Complex rational orthogonal wavelet and its application in communications

Copyright © 2006 IEEEThis letter proposes a generalized method to construct complex wavelets under the framework of rational multiresolution analysis, MRA(M), where is a rational number. Theorems and examples are given for the construction of complex rational orthogonal wavelets (CROWs) whose real and imaginary parts form an exact Hilbert transform pair. Since the classical Mallat’s MRA is a special case of the rational MRA(M) with = 2, the theorems hold for the construction of complex dyadic wavelets. Based on a rational MRA(M) with 1 < M < 2, the constructed CROWs not only achieve the benefit by capturing the phase information that the real-valued wavelets are lacking but also have the unique fine rational orthogonal property that is suited for specific application scenarios where binary orthogonality achieved by dyadic wavelets is not sufficient for the scale resolution. The CROWs’ application in communications as the modulation signal pulse for PSK/QAM signaling is discussed. Specific communication scenarios that could benefit from the properties of this class of complex wavelets include the communication through a multipath/Doppler channel and new CROW-based multicarrier modulation (MCM)/orthogonal frequency division multiplexing (OFDM) systems.Limin Yu and Langford B. Whit

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

A unified wavelet-based modelling framework for non-linear system identification: the WANARX model structure

A new unified modelling framework based on the superposition of additive submodels, functional components, and wavelet decompositions is proposed for non-linear system identification. A non-linear model, which is often represented using a multivariate non-linear function, is initially decomposed into a number of functional components via the wellknown analysis of variance (ANOVA) expression, which can be viewed as a special form of the NARX (non-linear autoregressive with exogenous inputs) model for representing dynamic input–output systems. By expanding each functional component using wavelet decompositions including the regular lattice frame decomposition, wavelet series and multiresolution wavelet decompositions, the multivariate non-linear model can then be converted into a linear-in-theparameters problem, which can be solved using least-squares type methods. An efficient model structure determination approach based upon a forward orthogonal least squares (OLS) algorithm, which involves a stepwise orthogonalization of the regressors and a forward selection of the relevant model terms based on the error reduction ratio (ERR), is employed to solve the linear-in-the-parameters problem in the present study. The new modelling structure is referred to as a wavelet-based ANOVA decomposition of the NARX model or simply WANARX model, and can be applied to represent high-order and high dimensional non-linear systems

Wavelets and their use

This review paper is intended to give a useful guide for those who want to apply discrete wavelets in their practice. The notion of wavelets and their use in practical computing and various applications are briefly described, but rigorous proofs of mathematical statements are omitted, and the reader is just referred to corresponding literature. The multiresolution analysis and fast wavelet transform became a standard procedure for dealing with discrete wavelets. The proper choice of a wavelet and use of nonstandard matrix multiplication are often crucial for achievement of a goal. Analysis of various functions with the help of wavelets allows to reveal fractal structures, singularities etc. Wavelet transform of operator expressions helps solve some equations. In practical applications one deals often with the discretized functions, and the problem of stability of wavelet transform and corresponding numerical algorithms becomes important. After discussing all these topics we turn to practical applications of the wavelet machinery. They are so numerous that we have to limit ourselves by some examples only. The authors would be grateful for any comments which improve this review paper and move us closer to the goal proclaimed in the first phrase of the abstract.Comment: 63 pages with 22 ps-figures, to be published in Physics-Uspekh

Characterizations of rectangular (para)-unitary rational Functions

We here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle: (i) Through the realization matrix of Schur stable systems. (ii) The Blaschke-Potapov product, which is then employed to introduce an easy-to-use description of all these functions with dimensions and McMillan degree as parameters. (iii) Through the (not necessarily reducible) Matrix Fraction Description (MFD). In cases (ii) and (iii) the poles of the rational functions involved may be anywhere in the complex plane, but the unit circle (including both zero and infinity). A special attention is devoted to exploring the gap between the square and rectangular cases.Comment: Improved versio