Multiple precision evaluation of the Airy Ai function with reduced cancellation

The series expansion at the origin of the Airy function Ai(x) is alternating and hence problematic to evaluate for x > 0 due to cancellation. Based on a method recently proposed by Gawronski, M\"uller, and Reinhard, we exhibit two functions F and G, both with nonnegative Taylor expansions at the origin, such that Ai(x) = G(x)/F(x). The sums are now well-conditioned, but the Taylor coefficients of G turn out to obey an ill-conditioned three-term recurrence. We use the classical Miller algorithm to overcome this issue. We bound all errors and our implementation allows an arbitrary and certified accuracy, that can be used, e.g., for providing correct rounding in arbitrary precision

Machine Precision Evaluation of Singular and Nearly Singular Potential Integrals by Use of Gauss Quadrature Formulas for Rational Functions

A new technique for machine precision evaluation of singular and nearly singular potential integrals with 1/R singularities is presented. The numerical quadrature scheme is based on a new rational expression for the integrands, obtained by a cancellation procedure. In particular, by using library routines for Gauss quadrature of rational functions readily available in the literature, this new expression permits the exact numerical integration of singular static potentials associated with polynomial source distributions. The rules to achieve the desired numerical accuracy for singular and nearly singular static and dynamic potential integrals are presented and discussed, and several numerical examples are provide

Calculation of MoM interaction integrals in highly conductive media

The construction of the impedance matrix in the method of moments requires the calculation of interaction integrals between the expansion functions, through the Green's function and its derivatives. The singular behavior of the Green's function poses considerable problems for an accurate numerical evaluation of these integrals, requiring techniques such as singularity extraction or cancellation. In this contribution we will show why these methods fail when the medium is highly conductive. A novel technique is proposed to handle these highly challenging integrals. The complexity of the new method is independent of the conductivity

Computing hypergeometric functions rigorously

We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions ${}_0F_1$, ${}_1F_1$, ${}_2F_1$ and ${}_2F_0$ (or the Kummer $U$-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function ${}_pF_q$ and computation of high-order parameter derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case E-G in section 8.5 (table 6, figure 2); adjusted paper siz

Multipath Parameter Estimation from OFDM Signals in Mobile Channels

We study multipath parameter estimation from orthogonal frequency division multiplex signals transmitted over doubly dispersive mobile radio channels. We are interested in cases where the transmission is long enough to suffer time selectivity, but short enough such that the time variation can be accurately modeled as depending only on per-tap linear phase variations due to Doppler effects. We therefore concentrate on the estimation of the complex gain, delay and Doppler offset of each tap of the multipath channel impulse response. We show that the frequency domain channel coefficients for an entire packet can be expressed as the superimposition of two-dimensional complex sinusoids. The maximum likelihood estimate requires solution of a multidimensional non-linear least squares problem, which is computationally infeasible in practice. We therefore propose a low complexity suboptimal solution based on iterative successive and parallel cancellation. First, initial delay/Doppler estimates are obtained via successive cancellation. These estimates are then refined using an iterative parallel cancellation procedure. We demonstrate via Monte Carlo simulations that the root mean squared error statistics of our estimator are very close to the Cramer-Rao lower bound of a single two-dimensional sinusoid in Gaussian noise.Comment: Submitted to IEEE Transactions on Wireless Communications (26 pages, 9 figures and 3 tables

Application of adaptive antenna techniques to future commercial satellite communication

The purpose of this contract was to identify the application of adaptive antenna technique in future operational commercial satellite communication systems and to quantify potential benefits. The contract consisted of two major subtasks. Task 1, Assessment of Future Commercial Satellite System Requirements, was generally referred to as the Adaptive section. Task 2 dealt with Pointing Error Compensation Study for a Multiple Scanning/Fixed Spot Beam Reflector Antenna System and was referred to as the reconfigurable system. Each of these tasks was further sub-divided into smaller subtasks. It should also be noted that the reconfigurable system is usually defined as an open-loop system while the adaptive system is a closed-loop system. The differences between the open- and closed-loop systems were defined. Both the adaptive and reconfigurable systems were explained and the potential applications of such systems were presented in the context of commercial communication satellite systems