Nyquist Limit Research Articles

Accurate power spectrum estimation toward Nyquist limit

The power spectrum, as a statistic in Fourier space, is commonly numerically calculated using the fast Fourier transform method to efficiently reduce the computational costs. To alleviate the systematic bias known as aliasing due to the insufficient sampling, the interlacing technique was proposed. We derive the analytical form of the shot noise under the interlacing technique, which enables the exact separation of the Poisson shot noise from the signal in this case. Thanks to the accurate shot noise subtraction, we demonstrate an enhancement in the accuracy of power spectrum estimation. For the three dark matter samples with number density ranging from 10-2 Mpc-3 h3 to 10-4 Mpc-3 h3 analyzed on a coarse mesh of size 3 h-1 Mpc with Cloud-in-Cell mass assignment scheme, the systematic bias is well under control up to the Nyquist frequency. On the contrary, the bias induced by the estimator that ignore the aliasing on the shot noise typically exceeds the statistical uncertainty on the frequency beyond 0.85 times the Nyquist frequency. The good performance of our estimation allows an abatement in the computational cost by using low resolution and low order mass assignment scheme in the analysis for huge surveys and mocks.

A Cyclic Spectroscopy Scintillation Study of PSR B1937+21. I. Demonstration of Improved Scintillometry

We use cyclic spectroscopy to perform high-frequency resolution analyses of multihour baseband Arecibo observations of the millisecond pulsar PSR B1937+21. This technique allows for the examination of scintillation features in far greater detail than is otherwise possible under most pulsar timing array observing setups. We measure scintillation bandwidths and timescales in each of eight subbands across a 200 MHz observing band in each observation. Through these measurements we obtain intra-epoch estimates of the frequency scalings for scintillation bandwidth and timescale. Thanks to our high-frequency resolution and the narrow scintles of this pulsar, we resolve scintillation arcs in the secondary spectra due to the increased Nyquist limit, which would not have been resolved at the same observing frequency with a traditional filterbank spectrum using NANOGrav’s current time and frequency resolutions, and the frequency-dependent evolution of scintillation arc features within individual observations. We observe the dimming of prominent arc features at higher frequencies, possibly due to a combination of decreasing flux density and the frequency dependence of the plasma refractive index of the interstellar medium. We also find agreement with arc curvature frequency dependence predicted by Stinebring et al. in some epochs. Thanks to the frequency-resolution improvement provided by cyclic spectroscopy, these results show strong promise for future such analyses with millisecond pulsars, particularly for pulsar timing arrays, where such techniques can allow for detailed studies of the interstellar medium in highly scattered pulsars without sacrificing the timing resolution that is crucial to their gravitational-wave detection efforts.

Real-Time Synthetic Aperture Radar Imaging with Random Sampling Employing Scattered Power Mapping.

A novel image-reconstruction method is proposed for the processing of data acquired at random spatial positions. The images are reconstructed and updated in real time concurrently with the measurements to produce an evolving image, the quality of which is continuously improving and converging as the number of data points increases with the stream of additional measurements. It is shown that the images converge to those obtained with data acquired on a uniformly sampled surface, where the sampling density satisfies the Nyquist limit. The image reconstruction employs a new formulation of the method of scattered power mapping (SPM), which first maps the data into a three-dimensional (3D) preliminary image of the target on a uniform spatial grid, followed by fast Fourier space image deconvolution that provides the high-quality 3D image.

Four-dimensional variational data assimilation of a turbulent jet for super-temporal-resolution reconstruction

The super-temporal-resolution (STR) reconstruction of turbulent flows is an important data augmentation application for increasing the data reach in measurement techniques and understanding turbulence dynamics. This paper proposes a data assimilation (DA) strategy based on weak-constraint four-dimensional variation to conduct an STR reconstruction in a turbulent jet beyond the Nyquist limit from given low-sampling-rate observations. Highly resolved large-eddy simulation (LES) data are used to produce synthetic measurements, which are used as observations and for validation. A segregated assimilation procedure is realised to assimilate the initial condition, inflow boundary condition and model error separately. Different types of observational data are tested. The first type is down-sampled LES data containing many small-scale turbulence structures with or without synthetic noise. The DA results show that the temporal variation of the small-scale structures is well recovered even with noise in the observations. The spectra are resolved to a frequency approximately one order of magnitude higher than what can be captured within the Nyquist limit. The second type of observation is low-sampling-rate tomographic particle image velocimetry (tomo-PIV) data with or without the injection of small-scale structures. The modulation between the large-scale structures contained in the tomo-PIV fields and the small scales injected from the observations is improved. The resultant small scales in the STR reconstruction have the characteristics of authentic turbulence to a considerable extent. Additionally, DA yields much smaller errors in the prediction of particle positions when compared with the Wiener filter, demonstrating the great potential for Lagrangian particle tracking in measurement techniques.

Compressed Sensing of Field-Resolved Molecular Fingerprints Beyond the Nyquist Frequency

Ultrashort time-domain spectroscopy and field-resolved spectroscopy of molecular fingerprints are gold standards for detecting samples’ constituents and internal dynamics. However, they are hindered by the Nyquist criterion, leading to prolonged data acquisition, processing times, and sizable data volumes. In this work, we present the first experimental demonstration of compressed sensing on field-resolved molecular fingerprinting by employing random scanning. Our measurements enable pinpointing the primary absorption peaks of atmospheric water vapor in response to terahertz light transients while sampling beyond the Nyquist limit. By drastically undersampling the electric field of the molecular response at a Nyquist frequency of 0.8 THz, we could successfully identify water absorption peaks up to 2.5 THz with a mean squared error of 12 × 10 −4 . To our knowledge, this is the first experimental demonstration of time-domain compressed sensing, paving the path toward real-time field-resolved fingerprinting and acceleration of advanced spectroscopic techniques.

Contributed Session III: The limits of resolution in the S-cone pathway.

The resolution of the S-cone pathway is first constrained by the density and arrangement of S-cones in the photoreceptor mosaic. Prior work comparing S-cone isolating visual acuity (sVA) to histological estimates of S-cone density has led to mixed conclusions, likely due to inter-individual differences in the S-cone sub-mosaic. We examined sVA in subjects with spectrally classified cone mosaics to test how the grain of the S-cone sub-mosaic limits resolution. Three observers whose cones were previously classified via adaptive optics (AO)-OCT based optoretinography participated in an sVA task at eccentricities ranging from 1.3° to 12.9° spread along all four cardinal meridians. Observers adapted to yellow light [CIE (0.45, 0.51); 200 cd/m2] for two minutes.Then, acuity was measured using a Tumbling 'E' task that showed blue [CIE (0.16, 0.044); 0.66 cd/m2) letters on the same yellow background, with an S-cone contrast of 0.93 (L/M-cone contrasts <0.01). Simultaneously recorded high-resolution AOSLO videos helped guide stimulus delivery to the spectrally classified retinal area. Measured sVAs were worse than predicted by the calculated S-cone Nyquist limit at all eccentricities, suggesting pooling of information from S-cones. Moreover, this pooling increased with eccentricity. While sVA does not follow the S-cone Nyquist limit, it is in good concordance with the Nyquist limit corresponding to known estimates of the sampling density of small bistratified retinal ganglion cells.

The m-dimensional spatial Nyquist limit using the wave telescope for larger numbers of spacecraft

Abstract. Spacecraft constellations consisting of multiple satellites are becoming more and more interesting not only for commercial use but also for space science missions. The proposed and accepted scientific multi-satellite missions that will operate within Earth's magnetospheric environment, like HelioSwarm, require researchers to extend established methods for the analysis of multi-spacecraft data to more than four spacecraft. The wave telescope is one of those methods. It is used to detect waves and characterize turbulence from multi-point magnetic field data, by providing spectra in reciprocal position space. The wave telescope can be applied to an arbitrary number of spacecraft already. However, the exact limits of the detection for such cases are not known if the spacecraft, acting as sampling points, are irregularly spaced. We extend the wave telescope technique to an arbitrary number of spatial dimensions and show how the characteristic upper detection limit in k space imposed by aliasing, the spatial Nyquist limit, behaves for irregularly spaced sampling points. This is done by analyzing wave telescope k-space spectra obtained from synthetic plane wave data in 1D up to 3D. As known from discrete Fourier transform methods, the spatial Nyquist limit can be expressed as the greatest common divisor in 1D. We extend this to arbitrary numbers of spatial dimensions and spacecraft. We show that the spatial Nyquist limit can be found by determining the shortest possible basis of the spacecraft distance vectors. This may be done using linear combination in position space and transforming the obtained shortest basis to k space. Alternatively, the shortest basis can be determined mathematically by applying the modified Lenstra–Lenstra–Lovász (MLLL) algorithm combined with a lattice enumeration algorithm. Thus, we give a generalized solution to the determination of the spatial Nyquist limit for arbitrary numbers of spacecraft and dimensions without any need of a priori knowledge of the measured data. Additionally, we give first insights into the application to real-world data incorporating spacecraft position errors and minimizing k-space aliasing. As the wave telescope is an estimator for a multi-dimensional power spectrum substituting spatial Fourier transform, the results of this analysis can be applied to power spectral density estimation via Fourier transform or other methods making use of irregular sampling points. Therefore, our findings are also of interest to other fields of signal processing.

Rotating machinery speed extraction through smartphone video acquisition from a radial viewpoint

This article investigates the viability of using a smartphone as a cost-effective tool for rotational speed extraction. The proposed method exploits the geometrical image deformations induced by a smartphone’s rolling shutter camera, which reads pixel lines sequentially with a delay equal to the rolling shutter period. This characteristic period is the basis for the developed methodology and allows measuring constant and varying speeds above the Nyquist limit related to the camera’s frame rate. With respect to the rotating shaft, the smartphone is pointed to the side surface of the shaft (radial viewpoint). A method is proposed to measure the speed from the deformation of a zebra pattern as a consequence of the sequential readout of a rolling shutter camera. A mathematical model, which also takes into account a possible misalignment between the shaft and the camera’s rolling shutter reading direction, is developed and validated on an in-house test rig. It is shown that the proposed method is robust against changing lighting conditions and a misalignment between the shaft and the smartphone camera. Normalized Root-Mean-Square Errors (NRMSE) of about 3% (or less) are reached for the smartphone aligned with the shaft with peaks in the percentage error of about 5% to 8%. Nevertheless, even for a misalignment of 60°, the NRMSE is only 4% with variations in the percentage error to about 10%.The proposed method shows promising results with room for improvements, making it possible to develop a smartphone application to measure a shaft’s rotational speed in the future.

Optical authentication method based on correspondence ghost imaging.

Ghost imaging technology has a great application potential in optical security because of its non-local characteristics. In this paper, on the basis of computational ghost imaging, an optical authentication scheme is proposed that utilizes the correspondence imaging technique for the preliminary reconstruction of the object image, and then authenticates the image by a nonlinear correlation algorithm. Different from the previous optical authentication schemes that usually adopted random selection of measurements, this authentication method consciously selects the bucket detector measurement values with large fluctuation and can achieve authentication using ultra-low data volumes less than 1% of the Nyquist limit. In brief, this scheme is easy to implement and has a simpler algorithm and higher stability, which is a tremendous advantage in practical optical authentication systems. The simulation and physical experimental results demonstrate the feasibility of the scheme.

Optical multiple-image authentication method based on Fourier single-pixel imaging and multiple logistic maps.

As a promising technique, the spatial information of an object can be acquired by employing active illumination of sinusoidal patterns in the Fourier single-pixel imaging. However, the major challenge in this field is that a large number of illumination patterns should be generated to record measurements in order to avoid the loss of object details. In this paper, an optical multiple-image authentication method is proposed based on sparse sampling and multiple logistic maps. To improve the measurement efficiency, object images to be authenticated are randomly sampled based on the spatial frequency distribution with smaller size, and the Fourier sinusoid patterns generated for each frequency are converted into binarized illumination patterns using the Floyd-Steinberg error diffusion dithering algorithm. In the generation process of the ciphertext, two chaotic sequences are used to randomly select spatial frequency for each object image and scramble all measurements, respectively. Considering initial values and bifurcation parameters of logistic maps as secret keys, the security of the cryptosystem can be greatly enhanced. For the first time to our knowledge, how to authenticate the reconstructed object image is implemented using a significantly low number of measurements (i.e., at a very low sampling ratio less than 5% of Nyquist limit) in the Fourier single-pixel imaging. The experimental results as well as simulations illustrate the feasibility of the proposed multiple-image authentication mechanism, which can provide an effective alternative for the related research.

Pixel Super‐Resolution Interference Pattern Sensing Via the Aliasing Effect for Laser Frequency Metrology

AbstractThe superposition of several optical beams with large mutual angles results in sub‐micrometer periodic patterns with a complex intensity, phase, and polarization structure. For high‐resolution imaging thereof, one often employs optical super‐resolution methods such as scanning nano‐particle imaging. Here, it is reported that by using a conventional arrayed image sensor in combination with 2D Fourier analysis, the periodicities of light fields much smaller than the pixel size can be resolved in a simple and compact setup, with a resolution far beyond the Nyquist limit set by the pixel size. The ability to resolve periodicities with spatial frequencies of ≈3 µm−1, 15 times higher than the pixel sampling frequency of 0.188 µm−1, is demonstrated. This is possible by analyzing high‐quality Fourier aliases in the first Brillouin zone. In order to obtain the absolute spatial frequencies of the interference patterns, simple rotation of the image sensor is sufficient, which modulates the effective pixel size and allows determination of the original Brillouin zone. Based on this method, wavelength sensing with a resolving power beyond 100,000 without any special equipment is demonstrated.

Pointing stabilization of 140 mJ, 10 Hz UV laser for Laser-Assisted Charge Exchange

This paper details a laser beam delivery and pointing stabilization system designed for the Laser-Assisted Charge Exchange (LACE) experiment at the Spallation Neutron Source (SNS) accelerator. The UV (355 nm) laser beam of 140 mJ pulses at 10 Hz must be transported over 65 m through a retrofitted laser transport line in a high-radiation area of the accelerator, causing the alignment of the beam to jitter and drift significantly on timescales of 0.1–1 s and >1 min, respectively, due to thermal effects and other disturbances. The laser pointing stabilization system is based on a novel technique which uses active feedback between CMOS cameras and a piezoelectric steering mirror controlled by feedback software to make corrections to the horizontal angle and vertical position of the laser beam at 10 Hz. Low frequency drift is easily controlled, and the long-term stability is limited by the amplitude of high frequency components of the spectrum near the Nyquist limit (5 Hz). With feedback turned on, the root-mean-square (RMS) deviations of the horizontal angle and vertical position over 20 min fell from 314 to 157 μrad and 309 to 167μm, respectively, which is below the pulse-to-pulse jitter at 10 Hz of 189 μrad RMS and 180μm RMS.

Distance recovery via swept frequency mixing for data-efficient FMCW LiDAR.

We propose a novel, to the best of our knowledge, distance recovery method via swept frequency mixing for frequency-modulated continuous wave (FMCW) light detection and ranging (LiDAR) to overcome the Nyquist limit and obtain high data efficiency. A one dimensional (1D) experiment was conducted to recover the optical fiber length; in addition, a 3D image was obtained by recovering the distances of several targets in free space. Compared to conventional methods based on fast Fourier transform (FFT), beat frequency up to 14 times the Nyquist limit for sampling frequency was successfully measured without aliasing. The proposed method dramatically increases the data efficiency in FMCW LiDAR by reducing the number of complex algorithms and experimental resources required.

Generalized central slice theorem perspective on Fourier-transform spectral imaging at a sub-Nyquist sampling rate.

Fourier-transform spectral imaging captures frequency-resolved images with high spectral resolution, broad spectral range, high photon flux, and low stray light. In this technique, spectral information is resolved by taking Fourier transformation of the interference signals of two copies of the incident light at different time delays. The time delay should be scanned at a high sampling rate beyond the Nyquist limit to avoid aliasing, at the price of low measurement efficiency and stringent requirements on motion control for time delay scan. Here we propose, what we believe to be, a new perspective on Fourier-transform spectral imaging based on a generalized central slice theorem analogous to computerized tomography, using an angularly dispersive optics decouples measurements of the spectral envelope and the central frequency. Thus, as the central frequency is directly determined by the angular dispersion, the smooth spectral-spatial intensity envelope is reconstructed from interferograms measured at a sub-Nyquist time delay sampling rate. This perspective enables high-efficiency hyperspectral imaging and even spatiotemporal optical field characterization of femtosecond laser pulses without a loss of spectral and spatial resolutions.

Unrecognized Patent Foramen Ovale in Patient With Sinus Venosus-Type Atrial Septal Defect With Partial Anomalous Pulmonary Venous Return.

Unrecognized Patent Foramen Ovale in Patient With Sinus Venosus-Type Atrial Septal Defect With Partial Anomalous Pulmonary Venous Return.

Recovering cutting tool modal parameters from randomly sampled signals using compressed sensing

A change in the modal parameters of cutting tools could signal tool wear, tool breakage, or other instabilities. The cutting process must be continuously monitored using vibration signals to detect such changes. Since tools vibrate with frequencies of up to a few kHz, continuous monitoring requires sampling at rates of tens of kHz to respect the Nyquist limit. Processing and storing such large data for decision making is cumbersome. To address this issue, this paper discusses the use of a compressed sensing framework that enables non-uniform random sampling at rates below the Nyquist limit. For cutting tools, we show for the first time using synthesized data that it is possible to reconstruct original signals from as few as 1% of the original data. We numerically test the method to characterize the influence of damping, noise, and multiple modes. Recovered modal parameters from the reconstructed signal agree with signals sampled properly.

Sub-wavelength passive single-shot computational super-oscillatory imaging

Super-oscillatory (SO) imaging presents a passive solution to the sub-diffraction imaging challenge. The point spread functions (PSFs) of SO systems are band-limited and locally oscillate faster than their highest Fourier frequency. Because of this property, SO imaging systems are not bound by the Nyquist limit, allowing them to acquire details finer than the diffraction limit. In this work, we present a comprehensive theoretical analysis of passive incoherent SO imaging, leading to two key results. First, we show that the SO property of an imaging system is preserved when the system is combined with a general system of lenses. This opens the door for integrating SO imaging into existing microscopes and telescopes. Second, we show that incoherent SO imaging cannot resolve feature sizes below λ / 2 because of the inability to achieve incoherence at finer scales. This establishes the operational limits of the approach. We demonstrate our theory experimentally with a compound SO system that achieves sub-wavelength resolutions using high-NA lenses and Fourier spectrum modulation.

On acoustically modulated jet shear layers and the Nyquist–Shannon sampling theorem

The goal of this paper is to present the behavior of a jet shear layer in response to acoustic excitation from a signal processing perspective. The main idea is that the vortices that roll-up in the jet shear layer are similar to the discrete samples of a digital control system, and, hence, that the Nyquist–Shannon sampling theorem should apply. We further hypothesize that the strength of a vortex is determined by the mean amplitude of the excitation waveform during its creation. We also argue that, at least in some cases, demodulation occurs as a result of the vorticity signal generated by the convection of discrete vortices past a point in the shear layer. This vorticity signal is related to the amplitude modulation (AM) excitation waveform by a half-wave rectification operation, a common implementation of an AM demodulator. To investigate these ideas, a free, round jet that is excited upstream of the nozzle is studied using particle image velocimetry. Experiments are conducted that confirm that the sampling theorem applies, and an aliased response is observed when the Nyquist limit is exceeded. Previous authors have attributed demodulation to a vortex merging mechanism, but we demonstrate that merging is not always required for demodulation and suggest that it is one of two mechanisms at play.

Tolerance For the Pixelation Effect in Shear Measurement

Images taken by space telescopes typically have a superb spatial resolution, but a relatively poor sampling rate due to the finite CCD pixel size. Beyond the Nyquist limit, it becomes uncertain how much the pixelation effect may affect the accuracy of galaxy shape measurement. It is timely to study this issue given that a number of space-based large-scale weak lensing surveys are planned. Using the Fourier_Quad method, we quantify the shear recovery error as a function of the sampling factor Q, i.e., the ratio between the FWHM of the point-spread function (PSF) and the pixel size of the CCD, for different PSFs and galaxies of different sizes and noise levels. We show that sub-percent-level accuracy in shear recovery is achievable with single-exposure images for Q ≲ 2. The conclusion holds for galaxies much smaller than the PSF, and those with a significant level of noise.

Design and Analysis of Multi-User Faster-Than-Nyquist-DCSK Communication Systems over Multi-Path Fading Channels.

In this paper, we present a new multi-user chaos-based communication system using Faster-than-Nyquist sampling to achieve higher data rates and lower energy consumption. The newly designed system, designated Multi-user Faster Than Nyquist Differential Chaos Shift Keying (MU-FTN-DCSK), uses the traditional structure of Differential Chaos Shift Keying (DCSK) communication systems in combination with a filtering system that goes below the Nyquist limit for data sampling. The system is designed to simultaneously enable transmissions from multiple users through multiple sampling rates resulting in semi-orthogonal transmissions. The design, performance analysis, and experimental results of the MU-FTN-DCSK system are presented to demonstrate the utility of the newly proposed system in enabling multi-user communications and enhancing the spectral efficiency of the basic DCSK design without the addition of new blocks. The MU-FTN-DCSK system presented in this paper demonstrates spectral gains for one user of up to 23% and a combined gain of 25% for four () users. In this paper, we present a proof of concept demonstrating a new degree of freedom in the design of Chaos-based communication systems and their improvement in providing wireless transmissions without complicated signal processing tools or advanced hardware designs.