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Synthesis of FIR Filters

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Digital Filters Using MATLAB

Abstract

In this chapter, we discuss the design of digital filters with finite-length impulse response using windowing and more important using iterative optimisation, i.e., mini-max design of linear-phase as well as nonlinear-phase filters, e.g., special cases as minimum-phase, half-band and Nyquist FIR filters, differentiators and Hilbert transformers. We discuss concepts like zero-phase response, delay- and allpass-complementary filter pairs, which have reduced realisation cost. Finally, we demonstrated the design of FIR filters with a least squares approximation using linear programming and quadratic programming. The chapter contains 27 solved examples.

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Notes

  1. 1.

    After the Austrian, nineteenth century meteorologist, Julius van Hann. He is seen as the father of modern meteorology.

  2. 2.

    Richard W. Hamming (1915–1998).

  3. 3.

    Note that \({\text{wT}}\) should be normalized with respect to half the sampling frequency in the function firpmord. Hence, \({\text{wT}}\) should be divided by π, i.e., \([{\text{N, Be, D, W}}] = {\text{firpmord}}\,\,({\text{wT}}/{\text{pi, b, d}})\) Alternatively we may write [N, Be, D, W] = firpmord (wT, b, d, 2*pi) where b specifies the desired magnitude in the different bands, e.g., b = [1 0] and d specifies the maximum ripples allowable in each band, e.g., d = [δc δs] = [0.01 0.001].

  4. 4.

    Note that in MATLAB the first element in the vector h is element 1. Hence, this element corresponds to the first value in the impulse response, i.e., h(1) ↔ h(0).

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Authors and Affiliations

  1. Linköping University, Linköping, Sweden

    Lars Wanhammar

  2. Tampere University of Technology, Tampere, Finland

    Tapio Saramäki

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  1. Lars Wanhammar
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  2. Tapio Saramäki
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Correspondence to Lars Wanhammar .

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Wanhammar, L., Saramäki, T. (2020). Synthesis of FIR Filters. In: Digital Filters Using MATLAB . Springer, Cham. https://doi.org/10.1007/978-3-030-24063-9_5

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  • DOI: https://doi.org/10.1007/978-3-030-24063-9_5

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