# [phenixbb] lower limits on grid spacing/binning

Peter Zwart PHZwart at lbl.gov
Mon Oct 29 19:10:25 PDT 2007

```There was a good answer to this question on  the CCP4BB recently (okt
11, Dale Tronrud), see below.

The FFT chapter in the international tables by Bricogne is of course
also a good thing to read.

HTH

Peter

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The choice of sampling rates for maps to be Fourier transformed is
a deep topic.  The mathematical law is that you have to sample the
map at, at least, twice the frequency of the highest Fourier component
in the map.  This is, unfortunately, often misinterpreted as twice
the frequency of the highest component you are interested in.

The fact that you are interested in, say, 2A structure factors
has nothing to do with the calculation of the Fourier transform of
your map.  All that matters is the frequencies that were present in
your map before you sampled it on the grid.

Ten Eyck, (1977) Acta Cryst A33, 800-804 has a discussion of this
and provided the classic solution to this problem when the map to be
transformed is a calculated electron density map.  I presume you have
an NCS averaged map and the required interpolations introduce needs
of their own that are significant.  Gerard Bricogne has written on that
topic, also back in the 1970's, but I don't have the reference at hand.

The manual for your NCS averaging program should give you guidance
on the choice of sampling rate based on its interpolation method.  If
you are not even sampling at twice the resolution you are interested
in you are sampling way too coarsely.

All FFT based structure factor programs require that the sampling
rates along each axis be even.  They may have other required factors
depending on the space group, but they will be happy to inform you
if you make a choice it doesn't like.  They are also more efficient
when the prime factors of the sampling rates are small numbers.  Try
to stick with multiples of 2,3, and 5 if possible.

Since the program has no way of knowing the highest resolution
component actually in the map before you sampled it on your grid,
it assumes that the map contains no components of higher resolution
then you asked it to produce.  All FFT programs will fail if you
sample your map courser than twice that frequency, as SFALL did for
you.

That does NOT mean that twice the frequency you are interested in
and if that doesn't tell you, complain the the program's author, and
read Gerard's papers on the matter.  NCS averaging a map that is only
sampled at twice the rate you are interested in will not be a useful

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2007/10/29, Terry Lang <terry at lego.berkeley.edu>:
> Hey Everyone,
>
>       I have been told by several crystallographers that the general
> rule of thumb for grid spacing when generating a map  is one third the
> resolution (eg If your crystal structure has a resolution of 1.2 A, your
> grid spacing should be 0.4).  Is there a mathematical reason why this
> ratio should be maintained or is this simply a holdover from slower
> computers with lower memory?  More importantly for my application, is
> there a lower limit on how fine the grid spacing can be?  In other
> words, is there a lower limit at which one is oversampling the data for
> a particular resolution?  Any information and/or references would be
> greatly appreciated.
>
> Sincerely,
> Terry
>
> --
> P. Therese Lang
> Post Doc
> Alber Lab, UC Berkeley
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>

```