Indirect Fourier transformation
Introduction 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 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  can thus be exploited to simplify the relations by:In 1977 Glatter proposed an IFT method to obtain  form , 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  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,):where  are scalar coefficients. The relation between the scattering intensity  and the  is: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 * Frequency spectrum* Least-squares spectral analysis
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