A Nonlinear Squeezing of the Continuous Wavelet Transform Based on Auditory Nerve Models

Daubechies, Ingrid; Maes, Stéphane

doi:10.1201/9780203734032-20

Cited by 170 publications

(167 citation statements)

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“…Instead of computing the IF's after the signal is decomposed as is done when applying EMD, the approach that Daubechies et al Daubechies and Maes 1996) proposed is first to estimate the IF's of the signal components, under the assumption that the signal satisfies certain strict properties of the AHM in (11), before recovering the signal components of the model. For this purpose, the notion of the synchrosqueezed wavelet transform was introduced to compute a single reference IF function through which the IF's of all the signal components are "squeezed out" from the input signal in the form of a digital image, allowing the estimation of the individual IF functions and the signal components themselves.…”

Section: Synchrosqueezed Wavelet Transformmentioning

confidence: 99%

“…Hence, a Meyer-like wavelet was used for the SST in Daubechies et al (2011) and Daubechies and Maes (1996), with off-line implementation. In our recent work (Chui et al 2014), the notion of compactly supported spline vanishing moment (VM) wavelets was introduced for real-time computation of the CWT as well as for avoiding numerical estimation of the derivative of the CWT in the FRA rule (18).…”

Section: Synchrosqueezed Wavelet Transformmentioning

confidence: 99%

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Signal analysis via instantaneous frequency estimation of signal components

Chui

Walt

2015

Int J Geomath

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The empirical mode decomposition (EMD) algorithm, introduced by Huang et al. (Proc Roy Soc Lond Ser A Math Phys Eng Sci 454(1971):903-995, 1998, is arguably the most popular mathematical scheme for non-stationary signal decomposition and analysis. The objective of EMD is to separate a given signal into a number of components, called intrinsic mode functions (IMF's) after which the instantaneous frequency (IF) and amplitude of each IMF are computed through Hilbert spectral analysis (HSA). On the other hand, the synchrosqueezed wavelet transform (SST), introduced by Daubechies and Maes (Wavelets in Medicine and Biology, pp. 527-546, 1996) and further developed by Daubechies et al. (Appl Comput Harmon Anal 30:243-261, 2011), is applied to estimate the IF's of all signal components of the given signal, based on one single reference "IF function", under the assumption that the signal components satisfy certain strict properties of a so-called adaptive harmonic model, before the signal components of the model are recovered. The objective of our paper is to develop a hybrid EMD-SST computational scheme by applying a "modified SST" to each IMF of the EMD, as an alternative approach to the original EMD-HSA method. While our modified SST assures non-negative instantaneous frequencies of the IMF's, application of the EMD scheme eliminates the dependence on a single reference IF value as well as the guessing work of the number of signal components in the original SST approach. Our modification of the SST consists of applying vanishing moment wavelets (introduced in a recent paper with stacked knots to process signals on bounded or half-infinite time intervals, and spline curve fitting with optimal smoothing parameter selection through generalized cross-validation. In addition, we formulate a local cubic spline interpolation scheme for real-time realization of the EMD sifting process that improves over the standard global cubic spline interpolation, both in quality and computational cost, particularly when applied to bounded and half-infinite time intervals.

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Section: Synchrosqueezed Wavelet Transformmentioning

confidence: 99%

Section: Synchrosqueezed Wavelet Transformmentioning

confidence: 99%

Signal analysis via instantaneous frequency estimation of signal components

Chui

Walt

2015

Int J Geomath

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“…In order to improve the wavelet transform, synchrosqueezing has recently been proposed [13,14]. Synchrosqueezing adds to an existing transform by examining the local oscillations with respect to time.…”

Section: Introductionmentioning

confidence: 99%

Synchrosqueezed wavelet transform for damping identification

Mihalec

Slavič

Boltežar

2016

Mechanical Systems and Signal Processing

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“…Daubechies et al [13] introduce the notion of synchrosqueezing transform (SST) for audio signal processing as follows: first they compute the frequency re-assignment (FRA) reference value: approximates the delta distribution when β tends to 0. In [12], Daubechies et al develop a rigorous theory of the SST in (1.6) for extracting more than one instantaneous frequencies of the given signal G(t) at any desired time instant t = b, by choosing some vector (a [1], .…”

Section: Introductionmentioning

confidence: 99%

Signal decomposition and analysis via extraction of frequencies

Chui

Mhaskar

2016

Applied and Computational Harmonic Analysis

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a r t i c l e i n f o a b s t r a c t Available online xxxx Communicated by the Editors Dedicated to Professor Ranko Bojanic on the occasion of his 90th birthday Keywords: Blind source signal separation Signal separation operator Adaptive harmonic model Instantaneous frequency Time series analysisTime-frequency analysis is central to signal processing, with standard adaptation to higher dimensions for imaging applications, and beyond. However, although the theory, methods, and algorithms for stationary signals are well developed, mathematical analysis of non-stationary signals is almost nonexistent. For a realvalued signal defined on the time-domain R, a classical approach to compute its instantaneous frequency (IF) is to consider the amplitude-frequency modulated (AM-FM) formulation of its complex (or analytic) signal extension, via the Hilbert transform. In a popular paper by Huang et al., the so-called empirical mode decomposition (EMD) scheme is introduced to separate such a signal as a sum of finitely many intrinsic mode functions (IMFs), with a slowly oscillating signal as the remainder, so that more than one IFs of the given signal can be computed by extending each IMF to an AM-FM signal component. Based on the continuous wavelet transform (CWT), the notion of synchrosqueezing transform (SST), introduced by Daubechies and Maes in 1996, and further developed by Daubechies, Lu, and Wu (DLW) in a 2011 paper, provides another approach to extract more than one IFs of the signal on R. Furthermore, by introducing a list of fairly restrictive conditions on the adaptive harmonic model (AHM), the DLW paper also derives a theory for estimating the signal components according to this model, by using the IFs with estimates from the SST. The objective of our present paper is to introduce another mathematical theory, along with rigorous methods and computational schemes, to achieve a more ambitious goal than the SST approach, first to extract the polynomial-like trend from the source signal, then to compute the exact number of signal components according to a less restrictive AHM model, then to obtain better estimates of the IFs and instantaneous amplitudes (IAs) of the signal components, and finally to separate the signal components from the (blind) source signal. Furthermore, our computational scheme can be realized in near-real-time, and our mathematical theory has direct extension to the multivariate setting.

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A Nonlinear Squeezing of the Continuous Wavelet Transform Based on Auditory Nerve Models

Cited by 170 publications

References 1 publication

Signal analysis via instantaneous frequency estimation of signal components

Signal analysis via instantaneous frequency estimation of signal components

Synchrosqueezed wavelet transform for damping identification

Signal decomposition and analysis via extraction of frequencies

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