Cormac Herley

Wavelets and filter banks: theory and design

The wavelet transform is compared with the more classical short-time Fourier transform approach to signal analysis. Then the relations between wavelets, filter banks, and multiresolution signal processing are explored. A brief review is given of perfect reconstruction filter banks, which can be used both for computing the discrete wavelet transform, and for deriving continuous wavelet bases, provided that the filters meet a constraint known as regularity. Given a low-pass filter, necessary and sufficient conditions for the existence of a complementary high-pass filter that will permit perfect reconstruction are derived. The perfect reconstruction condition is posed as a Bezout identity, and it is shown how it is possible to find all higher-degree complementary filters based on an analogy with the theory of Diophantine equations. An alternative approach based on the theory of continued fractions is also given. These results are used to design highly regular filter banks, which generate biorthogonal continuous wavelet bases with symmetries. >

Tilings of the time-frequency plane: construction of arbitrary orthogonal bases and fast tiling algorithms

The authors consider expansions which give arbitrary orthonormal tilings of the time-frequency plane. These differ from the short-time Fourier transform, wavelet transform, and wavelet packets tilings in that they change over time. They show how this can be achieved using time-varying orthogonal tree structures, which preserve orthogonality, even across transitions. The method is based on the construction of boundary and transition filters; these allow us to construct essentially arbitrary tilings. Time-varying modulated lapped transforms are a special case, where both boundary and overlapping solutions are possible with filters obtained by modulation. They present a double-tree algorithm which for a given signal decides on the best binary segmentation in both time and frequency. That is, it is a joint optimization of time and frequency splitting. The algorithm is optimal for additive cost functions (e.g., rate-distortion), and results in time-varying best bases, the main application of which is for compression of nonstationary signals. Experiments on test signals are presented. >

Wavelets and recursive filter banks

It is shown that infinite impulse response (IIR) filters lead to more general wavelets of infinite support than finite impulse response (FIR) filters. A complete constructive method that yields all orthogonal two channel filter banks, where the filters have rational transfer functions, is given, and it is shown how these can be used to generate orthonormal wavelet bases. A family of orthonormal wavelets that have a maximum number of disappearing moments is shown to be generated by the halfband Butterworth filters. When there is an odd number of zeros at pi it is shown that closed forms for the filters are available without need for factorization. A still larger class of orthonormal wavelet bases having the same moment properties and containing the Daubechies and Butterworth filters as the limiting cases is presented. It is shown that it is possible to have both linear phase and orthogonality in the infinite impulse response case, and a constructive method is given. It is also shown how compactly supported bases may be orthogonalized, and bases for the spline function spaces are constructed. >

Wavelets and filter banks: relationships and new results

The discrete version of the wavelet transform, which has recently emerged as a powerful tool for nonstationary signal analysis is closely related to filter banks, which have been studied in digital signal processing. Also, multiresolution signal analysis has been used in image processing. The relationship between these techniques is indicated. It is shown how to construct biorthogonal systems with linear-phase finite impulse response (FIR) filters and with regular analysis and synthesis. Some examples of practical interest are given. The complexity of the discrete wavelet transform is also discussed.<<ETX>>

Orthogonal time-varying filter banks and wavelet packets

Considers the construction of orthogonal time-varying filter banks. By examining the time domain description of the two-channel orthogonal filter bank the authors find it possible to construct a set of orthogonal boundary filters, which allows to apply the filter bank to one-sided or finite-length signals, without redundancy or distortion. The method is constructive and complete. There is a whole space of orthogonal boundary solutions, and there is considerable freedom for optimization. This may be used to generate subband tree structures where the tree varies over time, and to change between different filter sets. The authors also show that the iteration of discrete-time time-varying filter banks gives continuous-time bases, just as in the stationary case. This gives rise to wavelet, or wavelet packet, bases for half-line and interval regions. >

Boundary filters for finite-length signals and time-varying filter banks

We examine the question of how to construct time-varying filter banks in the most general M-channel nonorthogonal case. We show that by associating with both analysis and synthesis operators a set of boundary filters, it is possible to make the analysis structure vary arbitrarily in time, and yet reconstruct the input with a similarly time-varying synthesis section. There is no redundancy or distortion introduced. This gives a solution to the problem of applying filter banks to finite length signals; it suffices to apply the boundary filters at the beginning and end of the signal segment. This also allows the construction of orthogonal and nonorthogonal bases with essentially any prescribed time and frequency localization, but which, nonetheless, are based on structures with efficient filter bank implementations. >

Orthogonal time-varying filter banks and wavelets

The construction of time-varying orthogonal filter banks is considered. It is shown that implementing an orthogonal finite impulse response filter bank over a finite signal segment involves finding a set of orthogonal boundary filters, and that by carrying out a Gram-Schmidt orthogonalization procedure boundary filters are generated that necessarily remain localized in the region of the boundary. A complete constructive characterization of such boundaries is given for two-channel finite impulse response filter banks. These boundary constructions allow changing the topology of orthogonal subband trees at will, by growing or pruning branches at any time. The boundary filter case can be further generalized to give overlapping transition filters when changing between orthogonal structures. If the time-varying filter banks are used in an iterated scheme, they converge to continuous-time bases, much as in the non-time-varying case.<<ETX>>

Arbitrary orthogonal tilings of the time-frequency plane

Expansions which give arbitrarily orthonormal tilings of the time-frequency plane are considered. These differ from the short-time Fourier transform, wavelet transform, and wavelet packets tilings in that they change over time. It is shown how orthonormal tilings can be achieved using time-varying orthogonal tree structures, which preserve orthogonality, even across transitions. One method is based on lapped orthogonal transforms, which makes it possible to change the number of channels in the transform. A second method is based on the construction of boundary filters and gives arbitrary tilings. An algorithm is presented which for a given signal decides on the best binary segmentation and which tree split to use for each segment. It is optimal in a rate-distortion sense. The results of experiments on test signals are presented.<<ETX>>

Time-varying orthonormal tilings of the time-frequency plane

Expansions that give arbitrary orthonormal tilings of the time-frequency plane are considered. These differ from the short-time Fourier transform, wavelet transform, and wavelet packet tilings in that they change over time. It is shown how this can be achieved using time-varying orthogonal tree structures, which preserve orthogonality, even across transitions. One method is based on lapped orthogonal transforms, which makes it possible to change the number of channels in the transform. A second method is based on the construction of orthogonal boundary filters to construct essentially arbitrary tilings. A double-tree algorithm is presented that, for a given signal, decides on the best binary segmentation and on which tree split to use for each segment. That is, it is a joint optimization of time and frequency splitting. The algorithm is optimal for additive cost functions (e.g., rate distortion), which gives the best time-varying bases. Results of experiments on test signals are shown.<<ETX>>

Linear phase wavelets: theory and design

The authors present new theoretical results on FIR (finite impulse response) filter banks based on Diophantine equations and continued fraction expansions and use them to show how general wavelets may be designed. They further show that by considering a noncausal IIR (infinite impulse response) structure it is possible to have a linear phase paraunitary solution. A number of design examples illustrating the advantages of the new results are presented.<<ETX>>