Terms Of Meijer-G Function Research Articles

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.

Dual-Hop Mixed FSO-VLC Underwater Wireless Communication Link

Underwater optical wireless communications (UOWCs) are promising and potential wireless carriers to envisage underwater phenomenal activities for various applications towards the futuristic 5G and beyond (5GB) wireless systems. The main challenges to deploy underwater applications are the physicochemical properties and strong turbulence channel conditions. In this regard, the end-to-end (E2E) performance analysis of a dual-hop mixed FSO/UVLC system under the intensity modulation/direct detection (IM/DD) technique in consideration of pulse amplitude modulation (PAM) scheme is investigated. Throughout this study, to tackle the issues of moderate-to-strong turbulence channel conditions, this work deploys the Gamma-Gamma (GG) distribution fading model and the links are designed by unifying plane wave models in the corresponding links, respectively. This investigation outperforms higher achievable data rate with minimal delay response and enhance network connectivity in real-time monitoring scenarios as compared with the traditional underwater wireless communication technologies. In more contrast, the probability distribution function (PDF), cumulative distribution function (CDF), and closed-form expression of the system are derived and presented in terms of Meijer-G function as well as Extended Generalized Bivariate Meijer-G Function (EGBMGF). The significant E2E performance metrics are obtained by employing the decode-and-forward (DF) relay protocol in hostile channel conditions. In aggregating this work, we combine the analytical expressions that present an efficient tool to depict the impact of channel parameters on the system. The simulation results are plausible of the system performance metrics as average BER (ABER) and outage probability <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(P_{out})$ </tex-math></inline-formula> in the presence of pointing and without pointing error events. Finally, in this work, we use the Monte-Carlo approach for the best fitting curves and validate the numerical expression yields simulation results.

A vector Riemann-Hilbert approach to the Muttalib-Borodin ensembles

In this paper, we consider the Muttalib-Borodin ensemble of Laguerre type, a determinantal point process on [0,∞) which depends on the varying weights xαe−nV(x), α>−1, and a parameter θ. For θ being a positive integer, we derive asymptotics of the associated biorthogonal polynomials near the origin for a large class of potential functions V as n→∞. This further allows us to establish the hard edge scaling limit of the correlation kernel, which is previously only known in special cases and conjectured to be universal. Our proof is based on a Deift/Zhou nonlinear steepest descent analysis of two 1×2 vector-valued Riemann-Hilbert problems that characterize the biorthogonal polynomials and the explicit construction of (θ+1)×(θ+1)-dimensional local parametrices near the origin in terms of Meijer G-functions.

Change Detection in Multilook Polarimetric SAR Imagery With Determinant Ratio Test Statistic

In this article, we propose a determinant ratio test (DRT) statistic to measure the similarity of two covariance matrices for unsupervised change detection in polarimetric radar images. The multilook complex covariance matrix is assumed to follow a scaled complex Wishart distribution. In doing so, we provide the distribution of the DRT statistic that is exactly Wilks’s lambda of the second kind distribution, with density expressed in terms of Meijer G-functions. Due to this distribution, the constant false alarm rate (CFAR) algorithm is derived in order to achieve the required performance. More specifically, a threshold is provided by the CFAR to apply to the DRT statistic producing a binary change map. Finally, simulated and real multilook polarimetric SAR (PolSAR) data are employed to assess the performance of the method and is compared with the Hotelling–Lawley trace (HLT) statistic and the likelihood ratio test (LRT) statistic.

Threshold Setting of Generalized Likelihood Ratio Test for Impropriety of Complex Signals

Testing the impropriety of complex signals is of great importance in complex signal processing. Such test is often accomplished by the generalized likelihood ratio test (GLRT) in the literature. However, when the sample size is small, the associated decision threshold obtained via the existing methods is not accurate enough, which might produce incorrect decisions in practice. Moreover, few methods are available to deal with scenarios where the system dimensionality may be either low or high. To overcome these drawbacks, this letter develops two new methods to compute the threshold for a given nominal false alarm probability (FAP), without any assumptions on the sample size and system dimensionality. Specifically, the exact result of FAP in terms of MeijerG function is established, which enables us to accurately determine the threshold through numerical solution. We also develop an approximate method with low computational load and high precision, by means of box approximation as well as the properties of chi-square distribution. The effectiveness of our proposed methods is demonstrated via simulation results.

On the Performance Analysis of Higher Order QAM Schemes Over Mixed RF/FSO Systems

In this work, the performance of a dual-hop variable gain amplify-and-forward (AF) mixed radio frequency (RF)/free space optics (FSO) system is analyzed in details. As variable gain AF relay network is considered, processing delay results in outdated channel state information (CSI) during amplification at the relay. For general applicability, the RF link is modeled with the generalized Nakagami-m fading. The FSO link is modeled with the Gamma-Gamma distribution which is affected with the atmospheric turbulence and pointing error impairments, and both the intensity modulation with direct detection (IM/DD) and heterodyne detection are employed at the FSO receiver. In this context, analytical expressions of outage probability and ergodic capacity are derived in terms of Meijer-G function and extended generalized bivariate Meijer-G function. For diversity order, asymptotic outage probability expression is also derived. Further, cumulative distribution function based generalized average symbol error rate expressions for various quadrature amplitude modulation (QAM) schemes such as hexagonal QAM, rectangular QAM, and cross QAM are derived in terms of Meijer-G function. Furthermore, a detailed comparative study of various modulation schemes is presented and the impact of pointing error, atmospheric turbulence, outdated CSI, and Nakagami-m parameter are highlighted on the system performance. Finally, all the analytical results are verified through Monte-Carlo simulations.

Multiplicative convolution of real asymmetric and real anti-symmetric matrices

AbstractThe singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble. It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral. It has recently been shown that the Hermitised product {X_{M}\cdots X_{2}X_{1}AX_{1}^{T}X_{2}^{T}\cdots X_{M}^{T}}, where each {X_{i}} is a standard real Gaussian matrix and A is real anti-symmetric, exhibits analogous properties. Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case. As an example we show that the theory also allows for a treatment of this class of Hermitised product when the {X_{i}} are chosen as sub-blocks of Haar distributed real orthogonal matrices.

Product of Squared SR Random Variables: Application to Satellite Communication

We investigate the problem associated with the statistics of the product of two squared Shadowed Rician (SR) random variables (RVs). Both SR RVs are independent and nonidentically distributed. The distribution of the product of two squared SR RVs is very important from the satellite communication perspective. We derive approximate expressions of the probability density function (p.d.f.), the moment generating function (m.g.f.), and the cumulative distribution function (c.d.f.) in terms of Meijer-G functions. Furthermore, exact closed-form expressions for higher order moments, channel quality estimation index, and amount of fading are derived. The outage probability is also discussed by utilizing the expression of the c.d.f. An approximate closed-form expression of channel capacity is also derived with the help of the p.d.f. expression. Furthermore, the performance of satellite communication system is discussed in terms of the approximate symbol error rate by using the derived results. Specifically, the derived expression of the p.d.f. is used to develop an approximate closed-form expression of the m.g.f. for an amplify-and-forward-based satellite communication system.

Fingerprints of tilted Dirac cones on the RKKY exchange interaction in 8- Pmmn borophene

We theoretically investigate the indirect signatures of the tilted anisotropic Dirac cones on Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction in a two dimensional polymorph of boron atoms. Unlike the case of isotropic non-tilted Dirac material-graphene, here we observe that the tilting of the Dirac cones exhibits a significant impact on the RKKY exchange interaction in terms of the suppression of oscillation frequency. The reason can be attributed to the behavior of the Fermi level and the corresponding density of states with respect to the tilting parameter. The direct measurement of the period of the RKKY interaction can thus be a possible probe of the tilt parameter associated with the tilted Dirac cones. We also obtain the direction dependent analytical expressions of the RKKY exchange interaction, in terms of Meijer G-function. However, the effects due to tilting of the Dirac cones on the RKKY interaction depend on the spatial alignments of the two magnetic impurities with respect to the direction of tilting.

Relay-aided free-space optical communications using [formula omitted] distribution over atmospheric turbulence channels with misalignment errors

In this paper, we have analyzed the performance of dual hop radio frequency (RF)/free-space optical (FSO) fixed gain relay environment confined by atmospheric turbulence induced fading channel over FSO link and modeled using α−μ distribution. The RF hop of the amplify-and-forward scheme undergoes the Rayleigh fading and the proposed system model also considers the pointing error effect on the FSO link. A novel and accurate mathematical expression of the probability density function for a FSO link experiencing α−μ distributed atmospheric turbulence in the presence of pointing error is derived. Further, we have presented analytical expressions of outage probability and bit error rate in terms of Meijer-G function. In addition to this, a useful and mathematically tractable closed-form expression for the end-to-end ergodic capacity of the dual hop scheme in terms of bivariate Fox’s H function is derived. The atmospheric turbulence, misalignment errors and various binary modulation schemes for intensity modulation on optical wireless link are considered to yield the results. Finally, we have analyzed each of the three performance metrics for high SNR in order to represent them in terms of elementary functions and the achieved analytical results are supported by computer-based simulations.

Effects of the fermionic vacuum polarization in QED

Some effects of vacuum polarization in QED due to the presence of field sources are investigated. We focus on effects with no counter-part in Maxwell electrodynamics. The the Uehling interaction energy between two stationary point-like charges is calculated exactly in terms of Meijer-G functions. Effects induced on a hydrogen atom by the vacuum polarization in the vicinity of a Dirac string are considered. We also calculate the interaction between two parallel Dirac strings and corrections to the energy levels of a quantum particle constrained to move on a ring circumventing a solenoid.

Relay selection‐based hybrid satellite‐terrestrial communication systems

The problem of relay selection in amplify-and-forward (AF)-based hybrid satellite-terrestrial cooperative communication systems is considered. A partial relay selection scheme is studied in which satellite selects a relay earth station (ES) among multiple relay ESs (situated on ground) on the basis of maximum instantaneous signal-to-noise ratio (SNR). Satellite-relay ESs are assumed to follow Shadowed-Rician fading, whereas relay ESs-destination ES links (terrestrial links) are assumed to follow Nakagami- m distribution. First, the cumulative distribution function and probability density function (PDF) of the maximum instantaneous SNR of satellite-relay ESs links are derived; then by using this PDF, the expression of moment generating function (MGF) of the received instantaneous SNR at destination ES is obtained. The average error performance of the considered system with the proposed relay ES selection is derived in terms of Meijer- G functions by using MGF approach. In order to get the diversity order of the relay ES selection-based AF scheme, the asymptotic PDF of the considered scheme is derived, and then by using this asymptotic PDF, the analytical diversity order of the system is obtained. It is demonstrated by the analysis and simulation that the performance of the AF-based hybrid satellite-terrestrial communication systems can be significantly improved by deploying more relay ES at the ground in between satellite and destination ES.

Universality classes of fluctuation dynamics in hierarchical complex systems

A unified approach is proposed to describe the statistics of the short-time dynamics of multiscale complex systems. The probability density function of the relevant time series (signal) is represented as a statistical superposition of a large time-scale distribution weighted by the distribution of certain internal variables that characterize the slowly changing background. The dynamics of the background is formulated as a hierarchical stochastic model whose form is derived from simple physical constraints, which in turn restrict the dynamics to only two possible classes. The probability distributions of both the signal and the background have simple representations in terms of Meijer G functions. The two universality classes for the background dynamics manifest themselves in the signal distribution as two types of tails: power law and stretched exponential, respectively. A detailed analysis of empirical data from classical turbulence and financial markets shows excellent agreement with the theory.

Bulk and soft-edge universality for singular values of products of Ginibre random matrices

Il a été démontré par Akemann, Ipsen et Kieburg que les valeurs singulières carrées de produits de $M$ matrices aléatoires rectangulaires composées de variables aléatoires complexes gaussiennes, suivent un processus ponctuel déterminantal, avec un noyau de corrélation qui admet une représentation en fonction des fonctions G de Meijer. Nous démontrons l’universalité des statistiques locales des valeurs singulières carrées, c’est-à-dire, l’universalité dans le bulk donné par le noyau sinus et l’universalité au bord donné par le noyau Airy. La preuve est basée sur l’analyse asymptotique de la représentation du noyau de corrélation comme une double intégrale de contour. Notre stratégie peut être généralisée à d’autres modèles de produits de matrices aléatoires introduits récemment, afin d’obtenir des résultats d’universalité similaires. Deux autres exemples sont considérés, le premier étant le produit de $M$ matrices de Ginibre et l’inverse de $K$ matrices de Ginibre, étudié par Forrester, le deuxième étant le produit de $M-1$ matrices de Ginibre avec une troncation d’une matrice unitaire, considéré par Kuijlaars et Stivigny.

Efficient method for the computation of oscillatory Bessel transform and Bessel Hilbert transform

In this paper, we study the numerical methods for the evaluation of two kinds of highly oscillatory Bessel transforms. Firstly, we rewrite both integrals as the sum of two integrals. By rewriting the Bessel function as a linear combination of Whittaker W function, we then transform one of integrals into the Fourier type, which can be transformed into the integrals on [0,+∞), and can be computed by some proper Gaussian quadrature, which take into account the asymptotic property of Whittaker W function as x→0. The other can be efficiently computed based on the evaluation of special functions. In addition, error analysis for the presented methods is given. Moreover, we also give an explicit formula for the integral ⨍0+∞Jν(ωx)x−τdx in terms of Meijer G-function, and then apply the method for the oscillatory Bessel transforms to the computation of highly oscillatory Bessel Hilbert transforms. Theoretical results and numerical examples demonstrate the efficiency and accuracy of the proposed methods.

Performance evaluation of decode‐and‐forward dual‐hop asymmetric radio frequency‐free space optical communication system

In this study, the error performance and the capacity analysis is performed for the decode-and-forward based dual-hop asymmetric radio frequency-free space optical communication (RF-FSO) system. The RF link is characterised by Nakagami- m fading and the FSO link is characterised by path loss, Gamma-Gamma distributed turbulence and pointing error. For this mixed RF-FSO cooperative system, novel closed-form mathematical expressions are derived for cumulative distribution function, probability density function and moment generating function of the equivalent signal-to-noise ratio in terms of Meijer-G function. Using these channel statistics, new finite power series based analytical expressions are obtained for the outage probability, the average bit error rate for various binary and M -ary modulation techniques and the average channel capacity of the considered system in terms of Meijer-G function. As a special case, the analytical framework can also be obtained for channel statistics and performance metrics of dual-hop mixed Rayleigh-Gamma-Gamma system. Simulation results validate the proposed mathematical analysis. The effects of fading, turbulence and pointing error are studied on the outage probability, average bit error rate and channel capacity of the asymmetric RF-FSO system.

Universality Conjecture and Results for a Model of Several Coupled Positive-Definite Matrices

The paper contains two main parts: in the first part, we analyze the general case of \({p \geq 2}\) matrices coupled in a chain subject to Cauchy interaction. Similarly to the Itzykson-Zuber interaction model, the eigenvalues of the Cauchy chain form a multi level determinantal point process. We first compute all correlations functions in terms of Cauchy biorthogonal polynomials and locate them as specific entries of a \({(p+1) \times (p+1)}\) matrix valued solution of a Riemann–Hilbert problem. In the second part, we fix the external potentials as classical Laguerre weights. We then derive strong asymptotics for the Cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin. As a result, we obtain a new family of universality classes for multi-level random determinantal point fields, which include the Bessel\({_{\nu}}\) universality for 1-level and the Meijer-G universality for 2-level. Our analysis uses the Deift-Zhou nonlinear steepest descent method and the explicit construction of a \({(p+1) \times (p+1)}\) origin parametrix in terms of Meijer G-functions. The solution of the full Riemann–Hilbert problem is derived rigorously only for p = 3 but the general framework of the proof can be extended to the Cauchy chain of arbitrary length p.

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows us to find its scaling limits at the origin (hard edge). The limiting kernels generalize the classical Bessel kernels. For M = 2 they coincide with the scaling limits found by Bertola, Gekhtman, and Szmigielski in the Cauchy–Laguerre two-matrix model, which indicates that these kernels represent a new universality class in random matrix theory.

A Solution of Fractional Laplace's Equation by Modified Separation of Variables

This paper applies the Modified separation of variables method (MSV) suggested by Pishkoo and Darus towards obtaining a solution for fractional Laplace's equation. The closed form expression for potential function is formulated in terms of Meijer's G-functions (MGFs). Moreover, the relationship between fractional dimension and parameters of Meijer’s G-function for spherical Laplacian in R, for radial functions is derived.

Distribution of the Sample Correlation Matrix and Applications

For the case where the multivariate normal population does not have null correlations, we give the exact expression of the distribution of the sample matrix of correlations R, with the sample variances acting as parameters. Also, the distribution of its determinant is established in terms of Meijer G-functions in the null-correlation case. Several numerical examples are given, and applications to the concept of system de- pendence in Reliability Theory are presented.