Shannon Sampling Theory Research Articles

Edge Collaborative Compressed Sensing in Wireless Sensor Networks for Mechanical Vibration Monitoring

This paper proposes a novel edge collaborative compressed sensing for mechanical vibration monitoring (MVM) to address the severe shortage of storage and computational resources and long delay in transmitting massive vibration data in wireless sensor networks (WSN) for MVM. It first combines compressed sensing (CS) and edge computing (EC) into a WSN to effectively solve the above problems. Based on the self-developed acquisition node (AN) and edge computing node (ECN), the proposed approach is implemented in the AN and ECN respectively. This enhances the acquisition efficiency and computing capacity of the WSN. Meanwhile, the sparse pattern of a mechanical fault signal is analyzed. In addition, the adaptive parameter adjustment mechanism (APA) for the alternating direction of multiplier method based on Anderson acceleration is proposed to generate a convex optimization algorithm with fast convergence to realize data reconstruction in the ECN. The results of comprehensive experiments demonstrate that the proposed method can reduce the transmission data by 70%, while maintaining high-precision bearing fault feature detection, compared with the Shannon sampling theory. Furthermore, the acquisition and transmission efficiencies are improved significantly. This provides a potential solution for practical engineering applications.

Hilbert transform of unequally sampled data: Application to dispersion relations in magnetotellurics

The real and imaginary parts of the Fourier transform of a causal signal can be estimated from one another using the Hilbert transform. The numerical computation can be carried out by a fast Fourier transform or the convolution of the data with an appropriate Hilbert kernel. However, magnetotelluric (MT) data are unequally sampled because logarithmic frequency axes are conventionally used. A method has been developed for the Hilbert transformation of unequally sampled data and applied to the real and imaginary parts of the frequency normalized impedance to examine if the dispersion relations are satisfied for a given impedance tensor. A version of the Shannon sampling theory is generalized for the unequally sampled data. The method involves the reconstruction of the measured data by a linear combination of an analytic interpolation function distributed along the horizontal axis. Then, the decomposition coefficients that provide a fit between the measured and reconstructed data are solved using the linear least-squares method with singular-value decomposition. Finally, the discrete Hilbert transformed data are constructed at any desired abscissa values by the sum of the analytic Hilbert transform of the interpolation function using the decomposition coefficients solved in the previous step. Examples from the MT synthetic and field data, including out-of-quadrant phase cases, are given. In principle, the method also can be applied to other integral transforms.

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.

Short-Observation Measurement of Multiple Sinusoids With Multichannel Sub-Nyquist Sampling

Measurement of multiple sinusoidal signals (MSSs) is significant in instrumentation and measurement, radar, communications, and many other electric systems. Measurement methods based upon Shannon sampling theory require very high sampling rates and heavy processing to estimate the frequencies and amplitudes. In this article, we propose a multichannel time-staggered sampling system to measure MSSs with short-observation sub-Nyquist samples. The sampling scheme is composed of $N'$ time-staggered sampling channels, where the staggered time is allowed to be greater than the Nyquist sampling interval. The estimation method is based on the frequency domain sparse common support (SCS) model and the Chinese remainder theorem (CRT). To estimate $K$ frequency components in the signal without image frequency aliasing, $N' \geq 2$ sampling channels and $N\geq \lceil K+K/N' \rceil $ samples for each channel are enough. In the situation when image frequency aliasing exists, $N' \geq 2K$ and $N\geq 2K$ samples are sufficient. We further explore the analysis of noisy signals and generalize our proposal to real-valued signals. Simulation and hardware experimental results demonstrate the effectiveness and robustness of the proposed method.

A Note on Reconstruction of Bandlimited Signals of Several Variables Sampled at Nyquist Rate

A standard problem in certain applications requires one to find a reconstruction of an analogue signal f from a sequence of its samples $f{({t_{k}})_{k}}$. The great success of such a reconstruction consists, under additional assumptions, in the fact that an analogue signal f of a real variable $t\in \mathbb{R}$ can be represented equivalently by a sequence of complex numbers $f{({t_{k}})_{k}}$, i.e. by a digital signal. In the sequel, this digital signal can be processed and filtered very efficiently, for example, on digital computers. The sampling theory is one of the theoretical foundations of the conversion from analog to digital signals. There is a long list of impressive research results in this area starting with the classical work of Shannon. Note that the well known Shannon sampling theory is mainly for one variable signals. In this paper, we concern with bandlimited signals of several variables, whose restriction to Euclidean space ${\mathbb{R}^{n}}$ has finite p-energy. We present sampling series, where signals are sampled at Nyquist rate. These series involve digital samples of signals and also samples of their partial derivatives. It is important that our reconstruction is stable in the sense that sampling series converge absolutely and uniformly on the whole ${\mathbb{R}^{n}}$. Therefore, having a stable reconstruction process, it is possible to bound the approximation error, which is made by using only of the partial sum with finitely many samples.

Compressed-Sensing-Based Periodic Impulsive Feature Detection for Wind Turbine Systems

Identifying impulsive features from massive amounts of dynamic signals for wind turbine systems is like finding needles in haystacks, leading to a major challenge for the Shannon-sampling-theorem-based fault detection techniques. Therefore, this paper describes and analyzes a novel impulsive feature identification technique based on compressed sensing model and convex optimization techniques. One important point of this work is to establish the prior information that periodic impulsive component is frequency compressible. On the other hand, based on a small set of nonadaptive linear measurements, a convex optimization algorithm generated from a popular algorithmic framework (alternating direction of multiplier method) is developed to recover the impulsive features. Consequently, the main highlight of this work is to enable the high-accuracy recovery of impulsive feature signals from far few measurements than the Shannon sampling theory requires. Extensive numerical studies are implemented to quantitatively evaluate the performance of the proposed technique, and its feasibility and superiority are verified simultaneously. More importantly, the validity and applicability of our impulsive feature detection technique is comprehensively investigated and confirmed based on a practical engineering dataset from a wind turbine gearbox in a wind farm.

Analysis and Experiment of the Compressive Sensing Approach for Duct Mode Detection

This paper describes a new mode detection method based on the compressive sensing approach, which has been developed in information technology to reduce samples required by the classical Nyquist–Shannon sampling theory. A revised approach is proposed here to detect azimuthal spinning modes from aeroengine fan noise by using a microphone array outside the bypass duct. A number of numerical simulations is prepared to examine the associated test performance, such as the reconstruction accuracy, robustness, background noise interference, and coherence impact. The mode sparsity is usually much smaller than the highest order of the modes, so the azimuthal mode is sparse at the frequency of interest to ensure a satisfactory compressive sensing-based mode detection. The current simulations show log sensors shall be generally sufficient rather than sensors required by the Shannon–Nyquist sampling theory. An experimental system is designed and implemented to demonstrate the proposed method. The test system contains a straightened and rigid duct with an enclosed spinning mode synthesizer to emulate fan noise propagation inside and scattering from a bypass duct. An outer sensor array is employed to demonstrate that the compressive sensing-based mode detection method can significantly reduce the required number of sensors, especially for modes of high order, which results in a much simplified sensor array design for aerospace noise tests.

Compressed Sensing with Basis Mismatch: Performance Bounds and Sparse-Based Estimator

Compressed sensing (CS) is a promising emerging domain that outperforms the classical limit of the Shannon sampling theory if the measurement vector can be approximated as the linear combination of few basis vectors extracted from a redundant dictionary matrix. Unfortunately, in realistic scenario, the knowledge of this basis or equivalently of the entire dictionary is often uncertain, i.e., corrupted by a Basis Mismatch (BM) error. The consequence of the BM problem is that the estimation accuracy in terms of Bayesian Mean Square Error (BMSE) of popular sparse-based estimators collapses even if the support is perfectly estimated and in the high Signal to Noise Ratio (SNR) regime. This saturation effect considerably limits the effective viability of these estimation schemes. In the first part of this work, the Bayesian Cramer-Rao Bound (BCRB) is derived for CS model with unstructured BM. We show that the BCRB foresees the saturation effect of the estimation accuracy of standard sparse-based estimators as for instance the OMP, Cosamp or the BP. In addition, we provide an approximation of this BMSE threshold. In the second part and in the context of the structured BM model, a new estimation scheme called Bias-Correction Estimator (BiCE) is proposed and its statistical properties are studied. The BiCE acts as a post-processing estimation layer for any sparse-based estimator and mitigates considerably the BM degradation. Finally, the BiCE i) is a blind algorithm, i.e., is unaware of the uncorrupted dictionary matrix, ii) is generic since it can be associated to any sparse-based estimator, iii) is fast, i.e., the additional computational cost remains low, and iv) has good statistical properties. To illustrate our results and propositions, the BiCE is applied in the challenging context of the compressive sampling of non-bandlimited impulsive signals.

Feature Identification With Compressive Measurements for Machine Fault Diagnosis

Machine fault diagnosis collects massive amounts of vibration data about complex mechanical systems. Performing feature detection from these data sets has already led to a major challenge. Compressive sensing theory is a new sampling framework that provides an alternative to the well-known Shannon sampling theory. This theory enables the recovery of sparse or compressible signals from a small set of nonadaptive linear measurements. However, it is suboptimal to recover the whole signals from the compressive measurements and then solve feature identification problems through traditional DSP techniques. Thus, a novel mechanical feature identification method is proposed in this paper. Its main advantage is that fault features are extracted directly in the compressive measurement domain without sacrificing accuracy, while a significant reduction in the dimensionality of the measurement data is achieved. Moreover, Gaussian white noises are significantly alleviated, which dramatically enhances the reliability of machine fault diagnosis. Parameter analysis is also profoundly investigated through a set of numerical experiments. Numerical simulations and experiments are further performed to prove the reliability and effectiveness of the proposed method.

Sampling Conditions for the Circular Radon Transform.

Recovering a function from circular or spherical mean values is the basis of many modern imaging technologies, such as photo and thermoacoustic computed tomography or ultrasound reflection tomography. Recently, much progress has been made concerning the problem of recovering a function from its circular mean values (its circular Radon transform). In particular, theoretically exact inversion formulas of the back-projection type have been discovered using continuously sampled data. In practical applications, however, only a discrete number of circular mean values can be collected. In this paper, we address this issue in the context of the Shannon sampling theory. We derive sharp sampling conditions for the number of angular and radial samples, such that any essentially b0-bandlimited function can be recovered from a finite number of such circular mean values.

Reduced Image Aliasing With Microwave Radiometers and Weather Radar Through Windowed Spatial Averaging

Microwave remote sensing instruments detect and image physical phenomena such as brightness temperature and volume reflectivity. The spatial resolution of these measurements is limited by the physical properties of the instrument such as the antenna size, the spatial scan pattern, and temporal sampling. Analysis shows that common sampling schemes undersample the spatial information present at the antenna. Here, we address methods to better capture the spatial information available by applying the Nyquist–Shannon sampling theory to the spatial averaging and sampling of remote sensing data. The use of overlapping windows for spatial averaging rather than treating pixels independently improves the image fidelity while maintaining the system sensitivity. Additionally, the sensitivity to spatially small targets can be maximized by matching the window shape to the antenna pattern. The spatial imaging of scanning radiometers, radars, and phased-array systems is addressed. These principles are demonstrated with the theory and data from the National Aeronautics and Space Administration Goddard Space Flight Center's High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP) radar.

Are nonlinear discrete cellular automata compatible with quantum mechanics?

We consider discrete and integer-valued cellular automata (CA). A particular class of which comprises “Hamiltonian CA” with equations of motion that bear similarities to Hamilton's equations, while they present discrete updating rules. The dynamics is linear, quite similar to unitary evolution described by the Schrödinger equation. This has been essential in our construction of an invertible map between such CA and continuous quantum mechanical models, which incorporate a fundamental discreteness scale. Based on Shannon's sampling theory, it leads, for example, to a one-to-one relation between quantum mechanical and CA conservation laws. The important issue of linearity of the theory is examined here by incorporating higher-order nonlinearities into the underlying action. These produce inconsistent nonlocal (in time) effects when trying to describe continuously such nonlinear CA. Therefore, in the present framework, only linear CA and local quantum mechanical dynamics are compatible.

Recent Progress of Shannon Sampling Theory with Applications

Errors appear when the Shannon sampling series is applied to approximate a signal in practice since the measured sampled values are usually given by averages of a function or the sampled values with amplitude error and time-jitter error. In this paper we give a survey of recent progress of the study of error analysismulti-dimensional Whittaker-Shannon sampling expansion approximations with these measured values.

The Exponential Sampling Theorem of Signal Analysis and the Reproducing Kernel Formula in the Mellin Transform Setting

The Shannon sampling theory of signal analysis, the so-called WKS-sampling theorem, which can be established by methods of Fourier analysis, plays an essential role in many fields. The aim of this paper is to study the so-called exponential sampling theorem (ESF) of optical physics and engineering in which the samples are not equally spaced apart as in the Shannon case but exponentially spaced, using the Mellin transform methods. One chief aim of this paper is to study the reproducing kernel formula, not in its Fourier transform setting, but in that of Mellin, for Mellin-bandlimited functions as well as an approximate version for a less restrictive class, the MRKF, namely for functions which are only approximately Mellin-bandlimited. The rate of approximation for such signals will be measured in terms of the strong Mellin fractional differential operators recently studied by the authors. The final aim is to show that the exponential sampling theorem ESF is equivalent to the MRKF for Mellin-bandlimited functions (signals) in the sense that each is a corollary of the other. Three graphical examples are given, as illustration of the theory.

Advances in Shannon Sampling Theory

In this paper, the authors have provided a brief review of the recent advances in the Shannon sampling theory. In particular, they discuss the work related to 1-D signal reconstruction involving the samples taken below the Nyquist rate using nonlinear/time-variant systems. The extensions of the sampling theorems to the fractional Fourier and Linear canonical transform domains are also discussed. Defence Science Journal, 2013, 63(1), pp.41 -45 , DOI:http://dx.doi.org/10.14429/dsj. 63.3762

Analytic Kramer kernels, Lagrange-type interpolation series and de Branges spaces

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. In particular, when the involved kernel is analytic in the sampling parameter it can be stated in an abstract setting of reproducing kernel Hilbert spaces of entire functions which includes as a particular case the classical Shannon sampling theory. This abstract setting allows us to obtain a sort of converse result and to characterize when the sampling formula associated with an analytic Kramer kernel can be expressed as a Lagrange-type interpolation series. On the other hand, the de Branges spaces of entire functions satisfy orthogonal sampling formulas which can be written as Lagrange-type interpolation series. In this work some links between all these ideas are established.

Reconstruction of Noisy Electromagnetic Fields by Means of Compressive Sensing Theory

This paper deals with the emerging compressive sensing theory applied to reconstruction of the spatial distribution of a magnetic/electric field corrupted by additive noise. A typical application could be monitoring of the electromagnetic environment of a wide area where an electromagnetic source or susceptible electric/electronic apparatus are located. To this end, wireless field sensors can be assumed to be deployed over the monitored area and used to provide spatial samples of the field. The main advantages offered by compressive sensing include a number of sensors much smaller than the number foreseen by the traditional Shannon sampling theory, and the possibility to resort to nonuniform distribution of the sensors. A specific numerical analysis is devoted to investigate the effects of additive noise introduced by wireless technology, including quantization noise featuring low-cost sensors, environment and electronic noise.

Recovery of Missing Samples from Multi-Channel Oversampling in Shift-Invariant Spaces

It is well known that in the classical Shannon sampling theory on band-limited signals, any finitely many missing samples can be recovered when the signal is oversampled at a rate higher than the minimum Nyquist rate. In this work, we consider the problem of recovering missing samples from multi-channel oversampling in a general shift-invariant space. We find conditions under which any finite or infinite number of missing samples can be recovered when they are from a single or two channels.

The Link Dividing Method for Traffic Information Based on Floating Car

The technologies of traffic information collection based on GPS equipped floating car have been become one of the main important means for real-time collecting traffic information in intelligent transportation system. So, the electronic map of route network is important to collect traffic information. As present, the link dividing methods can’t take into account the different conditions of traffic flow and the different attributes of the route. This paper presents a novel link dividing method: Firstly, the speed of the floating car history track is regarded as a stochastic and its frequency spectrum is obtained using Fourier transform; Secondly, the optimal sampling intervals of the link dividing are obtained using the frequency spectrum of speed and Shannon sampling theory; Finally, the novel link dividing model is founded based on the optimal sampling intervals and the attributes of the routes. This method not only considers the different conditions of traffic flow, but also distinguishes the different traffic flow of the different places of a link, which can improve application effect of the route traffic conditions and vehicle dynamic navigation.

Small-angle solution scattering using the mixed-mode pixel array detector

Solution small-angle X-ray scattering (SAXS) measurements were obtained using a 128 × 128 pixel X-ray mixed-mode pixel array detector (MMPAD) with an 860 µs readout time. The MMPAD offers advantages for SAXS experiments: a pixel full-well of >2 × 10(7) 10 keV X-rays, a maximum flux rate of 10(8) X-rays pixel(-1) s(-1), and a sub-pixel point-spread function. Data from the MMPAD were quantitatively compared with data from a charge-coupled device (CCD) fiber-optically coupled to a phosphor screen. MMPAD solution SAXS data from lysozyme solutions were of equal or better quality than data captured by the CCD. The read-noise (normalized by pixel area) of the MMPAD was less than that of the CCD by an average factor of 3.0. Short sample-to-detector distances were required owing to the small MMPAD area (19.2 mm × 19.2 mm), and were revealed to be advantageous with respect to detector read-noise. As predicted by the Shannon sampling theory and confirmed by the acquisition of lysozyme solution SAXS curves, the MMPAD at short distances is capable of sufficiently sampling a solution SAXS curve for protein shape analysis. The readout speed of the MMPAD was demonstrated by continuously monitoring lysozyme sample evolution as radiation damage accumulated. These experiments prove that a small suitably configured MMPAD is appropriate for time-resolved solution scattering measurements.