Coxeter Groups and Wavelet Sets

A traditional wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a system of unitary operators defined in terms of translation and dilation operations. A Coxeter/fractal-surface wavelet is obtained by defining fractal surfaces on foldable figures, which tesselate the embedding space by reflections in their bounding hyperplanes instead of by translations along a lattice. Although both theories look different at their onset, there exist connections and communalities which are exhibited in this semi-expository paper. In particular, there is a natural notion of a dilation-reflection wavelet set. We prove that dilation-reflection wavelet sets exist for arbitrary expansive matrix dilations, paralleling the traditional dilation-translation wavelet theory. There are certain measurable sets which can serve simultaneously as dilation-translation wavelet sets and dilation-reflection wavelet sets, although the orthonormal structures generated in the two theories are considerably different

Optimum receiver design for broadband Doppler compensation in multipath/Doppler channels with rational orthogonal wavelet signaling

Copyright © 2007 IEEEIn this paper, we address the issue of signal transmission and Doppler compensation in multipath/Doppler channels. Based on a wavelet-based broadband Doppler compensation structure, this paper presents the design and performance characterization of optimum receivers for this class of communication systems. The wavelet-based Doppler compensation structure takes account of the coexistence of multiple Doppler scales in a multipath/Doppler channel and captures the information carried by multiple scaled replicas of the transmitted signal rather than an estimation of an average Doppler as in conventional Doppler compensation schemes. The transmitted signal is recovered by the perfect reconstruction (PR) wavelet analysis filter bank (FB). We demonstrate that with rational orthogonal wavelet signaling, the proposed communication structure corresponds to a Lth-order diversity system, where L is the number of dominant transmission paths. Two receiver designs for pulse amplitude modulation (PAM) signal transmission are presented. Both receiver designs are optimal under the maximum-likelihood (ML) criterion for diversity combination and symbol detection. Good performance is achieved for both receivers in combating the Doppler effect and intersymbol interference (ISI) caused by multipath while mitigating the channel noise. In particular, the second receiver design overcomes symbol timing sensitivities present in the first design at reasonable cost to performance.Limin Yu and Langford B. Whit

Multifractal analyses of row sum signals of elementary cellular automata

We first apply the WT-MFDFA, MFDFA, and WTMM multifractal methods to binomial multifractal time series of three different binomial parameters and find that the WTMM method indicates an enhanced difference between the fractal components than the known theoretical result. Next, we make use of the same methods for the time series of the row sum signals of the two complementary ECA pairs of rules (90,165) and (150,105) for ten initial conditions going from a single 1 in the central position up to a set of ten 1's covering the ten central positions in the first row. Since the members of the pairs are actually similar from the statistical point of view, we can check which method is the most stable numerically by recording the differences provided by the methods between the two members of the pairs for various important quantities of the scaling analyses, such as the multifractal support, the most frequent Holder exponent, and the Hurst exponent and considering as the better one the method that provides the minimum differences. According to this criterion, our results show that the MFDFA performs better than WT-MFDFA and WTMM in the case of the multifractal support, while for the other two scaling parameters the WT-MFDFA is the best. The employed set of initial conditions does not generate any specific trend in the values of the multifractal parametersComment: 23 pages including an appendix and 11 figures, extended version accepted for publication by Physica

Renormalization of Massless Feynman Amplitudes in Configuration Space

A systematic study of recursive renormalization of Feynman amplitudes is carried out both in Euclidean and in Minkowski configuration space. For a massless quantum field theory (QFT) we use the technique of extending associate homogeneous distributions to complete the renormalization recursion. A homogeneous (Poincare covariant) amplitude is said to be convergent if it admits a (unique covariant) extension as a homogeneous distribution. For any amplitude without subdivergences - i.e. for a Feynman distribution that is homogeneous off the full (small) diagonal - we define a renormalization invariant residue. Its vanishing is a necessary and sufficient condition for the convergence of such an amplitude. It extends to arbitrary - not necessarily primitively divergent - Feynman amplitudes. This notion of convergence is finer than the usual power counting criterion and includes cancellation of divergences.Comment: LaTeX, 64 page

A new class of wavelet networks for nonlinear system identification

A new class of wavelet networks (WNs) is proposed for nonlinear system identification. In the new networks, the model structure for a high-dimensional system is chosen to be a superimposition of a number of functions with fewer variables. By expanding each function using truncated wavelet decompositions, the multivariate nonlinear networks can be converted into linear-in-the-parameter regressions, which can be solved using least-squares type methods. An efficient model term selection approach based upon a forward orthogonal least squares (OLS) algorithm and the error reduction ratio (ERR) is applied to solve the linear-in-the-parameters problem in the present study. The main advantage of the new WN is that it exploits the attractive features of multiscale wavelet decompositions and the capability of traditional neural networks. By adopting the analysis of variance (ANOVA) expansion, WNs can now handle nonlinear identification problems in high dimensions