> Assume you have a one dimensional crystal with a 10 Angstrom repeat.
> Someone has told you the value of the electron density at 10 equally
> spaced points in this little unit cell, but you know nothing about the
> value of the function between those points. I could spend all night
> with a crayon drawing different functions that exactly hit all 10 points -
> They are infinite in number and each one has a different set of Fourier
> coefficients. How can I control this chaos and come up with a simple
> description, particularly of the reciprocal space view of these 10
> points?
>
> The Nyquist-Shannon sampling theorem simply means that if we assume
> that all Fourier coefficient of wave length shorter than 2 Angstrom/cycle
> (twice our sampling rate) are defined equal to zero we get only one
> function that will hit all ten points exactly. If we say that the 2
> A/cycle
> reflection has to be zero as well, there are no functions that hit all ten
> points (except for special cases) but if we allow the next reflection (the
> h=6 or 1.67 A/cycle wave) to be non-zero we are back to an infinite number
> of solutions.
>
Dear Dale,
I'm not sure that this is true. Let's assume that the Fourier transform of
the continuous function is band-limited, and the real-space sampling rate
is over twice the Shannon frequency. There are at least *two* different
mathematical functions that pass precisely through your sampled values:
1. the original continuous function, and
2. the sampled values themselves.
One could perfectly reconstruct the original continuous function using a
low pass "top-hat" filter of width +/-1/2q about the origin in reciprocal
space (where q is the real-space sampling interval), thus cutting out the
higher resolution "ghosts". In real space, this corresponds to convolution
of your samples with a sinc function (sinc(x/(q/2)) up to a multiplicative
constant). But you could also filter your samples using wider top hats to
include higher resolution ghosts (between +/-(2n+1)/2q, where n is an
integer), corresopnding to narrower sinc functions in the real-space
interplation and therefore resulting in different continuous functions.
All these functions will pass though the initial set of sampled values*,
but will differ inbetween. For example, in the limit of making your
reciprocal space top-hat filter very wide indeed, your sinc function in
the real-space interpolation will be delta function-like and will give you
a reconstructed continuous function that will look almost like your
sequence of sampled values. So I think that even if your function is
band-limited and is sampled at a rate greater than twice the Nyquist
frequency, there are still an infinite number of functions that can be
derived from the samples and that will pass through them.
Am I wrong?
Joe
*The transforms of these continuous functions will have local
translational symmetry in reciprocal space that is derived from the
periodicity of the transform of the original unfiltered samples. If you
now sample these functions at the same positions as with the original
function, their transform will be identical to the transform of the
original samples (because the periodicity imposed by the sampling will be
in register with the translational symmetry mentioned above). So the
values obtained from sampling functions derived from the different
interpolation schemes must be identical to the original set of samples.
> That's all it is - If you assume that all the Fourier coefficients of
> higher resolution than twice your sampling rate are zero you are
> guaranteed
> one, and only one, set of Fourier coefficients that hit the points and the
> Discrete Fourier Transform (probably via a FFT) will calculate that set
> for
> you.
>
> As usual, if your assumption is wrong you will not get the right
> answer.
> If you have a function that really has a non-zero 1.67 A/cycle Fourier
> coefficient but you sample your function at 10 points and calculate a
> FFT you will get a set of coefficients that hit the 10 points exactly
> (when back transformed) but they will not be equal to "true" values.
>
> The overlapping spheres that Gerard Bricogne described are simply the
> way of calculating the manor in which the coefficients are distorted by
> this bad assumption. Ten Eyck, L. F. (1977). Acta Cryst. A33, 486-492
> has an excellent description.
>
> If you are certain that your function has no Fourier components higher
> than your sampling rate can support then the FFT is your friend. If your
> function has high resolution components and you don't sample it finely
> enough then the FFT will give you an answer, but it won't be the correct
> answer. The answer will exactly fit the points you sampled but it will
> not correctly predict the function's behavior between the points.
>
> The principal situations where this is a problem are:
>
> Calculating structure factors (Fcalc) from a model electron density map.
> Calculating gradients using the Agarwal method.
> Phase extension via ncs map averaging (including cross-crystal averaging).
> Phase extension via solvent flattening (depending on how you do it).
>
> Thank you for your time,
> Dale Tronrud
>
> On 4/15/2011 6:34 AM, Dirk Kostrewa wrote:
>> Dear colleagues of the CCP4BB,
>>
>> many thanks for all your replies - I really got lost in the trees (or
>> wood?) and you helped me out with all your kind responses!
>>
>> I should really leave for the weekend ...
>>
>> Have a nice weekend, too!
>>
>> Best regards,
>>
>> Dirk.
>>
>> Am 15.04.11 13:20, schrieb Dirk Kostrewa:
>>> Dear colleagues,
>>>
>>> I just stumbled across a simple question and a seeming paradox for me
>>> in crystallography, that puzzles me. Maybe, it is also
>>> interesting for you.
>>>
>>> The simple question is: is the discrete sampling of the continuous
>>> molecular Fourier transform imposed by the crystal lattice
>>> sufficient to get the desired information at a given resolution?
>>>
>>> From my old lectures in Biophysics at the University, I know it has
>>> been theoretically proven, but I don't recall the argument,
>>> anymore. I looked into a couple of crystallography books and I couldn't
>>> find the answer in any of those. Maybe, you can help me out.
>>>
>>> Let's do a simple gedankenexperiment/thought experiment in the
>>> 1-dimensional crystal case with unit cell length a, and desired
>>> information at resolution d.
>>>
>>> According to Braggs law, the resolution for a first order reflection
>>> (n=1) is:
>>>
>>> 1/d = 2*sin(theta)/lambda
>>>
>>> with 2*sin(theta)/lambda being the length of the scattering vector |S|,
>>> which gives:
>>>
>>> 1/d = |S|
>>>
>>> In the 1-dimensional crystal, we sample the continuous molecular
>>> transform at discrete reciprocal lattice points according to the
>>> von Laue condition, S*a = h, which gives |S| = h/a here. In other
>>> words, the unit cell with length a is subdivided into h evenly
>>> spaced crystallographic planes with distance d = a/h.
>>>
>>> Now, the discrete sampling by the crystallographic planes a/h is only
>>> 1x the resolution d. According to the Nyquist-Shannon
>>> sampling theorem in Fourier transformation, in order to get a desired
>>> information at a given frequency, we would need a discrete
>>> sampling frequency of *twice* that frequency (the Nyquist frequency).
>>>
>>> In crystallography, this Nyquist frequency is also used, for instance,
>>> in the calculation of electron density maps on a discrete
>>> grid, where the grid spacing for an electron density map at resolution
>>> d should be <= d/2. For calculating that electron density
>>> map by Fourier transformation, all coefficients from -h to +h would be
>>> used, which gives twice the number of Fourier coefficients,
>>> but the underlying sampling of the unit cell along a with maximum index
>>> |h| is still only a/h!
>>>
>>> This leads to my seeming paradox: according to Braggs law and the von
>>> Laue conditions, I get the information at resolution d
>>> already with a 1x sampling a/h, but according to the Nyquist-Shannon
>>> sampling theory, I would need a 2x sampling a/(2h).
>>>
>>> So what is the argument again, that the sampling of the continuous
>>> molecular transform imposed by the crystal lattice is
>>> sufficient to get the desired information at a given resolution?
>>>
>>> I would be very grateful for your help!
>>>
>>> Best regards,
>>>
>>> Dirk.
>>>
>>
>
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