### Abstract

The objective of this paper is to present the subject of wavelets from a filter-theory perspective, which is quite familiar to electrical engineers. Such a presentation provides both physical and mathematical insights into the problem. It is shown that taking the discrete wavelet transform of a function is equivalent to filtering it by a bank of constant-Q filters, the non-overlapping bandwidths of which differ by an octave. The discrete wavelets are presented, and a recipe is provided for generating such entities. One of the goals of this tutorial is to illustrate how the wavelet decomposition is carried out, starting from the fundamentals, and how the scaling functions and wavelets are generated from the filter-theory perspective. Examples are presented to illustrate the class of problems for which the discrete wavelet techniques are ideally suited. It is interesting to note that it is not necessary to generate the wavelets or the scaling functions in order to implement the discrete wavelet transform. Finally, it is shown how wavelet techniques can be used to solve operator/matrix equations. It is shown that the "orthogonal-transform property" of the discrete wavelet techniques does not hold in numerical computations.

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

Pages (from-to) | 49-68 |

Number of pages | 20 |

Journal | IEEE Antennas and Propagation Magazine |

Volume | 40 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1998 |

### Keywords

- Filters
- Matrix equations
- Operators
- Wavelet transforms
- Wavelets

### ASJC Scopus subject areas

- Condensed Matter Physics
- Electrical and Electronic Engineering

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## Cite this

*IEEE Antennas and Propagation Magazine*,

*40*(5), 49-68. https://doi.org/10.1109/74.735965