two-dimensional Fourier transforms

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

,

000

⁢

M

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

⁢

(

ν

1

,

ν

2

)

=

𝐑𝐞

⁢

∫

∫

c

⁢

o

⁢

s

⁢

(

ν

1

⁢

t

1

)

⁢

e

⁢

x

⁢

p

(

-

i

⁢

ν

2

⁢

t

2

)

⁢

s

⁢

(

t

1

,

t

2

)

⁢

𝑑

t

1

⁢

𝑑

t

2

where

ν

1

and

ν

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

⁢

(

ν

2

′

,

ν

2

)

=

S

T

⁢

S

=

∑

ν

1

[

S

⁢

(

ν

1

,

ν

2

′

)

⁢

S

⁢

(

ν

1

,

ν

2

)

]

,

with

ν

2

,

ν

2

′

taking all possible single-quantum frequency values and with the

summation

carried out over all discrete, double quantum
frequencies

ν

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=uk

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

http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernst-lecture.pdf

Chemistry, 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.jpg

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.

http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernst-lecture.pdf

Nobel Lecture, on December 9, 1992.

3

Peter Mansfield. 2003.

http://www.parteqinnovations.com/pdf-doc/fandr-Gaz1006.pdf

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

.)

http://www.wpi.edu/Pubs/ETD/Available/etd-081707-080430/unrestricted/dbennett.pdf

PDF 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

π

/

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

2

⁢

D

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

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FourierTransform

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TableOfGeneralizedFourierAndMeasuredGroupoidTransforms

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SpinNetworksAndSpinFoams

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FourierSineAndCosineSeries

Related topic

DiscreteFourierTransform

Defines

2D-FT

Defines

2D-FT NMRI

Defines

MRI

Defines

NMRI

Defines

2D-FT NMR

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