Indirect Fourier Transform

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

:$\hat f(s) = \int_^\infty f(t)e^dt$

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

:$f(t) = \frac\int_^\infty \hat f(s)e^dt$

$s$ 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 $I(\mathbf)$ is measured and is a function of the magnitude of the scattering vector $q = , \mathbf, = 4\pi \sin(\theta)/\lambda$, 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:

:$I(q) = 4\pi n\int_^\infty p(r)e^dr$

:$p(r) = \frac\int_^\infty\hat (qr)^2 I(q)e^dq$

where $\phi$ is the angle between $\mathbf$ and $\mathbf$, 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

:$\langle e^\rangle = \langle e^\rangle = \frac$

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 a 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 $D_$ 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 ''p_i(r)'' (1) into (2) and using that the transformation from $p(r)$ to $I(q)$ is linear gives:

:$I(q) = 4\pi\sum_^N c_i\psi_i(q),$

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

:$\psi_i(q)=\int_0^\infty\phi_i(r)\frac\textr.$

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

:$\chi^2 = \sum_^\frac$

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:

:$S = \sum_^(c_-c_i)^2$.

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: $p_i(r) \rightarrow \text \rightarrow p_f(r)$.

See also

* Frequency spectrum
* Least-squares spectral analysis

References

{{DEFAULTSORT:Indirect Fourier Transform
Fourier analysis