Sorry Kay,
I completely agree with you and should have read your message more
carefully before jumping to conclusions. I thought you suggested the
ripples where not strong enough ... I'd better have my coffee now :)
Anyway, I don't think I wasted your time because your expanded
explanation of the convolution theorem on this particular case is very
useful as a reminder of this important concept.
Bart
Kay Diederichs wrote:
Bart Hazes schrieb:
...
W.r.t. Kay's reply I think the argument does not hold since it depends
on how badly the data is truncated. E.g. truncated near the limit of
diffraction will give few ripples whereas a data set truncated at
I/SigI of 5 will have much more servious effects.
Bart
Bart,
if you truncate at the limit of diffraction (i.e. where there is no more
signal) you will not get any ripple at all !
Of course, if you truncate at a resolution where there is significant
signal (and I do agree with you in that respect: many people truncate
their datasets at too low resolution) there _will_ be Fourier ripples.
However, a ripple is never as high than the peak itself.
To get a quantitative picture of the worst-case scenario, consider the
following: truncation means multiplication of the data with a Heaviside
function (that is 1 up to the chosen resolution limit, and 0 beyond). In
real space, this translates into a series of ripples, arising by
convolution of the true electron density with the Fourier transform of
the Heaviside function. The Fourier transform of a one-dimensional
Heaviside function is the function sin(x)/x . Convolution with sin(x)/x
has the effect of
a) broadening (or "smearing") the true electron density, resulting in a
low-resolution electron density map instead of the true one
b) adding ripples at certain distances (which can be calculated from the
resolution) around each peak. The first negative ripple has an absolute
value of less than 1/4 of the peak height, and the first positive ripple
about 1/8 of the peak height.
So in the worst case (one-dimensional truncation of data) my estimate of
12% was wrong - I estimated the height of the first positive ripple
whereas Klemens reported the first negative ripple!
On the other hand, if I remember correctly, the Fourier transform of the
3-dimensional Heaviside function (a filled sphere) is a Bessel function
that has ripples which (I think) are lower than those of the
one-dimensional Heaviside function. Surely somebody knows the function,
and its peak heights?
best,
Kay
--
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Bart Hazes (Assistant Professor)
Dept. of Medical Microbiology & Immunology
University of Alberta
1-15 Medical Sciences Building
Edmonton, Alberta
Canada, T6G 2H7
phone: 1-780-492-0042
fax: 1-780-492-7521
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