Indirect Fourier transformation

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In a Fourier transformation (FT), the Fourier transformed function is obtained from by:

where is defined as . can be obtained from by inverse FT:

and are inverse variables, e.g. frequency and time.

Obtaining directly requires that is well known from to , vice versa. In real experimental data this is rarely the case due to noise and limited measured range, say is known from to . Performing a FT on in the limited range may lead to systematic errors and overfitting.

An indirect Fourier transform (IFT) is a solution to this problem.

Indirect Fourier transformation in small-angle scattering

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In small-angle scattering on single molecules, an intensity   is measured and is a function of the magnitude of the scattering vector  , where   is the scattered angle, and   is the wavelength of the incoming and scattered beam (elastic scattering).   has units 1/length.   is related to the so-called pair distance distribution   via Fourier Transformation.   is a (scattering weighted) histogram of distances   between pairs of atoms in the molecule. In one dimensions (  and   are scalars),   and   are related by:

 
 

where   is the angle between   and  , and   is the number density of molecules in the measured sample. The sample is orientational averaged (denoted by  ), and the Debye equation [1] can thus be exploited to simplify the relations by

 

In 1977 Glatter proposed an IFT method to obtain   form  ,[2] and three years later, Moore introduced an alternative method.[3] Others have later introduced alternative methods for IFT,[4] and automatised the process [5][6]

The Glatter method of IFT

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This is an brief outline of the method introduced by Otto Glatter.[2] For simplicity, we use   in the following.

In indirect Fourier transformation, a guess on the largest distance in the particle   is given, and an initial distance distribution function   is expressed as a sum of   cubic spline functions   evenly distributed on the interval (0, ):

 
(1)

where   are scalar coefficients. The relation between the scattering intensity   and the   is:

 
(2)

Inserting the expression for pi(r) (1) into (2) and using that the transformation from   to   is linear gives:

 

where   is given as:

 

The  's are unchanged under the linear Fourier transformation and can be fitted to data, thereby obtaining the coefficients  . Inserting these new coefficients into the expression for   gives a final  . The coefficients   are chosen to minimise the   of the fit, given by:

 

where   is the number of datapoints and   is the standard deviations on data point  . The fitting problem is ill posed and a very oscillating function would give the lowest   despite being physically unrealistic. Therefore, a smoothness function   is introduced:

 .

The larger the oscillations, the higher  . Instead of minimizing  , the Lagrangian   is minimized, where the Lagrange multiplier   is denoted the smoothness parameter. The method is indirect in the sense that the FT is done in several steps:  .

See also

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References

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  1. ^ Scardi, P.; Billinge, S. J. L.; Neder, R.; Cervellino, A. (2016). "Celebrating 100 years of the Debye scattering equation". Acta Crystallogr A. 72 (6): 589–590. doi:10.1107/S2053273316015680. hdl:11572/171102. PMID 27809198.
  2. ^ a b O. Glatter (1977). "A new method for the evaluation of small-angle scattering data". Journal of Applied Crystallography. 10 (5): 415–421. doi:10.1107/s0021889877013879.
  3. ^ P.B. Moore (1980). "Small-angle scattering. Information content and error analysis". Journal of Applied Crystallography. 13 (2): 168–175. doi:10.1107/s002188988001179x.
  4. ^ S. Hansen, J.S. Pedersen (1991). "A Comparison of Three Different Methods for Analysing Small-Angle Scattering Data". Journal of Applied Crystallography. 24 (5): 541–548. doi:10.1107/s0021889890013322.
  5. ^ B. Vestergaard and S. Hansen (2006). "Application of Bayesian analysis to indirect Fourier transformation in small-angle scattering". Journal of Applied Crystallography. 39 (6): 797–804. doi:10.1107/S0021889806035291.
  6. ^ Petoukhov M. V. and Franke D. and Shkumatov A. V. and Tria G. and Kikhney A. G. and Gajda M. and Gorba C. and Mertens H. D. T. and Konarev P. V. and Svergun D. I. (2012). "New developments in the ATSAS program package for small-angle scattering data analysis". Journal of Applied Crystallography. 45 (2): 342–350. doi:10.1107/S0021889812007662. PMC 4233345. PMID 25484842.