Sinc Interpolation Research Articles

ФИЗИКАНЫ ЖАНА АСТРОНОМИЯНЫ ОКУТУУДА ЖАНА ИЗИЛДӨӨДӨ МАТЕМАТИКАНЫН ОРДУ

This scientific article is devoted to the ways of using mathematics in teaching and researching the subjects of physics and astronomy. Newton's first polynomial was used in mathematical modeling of the emission of electron and hole color centers in solid state physics. Shannon's formula was used to determine information entropy, and the logarithmic function of Boltzmann's formula was used to determine statistical entropy. In astronomy, trigonometric functions and stereographic transformations are used to determine celestial coordinates and transform very large angles. This research paper will help students in their study of these disciplines. Particular attention is paid to the close connection between mathematics and astronomy and physics.

Two‐dimensional adaptive Whittaker–Shannon Sinc‐based zooming

AbstractIn this work, we introduce a novel image zooming methodology that transitions from a nonadaptive Sin‐based approach to an adaptive Sinc‐based zooming technique. The two techniques base their theoretical foundation on the Whittaker–Shannon interpolation formula and the Nyquist theorem. The evolution into adaptive Sinc‐based zoom is accomplished through the use of two novel concepts: (1) the pixel‐local scaled k‐space and (2) the k‐space filtering sigmoidal function. The pixel‐local scaled k‐space is the standardized and scaled k‐space magnitude of the image to zoom. The k‐space filtering sigmoidal function scales the pixel‐local scaled k‐space values into the numerical interval [0, 1]. Using these two novel concepts, the Whittaker–Shannon interpolation formula is elaborated and used to zoom images. Zooming is determined by the shape of the Sinc functions in the Whittaker–Shannon interpolation formula, which, in turn, depends on the combined effect of the pixel‐local scaled k‐space, the sampling rate, and the k‐space filtering sigmoidal function. The primary outcome of this research demonstrates that the Whittaker–Shannon interpolation formula can achieve successful zooms for values of the sampling rate significantly greater than the bandwidth. Conversely, when the sampling rate is much greater than the bandwidth, the nonadaptive technique fails to perform the zoom correctly. The conclusion is that the k‐space filtering sigmoidal function is identified as the crucial parameter in the adaptive Sinc‐based zoom technique. The implications of this research extend to Sinc‐based image zooming applications.

Variable Doppler Starting Point Keystone Transform for Radar Maneuvering Target Detection

The Doppler band compensated by the keystone transform (KT) is limited. Therefore, it needs to be used in conjunction with the Doppler ambiguity compensation function to correct the range migration (RM) caused by maneuvering targets with Doppler ambiguity. However, the KT implemented by sinc interpolation suffers from significant performance loss at boundaries of compensation Doppler bands. Additionally, in a multi-target scenario, KT implementation methods occupy high complexity when the Doppler range of targets spans over two compensation Doppler bands. To address the aforementioned issues, this study presents a variable Doppler starting point keystone transform (VDSPKT) method, where a new form of ambiguity compensation function is constructed, turning the Doppler starting point of the compensation band in KT variable. Firstly, the position of the compensation Doppler band is changed from fixed to adjustable as needed, enhancing the flexibility of KT. Crucially, the connection points of the compensation Doppler bands in sinc interpolation are reset, avoiding performance loss at their boundaries. Also, the compensation band is adjusted to cover the narrow Doppler frequency range caused by targets, significantly improving computational efficiency. Finally, the simulation and real data experiments demonstrate that the proposed approach effectively addresses the performance degradation and high computational complexity of KT in the aforementioned scenarios, resulting in a computational load reduced by approximately 50% compared to traditional methods.

Randomized Nonuniform Sampling for Random Signals Bandlimited in the Special Affine Fourier Transform Domain

The nonuniform sampling and reconstruction of bandlimited random signals in the SAFT domain is discussed in the paper, where the nonuniform samples are obtained by randomly disturbing the uniform sampling. First, we prove that the concerned nonuniform problem is equivalent to the process of uniform sampling after a prefilter in the statistic sense. Then, an approximate reconstruction method based on sinc interpolation is proposed for the randomized nonuniform sampling of SAFT-bandlimited random signals. Finally, we offer the mean square error estimate for the corresponding approximate recovery approach. The results generalize the conclusions of nonuniform sampling of bandlimited random signals in the FrFT and LCT domains to the SAFT domain.

The Evaluation of Entropy-based Algorithm towards the Production of Closed-Loop Edge

This research concerns the common problem of edge detection that produces a disjointed and incomplete edge, leading to the misdetection of visual objects. The entropy-based algorithm can potentially solve this problem by classifying the pixel belonging to which objects in an image. Hence, the paper aims to evaluate the performance of entropy-based algorithm to produce the closed-loop edge representing the formation of object boundary. The research utilizes the concept of Entropy to sense the uncertainty of pixel membership to the existing objects to classify pixels as the edge or object. Six entropy-based algorithms are evaluated, i.e., the optimum Entropy based on Shannon formula, the optimum of relative-entropy based on Kullback-Leibler divergence, the maximum of optimum entropy neighbor, the minimum of optimum relative-entropy neighbor, the thinning of optimum entropy neighbor, and the thinning of optimum relative-entropy neighbor. The experiment is held to compare the developed algorithms against Canny as a benchmark by employing five performance parameters, i.e., the average number of detected objects, the average number of detected edge pixels, the average size of detected objects, the ratio of the number of edge pixel per object, and the average of ten biggest sizes. The experiment shows that the entropy-based algorithms significantly improve the production of closed-loop edges, and the optimum of relative-entropy neighbor based on Kullback-Leibler divergence becomes the most desired approach among others due to the production of more considerable closed-loop edge in the average. This finding suggests that the entropy-based algorithm is the best choice for edge-based segmentation. The effectiveness of Entropy in the segmentation task is addressed for further research.

The Evaluation of Entropy-based Algorithm towards the Production of Closed-Loop Edge

This research concerns the common problem of edge detection that produces a disjointed and incomplete edge, leading to the misdetection of visual objects. The entropy-based algorithm can potentially solve this problem by classifying the pixel belonging to which objects in an image. Hence, the paper aims to evaluate the performance of entropy-based algorithm to produce the closed-loop edge representing the formation of object boundary. The research utilizes the concept of Entropy to sense the uncertainty of pixel membership to the existing objects to classify pixels as the edge or object. Six entropy-based algorithms are evaluated, i.e., the optimum Entropy based on Shannon formula, the optimum of relative-entropy based on Kullback-Leibler divergence, the maximum of optimum entropy neighbor, the minimum of optimum relative-entropy neighbor, the thinning of optimum entropy neighbor, and the thinning of optimum relative-entropy neighbor. The experiment is held to compare the developed algorithms against Canny as a benchmark by employing five performance parameters, i.e., the average number of detected objects, the average number of detected edge pixels, the average size of detected objects, the ratio of the number of edge pixel per object, and the average of ten biggest sizes. The experiment shows that the entropy-based algorithms significantly improve the production of closed-loop edges, and the optimum of relative-entropy neighbor based on Kullback-Leibler divergence becomes the most desired approach among others due to the production of more considerable closed-loop edge in the average. This finding suggests that the entropy-based algorithm is the best choice for edge-based segmentation. The effectiveness of Entropy in the segmentation task is addressed for further research.

The Evaluation of Entropy-based Algorithm towards the Production of Closed-Loop Edge

This research concerns the common problem of edge detection that produces a disjointed and incomplete edge, leading to the misdetection of visual objects. The entropy-based algorithm can potentially solve this problem by classifying the pixel belonging to which objects in an image. Hence, the paper aims to evaluate the performance of entropy-based algorithm to produce the closed-loop edge representing the formation of object boundary. The research utilizes the concept of Entropy to sense the uncertainty of pixel membership to the existing objects to classify pixels as the edge or object. Six entropy-based algorithms are evaluated, i.e., the optimum Entropy based on Shannon formula, the optimum of relative-entropy based on Kullback-Leibler divergence, the maximum of optimum entropy neighbor, the minimum of optimum relative-entropy neighbor, the thinning of optimum entropy neighbor, and the thinning of optimum relative-entropy neighbor. The experiment is held to compare the developed algorithms against Canny as a benchmark by employing five performance parameters, i.e., the average number of detected objects, the average number of detected edge pixels, the average size of detected objects, the ratio of the number of edge pixel per object, and the average of ten biggest sizes. The experiment shows that the entropy-based algorithms significantly improve the production of closed-loop edges, and the optimum of relative-entropy neighbor based on Kullback-Leibler divergence becomes the most desired approach among others due to the production of more considerable closed-loop edge in the average. This finding suggests that the entropy-based algorithm is the best choice for edge-based segmentation. The effectiveness of Entropy in the segmentation task is addressed for further research.

Emergent Spatial–Temporal Patterns in a Ring of Locally Coupled Population Oscillators

A closed chain of oscillators can be considered a model for ring-shaped ecosystems, such as atolls or the coastal zones of inland reservoirs. We use the logistic map, which is often referred to as an archetypical example of how complex dynamics can arise from very simple nonlinear equations, as a model for a separate oscillator in the chain. We present an original algorithm that allows us to find solutions to the spatiotemporal logistic equation quite efficiently or to state with certainty that there are no such solutions. Based on the Shannon formula, we propose formulas for estimating the spatial and temporal entropy, which allow us to classify our solutions as regular or irregular. We show that regular solutions can occur within the Malthus parameter region that corresponds to the irregular dynamics of a solitary logistic map.

Moving Target Detection Algorithm for Millimeter Wave Radar Based on Keystone-2DFFT

Millimeter wave radar has the advantage of all-day and all-weather capability for detection, speed measurement. It plays an important role in urban traffic flow monitoring and traffic safety monitoring. The conventional 2-dimensional Fast Fourier Transform (2DFFT) algorithm is performed target detection in the range-Doppler domain. However, the target motion will induce the range walk phenomenon, which leads to a decrease in the target energy and the performance of the target detection and speed measurement. To solve the above problems, this paper proposes a moving vehicle detection algorithm based on Keystone-2DFFT for a traffic scene. Firstly, this paper constructs and analyzes the Frequency Modulated ContinuousWave (FMCW) moving target signal model under traffic monitoring scenario’s radar observation geometry. The traditional 2DFFT moving target detection algorithm is briefly introduced. Then, based on mentioned signal model, an improved moving vehicle detection algorithm based on Keystone-2DFFT transform is proposed. The method first input the echo, then the range walk is removed by keystone transformation. the keystone transformation is achieved via Sinc interpolation. Next is transform data into range-Doppler domain to perform detection and speed estimation. The algorithm is verified by simulation data and real data.

Signal Processing and Waveform Re-Tracking for SAR Altimeters on High Mobility Platforms with Vertical Movement and Antenna Mis-Pointing.

Synthetic aperture radar (SAR) altimeters can achieve higher spatial resolution and signal-to-noise ratio (SNR) than conventional altimeters by Doppler beam sharpening or focused SAR imaging methods. To improve the estimation accuracy of waveform re-tracking, several average echo power models for SAR altimetry have been proposed in previous works. However, these models were mainly proposed for satellite altimeters and are not applicable to high-mobility platforms such as aircraft, unmanned aerial vehicles (UAVs), and missiles, which may have a large antenna mis-pointing angle and significant vertical movement. In this paper, we propose a novel semi-analytical waveform model and signal processing method for SAR altimeters with vertical movement and large antenna mis-pointing angles. A new semi-analytical expression that can be numerically computed for the flat pulse response (FSIR) is proposed. The 2D delay-Doppler map is then obtained by numerical computation of the convolution between the proposed analytical function, the probability density function, and the time/frequency point target response of the radar. A novel delay compensation method based on sinc interpolation for SAR altimeters with vertical movement is proposed to obtain the multilook echo, which can optimally handle non-integer delays and maintain signal frequency characteristics. In addition, a height estimation method based on least squares (LS) estimation is proposed. The LS estimator does not have an analytical solution, and requires iterative solving through gradient descent. We evaluate the performance of the proposed estimation strategy using simulated data for typical airborne scenarios. When the mis-pointing angles are within 10 degrees, the normalized quadratic error (NQE) of the proposed model is less than 10-10 and the RMSE of τ obtained by the re-tracking method fitted by the proposed model is less than 0.2 m, which indicates the high applicability of the model and accuracy of the re-tracking method.

Spatiotemporal Dynamics of a Reaction Diffusive Predator-Prey Model: A Weak Nonlinear Analysis

In the realm of ecology, species naturally strive to enhance their own survival odds. This study introduces and investigates a predator-prey model incorporating reaction-diffusion through a system of differential equations. We scrutinize how diffusion impacts the model’s stability. By analysing the stability of the model’s uniform equilibrium state, we identify a condition leading to Turing instability. The study delves into how diffusion influences pattern formation within a predator-prey system. Our findings reveal that various spatiotemporal patterns, such as patches, spots, and even chaos, emerge based on species diffusion rates. We derive the amplitude equation by employing the weak nonlinear multiple scales analysis technique and the Taylor series expansion. A novel sinc interpolation approach is introduced. Numerical simulations elucidate the interplay between diffusion and Turing parameters. In a two-dimensional domain, spatial pattern analysis illustrates population density dynamics resulting in isolated groups, spots, stripes, or labyrinthine patterns. Simulation results underscore the method’s effectiveness. The article concludes by discussing the biological implications of these outcomes.

Shannon Entropy of Ramsey Graphs with up to Six Vertices.

Shannon entropy quantifying bi-colored Ramsey complete graphs is introduced and calculated for complete graphs containing up to six vertices. Complete graphs in which vertices are connected with two types of links, labeled as α-links and β-links, are considered. Shannon entropy is introduced according to the classical Shannon formula considering the fractions of monochromatic convex α-colored polygons with n α-sides or edges, and the fraction of monochromatic β-colored convex polygons with m β-sides in the given complete graph. The introduced Shannon entropy is insensitive to the exact shape of the polygons, but it is sensitive to the distribution of monochromatic polygons in a given complete graph. The introduced Shannon entropies Sα and Sβ are interpreted as follows: Sα is interpreted as an average uncertainty to find the green α-polygon in the given graph; Sβ is, in turn, an average uncertainty to find the red β-polygon in the same graph. The re-shaping of the Ramsey theorem in terms of the Shannon entropy is suggested. Generalization for multi-colored complete graphs is proposed. Various measures quantifying the Shannon entropy of the entire complete bi-colored graphs are suggested. Physical interpretations of the suggested Shannon entropies are discussed.

Implementation of scheduling in hybrid power domain non‐orthogonal multiple access system considering adaptive modulation and coding schemes selection

SummaryCommunication systems that simultaneously combine orthogonal (OMA) and nonorthogonal multiple access (NOMA) are currently under research. Such systems are called hybrid and have more flexibility in user multiplexing, which allows for increased system capacity. One of the main focus areas of research in such systems is radio resource scheduling. The existing research works on scheduling approaches are based on the Shannon's formula, which can be used to obtain the limit capacity by ideal modulation and coding schemes (MCS). However, it is more correct to consider scheduling based on the real capacity. To get it, one should consider the performance of real MCSs with finite code length; however, they are very rarely taken into account in NOMA research. The reason is that one of the basic unresolved problems in NOMA is MCS selection for users. To solve it, we have presented a table of joint pairwise MCS combinations with their adaptive selection for NOMA. In this work, we propose a radio resource scheduling approach for the OMA/NOMA system based on real MCSs that uses our own method of joint MCS selection for two NOMA users. By performing the simulation of the proposed scheduling approach, we have obtained both total and user capacity in the OMA/NOMA system, as well as the distribution of traffic among users, demonstrating the execution of the scheduling strategies.

Achievable Rate Region for uRLLC Interference Channel With Finite Blocklength Transmission

Finite blocklength is a favorable way to meet the extremely low-latency requirement, which leads to the classic Shannon's capacity formula no longer suitable. With the achievable rate region, optimization problems on three different scenarios are considered: (i) joint design of power and blocklength allocation for two transceiver pairs with single-antenna transmitter; (ii) beamforming and blocklength design for the two transceiver pairs with multi-antenna transmitter; (iii) beamforming and blocklength design for the multiple transceiver pairs with multi-antenna transmitter. Under the finite blocklength transmission mode, the achievable rate expression is a complicated function with respected to the blocklength and power allocation coefficients or beamforming vector, which makes the optimization problems difficult to solve. In this paper, by analyzing the monotonic and concaveconvex properties of achievable rate with interference channel (IFC), we formulate the achievable rate region as the convex hull of a series of power/beamforming and blocklength rate regions for two-user systems with IFC, as well as the Pareto boundary of achievable rate region on the blocklength and beamforming vector for the two-user systems, respectively. Sequentially, the optimal blocklength and power coefficients are derived in the closed-form and optimal beamforming vectors are also obtained by transforming the non-convex problem into convex one. Numerical results provide the perspective of looking at the rate region with finite blocklength.

CNN-based Robust Sound Source Localization with SRP-PHAT for the Extreme Edge

Robust sound source localization for environments with noise and reverberation are increasingly exploiting deep neural networks fed with various acoustic features. Yet, state-of-the-art research mainly focuses on optimizing algorithmic accuracy, resulting in huge models preventing edge-device deployment. The edge, however, urges for real-time low-footprint acoustic reasoning for applications such as hearing aids and robot interactions. Hence, we set off from a robust CNN-based model using SRP-PHAT features, Cross3D [ 16 ], to pursue an efficient yet compact model architecture for the extreme edge. For both the SRP feature representation and neural network, we propose respectively our scalable LC-SRP-Edge and Cross3D-Edge algorithms which are optimized towards lower hardware overhead. LC-SRP-Edge halves the complexity and on-chip memory overhead for the sinc interpolation compared to the original LC-SRP [ 19 ]. Over multiple SRP resolution cases, Cross3D-Edge saves 10.32%~73.71% computational complexity and 59.77%~94.66% neural network weights against the Cross3D baseline. In terms of the accuracy-efficiency tradeoff, the most balanced version ( EM ) requires only 127.1 MFLOPS computation, 3.71 MByte/s bandwidth, and 0.821 MByte on-chip memory in total, while still retaining competitiveness in state-of-the-art accuracy comparisons. It achieves 8.59 ms/frame end-to-end latency on a Rasberry Pi 4B, which is 7.26× faster than the corresponding baseline.

Energy efficient power allocation for ultra‐reliable and low‐latency communications via unsupervised learning

AbstractEnergy efficiency (EE) is an important indicator in ultra‐reliable and low‐latency communication (URLLC). Power allocation is considered as an effective method to achieve high EE in URLLC. However, since the EE optimization problem is non‐convex, it is difficult to obtain the analytical solution efficiently. Moreover, to ensure reliable and low‐latency communication within a finite blocklength, the Shannon formula becomes impractical for URLLC. Therefore, finite blocklength coding theory is used to meet the requirements of URLLC. In this paper, the EE problem of URLLC is formulated and the power allocation function is parameterized to be optimized through a deep neural network (DNN). The DNN is trained through the primal‐dual iterative algorithm offline in the unsupervised manner, and can be deployed online to achieve real time power allocation results. The numerical results show the effectiveness of the proposed method.

How to generate accurate continuous thermal regimes from sparse but regular temperature measurements

Abstract In ecology, there is an emerging emphasis on the importance of capturing temperature variation at relevant scales. Temperature fluctuates continuously in nature but is sampled at discrete time points, so how often should ecologists measure temperature to capture its variation? A recent development in thermal ecology is the use of spectral analysis of temperature time series to determine at what frequencies important temperature fluctuations occur. Building on this, I borrow from signal processing theory to show how continuous thermal regimes can be effectively reconstructed from discrete, regular, measurements, and provide a rule of thumb for designing temperature sampling schemes that capture ecologically relevant temporal variation. I introduce sinc interpolation, a method for reconstructing continuous waveforms from discrete samples. Furthermore, I introduce the Nyquist–Shannon sampling theorem, which states that continuous complex waveforms can be perfectly sinc‐interpolated from discrete, regular, samples if sampling intervals are sufficiently short. To demonstrate the power of these concepts in an ecological context, I apply them to several published high‐resolved (15‐min intervals) temperature time series used for ecological predictions of insect development times. First, I use spectral analysis to illuminate the fluctuation frequencies that dominate the temperature data. Second, I employ sinc interpolation over artificially thinned versions of the temperature time series. Third, I compare interpolated temperatures with observed temperatures to demonstrate the Nyquist–Shannon sampling theorem and its relation to spectral analysis. Last, I repeat the ecological predictions using sinc‐interpolated temperatures. Daily, and less frequent, fluctuations dominated the variation in all the temperature time series. Therefore, in accordance with the Nyquist–Shannon sampling theorem, 11‐h measurement intervals consistently retrieved most 15‐min temperature variation. Moreover, previous predictions of insect development times were improved by using sinc‐interpolated, rather than averaged, temperatures. By identifying the highest frequency at which ecologically (or otherwise) relevant temperature fluctuations occur and applying the Nyquist–Shannon sampling theorem, ecologists (or others doing climate‐related research) can use sinc interpolation to produce remarkably accurate continuous thermal regimes from sparse but regular temperature measurements. Surprisingly, these concepts have remained largely unexplored in ecology despite their applicability, not least in thermal ecology.

A Multiple Interpolation Algorithm to Improve Resampling Accuracy in Data Triggers

To address the problem of low trigger accuracy during trigger resampling and variable sampling rate trigger resampling using a fixed sampling rate analog-to-digital converter (ADC), this paper proposes an interpolation method combining sinc interpolation and linear interpolation to improve accuracy, based on a digital trigger. After behavior simulation verification and actual field programmable gate array (FPGA) test verification, the data collected by two 3GSps 12-bit ADCs were subjected to 8-times sinc interpolation followed by 16-times linear interpolation processing, after which the original trigger resampling accuracy was increased by 128 times and the sampling rate could be realized to vary between 100 MHz and 1 GHz. A signal–noise ratio (SNR) of 46.80 dBFS, a spurious free dynamic range (SFDR) of 45.91 dB, and an effective number of bits (ENOB) of 7.48 bits were obtained by direct trigger resampling without algorithm processing in the behavior simulation. Meanwhile, an SNR of 58.98 dBFS, an SFDR of 60.96 dB, and an ENOB of 9.42 bits were obtained by trigger resampling after algorithm processing. Due to the influence of analog link signal loss and signal interference on the development board, an SNR, SFDR and ENOB of 51.97 dBFS, 61.26 dB, and 8.32 bits, respectively, were obtained from the trigger resampling in the FPGA test. The experimental results show that the algorithm has not only improved the triggering accuracy but has also improved the SNR, SFDR, and ENOB parameters.

Deep Neural Networks With Data Rate Model: Learning Power Allocation Efficiently

Learning-based resource allocation can be implemented in real-time, but deep neural networks (DNNs) developed in other fields such as computer vision are with high training complexity and weak generalizability. Leveraging domain knowledge in communications is promising for learning wireless policies efficiently. In this paper, we propose a framework of integrating the Shannon formula with DNNs, and derive a data rate-based DNN (DRNN), for learning resource allocation by taking power allocation as an example. The DRNN is with an iterative structure with multiple update layers, each consisting of a pre-determined model function, an update network, and a dimension reduction network. To justify the iterative structure, we prove the existence of an iteration function that converges to the optimal policy for the update layer to learn. To justify the structure of each update layer, we provide the conditions for the iteration function to be a composite function of the model function. We further incorporate permutation equivariance properties into the DRNN. Simulation results show that the numbers of training samples and free parameters, and the training time to achieve a desired system performance can be reduced remarkably by harnessing the data rate model and PE prior.

Pipeline Leakage Detection Based on Secondary Phase Transform Cross-Correlation.

Leaks from pipes and valves are a reputational issue in industry. Maintenance of pipeline integrity is becoming a growing challenge due to the serious socioeconomic consequences. This paper presents a secondary phase transform (PHAT) cross-correlation method to improve the performance of the acoustic methods based on cross-correlation for pipeline leakage detection. Acoustic emission signals generated by pipe leakage are first captured by the sensors at different locations, and are subsequently analyzed using the cross-correlation curve to determine whether leakage is occurring. When leakage occurs, time delay estimation (TDE) is further carried out by peak search in the cross-correlation curve between the two sensor signals. In the analysis, the proposed method calculates the secondary cross-correlation function before the PHAT operation. A sinc interpolation method is then introduced for automatic searching the peak value of the cross-correlation curve. Numerical simulations and experimental results confirm the improved performance of the proposed method for noise suppression and accurate TDE compared to the basic cross-correlation method, which may be beneficial in engineering applications.