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Welch Apodization Function

\begin{figure}\begin{center}\BoxedEPSF{Welch.epsf scaled 800}\end{center}\end{figure}

The Apodization Function

A(x)=1-{x^2\over a^2}.

Its Full Width at Half Maximum is $\sqrt{2}\,a$. Its Instrument Function is
$\displaystyle I(k)$ $\textstyle =$ $\displaystyle 2a\sqrt{2\pi}\, {J_{3/2}(2\pi ka)\over (2\pi ka)^{3/2}}$  
  $\textstyle =$ $\displaystyle a{\sin(2\pi ka)-2\pi ak\cos(2\pi ak)\over 2a^3k^3\pi^3},$  

where $J_\nu(z)$ is a Bessel Function of the First Kind. It has a width of 1.59044, a maximum of ${\textstyle{4\over 3}}$, maximum Negative sidelobe of $-0.0861713$ times the peak, and maximum Positive sidelobe of 0.356044 times the peak.

See also Apodization Function, Instrument Function


Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, p. 547, 1992.

© 1996-9 Eric W. Weisstein