Quadratic Transformation Research Articles

Three families of q-supercongruences from a quadratic transformation of Rahman

We present three families of q-supercongruences modulo the square and cube of a cyclotomic polynomial from a quadratic transformation by Rahman. In particular, as a limiting case we obtain the following supercongruence: for 0<r<d, gcd⁡(d,r)=1, d+r odd, and any prime p≡d+r(mod2d),∑k=0(dp+p−r)/d(3dk+r)(r2d)k(rd)k2(d−rd)kk!3(d+2r2d)k4k≡0(modp3), where (x)n=x(x+1)⋯(x+n−1) is the rising factorial. We also put forward four related conjectures on q-supercongruences.

On the eighth power mean of the two-term exponential sums

The main purpose of this article is using the elementary methods and the quadratic residual transformation technique, and the properties of the solutions of the congruence equations modulo an odd prime p to study the calculating problem of the eighth power mean of the two-term exponential sums, and give an interesting calculating formula for it. At the same time, the paper also provides a new and effective method for the study of the high order power mean of the two-term exponential sums.

Active RIS Assisted Spectrum Sharing: Able to Achieve Energy-Efficient Notable Detection Performance Gains

Efficient radio resource utilization is crucially important for future wireless communication systems. The advanced reconfigurable intelligent surface (RIS) has recently been incorporated into the spectrum sharing mechanism, allowing for both spectrum efficiency (SE) and energy efficiency (EE). Spectrum sensing is the cornerstone of spectrum sharing. Nevertheless, due to the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">doubling fading</i> effect, the path loss experienced from the reflection link is thousands of times that from the direct link. Consequently, the detection performance gains obtained through passive RIS are negligible, especially in typical communication scenarios where the direct link is not weak. To address this issue, this paper proposes an active RIS assisted spectrum sensing mechanism that can notably improve the detection performance gains. Multiple active RISs are introduced and the closed-form expressions of the detection performance metrics are derived in this paper. More importantly, due to the extra energy consumption in amplifying reflected signals, it's necessary to investigate the EE maximization problem for the active RIS assisted mechanism by jointly optimizing the detection parameter and the reflecting precoding. Alternating optimization (AO) algorithm and quadratic transform (QT) method are utilized to decouple the multidimensional <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fractional programming</i> (FP) problem. Then, the optimal detection parameter is obtained with a closed-form solution based on the convex optimization theory. The phase shift and amplification coefficient of each active element are optimized through the sequential rotation (SR) method and an improved differential evolutionary (DE) algorithm, respectively. Simulation results show that the proposed active-RIS mechanism could outperform the conventional passive-RIS mechanism in achieving notably enhanced detection performance gains and higher EE by optimizing the reflecting precoding.

Joint MU-MIMO Precoding and Computation Optimization for Energy Efficient Industrial IoT With Mobile Edge Computing

In the fourth industrial revolution, the industrial Internet of things (IIoT) will bring fundamental changes to human communities. This paper proposes to make full use of the under-utilized computing resources of wired edge devices to alleviate the computing burden of the processing center in delay-constrained multi-user networks. Our goal is to minimize the weighted energy consumption while ensuring the required latency. The formulated problems are NP-hard involving joint optimization of the computation task assignment, transmit association design, multiple-input multiple-output (MIMO) precoding, and computing resource allocation with binary and partial offloading protocols. By utilizing the weighted minimum mean-squared-error method, quadratic transformation, and difference of convex functions algorithm, we propose two joint computation offloading and resource allocation algorithms for binary and partial offloading protocols, respectively. Simulation results confirm the efficiency of the proposed algorithms and demonstrate that the proposed algorithms achieve a significant computation performance enhancement.

On a Generalization of the Kummer’s Quadratic Transformation and a Resolution of an Isolated Case

The resolution of isolated cases of the well-known quadratic transformation due to Kummer was thoroughly examined in two recent publications by Atia as well as Atia and Al-Mohaimeed. The objective of this paper is twofold. We establish generalizations of the quadratic transformation due to Kummer in the most general case in the first section, and in the second section, an effort is made to discuss the resolution of an isolated case of a quadratic transformation contiguous to that of Kummer established by Choi and Rathie.

Cloaking Rayleigh waves via symmetrized elastic tensors

In this work we propose a strategy based on coordinate transformation to cloak Rayleigh waves. Rayleigh waves are in-plane elastic waves which propagate along the free surface of semi-infinite media. They are governed by Navier equations that retain their form for an in-plane arbitrary coordinate transformation x=Ξ(X), upon choosing the specific kinematic relation U(Ξ(X))=u(x) between displacement fields in virtual, i.e. reference, (U) and transformed, i.e. cloaked, (u) domains. However, the elasticity tensor of the transformed domain is no longer fully symmetric, and thus, it is difficult to design with common materials. Motivated by this issue, we propose a symmetrization technique, based on the arithmetic mean, to obtain anisotropic, yet symmetric, elastic tensors for Rayleigh wave near-cloaking. In particular, by means of time-harmonic numerical simulations and dispersion analyses, we compare the efficiency of triangular and semi-circular cloaks designed with the original non-symmetric tensors and the related symmetrized versions. In addition, different coordinate transformations, e.g. linear, quadratic and cubic, are adopted for the semi-circular cloaks. Through the analyses, we show that a symmetrized semi-circular cloak, obtained upon the use of a quadratic transformation, performs better than the other investigated designs. Our study provides a step towards the design of feasible and efficient broadband elastic metamaterial cloaks for surface waves.

Active RIS Assisted Rate-Splitting Multiple Access Network: Spectral and Energy Efficiency Tradeoff

With the increasing demand of high data rate and massive access in both ultra-dense and industrial Internet-of-things networks, spectral efficiency (SE) and energy efficiency (EE) are regarded as two important and inter-related performance metrics for future networks. In this paper, we investigate a novel integration of rate-splitting multiple access (RSMA) and reconfigurable intelligent surface (RIS) into cellular systems to achieve a desirable tradeoff between SE and EE. Different from the commonly used passive RIS, we adopt reflection elements with active load to improve a newly defined metric, called resource efficiency (RE), which is capable of striking a balance between SE and EE. This paper focuses on the RE optimization by jointly designing the base station (BS) transmit precoding and RIS beamforming (BF) while guaranteeing the transmit and forward power budgets of the BS and RIS, respectively. To efficiently tackle the challenges for solving the RE maximization problem due to its fractional objective function, coupled optimization variables, and discrete coefficient constraint, the formulated nonconvex problem is solved by proposing a two-stage optimization framework. For the outer stage problem, a quadratic transformation is used to recast the fractional objective into a linear form, and a closed-form solution is obtained by using auxiliary variables. For the inner stage problem, the system sum rate is approximated into a linear function. Then, an alternating optimization (AO) algorithm is proposed to optimize the BS precoding and RIS BF iteratively, by utilizing the penalty dual decomposition (PDD) method. Simulation results demonstrate the superiority of the proposed design compared to other benchmarks.

IRS-Assisted Wireless Powered IoT Network With Multiple Resource Blocks

In this paper, we investigate an intelligent reflecting surface (IRS)-assisted wireless powered Internet of Things (WP-IoT) network that operates in multiple resource blocks (RBs). Particularly, the IRS helps in both downlink wireless energy transfer (WET) and uplink wireless information transfer (WIT), in a way that it improves energy reflection in WET from a power station (PS) to various IoT devices and boosts information delivery in WIT from the IoT devices to an access point (AP). Those IoT devices are capable of utilizing the collected energy, and adopting the time-division multiple access (TDMA) or non-orthogonal multiple access (NOMA) scheme in the uplink WIT. Aiming to maximize the average throughput as the overall performance indicator of the considered network, we jointly optimize the transmit power allocation of the PS, the time scheduling, and the IRS phase shifts. These coupled variables lead to the non-convexity of this optimization problem, which cannot be solved directly. To address this problem, we first design the optimal PS’s transmit power allocation for each RB. For the TDMA-based scheme, we design the closed-form IRS beam pattern of the uplink WIT. Then, the closed-form downlink and uplink time allocations are derived by the Lagrange dual method and the Karush-Kuhn-Tucker (KKT) conditions. In addition, the quadratic transformation (QT)-based Alternating Direction Method of Multipliers (ADMM) approach is proposed to iteratively derive the sub-optimal IRS beam pattern of the downlink WET in an alternated fashion. For the NOMA-based scheme, we propose to apply an alternating optimization (AO) algorithm to iteratively optimize the IRS phase shifts, where the uplink IRS beam pattern is iteratively designed by the Riemannian Manifold Optimization (RMO) approach, and the QT-based ADMM method is adopted to alternately derive the sub-optimal downlink IRS phase shifts. Finally, numerical results demonstrate the improved performance of the proposed solution approaches compared to the benchmark schemes, also highlight advantages of the application of IRS in multiple RB scenarios.

Properties of the Quadratic Transformation of Dual Variables

We investigate a solution of a convex programming problem with a strongly convex objective function based on the dual approach. A dual optimization problem has constraints on the positivity of variables. We study the methods and properties of transformations of dual variables that enable us to obtain an unconstrained optimization problem. We investigate the previously known method of transforming the components of dual variables in the form of their modulus (modulus method). We show that in the case of using the modulus method, the degree of the degeneracy of the function increases as it approaches the optimal point. Taking into account the ambiguity of the gradient in the boundary regions of the sign change of the new dual function variables and the increase in the degree of the function degeneracy, we need to use relaxation subgradient methods (RSM) that are difficult to implement and that can solve non-smooth non-convex optimization problems with a high degree of elongation of level surfaces. We propose to use the transformation of the components of dual variables in the form of their square (quadratic method). We prove that the transformed dual function has a Lipschitz gradient with a quadratic method of transformation. This enables us to use efficient gradient methods to find the extremum. The above properties are confirmed by a computational experiment. With a quadratic transformation compared to a modulus transformation, it is possible to obtain a solution of the problem by relaxation subgradient methods and smooth function minimization methods (conjugate gradient method and quasi-Newtonian method) with higher accuracy and lower computational costs. The noted transformations of dual variables were used in the program module for calculating the maximum permissible emissions of enterprises (MPE) of the software package for environmental monitoring of atmospheric air (ERA-AIR).

Spectral-Efficiency of Cell-Free Massive MIMO With Multicarrier-Division Duplex

A multicarrier-division duplex (MDD)-based cell-free (CF) scheme, namely MDD-CF, is proposed, which enables downlink (DL) data and uplink (UL) data or pilots to be concurrently transmitted on mutually orthogonal subcarriers in distributed CF massive MIMO (mMIMO) systems. To demonstrate the advantages of MDD-CF, we firstly study the spectral-efficiency (SE) performance in terms of one coherence interval (CT) associated with access point (AP)-selection, power- and subcarrier-allocation. Since the formulated SE optimization is a mixed-integer non-convex problem that is NP-hard to solve, we leverage the inherent association between involved variables to transform it into a continuous-integer convex-concave problem. Then, a quadratic transform (QT)-assisted iterative algorithm is proposed to achieve SE maximization. Next, we extend our study to the case of one radio frame consisting of several CT intervals. In this regard, a novel two-phase CT interval (TPCT) scheme is designed to not only improve the SE in radio frame but also provide consistent data transmissions over fast time-varying channels. Correspondingly, to facilitate the optimization, we propose a two-step iterative algorithm by building the connections between two phases in TPCT through an iteration factor. Simulation results show that, MDD-CF can significantly outperform in-band full duplex (IBFD)-CF due to the efficient interference management. Furthermore, compared with time-division duplex (TDD)-CF, MDD-CF is more robust to high-mobility scenarios and achieves better SE performance.

Computer analysis of multiple measurements with the sensor's quadratic transformation function

The object of research is multiple measurements. The research aims to improve the accuracy of multiple measurements with a non-linear and unstable sensor transformation function. It is proved that the redundant measurement equation ensures the independence of the measurement result from the parameters of the transformation function and their deviations from the nominal values. It was found that the result of redundant measurements is affected by the reproduction errors of normalized temperatures T1 and T2. It is shown that the best accuracy results are obtained with a reproduction error of normalized temperature T2 within ±1.0 % and temperature T1 within ±0.1 %. This makes it possible to reduce the accuracy requirements for the source of reproduction of normalized temperature Т2. The possibility of processing the results of multiple measurements by two approaches is presented. Computer modeling using the first approach found that with a reproduction error of normalized temperature T2 within ±0.5 %, the relative measurement error is 0.003 %. When modeling the second approach, the relative error is 0.05 %. It was also found that with an increase in the reproduction error of normalized temperature T2 to ±1.0 %, the value of the relative error is 0.04 %. Due to this, when applying the second approach, it becomes possible to choose a non-high-precision source of reproduction of normalized temperature T2. In addition, the sensitivity of the second approach to the digit range of measuring devices was found, which leads to the dependence of the measurement result on their accuracy. There are reasons to assert the possibility of increasing the accuracy of multiple measurements by processing the results of intermediate measurements according to redundant measurement equations using two approaches

On a Resolution of Another Isolated Case of a Kummer’s Quadratic Transformation for 2F1

It is well-known that the Kummer quadratic transformation formula is valid provided that its parameters fulfill some specific conditions (see Gradshteyn, Ryzhik, Tables of Integrals, Series and Products, 9.130, 9.134.1). Very recently, one of us established a new identity when one of these conditions is not fulfilled. In this paper, we aim to discuss another isolated case which completely different from the first. Moreover, in the end, we mention two interesting consequences of these two new results.

RIS-Empowered Ambient Backscatter Communication Systems

Passive tags must harvest sufficient energy to operate. However, ensuring the tag’s energy needs and simultaneously suppressing the interference in ambient backscatter systems is challenging. To meet this, we employ a reconfigurable intelligent surface (RIS) to amplify the ambient source signal that reaches the tag. We maximize the achievable rate by optimizing the RIS phase-shifts subject to the tag’s energy harvesting constraint. Specifically, the phase-shifts are optimized to increase the tag’s received power while suppressing the direct-and-RIS-cascaded link interference. The optimization problem is non-convex, and we convexify it by means of fractional programming and a quadratic transformation. In comparison to a benchmark, our optimal phase-shift design provides ~150% and ~250% gains in harvested power and achieved rate, respectively, at 20 dB m transmit power.

Utility Maximization for IRS Assisted Wireless Powered Mobile Edge Computing and Caching (WP-MECC) Networks

This paper exploits an intelligent reflecting surface (IRS) assisted wireless powered mobile edge computing and caching (WP-MECC) network. In particular, an IRS is utilized to reflect energy signals from a power station (PS) to various IoT devices for energy harvesting during uplink wireless energy transfer (WET). These devices collect energy to support their own partially local computing for computational tasks and their offloading capabilities to an access point (AP), with the help of IRS via time or frequency division multiple access (TDMA or FDMA). The AP is equipped with a local cache connected with a MEC server via a backhaul link, which prefetches the data to facilitate edge computing capabilities. The maximization of a utility function is formulated to evaluate the overall network performance, which is defined as the difference between the sum of computational bits (offloading bits and local computing bits) and total backhaul cost. Due to multiple coupled variables, we first design the optimal caching strategy. Then, an auxiliary vector is introduced to coordinate the energy consumption of local computing and offloading, where its optimal solution can be achieved by an exhaustive search. Moreover, we utilize the Lagrange dual method and the Karush-Kuhn-Tucker (KKT) conditions to derive the optimal time scheduling for the TDMA scheme or the optimal bandwidth allocation for the FDMA counterpart in closed form. The IRS phase shifts are iteratively designed by employing the quadratic transformation (QT) and the Riemannian Manifold Optimization (RMO). Finally, simulation results are demonstrated to validate the network utility performance and confirm the advantage of the employment of IRS, the optimal IRS phase shift design and caching strategy, in comparison to the benchmark schemes.

Separability-based Quadratic Feature Transformation to Improve Classification Performance

Separability-based Quadratic Feature Transformation to Improve Classification Performance

An Extended-Matrix-Separation-Based Summation Inequality on Stability of Discrete-Time Delayed Lur'e System*

This paper investigates the stability of discrete-time Lur'e systems with a time-varying delay. Firstly, an extended-matrix-separated-based summation inequality is established to estimate the summation term containing the system state and the forward difference of the state. Then, an improved delay-product-type Lyapunov-Krasovskii functional (LKF) is proposed, which involves the characteristics of matrices refined and the information of delay squared. By using a quadratic function negative transformation lemma based on matrix-injection, a delay-dependent stability criterion is obtained. Finally, a numerical example demonstrates the advantages of the proposed criterion.

Characteristics of planar sextic indirect-PH curves

&lt;abstract&gt;&lt;p&gt;By a fractional quadratic transformation, an indirect-PH curve can have rational offsets. In this paper, I study properties of planar sextic indirect-PH curves, in terms of their Bézier control polygon legs. With our results, sextic Bézier curves can be efficiently tested whether they are indirect-PH curves. The main strategy to achieve our results is using complex representation of planar parametric curve. Sextic indirect-PH curves can be classified into three classes according to different factorizations of their hodographs. Necessary and sufficient conditions for all classes of sextic indirect-PH curves can be described by non-linear complex systems. By analyzing these non-linear systems, algebraic conditions for a sextic Bézier curve to be an indirect-PH curve are first discussed, then geometric characteristics in terms of legs of its control polygon are revealed.&lt;/p&gt;&lt;/abstract&gt;

Low-Complexity Two-Step Optimization in Active-IRS-Assisted Uplink NOMA Communication

Recently, the active intelligent reflecting surface (IRS) has been proposed to adjust the phase and amplify the magnitude of the incident signal simultaneously. It has a more reasonable hardware overhead compared with the conventional relay. This letter aims to maximize the sum rate of multiple users in the active-IRS-assisted uplink non-orthogonal multiple access (NOMA) system. We propose a low-complexity two-step optimization algorithm to decompose the original non-convex problem into two sub-problems. First, the extreme low-complexity fixed point iteration (FPI) method is proposed to optimize the phase shifts. Then, two algorithms are proposed to solve the amplification optimization problem: the convergence-guaranteed quadratic transform (QT) and the low-complexity generalized eigenvalue decomposition (GEVD) algorithms. Simulation results show that the performance can be enhanced significantly compared with the orthogonal multiple access (OMA) and the passive-IRS scheme.

Total Delay Optimization in Cache-Enabled C-RANs With Hierarchical Rate Splitting

This paper proposes a Hierarchical Rate Splitting (HRS)-based and Macro Base Station (MBS)-assisted cooperative content delivery scheme for Cache-enabled Cloud Radio Access Networks (C <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{2}$</tex-math></inline-formula> -RANs). In this scheme, a user is served by a cluster of Small Base Stations (SBSs) that cache contents. If the user-requested content is missing in its cache, the SBS retrieves it from the central processor (backhaul link) and the MBS (wireless broadcast). We minimize the total delay of this system. We thus optimize user grouping, decoding order, RS variables, BS clustering, and beamforming subject to the transmit power and backhaul capacity constraints. This process formulates a non-convex problem, which we decouple into three subproblems. First, we must pair users to reduce intra-group interference. We thus develop a Low-Complexity User Pairing (LCUP) algorithm. Second, we optimize the decoding order to minimize the broadcast link delay. We, therefore, propose a Dynamic Delay-based Decoding Order Update (D <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{3}$</tex-math></inline-formula> OU) algorithm. Third, we jointly design the RS variables, beamforming, and BS clustering. We use surrogate optimization via the Quadratic Transform (QT), Woodbury matrix identity, and Taylor expansion. We then recast this problem to a Two-Tier Alternating (TTA) problem, which we solve with closed-form (outer tier) and convex (inner tier) formulations. We integrate these algorithms to propose an overall algorithm. It achieves significant performance gains and a 3-fold reduction in time-complexity over the standard Semi-Definite Relaxation (SDR)-based algorithm and several benchmark schemes.

Unsupervised Learning-Inspired Power Control Methods for Energy-Efficient Wireless Networks Over Fading Channels

Energy-efficiency (EE) is a critical metric within wireless optimization. Power control over fading channels is considered as a promising EE-improving technique, but requires optimization of a series of fractional functional optimization problems which are hard to handle by existing optimization techniques. In this paper, we propose a novel EE power control method with unsupervised learning. Firstly, the original fractional problems are decomposed into sub-problems by Dinkelbach and quadratic transformations. Then, these sub-problems are reformulated into unconstrained forms through Lagrange dual formulation. Furthermore, unsupervised primal-dual learning method is applied to handle these unconstrained problems with strong duality. Finally, The unsupervised primal-dual learning is implemented by the deep neural network (DNN) with low computational complexity. Simulation results verify the effectiveness of the proposed approach on a number of typical wireless optimizing scenarios. It is shown that compared to conventional algorithms our method achieves better performance in cognitive radio networks, interference networks, and OFDM networks.