Some fun with Bessel functions

A theoretical investigation of generating functions is pursued using the theory of the Z-transform. Bessel functions are viewed as outputs from infinite impulse response (IIR) filters. It is shown that a single second-order linear ordinary differential equation with nonconstant coefficients defines a family of generating functions for all the Bessel functions of integral order. A numerical method that directly makes use of the three-term recurrence relation, as a tri-diagonal set of equations solved by Gaussian elimination, is proposed for their evaluation. What results is that both J_n and Y_n, for each n, are derived at the cost of one floating-point division and four floating-point multiplications, after initial set up, using an algorithm that has possibilities for parallelization     

Calculating time-dependent performance measures for the M/M/1 queue

Methods are discussed for computing transient performance measures for the M/M/1 queue. These performance measures are often expressed in terms of modified Bessel functions without any discussion about computation. In fact, a common expression for the probability transition function of the M/M/1 queue length process has an infinite sum of modified Bessel functions. For actually generating numbers, however, it is convenient to use numerical integration with associated integral representations, as was first pointed out by P.M. Morse (Oper. Res., vol.3, p.255-61, 1955)     

Multilayer approximation for plane wave propagation in a layered medium

Linear profile multilayer approximation for computing reflection or transmission coefficients of plane wave by a layered medium, which is formal extension of standard staircase stratification method, is studied theoretically and numerically. The problem is reduced to evaluate2 times 2matrix product whose elements include the Bessel functions of orderpm 1/3andpm 2/3. The polynomial approximation for the Bessel function is constructed by applying modified Chebyshev method to facilitate the numerical calculation.     

A procedure for determining the largest computable order of Besselfunctions of the second kind and Hankel functions

Routines for the computation of Bessel functions of the second kind, Y_n(z), and Hankel functions H_n^{(1/)(z) and H}sub n//sup (2/)(z), may exhibit numerical overflow problems for large order n and small complex or real arguments z. An upper limit for the largest calculable order of these functions for a given argument, is derived     

基于Bessel权向量的宽带波束形成器

针对均匀线阵接收宽带信号产生的方向图畸变问题, 提出了一种基于Bessel权向量的宽带波束形成器.从截断后的Bessel级数展开式与均匀线阵的方向图函数相似的特性出发, 推导Bessel级数的截断位置与阵列单元数的关系, 并确定使用第一类Bessel函数设计均匀线阵的权向量.仿真结果表明:应用Bessel权向量后, 在1~3 GHz频率范围内, 均匀线阵可以无失真地接收宽带信号, 不同频率的方向图函数幅值与期望方向图函数幅值的相对误差降低到10-3以下, 且单元越多相对误差越小.与现有方法相比, 本方法直接利用截断后的Bessel级数展开式系数设计权向量, 计算简单且精度高.     

New approximations to J0 and J1 Besselfunctions

New approximate solutions to the 0th- and 1st order Bessel functions of the first kind are derived. The formulations are based upon using a new integral with no previously known solution. The new integral in the limiting case is identical to the 0th-order Bessel function integral. It is solved in closed form, and the solution is expressed as a simple even order polynomial with integer coefficients. The polynomial coefficients are all of integer value. The 1st-order Bessel function approximation can then be found through a simple derivative. Comparisons are made between the exact solution, classic solutions, and the new approximation. The new approximation proves to be much more accurate than the classic small argument approximation. It is also sufficiently accurate to bridge the gap between the classic large and small argument approximations and has potential applications in allowing one to analytically evaluate integrals containing Bessel functions     

Evaluation of some algorithms and programs for the computation ofinteger-order Bessel functions of the first and second kind with complexarguments

The efficiency and accuracy of a few available algorithms for the computation of integer-order Bessel functions are considered. First, the computation of integer-order Bessel functions of the first kind, using the fast Fourier transform (FFT) algorithm as opposed to recurrence techniques, is investigated. It is shown that recurrence techniques are superior to the FFT technique, both in accuracy and speed. An algorithm suggested in the literature and used in commercially available software, specifically MATLAB 3.5 and MATHEMATICA 1.2, for computing integer-order Bessel functions of the second kind is revealed to be erroneous by comparing these routines with an algorithm developed by the author. It is shown that catastrophic errors result from using the erroneous algorithm to compute high-order Bessel functions with nonreal arguments     

Further Formulas for Calculating Approximate Values of the Zeros of Certain Combinations of Bessel Functions (Correspondence)

In a recent letter Gunston has presented a wonderfully simple approximate formula for the smallest z zero of the Bessel function equation J/sub p/(z) N/sub p/ (kz) - J/sub p/(kz)N/sub p/(z) =0(1) where J/sub p/ and N/sub p/ are, respectively, the Bessel functions of the first and second kinds of real-order p. This communication is intended to draw attention to the existence of similar approximate formulas for both the larger z zeros of (1) and the roots of the equally-important companion equationJ'/spl/sub p/(z) N'/sub p/ (kz) - J'/sub p/(kz)N'/sub p/(z) =0 (2) where ' indicates differentiation.     

Generation of Bessel functions of complex order and argument

Recurrence techniques are described for the generation of Bessel functions of the first and second kinds where both the argument and order may be complex. The method is shown to be accurate for several well known forms of functions, including Kelvin and spherical Bessel functions. The accuracy of the general case of complex order and argument is determined by computing the Wronskians and by verifying some addition theorems of the Bessel functions over wide ranges of order and argument. Procedures for the accurate generation of complex-argument gamma functions are also described.     

An Efficient Computational Procedure for the Evaluation of the M/M/I Transient State Occupancy Probabilities

In this note a procedure is given for the numerical evaluation of theM/M/1queue transient state occupancy probabilities which arise in the analysis of dynamic buffer behavior in store and forward networks. The procedure uses the circular coverage function of radar and communication theory to eliminate an infinite series of modified Bessel functions. Computational savings are illustrated by several numerical examples.     

航空时间域电磁法回线源有限差分初始场计算

针对时间域回线源航空电磁有限差分法初始场中双重贝塞尔函数的数值计算,通过分析回线积分法、宗量近似法和贝塞尔函数展开法的特点,提出将贝塞尔函数展开与小宗量近似相结合的混合法,并用共求解全部空间的初始电磁场,不仅计算速度快,而且精度高,满足有限差分对计算时间和计算精度的要求,为计算全空间复杂地质异常体电磁响应奠定了基础。     

Analysis of Acoustic Events in Speech Signals Using Bessel Series Expansion

In this paper, we propose an approach for the analysis and detection of acoustic events in speech signals using the Bessel series expansion. The acoustic events analyzed are the voice onset time (VOT) and the glottal closure instants (GCIs). The hypothesis is that the Bessel functions with their damped sinusoid-like basis functions are better suited for representing the speech signals than the sinusoidal basis functions used in the conventional Fourier representation. The speech signal is band-pass filtered by choosing the appropriate range of Bessel coefficients to obtain a narrow-band signal, which is decomposed further into amplitude modulated (AM) and frequency modulated (FM) components. The discrete energy separation algorithm (DESA) is used to compute the amplitude envelope (AE) of the narrow-band AM-FM signal. Events such as the consonant and vowel beginnings in an unvoiced stop consonant vowel (SCV) and the GCIs are derived by processing the AE of the signal. The proposed approach for the detection of the VOT using the Bessel expansion is shown to perform better than the conventional Fourier representation. The performance of the proposed GCI detection method using the Bessel series expansion is compared against some of the existing methods for various noise environments and signal-to-noise ratios.     

M/M/1 Transient State Occupancy Probabilities Via the Discrete Fourier Transform

A procedure is given for the computation of the transient state occupancy probabilities of theM/M/1queue. The method makes use of the inverse discrete Fourier transform, computed by means of the fast Fourier transform. It avoids the direct evaluation of modified Bessel functions and sidesteps difficulties due to the computation of very large and very small intermediate quantities.     

The numerical computation of Bessel functions of the first andsecond kind for integer orders and complex arguments

A rigorous algorithm for computing integer order Bessel functions of the first and second kind with complex arguments is discussed. The algorithm makes use of backward recurrence for computing Bessel functions of the first kind where applicable, and of Hankel's asymptotic expansion for large arguments. The argument is limited only by the ability of the computer to handle large numbers. Orders up to a few thousand have been computed to 13-digit accuracy     

A Simple Formula For Calculating Approximate Values of the First Zeros of a Combination Bessel Function Equation (Correspondence)

The solution of many problems in microwave theory, particularly those relating to waveguides having curved boundary surfaces, is dependent upon a determination of the zeros of the Bessel function equation J/sub p/(x) N/sub p/(kx) - J/sub p/(kx) N/sub p/(x) = 0 where J/sub p/ and N/sub p/ are respectively the Bessel functions of the first and second kinds, of order p. In a majority of the cases arising in waveguide theory, the parameters k and p are real and positive, but may be fractional or integral.     

Series Representations for Rice's Ie Function

Two new series representations for the Rice functionIe (k, x)are presented. One of the series involves the modified Struve functions and the other involves the modified Bessel functions. These two series complement each other in their convergence speeds as functions of the values ofkandx. The truncation error bounds are derived for both series. Therefore, they can be used alternatively with high efficiency and known precision.     

Impulse response of a lossy magnetoplasma half-space

An exact solution for the transient reflected wave from a lossy magnetoplasma half-space is derived for the case of a vertically incident impulsive plane wave. The solution is expressed as an infnite series of modified Bessel functions of the first kind with complex argument. For small values of the collision and magnetic parameters, the exact solution is approximated in terms of a single integer-order Bessel function.     

Bounds onR_1 (D)functions for speech probability models (Corresp.)

Upper and lower bounds on first-order rate-distortion functions are derived for some selected speech probability models and two different fidelity criteria. The bounds are computed numerically when the analytical procedures are intractable. The models employed involve the Laplacian density, the gamma density, and first- and second-order modified Bessel functionsK_{o}andK_{1}. For both the absolute difference and the mean-square difference distortion measures, theK_{o}model yields the tightest bounds onR_{1}(D)functions.     

Circular sectoral waveguides

Hollow cylindrical waveguides that have pie-shaped cross sections are described. The explicit equations for the field components of the TE (transverse electric) and TM (transverse magnetic) waves are given, and the modal patterns for a number of modes are plotted. The relation between the field distributions inside the circular and sectoral waveguides is discussed. It is shown that the electric and magnetic fields of the modes TE_1n and TM_1n display edge singularity when the sectoral angle is greater than π. Formulas convenient for the evaluation of Bessel functions of fractional order on a personal computer are given