# PlanetPhysics/Two Dimensional Fourier Transforms
## Two-dimensional Fourier transforms
### Introduction

A two-dimensional Fourier transform (2D-FT) is computed numerically or carried out in two stages, both involving 'standard', one-dimensional Fourier transforms. However, the second stage Fourier transform is not the inverse Fourier transform (which would result in the original function that was transformed at the first stage), but a Fourier transform in a second variable-- which is 'shifted' in value-- relative to that involved in the result of the first Fourier transform. Such 2D-FT analysis is a very powerful method for three-dimensional reconstruction of polymer and biopolymer structures by two-dimensional Nuclear Magnetic resonance (2D-NMR,) of solutions for molecular weights ($M_{w}$) of the dissolved polymers up to about 50,000 $M_{w}$. For larger biopolymers or polymers, more complex methods have been developed to obtain the desired resolution needed for the 3D-reconstruction of higher molecular structures, e.g. for $900,000M_{w}$, methods that can also be utilized in vivo . The 2D-FT method is also widely utilized in optical spectroscopy, such as 2D-FT NIR Hyperspectral Imaging}, or in MRI imaging for research and clinical, diagnostic applications in Medicine.

## Basic definition

A more precise mathematical definition of the 'double' Fourier transform involved is specified next.
A 2D-FT, or two-dimensional Fourier transform, is a standard Fourier transformation of a function of two variables, $f(x_{1},x_{2})$, carried first in the first variable $x_{1}$, followed by the Fourier transform in the second variable $x_{2}$ of the resulting function $F(s_{1},x_{2})$. (For further specific details and example for 2D-FT Imaging v. URLs provided in the following recent Bibliography).

### Examples

A 2D Fourier transformation and phase correction is applied to a set of 2D NMR (FID) signals $s(t_{1},t_{2})$ yielding a real 2D-FT NMR 'spectrum' (collection of 1D FT-NMR spectra) represented by a matrix $S$ whose elements are $S(\nu _{1},\nu _{2})='''Re'''\int \int cos(\nu _{1}t_{1})exp^{(-i\nu _{2}t_{2})}s(t_{1},t_{2})dt_{1}dt_{2}$ where $\nu _{1}$ and $\nu _{2}$ denote the discrete indirect double-quantum and single-quantum(detection) axes, respectively, in the 2D NMR experiments. Next, the \htmladdnormallink{covariance {http://planetphysics.us/encyclopedia/Covariance.html} matrix} is calculated in the frequency domain according to the following equation $C(\nu _{2}',\nu _{2})=S^{T}S=\sum _{\nu ^{1}}[S(\nu _{1},\nu _{2}')S(\nu _{1},\nu _{2})],$
with $\nu _{2},\nu _{2}'$ taking all possible single-quantum frequency values and with the summation carried out over all discrete, double quantum frequencies $\nu _{1}$.
2D-FT STEM Images (obtained at Cornell University) of electron distributions in a high-temperature cuprate superconductor 'paracrystal' reveal both the domains (or 'location') and the local symmetry of the "pseudo-gap" in the electron-pair correlation band responsible for the high--temperature superconductivity effect (a definite possibility for the next Nobel (?) iff the mathematical physics treatment is also developed to include also such results).
So far there have been three Nobel prizes awarded for 2D-FT NMR/MRI during 1992-2003, and an additional, earlier Nobel prize for 2D-FT of X-ray data (`CAT scans'); recently the advanced possibilities of 2D-FT techniques in Chemistry, Physiology and Medicine received very significant recognition.