Single-shot Phase Retrieval from a Fractional Fourier Transform Perspective

The realm of classical phase retrieval concerns itself with the arduous task of recovering a signal from its Fourier magnitude measurements, which are fraught with inherent ambiguities. A single-exposure intensity measurement is commonly deemed insufficient for the reconstruction of the primal signal, given that the absent phase component is imperative for the inverse transformation. In this work, we present a novel single-shot phase retrieval paradigm from a fractional Fourier transform (FrFT) perspective, which involves integrating the FrFT-based physical measurement model within a self-supervised reconstruction scheme. Specifically, the proposed FrFT-based measurement model addresses the aliasing artifacts problem in the numerical calculation of Fresnel diffraction, featuring adaptability to both short-distance and long-distance propagation scenarios. Moreover, the intensity measurement in the FrFT domain proves highly effective in alleviating the ambiguities of phase retrieval and relaxing the previous conditions on oversampled or multiple measurements in the Fourier domain. Furthermore, the proposed self-supervised reconstruction approach harnesses the fast discrete algorithm of FrFT alongside untrained neural network priors, thereby attaining preeminent results. Through numerical simulations, we demonstrate that both amplitude and phase objects can be effectively retrieved from a single-shot intensity measurement using the proposed approach and provide a promising technique for support-free coherent diffraction imaging

Radio Science Experiment Data Analysis in the framework of the ESA Missions “Venus Express" and “Rosetta"

Occultation measurements exploit an observational geometry in which the spacecraft to Earth communication link is interrupted by the planet itself. Coherent, high-rate (100 ksamples/s) sampling of the down-converted RF incoming signal enables the OL receiving system to safeguard the high dynamics (up to 2 kHz/s) of the weak signals (attenuation > 50dB) emerging from the deep layers of the Venus atmosphere. The purpose of the developed software package, the Open-Loop data processing software (OL SW), is to extract the information embedded in noise by means of an iterative strategy. Essential skill of the OL SW is the progressive reduction of the signal bandwidth while at the same time maintaining high time resolution of the data. This implies high spacial resolution of the sounded media (i.e., the Venus atmosphere) and the capability of resolving effects of multipath propagation

Stochastic response determination and spectral identification of complex dynamic structural systems

Uncertainty propagation in engineering mechanics and dynamics is a highly challenging problem that requires development of analytical/numerical techniques for determining the stochastic response of complex engineering systems. In this regard, although Monte Carlo simulation (MCS) has been the most versatile technique for addressing the above problem, it can become computationally daunting when faced with high-dimensional systems or with computing very low probability events. Thus, there is a demand for pursuing more computationally efficient methodologies. Further, most structural systems are likely to exhibit nonlinear and time-varying behavior when subjected to extreme events such as severe earthquake, wind and sea wave excitations. In such cases, a reliable identification approach is behavior and for assessing its reliability. Current work addresses two research themes in the field of stochastic engineering dynamics related to the above challenges. In the first part of the dissertation, the recently developedWiener Path Integral (WPI) technique for determining the joint response probability density function (PDF) of nonlinear systems subject to Gaussian white noise excitation is generalized herein to account for non-white, non-Gaussian, and non-stationary excitation processes. Specifically, modeling the excitation process as the output of a filter equation with Gaussian white noise as its input, it is possible to define an augmented response vector process to be considered in the WPI solution technique. A significant advantage relates to the fact that the technique is still applicable even for arbitrary excitation power spectrum forms. In such cases, it is shown that the use of a filter approximation facilitates the implementation of the WPI technique in a straightforward manner, without compromising its accuracy necessarily. Further, in addition to dynamical systems subject to stochastic excitation, the technique can also account for a special class of engineering mechanics problems where the media properties are modeled as non-Gaussian and non-homogeneous stochastic fields. Several numerical examples pertaining to both single- and multi-degree-of freedom systems are considered, including a marine structural system exposed to flow-induced non-white excitation, as well as a beam with a non-Gaussian and non-homogeneous Young’s modulus. Comparisons with MCS data demonstrate the accuracy of the technique. In the second part of the dissertation, a novel multiple-input/single-output (MISO) system identification technique is developed for parameter identification of nonlinear time-variant multi-degree-of-freedom oscillators with fractional derivative terms subject to incomplete non-stationary data. The technique utilizes a representation of the nonlinear restoring forces as a set of parallel linear subsystems. In this regard, the oscillator is transformed into an equivalent MISO system in the wavelet domain. Next, a recently developed L1-norm minimization procedure based on compressive sampling theory is applied for determining the wavelet coefficients of the available incomplete non-stationary input-output (excitation-response) data. Finally, these wavelet coefficients are utilized to determine appropriately defined time- and frequency-dependent wavelet based frequency response functions and related oscillator parameters. A nonlinear time-variant system with fractional derivative elements is used as a numerical example to demonstrate the reliability of the technique even in cases of noise corrupted and incomplete data

Condition Monitoring of Mechanical Faults in Variable Speed Induction Motor Drives. Application of Stator Current Time-Frequency Analysis and Parameter Estimation

Ce travail de thèse traite de la détection et du diagnostic de défaillances mécaniques par analyse du courant statorique dans les entraînements électriques à base de machine asynchrone. Deux effets d'un défaut mécanique, des oscillations de couple et une excentricité d'entrefer, sont supposés. La modélisation par approche des ondes de forces magnétomotrices et de perméance conduit à deux modèles analytiques du signal courant. La conséquence des défauts est soit une modulation de phase, soit une modulation d'amplitude du signal courant statorique. Ces phénomènes sont détectés par une analyse spectrale en régime permanent, ou des méthodes temps fréquence en régime transitoire. Les méthodes étudiées sont la fréquence instantanée, le spectrogramme et la représentation de Wigner-Ville. L'estimation paramétrique d'indices de modulation a également été traitée. Des résultats de simulation et expérimentaux permettent de valider les signatures et d'extraire de façon automatique des indicateurs de défaut. De plus, une méthode permettant la distinction des oscillations de couple d'une excentricité dynamique est proposée. L'étude est complétée par une implémentation sur DSP des méthodes temps-fréquence afin de démontrer la faisabilité d'une surveillance en ligne. ABSTRACT : This Ph.D. thesis deals with condition monitoring of mechanical failures in variable speed induction motor drives by stator current analysis. Two effects of a mechanical fault are considered: load torque oscillations and airgap eccentricity. The analytical modelling using the magnetomotive force and permeance wave approach leads to two stator current models. The fault provokes amplitude or phase modulation of the fundamental current component. Suitable detection methods are spectral analysis and parameter estimation in steady state whereas time-frequency analysis is required during transients. Instantaneous frequency estimation, the Wigner Distribution and the spectrogram are studied. Simulation and experimental results validate the theoretical approach. Automatic extraction of fault indicators is proposed for an unsupervised monitoring system. Moreover, load torque oscillations and dynamic eccentricity can be discriminated with the proposed methods. The feasibility of an on-line monitoring system is demonstrated by a DSP implementation of the time-frequency analysis including indicator extraction

Digital Signal Processing (Second Edition)

This book provides an account of the mathematical background, computational methods and software engineering associated with digital signal processing. The aim has been to provide the reader with the mathematical methods required for signal analysis which are then used to develop models and algorithms for processing digital signals and finally to encourage the reader to design software solutions for Digital Signal Processing (DSP). In this way, the reader is invited to develop a small DSP library that can then be expanded further with a focus on his/her research interests and applications. There are of course many excellent books and software systems available on this subject area. However, in many of these publications, the relationship between the mathematical methods associated with signal analysis and the software available for processing data is not always clear. Either the publications concentrate on mathematical aspects that are not focused on practical programming solutions or elaborate on the software development of solutions in terms of working ‘black-boxes’ without covering the mathematical background and analysis associated with the design of these software solutions. Thus, this book has been written with the aim of giving the reader a technical overview of the mathematics and software associated with the ‘art’ of developing numerical algorithms and designing software solutions for DSP, all of which is built on firm mathematical foundations. For this reason, the work is, by necessity, rather lengthy and covers a wide range of subjects compounded in four principal parts. Part I provides the mathematical background for the analysis of signals, Part II considers the computational techniques (principally those associated with linear algebra and the linear eigenvalue problem) required for array processing and associated analysis (error analysis for example). Part III introduces the reader to the essential elements of software engineering using the C programming language, tailored to those features that are used for developing C functions or modules for building a DSP library. The material associated with parts I, II and III is then used to build up a DSP system by defining a number of ‘problems’ and then addressing the solutions in terms of presenting an appropriate mathematical model, undertaking the necessary analysis, developing an appropriate algorithm and then coding the solution in C. This material forms the basis for part IV of this work. In most chapters, a series of tutorial problems is given for the reader to attempt with answers provided in Appendix A. These problems include theoretical, computational and programming exercises. Part II of this work is relatively long and arguably contains too much material on the computational methods for linear algebra. However, this material and the complementary material on vector and matrix norms forms the computational basis for many methods of digital signal processing. Moreover, this important and widely researched subject area forms the foundations, not only of digital signal processing and control engineering for example, but also of numerical analysis in general. The material presented in this book is based on the lecture notes and supplementary material developed by the author for an advanced Masters course ‘Digital Signal Processing’ which was first established at Cranfield University, Bedford in 1990 and modified when the author moved to De Montfort University, Leicester in 1994. The programmes are still operating at these universities and the material has been used by some 700++ graduates since its establishment and development in the early 1990s. The material was enhanced and developed further when the author moved to the Department of Electronic and Electrical Engineering at Loughborough University in 2003 and now forms part of the Department’s post-graduate programmes in Communication Systems Engineering. The original Masters programme included a taught component covering a period of six months based on two semesters, each Semester being composed of four modules. The material in this work covers the first Semester and its four parts reflect the four modules delivered. The material delivered in the second Semester is published as a companion volume to this work entitled Digital Image Processing, Horwood Publishing, 2005 which covers the mathematical modelling of imaging systems and the techniques that have been developed to process and analyse the data such systems provide. Since the publication of the first edition of this work in 2003, a number of minor changes and some additions have been made. The material on programming and software engineering in Chapters 11 and 12 has been extended. This includes some additions and further solved and supplementary questions which are included throughout the text. Nevertheless, it is worth pointing out, that while every effort has been made by the author and publisher to provide a work that is error free, it is inevitable that typing errors and various ‘bugs’ will occur. If so, and in particular, if the reader starts to suffer from a lack of comprehension over certain aspects of the material (due to errors or otherwise) then he/she should not assume that there is something wrong with themselves, but with the author

Phase extraction of non-stationary signals produced in dynamic interferometry involving speckle waves

It is now widely acknowledged, among communities of researchers and engineers of very different horizons, that speckle interferometry (SI) offers powerful techniques to characterize mechanical rough surfaces with a submicronic accuracy in static or quasi-static regime, when small displacements are involved (typically several microns or tens of microns). The issue of dynamic regimes with possibly large deformations (typically several hundreds of microns) is still topical and prevents an even more widespread use of speckle techniques. This is essentially due to the lack of efficient processing schemes able to cope with non-stationary AM-FM interferometric signals. In addition, decorrelation-induced phase errors represent an hindrance to accurate measurement when such large displacements and classical fringe analysis techniques are considered. This work is an attempt to address those issues and to endeavor to make the most of speckle interferometry signals. Our answers to those problems are located on two different levels. First of all, we adopt the temporal analysis approach, i.e. the analysis of the temporal signal of each pixel of the sensor area used to record the interferograms. A return to basics of phase extraction is operated to properly identify the conditions under which the computed phase is meaningful and thus give some insight on the physical phenomenon under analysis. Due to their intrinsic non-stationary nature, a preprocessing tool is missing to put the SI temporal signals in a shape which ensures an accurate phase computation, whichever technique is chosen. This is where the Empirical Mode Decomposition (EMD) intervenes. This technique, somehow equivalent to an adaptive filtering technique, has been studied and tailored to fit with our expectations. The EMD has shown a great ability to remove efficiently the random fluctuating background intensity and to evaluate the modulation intensity. The Hilbert tranform (HT) is the natural quadrature operator. Its use to build an analytical signal from the so-detrended SI signal, for subsequent phase computation, has been studied and assessed. Other phase extraction techniques have been considered as well for comparison purposes. Finally, our answer to the decorrelation-induced phase error relies on the well-known result that the higher the pixel modulation intensity, the lower the random phase error. We took benefit from this result – not only linked to basic SNR considerations, but more specifically to the intrinsic phase structure of speckle fields – with a novel approach. The regions within the pixel signal history classified as unreliable because under-modulated, are purely and simply discarded. An interpolation step with the Delaunay triangulation is carried out with the so-obtained non-uniformly sampled phase maps to recover a smooth phase which relies on the most reliable available data. Our schemes have been tested and discussed with simulated and experimental SI signals. We eventually have developed a versatile, accurate and efficient phase extraction procedure, perfectly able to tackle the challenge of dynamic behaviors characterization, even for displacements and/or deformations beyond the classical limit of the correlation dimensions

New approaches for EEG signal processing: artifact EOG removal by ICA-RLS scheme and tracks extraction method

Localizing the bioelectric phenomena originating from the cerebral cortex and evoked by auditory and somatosensory stimuli are clear objectives to both understand how the brain works and to recognize different pathologies. Diseases such as Parkinson’s, Alzheimer’s, schizophrenia and epilepsy are intensively studied to find a cure or accurate diagnosis. Epilepsy is considered the disease with major prevalence within disorders with neurological origin. The recurrent and sudden incidence of seizures can lead to dangerous and possibly life-threatening situations. Since disturbance of consciousness and sudden loss of motor control often occur without any warning, the ability to predict epileptic seizures would reduce patients’ anxiety, thus considerably improving quality of life and safety. The common procedure for epilepsy seizure detection is based on brain activity monitorization via electroencephalogram (EEG) data. This process consumes a lot of time, especially in the case of long recordings, but the major problem is the subjective nature of the analysis among specialists when analyzing the same record. From this perspective, the identification of hidden dynamical patterns is necessary because they could provide insight into the underlying physiological mechanisms that occur in the brain. Time-frequency distributions (TFDs) and adaptive methods have demonstrated to be good alternatives in designing systems for detecting neurodegenerative diseases. TFDs are appropriate transformations because they offer the possibility of analyzing relatively long continuous segments of EEG data even when the dynamics of the signal are rapidly changing. On the other hand, most of the detection methods proposed in the literature assume a clean EEG signal free of artifacts or noise, leaving the preprocessing problem opened to any denoising algorithm. In this thesis we have developed two proposals for EEG signal processing: the first approach consists in electrooculogram (EOG) removal method based on a combination of ICA and RLS algorithms which automatically cancels the artifacts produced by eyes movement without the use of external “ad hoc” electrode. This method, called ICA-RLS has been compared with other techniques that are in the state of the art and has shown to be a good alternative for artifacts rejection. The second approach is a novel method in EEG features extraction called tracks extraction (LFE features). This method is based on the TFDs and partial tracking. Our results in pattern extractions related to epileptic seizures have shown that tracks extraction is appropriate in EEG detection and classification tasks, being practical, easily applicable in medical environment and has acceptable computational cost

Development of new fault detection methods for rotating machines (roller bearings)

Abstract Early fault diagnosis of roller bearings is extremely important for rotating machines, especially for high speed, automatic and precise machines. Many research efforts have been focused on fault diagnosis and detection of roller bearings, since they constitute one the most important elements of rotating machinery. In this study a combination method is proposed for early damage detection of roller bearing. Wavelet packet transform (WPT) is applied to the collected data for denoising and the resulting clean data are break-down into some elementary components called Intrinsic mode functions (IMFs) using Ensemble empirical mode decomposition (EEMD) method. The normalized energy of three first IMFs are used as input for Support vector machine (SVM) to recognize whether signals are sorting out from healthy or faulty bearings. Then, since there is no robust guide to determine amplitude of added noise in EEMD technique, a new Performance improved EEMD (PIEEMD) is proposed to determine the appropriate value of added noise. A novel feature extraction method is also proposed for detecting small size defect using Teager-Kaiser energy operator (TKEO). TKEO is applied to IMFs obtained to create new feature vectors as input data for one-class SVM. The results of applying the method to acceleration signals collected from an experimental bearing test rig demonstrated that the method can be successfully used for early damage detection of roller bearings. Most of the diagnostic methods that have been developed up to now can be applied for the case stationary working conditions only (constant speed and load). However, bearings often work at time-varying conditions such as wind turbine supporting bearings, mining excavator bearings, vehicles, robots and all processes with run-up and run-down transients. Damage identification for bearings working under non-stationary operating conditions, especially for early/small defects, requires the use of appropriate techniques, which are generally different from those used for the case of stationary conditions, in order to extract fault-sensitive features which are at the same time insensitive to operational condition variations. Some methods have been proposed for damage detection of bearings working under time-varying speed conditions. However, their application might increase the instrumentation cost because of providing a phase reference signal. Furthermore, some methods such as order tracking methods still can be applied when the speed variation is limited. In this study, a novel combined method based on cointegration is proposed for the development of fault features which are sensitive to the presence of defects while in the same time they are insensitive to changes in the operational conditions. It does not require any additional measurements and can identify defects even for considerable speed variations. The signals acquired during run-up condition are decomposed into IMFs using the performance improved EEMD method. Then, the cointegration method is applied to the intrinsic mode functions to extract stationary residuals. The feature vectors are created by applying the Teager-Kaiser energy operator to the obtained stationary residuals. Finally, the feature vectors of the healthy bearing signals are utilized to construct a separating hyperplane using one-class support vector machine. Eventually the proposed method was applied to vibration signals measured on an experimental bearing test rig. The results verified that the method can successfully distinguish between healthy and faulty bearings even if the shaft speed changes dramatically