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Journal ArticleDOI

Localization of the complex spectrum: the S transform

01 Apr 1996-IEEE Transactions on Signal Processing (IEEE)-Vol. 44, Iss: 4, pp 998-1001
TL;DR: The S transform is shown to have some desirable characteristics that are absent in the continuous wavelet transform, and provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum.
Abstract: 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.
Citations
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Journal ArticleDOI
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


Cites background from "Localization of the complex spectru..."

  • ...Research activities reported in the literature can be summarized in the following four aspects: The first two deal with the development of new TFRs based on either signal decomposition or Cohen’s idea....

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  • ...The Fourier transform of the signal cannot depict how the spectral content of the signal changes with time, which is critical in many nonstationary signals in practice....

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Journal ArticleDOI
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


Cites background or methods from "Localization of the complex spectru..."

  • ...The expression of the S-transform given by Stockwell et al. (1996) is...

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  • ...The S-transform ( Stockwell et al., 1996 ) is a time-frequency spectral localization method, similar to the short-time Fourier transform (STFT), but with a Gaussian window whose width scales inversely, and whose height scales linearly, with the frequency....

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  • ...S-transform of Stockwell et al. (1996) and its relation to the Fourier spectrum....

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  • ...Specific cases of the generalized S-transform are the Gaussian S-transform of Stockwell et al. (1996) and the newly introduced hyperbolic S-transform, which is designed to improve the resolution of wave train initiation time over the Gaussian S-transform....

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Journal ArticleDOI
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


Cites background from "Localization of the complex spectru..."

  • ...This paper presents a probabilistic neural network (PNN) classifier based on S-Transform[9]where less numberof features i....

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  • ...In the discrete case, the S-Transform is the projection of the vector defined by the time series , onto a spanning set of vectors [9]....

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  • ...The S-Transform of a discrete time series , is given by [9]...

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Journal ArticleDOI
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

Journal ArticleDOI
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


Cites background or methods from "Localization of the complex spectru..."

  • ...The use of instantaneous frequency in a local spectral representation was introduced for the S-transform [19], and independently was employed by Nelson [47] to study nonstationary multicomponent FM signals....

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  • ...The continuous S-transform [19] of a function h(t) is...

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  • ...This is called “voice windowing” and is analogous to the “voice Gaussian” of the S-transform [19]....

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  • ...A more recent time-frequency representation, the S-transform [19], has found application in a range of fields....

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  • ...(11) of [19], the “n” voice S[jT ,n/NT ] has the same frequency translation applied by the shift of the spectrum H [(m + n)/NT ] by “n” which centers the spectrum around the n frequency....

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References
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Journal ArticleDOI
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


"Localization of the complex spectru..." refers background in this paper

  • ...Along with (1), there exists an admissibility condition on the mother wavelet w(t, d) [5] that w(t, d) must have zero mean....

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01 Jan 1946

5,910 citations


"Localization of the complex spectru..." refers methods in this paper

  • ...Several techniques of examining the time-varying nature of the spectrum have been proposed in the past; among them are the Gabor transform [7], the related short time Fourier transforms, the continuous wavelet transform [8] and the bilinear class of time-frequency distributions known as Cohen’s class [4], of which the Wigner distribution [9] is a member....

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Book
01 Jan 1965
TL;DR: In this paper, the authors provide a broad overview of Fourier Transform and its relation with the FFT and the Hartley Transform, as well as the Laplace Transform and the Laplacian Transform.
Abstract: 1 Introduction 2 Groundwork 3 Convolution 4 Notation for Some Useful Functions 5 The Impulse Symbol 6 The Basic Theorems 7 Obtaining Transforms 8 The Two Domains 9 Waveforms, Spectra, Filters and Linearity 10 Sampling and Series 11 The Discrete Fourier Transform and the FFT 12 The Discrete Hartley Transform 13 Relatives of the Fourier Transform 14 The Laplace Transform 15 Antennas and Optics 16 Applications in Statistics 17 Random Waveforms and Noise 18 Heat Conduction and Diffusion 19 Dynamic Power Spectra 20 Tables of sinc x, sinc2x, and exp(-71x2) 21 Solutions to Selected Problems 22 Pictorial Dictionary of Fourier Transforms 23 The Life of Joseph Fourier

5,714 citations

Journal ArticleDOI
Leon Cohen1
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


"Localization of the complex spectru..." refers background or methods in this paper

  • ...Several techniques of examining the time-varying nature of the spectrum have been proposed in the past; among them are the Gabor transform [7], the related short time Fourier transforms, the continuous wavelet transform [8] and the bilinear class of time-frequency distributions known as Cohen’s class [4], of which the Wigner distribution [9] is a member....

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  • ...This is an advantage over the bilinear class of TFRs (see “propagation of noise” in [4] ), where one £nds TFR{data} = TFR{signal} + TFR{noise} +2 ∗ TFR{signal} ∗ TFR{noise}....

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Journal ArticleDOI
Olivier Rioul1, Martin Vetterli
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


"Localization of the complex spectru..." refers background in this paper

  • ...The reader is referred to Rioul and Vetterli [10] and Young [11] for reviews of the literature....

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