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Hometwo-dimensional Fourier transforms

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# 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 $\nu_{1}$.

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).

# References

- 1
Kurt W'́utrich: 1986,
*NMR of Proteins and Nucleic Acids.*, J. Wiley and Sons: New York, Chichester, Brisbane, Toronto, Singapore. (Nobel Laureate in 2002 for 2D-FT NMR Studies of Structure and Function of Biological Macromolecules); 2D-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. Nobel Lecture, on December 9, 1992.
- 3 Peter Mansfield. 2003. Nobel 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.) PDF of 2D-FT Imaging Applications to MRI in Medical Research. - 5 Paul Lauterbur. 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.”

## Mathematics Subject Classification

81V80*no label found*81V55

*no label found*42B10

*no label found*

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