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 ^[1] 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 ${\displaystyle I(q)}$,^[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

This is an brief outline of the method introduced by Otto Glatter.^[2] 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,${\displaystyle p_{i}(r)}$):

${\displaystyle p_{i}(r)=\sum _{i=1}^{N}c_{i}\phi _{i}(r),}$

(1)

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

${\displaystyle I(q)=4\pi \int _{0}^{\infty }p(r){\frac {\sin(qr)}{qr}}{\text{d}}r.}$

(2)

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:

${\displaystyle \chi ^{2}=\sum _{k=1}^{M}{\frac {[I_{experiment}(q_{k})-I_{fit}(q_{k})]^{2}}{\sigma ^{2}(q_{k})}}}$

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