Meijer's G-function Research Articles

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.

BER performance analysis of FSO using hybrid-SIM technique with APD receiver over weak and strong turbulence channels

Abstract In this paper, a hybrid-subcarrier-intensity-modulation (hybrid-SIM) technique for the performance improvement of free-space-optical (FSO) communication system has been proposed. Subsequently, for further error performance improvement, avalanche photodiode (APD) based receiver is used in the proposed system. The system performance is analyzed at various atmospheric turbulence levels over weak and strong turbulence channels. The bit error rate (BER) is theoretically derived using Gauss–Hermite approximation and Meijer-G function and it is simulated in the MATLAB environment. The simulation result shows that the BER performance of hybrid-SIM is better than BPSK-SIM technique irrespective of the channel types and also the significant BER performance improvement is observed by APD receiver.

A Guide to Special Functions in Fractional Calculus

Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as a “Guide to the Functions”. This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to “new” classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag–Leffler type functions, among them the “Queen function of fractional calculus”, the Mittag–Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author’s works for more than 30 years, and support the wide spreading and important role of these functions by several examples.

Unified Performance Analysis of Near and Far User in Downlink NOMA System over η - μ Fading Channel

The non-orthogonal multiple access (NOMA) scheme is considered as a frontier technology to cater the requirements of 5G and beyond 5G (B5G) communication systems. To fully exploit the essence of NOMA, it is very important to explore the behavior of NOMA users over the most realistic nonhomogenous fading conditions. In this paper, we derive unified closed-form expressions of the basic performance metrics of the NOMA users. Their performance is evaluated in terms of average bit error rate (ABER), average channel capacity (ACC) and outage probability (OP) over η−μ fading channel. These expressions are in terms of popular functions, such as Meijer G-function and Gauss hypergeometric function, leading to their versatile use in analytical research. Unlike the existing outage probability expressions in terms of Yacoub integral, the derived expressions are easier to implement in software packages like MATLAB. Moreover, we compare the obtained results with a reference system, consisting of genie-aided NOMA system. We interpret that genie-aided performance results provide benchmark bounds for the metrics. Extensive simulations are carried out to validate the derived analytical expressions.

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.

Unified Approach to Fractional Calculus Images of Special Functions—A Survey

Evaluation of images of special functions under operators of fractional calculus has become a hot topic with hundreds of recently published papers. These are growing daily and we are able to comment here only on a few of them, including also some of the latest of 2019–2020, just for the purpose of illustrating our unified approach. Many authors are producing a flood of results for various operators of fractional order integration and differentiation and their generalizations of different special (and elementary) functions. This effect is natural because there are great varieties of special functions, respectively, of operators of (classical and generalized) fractional calculus, and thus, their combinations amount to a large number. As examples, we mentioned only two such operators from thousands of results found by a Google search. Most of the mentioned works use the same formal and standard procedures. Furthermore, in such results, often the originals and the images are special functions of different kinds, or the images are not recognized as known special functions, and thus are not easy to use. In this survey we present a unified approach to fulfill the mentioned task at once in a general setting and in a well visible form: for the operators of generalized fractional calculus (including also the classical operators of fractional calculus); and for all generalized hypergeometric functions such as pΨq and pFq, Fox H- and Meijer G-functions, thus incorporating wide classes of special functions. In this way, a great part of the results in the mentioned publications are well predicted and appear as very special cases of ours. The proposed general scheme is based on a few basic classical results (from the Bateman Project and works by Askey, Lavoie–Osler–Tremblay, etc.) combined with ideas and developments from more than 30 years of author’s research, and reflected in the cited recent works. The main idea is as follows: From one side, the operators considered by other authors are cases of generalized fractional calculus and so, are shown to be (m-times) compositions of weighted Riemann–Lioville, i.e., Erdélyi–Kober operators. On the other side, from each generalized hypergeometric function pΨq or pFq (p≤q or p=q+1) we can reach, from the final number of applications of such operators, one of the simplest cases where the classical results are known, for example: to 0Fq−p (hyper-Bessel functions, in particular trigonometric functions of order (q−p)), 0F0 (exponential function), or 1F0 (beta-distribution of form (1−z)αzβ). The final result, written explicitly, is that any GFC operator (of multiplicity m≥1) transforms a generalized hypergeometric function into the same kind of special function with indices p and q increased by m.

Nonlinear photon trident versus double Compton scattering and resummation of one-step terms

We study the photon trident process, where an initial photon turns into an electron-positron pair and a final photon under a nonlinear interaction with a strong plane-wave background field. We show that this process is very similar to double Compton scattering, where an electron interacts with the background field and emits two photons. We also show how the one-step terms can be obtained by resumming the small- and large-$\chi$ expansions. We consider a couple of different resummation methods, and also propose new resummations (involving Meijer-G functions) which have the correct type of expansions at both small and large $\chi$. These new resummations require relatively few terms to give good precision.

Applications of Meijer’s Factorization Theorems in Performance Analyses of All-Optical Multi-Hop FSO Systems

The use of bivariate Fox H-functions (BFHFs) in performance analyses of wireless communication systems has gained considerable attention in past few decades. However, the non-existence of robust built-in routines for evaluating such functions in standard computing systems poses numerous challenges in numerical experiments and simulations. Motivated by the apparent need to circumvent these difficulties in performance analyses of cooperative wireless communications, this work presents an alternative method for obtaining the exact, approximate and asymptotic BFHF-free cumulative distribution function (CDF) of the end-to-end (e2e) signal-to-noise ratio (SNR) of multi-hop amplify-and-forward (AF) relaying wireless communication systems. As an illustration, the e2e performance analysis of an all-optical dual-hop free-space optical (FSO) transmission system over Gamma-Gamma turbulence in the presence of pointing errors is revisited. Specifically, new mathematical formulae for the statistical characteristics of the e2e SNR for systems with AF fixed-gain relaying as well as channel-state-information(CSI)-assisted using heterodyne detection (HD) or intensity modulation with direct detection (IM/DD) are derived in terms of mathematically malleable and uniformly convergent infinite series of weighted Meijer G-functions. The usefulness of the derived CDFs is illustrated through derivation of traditional system performance metrics. The accuracy of the derived analytical formulae is verified via Monte Carlo simulations in MATLAB®. Finally, based on results observed in this paper, useful expansions of common BFHFs in terms of easily computable univariate hypergeometric functions are proposed.

Mellin–Meijer kernel density estimation on $${{\mathbb {R}}}^+$$

Kernel density estimation is a nonparametric procedure making use of the smoothing power of the convolution operation. Yet, it performs poorly when the density of a positive variable is estimated, due to boundary issues. So, various extensions of the kernel estimator allegedly suitable for $${\mathbb {R}}^+$$ -supported densities, such as those using asymmetric kernels, abound in the literature. Those, however, are not based on any valid smoothing operation. By contrast, in this paper a kernel density estimator is defined through the Mellin convolution, the natural analogue on $${\mathbb {R}}^+$$ of the usual convolution. From there, a class of asymmetric kernels related to Meijer G-functions is suggested, and asymptotic properties of this ‘Mellin–Meijer kernel density estimator’ are presented. In particular, its pointwise- and $$L_2$$ -consistency (with optimal rate of convergence) are established for a large class of densities, including densities unbounded at 0 and showing power-law decay in their right tail.

On the BEP analysis of M-QAM and R-QAM under cascaded double η-μ, κ-μ or α-μ fading channels

This article presents new expressions for the bit error probability (BEP) of the M-ary quadrature amplitude modulation (M-QAM) and rectangular QAM (R-QAM) scheme in a frequency non-selective channel model with cascaded double fading, characterized in the links by the distributions η-μ, κ-μ or α-μ. In this paper, the probability density function (PDF) of the signal-to-noise ratio (SNR) of the double link is written in terms of an integral of the product of the PDF of the envelope of the first link by the PDF of the SNR of the second link. With this procedure, expressions for the BEP of the M-QAM and R-QAM scheme, with and without a maximum ratio combiner (MRC) associated, are obtained by means of the general expressions of Cho and Yoon, calculated with the help of the Gauss-Laguerre quadrature. The new expressions are written in terms of the integral of the moment generating function (MGF) of the instantaneous SNR from the second link, which are simpler than using expressions involving infinite series in terms of special functions like hypergeometric functions and the Meijer G-function. In addition, insights regarding the expressions obtained and comments about their complexity are provided. Several curves for the BEP are presented, for different parameters that characterize the channel and different numbers of branches of the MRC. All analytical expressions obtained in this paper are corroborated by computer simulations performed with the Monte Carlo method.

Weak-coupling, strong-coupling and large-order parametrization of the hypergeometric-Meijer approximants

Without Borel or Padé techniques, we show that for a divergent series with n! large-order growth factor, the set of hypergeometric series pFp-2 represents suitable approximants. The divergent pFp-2 series are then resummed via their representation in terms of the Meijer-G function. The choice of pFp-2 accelerates the convergence even with only weak-coupling information as input. For more acceleration of the convergence, we employ the strong-coupling and large-order information. We obtained a new constraint that relates the difference between the sum of the numerator and the sum of denominator parameters in the hypergeometric approximant to one of the large-order parameters. To test the validity of that constraint, we employed it to obtain the exact partition function of the zero-dimensional ϕ4 scalar field theory. The algorithm is also applied for the resummation of the ground state energies of ϕ0+14 and iϕ0+13 scalar field theories. We get accurate results for the whole coupling space and the precision is improved systematically in using higher orders. Precise results for the critical exponents of the O(4)-symmetric field model in three dimensions have been obtained from resummation of the recent six-loops order of the corresponding perturbation series. The recent seven-loops order for the β-function of the ϕ3+14 field theory has been resummed which shows non-existence of fixed points. The first resummation result of the seven-loop series representing the fractal dimension of the two-dimensional self-avoiding polymer is presented here where we get a very accurate value of df=1.3307 compared to its exact value (4/3≈1.3333).

Secrecy Analysis Over Cascaded $\kappa -\mu$ Fading Channels With Multiple Eavesdroppers

In this paper, secrecy performance is investigated over the cascaded fading channel for the κ-μ distribution. The probability density function (PDF) and the cumulative distribution function (CDF) for the cascaded κ-μ fading channels as well as for some of its special cases, such as the Rayleigh and Rician fading channels are derived. A communication system operating over the cascaded κ-μ fading channel is analyzed in which the outage probability (P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> ), the average symbol error probability (ASEP), and the average channel capacity (ACC) are obtained in terms of the Meijer G-function. In addition, the secrecy is examined for a system model where we have a transmitter, a receiver, and multiple colluding eavesdroppers attempting to overhear the information received. The secrecy performance is investigated in terms of the secrecy outage probability (SOP) and the probability of non-zero secrecy capacity (P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nzc</sup> ). The analyses illustrate that increasing the number of eavesdroppers in the network degrades the secrecy of the system. Furthermore, increasing the cascade level in the channel degrades the system performance as well as the information security. In addition, the impact of the wiretap channel parameters and the average received signal-to-noise-ratio (SNR) at the eavesdroppers over the security are studied. To verify the accuracy of the derived analytical expressions, Monte-Carlo simulations are utilized.

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.

Precise critical exponents of the O(N) -symmetric quantum field model using hypergeometric-Meijer resummation

In this work, we show that one can select different types of Hypergeometric approximants for the resummation of divergent series with different large-order growth factors. Being of $n!$ growth factor, the divergent series for the $\varepsilon$-expansion of the critical exponents of the $O(N)$-symmetric model is approximated by the Hypergeometric functions $_{k+1}F_{k-1}$. The divergent $_{k+1}F_{k-1}$ functions are then resummed using their equivalent Meijer-G function representation. The convergence of the resummation results for the exponents $\nu$,\ $\eta$ and $\omega$ has been shown to improve systematically in going from low order to the highest known six-loops order. Our six-loops resummation results are very competitive to the recent six-loops Borel with conformal mapping predictions and to recent Monte Carlo simulation results. To show that precise results extend for high $N$ values, we listed the five-loops results for $\nu$ which are very accurate as well. The recent seven-loops order ($g$-series) for the renormalization group functions $\beta,\gamma_{\phi^2}$ and $\gamma_{m^2}$ have been resummed too. Accurate predictions for the critical coupling and the exponents $\nu$, $\eta$ and $\omega$ have been extracted from $\beta$,$\gamma_{\phi^2}$ and $\gamma_{m^2}$ approximants.

An Enriched α − μ Model as Fading Candidate

This paper introduces an enriched α − μ distribution which may act as fading model with its origins via the scale mixture construction. The distribution’s characteristics are visited and its feasibility as a fading candidate in wireless communications systems is investigated. The analysis of the system reliability and some performance measures of wireless communications systems over this enriched α − μ fading candidate are illustrated. Computable representations of the Laplace transform for this scale mixture construction are also provided. The derived expressions are explored via numerical investigations. Tractable results are computed in terms of the Meijer G-function. This unified scale mixture approach allows access to previously unconsidered underlying models that may yield improved fits to experimental data in practice.

Bootstrapping inflationary correlators in Mellin space

We develop a Mellin space approach to boundary correlation functions in anti-de Sitter (AdS) and de Sitter (dS) spaces. Using the Mellin-Barnes representation of correlators in Fourier space, we show that the analytic continuation between AdSd+1 and dSd+1 is encoded in a collection of simple relative phases. This allows us to determine the late-time tree-level three-point correlators of spinning fields in dSd+1 from known results for Witten diagrams in AdSd+1 by multiplication with a simple trigonometric factor. At four point level, we show that Conformal symmetry fixes exchange four-point functions both in AdSd+1 and dSd+1 in terms of the dual Conformal Partial Wave (which in Fourier space is a product of boundary three-point correlators) up to a factor which is determined by the boundary conditions. In this work we focus on late-time four-point correlators with external scalars and an exchanged field of integer spin-ℓ. The Mellin-Barnes representation makes manifest the analytic structure of boundary correlation functions, providing an analytic expression for the exchange four-point function which is valid for general d and generic scaling dimensions, in particular massive, light and (partially-)massless fields. It moreover naturally identifies boundary correlation functions for generic fields with multi-variable Meijer-G functions. When d = 3 we reproduce existing explicit results available in the literature for external conformally coupled and massless scalars. From these results, assuming the weak breaking of the de Sitter isometries, we extract the corresponding correction to the inflationary three-point function of general external scalars induced by a general spin- ℓ field at leading order in slow roll. These results provide a step towards a more systematic understanding of de Sitter observables at tree level and beyond using Mellin space methods.

Matrix model generating function for quantum weighted Hurwitz numbers

The KP -function of hypergeometric type serving as generating function for quantum weighted Hurwitz numbers is used to compute the Baker function and the corresponding adapted basis elements, expressed as absolutely convergent Laurent series in the spectral parameter. These are equivalently expressed as Mellin–Barnes integrals, analogously to Meijer G-functions, but with an infinite product of -functions as integral kernel. A matrix model representation is derived for the -function evaluated at trace invariants of an externally coupled matrix.

On the Physical Layer Security of a Decode and Forward Based Mixed FSO/RF Co-operative System

In this letter, the secrecy performance of a decode-and-forward (DF) principle based mixed free space optics (FSO) and radio frequency (RF) system is analyzed. The analysis is done from physical layer (PHY) security perspective. Herein, two different eavesdroppers are considered while signal is transmitting from source (S) to destination (D) through a relay (R) point. The FSO based and RF based eavesdroppers are trying to intercept the confidential signal from S-R link and R-D links respectively. The faded FSO links are modeled by Málaga (M) distribution and RF links are characterized by Nakagami-m distribution. Exact closed form expression for secrecy performance metric such as secrecy outage probability is derived and analyzed for the aforementioned system in terms of Meijer's G-function and bivariate Fox's H-function. Furthermore, the asymptotic expressions for these performance metrics are derived by employing high signal to noise ratio (SNR) approximation. Afterwards the results are verified by Monte-Carlo simulations. Finally, the secrecy threshold for the system is derived to get insight of the system reliability and secrecy.

OP Performance Prediction for Complex Mobile Multiuser Networks Based on Extreme Learning Machine

Due to the complex and variable environments of mobile communication, the mobile multiuser networks become a hot topic. To process active complex event in mobile multiuser networks, it is important to predict the system performance. In this work, the authors consider the multiuser networks which utilizes transmit antenna selection (TAS). We derive novel closed-form expressions for the outage probability (OP) in terms of the Meijer's G-function. Then, a extreme learning machine (ELM)-based OP performance prediction algorithm is proposed. We use the theoretical results to generate training data. We test back-propagation (BP) neural network, locally weighted linear regression (LWLR), wavelet neural network, ELM, and support vector machine (SVM) methods. Compared with wavelet neural network, SVM, BP neural network, and LWLR methods, the Monte-Carlo results shows that the proposed prediction algorithm can consistently achieve higher OP performance prediction results.

Hardware- and Interference-Limited Cognitive IoT Relaying NOMA Networks With Imperfect SIC Over Generalized Non-Homogeneous Fading Channels

Internet-of-Things (IoT) technology has received much attention due to its great potential to interconnect billions of devices in a broad range of applications. IoT networks can provide high-quality services for a large number of users and smart objects. On the other hand, massive connectivity in IoT networks brings problems associated with spectral congestion. This issue can be solved by applying cognitive radio (CR) and non-orthogonal multiple access (NOMA) techniques. In this respect, this paper studies the performance of cooperative CR-NOMA enabled IoT networks over a generalized α - μ fading channel model. Closed-form analytical expressions of the end-to-end outage probability (OP) for the secondary NOMA users are derived using the Meijer's G-function with a consideration of the impacts of the interference temperature constraint, primary interference, residual hardware impairments and imperfect successive interference cancellation. Moreover, to acquire some useful insights on the system performance, asymptotic closed-form OP expressions are provided. Additionally, the impact of α and μ fading parameters on the outage performance is examined and, as a result, it is concluded that the system performance sufficiently improves as α and/or α increase. Furthermore, the outage performance of the proposed system model is shown to outperform that of an identical IoT network operating on orthogonal multiple access. Finally, the provided closed-form OP expressions are validated with Monte Carlo simulations.