Array Theory Research Articles

Theory and generation of heterogeneous 2D arrays of optical vortices by using 2D fork-shaped gratings: topological charge and power sharing management.

In this work, by providing comprehensive theoretical foundations, we revisit and improve a simple and efficient method that has been used for generation of 2D orthogonal arrays of optical vortices with components having different topological charges (TCs). This method has been implemented by the diffraction of a plane wave from 2D gratings where the gratings' profiles are determined by iterative computational process. Here, based on the theoretical predictions, specifications of the diffraction gratings can be easily adjusted in a way to generate experimentally a heterogeneous vortex array with the desired power shares among different elements of the array. We use the diffraction of a Gaussian beam from a class of pure phase 2D orthogonal periodic structures having sinusoidal or binary profiles possessing a phase singularity, calling pure phase 2D fork-shaped gratings (FSGs). The transmittance of each of the introduced gratings is obtained by multiplying the transmittance of two pure phase 1D FSGs along x and y directions, having topological defect numbers lx and ly and phase variation amplitudes γx and γy, respectively. By solving the Fresnel integral, we show that the diffraction of a Gaussian beam from a pure phase 2D FSG leads to generation of a 2D array of vortex beams having different TCs and power shares. The power distribution among the generated optical vortices over the different diffraction orders can be adjusted by γx and γy, and it strongly depends on the profile of the grating. Meanwhile the TCs of the generated vortices depend on lx and ly and the corresponding diffraction orders, namely lm,n = -(mlx + nly) presents the TC of (m, n)th diffraction order. We recorded the intensity patterns of the experimentally generated vortex arrays which are fully consistent with the theoretically predicted results. Furthermore, the TCs of the experimentally generated vortices are measured individually by the diffraction of each of them through a pure amplitude quadratic curved-line (parabolic-line) grating. The absolute values and signs of the measured TCs are consistent with the theoretical prediction. The generated configuration of vortices with adjustable TC and power sharing features might find many applications such as non-homogeneous mixing of a solution consisting trapped particles.

Beampattern Synthesis Based on Novel Receive Delay Array for Mainlobe Interference Mitigation

This paper investigates beampattern synthesis methods based on a novel Receive Delay Array (RDA) to mitigate interferences located in the mainlobe of the beampattern. In our system design, the RDA is developed by transmitting a stepped frequency linear frequency modulation waveform and delaying the received echo with a small time offset between adjacent receive antenna elements. In doing so, a range-angle-dependent beampattern is obtained in the joint transmit-receive spatial domain. A two-stage beamforming scheme, including the equivalent data-independent transmit beamforming and dimension-reduced data-dependent receive beamforming, is then proposed to mitigate mainlobe interferences. Furthermore, according to the adaptive array theory, by designing a transmit weight vector dependent of the interference steering vector, two beampattern synthesis algorithms are developed respectively based on the maximum gain and minimum deviation to adjust multiple responses of the transmit beampattern precisely. The mainlobe of the receive beampattern is also broadened with the use of the derivative constraint. By combining the devised transmit and receive weight vectors, a two-dimensional joint transmit-receive beampattern with a flat-top mainlobe and broadened nulls can be formed, where the interferences are mitigated against the direction-of-arrival (DOA) mismatch. Theoretical and practical analysis as well as parametric studies are provided to demonstrate the effectiveness of the proposed beampattern synthesis methods in mitigating mainlobe interferences.

Optimized reduced order ARX modelling and robust control for multivariable uncertain systems

This paper deals with modelling and robust control of linear multivariable uncertain systems based on the new linear multivariable ARX (Auto-Regressive with eXogenous input)-Laguerre model and the loop-shaping design procedure with the relative gain array theory. In fact, to guarantee significant parameter number reduction of the multivariable ARX-Laguerre model compared with the classic multivariable ARX model, we propose the optimization of the Laguerre poles of the model using the genetic algorithm where the Fourier coefficients are identified using the recursive least square method. Later, the optimized multivariable ARX-Laguerre model is exploited to propose a robust control algorithm for linear multivariable uncertain systems. Indeed, we propose to combine the loop-shaping design procedure approach and the relative gain array theory to develop a simplified final robust controller guaranteeing reference tracking and robust stability against parametric uncertainties. This proposition aims for the simplification of the control’s scheme by adjusting the weighting matrix and the final robust controller calculated from the loop-shaping design procedure for a simpler and more experimentally applicable control scheme. The identification of Fourier coefficients and Laguerre poles as well as the robust multivariable control algorithm are experimentally validated on a laboratory coupled two-tank system showing good results in terms of modelling and attending desired control performances.

Interpolation Results for Arrays with Length and MaxDiff

In this paper, we enrich McCarthy’s theory of extensional arrays with a length and a maxdiff operation. As is well-known, some diff operation (i.e., some kind of difference function showing where two unequal array differ) is needed to keep interpolants quantifier-free in array theories; our maxdiff operation returns the max index where two arrays differ and so it has a univocally determined semantics. The length function is a natural complement of such a maxdiff operation and is needed to handle real arrays. Obtaining interpolation results for such a rich theory is a surprisingly hard task. We get such results via a thorough semantic analysis of the models of the theory and of their amalgamation and strong amalgamation properties. The results are modular with respect to the index theory and we show how to convert them into concrete interpolation algorithms via a hierarchical approach realizing a polynomial reduction to interpolation in linear arithmetics endowed with free function symbols. In this paper, we enrich McCarthy’s theory of extensional arrays with a length and a maxdiff operation. It is known from the literature that a diff operation is required in order for the theory of arrays to enjoy the Craig interpolation property at the quantifier-free level. However, the diff operation introduced in the literature is merely instrumental to this purpose and has only a purely formal meaning (it is obtained from the Skolemization of the extensionality axiom): instead, our maxdiff operation returns the max index where two arrays differ and so it is univocally determined at the semantic level. The length function is a natural complement of such a maxdiff operation and is needed to handle real arrays (which are defined in their specified allocation memory). Obtaining interpolation results for such a rich theory is a surprisingly hard task. We get such results via a thorough semantic analysis of the models of the theory and of their amalgamation and strong amalgamation properties. The results are modular with respect to the index theory and we show how to convert them into concrete interpolation algorithms via a hierarchical approach realizing a polynomial reduction to interpolation in linear arithmetics endowed with free function symbols. The array theory in the paper has been modified so as to model real arrays used in common programming languages (we now require them to be ‘contiguous’, i.e., not undefined in any of their allocation entries); moreover strong amalgamation and interpolation with free function symbols are proved. The interpolation algorithm avoids full instantiation routines and unbounded loops, thus achieving the above mentioned polynomial reduction complexity. The present paper is a substantially revised version of a previous conference paper presented at FoSSaCS 2021.

Linear Trees, Lattice Walks, and RNA Arrays

The leftmost column entries of RNA arrays I and II count the RNA numbers that are related to RNA secondary structures from molecular biology. RNA secondary structures sometimes have mutations and wobble pairs. Mutations are random changes that occur in a structure, and wobble pairs are known as non-Watson–Crick base pairs. We used topics from RNA combinatorics and Riordan array theory to establish connections among combinatorial objects related to linear trees, lattice walks, and RNA arrays. In this paper, we establish interesting new explicit bijections (one-to-one correspondences) involving certain subclasses of linear trees, lattice walks, and RNA secondary structures. We provide an interesting generalized lattice walk interpretation of RNA array I. In addition, we provide a combinatorial interpretation of RNA array II as RNA secondary structures with n bases and k base-point mutations where ω of the structures contain wobble base pairs. We also establish an explicit bijection between RNA structures with mutations and wobble bases and a certain subclass of lattice walks.

RCS reduction using grounded multi-height multi-dielectrics metasurfaces

In this paper, the PO method with the array theory are used to formulate the RCS of a grounded multi-height dielectric surface which can be employed in the design and optimization of a metasurface consisting of dielectric tiles with different heights and permittivities. The proposed closed form relations can be used instead of full wave simulation to design an optimized dielectric grounded metasurface, properly. Finally, three different RCS reducer metasurfaces are designed and optimized with three different dielectric tiles by using the proposed analytical relations. The results prove that the proposed ground dielectric metasurface achieve more than 10 dB RCS reduction at 4.4–16.3 GHz (114.9%) frequency range. This result proves the accuracy and effectiveness of the proposed analytical method which can be used in the RCS reducer metasurfaces design.

Transfer-Learning-Assisted Multielement Calibration for Active Phased Antenna Arrays

A transfer-learning-based method for accelerating power-only calibration of phased array antennas by combining the conventional array theory with deep learning is presented in this communication. The existing power-only calibration methods either require a significant number of measurement cycles or have restrictive phase shifter resolution requirements. The proposed array calibration method uses a surrogate model to calibrate all the array elements in one pass without restricting phase resolution requirements. We developed a novel feature extraction scheme (FES) that picks out the most important power features resulting in reduced measurement cycles. The burden of data acquisition for model training is further reduced by relational knowledge transfer learning. The surrogate model acquires its general calibration capability from massive theoretical data, which is easily collected by the radiation multiplication theorem, and captures the detailed nonideal response from a small number of simulations. The proposed methodology has been demonstrated and tested on several arrays for validation. The effectiveness and performance of the method have been verified, and hence, it can serve as a complementary tool to accelerate the calibration process of phased antenna arrays.

Analysis of Exponential Frequency-Diverse Array for Short-Range Beam-Focusing Technology

The frequency-diverse array (FDA) is a promising technology because of its angle- and range-dependent beam pattern. The FDA has unique properties, such as beam scanning and focusing, which are difficult to achieve using the conventional phased antenna array theory. In this study, the relationship between range and frequency increment is analyzed, and the fundamental operation principle of FDA is thoroughly investigated. The effect of coupling among antenna elements is also discussed to analyze the FDA theory. Furthermore, a novel exponential FDA method for short-range beam focusing is presented based on the proposed FDA analysis method. The proposed method has been investigated through various case studies and applied to research efforts on FDA methods. The FDA design procedures and analysis methods are also presented in this article.

The Density of Costas Arrays Decays Exponentially

Costas arrays are useful in radar and sonar engineering, and many other settings in which optimal 2-D autocorrelation is needed: they are permutation matrices in which the vectors joining different pairs of ones are all distinct. We prove that the density of Costas arrays among permutation matrices decays exponentially, solving a core problem in the theory of Costas arrays. The proof combines ideas from random graph theory with tools from probabilistic combinatorics.

Увеличение эффективности обнаружения воздушных судов на догонных курсах в импульсно-доплеровских бортовых РЛС с малой высотой полета носителя

Problem statement. The effectiveness of solving the problem of warning about possible collisions of small aircraft or unmanned aerial vehicles intended for the development of hard-to-reach territories depends on the time of early detection of another aircraft on intersecting trajectories. As a result of the comparative analysis of radar detection methods carried out in the article, taking into account the differences in the signal-interference situation characteristic of on-board pulse−Doppler radars for targets on catch-up courses in versions "multiple input - multiple output", "with joint a posteriori processing of results" and in traditional radar with active or passive phased antenna arrays have shown that the latter have energy advantages. Therefore, for this category of pulse-Doppler radars, a method for estimating the detection range on several frames is presented, which allows you to choose the logic of operation with an increase in the detection range. Research methods. The theory of antenna arrays and the theory of radar detection for pulse-Doppler radars were actively used in the work with the specified requirements for the probabilities of correct detection, false alarm and the selected model of fluctuations of the reflected useful signal. Purpose. To substantiate the advantages of technical solutions based on a traditional technical solution with phased antenna arrays for detection on catch-up courses and to develop recommendations for finding conditions that allow to increase the detection range. Results. The simulation results are presented, showing an improvement in the conditions for radar detection in onboard pulse-Doppler radars when finding the spectrum of the reflected radar signal in the Doppler frequency range, on which the spectrum of the reflected signal from the Earth's surface is superimposed when using radar with phased antenna arrays compared with spatial processing methods in MIMO and radars with joint a posteriori processing of results. A method for calculating the detection range on several adjacent frames has been developed and it is shown that in this case a longer range is achieved compared to the case of detection on a single frame. Practical significance. The results of the work can be used in small-sized onboard pulse radars of low-altitude carriers in order to increase the time of early warning of the presence of other aircraft on the flight path. This task seems to be especially relevant when flying radar carriers in hard-to-reach, developed territories.

РАЗРАБОТКА МИКРОПОЛОСКОВОЙ РЕШЕТКИ С НАКЛОННЫМ РАСКРЫВОМ И СНИЖЕННЫМ ЧИСЛОМ ЭЛЕМЕНТОВ РЕГУЛИРОВКИ АМПЛИТУДНО-ФАЗОВОГО РАСПРЕДЕЛЕНИЯ ДЛЯ МАЛОВЫСОТНОЙ МЕТЕОНАВИГАЦИИ БПЛА В АРКТИКЕ

Problem statement. The development of hard-to-reach regions in the absence of ground-based navigation systems and data sources on the meteorological situation can cause an increase in the number of aviation accidents. For small Arctic aviation, there is a need to introduce onboard radar stations that solve various tasks: from ensuring flight safety to analyzing the situation at sea in the area of the escorted vessel. In low-altitude flight conditions, re-reflections from the underlying surface coming along the side lobes of the antenna pattern lead to the identification of false areas dangerous for flight. The paper searches for the requirements for a directional pattern focused on solving the problem of meteorological navigation in polar latitudes. The stages of the development of an antenna with an inclined opening and a deflected main beam of the radiation pattern, which guarantees a very low level of lateral radiation in the lower hemisphere, are presented. Research methods. The mathematical apparatus of the theory of antenna arrays and detection of surface and volume extended radar targets was actively used in the work. Purpose. To substantiate the prospects of using antenna arrays with an inclined opening and a beam to solve meteorological navigation problems. To outline the main stages of the synthesis of a patch antenna, which is characterized by a low level of lateral radiation in the lower hemisphere. Results. The results of modeling are presented, which allow us to obtain requirements for the parameters of antenna systems for meteorological navigation onboard radars operating in polar latitudes. The results of a comparative analysis of an inclined patch antenna array and an antenna array with an opening slope and an inclination of the beam of the opposite sign to reduce the level of lateral radiation are presented. Practical significance. The results of the work can be used in determining the requirements for radiation characteristics, as well as in the development of original designs of patch antenna arrays of reduced cost for small-sized airborne radars of low-altitude carriers in order to increase the probability of correct detection of dangerous weather formations for flight.

Simulation and Experimental Study of a Linear Bifocal Antenna Array

A leaky-wave bifocal linear array focused in the Fresnel zone is considered. The structural parameters provide perfect focusing of the field at two points corresponding to two different frequencies. An approximate theory of antenna arrays is used to analyze the bifocal array, to study the shape of the frequency scanning arc, and to determine the field distribution in the focal spot. An electrodynamic simulation is performed for a strip array with slotted radiators. The scattering parameters of the array and its radiation characteristicsare presented. The design features of a strip antenna that suppress the effect of normal are discussed. The results of an experimental study of a strip bifocal array are presented. The possibility of implementation of a symmetrical scanning arc and a relatively low level of optical aberrations is shown.

Anomalous Reflection From a Phase Gradient Metasurface With Arbitrary Incident Angle

A planar phase gradient metasurface (PGM) with phase gradients in two in-plane directions will introduce two additional wave vectors to the reflected wave vector. For small incident angles, close to the normal vector, the magnitude of the reflected wave vector is smaller than that of the incident wave vector and the direction of the anomalous reflected wave can be determined from the incident wave vector components and additional phase gradient components. The expanded generalized Snell’s law, which includes diffraction order modes, is combined with array theory to accurately predict the directions of the reflected waves from a planar PGM for incident angles where the magnitude of the reflected wave vector is larger than that of the incident wave vector. The predicted directions of the reflected waves from a planar PGM are compared with simulated RCS results obtained with CST Studio Suite and measured results obtained in a compact range.

Riordan arrays and r-Stirling number identities

By using the theory of Riordan arrays, we establish four pairs of general r-Stirling number identities, which reduce to various identities on harmonic numbers, hyperharmonic numbers, the Stirling numbers of the first and second kind, the r-Stirling numbers of the first and second kind, and the r-Lah numbers. We further discuss briefly the connections between the r-Stirling numbers and the Cauchy numbers, the generalized hyperharmonic numbers, and the poly-Bernoulli polynomials. Many known identities are shown to be special cases of our results, and the combinatorial interpretations of several particular identities are also presented as supplements.

A low-cost digital coding metasurface applying modified ‘crusades-like’ cell topologies for broadband RCS reduction

The popularity of metasurfaces (MSs) has been continuously grown due to their powerful ability to manipulate electromagnetic waves. One of their important application areas is radar stealth technology, in particular, the realization of radar cross section (RCS) reduction. However, the high costs of substrate material limit its large-scale applications. In this paper, a binary digital coding metasurface (DCM) with novel modified ‘crusades-like’ cell topologies is proposed and implemented using a low-cost FR4 substrate to achieve broadband RCS reduction (RCSR). To realize the 1-bit DCM, initially, two elements with rotational symmetry are chosen for polarization insensitive properties while considering an unconventional phase deviation criteria. Next, the optimal hybrid coding layout is given out by using a genetic algorithm and antenna array theory. Last, the proposed novel MS prototype composed of 40 × 40 unit cells is fabricated and measured to validate the RCSR behaviour predicted by full-wave simulation. The results show a good consistency between the theoretical simulation and experimental measurement from 7.9 to 15.8 GHz. In addition, the simulation results indicate that the designed MS features high angular stability. Our work may provide a promising approach and good reference for the low-cost MS design in radar stealth applications.

A Comparison of MB-CBFM and Modified ASED Basis Function Method to Analyze Interconnected Finite Periodic Arrays

This letter compares the macro block-characteristic basis function method (MB-CBFM) with the modified accurate subentire-domain (ASED) basis function method to solve the interconnected finite periodic arrays. In the MB-CBFM, a compressed matrix equation whose size is less than that of the conventional CBFM can be obtained using the idea of macro blocks and the array theory. In the modified ASED method, a few sets of ASED bases on the cells are constructed by solving the subarray problem excited by the several plane waves. Afterwards, a reduced matrix equation will be obtained. The size of the matrix is greater than that in the MB-CBFM for the large-scale arrays. Numerical results obtained from the modified ASED basis function method are more accurate than those of the MB-CBFM.

Counterexample-Guided Prophecy for Model Checking Modulo the Theory of Arrays

We develop a framework for model checking infinite-state systems by automatically augmenting them with auxiliary variables, enabling quantifier-free induction proofs for systems that would otherwise require quantified invariants. We combine this mechanism with a counterexample-guided abstraction refinement scheme for the theory of arrays. Our framework can thus, in many cases, reduce inductive reasoning with quantifiers and arrays to quantifier-free and array-free reasoning. We evaluate the approach on a wide set of benchmarks from the literature. The results show that our implementation often outperforms state-of-the-art tools, demonstrating its practical potential.

A wireless signal-based sensing framework for robotics

In this paper, we develop the analytical framework for a novel Wireless signal-based Sensing capability for Robotics (WSR) by leveraging a robots’ mobility in 3D space. It allows robots to primarily measure relative direction, or Angle-of-Arrival (AOA), to other robots, while operating in non-line-of-sight unmapped environments and without requiring external infrastructure. We do so by capturing all of the paths that a wireless signal traverses as it travels from a transmitting to a receiving robot in the team, which we term as an AOA profile. The key intuition behind our approach is to enable a robot to emulate antenna arrays as it moves freely in 2D and 3D space. The small differences in the phase of the wireless signals are thus processed with knowledge of robots’ local displacement to obtain the profile, via a method akin to Synthetic Aperture Radar (SAR). The main contribution of this work is the development of (i) a framework to accommodate arbitrary 2D and 3D motion, as well as continuous mobility of both signal transmitting and receiving robots, while computing AOA profiles between them and (ii) a Cramer–Rao Bound analysis, based on antenna array theory, that provides a lower bound on the variance in AOA estimation as a function of the geometry of robot motion. This is a critical distinction with previous work on SAR-based methods that restrict robot mobility to prescribed motion patterns, do not generalize to the full 3D space, and require transmitting robots to be stationary during data acquisition periods. We show that allowing robots to use their full mobility in 3D space while performing SAR results in more accurate AOA profiles and thus better AOA estimation. We formally characterize this observation as the informativeness of the robots’ motion, a computable quantity for which we derive a closed form. All analytical developments are substantiated by extensive simulation and hardware experiments on air/ground robot platforms using 5 GHz WiFi. Our experimental results bolster our analytical findings, demonstrating that 3D motion provides enhanced and consistent accuracy, with a total AOA error of less than 10◦ for 95% of trials. We also analytically characterize the impact of displacement estimation errors on the measured AOA and validate this theory empirically using robot displacements obtained using an off-the-shelf Intel Tracking Camera T265. Finally, we demonstrate the performance of our system on a multi-robot task where a heterogeneous air/ground pair of robots continuously measure AOA profiles over a WiFi link to achieve dynamic rendezvous in an unmapped, 300 m2 environment with occlusions.

Learning populations with hubs govern the initiation and propagation of spontaneous bursts in neuronal networks after learning.

Spontaneous bursts in neuronal networks with propagation involving a large number of synchronously firing neurons are considered to be a crucial feature of these networks both in vivo and in vitro. Recently, learning has been shown to improve the association and synchronization of spontaneous events in neuronal networks by promoting the firing of spontaneous bursts. However, little is known about the relationship between the learning phase and spontaneous bursts. By combining high-resolution measurement with a 4,096-channel complementary metal-oxide-semiconductor (CMOS) microelectrode array (MEA) and graph theory, we studied how the learning phase influenced the initiation of spontaneous bursts in cultured networks of rat cortical neurons in vitro. We found that a small number of selected populations carried most of the stimulus information and contributed to learning. Moreover, several new burst propagation patterns appeared in spontaneous firing after learning. Importantly, these “learning populations” had more hubs in the functional network that governed the initiation of spontaneous burst activity. These results suggest that changes in the functional structure of learning populations may be the key mechanism underlying increased bursts after learning. Our findings could increase understanding of the important role that synaptic plasticity plays in the regulation of spontaneous activity.

Sound Field Modeling Method and Key Imaging Technology of an Ultrasonic Phased Array: A Review

An ultrasonic phased array consists of multiple ultrasonic probes arranged in a certain regular order, and the delay time of the excitation signal sent to each array element is controlled electronically. The testing system model based on ultrasonic propagation theory is established to obtain a controllable and focused sound field, which has theoretical and engineering guiding significance for the calculation and analysis of ultrasonic array sound fields. Perfecting array theory and exploring array imaging methods can obtain rich acoustic information, provide more intuitive and reliable research results, and further the development of ultrasonic phased-array systems. This paper reviews the progress of research on the application of ultrasound arrays for non-destructive testing (NDT) and brings together the most relevant published work on the application of simulation methods and popular imaging techniques for ultrasonic arrays. It mainly reviews the modeling approaches, including the angular spectrum method (ASM), multi-Gaussian beam method (MGB), ray tracing method, finite element method (FEM), finite difference method (FDM), and distributed point source method (DPSM), which have been used to assess the performance and inspection modality of a given array. In addition, the array of imaging approaches, including the total focusing method (TFM), compression sensing imaging (CSI), and acoustic nonlinearity imaging (ANI), are discussed. This paper is expected to provide strong technical support in related areas such as ultrasonic array testing theory and imaging methods.