Sampling Theorem Research Articles

Application of compressed sensing in the guided wave structural health monitoring of switch rails

Switch rails are weak but essential components of high-speed rail (HSR) systems. In the condition-based maintenance of HSR, ultrasonic guided wave (UGW) on-line monitoring technology is widely used in judging real-time operating conditions; however, it always generates large amounts of data. Too much data bring significant challenges, such as too many unnecessary costs of energy, storage, and network bandwidth in the structural health monitoring of switch rails, making it challenging to realize embedded sensor networks with high durability and low power consumption. Furthermore, the structural damage occurs relatively less over the long-term, indicating that these measurements are inherently sparse. The sparseness of structural damage makes them attractive for the compressed sensing technique. This study proposes a novel data compression and reconstruction method to meet the challenges and reduce the amount of data transmitted by sensor networks and maintain their accuracies simultaneously. First, a lightweight data dictionary is constructed to perform a sparse decomposition of UGW signals according to the characteristics of UGW propagation. Second, an effective sampling method based on a sparse random matrix is designed for sub-Nyquist sampling and compressing UGW signals. Third, a novel block adaptive matching pursuing algorithm is proposed to reconstruct UGW signals from compressed data. Finally, numerical signals, finite element simulation, and several actual monitoring experiments on the foot of a switch rail are conducted to verify the effectiveness and accuracy of the proposed method. The influence of different compression ratios and block sizes on the reconstruction performance of guided wave signals is investigated. The results indicate that the proposed method can sample UGW signals with much lower requirements than the Nyquist sampling theorem and is superior to other novel algorithms.

The SINC way: a fast and accurate approach to Fourier pricing

The goal of this paper is to investigate the method outlined by one of us (P. R.) in Cherubini, U., Della Lunga, G., Mulinacci, S. and Rossi, P. [Fourier Transform Methods in Finance, 2009 (John Wiley & Sons Inc.).] to compute option prices. We name it the SINC approach. While the COS method by Fang, F. and Oosterlee, C.W. [A novel pricing method for european options based on Fourier-cosine series expansions. SIAM. J. Sci. Comput., 2009, 31(2), 826–848.] leverages the Fourier-cosine expansion of truncated densities, the SINC approach builds on the Shannon Sampling Theorem revisited for functions with bounded support. We provide several results which were missing in the early derivation: (i) a rigorous proof of the convergence of the SINC formula to the correct option price when the support grows and the number of Fourier frequencies increases; (ii) ready to implement formulas for put, Cash-or-Nothing, and Asset-or-Nothing options; (iii) a systematic comparison with the COS formula for several log-price models; iv) a numerical challenge against alternative Fast Fourier specifications, such as Carr, P. and Madan, D. [Option valuation using the fast Fourier transform. J. Comput. Finance, 1999, 2(4), 61–73.] and Lewis, A.L. [Option Valuation Under Stochastic Volatility with Mathematica Code, 2000 (Newport Beach: Finance Press).]; (v) an extensive pricing exercise under the rough Heston model of Jaisson, T. and Rosenbaum, M. [Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Probab., 2015, 25(2), 600–631.]; (vi) formulas to evaluate numerically the moments of a truncated density. The advantages of the SINC approach are numerous. When compared to benchmark methodologies, SINC provides the most accurate and fast pricing computation. The method naturally lends itself to price all options in a smile concurrently by means of Fast Fourier techniques, boosting fast calibration. Pricing requires resorting only to odd moments in the Fourier space.

Audio Keyword Reconstruction from On-Device Motion Sensor Signals via Neural Frequency Unfolding

In this paper, we present a novel deep neural network architecture that reconstructs the high-frequency audio of selected spoken human words from low-sampling-rate signals of (ego-)motion sensors, such as accelerometer and gyroscope data, recorded on everyday mobile devices. As the sampling rate of such motion sensors is much lower than the Nyquist rate of ordinary human voice (around 6kHz+), these motion sensor recordings suffer from a significant frequency aliasing effect. In order to recover the original high-frequency audio signal, our neural network introduces a novel layer, called the alias unfolding layer, specialized in expanding the bandwidth of an aliased signal by reversing the frequency folding process in the time-frequency domain. While perfect unfolding is known to be unrealizable, we leverage the sparsity of the original signal to arrive at a sufficiently accurate statistical approximation. Comprehensive experiments show that our neural network significantly outperforms the state of the art in audio reconstruction from motion sensor data, effectively reconstructing a pre-trained set of spoken keywords from low-frequency motion sensor signals (with a sampling rate of 100-400 Hz). The approach demonstrates the potential risk of information leakage from motion sensors in smart mobile devices.

The Relationship Between Perifoveal L-Cone Isolating Visual Acuity and Cone Photoreceptor Spacing-Understanding the Transition Between Healthy Aging and Early AMD.

Background: Age-related macular degeneration (AMD) is a multifactorial degenerative disorder that can lead to irreversible loss of visual function, with aging being the prime risk factor. However, knowledge about the transition between healthy aging and early AMD is limited. We aimed to examine the relationship between psychophysical measures of perifoveal L-cone acuity and cone photoreceptor structure in healthy aging and early AMD.Methods and Results: Thirty-nine healthy participants, 10 with early AMD and 29 healthy controls were included in the study. Multimodal high-resolution retinal images were obtained with adaptive-optics scanning-light ophthalmoscopy (AOSLO), optical-coherence tomography (OCT), and color fundus photographs. At 5 degrees retinal eccentricity, perifoveal L-cone isolating letter acuity was measured with psychophysics, cone inner segment and outer segment lengths were measured using OCT, while cone density, spacing, and mosaic regularity were measured using AOSLO. The Nyquist sampling limit of cone mosaic (Nc) was calculated for each participant. Both L-cone acuity and photoreceptor inner segment length declined with age, but there was no association between cone density nor outer segment length and age. A multiple regression showed that 56% of the variation in log L-cone acuity was accounted for by Nc when age was taken into account. Six AMD participants with low risk of progression were well within confidence limits, while two with medium-to-severe risk of progression were outliers. The observable difference in cone structure between healthy aging and early AMD was a significant shortening of cone outer segments.Conclusion: The results underscore the resilience of cone structure with age, with perifoveal functional changes preceding detectable changes in the cone photoreceptor mosaic. L-cone acuity is a sensitive measure for assessing age-related decline in this region. The transition between healthy aging of cone structures and changes in cone structures secondary to early AMD relates to outer segment shortening.

A 3.66 μW 12-bit 1 MS/s SAR ADC with mismatch and offset foreground calibration

This paper presents a 12-bit 1 MS/s successive approximation register (SAR) analog-to-digital converter (ADC) with foreground calibration for digital-to-analog converter (DAC) mismatch and comparator static offset errors. The proposed foreground calibration utilizes the redundancy to facilitate the detection of DAC weight errors, and compensates for deviations of offset and DAC weights from the ideal value in the analog domain respectively. Benefit from the choice of 0.3 fF unit capacitance, the calibration achieves a 90% reduction in DAC power overhead over the conventional method. The effectiveness of this method is demonstrated by simulations in which differential non-linearity (DNL) is reduced from −1.28 LSB to −0.53 LSB and integral non-linearity (INL) is reduced from 2.20 LSB to 1.08 LSB. The ADC implemented in 40 nm CMOS consumes 3.66 μW from a 1 V supply, and achieves an improved signal-to-noise and distortion ratio (SNDR) of from 59.68 dB to 66.67 dB and a figure of merit (FoM) of 2.07 fJ/conversion-step at Nyquist rate.

Estimation of parameters of a sum of sinusoids sampled below the Nyquist rate with frequencies away from the grid

Abstract We are witnessing a growing interest in processing signals sampled below the Nyquist rate. The main limitation of current approaches considering estimation of multicomponent sinusoids parameters is the assumption of frequencies on the frequency grid. The sinusoids away from the frequency grid are considered in this paper. The proposed procedure has three stages. In the first two, a rough estimation of signal components is performed while in the third refinement in estimation is achieved in a component-by-component manner. We have tested the developed technique on an extended set of simulation examples showing excellent accuracy. Three scenarios are considered in experiments: missing samples, noisy environment, and non-uniform sampling below the Nyquist rate.

Compressive narrowband interference detection and parameter estimation in direct sequence spread spectrum communication

The Shannon-Nyquist sampling theorem that is based on narrowband interference (NBI) detection and parameter estimation methods in direct sequence spread spectrum (DSSS) communication is limited by the high sampling rate. Compressive sensing (CS) is adopted to address the problem. But it will change the signal nature, which leads to the unavailability of Shannon-Nyquist sampling theorem-based interference detection and parameter estimation methods. According to the different posterior probability distribution features of NBI, DSSS signal, and noise in compressed domain, a posterior probability model of whether NBI exists in the received signal is constructed by using the compressed measurements. The posterior probability that whether NBI exists in the received signal is employed as the feature parameters of NBI detection and parameter estimation. With the feature parameters detected, NBI detection can be achieved and the edge location of NBI components can be located. The relationship between the edge location of NBI components and the edge frequency of NBI is constructed, which will contribute to estimate the NBI edge frequency. The numerical simulation results demonstrate that the proposed method can effectively achieve NBI detection and parameter estimation in the compressed domain and it performs significantly better than the other existing methods.

Novel deep learning framework for wideband spectrum characterization at sub-Nyquist rate

Limited availability and high auction cost of the sub-6 GHz spectrum led to the introduction of spectrum-sharing in 5G networks. This demands base stations with the capability of automatic wideband spectrum characterization (AWSC) to identify available vacant spectrum and parameters (carrier frequency, modulation scheme, etc.) of occupied bands. Since WSC at Nyquist sampling (NS) is area and power-hungry and conventional statistical AWSC performs poorly at a low signal-to-noise ratio (SNR), we propose a novel sub-Nyquist sampling (SNS) based deep-learning framework. It is a single unified pipeline that accomplishes two tasks: (1) Reconstruct the signal directly from the sub-Nyquist samples, and (2) Identify the occupancy status and modulation scheme of all bands. The proposed non-iterative approach based reconstruction provides the occupancy status of all bands in a single forward pass leading to significant improvement in execution time over state-of-the-art iterative methods. In addition, the proposed approach does not need complex signal conditioning between reconstruction and characterization. We extensively compare the performance of our framework for a wide range of modulation schemes, SNR and channel conditions. We show that the proposed framework outperforms existing SNS based characterization and its performance approaches NS based framework with an increase in SNR without compromising computation complexity. A single unified deep-learning framework makes the proposed method a good candidate for reconfigurable platforms.

Average Sampling Theorems on Multidimensional Random Signals

In this paper, a lower bounded of sequences on classical sampling theorem for random signals is given and is used to extend the applicable range of frame. The convergence property of sampling series, the estimate of truncation error both in mean square sense and for almost sure results on sampling theorem for multidimensional random signals from asymmetrical average sampling are analyzed. Using the new results on frames recently, the famous Shannon sampling theorem on multidimensional random signals has been extended with a new idea.

Compressive sensing for perceptually correct reconstruction of music and speech signals

Compressive sensing (CS) is a technique that can achieve exact signal reconstruction by using fewer samples than those in the Nyquist theorem. In this study, listening tests were conducted to investigate the minimum number of samples needed for perceptually correct reconstruction by means of CS. The dictionary used as a sparsity domain for the signal reconstruction was the discrete cosine transform, and the reconstruction approach was the one provided by the L1-Magic. Three music signals and four speech signals were used as source signals. These signals were reconstructed by CS using different percentages of Nyquist samples. The results of the listening tests showed that, when 50% of the samples were used for the CS reconstruction, half of the test listeners judged the original and reconstructed signals to be perceptually the same. Listeners with musical training showed better sensitivity in distinguishing the original signals from the reconstructed signals than listeners without musical training. The log spectral distance between the original and reconstructed signals was a better objective index than the root mean square error and signal-to-noise ratio to evaluate the performance of the CS because inaccurate signal reconstruction mainly appeared at high frequencies.

Sampling Theorems with Nonlinear Signal Reconstruction Scheme

In this paper, sampling Theorems for perfect signal reconstruction using samples of a signal taken below the Nyquist rate involving nonlinear signal reconstruction technique are presented. The presented results are in contrast with the linear time-invariant filtering based signal reconstruction in the celebrated Shannon sampling theory. It is shown that using kth order nonlinearity in the signal reconstruction system, the required sampling rate for perfect signal reconstruction can be reduced by the same factor k for positive odd values of it. The extensions of the proposed sampling expansion to the fractional Fourier transform and linear canonical transform domains are also derived. Simulation results of the proposed technique are given to validate the proposed approach.

Evaluation of the Effects of Compressive Spectrum Sensing Parameters on Primary User Behavior Estimation

As the Internet of Things (IoT) technology is being deployed, the demand for radio spectrum is increasing. Cognitive radio (CR) is one of the most promising solutions to allow opportunistic spectrum access for IoT secondary users through utilizing spectrum holes resulting from the underutilization of frequency spectrum. A CR needs to frequently sense the spectrum to avoid interference with primary users (PUs). Compressive spectrum sensing techniques have been gaining increasing interest in wideband spectrum sensing, as they reduce the need for high-rate analog-to-digital converters, reducing the complexity and energy requirements of the CR. In order to enhance spectrum sensing performance, researchers proposed to incorporate PU spectrum usage information into the process of spectrum sensing. Spectrum usage information can be obtained through pilot signals, geo-locational databases or through evaluation of previous spectrum sensing results. In this paper, we are studying the effects of compressive sensing parameters namely compression ratio, sensing period, and sensing duration on the estimation of primary user behavior statistics. We achieved an accurate estimation of the primary user's behavior while saving 40% of the sampling rate by using compressive spectrum sensing compared to traditional spectrum sensing with Nyquist rate sampling.

Lattice-based multi-channel sampling theorem for linear canonical transform

The multi-channel sampling theorem has flourished as one of the nicest alternatives to the classical Shanon's sampling theorem which relies on non-uniform or multi-channel data acquisition. Although, a primary attempt to study the multi-channel sampling theorem for the multi-dimensional linear canonical transform (LCT) was made by D. Wei and Y. Li (2014) [16], however, it lacks the inherent multi-dimensional nature in the sense that the associated kernel is a product of N-copies of the usual one dimensional kernel, restricting the degrees of freedom only to three. The goal of this paper is to fill this gap by studying the lattice-based multi-channel sampling theorem for the multi-dimensional LCT with N(2N+1) degrees of freedom. The idea is accomplished in two steps: firstly, an elegant convolution structure is presented for constructing the multiplicative filters; secondly, the sampling lattices are constructed via general separable and non-separable matrices which often provide more flexibility and better performance in the context of multi-dimensional signal analysis. In the sequel, the Shanon's sampling theorem for the multi-dimensional LCT is deduced as a particular case of the proposed multi-channel sampling theorem. Finally, some potential applications including signal reconstruction and image super-resolution are presented to validate the obtained results.

A Micro-Doppler Frequency Ambiguity Resolution Method Based on Complex-Valued U-Net

In order to resolute the micro-Doppler frequency ambiguity caused by radar pulse repetition frequency not high enough (i.e., pulse dimension does not satisfy the requirement of Nyquist sampling theorem), this paper presents a micro-Doppler frequency ambiguity resolution method based on complex-valued U-net. The echo sequence is interpolated by zeros in the pulse dimension to increase the equivalent pulse repetition frequency, so that the echo sequence after zero interpolation contains the real micro-Doppler frequency; at the same time, some new frequency components are generated. The variation law of the echo sequence frequency after zero interpolation is analyzed. Then, the echo sequence in time domain after zero interpolation is transformed to the time-frequency domain by short-time Fourier transform (STFT). Finally, the time-frequency results can be segmented by the model, which is trained by complex-valued U-net to eliminate the redundant frequencies generated by zero interpolation; thus, the reconstruction of real micro-Doppler frequency is realized. Theoretical analysis and simulation results show that the proposed method can solve the problem of micro-Doppler frequency ambiguity. Compared with fully convolution network (FCN) and fully convolution residual network (FCRN), the proposed method has better performance and robustness.

Research Progress on On‐Chip Fourier Transform Spectrometer

AbstractA Fourier transform spectrometer (FTS) is recognized as a highly precise analytical instrument for analyzing the constituent elements of matter in the fields of physics, chemistry, aerospace, and so on. With the emergence and development of miniaturized and portable devices in numerous scientific and technological fields, there has been an urgent need for on‐chip FTSs due to their benefits of small size, portability, low energy consumption, and robustness. However, the small size of on‐chip FTSs hinders the acquisition of a large optical path difference, resulting in a low resolution of the spectrometer. In addition, the sampler spacing is not sufficiently small to satisfy Nyquist's sampling theorem, directly influencing the bandwidth of the spectrometer. Therefore, studies have been performed to investigate the trade‐off between the reduced size and performance of spectrometers. This paper aims to systematically review the progress in on‐chip FTSs, especially in on‐chip static FTSs, including spatially modulated, temporally modulated, and space‐time comodulated FTSs, from the aspects of theories, implementations, and performance indicators. Finally, the paper ends with a discussion of the challenges and a view of the prospective development of this exciting field.

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.

Probabilistic nilpotence in infinite groups

The ‘degree of k-step nilpotence’ of a finite group G is the proportion of the tuples (x1,…, xk+1 ∈ Gk+1 for which the simple commutator [x1, …, xk+1] is equal to the identity. In this paper we study versions of this for an infinite group G, with the degree of nilpotence defined by sampling G in various natural ways, such as with a random walk, or with a Følner sequence if G is amenable. In our first main result we show that if G is finitely generated, then the degree of k-step nilpotence is positive if and only if G is virtually k-step nilpotent (Theorem 1.5). This generalises both an earlier result of the second author treating the case k = 1 and a result of Shalev for finite groups, and uses techniques from both of these earlier results. We also show, using the notion of polynomial mappings of groups developed by Leibman and others, that to a large extent the degree of nilpotence does not depend on the method of sampling (Theorem 1.12). As part of our argument we generalise a result of Leibman by showing that if ϕ is a polynomial mapping into a torsion-free nilpotent group, then the set of roots of ϕ is sparse in a certain sense (Theorem 5.1). In our second main result we consider the case where G is residually finite but not necessarily finitely generated. Here we show that if the degree of k-step nilpotence of the finite quotients of G is uniformly bounded from below, then G is virtually k-step nilpotent (Theorem 1.19), answering a question of Shalev. As part of our proof we show that degree of nilpotence of finite groups is sub-multiplicative with respect to quotients (Theorem 1.21), generalising a result of Gallagher.

Random Acquisition in Compressive Sensing

Compressive sensing has the ability of reconstruction of signal/image from the compressive measurements which are sensed with a much lower number of samples than a minimum requirement by Nyquist sampling theorem. The random acquisition is widely suggested and used for compressive sensing. In the random acquisition, the randomness of the sparsity structure has been deployed for compressive sampling of the signal/image. The article goes through all the literature up to date and collects the main methods, and simply described the way each of them randomly applies the compressive sensing. This article is a comprehensive review of random acquisition techniques in compressive sensing. Theses techniques have reviews under the main categories of (1) random demodulator, (2) random convolution, (3) modulated wideband converter model, (4) compressive multiplexer diagram, (5) random equivalent sampling, (6) random modulation pre-integration, (7) quadrature analog-to-information converter, (8) randomly triggered modulated-wideband compressive sensing (RT-MWCS).

Error Self-Detection of Smart Electric Meter Base on DFT processing

The paper proposes a high frequency signal (975 HZ) through superposition method for error self-detection. The feasibility of 975Hz superposition signal is illustrated and verified. According to the Nyquist sampling theorem, the sampling data are processed by DFT theorem. And through the self-detection error and metering error analysis judgment, it is concluded that scheme design of the self-detection error can be used as report and the early warning mechanism of smart IoT electric meter. It lays a good foundation for realizing the transformation of smart electric meter from regular replacement to state replacement, and provides a theoretical basis for ensuring the accuracy of metering.

Generalizing the sampling theorem for a frequency-time domain to sample signals under the conditions of a priori uncertainty

The radio monitoring of radiation and interference with electronic means is characterized by the issue related to the structural-parametric a priori uncertainty about the type and parameters of the ensemble of signals by radio-emitting sources. Given this, it is a relevant task to devise a technique for the mathematical notation of signals in order to implement their processing, overcoming their a priori uncertainty in terms of form and parameters. A given problem has been solved by the method of generalization and proof for the finite signals of the Whittaker-Kotelnikov-Shannon sampling theorem (WKS) in the frequency-time domain. The result of proving it is a new discrete frequency-temporal description of an arbitrary finite signal in the form of expansion into a double series on the orthogonal functions such as sinx/x, or rectangular Woodward strobe functions, with an explicit form of the phase-frequency-temporal modulation function. The properties of the sampling theorem in the frequency-time domain have been substantiated. These properties establish that the basis of the frequency-time representation is orthogonal, the accuracy of approximation by the basic functions sinx/x and rectangular Woodward strobe functions are the same, and correspond to the accuracy of the UCS theorem approximation, while the number of reference points of an arbitrary, limited in the width of the spectrum and duration, signal, now taken by frequency and time, is determined by the signal base. The devised description of signals in the frequency-time domain has been experimentally investigated using the detection-recovery of continuous, simple pulse, and linear-frequency-modulated (LFM) radio signals. The constructive nature of the resulting description has been confirmed, which is important and useful when devising methods, procedures, and algorithms for processing signals under the conditions of structural-parametric a priori uncertainty.