# two-dimensional Fourier transforms

## 1 Introduction

A two-dimensional Fourier transform (2D-FT) is computed numerically, or carried out, in two stages that are 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, [1]) 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.

## 2 Basic definition

A more precise mathematical definition of the ‘double’ Fourier transform involved is specified next.

###### Definition 2.1.

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).

### 2.1 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

 $S(\nu_{1},\nu_{2})=\textbf{Re}\int\int cos(\nu_{1}t_{1})exp^{(-i\nu_{2}t_{2})}% s(t_{1},t_{2})dt_{1}dt_{2}$

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 covariance matrix is calculated in the frequency 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 $\nu_{1}$.

http://74.125.95.132/search?q=cache:x6OQWq_GVoYJ:www.physorg.com/multimedia/pix1815/+http://www.physorg.com/multimedia/pix1815/&hl=en&ct=clnk&cd=1&gl=uk2D-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 http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernst-lecture.pdfChemistry, Physiology and Medicine received very significant recognition.

## References

• 1 Kurt W’́utrich: 1986, NMR of Proteins and Nucleic Acids., J. Wiley and Sons: New York, Chichester, Brisbane, Toronto, Singapore. http://nobelprize.org/nobel_prizes/chemistry/laureates/2002/wutrich-lecture.pdf(Nobel Laureate in 2002 for 2D-FT NMR Studies of Structure and Function of Biological Macromolecules); http://upload.wikimedia.org/wikipedia/en/b/bf/HWB-NMRv900.jpg2D-FT NMR Instrument Image Example: a JPG color image of a 2D-FT NMR Imaging ‘monster’ Instrument
• 2 Richard R. Ernst. 1992. Nuclear Magnetic Resonance Fourier Transform (2D-FT) Spectroscopy. http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernst-lecture.pdfNobel Lecture, on December 9, 1992.
• 3 Peter Mansfield. 2003. http://www.parteqinnovations.com/pdf-doc/fandr-Gaz1006.pdfNobel Laureate in Physiology and Medicine for (2D and 3D) MRI.
• 4 D. Benett. 2007. PhD Thesis. Worcester Polytechnic Institute. (lots of 2D-FT images of mathematical, brain scans.) http://www.wpi.edu/Pubs/ETD/Available/etd-081707-080430/unrestricted/dbennett.pdfPDF of 2D-FT Imaging Applications to MRI in Medical Research.
• 5 Paul Lauterbur. 2003. http://nobelprize.org/nobel_prizes/medicine/laureates/2003/Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI.
• 6 Jean Jeener. 1971. Two-dimensional Fourier Transform NMR, presented at an Ampère International Summer School, Basko Polje, unpublished. A verbatim quote follows from Richard R. Ernst’s Nobel Laureate Lecture delivered on December 2nd, 1992, “A new approach to measure two-dimensional (2D) spectra has been proposed by Jean Jeener at an Ampere Summer School in Basko Polje, Yugoslavia, 1971 ([6]). He suggested a 2D Fourier transform experiment consisting of two $\pi/2$ pulses with a variable time $t_{1}$ between the pulses and the time variable $t_{2}$ measuring the time elapsed after the second pulse as shown in Fig. 6 that expands the principles of Fig. 1. Measuring the response $s(t_{1},t_{2})$ of the two-pulse sequence and Fourier-transformation with respect to both time variables produces a two-dimensional spectrum $S(O_{1},O_{2})$ of the desired form (62,63). This two-pulse experiment by Jean Jeener is the forefather of a whole class of $2D$ experiments (8,63) that can also easily be expanded to multidimensional spectroscopy.”
 Title two-dimensional Fourier transforms Canonical name TwodimensionalFourierTransforms Date of creation 2013-03-22 18:18:56 Last modified on 2013-03-22 18:18:56 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 57 Author bci1 (20947) Entry type Topic Classification msc 81V80 Classification msc 81V55 Classification msc 42B10 Synonym 2D-FT imaging Related topic FourierStieltjesTransform Related topic FourierTransform Related topic TableOfGeneralizedFourierAndMeasuredGroupoidTransforms Related topic SpinNetworksAndSpinFoams Related topic FourierSineAndCosineSeries Related topic DiscreteFourierTransform Defines 2D-FT Defines 2D-FT NMRI Defines MRI Defines NMRI Defines 2D-FT NMR