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# The Accurate Numerical Inversion of Laplace Transforms

IMA Journal of Applied Mathematics 01/1979; 23(1). DOI: 10.1093/imamat/23.1.97
Source: OAI

ABSTRACT

Inversion of almost arbitrary Laplace transforms is effected by trapezoidal integration along a special contour. The number n of points to be used is one of several parameters, in most cases yielding absolute errors of order 10-7 for n = 10, 10-11 for n = 20, 10-23 for n = 40 (with double precision working), and so on, for all values of the argument from 0+ up to some large maximum. The extreme accuracy of which the method is capable means that it has many possible applications of various kinds, and some of these are indicated.

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• "An analytical expression for F T |H (λ) appears intractable. Moreover, a direct computation of the above integral over B is impractical due to possible oscillations of e sλ (where s = c + ȷw) as |w| → ∞, and thus Talbot suggested a deformation of the contour [21] for evaluating such integrals. To improve the numerical stability of such evaluation in a fixed-precision computing environment, a multi-precision method termed the " fixed Talbot method " was proposed in [22], which suggests an alternative form of the integral as "
##### Article: Performance of -Norm Detector in AWGN, Fading, and Diversity Reception
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ABSTRACT: Performance analysis of the \$p\$-norm detector to date has been limited to ad hoc approximations, nonfading channels, and Rayleigh fading. To overcome these limitations, we develop several analytical/numerical solutions for detection probability \$P_{d}\$ and false alarm probability \$P_{f}\$, which are necessary to specify the receiver operating characteristic (ROC) curves of the \$p\$-norm detector. First, for nonfading channels (additive white Gaussian noise (AWGN) only), the moment-generating function (mgf) of the decision variable is derived in two forms: 1) closed form for even integer \$p\$ and 2) series form for arbitrary \$p\$. To evaluate \$P_{d}\$ and \$P_{f}\$, a numerical method utilizing the Talbot inversion is developed for case 1, and an infinite series expansion with convergence acceleration based on the \$epsilon\$-algorithm is derived for case 2. As an alternative to mgf-based analysis, a Laguerre polynomial series is also used to derive new \$P_{d}\$ and \$P_{f}\$ approximations. Second, series-form mgf-based \$P_{d}\$ expressions are derived for \$kappahbox{-}mu\$ and \$alphahbox{-}mu\$ fading channels. Third, for antenna diversity reception, new \$p\$-law combining (pLC) and \$p\$-law selection (pLS) schemes are proposed. The performance of these combiners with the \$p\$-norm detector is derived for Nakagami- \$m\$ fading and is compared with that of the classical maximal ratio combining (MRC) and selection combining (SC). Interestingly, both pLC and pLS perform similarly to SC at low signal-to-noise ratio (SNR) but outperform it at relatively high SNR, with pLC performing closer to the optimal MRC. Numerical results are presented to verify the derived results and to provide further insights.
IEEE Transactions on Vehicular Technology 09/2014; 63(7):3209-3222. DOI:10.1109/TVT.2014.2298395 · 1.98 Impact Factor
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• "In this section, we approximate J ε (t) on an interval [t 0 , t 1 ] following Talbot's approach, [11]. More precisely, we use two parametrized Bromwich contours proposed in [12], either the parabola γ(u) = µ(˙ ιu + 1) 2 + β, or the hyperbola γ(u) = µ(1 + sin(˙ ιu − α)) + β where u ∈] − ∞, ∞[, µ > 0 regulates the width of the contours , β determines their foci, and α defines the hyperbola's asymptotic angle. "
##### Article: Time-domain simulation of functions and dynamical systems of Bessel type
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ABSTRACT: Two methods are investigated for the time-domain si- mulation of functions and dynamical systems of Bessel type, involved in wave propagation (see e.g. (1), (8), (2)). Both are based on complex analysis and lead to finite- dimensional approximations. The first method relies on optimized parametric contours and provides asymptotic convergence rates. The second is based on cuts and in- tegral representations, whose approximations prove effi- cient, even at low orders, using ad hoc frequency criteria. 1 Model under study For ℜe(s) > −", let c J"(s) = ((s + ")2 + 1)−1/2 be the Laplace transform of J"(t) = e−"t J0(t) for t ≥ 0 (cf. (3)). The general formula can be derived: J"(t) = 1
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• "The most common approach seems to be the one based on the FFT ([17] Chapter 7.5), already employed in [9] in the case of α = 1/2. More recently, in [21] the authors introduce a new approach based on the computation of the Laplace transform representing the formal power series of the unknown coefficients, using a quadrature rule on Talbot contours (see [24]) and hyperbolas. Without the explicit computation of R m (A p )e 1 , the definition of the coefficients γ j , η j , j = 1, ..., m, in (5), allows to construct the polynomials p m and q m of degree m such that R m (z) = p m (z)/q m (z) and then A α p ≈ [q m (A p )] −1 p m (A p ). (6) "
##### Article: Numerical approximation to the fractional derivative operator
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ABSTRACT: In this paper we consider the numerical approximation of A α by con-tour integral. We are mainly interested to the case of A representing the discretization of the first derivative by means of a BDF formula, and 0 < α < 1. The computation of the contour integral yields a rational approximation to A α which can be used to define k-step formulas for the numerical integration of Fractional Differential Equations.
Numerische Mathematik 07/2013; 127(3). DOI:10.1007/s00211-013-0596-7 · 1.61 Impact Factor