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In a [[Fourier transformation]] (FT), the Fourier transformed function <math>\hat f(s)</math> is obtained from <math> f(t)</math> by:

:<math> \hat f(s) = \int_{-\infty}^\infty f(t)e^{-ist}dt</math>

where <math>i</math> is defined as <math>i^2=-1</math>. <math>f(t)</math> can be obtained from <math>\hat f(s)</math> by inverse FT:

:<math> f(t) = \frac{1}{2\pi}\int_{-\infty}^\infty \hat f(s)e^{ist}dt</math>

<math>s</math> and <math>t</math> are inverse variables, e.g. frequency and time.

Obtaining <math>\hat f(s)</math> directly requires that <math>f(t)</math> is well known from <math>t=-\infty</math> to <math>t=\infty</math>, vice versa. In real experimental data this is rarely the case due to noise and limited measured range, say <math>f(t)</math> is known from <math>a>-\infty</math> to <math>b<\infty</math>. Performing a FT on <math>f(t)</math> 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 <math>I(\mathbf{r})</math> is measured and is a function of the magnitude of the scattering vector <math>q = |\mathbf{q}| = 4\pi \sin(\theta)/\lambda</math>, where <math> 2\theta </math> is the scattered angle, and <math>\lambda</math> is the wavelength of the incoming and scattered beam ([[elastic scattering]]). <math>q</math> has units 1/length. <math>I(q)</math> is related to the so-called [[pair distribution function|pair distance distribution]] <math> p(r) </math> via Fourier Transformation. <math> p(r) </math> is a (scattering weighted) histogram of distances <math> r </math> between pairs of atoms in the molecule. In one dimensions (<math> r </math> and <math> q </math> are [[Scalar (mathematics)|scalars]]), <math> I(q) </math> and <math> p(r) </math> are related by:

:<math> I(q) = 4\pi n\int_{-\infty}^\infty p(r)e^{-iqr\cos(\phi)}dr</math>

:<math> p(r) = \frac{1}{2\pi^2n}\int_{-\infty}^\infty\hat (qr)^2 I(q)e^{-iqr\cos(\phi)}dq</math>

where <math> \phi </math> is the angle between <math> \mathbf{q} </math> and <math> \mathbf{r} </math>, and <math> n </math> is the number density of molecules in the measured sample. The sample is orientational averaged (denoted by <math> \langle .. \rangle </math>), and the Debye equation <ref>{{cite journal | first1 = P. | last1 = Scardi | first2 = S. J. L. | last2 = Billinge | first3 = R. | last3 = Neder | first4 = A. | last4 = Cervellino | title = Celebrating 100 years of the Debye scattering equation | journal=Acta Crystallogr A | year=2016| volume=72 | issue = 6 | pages=589–590 | doi=10.1107/S2053273316015680| pmid = 27809198 | doi-access=free | hdl = 11572/171102 | hdl-access = free }}</ref> can thus be exploited to simplify the relations by

:<math> \langle e^{-iqr\cos(\phi)}\rangle = \langle e^{iqr\cos(\phi)}\rangle = \frac{\sin(qr)}{qr} </math>

In 1977 Glatter proposed an IFT method to obtain <math> p(r) </math> form <math> I(q) </math>,<ref name=Glatter1977>{{cite journal |author=O. Glatter |title=A new method for the evaluation of small-angle scattering data |journal=Journal of Applied Crystallography |year=1977 |volume=10 |issue=5 |pages=415–421 |doi=10.1107/s0021889877013879}}</ref> and three years later, Moore introduced an alternative method.<ref>{{cite journal|year=1980|title=Small-angle scattering. Information content and error analysis|journal=Journal of Applied Crystallography|volume=13|issue=2|pages=168–175 | doi=10.1107/s002188988001179x | author=P.B. Moore}}</ref> Others have later introduced alternative methods for IFT,<ref name=hansenpedersen>{{cite journal |author=S. Hansen, J.S. Pedersen |title = A Comparison of Three Different Methods for Analysing Small-Angle Scattering Data |journal=Journal of Applied Crystallography |year=1991 |volume=24 |issue = 5 |pages=541–548 |doi=10.1107/s0021889890013322|doi-access=free }}</ref> and automatised the process <ref name=ifc>{{cite journal |author=B. Vestergaard and S. Hansen| title=Application of Bayesian analysis to indirect Fourier transformation in small-angle scattering|journal=Journal of Applied Crystallography|year=2006| volume=39| issue=6|pages=797&ndash;804|doi=10.1107/S0021889806035291}}</ref><ref name="autognom">{{cite journal|year=2012|title=New developments in the ATSAS program package for small-angle scattering data analysis|journal=Journal of Applied Crystallography | volume=45 |issue=2| pages=342&ndash;350 | doi=10.1107/S0021889812007662 | author=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.| pmc=4233345 | pmid=25484842}}</ref>

== The Glatter method of IFT==
This is an brief outline of the method introduced by Otto Glatter.<ref name=Glatter1977/> For simplicity, we use <math>n=1</math> in the following.

In indirect Fourier transformation, a guess on the largest distance in the particle <math> D_{max}</math> is given, and an initial distance distribution function <math>p_i(r)</math> is expressed as a sum of <math>N</math> cubic [[Spline (mathematics)|spline functions]] <math>\phi_i(r)</math> evenly distributed on the interval (0,<math>p_i(r)</math>):

{{NumBlk|:|
:<math>p_i(r) = \sum_{i=1}^N c_i\phi_i(r), </math>
|{{EquationRef|1}}}}

where <math>c_i</math> are [[Scalar (mathematics)|scalar]] coefficients. The relation between the scattering intensity <math>I(q)</math> and the <math>p(r)</math> is:

{{NumBlk|:|
:<math>I(q) = 4\pi\int_0^\infty p(r)\frac{\sin(qr)}{qr}\text{d}r.</math>
|{{EquationRef|2}}}}

Inserting the expression for ''p<sub>i</sub>(r)'' (1) into (2) and using that the transformation from <math>p(r)</math> to <math>I(q)</math> is linear gives:

:<math>I(q) = 4\pi\sum_{i=1}^N c_i\psi_i(q),</math>

where <math>\psi_i(q)</math> is given as:

:<math>\psi_i(q)=\int_0^\infty\phi_i(r)\frac{\sin(qr)}{qr}\text{d}r.</math>

The <math> c_i </math>'s are unchanged under the linear Fourier transformation and can be fitted to data, thereby obtaining the coefficients <math> c_i^{fit} </math>. Inserting these new coefficients into the expression for <math>p_i(r)</math> gives a final <math>p_f(r)</math>. The coefficients <math> c_i^{fit} </math> are chosen to minimise the <math> \chi^2 </math> of the fit, given by:

:<math> \chi^2 = \sum_{k=1}^{M}\frac{[I_{experiment}(q_k)-I_{fit}(q_k)]^2}{\sigma^2(q_k)}</math>

where <math>M</math> is the number of datapoints and <math> \sigma_k </math> is the standard deviations on data point <math>k</math>. The fitting problem is [[Well-posed problem|ill posed]] and a very oscillating function would give the lowest <math> \chi^2 </math> despite being physically unrealistic. Therefore, a smoothness function <math>S</math> is introduced:

:<math> S = \sum_{i=1}^{N-1}(c_{i+1}-c_i)^2 </math>.

The larger the oscillations, the higher <math>S</math>. Instead of minimizing <math> \chi^2 </math>, the [[Lagrange multipliers|Lagrangian]] <math> L = \chi^2 + \alpha S </math> is minimized, where the [[Lagrange multipliers|Lagrange multiplier]] <math> \alpha </math> is denoted the smoothness parameter. 
The method is indirect in the sense that the FT is done in several steps: <math> p_i(r) \rightarrow \text{fitting} \rightarrow p_f(r) </math>.

==See also==
* [[Frequency spectrum]]
* [[Least-squares spectral analysis]]

==References==
<references/>

{{DEFAULTSORT:Indirect Fourier Transform}}
[[Category:Fourier analysis]]