf
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    References listed on IDEAS

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    1. Alessandra Iacobucci & Alain Noullez, 2005. "A Frequency Selective Filter for Short-Length Time Series," Computational Economics, Springer;Society for Computational Economics, vol. 25(1), pages 75-102, February.
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    1. repec:prg:jnlpep:v:preprint:id:512:p:1-18 is not listed on IDEAS
    2. D.S.G. Pollock, 2017. "Stochastic processes of limited frequency and the effects of oversampling," Discussion Papers in Economics 17/03, Department of Economics, University of Leicester.
    3. D. S. G. Pollock, 2016. "Econometric Filters," Computational Economics, Springer;Society for Computational Economics, vol. 48(4), pages 669-691, December.
    4. repec:prg:jnlpep:v:2015:y:2015:i:5:id:512:p:1-18 is not listed on IDEAS
    5. Jitka Poměnková & Roman Maršálek, 2015. "Empirical Evidence of Ideal Filter Approximation: Peripheral and Selected EU Countries Application," Prague Economic Papers, University of Economics, Prague, vol. 2015(5), pages 485-502.

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    Keywords

    Signal extraction; Linear filtering; Frequency-domain analysis;

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