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

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.

The y function applied in the study of an anomalous diffusion

In this article, we propose a new family of the extended analogues to the Y function for the first time. The relationships among the Y function, Fox H function, Meijer G function, Wright generalized hypergeometric function, and Clausen hypergeometric function are discussed in detail. This result is used to represent the solutions for the anomalous diffusion problems.

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.

The generalized hypergeometric function as the Meijer G-function

In this paper, we use the representation of the generalized hypergeometric function

The singular values of the GOE

As a unifying framework for examining several properties that nominally involve eigenvalues, we present a particular structure of the singular values of the Gaussian orthogonal ensemble (GOE): the even-location singular values are distributed as the positive eigenvalues of a Gaussian ensemble with chiral unitary symmetry, while the odd-location singular values, conditioned on the even-location ones, can be algebraically transformed into a set of independent χ-distributed random variables. We discuss three applications of this structure: first, there is a pair of bidiagonal square matrices, whose singular values are jointly distributed as the even- and odd-location ones of the GOE; second, the magnitude of the determinant of the GOE is distributed as a product of simple independent random variables; third, on symmetric intervals, the gap probabilities of the GOE can be expressed in terms of the Laguerre unitary ensemble. We work specifically with matrices of finite order, but by passing to a large matrix limit, we also obtain new insight into asymptotic properties such as the central limit theorem of the determinant or the gap probabilities in the bulk-scaling limit. The analysis in this paper avoids much of the technical machinery (e.g. Pfaffians, skew-orthogonal polynomials, martingales, Meijer G-function, etc.) that was previously used to analyze some of the applications.