### Balanced Multiwavelet Bases Based on Symmetric FIR Filters

#### Abstract

This paper describes a basic difference between multiwavelets and scalar wavelets that explains, without using zero moment properties, why certain complications arise in the implementation of discrete multiwavelet transforms. Assuming one wishes to avoid the use of prefilters in implementing the discrete multiwavelet transform, it is suggested that the behavior of the iterated filter bank associated with a multiwavelet basis of multiplicity r is more fully revealed by an expanded set of r^2 scaling functions phi_{i,j}.

This paper also introduces new K-balanced orthogonal multiwavelet bases based on symmetric FIR filters. The nonlinear design equations arising in this work are solved using Gröbner bases. The K-balanced multiwavelet bases based on even-length symmetric FIR filters are shown to be superior than those based on odd-length symmetric FIR filters, as illustrated by special relations they satisfy and by the examples constructed.

Ivan W. Selesnick, Balanced multiwavelet bases based on symmetric FIR filters, IEEE Trans on Signal Processing, 48(1):184-191, January 2000.

#### Examples from the paper

K-balanced orthonormal multiwavelet bases based on:

- Even-length symmetric FIR filters
- Odd-length symmetric FIR filters

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