Welch Apodization Function

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

$\begin{displaymath} A(x)=1-{x^2\over a^2}. \end{displaymath}$

$\displaystyle I(k)$	$\textstyle =$	$\displaystyle 2a\sqrt{2\pi}\, {J_{3/2}(2\pi ka)\over (2\pi ka)^{3/2}}$
	$\textstyle =$	$\displaystyle a{\sin(2\pi ka)-2\pi ak\cos(2\pi ak)\over 2a^3k^3\pi^3},$

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

times the peak, and maximum Positive sidelobe of 0.356044 times the peak.

References

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.