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\begin{document}
\section{Two-dimensional Fourier transforms}
\subsection{Introduction}
A two-dimensional Fourier transform (2D-FT) is computed numerically or carried out in two stages, both
involving `standard', one-dimensional \htmladdnormallink{Fourier transforms}{http://planetphysics.us/encyclopedia/FourierTransforms.html}. However, the second stage
Fourier transform is \emph{not the inverse} Fourier transform (which would result in the original
\htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} 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 \htmladdnormallink{two-dimensional}{http://planetphysics.us/encyclopedia/CoriolisEffect.html} Nuclear Magnetic \htmladdnormallink{resonance}{http://planetphysics.us/encyclopedia/QualityFactorOfAResonantCircuit.html} (2D-NMR, \cite{KurtWutrich86})
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 \htmladdnormallink{molecular structures}{http://planetphysics.us/encyclopedia/FCS3.html}, e.g. for $900,000 M_w$,
methods that can also be utilized \emph{in vivo}. The 2D-FT method is also widely utilized in optical spectroscopy, such as \emph{2D-FT \htmladdnormallink{NIR}{http://planetphysics.us/encyclopedia/SpectralImaging.html} \htmladdnormallink{Hyperspectral Imaging}{http://planetphysics.us/encyclopedia/SpectralImaging.html}}, or in \emph{\htmladdnormallink{MRI}{http://planetphysics.us/encyclopedia/MolecularOrbitals.html} imaging} for research and clinical, diagnostic applications in Medicine.
\section{Basic definition}
A more precise mathematical definition of the `double' Fourier transform involved is specified next.
\begin{definition}
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).
\end{definition}
\subsection{Examples}
A 2D Fourier transformation and phase correction is applied to a set of 2D \htmladdnormallink{NMR}{http://planetphysics.us/encyclopedia/SpectralImaging.html} (FID) signals $s(t_1, t_2)$ yielding a real 2D-FT \htmladdnormallink{NMR `spectrum}{http://planetphysics.us/encyclopedia/MolecularOrbitals.html}' (collection of 1D \htmladdnormallink{FT-NMR}{http://planetphysics.us/encyclopedia/MolecularOrbitals.html} spectra) represented by a \htmladdnormallink{matrix}{http://planetphysics.us/encyclopedia/Matrix.html} $S$ 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(\htmladdnormallink{detection}{http://planetphysics.us/encyclopedia/MolecularOrbitals.html}) axes, respectively,
in the 2D NMR experiments. Next, the \emph{\htmladdnormallink{covariance}{http://planetphysics.us/encyclopedia/Covariance.html} matrix} is calculated in the frequency \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} 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 $\nu_1$.
\htmladdnormallink{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 \htmladdnormallink{superconductivity}{http://planetphysics.us/encyclopedia/LongRangeCoupling.html} effect (a definite
possibility for the next Nobel (?) iff the \htmladdnormallink{mathematical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html} 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 \htmladdnormallink{X-ray}{http://planetphysics.us/encyclopedia/FluorescenceCrossCorrelationSpectroscopy.html} data (`CAT scans'); recently the advanced possibilities
of 2D-FT techniques in
\htmladdnormallink{Chemistry}{http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernst-lecture.pdf}, Physiology and Medicine received very significant recognition.
\begin{thebibliography}{9}
\bibitem{KurtWutrich86}
Kurt W\''{u}trich: 1986, \emph{NMR of Proteins and Nucleic Acids.}, J. Wiley and Sons:
New York, Chichester, Brisbane, Toronto, Singapore.
\htmladdnormallink{(Nobel Laureate in 2002 for 2D-FT NMR Studies of Structure and Function of Biological Macromolecules)}{http://nobelprize.org/nobel_prizes/chemistry/laureates/2002/wutrich-lecture.pdf};
\htmladdnormallink{2D-FT NMR Instrument Image Example: a JPG color image of a 2D-FT NMR Imaging `monster' Instrument}{http://upload.wikimedia.org/wikipedia/en/b/bf/HWB-NMRv900.jpg}
\bibitem{RICHARDRERNST1992}
Richard R. Ernst. 1992. Nuclear Magnetic Resonance Fourier Transform (2D-FT) Spectroscopy.
\htmladdnormallink{Nobel Lecture}{http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernst-lecture.pdf}, on December 9, 1992.
\bibitem{PM2k3}
Peter Mansfield. 2003. \htmladdnormallink{Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI.}{http://www.parteqinnovations.com/pdf-doc/fandr-Gaz1006.pdf}
\bibitem{MRI-2DFT}
D. Benett. 2007. \emph{PhD Thesis}. Worcester Polytechnic Institute. ({\em lots of 2D-FT images of mathematical, brain scans}.)
\htmladdnormallink{PDF of 2D-FT Imaging Applications to MRI in Medical Research}{http://www.wpi.edu/Pubs/ETD/Available/etd-081707-080430/unrestricted/dbennett.pdf}.
\bibitem{PL2k3}
Paul Lauterbur. 2003.
\htmladdnormallink{Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI.}{http://nobelprize.org/nobel_prizes/medicine/laureates/2003/}
\bibitem{JeanJeneer1971}
Jean Jeener. 1971. Two-dimensional Fourier Transform NMR, presented at an Amp\`ere International Summer School, Basko Polje, \emph{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 Amp\`ere Summer School in Basko Polje, Yugoslavia, 1971 (\cite{JeanJeneer1971}). 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.''
\end{thebibliography}
\end{document}
