Evaluation of Integrals and Sums Involving (sin(Mx)/sin(X))n

Abstract

The response of equispaced arrays, either linear, planar, or volumetric, to distributed spatial fields, typically encounters, integrals which involve the kernel sin(Mx)/sin(x) or its square. Since this kernel oscillates rather fast with x for large M and does not decay with x, numerical integration of such functions can be very time consuming. By resorting to Parseval's theorem, such integrals can be significantly simplified, requiring only the Fourier transform of the complementary part of the integrand. This procedure is investigated and applied to several typical examples; programs for the examples are also included. Keywords: Mathematical models, Mathematical equations, Integration, Summation, Sinc function, Array response, Fourier transform, Parseval's theorem, Fast Fourier transform, Equispaced array.

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Document Details

Document Type
Technical Report
Publication Date
May 18, 1990
Accession Number
ADA224906

Entities

People

  • Albert H. Nuttall

Organizations

  • Naval Underwater Systems Center

Tags

Communities of Interest

  • Air Platforms
  • C4I

DTIC Thesaurus Topics

  • Autocorrelation
  • Availability
  • Boundary Layer
  • Convolution
  • Delta Functions
  • Distribution Functions
  • Equations
  • Fast Fourier Transforms
  • Frequency
  • Integrals
  • Intervals
  • Layers
  • Military Research
  • Numerical Integration
  • Test And Evaluation
  • Turbulent Boundary Layer
  • Two Dimensional

Fields of Study

  • Mathematics

Readers

  • Approximation Theory.
  • Calculus or Mathematical Analysis