Shannon Sampling Research Articles

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.

A Compressed Sensing Framework for Dynamic Sound-Field Measurements

The conventional sampling of sound fields by use of stationary microphones is impractical for large bandwidths. Satisfying the Nyquist–Shannon sampling theorem in three-dimensional space requires a huge number of sampling positions. Dynamic sound-field measurements with moving microphones together with a compressed-sensing recovery allow for weakening the spatial sampling problem. For bandlimited signals, the dynamic samples taken along the microphone trajectory may be related to the room impulse responses on a virtual grid in space via spatial interpolation. The tracking of the microphone positions and the knowledge of the excitation sequence allow for setting up a linear system of equations that can be solved for the room impulse responses on the modeled virtual grid. Nevertheless, there is still the necessity for recovering a huge number of sound-field variables, in order to ensure aliasing-free reconstruction. Thus, for practical applications, random or suboptimally chosen trajectories may be expected to lead to underdetermined sampling problems for a given volume of interest. In this paper, we present a compressed sensing framework that enables us to uniquely solve the dynamic sampling problem despite having underdetermined variables. The spatio-temporal sampling problem is integrated into compressed sensing models that allow for stable and robust sub-Nyquist sampling given incoherent measurements. For a modeled equidistant grid and sparse Fourier representations, the influence of the microphone trajectories on the compressed sensing problem is investigated and a simple expression is derived for evaluating trajectories with regard to compressed-sensing based recovery.

A dual-level method of fundamental solutions for three-dimensional exterior high frequency acoustic problems

• Physical essence of the fictitious boundary is explained. • The influence law of the fictitious boundary on numerical results is revealed. • A dual-level method of fundamental solutions is proposed. • A new strategy to solve the ill-conditioning of the radial basis functions methods is provided. • The Helmholtz problems with up to non-dimensional wavenumber of 600 is simulated. Physical essence of the fictitious boundary of the method of fundamental solutions has been a mystery for a long time. In this study, we attempt to explain the reason why fictitious boundary has such a dramatic effect on numerical results. The influence law of the fictitious boundary on numerical results is revealed. Based on this understanding, a dual-level method of fundamental solutions with self-adaptive adjustment coefficients is proposed. The competitive attributes of the method are that it inherits the high numerical efficiency of the method of fundamental solutions, and it improves numerical stability significantly. The effect of the fictitious boundary on numerical results is eliminated by introducing the concept of equivalent slope. It should be noted that the dual-level method of fundamental solutions can simulate exterior high frequency acoustic problems under the lowest sampling frequency allowed by the Shannon's sampling theorem. Numerical experiment with up to non-dimensional wavenumber of 600 has successfully been conducted on a single laptop when one uses 100,000 degrees of freedom.

Sampling and Super Resolution of Sparse Signals Beyond the Fourier Domain

Recovering a sparse signal from its low-pass projections in the Fourier domain is a problem of broad interest in science and engineering and is commonly referred to as super-resolution. In many cases, however, Fourier domain may not be the natural choice. For example, in holography, low-pass projections of sparse signals are obtained in the Fresnel domain. Similarly, time-varying system identification relies on low-pass projections on the space of linear frequency modulated signals. In this paper, we study the recovery of sparse signals from low-pass projections in the Special Affine Fourier Transform domain (SAFT). The SAFT parametrically generalizes a number of well known unitary transformations that are used in signal processing and optics. In analogy to the Shannon's sampling framework, we specify sampling theorems for recovery of sparse signals considering three specific cases: (1) sampling with arbitrary, bandlimited kernels, (2) sampling with smooth, time-limited kernels and, (3) recovery from Gabor transform measurements linked with the SAFT domain. Our work offers a unifying perspective on the sparse sampling problem which is compatible with the Fourier, Fresnel and Fractional Fourier domain based results. In deriving our results, we introduce the SAFT series (analogous to the Fourier series) and the short time SAFT, and study convolution theorems that establish a convolution--multiplication property in the SAFT domain.

Analog to information conversion for sparse signals band-limited in fractional Fourier transform domain

The classical Shannon sampling theorem has a profound influence on signal processing and communication. With the increasing contradiction between high rate sampling and conversion accuracy, the traditional analog to digital conversion technology, which is based on the Shannon sampling theorem, is facing a great challenge, especially for the bottleneck effect on reducing the sampling rate. In recent years, the analog-to-information conversion (AIC) technology, which is based on the theory of compressive sensing, provides an effective method to solve this problem. However, the signal model of the existing AIC is only suitable for sparse signals band-limited in the Fourier transform (FT) domain. It cannot be applied to non-bandlimited chirp signals which is widely used in electronic information systems, including radar and communications. Towards this end, we propose a new AIC based on the fractional Fourier transform (FRFT), which is not only the extension of the traditional AIC in the FRFT domain, but also can solve the problem as mentioned above. The theoretical derivation is presented, and the corresponding simulation analysis is also given. The simulation results are consistent with the theoretical analysis.

Comparing convenience and probability sampling for urban ecology applications

Abstract Urban forest ecosystems confer multiple ecosystem services. There is therefore a need to quantify ecological characteristics in terms of community structure and composition so that benefits can be better understood in ecosystem service models. Efficient sampling and monitoring methods are crucial in this process. Full tree inventories are scarce due to time and financial constraints, thus a variety of sampling methods exist. Modern vegetation surveys increasingly use a stratified‐random plot‐based sampling to reduce the bias associated with convenience sampling, even though the latter can save time and increase species richness scores. The urban landscape, with a high degree of conspecific clustering and high species diversity, provides a unique biogeographical case for comparing these two methodological approaches. We use two spatially extensive convenience samples of the urban forest of Meran (Italy), and compare the community structure, tree characteristics and ecosystem service provision with 200 random circular plots. The convenience sampling resulted in a higher species diversity, incorporating more rare species. This is a result of covering more area per unit sampling time. Pseudorandom subplots were compared to the random plots revealing similar Shannon diversity and sampling comparability indices. Measured tree variables (diameter at breast height, height, tree‐crown width, height to crown base) were similar between the two methods, as were ecosystem service model outputs. Synthesis and applications. Our results suggest that convenience sampling may be a time and money saving alternative to random sampling as long as stratification by land‐use type is incorporated into the design. The higher species richness can potentially improve the accuracy of urban ecological models, which rely on species‐specific functional traits.

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.

Dynamic SAX parameter estimation for time series

PurposeThe paper aims to estimate the segment size and alphabet size of Symbolic Aggregate approXimation (SAX). In SAX, time series data are divided into a set of equal-sized segments. Each segment is represented by its mean value and mapped with an alphabet, where the number of adopted symbols is called alphabet size. Both parameters control data compression ratio and accuracy of time series mining tasks. Besides, optimal parameters selection highly depends on different application and data sets. In fact, these parameters are iteratively selected by analyzing entire data sets, which limits handling of the huge amount of time series and reduces the applicability of SAX.Design/methodology/approachThe segment size is estimated based on Shannon sampling theorem (autoSAXSD_S) and adaptive hierarchical segmentation (autoSAXSD_M). As for the alphabet size, it is focused on how mean values of all the segments are distributed. The small number of alphabet size is set for large distribution to easily distinguish the difference among segments.FindingsExperimental evaluation using University of California Riverside (UCR) data sets shows that the proposed schemes are able to select the parameters well with high classification accuracy and show comparable efficiency in comparison with state-of-the-art methods, SAX and auto_iSAX.Originality/valueThe originality of this paper is the way to find out the optimal parameters of SAX using the proposed estimation schemes. The first parameter segment size is automatically estimated on two approaches and the second parameter alphabet size is estimated on the most frequent average (mean) value among segments.

Adaptive pixel-super-resolved lensfree in-line digital holography for wide-field on-chip microscopy

High-resolution wide field-of-view (FOV) microscopic imaging plays an essential role in various fields of biomedicine, engineering, and physical sciences. As an alternative to conventional lens-based scanning techniques, lensfree holography provides a new way to effectively bypass the intrinsical trade-off between the spatial resolution and FOV of conventional microscopes. Unfortunately, due to the limited sensor pixel-size, unpredictable disturbance during image acquisition, and sub-optimum solution to the phase retrieval problem, typical lensfree microscopes only produce compromised imaging quality in terms of lateral resolution and signal-to-noise ratio (SNR). Here, we propose an adaptive pixel-super-resolved lensfree imaging (APLI) method which can solve, or at least partially alleviate these limitations. Our approach addresses the pixel aliasing problem by Z-scanning only, without resorting to subpixel shifting or beam-angle manipulation. Automatic positional error correction algorithm and adaptive relaxation strategy are introduced to enhance the robustness and SNR of reconstruction significantly. Based on APLI, we perform full-FOV reconstruction of a USAF resolution target (~29.85 mm2) and achieve half-pitch lateral resolution of 770 nm, surpassing 2.17 times of the theoretical Nyquist–Shannon sampling resolution limit imposed by the sensor pixel-size (1.67µm). Full-FOV imaging result of a typical dicot root is also provided to demonstrate its promising potential applications in biologic imaging.

Shift-invariant and sampling spaces associated with the special affine Fourier transform

The Special Affine Fourier Transformation or the SAFT generalizes a number of well known unitary transformations as well as signal processing and optics related mathematical operations. Shift-invariant spaces also play an important role in sampling theory, multiresolution analysis, and many other areas of signal and image processing. Shannon's sampling theorem, which is at the heart of modern digital communications, is a special case of sampling in shift-invariant spaces. Furthermore, it is well known that the Poisson summation formula is equivalent to the sampling theorem and that the Zak transform is closely connected to the sampling theorem and the Poisson summation formula. These results have been known to hold in the Fourier transform domain for decades and were recently shown to hold in the Fractional Fourier transform domain by A. Bhandari and A. Zayed. The main goal of this article is to show that these results also hold true in the SAFT domain. We provide a short, self-contained proof of Shannon's theorem for functions bandlimited in the SAFT domain and then show that sampling in the SAFT domain is equivalent to orthogonal projection of functions onto a subspace of bandlimited basis associated with the SAFT domain. This interpretation of sampling leads to least-squares optimal sampling theorem. Furthermore, we show that this approximation procedure is linked with convolution and semi-discrete convolution operators that are associated with the SAFT domain. We conclude the article with an application of fractional delay filtering of SAFT bandlimited functions.

Spaceability and strong divergence of the Shannon sampling series and applications

In this paper the structure of the set of functions for which the peak value of the Shannon sampling series is strongly divergent is analyzed. Strong divergence is closely linked to the non-existence of adaptive reconstruction methods. Signals in the Paley–Wiener space PWπ1 of bandlimited functions with absolutely integrable Fourier transform are considered, and it is shown that the set of strong divergence is spaceable and dense-lineable, i.e., that there exist an infinite dimensional closed subspace and an infinite dimensional dense subspace, such that we have strong divergence of the peak value of the Shannon sampling series for all functions from these sets, except the zero function. Further, it is proved that this result is not restricted to the Shannon sampling series, but rather holds for an entire class of reconstruction processes.

Reconstruction from limited single-particle diffraction data via simultaneous determination of state, orientation, intensity, and phase

Free-electron lasers now have the ability to collect X-ray diffraction patterns from individual molecules; however, each sample is delivered at unknown orientation and may be in one of several conformational states, each with a different molecular structure. Hit rates are often low, typically around 0.1%, limiting the number of useful images that can be collected. Determining accurate structural information requires classifying and orienting each image, accurately assembling them into a 3D diffraction intensity function, and determining missing phase information. Additionally, single particles typically scatter very few photons, leading to high image noise levels. We develop a multitiered iterative phasing algorithm to reconstruct structural information from single-particle diffraction data by simultaneously determining the states, orientations, intensities, phases, and underlying structure in a single iterative procedure. We leverage real-space constraints on the structure to help guide optimization and reconstruct underlying structure from very few images with excellent global convergence properties. We show that this approach can determine structural resolution beyond what is suggested by standard Shannon sampling arguments for ideal images and is also robust to noise.

Variations on the Donoho–Stark Uncertainty Principle Estimate

We examine estimates of the form $$\begin{aligned} \Vert f\Vert _{L^2}\le \mathcal {C}(\Vert f-P_Af\Vert _{L^2}+\Vert f-Q_Bf\Vert _{L^2}), \end{aligned}$$ for \(f\in L^2(\mathbb {R}^n)\), where \(P_A\) multiplies f by the characteristic function \(\chi _A\) of a measurable set \(A\subset \mathbb {R}^n\) and \(Q_B\) multiplies its Fourier transform \(\hat{f}\) by \(\chi _B\). We discuss a hierarchy of estimates, starting with a classical estimate of Donoho and Stark. For comparison, we look at estimates of the form above that follow from identities related to the Shannon sampling theorem.

Spaceability for sets of bandlimited input functions and stable linear time-invariant systems with finite time blowup behavior

The approximation of linear time-invariant systems by sampling series is studied for bandlimited input functions in the Paley–Wiener space PW π 1 , i.e., bandlimited signals with absolutely integrable Fourier transform. It has been known that there exist functions and systems such that the approximation process diverges. In this paper we identify a signal set and a system set with divergence, i.e., a finite time blowup of the Shannon sampling expression. We analyze the structure of these sets and prove that they are jointly spaceable, i.e., each of them contains an infinite-dimensional closed subspace such that for any function and system pair from these subspaces, except for the zero elements, we have divergence.

Bayesian Refinement of Accelerated Molecular Dynamics Simulations for Interpreting SAXS Experiments

Small angle X-ray scattering (SAXS) is an experimental method that is useful for capturing the full ensemble of flexible biomolecules, but it typically results in low-dimensional data that is difficult to interpret without additional structural knowledge. In principle, this information may be provided by molecular dynamics (MD) simulation, but conventional MD trajectories rarely represent the complete ensemble. Accelerated MD (aMD) was developed to overcome these sampling inadequacies by introducing a bias to the underlying energy landscape, which distorts the observed ensemble as a result. Here, we present a method for fitting aMD simulations with experimental SAXS data to accurately model the relative populations of representative solution states. Scattering states are first identified from MD trajectories, and then their populations are re-weighted against empirical data through a Bayesian Monte Carlo approach. Resistance to ensemble over-fitting is achieved by iteratively considering increasing subsets of scattering states and by reducing experimental data to the Shannon sampling limit. We apply this technique to several ubiquitin trimers and find that aMD models converge upon agreement with the observed SAXS profiles up to an order of magnitude faster than conventional simulations.

A backward approach to certain class of transport equations in any dimension based on the Shannon sampling theorem

ABSTRACTA backward method is proposed to compute the solutions to some class of transport equations at any temporal instant regardless of the dimension. The widely adopted Shannon sampling in information theory and signal processing is employed for the reconstruction of solutions through truncated cardinal series, citing its properties of accuracy in approximation and convenience in construction. With the method of characteristics, approximation coefficients at sampling nodes are obtained via backward tracking along the characteristics. This approach, due to Gobbi et al. [Numerical solution of certain classes of transport equations in any dimension by Shannon sampling, J. Comput. Phys. 229 (2010), pp. 3502–5322], can be considered as either a spectral or a wavelet method. The proposed method is further extended to a backward–forward scheme to solve Cauchy problems by employing a forward evolution along the characteristics. Numerical experiments are presented to verify the effectiveness, efficiency and high accuracy of the proposed method.

A method for measuring dynamic radiation patterns of passive and active phased antenna arrays

The study deals with a method for measuring dynamic radiation patterns of phased antenna arrays and active phased antenna arrays. The method consists of measuring the signal during electronically scanning the beam of a static antenna. We show the main differences between dynamic and static radiation patterns. The Whittaker Nyquist – Kotelnikov – Shannon sampling theorem forms the basis of our recommendations for selecting the number of measurement points in the case of a dynamic radiation pattern. We supply a diagram of a work station dedicated to measuring dynamic radiation patterns using the workbench developed by the Joint-stock company “V. Tikhomirov Scientific Research Institute of Instrument Design”. We present measured static and dynamic radiation patterns of a real-world active phased antenna array.Our method for measuring dynamic radiation patterns significantly decreases the time required for determining radiation characteristics of phased antenna arrays and active phased antenna arrays for separate angular cross-sections and over the whole visibility scope, and increases their information content. Moreover, measuring dynamic radiation patterns does not require a rotating workbench. It is possible to use dynamic radiation pattern to determine static (as measured by turning the antenna) far-field patterns, to restore electric current magnitude and phase distribution over the aperture and to troubleshoot the antenna.

Attitude Jitter Detection Based on Remotely Sensed Images and Dense Ground Controls: A Case Study for Chinese ZY-3 Satellite

Satellite platform vibration induced by the onboard dynamic components and exterior perturbation deteriorates platform stability and causes attitude jitter, resulting in image distortion and geometric accuracy degradation. This paper presents an attitude jitter detection method based on images and dense ground controls, which requires neither high-performance attitude measurement devices nor specific sensor configuration like parallax observation. Attitude variations will result in image space discrepancies at control points, from which the attitude jitter can be retrieved. The method was validated by a case study with ZY-3 satellite, which is the first civilian high-resolution stereo mapping satellite in China. The experimental results show that an almost constant jitter frequency of about 0.65 Hz was detected, which was consistent with the direct attitude observations. The photogrammetric method is theoretically capable of detecting attitude jitter up to half of the image line scanning rate according to Shannon's sampling theorem, which is far beyond the detectable range by direct attitude observations at a frequency usually not higher than 4 Hz. Currently, the attitude jitter of the roll angle and the pitch angle can be effectively detected and accurately estimated for the nadir camera. The pitch angle jitter is also well estimable for the forward and the backward cameras. With approximation, the roll angle jitter for the two off-nadir cameras can also be estimated, though with some deviation caused by the influence of the yaw angle jitter.

Noisy Compressive Sampling Based on Block-Sparse Tensors: Performance Limits and Beamforming Techniques

Compressive sampling (CS) is an emerging research area for the acquisition of sparse signals at a rate lower than the Shannon sampling rate. Recently, CS has been extended to the challenging problem of multidimensional data acquisition. In this context, block-sparse core tensors have been introduced as the natural multidimensional extension of block-sparse vectors. The $(M_1,\ldots,M_Q)$ block sparsity for a tensor assumes that $Q$ support sets, characterized by $M_q$ indices corresponding to the nonzero entries, fully describe the sparsity pattern of the considered tensor. In the context of CS with Gaussian measurement matrices, the Cramer–Rao bound (CRB) on the estimation accuracy of a Bernoulli-distributed block-sparse core tensor is derived. This prior assumes that each entry of the core tensor has a given probability to be nonzero, leading to random supports of truncated Binomial-distributed cardinalities. Based on the limit form of the Poisson distribution, an approximated CRB expression is given for large dictionaries and a highly block-sparse core tensor. Using the property that the mode unfolding matrices of a block-sparse tensor follow the multiple-measurement vectors (MMV) model with a joint sparsity pattern, a fast and accurate estimation scheme, called Beamformed mOde based Sparse Estimator (BOSE), is proposed in the second part of this paper. The main contribution of the BOSE is to “map” the MMV model onto the single MV model, thanks to beamforming techniques. Finally, the proposed performance bounds and the BOSE are applied in the context of CS to $\text{1}\text{)}$ nonbandlimited multidimensional signals with separable sampling kernels and $\text{2)}$ for multipath channels in a multiple-input multiple-output wireless communication scheme.