### Abstract

In this paper, a general structure of linear-phase finite impulse response filters, whose frequency responses satisfy given derivative constraints imposed upon an arbitrary frequency, is proposed. It is comprised of a linear combination of parallelly connected subfilters, called the cardinal filters, with weighted coefficients being the successive derivatives of the desired frequency response at the constrained frequency. An advantage of such a cardinal filters design is that only the weighted coefficients are relevant to the desired frequency response but not the cardinal filters; hence, a dynamic adjustment of the filter system becomes feasible. The key to derive the coefficients of cardinal filters is the determination of the power series expansion of certain trigonometric-related functions. By showing the elaborately chosen trigonometric-related functions satisfy specific differential equations, recursive formulas for the coefficients of cardinal filters are subsequently established, which make efficient their computations. At last, a simple enhancement of the cardinal filters design by incorporating the mean square error minimization is presented through examples.

Original language | English |
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Article number | 7865939 |

Pages (from-to) | 1839-1852 |

Number of pages | 14 |

Journal | IEEE Transactions on Circuits and Systems I: Regular Papers |

Volume | 64 |

Issue number | 7 |

DOIs | |

State | Published - 1 Jul 2017 |

### Keywords

- Finite impulse response (FIR) filters
- Taylor interpolation
- linear phase filters
- maximally flat (MF) filters

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## Cite this

*IEEE Transactions on Circuits and Systems I: Regular Papers*,

*64*(7), 1839-1852. [7865939]. https://doi.org/10.1109/TCSI.2017.2665667