Indirect Fourier transformation

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


 * $$ \hat f(s) = \int_{-\infty}^\infty f(t)e^{-ist}dt$$

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


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

$$s$$ and $$t$$ are inverse variables, e.g. frequency and time.

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


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


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

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


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

In 1977 Glatter proposed an IFT method to obtain $$ p(r) $$ form $$ I(q) $$, 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 $$n=1$$ in the following.

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

where $$c_i$$ are scalar coefficients. The relation between the scattering intensity $$I(q)$$ and the $$p(r)$$ is:

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


 * $$I(q) = 4\pi\sum_{i=1}^N c_i\psi_i(q),$$

where $$\psi_i(q)$$ is given as:


 * $$\psi_i(q)=\int_0^\infty\phi_i(r)\frac{\sin(qr)}{qr}\text{d}r.$$

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


 * $$ \chi^2 = \sum_{k=1}^{M}\frac{[I_{experiment}(q_k)-I_{fit}(q_k)]^2}{\sigma^2(q_k)}$$

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


 * $$ S = \sum_{i=1}^{N-1}(c_{i+1}-c_i)^2 $$.

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