### Abstract

Explicit solutions are given for the rational function P(z) for two classes of IIR orthogonal two-band wavelet bases, for which the scaling filter is maximally flat. P(z) denotes the rational transfer function H(z)H(l/z), where H(z) is the (lowpass) scaling filter. The first is the class of solutions that are intermediate between the Daubechies and the Butterworth wavelets. It is found that the Daubechies, the Butterworth, and the intermediate solutions are unified by a single formula. The second is the class of scaling filters realizable as a parallel sum of two allpass filters (a particular case of which yields the class of symmetric IIR orthogonal wavelet bases). For this class, a closed-form solution is provided by the solution to an older problem in group delay approximation by digital allpole filters.

Original language | English (US) |
---|---|

Pages (from-to) | 1138-1141 |

Number of pages | 4 |

Journal | IEEE Transactions on Signal Processing |

Volume | 46 |

Issue number | 4 |

DOIs | |

State | Published - 1998 |

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### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Signal Processing

### Cite this

**Formulas for orthogonal iir wavelet filters.** / Selesnick, Ivan.

Research output: Contribution to journal › Article

*IEEE Transactions on Signal Processing*, vol. 46, no. 4, pp. 1138-1141. https://doi.org/10.1109/78.668565

}

TY - JOUR

T1 - Formulas for orthogonal iir wavelet filters

AU - Selesnick, Ivan

PY - 1998

Y1 - 1998

N2 - Explicit solutions are given for the rational function P(z) for two classes of IIR orthogonal two-band wavelet bases, for which the scaling filter is maximally flat. P(z) denotes the rational transfer function H(z)H(l/z), where H(z) is the (lowpass) scaling filter. The first is the class of solutions that are intermediate between the Daubechies and the Butterworth wavelets. It is found that the Daubechies, the Butterworth, and the intermediate solutions are unified by a single formula. The second is the class of scaling filters realizable as a parallel sum of two allpass filters (a particular case of which yields the class of symmetric IIR orthogonal wavelet bases). For this class, a closed-form solution is provided by the solution to an older problem in group delay approximation by digital allpole filters.

AB - Explicit solutions are given for the rational function P(z) for two classes of IIR orthogonal two-band wavelet bases, for which the scaling filter is maximally flat. P(z) denotes the rational transfer function H(z)H(l/z), where H(z) is the (lowpass) scaling filter. The first is the class of solutions that are intermediate between the Daubechies and the Butterworth wavelets. It is found that the Daubechies, the Butterworth, and the intermediate solutions are unified by a single formula. The second is the class of scaling filters realizable as a parallel sum of two allpass filters (a particular case of which yields the class of symmetric IIR orthogonal wavelet bases). For this class, a closed-form solution is provided by the solution to an older problem in group delay approximation by digital allpole filters.

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U2 - 10.1109/78.668565

DO - 10.1109/78.668565

M3 - Article

VL - 46

SP - 1138

EP - 1141

JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

SN - 1053-587X

IS - 4

ER -