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 [[../FourierTransforms/|Fourier transforms]]. However, the second stage Fourier transform is not the inverse Fourier transform (which would result in the original [[../Bijective/|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 [[../CoriolisEffect/|two-dimensional]] Nuclear Magnetic [[../QualityFactorOfAResonantCircuit/|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 [[../FCS3/|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 [[../SpectralImaging|NIR]] [[../SpectralImaging/|Hyperspectral Imaging]]}, or in [[../MolecularOrbitals|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 [[../SpectralImaging/|NMR]] (FID) signals $s(t_1,t_2)$ yielding a real 2D-FT [[../MolecularOrbitals/|NMR 'spectrum]]' (collection of 1D [[../MolecularOrbitals/|FT-NMR]] spectra) represented by a [[../Matrix/|matrix]] $S$ whose elements are $S({\nu }_1,{\nu }_2){=}^{‴}R{e}^{‴}\int \int cos({\nu }_1{t}_1)ex{p}^{(-i{\nu }_2{t}_2)}s(t_1,t_2)d{t}_{1}d{t}_2$ where ${\nu }_1$ and ${\nu }_2$ denote the discrete indirect double-quantum and single-quantum([[../MolecularOrbitals/|detection]]) axes, respectively, in the 2D NMR experiments. Next, the \htmladdnormallink{covariance {http://planetphysics.us/encyclopedia/Covariance.html} matrix} is calculated in the frequency [[../Bijective/|domain]] according to the following equation $C({{\nu }_2}^{\prime },{\nu }_2)=S^{T}S=\sum _{{\nu }^1}[S({\nu }_1,{{\nu }_2}^{\prime })S({\nu }_1,{\nu }_2)],$

with ${\nu }_2,{{\nu }_2}^{\prime }$ 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 [[../LongRangeCoupling/|superconductivity]] effect (a definite possibility for the next Nobel (?) iff the [[../PhysicalMathematics2/|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 [[../FluorescenceCrossCorrelationSpectroscopy/|X-ray]] data (`CAT scans'); recently the advanced possibilities of 2D-FT techniques in Chemistry, Physiology and Medicine received very significant recognition.