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Intelligent fault diagnosis methods based on signal analysis have been widely used for bearing fault diagnosis. These methods use a pre-determined transformation (such as empirical mode decomposition, fast Fourier transform, discrete wavelet transform) to convert time-series signals into frequency domain signals, the performance of dignostic system is significantly rely on the extracted features. However, extracting signal characteristic is fairly time consuming and depends on specialized signal processing knowledge. Although some studies have developed highly accurate algorithms, the diagnostic results rely heavily on large data sets and unreliable human analysis. This study proposes an automatic feature learning neural network that utilizes raw vibration signals as inputs, and uses two convolutional neural networks with different kernel sizes to automatically extract different frequency signal characteristics from raw data. Then long short-term memory was used to identify the fault type according to learned features. The data is down-sampled before inputting into the network, greatly reducing the number of parameters. The experiment shows that the proposed method can not only achieve 98.46% average accuracy, exceeding some state-of-the-art intelligent algorithms based on prior knowledge and having better performance in noisy environments.

Since their inception, computational homogenization methods based on the fast Fourier transform (FFT) have grown in popularity, establishing themselves as a powerful tool applicable to complex, digitized microstructures. At the same time, the understanding of the underlying principles has grown, in terms of both discretization schemes and solution methods, leading to improvements of the original approach and extending the applications. This article provides a condensed overview of results scattered throughout the literature and guides the reader to the current state of the art in nonlinear computational homogenization methods using the fast Fourier transform.

Positive definiteness is determined for a wide class of functions relevant in the study of operator means and their norm comparisons. Then, this information is used to obtain an abundance of new sharp (unitarily) norm inequalities comparing various operator means and sometimes other related operators.

Using a codimension-1 algebraic cycle obtained from the Poincaré line bundle, Beauville defined the Fourier transform on the Chow groups of an abelian variety A and showed that the Fourier transform induces a decomposition of the Chow ring \\mathrm{CH}^*(A). By using a codimension-2 algebraic cycle representing the Beauvilleâe\"Bogomolov class, the authors give evidence for the existence of a similar decomposition for the Chow ring of Hyperkähler varieties deformation equivalent to the Hilbert scheme of length-2 subschemes on a K3 surface. They indeed establish the existence of such a decomposition for the Hilbert scheme of length-2 subschemes on a K3 surface and for the variety of lines on a very general cubic fourfold.

Ball bearing monitoring employs time-frequency techniques to facilitate the early detection of faults; however, the presence of non-stationary or noisy signals can limit the effectiveness of these techniques, requiring advanced methods for reliable predictive maintenance. This study proposes a methodology for fault detection in complex systems,utilising Principal Component Analysis (PCA) to identify indicators with a higher probability of fault. Subsequent to this, the signal characteristics are decomposed using the Fast Fourier Transform (FFT). This technique is employed to identify the Hotelling component and the SPE (quadratic prediction error), with the objective of determining the state of health of the rolling bearings. This is achieved by extracting the frequencies and harmonics that characterise the fault. The Hotelling component considers elements in the main space with a higher energy representation for evaluation, while the SPE considers elements in the residual space. The results demonstrate a rapidly appreciable range of detection and dispersion of faulty signals. A comparative analysis of the KPCA-FFT and PCA-FFT results is performed. However, this study demonstrates that the combination of PCA-FFT with the Hotelling index test and SPE is more suitable for evaluating signals with defects.

The fast Fourier transform (FFT) algorithm was developed by Cooley and Tukey in 1965. It could reduce the computational complexity of discrete Fourier transform significantly from \\[O(N^2)\\] to \\[O(N\\log _2 {N})\\]. The invention of FFT is considered as a landmark development in the field of digital signal processing (DSP), since it could expedite the DSP algorithms significantly such that real-time digital signal processing could be possible. During the past 50 years, many researchers have contributed to the advancements in the FFT algorithm to make it faster and more efficient in order to match with the requirements of various applications. In this article, we present a brief overview of the key developments in FFT algorithms along with some popular applications in speech and image processing, signal analysis, and communication systems.

Gratings 1 and holograms 2 use patterned surfaces to tailor optical signals by diffraction. Despite their long history, variants with remarkable functionalities continue to be developed 3 , 4 . Further advances could exploit Fourier optics 5 , which specifies the surface pattern that generates a desired diffracted output through its Fourier transform. To shape the optical wavefront, the ideal surface profile should contain a precise sum of sinusoidal waves, each with a well defined amplitude, spatial frequency and phase. However, because fabrication techniques typically yield profiles with at most a few depth levels, complex ‘wavy’ surfaces cannot be obtained, limiting the straightforward mathematical design and implementation of sophisticated diffractive optics. Here we present a simple yet powerful approach to eliminate this design–fabrication mismatch by demonstrating optical surfaces that contain an arbitrary number of specified sinusoids. We combine thermal scanning-probe lithography 6 , 7 – 8 and templating 9 to create periodic and aperiodic surface patterns with continuous depth control and sub-wavelength spatial resolution. Multicomponent linear gratings allow precise manipulation of electromagnetic signals through Fourier-spectrum engineering 10 . Consequently, we overcome a previous limitation in photonics by creating an ultrathin grating that simultaneously couples red, green and blue light at the same angle of incidence. More broadly, we analytically design and accurately replicate intricate two-dimensional moiré patterns 11 , 12 , quasicrystals 13 , 14 and holograms 15 , 16 , demonstrating a variety of previously unattainable diffractive surfaces. This approach may find application in optical devices (biosensors 17 , lasers 18 , 19 , metasurfaces 4 and modulators 20 ) and emerging areas in photonics (topological structures 21 , transformation optics 22 and valleytronics 23 ). Combining thermal scanning-probe lithography with templating enables the production of high-quality gratings that manipulate light through Fourier-spectrum engineering in ways that are not achievable with conventional gratings.

Die Dissertation beschreibt die Synthese und Charakterisierung verschiedener Münzmetallkomplexe an einem strukturgebenden tetradentaten monoanionischen PNNP Liganden. Von Interesse war die Frage nach einer möglichen Korrelation der lumineszenten Eigenschaften mit ausgewählten strukturellen Parametern. Es wurden daher selektiv Komplexe mit verschiedenen Kombinationen aus Art und Anzahl an Metallatomen innerhalb des Ligandensystems realisiert. Nach der Vorstellung von homometallischen zwei- und dreikernigen Komplexen, konnten, ausgehend von der zweikernigen Goldverbindung, verschiedene vierkernige und ein sechskerniger Goldkomplex synthetisiert werden. Die Verbindungen wurden intensiv bezüglich ihrer lumineszenten Eigenschaften, einige davon in verschiedenen Phasen, und zudem teilweise theoretisch untersucht. Aufbauend darauf gelang die Darstellung mehrerer Lanthanoid-Münzmetallkomplexe, die ebenfalls bezüglich ihrer lumineszenten Eigenschaften analysiert wurden. Außerdem wurden in einem kleinen Teil der Arbeit Münzmetallkomplexe mit einem Ferrocenylbisamidinatliganden vorgestellt, die teilweise eine kurze Fe M Distanz aufweisen.

It's challenging to measure non-repetitive events in real time in the field of instrumentation and measurement. Dispersive Fourier transformation is an emerging method that permits capture of rare events, such as optical rogue waves and rare cancer cells in blood. This Review article covers the principle of dispersive Fourier transformation and its implementation in diverse applications. Dispersive Fourier transformation is an emerging measurement technique that overcomes the speed limitations of traditional optical instruments and enables fast continuous single-shot measurements in optical sensing, spectroscopy and imaging. Using chromatic dispersion, dispersive Fourier transformation maps the spectrum of an optical pulse to a temporal waveform whose intensity mimics the spectrum, thus allowing a single-pixel photodetector to capture the spectrum at a scan rate significantly beyond what is possible with conventional space-domain spectrometers. Over the past decade, this method has brought us a new class of real-time instruments that permit the capture of rare events such as optical rogue waves and rare cancer cells in blood, which would otherwise be missed using conventional instruments. In this Review, we discuss the principle of dispersive Fourier transformation and its implementation across a wide range of diverse applications.