Sampling Theorem Research Articles

Predictive coding compressive sensing optical coherence tomography hardware implementation

Compressed sensing (CS) is an approach that enables comprehensive imaging by reducing both imaging time and data density, and is a theory that enables undersampling far below the Nyquist sampling rate and guarantees high-accuracy image recovery. Prior efforts in the literature have focused on demonstrations of synthetic undersampling and reconstructions enabled by compressed sensing. In this paper, we demonstrate the first physical, hardware-based sub-Nyquist sampling with a galvanometer-based OCT system with subsequent reconstruction enabled by compressed sensing. Acquired images of a variety of samples, with volume scanning time reduced by 89% (12.5% compression rate), were successfully reconstructed with relative error (RE) of less than 20% and mean square error (MSE) of around 1%.

A Comparative Study of Generalized Sampling Theorem based Digital Superresolution for 2D Data.

A Comparative Study of Generalized Sampling Theorem based Digital Superresolution for 2D Data.

A vibration transducer framework for the measurement and reconstruction of vibration signals via low-rate sampling

This study presents a novel, cost-effective, and highly accurate computer vision-based vibration transducer framework, enabling the measurement and reconstruction of vibration signals using video frames captured at a low frame per second (fps) relative to the Nyquist rate of the measured signal. In the proposed framework, an assembly consisting of several spring-mass systems serves as the primary sensor, while a camera module tracks the motion of the seismic masses as the secondary sensor. The setup, along with the modified compressive sensing equation (CS) equation, results in a transducer scheme that can recover the base motion signals from the displacement history of seismic masses sampled at rates much lower than the maximum predominant frequency of the original signal. The performance of the proposed transducer framework has been successfully verified by numerical simulations. In addition, an experimental study has been conducted in order to validate the concept under modelling and measurement error.

High sensitivity cameras can lower spatial resolution in high-resolution optical microscopy

High-resolution optical fluorescence microscopies and, in particular, super-resolution fluorescence microscopy, are rapidly adopting highly sensitive cameras as their preferred photodetectors. Camera-based parallel detection facilitates high-speed live cell imaging with the highest spatial resolution. Here, we show that the drive to use ever more sensitive, photon-counting image sensors in cameras can, however, have detrimental effects on the spatial resolution of the resulting images. This is particularly noticeable in applications that demand a high space-bandwidth product, where the image magnification is close to the Nyquist sampling limit of the sensor. Most scientists will often select image sensors based on parameters such as pixel size, quantum efficiency, signal-to-noise performance, dynamic range, and frame rate of the sensor. A parameter that is, however, typically overlooked is the sensor’s modulation transfer function (MTF). We have determined the wavelength-specific MTF of front- and back-illuminated image sensors and evaluated how it affects the spatial resolution that can be achieved in high-resolution fluorescence microscopy modalities. We find significant differences in image sensor performance that cause the resulting spatial resolution to vary by up to 28%. This result shows that the choice of image sensor has a significant impact on the imaging performance of all camera-based optical microscopy modalities.

Unlimited sampling beyond modulo

Analog-to-digital converters (ADCs) act as a bridge between the analog and digital domains. Two important attributes of any ADC are sampling rate and its dynamic range. For bandlimited signals, the sampling should be above the Nyquist rate. It is also desired that the signals' dynamic range should be within that of the ADC's; otherwise, the signal will be clipped. Nonlinear operators such as modulo or companding can be used prior to sampling to avoid clipping. To recover the true signal from the samples of the nonlinear operator, either high sampling rates are required, or strict constraints on the nonlinear operations are imposed, both of which are not desirable in practice. In this paper, we propose a generalized flexible nonlinear operator which is sampling efficient. Moreover, by carefully choosing its parameters, clipping, modulo, and companding can be seen as special cases of it. We show that bandlimited signals are uniquely identified from the nonlinear samples of the proposed operator when sampled above the Nyquist rate. Furthermore, we propose a robust algorithm to recover the true signal from the nonlinear samples. Compared to the existing methods, our approach has a lower mean-squared error for a given sampling rate, noise level, and dynamic range. Our results lead to less constrained hardware design to address the dynamic range issues while operating at the lowest rate possible.

Sampling Theorems Associated with Offset Linear Canonical Transform by Polar Coordinates

The sampling theorem for the offset linear canonical transform (OLCT) of bandlimited functions in polar coordinates is an important signal analysis tool in many fields of signal processing and optics. This paper investigates two sampling theorems for interpolating bandlimited and highest frequency bandlimited functions in the OLCT and offset linear canonical Hankel transform (OLCHT) domains by polar coordinates. Based on the classical Stark’s interpolation formulas, we derive the sampling theorems for bandlimited functions in the OLCT and OLCHT domains, respectively. The first interpolation formula is concise and applicable. Due to the consistency of the OLCHT order, the second interpolation formula is superior to the first interpolation formula in computational complexity.

Shannon’s Sampling Theorem for Set-Valued Functions with an Application

In this study, we defined a kind of Fourier expansion of set-valued square-integrable functions. In fact, we have seen that the classical Fourier basis also constitutes a basis for the Hilbert quasilinear space L2(−π,π,Ω(C)) of Ω(C)-valued square-integrable functions, where Ω(C) is the class of all compact subsets of complex numbers. Furthermore, we defined the quasi-Paley–Wiener space, QPW, using the Fourier transform defined for set-valued functions and thus we showed that the sequence sinc.−kk∈Z form also a basis for QPW. We call this result Shannon’s sampling theorem for set-valued functions. Finally, we gave an application based on this theorem.

Explore random music generator based on Short-Time Fourier Transform

This paper delves into the mechanism of how the Short-Time Fourier Transform (STFT) is used for generating random music. A brief review of the history of stochastic music development is at the outset. The paper contains the principle of audio digitization. This starts with how to convert a continuous audio signal into discrete samples. The Nyquist Theorem plays an important role in that process to preserve signal integrity. The Heisenberg uncertainty principle takes effect as the STFT is applied to covert these samples. It states that when converting the audio signal between the time domain and the frequency domain, the audio signal can only have the properties of one or the other. This paper categorized the audio signals into three categories based on their spectral characteristics. The paper also points out the reason why natural sound effects are always chosen to generate random music due to their inherent complexity and randomness. This paper demonstrates the detail of STFT in generating random music, explaining how to specifically manipulate spectrum analysis to generate random music with practical examples. The paper concludes by discussing the possible future role and direction of STFT in the field of stochastic music.

Super-resolution techniques to simulate electronic spectra of large molecular systems

An accurate treatment of electronic spectra in large systems with a technique such as time-dependent density functional theory is computationally challenging. Due to the Nyquist sampling theorem, direct real-time simulations must be prohibitively long to achieve suitably sharp resolution in frequency space. Super-resolution techniques such as compressed sensing and MUSIC assume only a small number of excitations contribute to the spectrum, which fails in large molecular systems where the number of excitations is typically very large. We present an approach that combines exact short-time dynamics with approximate frequency space methods to capture large narrow features embedded in a dense manifold of smaller nearby peaks. We show that our approach can accurately capture narrow features and a broad quasi-continuum of states simultaneously, even when the features overlap in frequency. Our approach is able to reduce the required simulation time to achieve reasonable accuracy by a factor of 20-40 with respect to standard Fourier analysis and shows promise for accurately predicting the whole spectrum of large molecules and materials.

Ray-Tracing-Assisted SAR Image Simulation under Range Doppler Imaging Geometry

In order to achieve an effective balance between SAR image simulation fidelity and efficiency, we proposed a ray-tracing-assisted SAR image simulation method under range doppler (RD) imaging geometry. This method utilizes the spatial traversal mode of RD imaging geometry to transmit discrete electromagnetic (EM) waves into the SAR radiation area and follows the Nyquist sampling law to set the density of transmitted EM waves to effectively identify the beam radiation area. The ray-tracing algorithm is used to obtain the backscatter amplitude and real-time slant range of the transmitted EM wave, which can effectively record the multiple backscattering among the components of the distributed target so that the backscattering subfields of each component can be correlated. According to the RD condition equation, the backscattering amplitude is assigned to the corresponding range gate, and the three-dimensional (3D) target is mapped into the two-dimensional (2D) SAR slant-range coordinate system, and the SAR target simulated image is directly obtained. Finally, the simulation images of the proposed method are compared qualitatively and quantitatively with those obtained by commercial simulation software, and the effectiveness of the proposed method is verified.

Adaptive Sampling for Signals Associated with the Special Affine Fourier Transform

Adaptive sampling is inspired by the working principle of biological neurons, which converts the amplitude information of the received signal into a time sequence through a time encoding machine (TEM). The article mainly studies uniform sampling and adaptive sampling of non-bandlimited signals in two kinds of function spaces associated with the special affine Fourier transform (SAFT). Firstly, based on the stability characterization of the basis functions, we prove the existence of orthogonal projection operators in function spaces associated with the canonical convolution and the SAFT-convolution, respectively. Secondly, uniform sampling theorems are established for two kinds of signals living in the proposed function spaces. Finally, adaptive sampling schemes based on the crossing TEM and the integrate-and-fire TEM are studied. Moreover, the corresponding iterative reconstruction algorithms with exponential convergence are provided.

Active control of noise radiated from a long cylinder: Can the Nyquist limitation be broken?

In the present paper, we consider active control of noise propagating from a long cylinder. For that purpose control sources are distributed on the external surface of the cylinder. They provide a secondary sound field which enables them to compensate for the noise field propagating outside. To implement active noise control, we apply an approach based on the Calderón potentials that have the projection property. Numerical simulations to attenuate broadband noise with different numbers of control sources per wavelength are carried out. The effect of the cylinder on the Green's function and consequent noise attenuation is studied. There are two key findings in this paper. First, the results demonstrate that the level of relative noise attenuation remarkably increases along with the growth of the distance. Second, it is shown that even less than two control sources per wavelength can provide essential noise cancelation. At first glance, this result contradicts the well-known Nyquist-Shannon sampling theorem. A theoretical explanation of this result is provided.

A 2D Ultrasonic Phased Array Optimization Framework Enabled by Reconfigurable Laser Induced Phased Arrays

Abstract Three-dimensional (3D) ultrasonic imaging enables the viewing of internal features in a more accurate way than cross-sectional imaging. 3D imaging requires two-dimensional (2D) ultrasonic phased arrays to resolve all three spatial dimensions through beamforming along azimuthal and elevation angles. 2D phased arrays pose a manufacturing and control challenge due to the requirement of large number of elements to satisfy the Nyquist sampling limit. This problem can be alleviated through the use of sparse ultrasonic array designs. The optimization process is critical, as this dictates the efficiency of suppression of grating lobes. The aim of this work is to establish a framework for design of optimized sparse 2D phased arrays. This is enabled by exploiting the reconfigurability of Laser Induced Phased Arrays (LIPAs). LIPAs are synthetic arrays produced by scanning one laser for ultrasound generation and another for ultrasound detection, independently of each other. The reconfigurability and decoupled ultrasonic generation and detection of this systems enables easy and fast implementation of any arbitrary array layout. The framework consists of an analytical model for parametric evaluation of designs. The model performs a parametric sweep on the generation and detection element positions for each array design in order to identify the optimal parameters, based on mean and maximum side-lobe level. In the presented framework, four array designs are utilized for the optimization process: matrix, random, Vernier and a novel array design the Rotated Array (RA).

Compressive stochastic subspace identification: A modal parameter identification method suitable for compressive sampling

Stochastic subspace identification (SSI) and its variants are widely used in conventional structural health monitoring (SHM) owing to their high computation robustness and estimation accuracy. Conventional SHM is based on a Nyquist (high-rate and uniform) sampling dictated by Shannon–Nyquist sampling theorem. As data quantity of SHM is increasing with duration and equipment at an ever-growing rate and new applications of SHM with strict constricts on sampling are emerging one by one, sub-Nyquist sampling, also known as compressive sampling has increasingly been used in SHM in recent years. Therefore, despite the success of SSI method and its algorithmic variants, a significant drawback lies in their handling of sub-Nyquist data. To overcome this drawback, we set out to extend the applicable scenarios of SSI from Nyquist (high-rate/uniform) sampling to sub-Nyquist (low-rate/nonuniform) sampling. In this paper, we propose a compressive stochastic subspace identification (CSSI) which can identify modal parameters from sub-Nyquist/compressed data. Specifically, CSSI involves two aspects: covariance matrix reconstruction and system parameter identification. We find the complete covariance matrix in classic SSI can also be estimated/reconstructed by sub-Nyquist samples. More importantly, this reconstruction can be physically guaranteed through special sampling patterns without sparsity prior. Based on the above findings, we develop a compressed covariance sensing-based covariance matrix reconstruction technique and derive the optimal sub-Nyquist sampling pattern such that the reconstruction is hassle-free and efficient. Then, we perform modal parameter identification on the basis of the reconstructed covariance matrix. Finally, the effectiveness of CSSI is validated by simulations and experiments. In short, CSSI, as a new extension of SSI, can extract modal parameters at lower sampling rates than what is prescribed by the classic SSI. The implication of this is the possibility that CSSI for operational modal analysis will require fewer measurements thus it would reduce the storage requirement and the bandwidth for transmitting data.

Review on high spatial resolution dosimetry with pixelated semiconductor detectors for radiation therapy

Current and emerging radiation therapy treatment techniques are characterized by complex and spatially non-uniform dose distributions with steep dose gradients delivered through high-precision dose placement with small radiation fields. Such approaches necessitate advanced verification tools. For accurate dose profile representation in stereotactic radiation therapy using megavoltage X-rays, the Nyquist theorem places the minimum sampling step as 2.5 mm that determines the required spatial resolution of the detectors for dose verification. For techniques such as micro- and mini-beam radiotherapy, micron-scale sampling is required, and sub-millimeter for particle therapy. Silicon semiconductor detectors provide the possibility of producing small sensitive volumes (including in arrays) without compromising mechanical robustness or sensitivity, therefore allowing for high spatial resolution dosimetry. They exhibit many properties of an ideal detector, while the properties of silicon detectors known to negatively influence response can be compensated for and minimized. As such, this work reviews the current status of semiconductor radiation detectors providing dose sampling by diodes with submillimeter sensitive volumes and a pitch of 2.5 mm or smaller for relative dosimetry in X-ray and electron external beam radiation therapy, brachytherapy, microbeam radiation therapy, proton, and heavy ion therapy. The following types of devices are discussed in this work, including commercial systems and those under development: one-dimensional and two-dimensional pixelated arrays (monolithic and single diode based).

Energy Efficiency for Faster-than-Nyquist Data Transmission Using Processing Algorithms with Decision Feedback

One of the ways to increase the volume of transmitted information is to increase the bit rate above the Nyquist barrier. However, an increase in bit rate in the case of FTN (Faster-Than-Nyquist) signals leads to an increase in energy costs for receiving information on channels with limited bandwidth, for example, in Digital Video Broadcasting satellite systems like DVB-S2/S2X. It is possible to minimize energy losses by using the processing algorithm “maximum likelihood sequence estimation”. However, the computational complexity of this algorithm is extremely high, which limits its use, especially in terrestrial mobile satellite terminals. We propose a new bit-by-bit decision feedback algorithm with maximum likelihood ratio estimation of subsequent symbols in the observation interval. This algorithm provides minimal energy costs comparable to the method “maximum likelihood sequence estimation” at speeds 2–3 times higher than the Nyquist barrier. At the same time, the complexity is two orders of magnitude less. It is shown by simulation for a channel with additive noise that energy losses in relation to the potential bit error rate (BER) are less than 4.5 dB. In the presence of Rayleigh fading, the application of the proposed algorithm makes it possible to provide the processing of FTN signals for double bit rates in urban areas with energy costs equal to 12 dB when using an equalizer. We give numerical estimations of the increase in computational complexity for the proposed processing algorithm. It is shown that an increase in the bit rate by 1.5 times leads to an increase in the computational complexity by more than an order of magnitude. The same conclusion can be reformulated in another form: for the proposed algorithm, each decibel of energy gain is achieved by increasing the number of computational operations by 1.5×105. It is experimentally shown that additional energy losses due to non-ideal phase and timing synchronization are no more than 1 dB when the proposed algorithm is applied in a fading channel. The energy costs in fading channels relative to a stationary channel for twice the Nyquist rate are equal to 13.8 dB when using an equalizer.

Blind power spectrum reconstruction for multi-sinusoid signals with generative multicoset sampling

As the target frequency increases, the Nyquist rate is hard to reach due to hardware limitations. Alternatives to high-rate sampling have attracted considerable research interest. Compressed covariance sensing based on multicoset sampling (MCS) is a prevalent sub-Nyquist sampling strategy for blind power spectrum reconstruction. However, it requires the sampling pattern of MCS to be a minimal sparse ruler. In this paper, a generative multicoset sampling (GMCS) is proposed to minimize the number of cosets. In the sampling stage, a minimal MCS, delay coprime scheme is proposed to ensure the recoverability of the power spectrum. In the recovery stage, a new function, i.e., multifold correlation is constructed for blind power spectrum reconstruction. We prove that the power spectrum of a multi-sinusoid signal (MSS) can be estimated by its multifold correlation and the sample density of multifold correlation can be increased by correlation operations. Owing to multifold correlation, GMCS can generate virtual cosets satisfying the sparse ruler criterion through only two physical cosets and then efficiently recover the power spectrum by least-squares estimation. Moreover, we demonstrate the feasibility of GMCS for linear frequency modulation (LFM) signals. Finally, the effectiveness of GMCS are verified by numerical simulations and blade tip timing experiments (a non-contact vibration measurement). By reducing the average sampling rate and hardware complexity, GMCS opens new avenues for blind sampling and power spectrum reconstruction.

Spatial-spectral encoding and dictionary optimization in compressive single-pixel hyperspectral imaging based on mutual coherence minimization

A single-pixel detector based hyperspectral system provides an effective way to obtain the spatial-spectral information of target scenes. However, complex spectral dispersion and the substantial number of measurements not only increase the complexity of the system but also decrease the sampling efficiency and the reconstruction accuracy. In this paper, we propose a compressive sensing (CS) theory based single-pixel hyperspectral imaging system. Based on structured illumination, the spatial information is modulated by binary spatial patterns displayed on a liquid crystal on silicon (LCoS), while polarizing elements at specific angles, acting as a serious of filters, modulate the spectral dimension, effectively avoiding spectral dispersion. In terms of sampling efficiency, the application of CS significantly decreases the number of measurements required compared to the Nyquist-Shannon sampling theorem. Besides, to improve the reconstruction accuracy, mutual coherence minimization is employed to optimize the pre-trained dictionary, spatial patterns and filters. Furthermore, a two-step encoding method based on macro-pixel segmentation is proposed to address the issue of low resolution constrained by the size of the dictionary. Compared to the unoptimized system and dictionary, the proposed method achieves more accurate reconstruction results in both spectral and spatial dimensions. This work may provide opportunities for high-resolution single-pixel hyperspectral imaging systems based on CS.

Range-Velocity Measurement Accuracy Improvement Based on Joint Spatiotemporal Characteristics of Multi-Input Multi-Output Radar

For time division multiplexing multiple input multiple output (TDM MIMO) millimeter wave radar, the measurement of target range, velocity and other parameters depends on the phase of the received Intermediate Frequency (IF) signal. The coupling between range and velocity phases occurs when measuring moving targets, leading to inevitable errors in calculating range and velocity from the phase, which in turn affects measurement accuracy. Traditional two-dimensional fast fourier transform (2D FFT) estimation errors are particularly pronounced at high velocity, significantly impacting measurement accuracy. Additionally, due to limitations imposed by the Nyquist sampling theorem, there is a restricted range for velocity measurements that can result in aliasing. In this study, we propose a method to address the coupling of range and velocity based on the original signal as well as a method for velocity compensation to resolve aliasing issues. Our research findings demonstrate that this approach effectively reduces errors in measuring ranges and velocities of high-velocity moving targets while efficiently de-aliasing velocities.

LD-CSNet: A latent diffusion-based architecture for perceptual Compressed Sensing

Compressed Sensing (CS) is a groundbreaking paradigm in image acquisition, challenging the constraints of the Nyquist–Shannon sampling theorem. This enables high-quality image reconstruction using a minimal number of measurements. Neural Networks’ potent feature induction capabilities enable advanced data-driven CS methods to achieve high-fidelity image reconstruction. However, achieving satisfactory reconstruction performance, particularly in terms of perceptual quality, remains challenging at extremely low sampling rates. To tackle this challenge, we introduce a novel two-stage image CS framework based on latent diffusion, named LD-CSNet. In the first stage, we utilize an autoencoder pre-trained on a large dataset to represent natural images as low-dimensional latent vectors, establishing prior knowledge distinct from sparsity and effectively reducing the dimensionality of the solution space. In the second stage, we employ a conditional diffusion model for maximum likelihood estimates in the latent space. This is supported by a measurement embedding module designed to encode measurements, making them suitable for a denoising network. This guides the generation process in reconstructing low-dimensional latent vectors. Finally, the image is reconstructed using a pre-trained decoder. Experimental results across multiple public datasets demonstrate LD-CSNet’s superior perceptual quality and robustness to noise. It maintains fidelity and visual quality at lower sampling rates. Research findings suggest the promising application of diffusion models in image CS. Future research can focus on developing more appropriate models for the first stage.