Discrete sampling theorem to Shannon’s sampling theorem using the hyperreal numbers R

Shannon’s sampling theorem is one of the most important results of modern signal theory. It describes the reconstruction of any band-limited signal from a finite number of its samples. On the other hand, although less well known, there is the discrete sampling theorem, proved by Cooley while he was working on the development of an algorithm to speed up the calculations of the discrete Fourier transform. Cooley showed that a sampled signal can be resampled by selecting a smaller number of samples, which reduces computational cost. Then it is possible to reconstruct the original sampled signal using a reverse process. In principle, the two theorems are not related. However, in this paper we will show that in the context of Non Standard Mathematical Analysis (NSA) and Hyperreal Numerical System R, the two theorems are equivalent. The difference between them becomes a matter of scale. With the scale changes that the hyperreal number system allows, the discrete variables and functions become continuous, and Shannon’s sampling theorem emerges from the discrete sampling theorem.

Sampling Error of Continuous Periodic Data and its Application for Geodesy

Data acquisition for geoinformatics cannot be done continuously, but by discrete sampling of the object or phenomenon. The sampling involves errors on the knowledge of the continuous signal due to the loss of information in the sampling procedure. In the present study, an analytical formulation of the sampling error is provided, which embodies the amplitude, phase, bias and periodicity of the sampling error. The analysis is then subsequently applied for case studies: for the GRACE and GRACE-FO monthly solutions, and for different realizations of the Hungarian Gravimetric Network.

Investigating Fracture Network Deformation Using Noble Gas Release

We investigate deformation mechanics of fracture networks in unsaturated fractured rocks from subsurface conventional detonation using dynamic noble gas measurements and changes in air permeability. We dynamically measured the noble gas isotopic composition and helium exhalation of downhole gas before and after a large subsurface conventional detonation. These noble gas measurements were combined with measurements of the subsurface permeability field from 64 discrete sampling intervals before and after the detonation and subsurface mapping of fractures in borehole walls before well completion. We saw no observable increase in radiogenic noble gas release from either an isotopic composition or a helium exhalation point of view. Large increases in permeability were observed in 13 of 64 discrete sampling intervals. Of the sampling intervals which saw large increases in flow, only two locations did not have preexisting fractures mapped at the site. Given the lack of noble gas release and a clear increase in permeability, we infer that most of the strain accommodation of the fractured media occurred along previously existing fractures, rather than the creation of new fractures, even for a high strain rate event. These results have significant implications for how we conceptualize the deformation of rocks with fracture networks above the percolation threshold, with application to a variety of geologic and geological engineering problems.

Sensitivity of migratory connectivity estimates to spatial sampling design

Background The use of statistical methods to quantify the strength of migratory connectivity is commonplace. However, little attention has been given to their sensitivity to spatial sampling designs and scales of inference. Methods We examine sources of bias and imprecision in the most widely used methodology, Mantel correlations, under a range of plausible sampling regimes using simulated migratory populations. Results As Mantel correlations depend fundamentally on the spatial scale and configuration of sampling, unbiased inferences about population-scale connectivity can only be made under certain sampling regimes. Within a contiguous population, samples drawn from smaller spatial subsets of the range generate lower connectivity metrics than samples drawn from the range as a whole, even when the underlying migratory ecology of the population is constant across the population. Random sampling of individuals from contiguous subsets of species ranges can therefore underestimate population-scale connectivity. Where multiple discrete sampling sites are used, by contrast, overestimation of connectivity can arise due to samples being biased towards larger between-individual pairwise distances in the seasonal range where sampling occurs (typically breeding). Severity of all biases was greater for populations with lower levels of true connectivity. When plausible sampling regimes were applied to realistic simulated populations, accuracy of connectivity measures was maximised by increasing the number of discrete sampling sites and ensuring an even spread of sites across the full range. Conclusions These results suggest strong potential for bias and imprecision when making quantitative inferences about migratory connectivity using Mantel statistics. Researchers wishing to apply these methods should limit inference to the spatial extent of their sampling, maximise their number of sampling sites, and avoid drawing strong conclusions based on small sample sizes.

Teaching advanced methods of signal processing and transmission at universities of civil engineering

Introduction. Development and widespread dissemination of information technologies, including the Internet, mobile communications, cloud computing, Big Data, the Internet of Things, digital twin, etc. are being proactively introduced into versatile production and business processes of the construction industry. Therefore, a graduate of a civil engineering university must master the fundamentals of the above-listed technologies and know how to use them in his/her practical activities. It’s particularly important for those specialities that deal with information technologies in civil engineering. However, a number of important areas of knowledge that serve as the basis for mobile communications, the Internet, and wireless technologies, are only taught at specialized universities and schools. Nevertheless, students of civil engineering universities need to understand the theoretical provisions and processes of information technologies. Materials and methods. Systematization, the benchmark method, theoretical generalization of data extracted from literary sources were applied. Results. The founding notions, needed to discuss digital technologies, encompass temporary signal notation, spectral characteristics, spectral bandwidth, time and frequency division, multiplexing, discrete sampling and quantization of a signal. It is impossible to successfully study and effectively use these advanced technologies without having understood these fundamental technologies. The author offers a simplified explanation of these notions and principal processes in terms of their application in the construction industry. Conclusions. The author addresses complicated issues of the theory of signals and their transmission over communication lines. He presents the three key ideas that serve as the basis for multiple advanced information technologies, including time and spectrum signal notation, time and frequency division, multiplexing, discrete sampling and quantization of a signal. The author’s ideas can be used to teach mobile technologies, the Internet, the Internet of Things, cloud and edge computing, digital twin, etc.

No Evidence for a Single Oscillator Underlying Discrete Visual Percepts

Theories of perception based on discrete sampling posit that visual consciousness is reconstructed based on snapshot-like perceptual moments, as opposed to being updated continuously. According to a model proposed by Schneider (2018), discrete sampling can explain both the flash-lag and the Fröhlich illusion, whereby a lag in the conscious updating of a moving stimulus alters its perceived spatial location in comparison to a stationary stimulus. The alpha-band frequency, which is associated with phasic modulation of stimulus detection and the temporal resolution of perception, has been proposed to reflect the duration of perceptual moments. The goal of this study was to determine whether a single oscillator (e.g., alpha) is underlying the duration of perceptual moments, which would predict that the point of subjective equality (PSE) in the flash-lag and Fröhlich illusions are positively correlated across individuals. Although our displays induced robust flash-lag and Fröhlich effects, virtually zero correlation was seen between the PSE in the two illusions, indicating that the illusion magnitudes are unrelated across observers. These findings suggest that, if discrete sampling theory is true, these illusory percepts either rely on different oscillatory frequencies or not on oscillations at all. Alternatively, discrete sampling may not be the mechanism underlying these two motion illusions or our methods were ill-suited to test the theory.

Boundedness properties of semi-discrete sampling operators in Mellin–Lebesgue spaces

<p style='text-indent:20px;'>In this paper we study boundedness properties of certain semi-discrete sampling series in Mellin–Lebesgue spaces. Also we examine some examples which illustrate the theory developed. These results pave the way to the norm-convergence of these operators</p>