This advanced monograph is concerned with modern treatments of central problems in harmonic analysis. The main theme of the book is the interplay between ideas used to study the propagation of ...singularities for the wave equation and their counterparts in classical analysis. In particular, the author uses microlocal analysis to study problems involving maximal functions and Riesz means using the so-called half-wave operator. To keep the treatment self-contained, the author begins with a rapid review of Fourier analysis and also develops the necessary tools from microlocal analysis. This second edition includes two new chapters. The first presents Hörmander's propagation of singularities theorem and uses this to prove the Duistermaat–Guillemin theorem. The second concerns newer results related to the Kakeya conjecture, including the maximal Kakeya estimates obtained by Bourgain and Wolff.
Herein we report on a deep-learning method for the removal of instrumental noise and unwanted spectral artifacts in Fourier transform infrared (FTIR) or Raman spectra, especially in automated ...applications in which a large number of spectra have to be acquired within limited time. Automated batch workflows allowing only a few seconds per measurement, without the possibility of manually optimizing measurement parameters, often result in challenging and heterogeneous datasets. A prominent example of this problem is the automated spectroscopic measurement of particles in environmental samples regarding their content of microplastic (MP) particles. Effective spectral identification is hampered by low signal-to-noise ratios and baseline artifacts as, again, spectral post-processing and analysis must be performed in automated measurements, without adjusting specific parameters for each spectrum. We demonstrate the application of a simple autoencoding neural net for reconstruction of complex spectral distortions, such as high levels of noise, baseline bending, interferences, or distorted bands. Once trained on appropriate data, the network is able to remove all unwanted artifacts in a single pass without the need for tuning spectra-specific parameters and with high computational efficiency. Thus, it offers great potential for monitoring applications with a large number of spectra and limited analysis time with availability of representative data from already completed experiments.
Being an inter-disciplinary subject, Signal Processing has application in almost all scientific fields. Applied Signal Processing tries to link between the analog and digital signal processing ...domains. Since the digital signal processing techniques have evolved from its analog counterpart, this book begins by explaining the fundamental concepts in analog signal processing and then progresses towards the digital signal processing. This will help the reader to gain a general overview of the whole subject and establish links between the various fundamental concepts. While the focus of this book is on the fundamentals of signal processing, the understanding of these topics greatly enhances the confident use as well as further development of the design and analysis of digital systems for various engineering and medical applications. Applied Signal Processing also prepares readers to further their knowledge in advanced topics within the field of signal processing.
for many decades, there has been a general perception in the literature that Fourier methods are not suitable for the analysis of nonlinear and non-stationary data. In this paper, we propose a novel ...and adaptive Fourier decomposition method (FDM), based on the Fourier theory, and demonstrate its efficacy for the analysis of nonlinear and non-stationary time series. The proposed FDM decomposes any data into a small number of ‘Fourier intrinsic band functions’ (FIBFs). The FDM presents a generalized Fourier expansion with variable amplitudes and variable frequencies of a time series by the Fourier method itself. We propose an idea of zero-phase filter bank-based multivariate FDM (MFDM), for the analysis of multivariate nonlinear and non-stationary time series, using the FDM. We also present an algorithm to obtain cut-off frequencies for MFDM. The proposed MFDM generates a finite number of band-limited multivariate FIBFs (MFIBFs). The MFDM preserves some intrinsic physical properties of the multivariate data, such as scale alignment, trend and instantaneous frequency. The proposed methods provide a time–frequency–energy (TFE) distribution that reveals the intrinsic structure of a data. Numerical computations and simulations have been carried out and comparison is made with the empirical mode decomposition algorithms.
Segmentation of roads in remote sensing images is a challenging task due to the inhomogeneous intensity, non-consistent contrast, and very cluttered background in remote sensing images. Recent ...approaches, mostly relying on convolutions or self-attention, make it difficult to extract weak and continuous road objects. Fourier neural operators provide another novel mechanism for capturing long-range and fine-grained features beyond self-attention. Based on it, we propose an adaptive Fourier convolution network (AFCNet) on the spatial-spectral domain for road segmentation in this paper. The AFCNet is built on the pipeline of the classical U-Net model and its core is the proposed Fourier neural encoder (FNE), which is built on a feed-forward layer and a flexible Fourier convolutional structure composed of Fourier-domain pooling layers, asymmetric convolutions, squeeze-excitation inspired self-attention and adaptive multiscale fusion layers. Furthermore, we combine the FNE and bottleneck in ResNet to form a hybrid global-local feature representation scheme to capture the long and weak road objects in remote sensing images. The experiments on two public datasets, the Massachusetts Roads and DeepGlobe Road Datasets, have shown that AFCNet worked with fewer parameters and outperformed most previous methods in terms of accuracy, precision, recall, and mean intersection over union (mIoU), etc.
Fourier Transform Spectrometry is of immediate use to those who use Fourier transform spectrometers in their research, or are considering their use. The authors' presentations enable readers to ...obtain a clear understanding of FTS, which is crucial to their studies and research.Due to the increasing complexity and commercialization of instrumentation, achieving optimum performance in research applications and automated usage can be challenging. For example, a thorough understanding of the instrument can dramatically affect the outcome of the experiment and the generation of reliable data in applications where conditions are not ideal and resulting signals are weak. This book provides a comprehensive discussion of FTS from the ground up, covering basic concepts, instrumentation, data-processing algorithms, and techniques for computerized spectral analysis.
In this paper, we propose the notion of Fourier frequency vector (FFV) which is inherently associated with the multidimensional (MD) Fourier representation (FR) of a signal. The proposed FFV provides ...physical meaning to the so-called negative frequencies in the MD-FR that in turn yields MD spatial and MD space-time series analysis. The one-dimensional Hilbert transform (1D-HT) and associated 1D analytic signal (1D-AS) of an 1D signal are well established; however, their true generalization to an MD signal, which possess all the properties of 1D case, are not available in the literature. To achieve this, we observe that in MD-FR the complex exponential representation of a sinusoidal function always yields two frequencies, namely negative frequency corresponding to positive frequency and vice versa. Thus, using the MD-FR, we propose MD-HT and associated MD analytic signal (AS) as a true generalization of the 1D-HT and 1D-AS, respectively, and obtain an explicit expression for the analytic image computation by 2D discrete Fourier transform (2D-DFT). We also extend the Fourier decomposition method for 2D signals that decomposes an image into a set of amplitude-modulated and frequency-modulated (AM–FM) image components. We finally propose a single-orthant Fourier transform (FT) of real MD signals which computes FT in the first orthant, and values in rest of the orthants are obtained by simple conjugation defined in this study.
Raman and Fourier transform IR (FTIR) microspectroscopic images of biological material (tissue sections) contain detailed information about their chemical composition. The challenge lies in ...identifying changes in chemical composition, as well as locating and assigning these changes to different conditions (pathology, anatomy, environmental or genetic factors). Multivariate data analysis techniques are ideal for decrypting such information from the data. This protocol provides a user-friendly pipeline and graphical user interface (GUI) for data pre-processing and unmixing of pixel spectra into their contributing pure components by multivariate curve resolution-alternating least squares (MCR-ALS) analysis. The analysis considers the full spectral profile in order to identify the chemical compounds and to visualize their distribution across the sample to categorize chemically distinct areas. Results are rapidly achieved (usually <30-60 min per image), and they are easy to interpret and evaluate both in terms of chemistry and biology, making the method generally more powerful than principal component analysis (PCA) or heat maps of single-band intensities. In addition, chemical and biological evaluation of the results by means of reference matching and segmentation maps (based on k-means clustering) is possible.
In this paper, the design of frequency-locked loop (FLL) is proposed based on computationally efficient discrete Fourier transform (DFT) structures. In recent years, the DFT structures are evolved as ...sliding DFT (SDFT), modulated SDFT, hopping DFT (HDFT), modulated HDFT, and sliding-windowed infinite Fourier transform. Considering their tuned filter characteristics, an attempt has been made to obtain a solution for the instantaneous frequency estimation problem of the input signal under varying center frequency condition. In each DFT structure, the <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>th bin in-phase and quadrature components are separated for instantaneous signal extraction. The feedback loop is designed around these DFT structures, and it was observed that the frequency responses exhibit flat magnitude and phase interestingly when compared with the open-loop structures. Hence, an adaptive sampling frequency adjustment scheme is proposed for these structures as FLL to estimate the instantaneous frequency of the input signal for the wide variation in center frequency. These FLLs with different DFT structures are tested for dynamic performance and wide operating range. The proposed FLLs are implemented in field-programmable gate array (FPGA), and the experimental investigations have been carried out for frequency estimation. Further experimental investigations on these FLLs as a system on chip were carried out with area and power analysis.
We investigate the spectral efficiency of non-orthogonal frequency division multiplexing (NOFDM). NOFDM is regarded as a frequency-domain faster-than-Nyquist (FTN) signaling with block transmission, ...where each subcarrier is compressed in the frequency domain by utilizing the discrete fractional Fourier transform (DFRFT). Since pulse shaping filters are not employed, the effective bandwidth of the NOFDM signal should take into account the resulting out-of-band (OOB) radiation. Therefore, we derive a closed-form expression for the power spectral density (PSD) of the NOFDM signal with windowing for OOB reduction. Through numerical comparison, it is shown that NOFDM outperforms OFDM in view of the effective signal bandwidth if they are compared under the same number of subcarriers. On the other hand, by the capacity analysis based on the asymptotic spectral efficiency with a large number of subcarriers, we elucidate that NOFDM may not outperform OFDM from an information-theoretic perspective.
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