The logarithmic concavity of modified Bessel functions of the first kind and its related functions

Nanthanasub, Thanit; Novaprateep, Boriboon; Wichailukkana, Narongpol

doi:10.1186/s13662-019-2309-8

Cited by 5 publications

(2 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now substituting the NCX distribution ( A2 ) and its corresponding Gaussian distribution ( A3 ) into ( A1 ), the KL divergence can be expressed as

where

denotes the differential entropy of the NCX distribution ( A2 ) given parameter x , and

denotes the expectation over the NCX distribution ( A2 ). The first term can be expressed as

while the second term can be written as

Since function

where x is a given non-negative constant and function

are concave functions [ 39 ], Jensen’s inequality is applied to obtain an upperbound on the KL divergence as

Next, we find the limit of the upper bound of KL divergence

using the limits of mean

and variance

already calculated in ( A4 ) and ( A5 ), that is

At last, using the non-negativity of KL divergence, we have

, i.e.,

. Therefore, we can conclude that the KL divergence between ( A2 ) and ( A3 ) goes to zero when x is sufficiently large and this concludes the proof.…”

Section: Proof Of Propositionmentioning

confidence: 99%

See 1 more Smart Citation

On the Capacity of Amplitude Modulated Soliton Communication over Long Haul Fibers

Chen

Tavakkolnia

Alvarado

et al. 2020

Entropy

View full text Add to dashboard Cite

The capacity limits of fiber-optic communication systems in the nonlinear regime are not yet well understood. In this paper, we study the capacity of amplitude modulated first-order soliton transmission, defined as the maximum of the so-called time-scaled mutual information. Such definition allows us to directly incorporate the dependence of soliton pulse width to its amplitude into capacity formulation. The commonly used memoryless channel model based on noncentral chi-squared distribution is initially considered. Applying a variance normalizing transform, this channel is approximated by a unit-variance additive white Gaussian noise (AWGN) model. Based on a numerical capacity analysis of the approximated AWGN channel, a general form of capacity-approaching input distributions is determined. These optimal distributions are discrete comprising a mass point at zero (off symbol) and a finite number of mass points almost uniformly distributed away from zero. Using this general form of input distributions, a novel closed-form approximation of the capacity is determined showing a good match to numerical results. Finally, mismatch capacity bounds are developed based on split-step simulations of the nonlinear Schro¨dinger equation considering both single soliton and soliton sequence transmissions. This relaxes the initial assumption of memoryless channel to show the impact of both inter-soliton interaction and Gordon–Haus effects. Our results show that the inter-soliton interaction effect becomes increasingly significant at higher soliton amplitudes and would be the dominant impairment compared to the timing jitter induced by the Gordon–Haus effect.

show abstract

“…Now substituting the NCX distribution ( A2 ) and its corresponding Gaussian distribution ( A3 ) into ( A1 ), the KL divergence can be expressed as

where

denotes the differential entropy of the NCX distribution ( A2 ) given parameter x , and

denotes the expectation over the NCX distribution ( A2 ). The first term can be expressed as

while the second term can be written as

Since function

where x is a given non-negative constant and function

are concave functions [ 39 ], Jensen’s inequality is applied to obtain an upperbound on the KL divergence as

Next, we find the limit of the upper bound of KL divergence

using the limits of mean

and variance

already calculated in ( A4 ) and ( A5 ), that is

At last, using the non-negativity of KL divergence, we have

, i.e.,

. Therefore, we can conclude that the KL divergence between ( A2 ) and ( A3 ) goes to zero when x is sufficiently large and this concludes the proof.…”

Section: Proof Of Propositionmentioning

confidence: 99%

“…Since function

where x is a given non-negative constant and function

are concave functions [ 39 ], Jensen’s inequality is applied to obtain an upperbound on the KL divergence as

…”

Section: Proof Of Propositionmentioning

confidence: 99%

On the Capacity of Amplitude Modulated Soliton Communication over Long Haul Fibers

Chen

Tavakkolnia

Alvarado

et al. 2020

Entropy

View full text Add to dashboard Cite

show abstract

Probabilistic and Analytical Aspects of the Symmetric and Generalized Kaiser–Bessel Window Function

Baricz

Pogány

2023

Constr Approx

View full text Add to dashboard Cite

The generalized Kaiser–Bessel window function is defined via the modified Bessel function of the first kind and arises frequently in tomographic image reconstruction. In this paper, we study in details the properties of the Kaiser–Bessel distribution, which we define via the symmetric form of the generalized Kaiser–Bessel window function. The Kaiser–Bessel distribution resembles to the Bessel distribution of McKay of the first type, it is a platykurtic or sub-Gaussian distribution, it is not infinitely divisible in the classical sense and it is an extension of the Wigner’s semicircle, parabolic and n-sphere distributions, as well as of the ultra-spherical (or hyper-spherical) and power semicircle distributions. We deduce the moments and absolute moments of this distribution and we find its characteristic and moment generating function in two different ways. In addition, we find its cumulative distribution function in three different ways and we deduce a recurrence relation for the moments and absolute moments. Moreover, by using a formula of Ismail and May on quotient of modified Bessel functions of the first kind, we deduce a closed-form expression for the differential entropy. We also prove that the Kaiser–Bessel distribution belongs to the family of log-concave and geometrically concave distributions, and we study in details the monotonicity and convexity properties of the probability density function with respect to the argument and each of the parameters. In the study of the monotonicity with respect to one of the parameters we complement a known result of Gronwall concerning the logarithmic derivative of modified Bessel functions of the first kind. Finally, we also present a modified method of moments to estimate the parameters of the Kaiser–Bessel distribution, and by using the classical rejection method we present two algorithms for sampling independent continuous random variables of Kaiser–Bessel distribution. The paper is closed with conclusions and proposals for future works.

show abstract

Expressions for the calculation of isotropic Gaussian filter kernels in the spherical harmonic domain

Piretzidis

Sideris

2022

Stud Geophys Geod

View full text Add to dashboard Cite

The logarithmic concavity of modified Bessel functions of the first kind and its related functions

Cited by 5 publications

References 15 publications

On the Capacity of Amplitude Modulated Soliton Communication over Long Haul Fibers

On the Capacity of Amplitude Modulated Soliton Communication over Long Haul Fibers

Probabilistic and Analytical Aspects of the Symmetric and Generalized Kaiser–Bessel Window Function

Expressions for the calculation of isotropic Gaussian filter kernels in the spherical harmonic domain

Contact Info

Product

Resources

About