Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.

http://ahvazdownload.persiangig.com/document/article/39.pdf

A wavelet tour of signal processing

A review and tutorial of the fundamental ideas and methods of joint time-frequency distributions is presented. The objective of the field is to describe how the spectral content of a signal changes in time and to develop the physical and mathematical ideas needed to understand what a time-varying spectrum is. The basic gal is to devise a distribution that represents the energy or intensity of a signal simultaneously in time and frequency. Although the basic notions have been developing steadily over the last 40 years, there have recently been significant advances. This review is intended to be understandable to the nonspecialist with emphasis on the diversity of concepts and motivations that have gone into the formation of the field. >

/pdf/time-frequency-distributions-a-review-3lxbcw4fg4.pdf

Time-frequency distributions-a review

A simple, nonrigorous, synthetic view of wavelet theory is presented for both review and tutorial purposes. The discussion includes nonstationary signal analysis, scale versus frequency, wavelet analysis and synthesis, scalograms, wavelet frames and orthonormal bases, the discrete-time case, and applications of wavelets in signal processing. The main definitions and properties of wavelet transforms are covered, and connections among the various fields where results have been developed are shown. >

/pdf/wavelets-and-signal-processing-1gyx22a8e3.pdf

Wavelets and signal processing

First published in 1995, Wavelets and Subband Coding offered a unified view of the exciting field of wavelets and their discrete-time cousins, filter banks, or subband coding. The book developed the theory in both continuous and discrete time, and presented important applications. During the past decade, it filled a useful need in explaining a new view of signal processing based on flexible time-frequency analysis and its applications. Since 2007, the authors now retain the copyright and allow open access to the book.

Wavelets and Subband Coding

The S transform, which is introduced in the present correspondence, is an extension of the ideas of the continuous wavelet transform (CWT) and is based on a moving and scalable localizing Gaussian window. It is shown to have some desirable characteristics that are absent in the continuous wavelet transform. The S transform is unique in that it provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum. These advantages of the S transform are due to the fact that the modulating sinusoids are fixed with respect to the time axis, whereas the localizing scalable Gaussian window dilates and translates.

Localization of the complex spectrum: the S transform

A tutorial review of both linear and quadratic representations is given. The linear representations discussed are the short-time Fourier transform and the wavelet transform. The discussion of quadratic representations concentrates on the Wigner distribution, the ambiguity function, smoothed versions of the Wigner distribution, and various classes of quadratic time-frequency representations. Examples of the application of these representations to typical problems encountered in time-varying signal processing are provided. >

Linear and quadratic time-frequency signal representations

In this paper, the authors describe a QRS complex detector based on the dyadic wavelet transform (D/sub y/WT) which is robust to time-varying QRS complex morphology and to noise. They design a spline wavelet that is suitable for QRS detection. The scales of this wavelet are chosen based on the spectral characteristics of the electrocardiogram (ECG) signal. They illustrate the performance of the D/sub y/WT-based QRS detector by considering problematic ECG signals from the American Heart Association (AHA) database. Seventy hours of data was considered. The authors also compare the performance of D/sub y/WT-based QRS detector with detectors based on Okada, Hamilton-Tompkins, and multiplication of the backward difference algorithms. From the comparison, results the authors observed that although no one algorithm exhibited superior performance in all situations, the D/sub y/WT-based detector compared well with the standard techniques. For multiform premature ventricular contractions, bigeminy, and couplets tapes, the D/sub y/WT-based detector exhibited excellent performance.

Wavelet transform-based QRS complex detector

An event-detection pitch detector based on the dyadic wavelet transform is described. The proposed pitch detector is suitable for both low-pitched and high-pitched speakers and is robust to noise. Examples are provided that demonstrate the superior performance of the pitch detector in comparison with classical pitch detectors that use the autocorrelation and the cepstrum methods to estimate the pitch period. >

Application of the wavelet transform for pitch detection of speech signals

The short-time Fourier transform (STFT), the ambiguity function (AF), and the Wigner distribution (WD) are mixed time-frequency signal representations that use Fourier transform techniques to map a one-dimensional function of time into a two-dimensional function of time and frequency. These mixed time-frequency mappings have been used to analyze the local frequency characteristics of a variety of signals and systems. Although much work has also been done to develop STFT and AF synthesis algorithms that can be used to implement a variety of time-varying signal processing operations, no such synthesis techniques have thus far been developed for the WD. In this paper, a signal synthesis algorithm that works directly with the real-valued high-resolution WD will be derived. Examples of how this WD synthesis procedure can be used to perform time-varying filtering operations or signal separation will be given.

Time-varying filtering and signal estimation using Wigner distribution synthesis techniques

Using operator theory methods together with our previously introduced unitary fractional operator, we derive explicit definitions of fractional convolution and correlation operations in a systematic and comprehensive manner. Via operator manipulations, we also provide alternative formulations of those fractional operations that suggest efficient algorithms for discrete implementation. Through simulation examples, we demonstrate how well the proposed efficient method approximates the continuous formulation of fractional autocorrelation. It is also shown that the proposed fractional autocorrelation corresponds to radial slices of the ambiguity function (AF). We also suggest an application of the fast fractional autocorrelation for detection and parameter estimation of linear FM signals.

G.F. Boudreaux-Bartels

Papers

Linear and quadratic time-frequency signal representations

Wavelet transform-based QRS complex detector

Application of the wavelet transform for pitch detection of speech signals

Time-varying filtering and signal estimation using Wigner distribution synthesis techniques

Fractional convolution and correlation via operator methods and an application to detection of linear FM signals