High-speed nonlinear frequency sweeping signal distance extraction algorithm based on the table lookup method.

Sampling theorem for bandlimited Hardy space functions generated by Regge problem

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

The sampling theorem can be described as a economic way of representing a limited bandwidth function. A sample of points is chosen and an interpolation function of these points is used to represent the function. The great importance of this fact is that paves the way to discrete computation. The sample points act as a sort of “highlight” points of the original function, and computation involving the entire function may be restricted to calculation using only the sample points. Moreover we were able to find some refining of the classical outline of the sampling theorem that improves its precision bringing also some physical insight into the core of the theorem. Of course also these consideration are not restricted to optics, they can be construed as general properties of the signal theory. But we were optically minded at all times and most of the applications are in optics.

In 1933 the young Russian communications engineer Vladimir Aleksandrovich Kotelnikov published a paper in which he formulated, for the first time in an engineering context, the sampling theorem for lowpass and bandpass signals. He also considered the bandwidth requirements of discrete signal transmission for telegraphy and images. Kotelnikov subsequently worked on scrambling, cryptography, optimal detection, and planetary radar (including the radar-assisted cartography of Venus). He was awarded numerous Soviet and international prizes, and played a major role in Soviet academic and professional life in the field of radio engineering. Yet his achievements are still comparatively little known outside Russia.

Traditionally, discrete images are assumed to be sampled on a square grid and from a special kind of band-limited continuous image, namely one whose Fourier spectrum is contained within the rectangular "reciprocal cell" associated with the sampling grid. With such a simplistic model, resolution is just given by the distance between sample points. Whereas this model matches to some extent the characteristics of traditional acquisition systems, it doesn't explain aliasing problems, and it is no longer valid for certain modern ones, where the sensors may show a heavily anisotropic transfer function, and may be located on a non-square (in most cases hexagonal) grid. In this work we first summarize the generalizations of Fourier theory and of Shannon's sampling theorem, that are needed for such acquisition devices. Then we explore its consequences: (i) A new way of measuring the effective resolution of an image acquisition system; (ii) A more accurate way of restoring the original image which is represented by the samples. We show on a series of synthetic and real images, how the proposed methods make a better use of the information present in the samples, since they may drastically reduce the amount of aliasing with respect to traditional methods. Finally we show how in combination with Total Variation minimization, the proposed methods can be used to extrapolate the Fourier spectrum in a reasonable manner, visually increasing image resolution.

In scientific and engineering scenarios we can notice the predominance of explicit knowledge being manipulated and distributed, which makes recommender systems very useful in this environment. But along with a knowledge management approach this kind of system can support the organization in better identifying competences, help engage users in a continuous and dynamic knowledge exchange, and customize knowledge dissemination as much as possible. In this work we detail a collaborative recommender system which is used in a Knowledge Management Environment for Scientific and Engineering contexts; we show how this approach can be aimed at a KM process and how this approach can deal with other kinds of knowledge used in research centers and universities.

Let Bσ,p,1 <-p<-∞, be the set of all functions from L8(R) which can be continued to entire functions of exponential type <-σ. The well known Shannon sampling theorem and its generalization [1] state that every f∈Bσ,p, 1<p<∞, can be represented as $$f(x) = \mathop \Sigma \limits_{j \in z} f(j\pi /\sigma )\tfrac{{sin\sigma (x - j\pi /\sigma )}}{{\sigma (x - j\pi /\sigma )}}, \sigma > 0$$ It is well known that the Shannon sampling theorem plays an important role in many applied fields [1,2,7]. In this paper we give an application of sampling theorem to approximation theory.

High-resolution alignment of action potential waveforms using cubic spline interpolation

This paper describes a 28-Gb/s receiver IC with self-contained adaptive equalization and sampling point control using an on-chip stochastic sigma-tracking eye-opening monitor (SSEOM). The proposed SSEOM accurately detects the bit-error-rate (BER)-related eye contour efficiently without the use of an external microcontroller. The SSEOM determines the BER-optimal sampling point and equalizer coefficients on the basis of pattern-filtered eye diagrams. It also features a background adaptation scheme for robust long-term operation by tracking temperature variations and device aging. The proposed SSEOM is integrated in a 28-Gb/s receiver that is designed to compensate for channel loss up to 25 dB at the Nyquist rate by using a continuous time linear equalizer (CTLE) and a one-tap decision feedback equalizer (DFE) together with an one-tap pre-emphasis at a transmitter. The time required for complete adaptation and the total power consumption are 364 ms and 43.9 mW, respectively. The proposed 28-Gb/s receiver is fabricated in 40 nm CMOS.

This paper introduces a single-shot ultrafast imaging technique termed wavelength and polarization time-encoded ultrafast raster imaging (WP-URI). By integrating raster imaging principles with wavelength- and polarization-based temporal encoding, the system uses a spatial raster mask and time–space mapping to aggregate multiple two-dimensional temporal raster images onto a single detector plane, thereby enabling the effective spatial separation and extraction of target information. Finally, the target dynamics are recovered using a reconstruction algorithm based on the Nyquist–Shannon sampling theorem. Numerical simulations demonstrate the single-shot acquisition of four dynamic frames at 25 trillion frames per second (Tfps) with an intrinsic spatial resolution of 50 line pairs per millimeter (lp/mm) and a wide field of view. The WP-URI technique achieves unparalleled spatio-temporal resolution and frame rates, offering significant potential for investigating ultrafast phenomena such as matter interactions, carrier dynamics in semiconductor devices, and femtosecond laser–matter processes.

The sampling theorem enables a band-limited signal to be expressed in terms of a set of sample point values, which occur at the Nyquist rate. The sampling theorem has been generalized to include nonuniform sampling and the use of derivatives of the signal. In the present paper, a sampling theorem has been developed which utilizes information related to the zeros of the signal. The concept of complex zeros is introduced to show that the zeros occur at the Nyquist rate. This sampling theorem can be of use for enabling the transmission of binary signals, such as facsimile and infinitely-clipped speech, over a continuous band-limited channel. The result indicates the desirability of developing a completely general theory of sampling applicable to the various situations which may arise in practice.

This work includes some new aspects of motion estimation by the optic flow method in scale spaces. The usual techniques for motion estimation are limited to the application of coarse to fine strategies. The coarse to fine strategies can be successful only if there is enough information in every scale. In this work we investigate the motion estimation in the scale space more basically. The wavelet choice for scale space decomposition of image sequences is discussed in the first part of this work. We make use of the continuous wavelet transform with rotationally symmetric wavelets. Bandpass decomposed sequences allow the replacement of the structure tensor by the phase invariant energy operator. The structure tensor is computationally more expensive because of its spatial or spatio-temporal averaging. The energy operator needs in general no further averaging. The numerical accuracy of the motion estimation with the energy operator is compared to the results of usual techniques, based on the structure tensor. The comparison tests are performed on synthetic and real life sequences. Another practical contribution is the accuracy measurement for motion estimation by adaptive smoothed tensor fields. The adaptive smoothing relies on nonlinear anisotropic diffusion with discontinuity and curvature preservation. We reached an accuracy gain under properly chosen parameters for the diffusion filter. A theoretical contribution from mathematical point of view is a new discontinuity and curvature preserving regularization for motion estimation. The convergence of solutions for the isotropic case of the nonlocal partial differential equation is shown. For large displacements between two consecutive frames the optic flow method is systematically corrupted because of the violence of the sampling theorem. We developed a new method for motion analysis by scale decomposition, which allows to circumvent the systematic corruption without using the coarse to fine strategy. The underlying assumption is, that in a certain neighborhood the grey value undergoes the same displacement. If this is fulfilled, then the same optic flow should be measured in all scales. If there arise inconsistencies in a pixel across the scale space, so they can be detected and the scales containing this inconsistencies are not taken into account.

Compressed sensing theory by developing a signal sparse features, under the condition of far less than the Nyquist sampling rate, the correct signal is acquired with random sampling the discrete samples, and then through the nonlinear reconstruction algorithm reconstruction signal of high probability.

A method is proposed for superresolving multichannel data with application to video sequences. Based on a generalization of Papoulis’ sampling theorem, nonuniform samples of multiple channels are merged to generate high-resolution data. To overcome sampling ill posedness in the presence of noise, image frames are projected from standard orthonormal bases onto optimal Riesz bases defined by channel point spread functions (PSF’s). The method is therefore designed to perform under practical conditions of noise and other degradations. Unlike existing methods, where empirical models such as Gaussian, sinc, etc., are commonly used for characterizing channel PSF’s, the PSF’s are assumed unknown and possibly different and hence are blindly estimated from the observed data. The estimated PSF’s are then used to construct biorthogonal projection filters for the superresolution algorithm. This approach gives rise to a closed-form solution leading to a high-speed algorithm. The method has been tested and verified on PREDATOR video sequences (PREDATOR data are airborne video sequences of the Defense Advanced Research Projects Agency obtained by unmanned aircrafts).

&lt;abstract&gt;&lt;p&gt;This paper introduces a novel numerical approach for tackling the nonlinear fractional Phi-four equation by employing the Homotopy perturbation method (HPM) and the Adomian decomposition method (ADM), augmented by the Shehu transform. These established techniques are adept at addressing nonlinear differential equations. The equation's complexity is reduced by applying the Shehu Transform, rendering it amenable to solutions via HPM and ADM. The efficacy of this approach is underscored by conclusive results, attesting to its proficiency in solving the equation. With extensive ramifications spanning physics and engineering domains like fluid dynamics, heat transfer, and mechanics, the proposed method emerges as a precise and efficient tool for resolving nonlinear fractional differential equations pervasive in scientific and engineering contexts. Its potential extends to analogous equations, warranting further investigation to unravel its complete capabilities.&lt;/p&gt;&lt;/abstract&gt;