Journal Article•
Localisation of the complex spectrum : The S transform
01 Jan 1996-Journal of Association of Exploration Geophysicists (Association of Exploration Geophysicists)-Vol. 17, Iss: 3, pp 99-114
TL;DR: The S transform as discussed by the authors is an extension to the ideas of the Gabor transform and the Wavelet transform, based on a moving and scalable localising Gaussian window and is shown here to have characteristics that are superior to either of the transforms.
Abstract: The S transform, an extension to the ideas of the Gabor transform and the Wavelet transform, is based on a moving and scalable localising Gaussian window and is shown here to have characteristics that are superior to either of the transforms. The S transform is fully convertible both forward and inverse from the time domain to the 2-D frequency translation (time) domain and to the familiar Fourier frequency domain. Parallel to the translation (time) axis, the S transform collapses as the Fourier transform. The amplitude frequency-time spectrum and the phase frequency-time spectrum are both useful in defining local spectral characteristics. The superior properties of the S transform are due to the fact that the modulating sinusoids are fixed with respect to the time axis while the localising scalable Gaussian window dilates and translates. As a result, the phase spectrum is absolute in the sense that it is always referred to the origin of the time axis, the fixed reference point. The real and imaginary spectrum can be localised independently with a resolution in time corresponding to the period of the basis functions in question. Changes in the absolute phase ofa constituent frequency can be followed along the time axis and useful information can be extracted. An analysis of a sum of two oppositely progressing chirp signals provides a spectacular example of the power of the S transform. Other examples of the applications of the Stransform to synthetic as well as real data are provided.
Citations
More filters
TL;DR: Time-frequency domain signal processing using energy concentration as a feature is a very powerful tool and has been utilized in numerous applications and the expectation is that further research and applications of these algorithms will flourish in the near future.
646 citations
TL;DR: In this article, a generalized S-transform is presented, in which two prescribed functions of frequency control the scale and shape of the analyzing window, and apply it to determining P-wave arrival time in a noisy seismogram.
Abstract: The S-transform is an invertible time-frequency spectral localization technique which combines elements of wavelet transforms and short-time Fourier transforms. In previous usage, the frequency dependence of the analyzing window of the S-transform has been through horizontal and vertical dilations of a basic functional form, usually a Gaussian. In this paper, we present a generalized S-transform in which two prescribed functions of frequency control the scale and the shape of the analyzing window, and apply it to determining P-wave arrival time in a noisy seismogram. The S-transform is also used as a time-frequency filter; this helps in determining the sign of the P arrival.
452 citations
TL;DR: The simulation results reveal that the combination of S-Transform and PNN can effectively detect and classify different PQ events and it is found that the classification performance of PNN is better than both FFML and LVQ.
Abstract: This paper presents an S-Transform based probabilistic neural network (PNN) classifier for recognition of power quality (PQ) disturbances. The proposed method requires less number of features as compared to wavelet based approach for the identification of PQ events. The features extracted through the S-Transform are trained by a PNN for automatic classification of the PQ events. Since the proposed methodology can reduce the features of the disturbance signal to a great extent without losing its original property, less memory space and learning PNN time are required for classification. Eleven types of disturbances are considered for the classification problem. The simulation results reveal that the combination of S-Transform and PNN can effectively detect and classify different PQ events. The classification performance of PNN is compared with a feedforward multilayer (FFML) neural network (NN) and learning vector quantization (LVQ) NN. It is found that the classification performance of PNN is better than both FFML and LVQ.
444 citations
TL;DR: In this paper, a modified wavelet transform known as the S-transform is used for power quality analysis with very good time resolution. But, the amplitude peaks are regions of stationary phase.
Abstract: This paper presents a new approach for power quality analysis using a modified wavelet transform known as the S-transform. The local spectral information of the wavelet transform can, with slight modification, be used to perform local cross spectral analysis with very good time resolution. The "phase correction" absolutely references the phase of the wavelet transform to the zero time point, thus assuring that the amplitude peaks are regions of stationary phase. The excellent time-frequency resolution characteristic of the S-transform makes it an attractive candidate for analysis of power system disturbance signals. Several power quality problems are analyzed using both the S-transform and discrete wavelet transform, showing clearly the advantage of the S-transform in detecting, localizing, and classifying the power quality problems.
441 citations
TL;DR: A more efficient representation is introduced here as a orthogonal set of basis functions that localizes the spectrum and retains the advantageous phase properties of the S-transform, and can perform localized cross spectral analysis to measure phase shifts between each of multiple components of two time series.
363 citations
References
More filters
TL;DR: Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied and the notion of time-frequency localization is made precise, within this framework, by two localization theorems.
Abstract: Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied. The first procedure is the short-time or windowed Fourier transform; the second is the wavelet transform, in which high-frequency components are studied with sharper time resolution than low-frequency components. The similarities and the differences between these two methods are discussed. For both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time-frequency density. Finally, the notion of time-frequency localization is made precise, within this framework, by two localization theorems. >
6,180 citations
01 Jul 1989
TL;DR: A review and tutorial of the fundamental ideas and methods of joint time-frequency distributions is presented with emphasis on the diversity of concepts and motivations that have gone into the formation of the field.
Abstract: 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. >
3,568 citations
CNET1
TL;DR: A simple, nonrigorous, synthetic view of wavelet theory is presented for both review and tutorial purposes, which 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.
Abstract: 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. >
2,945 citations
TL;DR: Wavelet transforms are recent mathematical techniques, based on group theory and square integrable representations, which allows one to unfold a signal, or a field, into both space and scale, and possibly directions.
Abstract: Wavelet transforms are recent mathematical techniques, based on group theory and square integrable representations, which allows one to unfold a signal, or a field, into both space and scale, and possibly directions. They use analyzing functions, called wavelets, which are localized in space. The scale decomposition is obtained by dilating or contracting the chosen analyzing wavelet before convolving it with the signal. The limited spatial support of wavelets is important because then the behavior of the signal at infinity does not play any role. Therefore the wavelet analysis or syn thesis can be performed locally on the signal, as opposed to the Fourier transform which is inherently nonlocal due to the space-filling nature of the trigonometric functions. Wavelet transforms have been applied mostly to signal processing, image coding, and numerical analysis, and they are still evolving. So far there are only two complete presentations of this topic, both written in French, one for engineers (Gasquet & Witomski 1 990) and the other for mathematicians (Meyer 1 990a), and two conference proceedings, the first in English (Combes et al 1 989), the second in French (Lemarie 1 990a). In preparation are a textbook (Holschneider 199 1 ), a course (Dau bee hies 1 99 1), three conference procecdings (Mcyer & Paul 199 1 , Beylkin et al 199 1b, Farge et al 1 99 1), and a special issue of IEEE Transactions
2,770 citations