Topic
Fourier series
About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.
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TL;DR: A solution for the stationary queue length distribution that does not depend on roots is presented, and is well-suited for determining tail probabilities up to a high accuracy, as demonstrated by some numerical examples.
Abstract: In commonly used root-finding approaches for the discrete-time bulk service queue, the stationary queue length distribution follows from the roots inside or outside the unit circle of a characteristic equation. We present analytic representations of these roots in the form of sample values of periodic functions with analytically given Fourier series coefficients, making these approaches more transparent and explicit. The resulting computational scheme is easy to implement and numerically stable. We also discuss a method to determine the roots by applying successive substitutions to a fixed point equation. We outline under which conditions this method works, and compare these conditions with those needed for the Fourier series representation. Finally, we present a solution for the stationary queue length distribution that does not depend on roots. This solution is explicit and well-suited for determining tail probabilities up to a high accuracy, as demonstrated by some numerical examples.
66 citations
TL;DR: Lattice analogs of fractional differential equations that include suggested lattice differential and integral operators can serve as an important element of microscopic approach to nonlocal continuum models in mechanics and physics.
66 citations
TL;DR: In this article, the backscatter from conducting and from dielectric spheres is considered in the time domain, and the resulting response is examined as a function of time, and individually resolvable returns are compared with some approximate theories.
Abstract: The backscatter from conducting and from dielectric spheres is considered in the time domain. Utilizing rigorously computed values of the amplitude and phase of the continuous wave backscatter, short pulses of electromagnetic waves are synthesized by Fourier series. The resultant response is examined as a function of time, and the individually resolvable returns are compared with some approximate theories.
66 citations
01 Mar 1974
TL;DR: The principal aim of the paper is to investigate the truncation and roundoff errors associated with the use of Fourier and of Walsh series.
Abstract: In most of the applications contemplated for Walsh functions these binary waveforms would replace the more usual sinusoids, as the fast-Walsh-transform algorithm appears to make them very attractive for many kinds of signal processing. This paper begins with a brief review of the characteristics of Walsh functions and of their applications. Some old and some new interrelations are presented between sinusoids and Walsh functions, but the principal aim of the paper is to investigate the truncation and roundoff errors associated with the use of Fourier and of Walsh series. By employing simplifying approximations it is found that, for long samples of smooth signals, far more terms are required in the Walsh-series representation and greater accuracy is required of their coefficients for a given rms total error. Even for discontinuous signals the Walsh series may require substantially more terms, thus counterbalancing the computational advantage of the fast Walsh transform. This relative inefficiency of the Walsh-series representation of long waveforms may explain why it has not proven particularly effective in applications.
66 citations
TL;DR: In this paper, the authors derived expressions for the external field produced by a slot of arbitrary shape in the wall of a circular wave guide (of infinite extent and infinite conductivity), the tangential components of the electric field in the slot being assumed to have been prescribed.
Abstract: Expressions are derived for the external field produced by a slot of arbitrary shape in the wall of a circular wave guide (of infinite extent and infinite conductivity), the tangential components of the electric field in the slot being assumed to have been prescribed. This is accomplished by matching a Fourier representation of the external field, built up by superposition of basic sets of cylindrical waves, to a Fourier expansion of the prescribed field over the surface of the cylindrical wave guide. The far‐zone field is obtained by applying the method of steepest descent to the Fourier integrals that constitute the coefficients in the expansion for the external field. The results satisfy the radiation conditions for far‐zone fields.
66 citations