Wavelets are of wide potential use in statistical contexts. The basics of the discrete wavelet transform are reviewed using a filter notation that is useful subsequently in the paper. A `stationary wavelet transform', where the coefficient sequences are not decimated at each stage, is described. Two different approaches to the construction of an inverse of the stationary wavelet transform are set out. The application of the stationary wavelet transform as an exploratory statistical method is discussed, together with its potential use in nonparametric regression. A method of local spectral density estimation is developed. This involves extensions to the wavelet context of standard time series ideas such as the periodogram and spectrum. The technique is illustrated by its application to data sets from astronomy and veterinary anatomy. 1 Introduction In this paper we discuss some aspects of wavelets with a particular view to their statistical application. In particular we shall be conce...

During the last decade, CD-quality digital audio has essentially replaced analog audio. Emerging digital audio applications for network, wireless, and multimedia computing systems face a series of constraints such as reduced channel bandwidth, limited storage capacity, and low cost. These new applications have created a demand for high-quality digital audio delivery at low bit rates. In response to this need, considerable research has been devoted to the development of algorithms for perceptually transparent coding of high-fidelity (CD-quality) digital audio. As a result, many algorithms have been proposed, and several have now become international and/or commercial product standards. This paper reviews algorithms for perceptually transparent coding of CD-quality digital audio, including both research and standardization activities. The paper is organized as follows. First, psychoacoustic principles are described with the MPEG psychoacoustic signal analysis model 1 discussed in some detail. Next, filter bank design issues and algorithms are addressed, with a particular emphasis placed on the Modified Discrete Cosine Transform (MDCT), a perfect reconstruction (PR) cosine-modulated filter bank that has become of central importance in perceptual audio coding. Then, we review methodologies that achieve perceptually transparent coding of FM- and CD-quality audio signals, including algorithms that manipulate transform components, subband signal decompositions, sinusoidal signal components, and linear prediction (LP) parameters, as well as hybrid algorithms that make use of more than one signal model. These discussions concentrate on architectures and applications of

The theory of wavelets has recently undergone a period of rapid development. We introduce a software package called wavethresh that works within the statistical language S to perform one- and two-dimensional discrete wavelet transforms. The transforms and their inverses can be computed using any particular wavelet selected from a range of different families of wavelets. Pictures can be drawn of any of the one- or twodimensional wavelets available in the package. The wavelet coefficients can be presented in a variety of ways to aid in the interpretation of data. The package's wavelet transform "engine" is written in C for speed and the object-orientated functionality of S makes wavethresh easy to use. We provide a tutorial introduction to wavelets and the wavethresh software. We also discuss how the software may be used to carry out nonlinear regression and image compression. In particular, thresholding of wavelet coefficients is a method for attempting to extract signal from noise and ...

This paper constructs K-regular M-band orthonormal wavelet bases. K-regularity of the wavelet basis is known to be useful in numerical analysis applications and in image coding using wavelet techniques. Several characterizations of K-regularity and their importance are described. An explicit formula is obtained for all minimal length M-band scaling filters. A new state-space approach to constructing the wavelet filters from the scaling filters is also described. When M-band wavelets are constructed from unitary filter banks they give rise to wavelet tight frames in general (not orthonormal bases). Conditions on the scaling filter so that the wavelet bases obtained from it is orthonormal is also described.

Abstract. This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The “wavelet transform ” maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higher-order wavelets are constructed, and it is surprisingly quick to compute with them — always indirectly and recursively. We comment informally on the contest between these transforms in signal processing, especially for video and image compression (including high-definition television). So far the Fourier Transform — or its 8 by 8 windowed version, the Discrete Cosine Transform — is often chosen. But wavelets are already competitive, and they are ahead for fingerprints. We present a sample of this developing theory. 1. The Haar wavelet To explain wavelets we start with an example. It has every property we hope for, except one. If that one defect is accepted, the construction is simple and the computations are fast. By trying to remove the defect, we are led to dilation equations and recursively defined functions and a small world of fascinating new problems — many still unsolved. A sensible person would stop after the first wavelet, but fortunately mathematics goes on. The basic example is easier to draw than to describe: Figure 1. Scaling function φ(x), wavelet W(x), and the next level of detail.

Abstract--M channel maximally decimated filter banks have been used in the past to decompose signals into subbands. The theory of perfect-reconstruction filter banks has also been studied extensively. Nonparaunitary systems with linear phase filters have also been designed. In this paper, we study paraunitary systems in which each individual filter in the analysis synthesis banks has linear phase. Specific instances of this problem have been addressed by other authors, and linear phase paraunitary systems have been shown to exist. This property is often desirable for several applications, particularly in image processing. We begin by answering several theoretical questions pertaining to linear phase paraunitary systems. Next, we develop a minimal factorizdion for a large class of such systems. This factorization will be proved to be complete for even M. Further, we structurally impose the additional condition that the filters satisfy pairwise mirror-image symmetry in the frequency domain. This significantly reduces the number of parameters to be optimized in the design process. We then demonstrate the use of these filter banks in the generation of M-band orthonormal wavelets. Several design examples are also given to validate the theory. I.

Filterbanks for digital hearing aids must use significantly different criteria than those designed for coding applications. For digital hearing aids, the filterbank channel gains must be adjustable over a large dynamic range to compensate for the hearing loss. This adjustability violates the alias cancellation properties of critically sampled filterbanks designed for coding. This paper describes a filterbank designed exclusively for hearing aid applications. Consideration will be given to the extremely limited memory, low delay and low power requirements that must be met in a typical hearing aid application.

This paper solves an open problem, namely how to construct perfect reconstruction filter banks with rational sampling factors. Such filter banks have N branches, each one having a sampling factor of p i q i and their sum equals to one. In this way, the well-known theory of filter banks with uniform band splitting is extended to allow for non-uniform divisions of the spectrum. This can be very useful in the analysis of speech and music. The theory relies on two transforms, 1 and 2. While Transform 1, when applied, leads to uniform filter banks having polyphase components as individual filters, Transform 2 results in a uniform filter bank containing shifted versions of same filters. This, in turn, introduces dependencies in design, and is left for future work. As an illustration, several design examples for the ( 2 3 ; 1 3 ) are given. Filter banks are then classified according to the possible ways in which they can be built. It is also shown that some cases cannot be solved even wi...

Recently there has been significant development in the use of wavelet methods in various data mining processes. However, there has been written no comprehensive survey available on the topic. The goal of this is paper to fill the void. First, the paper presents a high-level data-mining framework that reduces the overall process into smaller components. Then applications of wavelets for each component are reviewd. The paper concludes by discussing the impact of wavelets on data mining research and outlining potential future research directions and applications.

When expanding digital signal processing of mobile communications terminals toward the antenna while making the terminal more wideband in order to be able to cope with different mobile communications standards in a software-radio-based terminal, the designer is faced with strong requirements such as bandwidth and dynamic range. Many publications claim that only reconfigurable hardware such as FPGAs can simultaneously cope with such diversity and requirements. Starting with considerations of the receiver architecture, we describe key functionalities of the digital front-end and highlight how signal characteristics of mobile communications signals and commonalities among different signal processing operations can be exploited to great advantage, eventually enabling implementations on an ASIC that, although not reconfigurable, would empower the software radio concept.