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
--
=============================================================================
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
=============================================================================
|