[Ifeffit] more fun with FFTs

[Scott Calvin]

First of all, thanks to Matt for the info--this seems to me to be a very
sensible way for Ifeffit to handle the issue.

In terms of Grant's comments, the interesting part to me (and I admit I am
getting esoteric here) is that zero-padding has the potential to change the
cutoff effects in some circumstances. If you're using a window that goes to
zero anyway, then I think it has exactly the effect that Grant describes.
But not all windows do this. As a particularly simple example, consider a
perfect cosine function over an integer number of periods, with no window.
Even though the transform is applied to a finite interval, the result is a
spike to within the resolution discussed in Grant's post. (As I recall, the
finite sampling also causes the transform to be periodic, but the period is
large enough to be disregarded in EXAFS analysis.) But now consider the
identical function over the identical interval with zero-padding. This
introduces a sharp discontinuity, and causes lots of sidebands on the
Fourier transform.

Of course, I chose a very special case. Windows reduce this effect
considerably, so I would certainly agree that zero-padding does not "botch
up" the data. But I wonder if some reduction in sidebands could be achieved
by, for example, using a window that goes to the average of the values of
the function at the two ends of the interval rather than to 0 (and
adjusting the windowing function accordingly). It's on my list of things to
try when I get the chance...I don't expect it will make a dramatic
difference, but it might make an incremental one. 

--Scott Calvin
Sarah Lawrence College

At 12:34 PM 6/8/2004 -0500, Grant wrote:
>You might ask whether zero padding botches up the data. It can be shown
>(I worked it out once) that zero padding is exactly equivalent to
>an interpolation of the unpadded data witha specific interpolation kernel
>(a sinc function or something as I recall). What happens is that as you
>pad with more zeros into a longer array, the sampling in r-space increases,
>and ultimately the curve looks just like what you would get with a continuous
>fourier transform (with cutoff effects from the finite k-space window).
>