Integral Transforms and Their Applications

“…where equality in (23) holds only at the points of increase of F opt X (see Appendix VII-E). At this point, as it is usual in these types of problems [5], the proof follows by ruling out different types of distributions.…”

Section: Resultsmentioning

confidence: 99%

On the capacity of the AWGN channel with additive radar interference

Shahi

Tuninetti

Devroye

2016

2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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This paper investigates the capacity of a communications channel that, in addition to additive white Gaussian noise, also suffers from interference caused by a co-existing radar transmission. The radar interference (of short duty-cycle and of much wider bandwidth than the intended communication signal) is modeled as an additive term whose amplitude is known and constant, but whose phase is independent and identically uniformly distributed at each channel use. The capacity achieving input distribution, under the standard average power constraint, is shown to have independent modulo and phase. The phase is uniformly distributed in [0, 2π]. The modulo is discrete with countably infinitly many mass points, but only finitely many in any bounded interval. From numerical evaluations, a proper-complex Gaussian input is seen to perform quite well for weak radar interference. We also show that for very large radar interference, capacity is equal to 1 2 log (1 + S) and a proper-complex Gaussian input achieves it. It is concluded that the presence of the radar interference results in a loss of half of the degrees of freedom compared to an AWGN channel without radar interference.

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“…The well‐known Mellin transform of a function

f false(z false)

is defined by, p340, eq (8.2.5)

frakturM ({}, f false(z false); frakturp) = \int_{0}^{\infty} z^{frakturp - 1} f false(z false) d z, frakturR false(frakturp false) > 0

provided that the improper integral exists.…”

Section: Classical Integral Transforms Of False(normalγfalse)ℵpiqiunclassified

Certain fractional calculus and integral transform results of incomplete ℵ‐functions with applications

Bansal

Kumar

Nisar

et al. 2020

Math Methods in App Sciences

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In this paper, we study certain interesting and useful properties of incomplete ℵ-functions. The incomplete ℵ-function is an extension of the ℵ-function. We find several useful classical integral transforms of these functions. Further, we examine the fractional calculus with the incomplete ℵ-functions and point out several special cases. Finally, we give the applications of incomplete ℵ-functions in detecting glucose supply in human blood. KEYWORDS Fox's H-function, fractional integral operators, incomplete ℵ-functions, integral transform, Mellin-Barnes type contour integral MSC CLASSIFICATION 26A33; 33C60; 44A10; 34A08; 33B20 INTRODUCTION, DEFINITIONSSeveral authors have introduced and investigated the various special functions (see, eg, gamma function, beta function, Bessel function, Mittag-Leffler function, hypergeometric function, and Fox's H-function). They have also done the systematic study of these functions and also given the applications in engineering, applied sciences, etc.In the 18th century, Legendre 1 and P. Schlömilch introduced the incomplete gamma functions (IGFs) and Tannery (1882), F. E. Prym (1877), and M. Lerch gave the series representation of incomplete gamma function. After that several researchers work deeply on incomplete gamma functions.In 19th Century, by Südland et al 2,3 introduced and investigated the Aleph function, after that many authors established interesting results and also gave the potential applications in Physics, Applied Mathematics, and Engineering disciplines, etc.Recently, Srivastava et al 4 investigate and systematic study of the incomplete Pochhammer symbols and also introduced generalized incomplete hypergeometric functions (GIHF). They also discussed the integral representation, derivative formulas, Mellin-Barnes contour integral of these GIHF and also gave the potential applications in the field of communication theory, groundwater pumping model and probability theory.Very recently, Srivastava et al 5 introduce and investigate the incomplete H-functions and incompleteH-functions, which is the extension of Fox's H-function. They established decomposition formula, derivative formula, classical integral transforms, and fractional calculus and also gave some applications of the incomplete H-functions.Motivated by the ongoing research work on incomplete special functions, we introduce and investigate the incomplete Aleph (ℵ)-functions with the help of the IGFs and also correlate with the other special functions. We also find the classical Math Meth Appl Sci. 2020;43:5602-5614. wileyonlinelibrary.com/journal/mma

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Integral Transforms and Their Applications

Cited by 413 publications

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A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow

A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow

On the capacity of the AWGN channel with additive radar interference

Certain fractional calculus and integral transform results of incomplete ℵ‐functions with applications

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