This book provides an overview of the theory and practice of continuous and discrete wavelet transforms. Divided into seven chapters, the first three chapters of the book are introductory, describing the various forms of the wavelet transform and their computation, while the remaining chapters are devoted to applications in fluids, engineering, medicine and miscellaneous areas. Each chapter is well introduced, with suitable examples to demonstrate key concepts. Illustrations are included where appropriate, thus adding a visual dimension to the text. A noteworthy feature is the inclusion, at the end of each chapter, of a list of further resources from the academic literature which the interested reader can consult. The first chapter is purely an introduction to the text. The treatment of wavelet transforms begins in the second chapter, with the definition of what a wavelet is. The chapter continues by defining the continuous wavelet transform and its inverse and a description of how it may be used to interrogate signals. The continuous wavelet transform is then compared to the short-time Fourier transform. Energy and power spectra with respect to scale are also discussed and linked to their frequency counterparts. Towards the end of the chapter, the two-dimensional continuous wavelet transform is introduced. Examples of how the continuous wavelet transform is computed using the Mexican hat and Morlet wavelets are provided throughout. The third chapter introduces the discrete wavelet transform, with its distinction from the discretized continuous wavelet transform having been made clear at the end of the second chapter. In the first half of the chapter, the logarithmic discretization of the wavelet function is described, leading to a discussion of dyadic grid scaling, frames, orthogonal and orthonormal bases, scaling functions and multiresolution representation. The fast wavelet transform is introduced and its computation is illustrated with an example using the Haar wavelet. The second half of the chapter groups together miscellaneous points about the discrete wavelet transform, including coefficient manipulation for signal denoising and smoothing, a description of Daubechies' wavelets, the properties of translation invariance and biorthogonality, the two-dimensional discrete wavelet transforms and wavelet packets. The fourth chapter is dedicated to wavelet transform methods in the author's own specialty, fluid mechanics. Beginning with a definition of wavelet-based statistical measures for turbulence, the text proceeds to describe wavelet thresholding in the analysis of fluid flows. The remainder of the chapter describes wavelet analysis of engineering flows, in particular jets, wakes, turbulence and coherent structures, and geophysical flows, including atmospheric and oceanic processes. The fifth chapter describes the application of wavelet methods in various branches of engineering, including machining, materials, dynamics and information engineering. Unlike previous chapters, this (and subsequent) chapters are styled more as literature reviews that describe the findings of other authors. The areas addressed in this chapter include: the monitoring of machining processes, the monitoring of rotating machinery, dynamical systems, chaotic systems, non-destructive testing, surface characterization and data compression. The sixth chapter continues in this vein with the attention now turned to wavelets in the analysis of medical signals. Most of the chapter is devoted to the analysis of one-dimensional signals (electrocardiogram, neural waveforms, acoustic signals etc.), although there is a small section on the analysis of two-dimensional medical images. The seventh and final chapter of the book focuses on the application of wavelets in three seemingly unrelated application areas: fractals, finance and geophysics. The treatment on wavelet methods in fractals focuses on stochastic fractals with a short section on multifractals. The treatment on finance touches on the use of wavelets by other authors in studying stock prices, commodity behaviour, market dynamics and foreign exchange rates. The treatment on geophysics covers what was omitted from the fourth chapter, namely, seismology, well logging, topographic feature analysis and the analysis of climatic data. The text concludes with an assortment of other application areas which could only be mentioned in passing. Unlike most other publications in the subject, this book does not treat wavelet transforms in a mathematically rigorous manner but rather aims to explain the mechanics of the wavelet transform in a way that is easy to understand. Consequently, it serves as an excellent overview of the subject rather than as a reference text. Keeping the mathematics to a minimum and omitting cumbersome and detailed proofs from the text, the book is best-suited to those who are new to wavelets or who want an intuitive understanding of the subject. Such an audience may include graduate students in engineering and professionals and researchers in engineering and the applied sciences.

The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance

A review of stochastic resonance in rotating machine fault detection

This paper presents a novel signal processing scheme, bandwidth empirical mode decomposition, and adaptive multiscale morphological analysis (BEMD-AMMA) for early fault diagnosis of rolling bearings. In this scheme, we propose a bandwidth based method to select the best envelope interpolation method. First, multiple envelope algorithms are defined and separately subtracted from the original data to obtain the preintrinsic mode functions (PIMFs). Second, an IMF with the smallest frequency bandwidth is selected to be the optimal IMF (OIMF). Third, this OIMF is subtracted from the original signal, and then repeat the sifting process until the residual is a constant or monotonic. Since the OIMF has the smallest frequency bandwidth, the mode mixing phenomenon can be significantly weakened. After that the OIMFs with clear fault information are used to construct the main component of the original signal. Then, the AMMA is introduced to demodulate the constructed main component. Simulation and experimental vibration signals are employed to evaluate the effectiveness of the proposed method. Results show that the proposed method outperforms EMD-AMMA, ensemble empirical mode decomposition-AMMA, and generalized empirical mode decomposition-empirical envelope demodulation in detecting early inner race fault.

Application of Bandwidth EMD and Adaptive Multiscale Morphology Analysis for Incipient Fault Diagnosis of Rolling Bearings

A fault diagnosis scheme for planetary gearboxes using adaptive multi-scale morphology filter and modified hierarchical permutation entropy

Fault diagnosis of rotating machinery plays a significant role in the industrial production and engineering field. Owing to the drawbacks of traditional fault diagnosis methods, such as heavily dependence on human knowledge and professional experience, intelligent fault diagnosis based on deep learning (DL) has aroused the interest of researchers. DL achieves the desirable automatic feature learning and fault classification. Therefore, in this review, DL and DL-based intelligent fault diagnosis techniques are overviewed. DL-based fault diagnosis approaches for rotating machinery are summarized and discussed, primarily including bearing, gear/gearbox and pumps. Finally, with respect to modern intelligent fault diagnosis, the existing challenges and possible future research orientations are prospected and analyzed.

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Deep Learning-Based Intelligent Fault Diagnosis Methods Toward Rotating Machinery

Demodulation for hydraulic pump fault signals based on local mean decomposition and improved adaptive multiscale morphology analysis

Gear fault diagnosis method based on local mean decomposition and generalized morphological fractal dimensions

Data processing is widely used to extract effective component from original signal, which is essential in mechanical condition monitoring and fault diagnosis. In order to solve the invalid component and non-stationary feature in the measured signal, the extraction method for effective signal component is proposed based on extreme-point symmetric mode decomposition (ESMD) and Kullback–Leibler (K–L) divergence. This method fully integrates the characteristics of ESMD in self-adaptive decomposition and the advantages of K–L divergence in measuring the distance between different signals. The effective and invalid components of non-stationary signal are automatically separated by ESMD, and the effective components are further identified through K–L divergence calculation. Some analyses of simulated data and experimental data were investigated. And the effect of the proposed method in effective component extraction was emphatically explored. Research results indicate that the proposed method can adaptively acquire effective signal components with higher accuracy. Moreover, compared with the classic method, it is more efficient in the extraction of effective components from complex signal. In addition, this research solves the interference problem of invalid signals and accurately reconstructs the desired useful signal.

Extraction method for signal effective component based on extreme-point symmetric mode decomposition and Kullback–Leibler divergence

Hydraulic automatic gauge control (AGC) system is the core control system to ensure the thickness accuracy of cold-rolled plate, and the reliability of its load roll system is the key to guarantee the rolling process with high precision, high speed, continuity and stability. However, the working mechanism of hydraulic AGC system is complex. And it possesses some features such as high nonlinearity, time variability and strong coupling. The vertical vibration easily happens in its working process. Moreover, the stability of system is seriously affected. Nevertheless, the incentive is difficult to determine. In this paper, the theory and method of nonlinear dynamics were introduced to establish the load vertical vibration (LVV) model of HAGC system under nonlinear action. The model was solved with multi-scale method, and the amplitude-frequency characteristic equation of system was obtained. Moreover, the research object of this article also focused on the nonlinear elastic force, nonlinear damping force and nonlinear excitation force, and deeply explored the effect of nonlinear parameters change on the amplitude-frequency characteristics of LVV system. The mechanism and incentive for LVV of hydraulic AGC system were further revealed. The research results lay a theoretical foundation for the vibration inducement and suppression of hydraulic AGC system.

Amplitude-frequency characteristics analysis for vertical vibration of hydraulic AGC system under nonlinear action

In the metallurgical industry, hydraulic automatic gauge control (HAGC) is a core mechanism for thickness control of plates used in the rolling process. The stability of the HAGC system’s kernel position closed-loop is key to ensuring a process with high precision, speed and reliability. However, the closed-loop position control system is typically nonlinear, and its stability is affected by several factors, making it difficult to analyze instability in the system. This paper describes in detail the functioning of the position closed-loop system. A mathematical model of each component was established using theoretical analysis. An incremental transfer model of the position closed-loop system was also derived by studying the connections between each component. In addition, based on the derived information transfer relationship, a transfer block diagram of disturbance quantity of the system was established. Furthermore, the Popov frequency criterion method was introduced to ascertain its absolute stability. The results indicate that the absolute stability conditions of the position closed-loop system are derived in two situations: when spool displacement is positive or negative. This study lays a theoretical foundation for research on the instability mechanism of an HAGC system.

https://www.mdpi.com/2227-9717/7/10/766/pdf

Wanlu Jiang

Papers

Demodulation for hydraulic pump fault signals based on local mean decomposition and improved adaptive multiscale morphology analysis

Gear fault diagnosis method based on local mean decomposition and generalized morphological fractal dimensions

Extraction method for signal effective component based on extreme-point symmetric mode decomposition and Kullback–Leibler divergence

Amplitude-frequency characteristics analysis for vertical vibration of hydraulic AGC system under nonlinear action

Absolute Stability Condition Derivation for Position Closed-Loop System in Hydraulic Automatic Gauge Control