Nyquist Limit Research Articles

Frequency-Packed Faster-Than-Nyquist Signaling via Symbol-Level Precoding for Multiuser MISO Redundant Transmissions

This work addresses the issue of interference generated by co-channel users in downlink multi-antenna multicarrier systems with frequency-packed faster-than-Nyquist (FTN) signaling. The resulting interference stems from an aggressive strategy for enhancing the throughput via frequency reuse across different users and the squeezing of signals in the time-frequency plane beyond the Nyquist limit. The spectral efficiency is proved to be increasing with the frequency packing and FTN acceleration factors. The lower bound for the FTN sampling period that guarantees information losslesness is derived as a function of the transmitting-filter roll-off factor, the frequency-packing factor, and the number of subcarriers. Space-time-frequency symbol-level precoders (SLPs) that trade off constructive and destructive interblock interference (IBI) at the single-antenna user terminals are proposed. Redundant elements are added as guard interval to cope with vestigial destructive IBI effects. The proposals can handle channels with delay spread longer than the multicarrier-symbol duration. The receiver architecture is simple, for it does not require digital multicarrier demodulation. Simulations indicate that the proposed SLP outperforms zero-forcing precoding and achieves a target balance between spectral and energy efficiencies by controlling the amount of added redundancy from zero (full IBI) to half (destructive IBI-free) the group delay of the equivalent channel.

Performance of QR Code Detectors near Nyquist Limits.

For the interacting with real world, augmented reality devices need lightweight yet reliable methods for recognition and identification of physical objects. In that regard, promising possibilities are offered by supporting computer vision with 2D barcode tags. These tags, as high contrast and visually well-defined objects, can be used for finding fiducial points in the space or to identify physical items. Currently, QR code readers have certain demands towards the size and visibility of the codes. However, the increase of resolution of built-in cameras makes it possible to identify smaller QR codes in the scene. On the other hand, growing resolutions cause the increase to the computational effort of tag location. Therefore, resolution reduction in decoders is a common trade-off between processing time and recognition capabilities. In this article, we propose the simulation method of QR codes scanning near limits that stem from Shannon’s theorem. We analyze the efficiency of three publicly available decoders versus different size-to-sampling ratios (scales) and MTF characteristics of the image capture subsystem. The MTF we used is based on the characteristics of real devices, and it was modeled using Gaussian low-pass filtering. We tested two tasks—decoding and locating-and-decoding. The findings of the work are several-fold. Among others, we identified that, for practical decoding, the QR-code module should be no smaller than 3–3.5 pixels, regardless of MTF characteristics. We confirmed the superiority of Zbar in practical tasks and the worst recognition capabilities of OpenCV. On the other hand, we identified that, for borderline cases, or even below Nyquist limit where the other decoders fail, OpenCV is still capable of decoding some information.

Performance of QR Code Detectors near Nyquist Limits

Performance of QR Code Detectors near Nyquist Limits

Spatial- and temporal-interpolations for efficient hybrid wave numerical simulations

The hybrid simulation method is developed for simulating wave propagation only in a localized heterogeneous media with hybrid inputs obtained once for all from a known reference model. Despite the fact that the hybrid simulation method has a wide range of applications in computational seismology, the associated error control of this method has received relatively little attention in previous research works. We quantitatively discuss the error of the two-step hybrid method in acoustic wave cases and propose a spatial refinement scheme to compute hybrid inputs based on the multi-elements spline interpolation, which is preferable to traditional Lagrange interpolation since it uses more polydirectional interpolated points. This method can also be used for local refinement of wavefield in more general applications, such as saving smooth wavefield in the full-waveform inversion framework. Furthermore, to save memory requirements, hybrid inputs are proposed to be sparsely stored with a high upsampling ratio during the global simulation, and the Fourier interpolation method is introduced to recover them to their original time series. To demonstrate the effect of the proposed methods, we perform several 2D and 3D hybrid wave numerical simulations using the spectral element method. We find that when the global and local meshing differs, the proposed spatial interpolation method can appreciably reduce the error of the hybrid waveforms caused by inaccurate hybrid inputs. We also point out that the Fourier interpolation can efficiently recover the original waveform, allowing hybrid inputs to be stored with time steps toward the Nyquist limit. Our method is expected to become a standard method to reduce the error of hybrid waveforms and save the memory requirements during hybrid simulations and has potential implications for further improving the accuracy of the so-called box tomography.

Recovering cutting tool modal parameters from fractionally uncorrelated and potentially aliased signals

Modal parameters characterize how tools vibrate. Correct evaluation of modal parameters depends on how signals are sampled. Since tools can vibrate with frequencies of up to several kHz, respecting the Nyquist criterion requires sampling potentially at tens of kHz. This is easy enough with modern data acquisition systems. However, if/when using modal parameters to monitor condition of tools, transmitting, storing, and processing large data sets becomes difficult. Moreover, when extracting modal parameters using newer vision-based methods, it may not always be possible to acquire high resolution images at rates that avoid aliasing. This paper present solutions to address such cases by recovering modal parameters from signals sampled potentially below the Nyquist limit. No a priori knowledge of the system order is assumed, and folding properties of signals are leveraged to recover parameters from fractionally uncorrelated signals using notions of set theory. To aid recovery, we suggest formal procedures to group candidates of likely modes together and resolve the case of modes being potentially confounded. Special spatiotemporal decoupling properties inherent to modal analysis are leveraged to recover eigenvectors from potentially aliased signals. Recovery is illustrated using the eigensystem realization algorithm. Numerical experiments with systems of different orders, of signals with noise, and of systems in which likely modes can be confounded with other likely candidates are designed to test robustness of the method. Those findings guide experiments with measured accelerations and video of an end mill. Results confirm that parameters recovered using proposed methods agree with those extracted from accelerations and videos sampled properly.

Universal photonics tomography.

3D imaging is essential for the study and analysis of a wide variety of structures in numerous applications. Coherent photonic systems such as optical coherence tomography (OCT) and light detection and ranging (LiDAR) are state-of-the-art approaches, and their current implementation can operate in regimes that range from under a few millimeters to over more than a kilometer. We introduce a general method, which we call universal photonics tomography (UPT), for analyzing coherent tomography systems, in which conventional methods such as OCT and LiDAR may be viewed as special cases. We demonstrate a novel approach (to our knowledge) based on the use of phase modulation combined with multirate signal processing to collect positional information of objects beyond the Nyquist limits.

Higher-Order Modulation vs Faster-Than-Nyquist PAM-4 for Datacenter IM-DD Optics: An AIR Comparison Under Practical Bandwidth Limits

The 4-ary pulse amplitude modulation (PAM-4) has been widely deployed to both electrical and optical interfaces in datacenter applications. Based on 100-GBd PAM-4, the 200G per lane optical interface is being actively developed in both industrial interoperability groups and standard bodies. Going beyond 200G, it is questionable if PAM-4 could support an even faster interface with higher baud-rate, considering the bandwidth growth for both electronics and optical components has started lagging the ever-increasing capacity demand. Under limited bandwidth, one option is to improve the spectral efficiency by higher order formats like PAM-6 and PAM-8, but this induces backward incompatibility as well as potential modulation mismatch between electrical and optical interfaces that requires additional energy for gearboxing. This revives the study of faster than Nyquist (FTN) signaling to pursue higher PAM-4 baud-rate breaking the bandwidth limit. In ECOC 2021, we demonstrated a reduced-complexity maximum a posteriori probability (MAP) decoder in intensity-modulation (IM) direct-detection (DD) systems with superior FTN performance. In this paper, we extend our work by adding more algorithm details and experimental results. Moreover, we perform the achievable information rate (AIR) comparison between the FTN PAM-4 and the higher-order modulation within the Nyquist limit in different channel conditions. Remarkably, in a practical IM-DD channel with gradually decayed frequency response, FTN shows AIR gain over the Nyquist approach even at a low FTN decoding complexity. The finding suggests FTN PAM could be a promising candidate at post 200G era to offer both AIR and implementation advantages.

Model order reduction of layered waveguides via rational Krylov fitting

Rational approximation recently emerged as an efficient numerical tool for the solution of exterior wave propagation problems. Currently, this technique is limited to wave media which are invariant along the main propagation direction. We propose a new model order reduction-based approach for compressing unbounded waveguides with layered inclusions. It is based on the solution of a nonlinear rational least squares problem using the RKFIT method. We show that approximants can be converted into an accurate finite difference representation within a rational Krylov framework. Numerical experiments indicate that RKFIT computes more accurate grids than previous analytic approaches and even works in the presence of pronounced scattering resonances. Spectral adaptation effects allow for finite difference grids with dimensions near or even below the Nyquist limit.

Technical loss estimation approach in power distribution systems using load model in frequency domain

Electricity losses have been a focus of attention for Distribution System Operators (DSOs) and regulatory bodies. To define non-technical loss reduction policies is necessary to determine their amount. The strategy adopted is to calculate the technical losses to obtain the non-technical losses through the total losses difference. This paper presents an approach to estimate electric technical losses in distribution systems, showing that the load model in the frequency domain calculation is robust. Our analyses examine a model for calculating the loss coefficient in distribution systems due to dynamic energy consumption characteristics. We decomposed the load profile by Discrete Fourier Transform (DFT) and applied it to a model adapted to compute the losses spectral analysis, using the Parseval Identity. This paper examines how to estimate the coefficient up to the Nyquist Limit, considering the losses contributions to high frequencies. We evaluate this impact using a practical calculation in the load frequency domain. In this context, it would be interesting to reduce the computational cost and increase the accuracy estimate. Load flow simulations show that the model in the frequency domain presents an error of 0.1514% when compared with the model in the time domain. We obtained an error of 1.8718% using a correlation model between the loss simulations with the low-frequency components and the resolution limit. The model in the frequency domain has greater predictability, considering the load harmonic components' exponential decay characteristic. The intended purpose of this paper is defining regulatory losses in distribution systems.

High-fidelity sub-Nyquist ghost imaging with tri-directional probing

Ghost imaging is an unconventional imaging method, which has invoked many applications in various fields. However, it is still a major challenge to achieve high-fidelity high-resolution images at a sub-Nyquist sampling rate. Here, we present a ghost imaging method that illuminates an object with three directional Tetris-like patterns, which can greatly trade off the contradiction between the high resolution and high detection signal-to-noise ratio. As the projected patterns gradually shrink during the detection, the image is also gradually recovered from low to high resolution. In addition, this method can recover complex chromatic objects without any compromising image quality by adaptively abandoning unnecessary patterns at sampling rates well below the Nyquist limit. Meanwhile, the dynamic probing scheme has an excellent noise-removal capability. The simulation and experiment demonstrate that the sampling rate to recover a high-fidelity image is only 10.7% for a scene of a 9.7% duty cycle. For a very noisy scene whose peak signal–noise rate (PSNR) is 10.18 dB [the structural similarity index (SSIM) is 0.068], this scheme increases the PSNR to 18.63 dB [structural similarity index (SSIM) to 0.73]. Therefore, the proposed method may be useful for ghost imaging in the low sampling rate regime or complex chromatic objects reconstruction.

Polarization-based Speckle Nulling Using a Spatial Light Modulator to Generate a Wide-field Dark Hole

Abstract Direct detection of exoplanets requires a high-contrast instrument called a coronagraph to reject bright light from the central star. However, a coronagraph cannot perfectly reject the starlight if the incoming stellar wave front is distorted by aberrations due to the Earth’s atmospheric turbulence and/or the telescope instrumental optics. Wave-front aberrations cause residual stellar speckles that prevent detection of faint planetary light. In this paper, we report a laboratory demonstration of a speckle-nulling wave-front control using a spatial light modulator (SLM) to suppress the residual speckles of a common-path visible nulling coronagraph. Because of its large format, the SLM potentially has the ability to generate a dark hole over a large region or at a large angular distance from a star of up to hundreds of λ/D. We carry out a laboratory demonstration for three cases of dark hole generation: (1) in an inner region (3–8 λ/D in horizontal and 5–15 λ/D in vertical directions), (2) in an outer region (70–75 λ/D in horizontal and 65–75 λ/D in vertical directions), and (3) in a large region (5–75 λ/D in both directions). As a result, the residual speckles are rejected to contrast levels on the order of 10−8 in cases 1 and 2. In cases 2 and 3, we can generate dark holes at a large distance (up to >100 λ/D) and with a large size (70 λ/D square), both of which are out of the Nyquist limit of currently available deformable mirrors.

Tunable tiling selective plane illumination microscopy with 3D Nyquist sampling

Selective plane illumination microscopy (SPIM) is a volumetric fluorescent imaging method based on orthogonal light pathways for excitation and detection. A common approach to rapidly image large samples is to reduce the spatial sampling rate below the Nyquist limit to acquire a large field of view (FOV). One consequence of extending the lateral FOV is a loss of axial resolution as the excitation light sheet properties are tuned to match the FOV. It is possible to maintain high axial resolution across large FOVs by translating the excitation light sheet through a static sample.

Application of super-resolution and correlative double sampling in cryo-electron microscopy.

Developments in cryo-EM have allowed atomic or near-atomic resolution structure determination to become routine in single particle analysis (SPA). However, near-atomic resolution structures determined using cryo-electron tomography and sub-tomogram averaging (cryo-ET STA) are much less routine. In this paper, we show that collecting cryo-ET STA data using the same conditions as SPA, with both correlated double sampling (CDS) and the super-resolution mode, allowed apoferritin to be reconstructed out to the physical Nyquist frequency of the images. Even with just two tilt series, STA yields an apoferritin map at 2.9 Å resolution. These results highlight the exciting potential of cryo-ET STA in the future of protein structure determination. While processing SPA data recorded in super-resolution mode may yield structures surpassing the physical Nyquist limit, processing cryo-ET STA data in the super-resolution mode gave no additional resolution benefit. We further show that collecting SPA data in the super-resolution mode, with CDS activated, reduces the estimated B-factor, leading to a reduction in the number of particles required to reach a target resolution without compromising the data size on disk and the area imaged in SerialEM. However, collecting SPA data in CDS does reduce throughput, given that a similar resolution structure, with a slightly larger B-factor, is achievable with optimised parameters for speed in EPU (without CDS).

Pixel MTF response effect on non-null interferometry.

The pixel modulation transfer function response degrades the contrast of non-null interferometric surface figure measurements. We experimentally quantify this effect for spatial frequencies ranging from 0 to 363 lp/mm (≈3.33 times the Nyquist limit). Our results show a low SNR spatial frequency band that behaves like a low-pass filter for sub-Nyquist interferometry and a stop-band filter for multiple-wavelength phase-shifting interferometry. We also introduce a multiple-mode, multiple-wavelength interferometry approach to measure optical surfaces with slope departure angles mapping to spatial frequencies in this low SNR band. The extended measurement range of this approach is achieved without using a sparse-array detector.

Removing the Courant-Friedrichs-Lewy stability criterion of the explicit time-domain very high degree spectral-element method with eigenvalue perturbation

The explicit time-domain spectral-element method (SEM) for synthesizing seismograms has gained tremendous credibility within the seismological community at all scales. Although the recent introduction of nonperiodic homogenization has addressed the spatial meshing difficulty of the mechanical discontinuities, the Courant-Friedrichs-Lewy (CFL) stability criterion strictly constrains the maximum time step, which still puts a great burden on the numerical simulation. In the explicit time-domain SEM, the source of instability of using a time step beyond the stability criterion is that some unstable eigenvalues of the updated matrix are larger than what can be accurately simulated. We have succeeded in removing the CFL stability condition in explicit time-domain SEM by combining the forward time dispersion-transform method, the eigenvalue perturbation, and the inverse time dispersion-transform method. Our theoretical analyses and numerical experiments in the homogeneous, moderate, and strong heterogeneous models show that this combination can accurately simulate waveforms with time steps several times the size of the CFL limit even toward the Nyquist limit especially for the efficient very high degree SEM, which abundantly saves iteration times without suffering from time-dispersion error. It demonstrates a potential application prospect in some situations such as full-waveform inversion that require multiple numerical simulations for the same model.

Sub-Nyquist ghost imaging by optimizing point spread function.

Point spread function (PSF) of ghost imaging (GI) with pseudo-thermal light source doesn't satisfy the property of space translation invariance and existing GI linear reconstruction algorithms offer images with low quality when the measurement process doesn't reach ergodic. By modifying the intensity value of the speckle patterns recorded by the camera in the reference path, the property of PSF can be optimized and a linear reconstruction method called optimized ghost imaging (OGI) is proposed to stably recover the object's image even in the measurements below Nyquist limit. In comparison with existing GI linear reconstruction algorithms, both the simulated and experimental results demonstrate that the image's SNR can be significantly enhanced by OGI especially when the sampling ratio is larger than 0.68 and the detection SNR is greater than 20 dB.

Practical considerations for using K3 cameras in CDS mode for high-resolution and high-throughput single particle cryo-EM

Detector technology plays a pivotal role in high-resolution and high-throughput cryo-EM structure determination. Compared with the first-generation, single-electron counting direct detection camera (Gatan K2), the latest K3 camera is faster, larger, and now offers a correlated-double sampling mode (CDS). Importantly this results in a higher DQE and improved throughput compared to its predecessor. In this study, we focused on optimizing camera data collection parameters for daily use within a cryo-EM facility and explored the balance between throughput and resolution. In total, eight data sets of murine heavy-chain apoferritin were collected at different dose rates and magnifications, using 9-hole image shift data collection strategies. The performance of the camera was characterized by the quality of the resultant 3D reconstructions. Our results demonstrated that the Gatan K3 operating in CDS mode outperformed standard (nonCDS) mode in terms of reconstruction resolution in all tested conditions with 8 electrons per pixel per second being the optimal dose rate. At low magnification (64kx) we were able to achieve reconstruction resolutions of 149% of the physical Nyquist limit (1.8 Å with a 1.346 Å physical pixel size). Low magnification allows more particles to be collected per image, aiding analysis of heterogeneous samples requiring large data sets. At moderate magnification (105kx, 0.834 Å physical pixel size) we achieved a resolution of 1.65 Å within 8-h of data collection, a condition optimal for achieving high-resolution on well behaved samples. Our results also show that for an optimal sample like apoferritin, one can achieve better than 2.5 Å resolution with 5 min of data collection. Together, our studies validate the most efficient ways of imaging protein complexes using the K3 direct detector and will greatly benefit the cryo-EM community.

Micro two-dimensional slit-array for super resolution beyond pixel Nyquist limits in grating spectrometers.

This paper presents a new micro 2-D slit-array device for spectral resolution enhancement in grating spectrometers. The 2-D slit-array is encoded in Hadamard matrix and the device is fabricated based on the micro-electromechanical system (MEMS) technology. By just using this 2-D slit-array to replace the single slit in the conventional grating spectrometer, real-time super spectral resolution detection beyond the pixel Nyquist limit, which is determined by the size of the detector pixel, can be realized. Furthermore, no other configuration of the spectrometer is changed, no movable parts are used, and the spectral range and instrument size remain almost unchanged while the resolution is improved. A series of experimental verifications for the feasibility of this design are included in this work.

Electronically steered metasurface antenna

Mobile devices, climate science, and autonomous vehicles all require advanced microwave antennas for imaging, radar, and wireless communications. We propose a waveguide-fed metasurface antenna architecture that enables electronic beamsteering from a lightweight circuit board with varactor-tuned elements. Our approach uses a unique feed structure and layout that enables spatial sampling at the Nyquist limit of half a wavelength. We detail the design of this Nyquist metasurface antenna and experimentally demonstrate electronic beamsteering in two directions. Nyquist metasurface antennas can realize high performance without costly and power hungry phase shifters, making them a compelling technology for future antenna hardware.

Iterative filtered ghost imaging

It is generally believed that, in ghost imaging, there has to be a compromise between resolution and visibility. Here we propose and demonstrate an iterative filtered ghost imaging scheme whereby a super-resolution image of a grayscale object is achieved, while at the same time the signal-to-noise ratio (SNR) and visibility are greatly improved, without adding complexity. The dependence of the SNR, visibility, and resolution on the number of iterations is also investigated and discussed. Moreover, with the use of compressed sensing the sampling number can be reduced to less than 1% of the Nyquist limit, while maintaining image quality with a resolution that can exceed the Rayleigh diffraction bound by more than a factor of 10.