Quadratic Transformation Research Articles

Accelerating Quadratic Transform and WMMSE

Accelerating Quadratic Transform and WMMSE

Low-complexity cooperative active and passive beamforming multi-RIS-assisted communication networks

Reconfigurable intelligent surface (RIS) is a groundbreaking technology that has a significant potential for sixth generation (6G) networks. Its unique capability to control wireless environments makes it an attractive option. However, the spatial diversity increased by assisting users with all deployed RISs, this investigation has two drawbacks the high complexity design, and the received signals by the far RISs are severely attenuated. Therefore, we propose a RIS selection strategy to select the proper RISs as a pre-stage before the joint beamforming between the base station (BS) and RISs to reduce the high complexity of joint beamforming optimization. Furthermore, the joint active and passive beamforming problem based on the selection is formulated. Hence, achieving spatial diversity by examining cooperation between passive beamforming of multi-hop RIS, leads to a challenging problem. To tackle this issue, we design an algorithm for the RIS selection scheme. Also, to relax the non-convexity of the proposed problem, we decouple the problem into solvable subproblems by utilizing the fractional programming (FP) and quadratic transform (QT) optimization methods. Simulation results have demonstrated through different user locations the effectiveness of the selection strategy in performance enhancement by 30% in the sum rate, besides an obvious reduction in the complexity cost than other techniques.

A quadratic transformation for a special confluent Heun function

We report a functional identity involving a quadratic transformation of the argument for the confluent Heun function. This identity is the first of its kind to be discovered for the confluent Heun function, and it has the potential to be useful in a variety of applications.

Applications of quadratic and cubic hypergeometric transformations

The purpose of this paper is to consider five classes of quadratic and cubic hypergeometric transformations in the spirit of Bailey and Whipple. We shall successfully evaluate several hypergeometric functions, of the types 2F1(x), 3F2(x), and 4F3(x), with each function having one or more free parameters, and with the argument $x$ chosen to equal such unusual values as x=±1,−8,14,−18, (these four values having been explored initially by Gessel and Stanton). In each case, companion identities and/or inverse transformations are given, which are sometimes proved by a limiting process for a divergent hypergeometric series. Some of the proofs use the Clausen quadratic formula, Euler reflection formula, Legendre duplication, Gauss multiplication formula, Euler transformation, hypergeometric reversion formula and known hypergeometric summation formulas. The proofs in the terminating case are simpler and can lead to mixed summation formulas, which depend on values of a negative integer. Some of the formulas use the Digamma function and a dimension formula is referred to.

Proof of two conjectures of Guo and of Tang

Recently, Guo put forward two q-congruence conjectures, and Tang later proposed a generalized form for one of them. In this paper, we successfully confirm both of Guo's conjectures and Tang's conjecture. The main tools we employ include Gasper and Rahman's quadratic summation, and Rahman's quadratic transformation.

AI-driven generalized polynomial transformation models for unsupervised fundus image registration.

We introduce a novel AI-driven approach to unsupervised fundus image registration utilizing our Generalized Polynomial Transformation (GPT) model. Through the GPT, we establish a foundational model capable of simulating diverse polynomial transformations, trained on a large synthetic dataset to encompass a broad range of transformation scenarios. Additionally, our hybrid pre-processing strategy aims to streamline the learning process by offering model-focused input. We evaluated our model's effectiveness on the publicly available AREDS dataset by using standard metrics such as image-level and parameter-level analyzes. Linear regression analysis reveals an average Pearson correlation coefficient (R) of 0.9876 across all quadratic transformation parameters. Image-level evaluation, comprising qualitative and quantitative analyzes, showcases significant improvements in Structural Similarity Index (SSIM) and Normalized Cross Correlation (NCC) scores, indicating its robust performance. Notably, precise matching of the optic disc and vessel locations with minimal global distortion are observed. These findings underscore the potential of GPT-based approaches in image registration methodologies, promising advancements in diagnosis, treatment planning, and disease monitoring in ophthalmology and beyond.

Optimisation of energy efficiency of ambient backscatter communication and reconfigurable intelligent surfaces in non‐orthogonal multiple access downlink

AbstractThe authors study the energy efficiency (EE) of ambient backscatter communication (AmBC) device‐assisted and reconfigurable intelligent surfaces (RIS)‐assisted non‐orthogonal multiple access (NOMA) downlinks. The authors establish two optimisation problems based on the two collaborative devices (AmBC devices, RIS) with the objective of maximising the EE of the system, taking into account the requirements of power limitation and rate limitation, etc. and also obtain the solutions of two problems by optimising the relevant performance metrics based on the alternating optimisation algorithm. For the backscatter device (BD)‐aided downlink NOMA network, the problem is first decoupled into three subproblems, where the power allocation optimisation subproblem is solved by using the quadratic transformation method and the subgradient algorithm. The maximum EE is obtained by iterating according to the Dinkelbach's algorithm. For the RIS‐aided downlink NOMA network, the power allocation problem is solved by the same method as above and the phase optimisation problem is solved by the successive convex approximation method. Numerical results show that the proposed algorithm can achieve convergence after several iterations, and the EE of systems with BD‐assisted and RIS‐assisted have different levels of sensitivity to different influencing factors.

Proof of some conjectures of Guo and of Tang

Abstract Recently, Guo and Tang independently established some q-supercongruences from Rahman’s quadratic transformation. In this paper, by applying the method of creative microscoping devised by Guo and Zudilin together with Rahman’s quadratic transformation again, we provide proofs for eight conjectures on q-supercongruences proposed by Guo and by Tang.

Energy efficiency enhancement in millimetre-wave MIMO-NOMA using three layer user grouping and adaptive power allocation algorithm

Massive multi-input multi-output (MIMO) is realized as the principal technology in the emerging fifth generation communication network system. Hybrid structure uplink communication is considered for the MIMO Non-orthogonal multiple access (MIMO-NOMA) system’s beam forming and power efficiency improvement through the novel three-layer user grouping. In the three-layer user grouping, the K-means algorithm is adopted in the initial layer for grouping users among different clusters and rectifying clustering errors in the third layer. The second layer used the agglomerative nesting (AGNES) algorithm for merging smaller clusters based on the channel correlation and angles of arrival similarity. The beam selection is carried out to minimize the intrusion of defined beam elements and to overcome beam overlapping problems. The non-convex optimization of the power allocating problem is modified as a convex problem by introducing a Quadratic transform (QT) to minimize each user’s data rate requirement. The algorithm of coati optimization is proposed to iteratively optimize the power allocation problem. The simulation results show that our proposed methodology goes beyond the existing schemes in terms of energy efficiency beyond the maximum power and achievable sum rate can be achieved.

The Jacobi-Sobolev, Laguerre-Sobolev, and Gegenbauer-Sobolev differential equations and their interrelations

Recently, the author determined the higher-order differential operator having the Jacobi-Sobolev polynomials as its eigenfunctions for certain eigenvalues. These polynomials form an orthogonal system with respect to an inner product equipped with the Jacobi measure on the interval [ − 1 , 1 ] with parameters α ∈ N 0 , β > − 1 and two point masses N, S>0 at the right end point of the interval involving functions and their first derivatives. The first purpose of the present paper is to reveal how the Jacobi-Sobolev equation reduces to the differential equation satisfied by the Laguerre-Sobolev polynomials on the positive half line via a confluent limiting process as β → ∞ . Secondly, we explicitly establish the differential equation for the symmetric Gegenbauer-Sobolev polynomials by employing a quadratic transformation of the argument. Each of the three differential operators involved is of order 4 α + 10 and symmetric with respect to the corresponding Sobolev inner product.

Connecting exceptional orthogonal polynomials of different kind

The known asymptotic relations interconnecting Jacobi, Laguerre, and Hermite classical orthogonal polynomials are generalized to the corresponding exceptional orthogonal polynomials of codimension m. It is proved that Xm-Laguerre exceptional orthogonal polynomials of type I, II, or III can be obtained as limits of Xm-Jacobi exceptional orthogonal polynomials of the same type. Similarly, Xm-Hermite exceptional orthogonal polynomials of type III can be derived from Xm-Jacobi or Xm-Laguerre ones. The quadratic transformations expressing Hermite classical orthogonal polynomials in terms of Laguerre ones is also extended to even X2m-Hermite exceptional orthogonal polynomials.

Physical‐layer security enhancement in wireless sensor networks through artificial noise optimization

AbstractEnsuring network security is of utmost importance, especially in wireless sensor networks (WSNs), where the confidentiality of data is at risk due to eavesdropping. This paper introduces an artificial‐noise‐aided secure communication scheme for WSNs. This scheme comprises numerous sensor nodes, the sink, a friendly jammer, and an eavesdropper. The primary aim is to enhance the secrecy rate of the network by optimizing the jamming power. The friendly jammer plays a crucial role in degrading the wiretap channel between the sensor nodes and the eavesdropper. The optimization process revolves around maximizing the secrecy rate, while carefully managing the jamming power to prevent any negative impact on the main transmission link between the sink node and the sensor nodes. Due to the nonconvexity of the secrecy rate problem, the convex approximation‐based algorithm is recommened to get a safe convex approximation of the original solution. To address this optimization problem, an iterative algorithm based on quadratic transformation is proposed. Afterwards, the optimum jamming power to achieve high security is obtained. The simulation outcomes underscore the dual impact of the jammer in WSNs. The jammer diminishes the eavesdropper channel. Unfortunately, it introduces interference that consequently affects the security enhancement. In response, we concentrate on the optimization of jamming power. The proposed scheme consistently elevates secrecy rates across a spectrum of eavesdropper positions, adeptly addressing worst‐case scenarios. Notably, the algorithm showcases resilience across diverse node distributions.

General q-series transformations based on Abel's lemma on summation by parts and their applications

In this paper, we establish three new and general transformations with sixteen parameters and bases via Abel's lemma on summation by parts. As applications, we set up a lot of new transformations of basic hypergeometric series. Among include some new quadratic, cubic, and quartic transformations. Furthermore, we put forward the so-called ( R , S ) -type transformation with arbitrary degree to unify such multibasic transformations. A special ( 3 , 3 ) -type transformation and its example are presented.

Yabu's formulae for hypergeometric $ _3F_2 $-series through Whipple's quadratic transformations

<p>By means of Whipple's quadratic transformations, two classes of hypergeometric $ _3F_2 $-series are expressed in terms of the Lerch transcendent function. Several difficult series with a free variable are explicitly evaluated in closed form, including Yabu's three remarkable identities.</p>

Precoding-Based Downlink OAM-MIMO Communications With Rate Splitting

Orbital angular momentum (OAM) and rate splitting (RS) are the potential key techniques for the future wireless communications. As a new orthogonal resource, OAM can achieve the multifold increase of spectrum efficiency to relieve the scarcity of the spectrum resource, but how to enhance the privacy performance imposes crucial challenge for OAM communications. RS technique divides the information into private and common parts, which can guarantee the privacies for all users. In this paper, we integrate the RS technique into downlink OAM-MIMO communications, and study the precoding optimization to maximize the sum capacity. First, the concentric uniform circular arrays (UCAs) are utilized to construct the downlink transmission framework of OAM-MIMO communications with RS. Particularly, users in the same user pair utilize RS technique to obtain the information and different user pairs use different OAM modes. Then, we derive the OAM-MIMO channel model, and formulate the sum capacity maximization problem. Finally, based on the fractional programming, the optimal precoding matrix is obtained to maximize the sum capacity by using quadratic transformation. Extensive simulation results show that by using the proposed precoding optimization algorithm, OAM-MIMO communications with RS can achieve higher sum capacity than the traditional communication schemes.

Resformer: Combine quadratic linear transformation with efficient sparse Transformer for long-term series forecasting

With the continuous development of deep learning, long sequence time-series forecasting (LSTF) has attracted more and more attention in power consumption prediction, traffic prediction and stock prediction. In recent studies, various improved models of Transformer are favored. While these models have made breakthroughs in reducing the time and space complexity of Transformer, there are still some problems, such as the predictive power of the improved model being slightly lower than that of Transformer. And these models ignore the importance of special values in the time series. To solve these problems, we designed a more concise network named Resformer, which has four significant characteristics: (1) The fully sparse self-attention mechanism achieves O⁢(𝐿𝑙𝑜𝑔𝐿) time complexity. (2) The AMS module is used to process the special values of time series and has comparable performance on sequences dependency alignment. (3) Using quadratic linear transformation, a simple LT module is designed to replace the self-attention mechanism. It effectively reduces redundant information. (4) The DistPooling method based on data distribution is proposed to suppress redundant information and noise. A large number of experiments on real data sets show that the Resformer method is superior to the existing improved model and standard Transformer method.

A reflection formula for the Gaussian hypergeometric function of matrix argument

We obtain a reflection formula for the Gaussian hypergeometric function of real symmetric matrix argument. We also show that this result extends to the Gaussian hypergeometric function defined over the symmetric cones, and even to generalizations of the Gaussian hypergeometric function defined in terms of series of Jack polynomials. Finally, we obtain a quadratic transformation formula for the Gaussian hypergeometric function of Hermitian 2×2 matrix argument.

Ramanujan-type series for , revisited

AbstractIn this note, we revisit Ramanujan-type series for $\frac {1}{\pi }$ and show how they arise from genus zero subgroups of $\mathrm {SL}_{2}(\mathbb {R})$ that are commensurable with $\mathrm {SL}_{2}(\mathbb {Z})$ . As illustrations, we reproduce a striking formula of Ramanujan for $\frac {1}{\pi }$ and a recent result of Cooper et al., as well as derive a new rational Ramanujan-type series for $\frac {1}{\pi }$ . As a byproduct, we obtain a Clausen-type formula in some general sense and reproduce a Clausen-type quadratic transformation formula closely related to the aforementioned formula of Ramanujan.

Wireless Information and Power Transfer Enabled Massive MIMO Multi-Way Relaying

This work considers a multi-way massive multi-input multi-output (mMIMO) relaying system wherein several wireless information and power transfer capable users, first harvest energy, and then exchange information via a relay. The mMIMO relaying system is practically modelled by considering spatially-correlated relay-user channels, and by using a non-linear energy harvesting (EH) model. A closed-form spectral efficiency expression is derived for this system, which is then used to jointly allocate the relay and users transmit powers, and the charging time to optimize the weighted sum energy efficiency (WSEE) metric. The spatially-correlated channel and the non-linear EH model not only couple the optimization variables, but also introduce nested scalar fractional optimization functions. This makes the WSEE metric non-convex. The proposed solution first uses the existing quadratic transformation (QT) to decouple the scalar fractional forms. It then applies novel results, which are developed by extending the QT framework, to decouple the optimization variables. The proposed WSEE optimization algorithm is shown to provide close-to-optimal WSEE. It is also shown that channel spatial correlation increases the amount of energy harvested by the users, but reduces their WSEE.

Effective and efficient crowd spectrum detection with active reconfigurable intelligent surface

The reconfigurable intelligent surface (RIS) assisted crowd spectrum detection (CSD) has emerged as a promising approach for achieving higher enhancements in both spectrum efficiency (SE) and energy efficiency (EE). Nevertheless, due to the double fading effect in reflecting links, the detection performance gains obtained through current passive RIS are negligible, especially in typical communication scenarios. As a result, the performance enhancement in passive RIS-assisted CSD is ineffective. To address the issue, this paper incorporates the advanced active RIS into CSD to assist the spectrum requestor (RU) in achieving notable and effective improvement in detection performance gains. Through amplifying the amplitude of reflected signals, active RIS can further enhance the received signal power at the RU while consuming more energy than the passive RIS. Therefore, for enabling both effective and efficient CSD with the assistance of active RIS, this paper investigates the detection efficiency (DE) maximization problem by jointly optimizing the RU’s sampling number and reward budget that is paid to compensate active RIS controllers for amplifying reflected signals. In a word, the key point of this paper is to evaluate whether the active RIS-assisted CSD can outperform the passive RIS-assisted CSD in both effective and efficient detection performance gains by joint parameter optimization. Quadratic transform (QT) and particle swarm optimization (PSO) methods are leveraged to solve the DE maximization problem. Finally, extensive simulation results demonstrate that, when taking the no-RIS mechanism as the benchmark, the proposed active RIS-assisted CSD can outperform the conventional passive-RIS mechanism in terms of significantly improved detection performance gains and a higher DE in most simulation scenarios.