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In a Fourier transformation (FT), the Fourier transformed function ${\displaystyle {\hat {f}}(s)}$ is obtained from ${\displaystyle f(t)}$ by:

${\displaystyle {\hat {f}}(s)=\int _{-\infty }^{\infty }f(t)e^{-ist}dt}$

where ${\displaystyle i}$ is defined as ${\displaystyle i^{2}=-1}$. ${\displaystyle f(t)}$ can be obtained from ${\displaystyle {\hat {f}}(s)}$ by inverse FT:

${\displaystyle f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\hat {f}}(s)e^{ist}dt}$

${\displaystyle s}$ and ${\displaystyle t}$ are inverse variables, e.g. frequency and time.

Obtaining ${\displaystyle {\hat {f}}(s)}$ directly requires that ${\displaystyle f(t)}$ is well known from ${\displaystyle t=-\infty }$ to ${\displaystyle t=\infty }$, vice versa. In real experimental data this is rarely the case due to noise and limited measured range, say ${\displaystyle f(t)}$ is known from ${\displaystyle a>-\infty }$ to ${\displaystyle b<\infty }$. Performing a FT on ${\displaystyle f(t)}$ 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

In small-angle scattering on single molecules, an intensity ${\displaystyle I(\mathbf {r} )}$ is measured and is a function of the magnitude of the scattering vector ${\displaystyle q=|\mathbf {q} |=4\pi \sin(\theta )/\lambda }$, where ${\displaystyle 2\theta }$ is the scattered angle, and ${\displaystyle \lambda }$ is the wavelength of the incoming and scattered beam (elastic scattering). ${\displaystyle q}$ has units 1/length. ${\displaystyle I(q)}$ is related to the so-called pair distance distribution ${\displaystyle p(r)}$ via Fourier Transformation. ${\displaystyle p(r)}$ is a (scattering weighted) histogram of distances ${\displaystyle r}$ between pairs of atoms in the molecule. In one dimensions (${\displaystyle r}$ and ${\displaystyle q}$ are scalars), ${\displaystyle I(q)}$ and ${\displaystyle p(r)}$ are related by:

${\displaystyle I(q)=4\pi n\int _{-\infty }^{\infty }p(r)e^{-iqr\cos(\phi )}dr}$

${\displaystyle p(r)={\frac {1}{2\pi ^{2}n}}\int _{-\infty }^{\infty }{\hat {(}}qr)^{2}I(q)e^{-iqr\cos(\phi )}dq}$

where ${\displaystyle \phi }$ is the angle between ${\displaystyle \mathbf {q} }$ and ${\displaystyle \mathbf {r} }$, and ${\displaystyle n}$ is the number density of molecules in the measured sample. The sample is orientational averaged (denoted by ${\displaystyle \langle ..\rangle }$), and the Debye equation can thus be exploited to simplify the relations by

${\displaystyle \langle e^{-iqr\cos(\phi )}\rangle =\langle e^{iqr\cos(\phi )}\rangle ={\frac {\sin(qr)}{qr}}}$

In 1977 Glatter proposed an IFT method to obtain ${\displaystyle p(r)}$ form ${\displaystyle I(q)}$, and three years later, Moore introduced an alternative method. Others have later introduced alternative methods for IFT, and automatised the process

The Glatter method of IFT

This is an brief outline of the method introduced by Otto Glatter. For simplicity, we use ${\displaystyle n=1}$ in the following.

In indirect Fourier transformation, a guess on the largest distance in the particle ${\displaystyle D_{max}}$ is given, and an initial distance distribution function ${\displaystyle p_{i}(r)}$ is expressed as a sum of ${\displaystyle N}$ cubic spline functions ${\displaystyle \phi _{i}(r)}$ evenly distributed on the interval (0,${\displaystyle p_{i}(r)}$):

where ${\displaystyle c_{i}}$ are scalar coefficients. The relation between the scattering intensity ${\displaystyle I(q)}$ and the ${\displaystyle p(r)}$ is:

Inserting the expression for p_i(r) (1) into (2) and using that the transformation from ${\displaystyle p(r)}$ to ${\displaystyle I(q)}$ is linear gives:

${\displaystyle I(q)=4\pi \sum _{i=1}^{N}c_{i}\psi _{i}(q),}$

where ${\displaystyle \psi _{i}(q)}$ is given as:

${\displaystyle \psi _{i}(q)=\int _{0}^{\infty }\phi _{i}(r){\frac {\sin(qr)}{qr}}{\text{d}}r.}$

The ${\displaystyle c_{i}}$'s are unchanged under the linear Fourier transformation and can be fitted to data, thereby obtaining the coefficients ${\displaystyle c_{i}^{fit}}$. Inserting these new coefficients into the expression for ${\displaystyle p_{i}(r)}$ gives a final ${\displaystyle p_{f}(r)}$. The coefficients ${\displaystyle c_{i}^{fit}}$ are chosen to minimise the ${\displaystyle \chi ^{2}}$ of the fit, given by:

${\displaystyle \chi ^{2}=\sum _{k=1}^{M}{\frac {^{2}}{\sigma ^{2}(q_{k})}}}$

where ${\displaystyle M}$ is the number of datapoints and ${\displaystyle \sigma _{k}}$ is the standard deviations on data point ${\displaystyle k}$. The fitting problem is ill posed and a very oscillating function would give the lowest ${\displaystyle \chi ^{2}}$ despite being physically unrealistic. Therefore, a smoothness function ${\displaystyle S}$ is introduced:

${\displaystyle S=\sum _{i=1}^{N-1}(c_{i+1}-c_{i})^{2}}$.

The larger the oscillations, the higher ${\displaystyle S}$. Instead of minimizing ${\displaystyle \chi ^{2}}$, the Lagrangian ${\displaystyle L=\chi ^{2}+\alpha S}$ is minimized, where the Lagrange multiplier ${\displaystyle \alpha }$ is denoted the smoothness parameter. The method is indirect in the sense that the FT is done in several steps: ${\displaystyle p_{i}(r)\rightarrow {\text{fitting}}\rightarrow p_{f}(r)}$.

See also

• Frequency spectrum
• Least-squares spectral analysis

References

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. ^ 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.

• Fourier analysis