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Because a finite set of measurements is limited in the amount of spectral content it can represent, the reconstruction process from discrete samples is inherently band-limited. In the case of 1D sampling using ideal measurements, the maximum bandwidth of regular and irregular sampling is well known using Nyquist and Gröchenig sampling theorems and lemmas, respectively. However, determining the appropriate reconstruction bandwidth becomes difficult when considering 2D sampling geometries, samples with variable apertures, or signal to noise ratio limitations. Instead of determining the maximum bandwidth a priori, we derive an inverse method to simultaneously reconstruct a signal and determine its effective bandwidth. This inverse method is equivalent to incrementally computing a band-limited inverse using a frequency-constrained QR decomposition (FQR). Comparisons between reconstruction results using FQR and QR decompositions illustrate how FQR is less sensitive to noisy measurement errors, but it is more sensitive to high-frequency components. These methods are particularly useful in the reconstruction of remote sensing images from such as microwave radiometers and scatterometers. [ABSTRACT FROM AUTHOR]

The short-time Fourier transform phase retrieval problem is reconstructing a signal from its short-time Fourier transform magnitude. This phase retrieval approach has a wide range of applications across various fields, such as ptychography and frequency-resolved optical gating. A recent contribution by Wellershof established that the complex-valued signals in the Paley–Wiener space can be uniquely recovered (up to global phase) by their short-time Fourier transform magnitudes sampled at 1 4 Ω Z × { ω 0 , ω 1 } . In this paper, we generalize Wellershof's findings to the case of short-time linear canonical transform phase retrieval. Specifically, we demonstrate that complex-valued signals in the generalized Paley–Wiener space can be uniquely recovered (up to global phase) by their short-time linear canonical transform magnitudes sampled at b 4 Ω Z × ω 0 , ω 1 . Since the generalized Paley–Wiener space includes a broader class of signals, our results expand the application scope of the phase retrieval. Finally, we apply short-time linear canonical transform and short-time Fourier transform to the bandlimited signals and compare their standard deviation and energy concentration rate. [ABSTRACT FROM AUTHOR]

We extend a classical result by Triebel on boundedness of bandlimited multipliers on L p (R n) , 0 < p ≤ 1 , to a vector-valued and matrix-weighted setting with boundedness of the bandlimited multipliers obtained on L p (W) , 0 < p ≤ 1 , for matrix-weights W : R n → C N × N that satisfy a matrix Muckenhoupt A p -condition. [ABSTRACT FROM AUTHOR]

Sampling is a pivotal element in the design of metasurfaces, enabling a broad spectrum of applications. Despite its flexibility, sampling can result in reduced efficiency and unintended diffractions, which are more pronounced at high numerical aperture or shorter wavelengths, e.g. ultraviolet spectrum. Prevailing metasurface research has often relied on the conventional Nyquist sampling theorem to assess sampling appropriateness, however, our findings reveal that the Nyquist criterion is insufficient guidance for sampling in metasurface. Specifically, we find that the performance of a metasurface is significantly correlated to the geometric relationship between the spectrum morphology and sampling lattice. Based on lattice-based diffraction analysis, we demonstrate several anti-aliasing strategies from visible to ultraviolet regimes. These approaches significantly reduce aliasing phenomena occurring in high numerical aperture metasurfaces. Our findings not only deepen the understanding in phase gradient metasurface but also pave the way for high numerical aperture operation down to the ultraviolet spectrum. Sampling is a key for metasurface design but causes aliasing at high numerical apertures or short wavelengths. The authors identify appropriateness of the Nyquist criterion and propose anti-aliasing strategies for metasurface design. [ABSTRACT FROM AUTHOR]

Grey Fourier model has been successfully applied in seasonal time series forecasting, but its performance in handling nonlinear seasonal time series may still require further improvement. To describe the nonlinear characteristics, a flexible time power grey Fourier model (TPGFM(1,1,N,r)) is proposed by introducing nonlinear time power terms to the grey action of grey Fourier model. The hyperparameters, the truncated Fourier order N and time power r are initially selected by the Nyquist-Shannon sampling theorem and the principle of simplicity, then the optimal parameters are determined by the hold-out method. To further improve the prediction accuracy for nonlinear time sequences, combination models based on the proposed grey model, statistical models and artificial intelligence models are designed. The variable weights are assigned by the inverse variance weighting method. Afterward, the results of the designed experiments based on numerical experiment verify the validity of the Fourier order and time power selection, illustrating the superior performances over benchmark models. Finally, the proposed model is applied for monthly PM2.5 forecasting and quarterly wind power generation forecasting, outperforming other benchmark models in prediction, including seasonal grey models, artificial intelligence models and statistical models. Moreover, the combination models, developed based on TPGFM(1,1,N,r) model, have achieved higher prediction accuracy. [ABSTRACT FROM AUTHOR]

The amount of data collected for sensing tasks in scientific computing is based on the Shannon-Nyquist sampling theorem proposed in the 1940s. Sensor data generation will surpass 73 trillion GB by 2025 as we increase the high-fidelity digitization of the physical world. Skyrocketing data infrastructure costs and time to maintain and compute on all this data are increasingly common. To address this, we introduce a selective learning approach, where the amount of data collected is problem dependent. We develop novel shift-invariant and spectrally stable neural networks to solve real-time sensing problems formulated as classification or regression problems. We demonstrate that (i) less data can be collected while preserving information, and (ii) test accuracy improves with data augmentation (size of training data), rather than by collecting more than a certain fraction of raw data, unlike information theoretic approaches. While sampling at Nyquist rates, every data point does not have to be resolved at Nyquist and the network learns the amount of data to be collected. This has significant implications (orders of magnitude reduction) on the amount of data collected, computation, power, time, bandwidth, and latency required for several embedded applications ranging from low earth orbit economy to unmanned underwater vehicles. [ABSTRACT FROM AUTHOR]

In power systems, the application of wind power generation equipment and power electronic devices leads to an increased frequency of broadband oscillation events, and the detection of oscillation information becomes extremely difficult, due to the limitations of communication bandwidth and the sampling theorem. To ensure the safety and stability of a power system, this paper presents a new recognition method of broadband oscillation information, which combines compressed sensing (CS) technology and an ensemble empirical mode decomposition (EEMD) algorithm to solve the problem of wideband oscillation recognition. Firstly, the broadband oscillation signal data collected by the phasor measuring unit (PMU) is compressed and sampled by a Gaussian random matrix in the substation, then the low-dimensional data obtained is uploaded to the main station. Secondly, in the main station, the subspace pursuit (SP) algorithm is used to reconstruct the low-dimensional signal; the broadband oscillation signal is recovered without losing the main features of the signal. Finally, we use the EEMD algorithm to decompose the reconstructed signal; the intrinsic mode function (IMF) components containing wideband oscillation information are screened by the energy coefficient, and the wideband oscillation information is identified. [ABSTRACT FROM AUTHOR]

Due to the application of power electronics and wind power generation equipment in power systems, broadband oscillation events constantly appear, which makes broadband oscillation difficult to detect due to the limitations of communication bandwidth and the sampling theorem. To ensure the safety and stability of the system, and to detect and recognize the broadband oscillation information timely and accurately, this paper presents a multi-mode recognition method of broadband oscillation based on compressed sensing (CS) and the adaptive Variational Mode Decomposition (VMD) algorithm. Firstly, the high-dimensional oscillation signal data collected by the Phasor Measurement Unit (PMU) is compressed and sampled by a Gaussian random matrix, and the obtained low-dimensional data are uploaded to the main station. Secondly, the orthogonal matching pursuit (OMP) algorithm of the master station is used to reconstruct the low-dimension signal, and the original high-dimension signal data are recovered without losing the main features of the signal. Finally, an adaptive VMD algorithm with energy loss minimization as a threshold is used to decompose the reconstructed signal, and the Intrinsic Mode Function (IMF) components with broadband oscillation information are obtained. By constructing oscillating signals with different frequencies, Gaussian white noise with a signal-to-noise ratio of 10 dB to 30 dB is added successively. After the signal is compressed and reconstructed by the proposed method, the signal-to-noise ratio can reach 18.8221 dB to 40.0794 dB, etc., and the oscillation frequency and amplitude under each signal-to-noise ratio can be accurately identified. The results show that the proposed method not only has good robustness to noise, but also has good denoising effect to noise. By using the simulation measurement model, the original oscillation signal is compressed and reconstructed, and the reconstruction error is 0.1263. The basic characteristics of the signal are restored, and the frequency and amplitude of the oscillation mode are accurately identified, which proves that the method is feasible and accurate. [ABSTRACT FROM AUTHOR]

Given M ≥ 2 distributions defined on a general measurable space, we introduce a nonparametric (kernel) measure of multi-sample dissimilarity (KMD)—a parameter that quantifies the difference between the M distributions. The population KMD, which takes values between 0 and 1, is 0 if and only if all the M distributions are the same, and 1 if and only if all the distributions are mutually singular. Moreover, KMD possesses many properties commonly associated with f-divergences such as the data processing inequality and invariance under bijective transformations. The sample estimate of KMD, based on independent observations from the M distributions, can be computed in near linear time (up to logarithmic factors) using k-nearest neighbor graphs (for k ≥ 1 fixed). We develop an easily implementable test for the equality of M distributions based on the sample KMD that is consistent against all alternatives where at least two distributions are not equal. We prove central limit theorems for the sample KMD, and provide a complete characterization of the asymptotic power of the test, as well as its detection threshold. The usefulness of our measure is demonstrated via real and synthetic data examples; our method is also implemented in an R package. for this article are available online. [ABSTRACT FROM AUTHOR]

Stochastic root-finding problems are fundamental in the fields of operations research and data science. However, when the root-finding problem involves rare events, crude Monte Carlo can be prohibitively inefficient. Importance sampling (IS) is a commonly used approach, but selecting a good IS parameter requires knowledge of the problem's solution, which creates a circular challenge. In "Adaptive Importance Sampling for Efficient Stochastic Root Finding and Quantile Estimation," He, Jiang, Lam, and Fu propose an adaptive IS approach to untie this circularity. The adaptive IS simultaneously estimates the root and the IS parameters, and can be embedded in sample average approximation–type algorithms and stochastic approximation–type algorithms. They provide theoretical analysis on strong consistency and asymptotic normality of the resulting estimators, and show the benefit of adaptivity from a worst-case perspective. They also provide specialized analyses on extreme quantile estimation under milder conditions. In solving simulation-based stochastic root-finding or optimization problems that involve rare events, such as in extreme quantile estimation, running crude Monte Carlo can be prohibitively inefficient. To address this issue, importance sampling can be employed to drive down the sampling error to a desirable level. However, selecting a good importance sampler requires knowledge of the solution to the problem at hand, which is the goal to begin with and thus forms a circular challenge. We investigate the use of adaptive importance sampling to untie this circularity. Our procedure sequentially updates the importance sampler to reach the optimal sampler and the optimal solution simultaneously, and can be embedded in both sample-average-approximation-type algorithms and stochastic-approximation-type algorithms. Our theoretical analysis establishes strong consistency and asymptotic normality of the resulting estimators. We also demonstrate, via a minimax perspective, the key role of using adaptivity in controlling asymptotic errors. Finally, we illustrate the effectiveness of our approach via numerical experiments. Funding: This work was supported by the National Natural Science Foundation of China [Grants 72293562, 72121001, and 72171060], the National Science Foundation [Grants CAREER CMMI-1834710 and IIS-1849280], and the Air Force Office of Scientific Research [Grant FA95502010211]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2023.2484. [ABSTRACT FROM AUTHOR]

Adaptive sampling is inspired by the working principle of biological neurons, which converts the amplitude information of the received signal into a time sequence through a time encoding machine (TEM). The article mainly studies uniform sampling and adaptive sampling of non-bandlimited signals in two kinds of function spaces associated with the special affine Fourier transform (SAFT). Firstly, based on the stability characterization of the basis functions, we prove the existence of orthogonal projection operators in function spaces associated with the canonical convolution and the SAFT-convolution, respectively. Secondly, uniform sampling theorems are established for two kinds of signals living in the proposed function spaces. Finally, adaptive sampling schemes based on the crossing TEM and the integrate-and-fire TEM are studied. Moreover, the corresponding iterative reconstruction algorithms with exponential convergence are provided. [ABSTRACT FROM AUTHOR]

In this study, we defined a kind of Fourier expansion of set-valued square-integrable functions. In fact, we have seen that the classical Fourier basis also constitutes a basis for the Hilbert quasilinear space L 2 (− π , π , Ω (C)) of Ω (C) -valued square-integrable functions, where Ω (C) is the class of all compact subsets of complex numbers. Furthermore, we defined the quasi-Paley–Wiener space, Q P W , using the Fourier transform defined for set-valued functions and thus we showed that the sequence s i n c. − k k ∈ Z form also a basis for Q P W . We call this result Shannon's sampling theorem for set-valued functions. Finally, we gave an application based on this theorem. [ABSTRACT FROM AUTHOR]

The sampling theorem for the offset linear canonical transform (OLCT) of bandlimited functions in polar coordinates is an important signal analysis tool in many fields of signal processing and optics. This paper investigates two sampling theorems for interpolating bandlimited and highest frequency bandlimited functions in the OLCT and offset linear canonical Hankel transform (OLCHT) domains by polar coordinates. Based on the classical Stark's interpolation formulas, we derive the sampling theorems for bandlimited functions in the OLCT and OLCHT domains, respectively. The first interpolation formula is concise and applicable. Due to the consistency of the OLCHT order, the second interpolation formula is superior to the first interpolation formula in computational complexity. [ABSTRACT FROM AUTHOR]

An accurate treatment of electronic spectra in large systems with a technique such as time-dependent density functional theory is computationally challenging. Due to the Nyquist sampling theorem, direct real-time simulations must be prohibitively long to achieve suitably sharp resolution in frequency space. Super-resolution techniques such as compressed sensing and MUSIC assume only a small number of excitations contribute to the spectrum, which fails in large molecular systems where the number of excitations is typically very large. We present an approach that combines exact short-time dynamics with approximate frequency space methods to capture large narrow features embedded in a dense manifold of smaller nearby peaks. We show that our approach can accurately capture narrow features and a broad quasi-continuum of states simultaneously, even when the features overlap in frequency. Our approach is able to reduce the required simulation time to achieve reasonable accuracy by a factor of 20-40 with respect to standard Fourier analysis and shows promise for accurately predicting the whole spectrum of large molecules and materials. Calculating electronic spectra of large systems is computationally challenging. Here, the authors combine exact short-time dynamics with approximate frequency space methods to capture narrow features embedded in a dense manifold of smaller peaks. [ABSTRACT FROM AUTHOR]

Distance covariance is a popular dependence measure for two random vectors |$X$| and |$Y$| of possibly different dimensions and types. Recent years have witnessed concentrated efforts in the literature to understand the distributional properties of the sample distance covariance in a high-dimensional setting, with an exclusive emphasis on the null case that |$X$| and |$Y$| are independent. This paper derives the first non-null central limit theorem for the sample distance covariance, and the more general sample (Hilbert–Schmidt) kernel distance covariance in high dimensions, in the distributional class of |$(X,Y)$| with a separable covariance structure. The new non-null central limit theorem yields an asymptotically exact first-order power formula for the widely used generalized kernel distance correlation test of independence between |$X$| and |$Y$|⁠. The power formula in particular unveils an interesting universality phenomenon: the power of the generalized kernel distance correlation test is completely determined by |$n\cdot \operatorname{dCor}^{2}(X,Y)/\sqrt{2}$| in the high-dimensional limit, regardless of a wide range of choices of the kernels and bandwidth parameters. Furthermore, this separation rate is also shown to be optimal in a minimax sense. The key step in the proof of the non-null central limit theorem is a precise expansion of the mean and variance of the sample distance covariance in high dimensions, which shows, among other things, that the non-null Gaussian approximation of the sample distance covariance involves a rather subtle interplay between the dimension-to-sample ratio and the dependence between |$X$| and |$Y$|⁠. [ABSTRACT FROM AUTHOR]

On finite time interval [ 0 , T ] , let X be an optional semimartingale of class (D) , and Z its strong Snell envelope, which is the smallest optional strong supermartingale bounding X above (except on evanescent set). In this article, we provide the several characterizations of strong Snell envelopes to establish the main result which is the following inequality: ∥ Z 1 - Z 2 ∥ ℍ p ≤ C p ⁢ ∥ X 1 - X 2 ∥ ℍ p , where X 1 and X 2 are two optional semimartingales of class (D) belonging to the space ℍ p ( p > 1 ), Z 1 and Z 2 are their Snell envelopes, C p is an absolute constant. [ABSTRACT FROM AUTHOR]

This paper discusses an abstract Kramer sampling theorem for functions within a reproducing kernel Hilbert space (RKHS) of vector valued holomorphic functions. Additionally, we extend the concept of quasi Lagrange-type interpolation for functions within a RKHS of vector valued entire functions. The dependence of having quasi Lagrange-type interpolation on an invariance condition under the generalized backward shift operator has also been discussed. Furthermore, the paper establishes the connection between quasi Lagrange-type interpolation, operator of multiplication by the independent variable, and de Branges spaces of vector valued entire functions. [ABSTRACT FROM AUTHOR]

For time division multiplexing multiple input multiple output (TDM MIMO) millimeter wave radar, the measurement of target range, velocity and other parameters depends on the phase of the received Intermediate Frequency (IF) signal. The coupling between range and velocity phases occurs when measuring moving targets, leading to inevitable errors in calculating range and velocity from the phase, which in turn affects measurement accuracy. Traditional two-dimensional fast fourier transform (2D FFT) estimation errors are particularly pronounced at high velocity, significantly impacting measurement accuracy. Additionally, due to limitations imposed by the Nyquist sampling theorem, there is a restricted range for velocity measurements that can result in aliasing. In this study, we propose a method to address the coupling of range and velocity based on the original signal as well as a method for velocity compensation to resolve aliasing issues. Our research findings demonstrate that this approach effectively reduces errors in measuring ranges and velocities of high-velocity moving targets while efficiently de-aliasing velocities. [ABSTRACT FROM AUTHOR]

As one of the most widely used spread spectrum techniques, the frequency-hopping spread spectrum (FHSS) has been widely adopted in both civilian and military secure communications. In this technique, the carrier frequency of the signal hops pseudo-randomly over a large range, compared to the baseband. To capture an FHSS signal, conventional non-cooperative receivers without knowledge of the carrier have to operate at a high sampling rate covering the entire FHSS hopping range, according to the Nyquist sampling theorem. In this paper, we propose an adaptive compressed method for joint carrier and direction of arrival (DOA) estimations of FHSS signals, enabling subsequent non-cooperative processing. The compressed measurement kernels (i.e., non-zero entries in the sensing matrix) have been adaptively designed based on the posterior knowledge of the signal and task-specific information optimization. Moreover, a deep neural network has been designed to ensure the efficiency of the measurement kernel design process. Finally, the signal carrier and DOA are estimated based on the measurement data. Through simulations, the performance of the adaptively designed measurement kernels is proved to be improved over the random measurement kernels. In addition, the proposed method is shown to outperform the compressed methods in the literature. [ABSTRACT FROM AUTHOR]

We study the problem of recovering a signal from magnitudes of its wavelet frame coefficients when the analyzing wavelet is real-valued. We show that every real-valued signal can be uniquely recovered, up to global sign, from its multiwavelet frame coefficients \[ \{\lvert \mathcal {W}_{\phi _i} f(\alpha ^{m}\beta n,\alpha ^{m}) \rvert : i\in \{1,2,3\}, m,n\in \mathbb {Z}\} \] for every \alpha >1,\beta >0 with \beta \ln (\alpha)\leq 4\pi /(1+4p), p>0, when the three wavelets \phi _i are suitable linear combinations of the Poisson wavelet P_p of order p and its Hilbert transform \mathscr {H}P_p. For complex-valued signals we find that this is not possible for any choice of the parameters \alpha >1,\beta >0, and for any window. In contrast to the existing literature on wavelet sign retrieval, our uniqueness results do not require any bandlimiting constraints or other a priori knowledge on the real-valued signals to guarantee their unique recovery from the absolute values of their wavelet coefficients. [ABSTRACT FROM AUTHOR]

Specification of the observation‐error covariance matrix for data assimilation systems affects the observation information content retained by the analysis, particularly for observations known to have correlated observation errors (e.g., geostationary satellite and Doppler radar data). A widely adopted approach for estimating observation‐error covariance matrices uses observation‐minus‐background and observation‐minus‐analysis residuals, which are routinely produced by most data assimilation systems. Although this approach is known to produce biased and noisy estimates, due to sampling and misspecification errors, there has been no systematic study of sampling errors with this approach to date. Furthermore, the eigenspectrum of the estimated observation‐error covariance matrix is known to influence the analysis information content and numerical convergence of variational assimilation schemes. In this work, we provide new theorems for the sampling error and eigenvalues of the estimated observation‐error covariance matrices with this approach. We also conduct numerical experiments to illustrate our theoretical results. We find that this method produces large sampling errors if the true observation‐error standard deviation is large, while the other error characteristics, including the true background‐error standard deviation and observation‐ and background‐error correlation length‐scales, have a relatively small effect. We also find that the smallest eigenvalues of the estimated matrices may be smaller or larger than the true eigenvalues, depending on the assumed and true observation‐ and background‐error statistics. These results may provide insights for practical applications: for example, in deciding on appropriate sample sizes and choosing parameters for matrix reconditioning techniques. [ABSTRACT FROM AUTHOR]

Abstract We consider a general case of a magnetotelluric (MT) study to reveal three-dimensional (3D) distribution of the electrical conductivity within the Earth based on measurements of electromagnetic (EM) fields by a two-dimensional (2D) array. Such an MT array observation can be regarded as a spatially discrete sampling of the MT responses (impedances), and each observation site can be regarded as a sampling point. This means that MT array measurements must follow the Nyquist–Shannon sampling theorem. This paper discusses how the sampling theorem is applied to MT array studies and what kind of consideration is required in the application on the basis of synthetic model calculations, with special attention to spatial resolutions. With an aid of the EM scattering theory and the sampling theorem, we can show that an observation array resolves some features of the MT impedance but does not others. We call the resolvable and unresolvable features the MT signal and noise, respectively. This study introduces the spatial Fourier transform of array MT data (impedances) which helps us investigating sampling effects of lateral heterogeneity from a different angle (in the wavenumber domain). Shallow heterogeneities cause a sharp spatial change of impedance elements near structural boundaries. High wavenumber Fourier components are required to describe such a feature, which means the site spacing must be sufficiently short to be able to resolve such features. Otherwise, a set of array MT data will suffer from aliasing, which is one of the typical causes of MT distortion (MT geologic noise). Conversely, a signal due to a deep-seated conductivity anomaly will have more reduced amplitude at higher wavenumbers, which means focused imaging of such an anomaly is generally difficult. Finally, it is suggested to properly consider the sampling theorem in an observation array design, so as to have best performance in resolving MT signals. Graphical abstract

Introduction: In the 1950s and 1960s, John Senders carried out a number of influential experiments on the monitoring of multidegree-of-freedom systems. In these experiments, participants were tasked with detecting events (threshold crossings) for multiple dials, each presenting a signal with different bandwidth. Senders’ analyses showed a nearly linear relationship between signal bandwidth and the amount of attention paid to the dial, and he argued that humans sample according to bandwidth, in line with the Nyquist–Shannon sampling theorem. Objective: The current study tested whether humans indeed sample the dials based on bandwidth alone or whether they also use salient peripheral cues. Methods: A dial-monitoring task was performed by 33 participants. In half of the trials, a gaze-contingent window was used that blocked peripheral vision. Results: The results showed that, without peripheral vision, humans do not effectively distribute their attention across the dials. The findings also suggest that, when given full view, humans can detect the speed of the dial using their peripheral vision. Conclusion: It is concluded that salience and bandwidth are both drivers of distributed visual attention in a dialmonitoring task. [ABSTRACT FROM AUTHOR]

Objectives of our study are to develop a procedure for intertemporal comparison of both technical and cost efficiency and estimate farm efficiency for the Korean rice industry from 2003 to 2017. The newly developed base‐year procedure excludes frontier shift and price effects from the standard procedure for intertemporal comparison. An adjusted central limit theorem for sample T‐tests is applied to avoid potential bias from efficiency scores by the Data Envelope Analysis. Our empirical results show that the two procedures yield different scores and trends. The standard approach indicates declining efficiency, while the base‐year method shows overall improvement in farm efficiency. [ABSTRACT FROM AUTHOR]

The paper is concerned with the sampling and reconstruction of bandlimited signals in the multi-dimensional special affine Fourier transform domain. First, we define the multi-dimensional special affine Fourier transform and establish the convolution theorem. Then, sampling and reconstruction of the multi-dimensional deterministic bandlimited signals in the special affine Fourier transform domain is studied. Finally, the definition of multi-dimensional random bandlimited signals in the special affine Fourier transform domain is given and a sampling theorem is demonstrated. [ABSTRACT FROM AUTHOR]

We consider a general case of a magnetotelluric (MT) study to reveal three-dimensional (3D) distribution of the electrical conductivity within the Earth based on measurements of electromagnetic (EM) fields by a two-dimensional (2D) array. Such an MT array observation can be regarded as a spatially discrete sampling of the MT responses (impedances), and each observation site can be regarded as a sampling point. This means that MT array measurements must follow the Nyquist–Shannon sampling theorem. This paper discusses how the sampling theorem is applied to MT array studies and what kind of consideration is required in the application on the basis of synthetic model calculations, with special attention to spatial resolutions. With an aid of the EM scattering theory and the sampling theorem, we can show that an observation array resolves some features of the MT impedance but does not others. We call the resolvable and unresolvable features the MT signal and noise, respectively. This study introduces the spatial Fourier transform of array MT data (impedances) which helps us investigating sampling effects of lateral heterogeneity from a different angle (in the wavenumber domain). Shallow heterogeneities cause a sharp spatial change of impedance elements near structural boundaries. High wavenumber Fourier components are required to describe such a feature, which means the site spacing must be sufficiently short to be able to resolve such features. Otherwise, a set of array MT data will suffer from aliasing, which is one of the typical causes of MT distortion (MT geologic noise). Conversely, a signal due to a deep-seated conductivity anomaly will have more reduced amplitude at higher wavenumbers, which means focused imaging of such an anomaly is generally difficult. Finally, it is suggested to properly consider the sampling theorem in an observation array design, so as to have best performance in resolving MT signals. [ABSTRACT FROM AUTHOR]

Photoacoustic imaging (PAI) is a rapidly developing emerging non-invasive biomedical imaging technique that combines the strong contrast from optical absorption imaging and the high resolution from acoustic imaging. Abnormal biological tissues (such as tumors and inflammation) generate different levels of thermal expansion after absorbing optical energy, producing distinct acoustic signals from normal tissues. This technique can detect small tissue lesions in biological tissues and has demonstrated significant potential for applications in tumor research, melanoma detection, and cardiovascular disease diagnosis. During the process of collecting photoacoustic signals in a PAI system, various factors can influence the signals, such as absorption, scattering, and attenuation in biological tissues. A single ultrasound transducer cannot provide sufficient information to reconstruct high-precision photoacoustic images. To obtain more accurate and clear image reconstruction results, PAI systems typically use a large number of ultrasound transducers to collect multi-channel signals from different angles and positions, thereby acquiring more information about the photoacoustic signals. Therefore, to reconstruct high-quality photoacoustic images, PAI systems require a significant number of measurement signals, which can result in substantial hardware and time costs. Compressed sensing is an algorithm that breaks through the Nyquist sampling theorem and can reconstruct the original signal with a small number of measurement signals. PAI based on compressed sensing has made breakthroughs over the past decade, enabling the reconstruction of low artifacts and high-quality images with a small number of photoacoustic measurement signals, improving time efficiency, and reducing hardware costs. This article provides a detailed introduction to PAI based on compressed sensing, such as the physical transmission model-based compressed sensing method, two-stage reconstruction-based compressed sensing method, and single-pixel camera-based compressed sensing method. Challenges and future perspectives of compressed sensing-based PAI are also discussed. [ABSTRACT FROM AUTHOR]

This paper studies the continuous-time behavior of sampled-data control systems. In these systems, a discrete-time control law is applied to a continuous-time system model using sampling and zero-order hold. Traditionally, sampled-data systems are modeled in discrete-time, and thus, the intersample dynamics cannot be examined. This paper shows that the classical methods (such as the d2c function of MATLAB) for obtaining continuous-time models can be inaccurate between the sampling instants, even when Shannon's sampling theorem is satisfied. In this paper, a new method is also presented which can be used to characterize the continuous-time dynamics of sampled-data control systems. The new continuous-time system model including the intersample behavior is derived analytically from the discrete-time model, and the results are verified by simulations. [ABSTRACT FROM AUTHOR]

One of the most important issues in the linear control subject is to obtain eigenvalues of the system to study the stability of a system. It is needed to identify the sign of eigenvalues but not the value of it. To fulfil this, there are different methods such as Routh stability criterion, Lyapunov's method and Nyquist stability criterion. In this research, we will present the most simple one to determine the sign of eigenvalues and we shall discuss and explain different types of stability. Also, we will discuss some special cases that cross a controversial mission, and a new method is proposed to calculate the stability of systems, which we call the Lyapunov-Routh method (composition of Lyapunov and Routh method). [ABSTRACT FROM AUTHOR]

The forthcoming wireless networks must integrate to a high density of end-user devices while ensuring a superior quality of service. It is imperative to devise solutions that reduce the density of wireless access points and associated costs, while simultaneously ensuring security and privacy throughout propagation. Herein, a novel dual-band photodetector with high selective responsivities in distinct visible and near-infrared (NIR) band is developed as an optical signal receiver. The receiver has an unprecedented fast switch speed > 500 kHz between different operation modes. Leveraging this fast switch speed, a high-speed Multiple-Input-Single-Output (MISO) system is developed, fusing spectrum resources from visible to infrared bands. The system achieves a data rate of 600 Kbps in the visible channel and 500 Kbps in the NIR channel, with a sum rate of 1.1 Mbps. This is the highest single-channel data rate and sum rate among reported dual-band photodetectors. Additionally, two previously overlooked mechanisms, the Subtraction and Addition modes, are studied in detail. The four operation modes are developed to establish a novel physical encryption method in terms of arithmetic relation. The dual-channel coupled link can support a data rate of 200 Kbps, significantly enhancing communication security. [ABSTRACT FROM AUTHOR]

High-pixel-number synchrotron radiation computed tomography (CT) has the advantages of high sensitivity, high resolution, and a large field of view. It has been widely used in biomedicine, cultural heritage research, non-destructive testing, and other fields. The Nyquist sampling theorem states that when the detector's pixels per row are increased, it requires more CT projections, resulting in a lengthened CT scan time and increased radiation damage. Sparse-view CT can significantly reduce radiation damage and improve the projection data acquisition speed. However, there is insufficient sparse projection data, and the slices reconstructed show aliasing artifacts. Currently, aliasing artifact correction processes more medical CT images, and the number of pixels of such images is small (mainly 512 × 512 pixels). This paper presents an aliasing artifact correction algorithm based on deep learning for synchrotron radiation CT with a high pixel number ( 1728 × 1728 pixels). This method crops high-pixel-number CT images with aliasing artifacts into patches with overlapping features. During the network training process, a convolutional neural network is utilized to enhance the details of the patches, after which the patches are reintegrated into a new CT slice. Subsequently, the network parameters are updated to optimize the new CT slice that closely approximates the full-view slice. To align with practical application requirements, the neural network is trained using only three samples to optimize network parameters and applied successfully to untrained samples for aliasing artifact correction. Comparative analysis with typical deep learning aliasing artifact correction algorithms demonstrates the superior ability of our method to correct aliasing artifacts while preserving image details more effectively. Furthermore, the effect of aliasing artifact correction at varying levels of projection sparsity is investigated, revealing a positive correlation between image quality after deep learning processing and the number of projections. However, the trade-off between rapid experimentation and artifact correction remains a critical consideration. [ABSTRACT FROM AUTHOR]

Traditional bridge health monitoring methods that necessitate sensor installation are not only costly but also time-consuming. In contrast, utilizing smartphone data collected from vehicles as they traverse bridges offers an efficient and cost-effective alternative. This paper introduces a cutting-edge damage detection framework for indirect monitoring of bridge structures, leveraging a substantial volume of acceleration data collected from smartphones in vehicles passing over the bridge. Our innovative approach addresses the challenge of collecting and transmitting high-frequency data while preserving smartphone battery life and data plans through the integration of compressed sensing (CS) into the crowdsensing-based monitoring framework. CS employs random sampling and signal recovery from a significantly reduced number of samples compared to the requirements of the Nyquist–Shannon sampling theorem. In the proposed framework, acceleration signals from vehicles are initially acquired using smartphone sensors, undergo compression, and are then transmitted for signal reconstruction. Subsequently, feature extraction and dimensionality reduction are performed using Mel-frequency cepstral coefficients and principal component analysis. Damage indexes are computed based on the dissimilarity between probability distribution functions utilizing the Wasserstein distance metric. The efficacy of the proposed methodology in bridge monitoring has been substantiated through the utilization of numerical models and a lab-scale bridge. Furthermore, the feasibility of implementing the framework in a real-world application has been investigated, leveraging the smartphone data from 102 vehicle trips on the Golden Gate Bridge. The results demonstrate that damage detection using the reconstructed signals obtained through compressed sensing achieves comparable performance to that obtained with the original data sampled at the Nyquist measurement sampling rate. However, it is observed that to retain severity information within the signals for accurate damage severity identification, the compression level should be limited to 20%. These findings affirm that compressed sensing significantly reduces the data collection requirements for crowdsensing-based monitoring applications, without compromising the accuracy of damage detection while preserving essential damage-sensitive information within the dataset. [ABSTRACT FROM AUTHOR]

Abstract In this paper, fast algorithms for the extrapolation of band-limited signals are presented by the sampling theorem and Fourier series in the case of over sampling. Assume the band-limited signal is known in a finite interval. We update the signal outside the interval by the Shannon sampling theorem in the case of over sampling. Then we obtain a fast algorithm in the form of Fourier series instead of the Fourier transform in the Papoulis–Gerchberg algorithm. Gibbs phenomena is analyzed in the method. An algorithm is presented to control the Gibbs phenomena, and some examples are given in the experimental results.

The fractional Fourier transform (FRFT) is a generalized form of the Fourier transform (FT), it is another important class of time–frequency analysis tool in signal processing. In this paper, we study the two-dimensional (2D) FRFT in the polar coordinates setting. First, Parseval theorem of the 2D FRFT in the polar coordinates is obtained. Then, according to the relationship between 2D FRFT and fractional Hankel transform (FRHT), the convolution theorem for the 2D FRFT in polar coordinates is obtained. It shows that the FRFT of the convolution of two functions is the product of their respective FRFTs. Moreover, the fast algorithm for the convolution theorem of the 2D FRFT is discussed. Finally, the sampling theorem for signal is explored. [ABSTRACT FROM AUTHOR]

Sound field separation techniques are an advancing tool for extracting a target sound field from a mixed sound field. However, the methods bear a high measurement cost due to the restriction of the sampling theorem. In this study, a sound field separation method based on sparse sampling is established. The method initially utilizes dictionary learning to generate a sparse basis of the sound field. Then, a mixed sound field can be precisely recovered from sparse sampling of sound pressure and the target sound field can be extracted based on the recovered sound field by means of the theory of equivalent source method. The method is validated by numerical simulations. Compared to sound field separation based on the equivalent source method, the proposed method has advantage in terms of both the accuracy and the stability for sparse sampling. [ABSTRACT FROM AUTHOR]

This work concerns Markov decision chains on a denumerable state space endowed with a bounded cost function. The performance of a control policy is assessed by a long-run average criterion as measured by a risk-seeking decision maker with constant risk-sensitivity. Besides standard continuity–compactness conditions, the framework of the paper is determined by the following conditions: (i) the state process is communicating under each stationary policy, and (ii) the simultaneous Doeblin condition holds. Within this framework it is shown that (i) the optimal superior and inferior limit average value functions coincide and are constant, and (ii) the optimal average cost is characterized via an extended version of the Collatz–Wielandt formula in the theory of positive matrices. [ABSTRACT FROM AUTHOR]

The approximation behavior of multivariate max-product Kantorovich exponential sampling operators has been analyzed. The point-wise and uniform approximation theorem for these sampling series I w , j χ , (M) is proved. The degree of approximation in-terms of logarithmic modulus of smoothness is studied. For the class of log-Hölderian functions, the order of uniform norm convergence is established. The norm-convergence theorems for the multivariate max-product Kantorovich exponential sampling operators in Mellin–Lebesgue spaces is studied. [ABSTRACT FROM AUTHOR]

A novel compressive sensing (CS) synthetic-aperture radar (SAR) called AgileSAR has been proposed to increase swath width for sparse scenes while preserving azimuthal resolution. AgileSAR overcomes the limitation of the Nyquist sampling theorem so that it has a small amount of data and low system complexity. However, traditional CS optimization-based algorithms suffer from manual tuning and pre-definition of optimization parameters, and they generally involve high time and computational complexity for AgileSAR imaging. To address these issues, a pseudo- L 0 -norm fast iterative shrinkage algorithm network (pseudo- L 0 -norm FISTA-net) is proposed for AgileSAR imaging via the deep unfolding network in this paper. Firstly, a pseudo- L 0 -norm regularization model is built by taking an approximately fair penalization rule based on Bayesian estimation. Then, we unfold the operation process of FISTA into a data-driven deep network to solve the pseudo- L 0 -norm regularization model. The network's parameters are automatically learned, and the learned network significantly increases imaging speed, so that it can improve the accuracy and efficiency of AgileSAR imaging. In addition, the nonlinearly sparsifying transform can learn more target details than the traditional sparsifying transform. Finally, the simulated and data experiments demonstrate the superiority and efficiency of the pseudo- L 0 -norm FISTA-net for AgileSAR imaging. [ABSTRACT FROM AUTHOR]

In this paper, combining the non-decaying properties with the mixed-norm properties, the revelent sampling problems are studied under the target space of L p → , 1 ω (R d) . Firstly, we will give a stability theorem for the shift-invariant subspace V p → , 1 ω (φ) . Secondly, an ideal sampling in W p → , 1 ω s (R d) is proved, and thirdly, a convergence theorem (or algorithm) is shown for V p → , 1 ω (φ) . It should be pointed out that the auxiliary function φ enjoys the membership in a Wiener amalgam space. [ABSTRACT FROM AUTHOR]

Pulse Wave Velocity (PWV) analysis is valuable for assessing arterial stiffness and cardiovascular health and potentially for estimating blood pressure cufflessly. However, conventional PWV analysis from two transducers spaced closely poses challenges in data management, battery life, and developing the device for continuous real-time applications together along an artery, which typically need data to be recorded at high sampling rates. Specifically, although a pulse signal consists of low-frequency components when used for applications such as determining heart rate, the pulse transit time for transducers near each other along an artery takes place in the millisecond range, typically needing a high sampling rate. To overcome this issue, in this study, we present a novel approach that leverages the Nyquist–Shannon sampling theorem and reconstruction techniques for signals produced by bioimpedance transducers closely spaced along a radial artery. Specifically, we recorded bioimpedance artery pulse signals at a low sampling rate, reducing the data size and subsequently algorithmically reconstructing these signals at a higher sampling rate. We were able to retain vital transit time information and achieved enhanced precision that is comparable to the traditional high-rate sampling method. Our research demonstrates the viability of the algorithmic method for enabling PWV analysis from low-sampling-rate data, overcoming the constraints of conventional approaches. This technique has the potential to contribute to the development of cardiovascular health monitoring and diagnosis using closely spaced wearable devices for real-time and low-resource PWV assessment, enhancing patient care and cardiovascular disease management. [ABSTRACT FROM AUTHOR]

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An effective near-field to far-field transformation using a reduced number of near-field measurements collected via a spherical scan over the upper hemisphere, due to the presence of a flat metallic ground, is devised in this paper. Such a transformation relies on the non-redundant sampling representations of electromagnetic fields and exploits the image principle to properly account for the metallic ground, supposed to be of infinite extent and realised by perfectly conducting material. The sampling representation of the probe voltage over the upper hemisphere is developed by modelling the antenna under test and its image by a very adaptable convex surface, which is able to fit as much as possible the geometry of any kind of antenna, thus minimising the volumetric redundancy and, accordingly, the number of required samples as well as the measurement time. Then, the use of a two-dimensional optimal sampling interpolation algorithm allows the reconstruction of the voltage value at each sampling point of the spherical grid required by the classical near-field-to-far-field transformation developed by Hansen. Numerical examples proving the effectiveness of the developed sampling representation and related near-field-to-far-field transformation techniques are reported. [ABSTRACT FROM AUTHOR]

Data reconstruction is the most essential step in seismic data processing. Although the compressed sensing (CS) theory breaks through the Nyquist sampling theorem, we previously proved that the CS-based reconstruction of spatially irregular seismic data could not fully meet the theoretical requirements, resulting in low reconstruction accuracy. Although deep learning (DL) has great potential in mining features from data and accelerating the process, it faces challenges in earth science such as limited labels and poor generalizability. To improve the generalizability of deep neural network (DNN) in reconstructing seismic data in the actual situation of limited labeling, this paper proposes a method called CSDNN that combines model-driven CS and data-driven DNN to reconstruct the spatially irregular seismic data. By physically constraining neural networks, this method increases the generalizability of the network and improves the insufficient reconstruction caused by the inability to sample randomly in the whole data definition domain. Experiments on the synthetic and field seismic data show that the CSDNN reconstruction method achieves better performance compared with the conventional CS method and DNN method, including those with low sampling rates, which verifies the feasibility, effectiveness and generalizability of this approach. [ABSTRACT FROM AUTHOR]

Gabor phase retrieval is the problem of recovering a signal from the absolute values of Gabor transform, which has turned into a very active research area. In the paper, we define a generalized Paley–Wiener space, which is obtained by applying the special affine Fourier transform to L p ([ - B , B ]) (2 ≤ p ≤ ∞) and provide the uniqueness of the Gabor phase retrieval in the generalized Paley–Wiener space. When the window function ψ is the inverse special affine Fourier transform of ϕ = 2 1 4 e - π x 2 , we show that every signal in the generalized Paley–Wiener space can be uniquely recovered, up to a global sign, from its the magnitudes of Gabor measurement sampled on a rectangular lattice. Recently, Grohs, Liehr and Wellershoff studied the Gabor phase retrieval problems in the conventional Paley–Wiener space. The paper extends their results to the generalized Paley–Wiener space. Since the generalized Paley–Wiener space includes a broader class of signals, our results expand the applied range of the Gabor phase retrieval. [ABSTRACT FROM AUTHOR]

The quantum-mechanical expression for the polarization of a crystalline solid does not bear any resemblance to the (trivial) expression for the dipole of a bounded crystallite; in fact, it has been proved via a conceptually different path. Here, I show how to alternatively define the dipole of a bounded sample in a somewhat unconventional way; from such a formula, the crystalline polarization formula—as routinely implemented in electronic-structure codes—follows almost seamlessly. [ABSTRACT FROM AUTHOR]