Meijer's G-function Research Articles

Performance analysis of fixed gain relay-assisted vertical UWOC systems with generalized misalignment fading model

Underwater wireless optical communication (UWOC) as a potential underwater transmission technology has attracted increasing attention due to its high speed, low delay, and wide range of applications. In addition to absorption and scattering, ocean turbulence fading and misalignment fading seriously affect system performance. Thus, the relay assistance methods have been presented as an effective technique for fading mitigation in optical channels. In this paper, we propose a fixed gain amplify-and-forward (AF) relay-assisted vertical UWOC system with a generalized misalignment fading model over multi-layer cascaded gamma-gamma turbulence channels. For the relay-assisted system, the cumulative distribution function (CDF) and the probability density function (PDF) of the end-to-end instantaneous signal-to-noise ratio (SNR) are derived with the help of the Meijer-G function and the bivariate Fox-H function. Subsequently, based on these statistical analyses of SNR, we derive the closed-form expressions of the outage probability and ergodic capacity. Furthermore, we provide a very tight asymptotic expression for the outage probability in terms of simple functions under high SNR, and the diversity order is analyzed. Finally, results from Monte Carlo simulations are used to validate our derived results. Numerical results for different relay position deployments and alignment error levels reveal that the relay-assisted vertical UWOC system performs better when the relay node is located near the source, while the increase of alignment error level can damage the system performance.

The distribution of the product of independent variance-gamma random variables

Let X and Y be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the product XY is derived. Some basic distributional properties are also derived, including formulas for the cumulative distribution function and the characteristic function, as well as asymptotic approximations for the density, tail probabilities and the quantile function. As special cases, we deduce some key distributional properties for the product of two independent asymmetric Laplace random variables as well as the product of four jointly correlated zero mean normal random variables with a particular block diagonal covariance matrix. As a by-product of our analysis, we deduce some new reduction formulas for the Meijer G-function.

Surface Green’s functions of a horizontally graded elastic half-plane

High-throughput mechanical testing based on functionally graded specimens is very promising for accelerating the development of new materials. However, due to the inhomogeneity-induced complexity, most existing analyses on functionally graded materials have recourse to numerical methods to predict their mechanical responses in reaction to external stimuli. This work investigates the surface Green’s functions for an inhomogeneous half-plane with horizontal exponential material gradient subject to both normal and tangential concentrated forces acting on the surface. The governing equations are first simplified by introducing appropriate potential functions, which facilitates the mathematical derivation of displacements via the Fourier transform technique. In the case of normal force, the vertical surface displacement is derived explicitly under the weak gradient assumption while the horizontal surface displacement is derived directly without the same assumption. In particular, the Meijer G-function and Fox H-function are introduced to express and simplify the vertical displacement. In the case of tangential force, the analytical expressions of surface displacements are also derived similarly. It is noted that the surface Green’s functions not only exhibit singularity and asymmetry properties as expected, but also can be reduced to the classical Boussinesq-Flamant solutions for a homogeneous half-plane. In addition, the analytical results are verified through comparison with the finite element analyses. The surface Green’s functions derived here could be a theoretical basis for developing high-throughput mechanical testing methods which use specimens made of functionally graded materials.

Integral Representations over Finite Limits for Quantum Amplitudes

We extend previous research to derive three additional M-1-dimensional integral representations over the interval [0,1]. The prior version covered the interval [0,∞]. This extension applies to products of M Slater orbitals, since they (and wave functions derived from them) appear in quantum transition amplitudes. It enables the magnitudes of coordinate vector differences (square roots of polynomials) |x1−x2|=x12−2x1x2cosθ+x22 to be shifted from disjoint products of functions into a single quadratic form, allowing for the completion of its square. The M-1-dimensional integral representations of M Slater orbitals that both this extension and the prior version introduce provide alternatives to Fourier transforms and are much more compact. The latter introduce a 3M-dimensional momentum integral for M products of Slater orbitals (in M separate denominators), followed in many cases by another set of M-1-dimensional integral representations to combine those denominators into one denominator having a single (momentum) quadratic form. The current and prior methods are also slightly more compact than Gaussian transforms that introduce an M-dimensional integral for products of M Slater orbitals while simultaneously moving them into a single (spatial) quadratic form in a common exponential. One may also use addition theorems for extracting the angular variables or even direct integration at times. Each method has its strengths and weaknesses. We found that these M-1-dimensional integral representations over the interval [0,1] are numerically stable, as was the prior version, having integrals running over the interval [0,∞], and one does not need to test for a sufficiently large upper integration limit as one does for the latter approach. For analytical reductions of integrals arising from any of the three, however, there is the possible drawback for large M of there being fewer tabled integrals over [0,1] than over [0,∞]. In particular, the results of both prior and current representations have integration variables residing within square roots asarguments of Macdonald functions. In a number of cases, these can be converted to Meijer G-functions whose arguments have the form (ax2+bx+c)/x, for which a single tabled integral exists for the integrals from running over the interval [0,∞] of the prior paper, and from which other forms can be found using the techniques given therein. This is not so for integral representations over the interval [0,1]. Finally, we introduce a fourth integral representation that is not easily generalizable to large M but may well provide a bridge for finding the requisite integrals for such Meijer G-functions over [0,1].

Energy efficiency optimization in a parallel relay-assisted UWOC system with simultaneous lightwave information and power transfer.

In this paper, we investigate the energy efficiency optimization for a parallel relay-assisted underwater wireless optical communication (UWOC) system with simultaneous lightwave information and power transfer (SLIPT) over an aggregate channel. In this system, relay nodes are equipped with energy harvesting devices, getting energy from the direct current component of the received signal transmitted by the source node. These nodes utilize the harvested energy to transmit the signal to the destination node with the decoding and forwarding strategy. The harvested energy for each relay node is derived by the Gauss-Laguerre quadrature formula and the outage probability is deduced by the Meijer-G function. Then, the system's energy efficiency can be calculated and an energy efficiency maximization problem is built up with respect to the bias current. We propose a three-level-iteration algorithm to solve this problem. In the first level, the Dinkelbach method is used to represent energy efficiency in a parametric subtractive form. In the second level, we use the penalty function method to convert the object function and constraint. In the third level, the objective function is transformed into a quadratic function by using a successive convex approximate method, thereby solving for the bias current. The effects of system parameters on energy efficiency are also analyzed. Theoretical results and Monte Carlo simulations suggest that employing the solved bias current can significantly improve the system's energy efficiency.

New algorithms for approximating oscillatory Bessel integrals with Cauchy-type singularities

In this paper, we present an efficient numerical algorithm for approximating integrals involving highly oscillatory Bessel functions with Cauchy-type singularities. By employing the technique of complex line integration, the highly oscillatory Bessel integrals are transformed into oscillatory integrals with a Fourier kernel. When the integration interval does not contain zeros, we use Cauchy’s theorem to transform the integration path to the complex plane and then use the Gaussian–Laguerre formula to compute the integral. For cases in which the integration interval contains zeros, we decompose the integral into two parts: the ordinary and the singular integral. We give a stable recursive formula based on Chebyshev polynomials and Bessel functions for ordinary integrals. For singular integrals, we utilize the MeijerG function for efficient computation. Numerical examples verify the effectiveness of the new algorithm and the fast convergence.

Study on performance of a relay-assisted UWOC system based on adaptive optics

To improve the performance of underwater wireless optical communication (UWOC) systems, we propose a relay-assisted UWOC system model based on adaptive optics (AO). The closed expressions of the scintillation index, composite channel probability density function, and outage probability of the Gaussian beam before and after AO compensation are derived using the extended Rytov theory and Meijer G-function. The performance variation of an UWOC system with different parameters is analyzed by simulation. The results show that AO correction can compensate for the distorted wavefront and significantly reduce the intensity fluctuation at the receiving end. The proposed system can efficiently alleviate channel fading, improving the outage probability performance of the UWOC system.

UAV-assisted FSO communication system with amplify-and-forward protocol under AOA fluctuations: A performance analysis

Free space optical (FSO) communication has recently aroused great interest in academia due to its unique features, such as large transmission band, high data rates, and strong anti-electromagnetic interference. With the aim of evaluating the performance of an FSO communication system and extending the line-of-sight transmission distance, we propose an unmanned aerial vehicle (UAV)-assisted dual-hop FSO communication system equipped with amplify-and-forward protocol at the relay node. Specifically, we consider impairments of atmospheric absorption, pointing errors, atmospheric turbulence, and link interruptions due to angle-of-arrival fluctuations in the relay system. The Gamma-Gamma and Malaga distributions are used to model the influence of atmospheric turbulence on the source-to-UAV and UAV-to-destination links, respectively. We derive closed-form expressions of the probability density function (PDF) and cumulative distribution function (CDF) for the proposed communication system, in terms of the Meijer-G function. Based on the precise PDF and CDF, analytical expressions for the outage probability, average bit error rate, and ergodic capacity are proposed with the aid of the extended generalized bivariate Fox's H function. Finally, we show that there is a match between the analytical results and numerical results, and we analyze the influence of the system and channel parameters on the performance.

An integral representation for quantum amplitudes

The central impediment to reducing multidimensional integrals of transition amplitudes to analytic form, or at least to a fewer number of integral dimensions, is the presence of magnitudes of coordinate vector differences (square roots of polynomials) in disjoint products of functions. Fourier transforms circumvent this by introducing a three-dimensional momentum integral for each of those products, followed in many cases by another set of integral representations to move all of the resulting denominators into a single quadratic form in one denominator whose square my be completed. Gaussian transforms introduce a one-dimensional integral for each such function while squaring the square roots of coordinate vector differences and moving them into a common exponential. Addition theorems may also be used for extracting the angular variables, and sometimes direct integration is even possible. Each method has its strengths and weaknesses. An integral representation is derived herein that stands as an alternative to these four approaches. A number of consequent integrals of Macdonald functions, hypergeometric functions, and Meijer G-functions with complicated arguments are given.

Bayesian interpolation with deep linear networks

Characterizing how neural network depth, width, and dataset size jointly impact model quality is a central problem in deep learning theory. We give here a complete solution in the special case of linear networks with output dimension one trained using zero noise Bayesian inference with Gaussian weight priors and mean squared error as a negative log-likelihood. For any training dataset, network depth, and hidden layer widths, we find non-asymptotic expressions for the predictive posterior and Bayesian model evidence in terms of Meijer-G functions, a class of meromorphic special functions of a single complex variable. Through novel asymptotic expansions of these Meijer-G functions, a rich new picture of the joint role of depth, width, and dataset size emerges. We show that linear networks make provably optimal predictions at infinite depth: the posterior of infinitely deep linear networks with data-agnostic priors is the same as that of shallow networks with evidence-maximizing data-dependent priors. This yields a principled reason to prefer deeper networks when priors are forced to be data-agnostic. Moreover, we show that with data-agnostic priors, Bayesian model evidence in wide linear networks is maximized at infinite depth, elucidating the salutary role of increased depth for model selection. Underpinning our results is a novel emergent notion of effective depth, given by the number of hidden layers times the number of data points divided by the network width; this determines the structure of the posterior in the large-data limit.

Riesz-type criteria for L-functions in the Selberg class

AbstractWe formulate a generalization of Riesz-type criteria in the setting of L-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the grand Riemann hypothesis (GRH) for L-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for L-functions in this subclass. Identities of Ramanujan–Hardy–Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer G-function of the type ${G^{n , 0}_{0 , n}}$ .

SIMO-Underwater Visible Light Communication (UVLC) system

The advent of visible light communication (VLC) towards 5G and beyond (5GB) networks opened new doors to unveil the mysteries of unguided mediums such as ocean. The varying physicochemical properties of water introduces the underwater optical turbulence (UOT). The UOT of UVLC link is modelled as unified Gamma-Gamma (GG) distribution fading under moderate-to-strong turbulence channel conditions as well as the misalignment of transceivers following exponential fading are included and studied. This study also contains the effect of UOT which is characterized by multiple water layers within the Monte Carlo simulation framework. The end-to-end (E2E) performances of a Single-Input and Multi-Output (SIMO) underwater VLC (UVLC) system is in consideration of Intensity-Modulation/Direct-Detection (IM/DD) technique which is obtained to fulfill the current communication requirements in hostile aqueous channels. More specifically, the probability distribution function (PDF) and cumulative distribution function (CDF) are derived to obtain the E2E performance metrics of the SIMO-UVLC system in terms of Meijer-G function. Consequently, novel closed-form expressions for average bit-error-rate (ABER) and outage probability of the system are formulated. Finally, the numerical and simulation results are obtained based on the experimental data in mixed water layers of the oceanic environments, and analytically verified through the Monte Carlo simulation approach in harsh channel conditions.

Performance of IRS-Aided FD Two-Way Communication Network With Imperfect SIC

Intelligent reflecting surface (IRS) is a promising technology to deploy reconfigurable and cost-effective next-generation wireless communication systems. In this work, we investigate the performance of an IRS-aided two-way communication network over Rayleigh fading channels, where downlink and uplink transmission between the base station and the user equipment is established in full-duplex mode. We take into account the impact of residual self-interference (RSI), which results from imperfect self-interference cancellation. We derive the analytical expressions of outage probability (OP), asymptotic OP, and obtain diversity order of the system. We also derive ergodic capacity in terms of the Meijer-G function. Further, we derive the analytical expressions of average symbol error rate (ASER) for general order rectangular quadrature amplitude modulation (QAM), cross QAM, and hexagonal QAM. We highlight the impact of reflecting elements, RSI, and other system parameters on the performance of the considered network and also compare the ASER of various QAM constellations. Finally, Monte-Carlo simulations validate the derived analytical expressions.

The indefinite integrals of Meijer's G-functions from integrating factors

Conway [A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec Funct. 2015;26:812–824] introduced a new and simple method named the ‘Lagrangian method’ for deriving indefinite integrals of both elementary and special functions, provided the function satisfies the second-order linear differential equation. In this paper, different Meijer's G-functions have been used for deriving indefinite integrals, which satisfy second-order differential equations. We have derived recurrence relations and the integration of those recurrence relations by the Lagrangian method. We have discussed the integration of Euler identity, which is given in terms of Meijer's G-function. Different additional relations of Meijer's G-functions have also been discussed.

Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics

A framework to study the eigenvalue probability density function for products of unitary random matrices with an invariance property is developed. This involves isolating a class of invariant unitary matrices, to be referred to as cyclic Pólya ensembles, and examining their properties with respect to the spherical transform on $$\mathrm U(N)$$ . Included in the cyclic Pólya ensemble class are Haar invariant unitary matrices, the circular Jacobi ensemble, known in relation to the Fisher-Hartwig singularity in the theory of Toeplitz determinants, as well as the heat kernel for Brownian motion on the unitary group. We define cyclic Pólya frequency functions and show their relation to the cyclic Pólya ensembles, and give a uniqueness statement for the corresponding weights. The natural appearance of bilateral hypergeometric series is highlighted, with this special function playing the role of the Meijer G-function in the transform theory of unitary invariant product of positive definite matrices. We construct a family of functions forming bi-orthonormal pairs which underly the correlation kernel of the corresponding determinantal point processes, and furthermore obtain an integral formula for the correlation kernel involving just two of these functions.

Performance analysis of adaptive transmission based on power control over atmosphere composite channel

In free space optical (FSO) communications systems, the signal fading caused by atmospheric turbulence is a slow fading, so adaptive transmission technology can effectively mitigate the problem of channel fading. Considering the combined effects of turbulence, attenuation, and pointing errors on the atmospheric transmission of laser signals, we proposed an adaptive M-ary phase shift keying subcarrier modulation algorithm based on power control and studied the combined adaptation of modulation size and average transmit power. Karush-Kuhn-Tucker conditions are used to solve the optimization problem of maximizing the throughput under the constraint of average transmit power, the expressions of throughput and outage probability are derived based on Meijer G-function. We finally evaluated the effects of turbulence intensity, beam waist radius, jitter standard deviation, weather conditions, and maximum modulation order on the throughput and outage probability of adaptive system.

Efficient computation of oscillatory Bessel transforms with a singularity of Cauchy type

In this article, we present and analyze efficient methods for computing the Cauchy principal value integral of the oscillatory Bessel function ∫abf(x)x−cJm(ωx)dx, where 0≤a<c<b. For the two cases of a>0 and a=0, we establish new steepest descent integration paths, and employ the complex line integration technique to obtain affordable quadrature rules. According to the relation formula between the Meijer G-function and the Bessel function, a crucial explicit expression of the required integral ∫0+∞xjx−cJm(ωx)dx can be derived in terms of the Meijer G-function. Furthermore, for the proposed quadrature formulae, we perform rigorous error analysis in inverse powers of ω. Particularly, compared with a combination method given in the work (Kang et al., 2022), the proposed method takes less time and produces more accurate results. We verify error analysis of the proposed method by experimental results. Numerical examples are presented to illustrate the accuracy and efficiency of the proposed numerical methods.

Optimized performance analysis of nonrecursive low-density parity-check-coded Gaussian minimum shift keying in 5G-application

We present the low-density parity-check-coded nonrecursive Gaussian minimum shift keying-based hybrid modulation scheme for free-space optical (FSO) communications under optimal signal-to-noise ratio, limited power, and spectrum resources to support 5G and beyond applications with high-throughput efficiency. Our work investigates and approximates the error rate of the proposed scheme across a negative exponential stochastic FSO channel model with misalignment-induced fading. Using the Meijer-G function over the combined channel probability density function, we develop a unique closed-form formulation of outage probability. The potential benefits of utilizing the normalized beamwidth to achieve the desired error rate ( ≈ 10 − 12 ) is discussed. An improvement of 88.64% in the case of narrow beamwidth is perceived which collaborates with the proposed analytical expressions. The precision of our study is supported by numerical data from Monte Carlo simulation.

Multiple orthogonal polynomials associated with branched continued fractions for ratios of hypergeometric series

The main objects of the investigation presented in this paper are branched-continued-fraction representations of ratios of contiguous hypergeometric series and type II multiple orthogonal polynomials on the step-line with respect to linear functionals or measures whose moments are ratios of products of Pochhammer symbols. This is an interesting case study of the recently found connection between multiple orthogonal polynomials and branched continued fractions that gives a clear example of how this connection leads to considerable advances on both topics.We start by obtaining new results about generating polynomials of lattice paths and total positivity of matrices and giving new contributions to the general theory of the connection between multiple orthogonal polynomials and branched continued fractions with emphasis on its application to the analysis of multiple orthogonal polynomials. Then, we construct new branched continued fractions for ratios of contiguous hypergeometric series. We give conditions for positivity of the coefficients of these branched continued fractions and we show that the ratios of products of Pochhammer symbols are generating polynomials of lattice paths for a special case of the branched continued fractions under study. Next, we introduce a family of type II multiple orthogonal polynomials on the step-line associated with those branched continued fractions. We present a formula as terminating hypergeometric series for these polynomials, we study their differential properties, and we find an explicit recurrence relation satisfied by them. Finally, we focus the analysis of the multiple orthogonal polynomials to the cases where the corresponding branched-continued-fraction coefficients are all positive. In those cases, the orthogonality conditions can be written using measures on the positive real line involving Meijer G-functions and we obtain results about the location of the zeros and the asymptotic behaviour of the polynomials.Specialisations of the multiple orthogonal polynomials studied here include the classical Laguerre, Jacobi, and Bessel orthogonal polynomials, multiple orthogonal polynomials with respect to Nikishin systems of two measures involving modified Bessel functions, confluent hypergeometric functions, and Gauss' hypergeometric function, and multiple orthogonal polynomials with respect to Meijer G-functions used to investigate the singular values of products of Ginibre random matrices as well as a r-orthogonal polynomial sequence with constant recurrence coefficients for any positive integer r and particular instances of the Jacobi-Piñeiro polynomials.

Direct Air-to-Underwater Optical Wireless Communication: Statistical Characterization and Outage Performance

In general, a buoy relay is used to connect the underwater communication to the terrestrial network over a radio or optical wireless communication (OWC) link. The use of relay deployment may pose security and deployment issues. This paper investigates the feasibility of direct air-to-underwater (A2UW) communication from an over-the-sea OWC system to an underwater submarine without deploying a relaying node. We analyze the statistical performance of the direct transmission over the combined channel fading effect of atmospheric turbulence, random fog, air-to-water interface, oceanic turbulence, and pointing errors. We develop novel analytical expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the resultant signal-to-noise ratio (SNR) in terms of bivariate Meijer-G and Fox-H functions. We use the derived statistical results to analyze the system performance by providing exact and asymptotic results of the outage probability in terms of system parameters. We use computer simulations to demonstrate the performance of direct A2UW transmissions compared to the relay-assisted system.