Gaussian Fourier transform approximations?
- To: mathgroup at smc.vnet.net
- Subject: [mg120901] Gaussian Fourier transform approximations?
- From: AES <siegman at stanford.edu>
- Date: Sun, 14 Aug 2011 20:19:23 -0400 (EDT)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
[This is really a numerical analysis query, but since the results will
be evaluated using Mathematica, perhaps I can post it here.]
I want to develop a closed-form approximation for the inverse
transform f[x] of a gaussian spectrum g[s] = Exp[ - s^2 / w^2 ]
multiplied by a transfer function h[s] which is a reasonably smooth
function about s=0, and use it to make reasonably accurate numerical
plots of f[x] . (Think of f[x] as a spatial profile, g[s] as its
spatial frequency transform.)
That is, I want a closed form approximation to
f[x] = Integrate[ h[s] Exp[ - (s/w)^2 - I s x ],
{x, -Infinity, Infinity}]
where s, x and w are real; and f and h potentially complex-valued.
Evidently I can approximate the transfer function by expanding it as
h[s] ? h0 Exp[ h1 s + h2 s^2] , in which case the integral above
becomes closed form, giving me just a single complex gaussian with a
somewhat complicated argument.
Or I can approximate h[s] by h[s] ? h0 (1 + h11 s + h22 s^2 ) in
which case I get three closed-form integrals involving a gaussian
times Hermite polynomials of order n=0 up to n=2.
Either of these should be adequately fast for numerical calculations,
but I somehow think the second should be more accurate (and maybe also
give more physical insight).
Comments?
- Follow-Ups:
- Re: Gaussian Fourier transform approximations?
- From: Daniel Lichtblau <danl@wolfram.com>
- Re: Gaussian Fourier transform approximations?