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     

任意实数阶复宗量第一类和第二类Bessel函数的精确计算

本文介绍了精确计算任意实数阶复宗量第一类和第二类Ｂｅｓｓｅｌ函数Ｊｕ（ｚ）和Ｙｕ（ｚ）的递推算法；给出了计算函数值时要用到的数学公式和典型数值计算结果，并对如何验证程序的正确性和分析计算结果的精度以及编程技巧作了较详细的讨论。     

Rapid evaluation of integer-order Bessel functions using an FFTalgorithm

A rapid, accurate, and stable method for computing integer-order Bessel functions is presented. In contrast to the iterative procedures normally used in the numerical evaluation of Bessel functions, this approach requires little more than a commonly available fast Fourier transform algorithm, and yet, for a given argument, can provide many J_ns in a single pass     

基于FFT的阵列方向图快速计算

分析了等距线阵方向图与傅立叶变换的一致性,用FFT算法进行阵列方向图快速计算。通过严格的数学推导,揭示变换序列与可见空间的关系,讨论了不同单元间距情况下的处理方法。文中结合实例,阐述了FFT算法在阵列方向图计算中的应用技巧,如扫描处理方法、指向分辨精度和边缘数据点估计,并指出用FFT算法计算阵列方向图其运算速度可提高一个数量级以上。     

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.     

Fast computation of modified Mathieu functions applied toelliptical waveguide problems

Modified Mathieu functions of the first kind and integer order are expanded in power series, which allow an accurate and timesaving computation. The recurrence relations for the computation of the expansion coefficients are derived and can be easily implemented in computer code. Compared to the calculation with series of Bessel function products, the method presented here is simpler and faster. The efficiency of the method is demonstrated by the computation of eigenmodes of elliptical waveguides     

An exact solution for the impulse response of a uniform plasma half space

A horizontally polarized impulsive plane wave is incident on a uniform half space of cold lossless plasma. The exact form of the resultant impulse response is found for any depth in the plasma in terms of two integer-order Bessel functions by inverting an appropriate Laplace transform. It is shown that as the observer's position in the plasma is increased, the frequency and relative amplitudes of the oscillations of the impulse response increase in a corresponding fashion. Numerical results are presented to illustrate these effects.     

有耗电磁场问题中复宗量整数阶圆柱函数的计算

本文介绍了复宗量整数阶圆柱函数数值计算的递推算法。在计算机允许的数值范围内对圆柱函数的阶数及宗量范围均没有限制。试算结果经加法定理和Wronskian关系式的验算表明准确度很高。     

整数阶复宗量圆柱函数的数值计算

提出了一种复宗量圆柱函数数值计算的新方法。该方法通过研究各类圆柱函数模值随其阶数ｎ的变化关系，提出了首先解决复宗量第二类修正Ｂｅｓｓｅｌ函数Ｋｎ（ｚ）的数值计算，然后应用Ｊｎ（ｚ）、Ｋｎ（ｚ）与其它圆柱函数的关系，解决Ｙｎ（Ｚ）、（ｚ）的数值计算方法。在计算机允许的数值范围内，该方法对圆柱函数的阶数及宗量范围均没有限制，而且计算精度高、编程简单。     

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.     

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     

Fast, accurate interpolation with B-spline scaling functions

A new method for interpolation using B-spline scaling functions is presented which provides significant advantages compared with previously published approaches using the FFT or DST. Periodisation of the sequence is avoided and constraints on the boundary conditions are relaxed. A simple algorithm is given which yields high accuracy with lower computational complexity than FFT or DST techniques     

Digital TDM-FDM Translator with Multistage Structure

In this paper, a new digital signal processing algorithm for the digital TDM-FDM translator is proposed. The digital TDM-FDM translator, which performs a direct translation between two multiplex formats in the telephone network (time-division-multiplexing (TDM) and frequency-division-multiplexing (FDM)) by using digital techniques, has advantages in accuracy and stability of characteristics over equivalent analog equipments. However from the economical point of view, it largely depends on the cost reduction of semiconductor devices and LSI technologies. The proposed algorthm can be realized using only two digital filters and does not require product modulators or Fast Fourier Transform (FFT) processors. The required number of multiplications, which is closely related to the quantity of hardware, is considerably reduced by the multistage structure of this algorithm. The reduction in the kind of required digital hardware and the required number of multiplications makes it possible to efficiently utilize the new hardware realization techniques of digital filters or multipliers using read-only memories and simple logic devices. Since it is foreseen that cost reduction of memory devices will be more rapid than that of logic devices, the proposed algorithm is expected to be advantageous with regard to cost over existing algorithms where complex multiplier logic is required. The estimation of the computation rate is carried out with reference to a practical case. The computer simulation results are also shown.     

The transmitted impulse response of a lossy magnetoplasma by a convolution method

In this note a convolution method is used to find an exact representation in terms of special functions for the transmitted impulse response of a lossy magnetoplasma half space. For an observer deep in the plasma it is shown that the impulse response reduces to a fairly simple expression in terms of integer-order Bessel functions. Also, a comparison is made between the convolution method and the Fourier transform approach to the solution.     

小变量情况下第一类整数阶Bessel函数的计算

在计算第一类整数阶Bessel函数时，后向递推算法稳定高效.然而，起始点的选取必须有足够高的阶数，并且需要进行归一化处理.本文对Taylor级数展开算法进行研究，并对其级数展开规律、计算精度，以及求和项与参数间的关系进行了讨论.此外，本文利用指数形式，极大扩展了该算法的可计算范围.与du Toit算法、MATLAB和Mathematica应用软件的计算结果比较显示，本文的算法具有较高的准确性和稳定性.     

Accurate evaluation of Sommerfeld integrals using the fast Fouriertransform

It is shown that the fast Fourier transform (FFT) combines naturally with Simpson's rule for Sommerfeld-type integral computation. The principal advantage of using the FFT is that a single subroutine call yields a set of sample values of an integral (i.e. the integral for various values of an integrand parameter). Such samples could be useful in themselves. In other applications Sommerfeld integrals represent Green's functions nested within other spatial integrals, so samples from the FFT might be useful in approximating the outernested integral. Several examples are provided to illustrate the process     

基于FFT滑动平均极大似然法的正弦信号频率估计

该文基于正弦信号采样序列的FFT频谱,利用谱图上多条显著谱线与峰值谱线实部之间的关系,推导建立了一用于信号频率估计的滑动平均模型,基于此模型得出的极大似然频率估计器结合传统的Quinn方法后得到一种新的基于FFT谱滑动平均极大似然估计算法。仿真实验表明该算法精确有效,估计性能优于Rife,Quinn法,十分接近CRLB下限,计算量不大且信噪比门限要求可降至-9dB左右。     

虚阶变型Bessel函数的计算

给出虚阶变型Ｂｅｓｓｅｌ函数及其导数的计算公式和计算方法。数值结果表明，计算精度可达到１０位以上有效数字。     

Fast Fourier transform for discontinuous functions

In computational electromagnetics and other areas of computational science and engineering, Fourier transforms of discontinuous functions are often required. We present a fast algorithm for the evaluation of the Fourier transform of piecewise smooth functions with uniformly or nonuniformly sampled data by using a double interpolation procedure combined with the fast Fourier transform (FFT) algorithm. We call this the discontinuous FFT algorithm. For N sample points, the complexity of the algorithm is O(/spl nu/Np+/spl nu/Nlog(N)) where p is the interpolation order and /spl nu/ is the oversampling factor. The method also provides a new nonuniform FFT algorithm for continuous functions. Numerical experiments demonstrate the high efficiency and accuracy of this discontinuous FFT algorithm.     

Design and implementation of a fast algorithm for modulated lappedtransform

A new fast algorithm for the computation of the modulated lapped transform (MLT) is proposed and its efficient implementation using pipelining techniques and complex programmable logic device (CPLD) is presented. The new algorithm computes a length-M MLT via the length-M/2 fast Fourier transform (FFT). Computational overhead due to data shuffling in pre-processing and post-processing is offset in hardware realisation. Hence the overall throughput of the MLT computation for real-time applications is significantly improved. The pipelined CPLD architecture and circuitry are described in detail. Computational complexity of the proposed algorithm is analysed, and throughput improvement is verified by experimental results