Faculty Bibliography

This paper considers the classical Shannon sampling theorem in multiresolution spaces with scaling functions as interpolants. As discussed by Xia and Zhang, for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal. They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar function, which has only 1 vanishing moment and is not continuous. This paper addresses the same question, but in the multiwavelet context, where the situation is different. This paper presents the construction of orthogonal multiscaling functions that are simultaneously cardinal, of compact support, and have more than one vanishing moment The scaling functions thereby support a Shannon-like sampling theorem. Such wavelet bases are appealing because the initialization of the discrete wavelet transform (prefiltering) is the identity operator - the projection of a function onto the scaling space is given by its samples.

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 IC-balanced orthogonal multiwavelet bases based on symmetric FIR filters. The nonlinear design equations arising in this work are solved using Grobner bases. The K-balanced multiwavelet bases based on even-length symmetric FIR filters are shown to be particularly well behaved, as illustrated by special relations they satisfy and by the examples constructed.

This paper highlights the differences between traditional wavelet and multiwavelet bases with equal approximation order. Because multiwavelet bases normally lack important properties that traditional wavelet bases (of equal approximation order) possess, the associated discrete multiwavelet transform is less useful for signal processing unless preceded by a preprocessing step (preflltering). This paper examines the properties and design of orthogonal multiwavelet bases with approximation order > 1 that possess those properties that are normally absent. For these balanced bases (so named by Lebrun and Vetterli), prefltering can be avoided. By reorganizing the multiwavelet (vector) filter bank as a multichannel scalar filter bank, the development in this paper draws from results regarding the approximation order of M-band wavelet bases. A main result thereby obtained is a characterization of balanced multiwavelet bases in terms of the divisibility of certain transfer functions by 0~2r - i)/(z-1 - 1). ©1998 IEEE.

The discrete wavelet transform is usually carried out by filter bank iteration; however, for a fixed approximation order, this does not yield a discrete-time basis that is optimal with respect to time-localization. This paper describes new orthogonal discrete wavelet transform with approximation order two and improved time-localization. It is not based on filter bank iteration; instead, new filters are used for each scale. For coarse scales, the support of the discrete-time basis functions approaches two-thirds that of the corresponding functions obtained by filter bank iteration. The new basis retains the octave-band characteristic and leads cleanly to a DWT for finite-length signals. Closed-form expressions for the filters are given, an efficient implementation of the transform is described, and improvement in a denoising example is shown. The basis, being piecewise linear, is reminiscent of the slant transform with which it is compared. ©199g IEEE.

This paper highlights the differences between traditional wavelet and multiwavelet bases with equal approximation order, Because multiwavelet bases normally lack important properties that traditional wavelet bases (of equal approximation order) possess, the associated discrete multiwavelet transform is less useful for signal processing unless it is preceded by a preprocessing step (prefiltering). This paper examines the properties and design of orthogonal multiwavelet bases, with approximation order >1 that possess those properties that are normally absent. For these "balanced" bases (so named by Lebrun and Vetterli), prefiltering can be avoided. By reorganizing the multiwavelet filter bank, the development in this paper draws from results regarding the approximation order of M-band wavelet bases. The main result thereby obtained is a characterization of balanced multiwavelet bases in terms of the divisibility of certain transfer functions by powers of (z(-2r) - 1)1(z(-1) - I). For traditional wavelets (r - 1), this specializes to the usual factor (z + 1)(K).

This correspondence introduces a new class of infinite impulse response (IIR) digital filters that unifies the classical digital Butterworth filter and the well-known maximally flat FIR filter. New closed-form expressions are provided, and a straightforward design technique is described. The new IIR digital filters have more zeros than poles (away from the origin), and their (monotonic) square magnitude frequency responses are maximally nat at omega = 0 and at omega = pi. Another result of the correspondence is that for a specified cut-off frequency and a specified number of zeros, there is only one valid way in which to split the zeros between z = -1 and the passband, This technique also permits continuous variation of the cutoff frequency. IIR filters having more zeros than poles are of interest because often, to obtain a good tradeoff between performance and implementation complexity, just a few poles are best.

Explicit solutions are given for the rational function P(z) for two classes of IIR orthogonal two-hand wavelet bases, for which the scaling filter is maximally flat. P(z) denotes the rational transfer function H(z)H(1/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 with 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.

In a previous paper, we described a constrained least square approach to FIR filter design that does not use "don't care" regions, In that paper, we described a simple algorithm for the design of lowpass filters according to that approach, In this correspondence, we describe a modification of that algorithm that makes it converge for many multiband filter designs, Although no proof of convergence is given, the modified algorithm remains simple and converges rapidly in many cases. In this approach, the user supplies a lower and upper bound constraint that is exactly satisfied by the local minima and maxima of the frequency response amplitude. Yet, the constraints can be made as tight as desired-the transition band automatically adjusts (widens) to accommodate the constraints.

This paper describes a new class of nonsymmetric maximally flat low-pass finite impulse response (FIR) filters, By subjecting the magnitude and group delay responses (individually) to differing numbers of flatness constraints, the new filters are obtained, It is found that these filters achieve a smaller delay than symmetric filters while maintaining relatively constant group delay around omega = 0, with no degradation of the frequency response magnitude, The design of these filters is initially investigated using Grobner bases, An analytic design technique, applicable to a subset of the forgoing filters, is provided that does not depend on Grobner basis computations.