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Magnetic Resonance
Spectroscopy and
lmaging:NMR, MRI and ESR

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Contents

Articles

Nuclear Magnetic Resonance x

Relaxation 1

Chemical Shift 2

Knight shift 6

Robinson oscillator 7

Relaxation 7

Chemical Shift 8

Fourier transform 12

Discrete Fourier transform 34

Fast Fourier transform 47

Fourier transform spectroscopy 55

NMR Spectroscopy 59

2D-NMR 65

User:Bci2/2D-FT NMRI and Spectroscopy 67

Solid-state nuclear magnetic resonance 73

Magnetic resonance microscopy 79

Imaging 81

Medical imaging 8 1

MRI 89

ESR Spectroscopy and Microspectroscopy 10 8

ESR 108

References

Article Sources and Contributors

Image Sources, Licenses and Contributors

Article Licenses

Nuclear Magnetic Resonance

Relaxation

Relaxation stands quite generally for a release of tension, a return to equilibrium.
In the sciences, the term is used in the following ways:

• Relaxation (physics), and more in particular:

• Relaxation (NMR), processes by which nuclear magnetization returns to the equilibrium distribution

• Dielectric relaxation, the delay in the dielectric constant of a material

• Structural relaxation, responsible for the glass transition

• In mathematics:

• Relaxation technique (mathematics), a technique for transforming hard constraints into easier ones

• Relaxation method, for numerically solving elliptic partial differential equations

• In computer science:

• Relaxation, the act of substituting alternative program code during linking
In Physiology, Hypnotism, Meditation, Recreation:

• Relaxation technique, an activity that helps a person to relax

• Relaxed in Flow (psychology), a state of arousal, flow, over-learned self-control and relaxation

• Relaxation (psychology), the emotional state of low tension
See also:

• Tension (music)

Chemical Shift

In nuclear magnetic resonance (NMR), the chemical shift describes the dependence of nuclear magnetic energy
levels on the electronic environment in a molecule. Chemical shifts are relevant in NMR spectroscopy

techniques such as proton NMR and carbon- 13 NMR.

An atomic nucleus can have a magnetic moment (nuclear spin), which gives rise to different energy levels and
resonance frequencies in a magnetic field. The total magnetic field experienced by a nucleus includes local magnetic
fields induced by currents of electrons in the molecular orbitals (note that electrons have a magnetic moment
themselves). The electron distribution of the same type of nucleus (e.g. H, C, N) usually varies according to the
local geometry (binding partners, bond lengths, angles between bonds, ...), and with it the local magnetic field at
each nucleus. This is reflected in the spin energy levels (and resonance frequencies). The variations of nuclear
magnetic resonance frequencies of the same kind of nucleus, due to variations in the electron distribution, is called
the chemical shift. The size of the chemical shift is given with respect to a reference frequency or reference sample
(see also chemical shift rcfercm lug), usually a molecule with a barely distorted electron distribution.
The chemical shift is of great importance for NMR spectroscopy, a technique to explore molecular properties by
looking at nuclear magnetic resonance phenomena.

Operating frequency

The operating frequency C^oof a magnet is calculated from the Larmor equation

u Q = 7 * B
where Bois the actual strength of the magnet in units like teslas or gauss, and 7 is the gyromagnetic ratio of the
nucleus being tested which is in turn calculated from its magnetic moment fl and spin number / with the nuclear
magneton /ijv and the Planck constant h:

Thus, the proton operating frequency for a 1 T magnet is calculated as:
2.79 x 5.05 x 10- 27 J/T
"■ = ^ = 6.62 xlO^J.* (1/2) X ' T = 425MHZ

Chemical shift referencing

Chemical shift 5 is usually expressed in parts per million (ppm) by frequency, because it is calculated from:

difference in precession frequency between two nuclei

= :

operating frequency of the magnet

Since the numerator is usually in hertz, and the denominator in megahertz, delta is expressed in ppm.
The detected frequencies (in Hz) for H, C, and Si nuclei are usually referenced against TMS (tetramethylsilane)
or DSS, which is assigned the chemical shift of zero. Other standard materials are used for setting the chemical shift
for other nuclei.

Thus, an NMR signal at 300 Hz from TMS at an applied frequency of 300MHz has a chemical shift of:
300Hz ,..„„.,

300X10° Hz =1Xl ° =lppm

Although the frequency depends on the applied field the chemical shift is independent of it. On the other hand the
resolution of NMR will increase with applied magnetic field resulting in ever increasing chemical shift changes.

Chemical Shift

The induced magnetic field

The electrons around a nucleus will circulate in a magnetic field and create a secondary induced magnetic field. This
field opposes the applied field as stipulated by Lenz's law and atoms with higher induced fields (i.e., higher electron
density) are therefore called shielded, relative to those with lower electron density. The chemical milieu of an atom
can influence its electron density through the polar effect. Electron-donating alkyl groups, for example, lead to
increased shielding while electron-withdrawing substituents such as nitro groups lead to deshielding of the nucleus.
Not only substituents cause local induced fields. Bonding electrons can also lead to shielding and deshielding effects.
A striking example of this are the pi bonds in benzene. Circular current through the hyperconjugated system causes a
shielding effect at the molecule's center and a deshielding effect at its edges. Trends in chemical shift are explained
based on the degree of shielding or deshielding.

Nuclei are found to resonate in a wide range to the left (or more rare to the right) of the internal standard. When a
signal is found with a higher chemical shift:

• the applied effective magnetic field is lower, if the resonance frequency is fixed, (as in old traditional CW
spectrometers)

• the frequency is higher, when the applied magnetic field is static, (normal case in FT spectrometers)

• the nucleus is more deshielded

• the signal or shift is downfield or at low field or paramagnetic

Conversely a lower chemical shift is called a diamagnetic shift, and is upfield and more shielded.

Diamagnetic shielding

In real molecules protons are surrounded by a cloud of charge due to adjacent bonds and atoms. In an applied
magnetic field (B ) electrons circulate and produce an induced field (B.) which opposes the applied field. The
effective field at the nucleus will be B = B - B . The nucleus is said to be experiencing a diamagnetic shielding

Factors causing chemical shifts

Important factors influencing chemical shift are electron density, electronegativity of neighboring groups and
anisotropic induced magnetic field effects.

Electron density shields a nucleus from the external field. For example in proton NMR the electron-poor tropylium
ion has its protons downfield at 9.17 ppm, those of the electron-rich cyclooctatetraenyl anion move upfield to 6.75
ppm and its dianion even more upfield to 5.56 ppm.

A nucleus in the vicinity of an electronegative atom experiences reduced electron density and the nucleus is therefore
deshielded. In proton NMR of methyl halides (CH X) the chemical shift of the methyl protons increase in the order I
< Br < CI < F from 2.16 ppm to 4.26 ppm reflecting this trend. In carbon NMR the chemical shift of the carbon
nuclei increase in the same order from around -10 ppm to 70 ppm. Also when the electronegative atom is removed
further away the effect diminishes until it can be observed no longer.

Anisotropic induced magnetic field effects are the result of a local induced magnetic field experienced by a nucleus
resulting from circulating electrons that can either be paramagnetic when it is parallel to the applied field or
diamagnetic when it is opposed to it. It is observed in alkenes where the double bond is oriented perpendicular to the
external field with pi electrons likewise circulating at right angles. The induced magnetic field lines are parallel to
the external field at the location of the alkene protons which therefore shift downfield to a 4.5 ppm to 7.5 ppm range.
The three-dimensional space where a nucleus experiences diamagnetic shift is called the shielding zone with a
cone-like shape aligned with the external field.

Chemical Shift

The protons in aromatic compounds are shifted downfield even further with a signal for benzene at 7.73 ppm as a
consequence of a diamagnetic ring current.

Alkyne protons by contrast resonate at high field in a 2-3 ppm range. For alkynes the most effective orientation is
the external field in parallel with electrons circulation around the triple bond. In this way the acetylenic protons are
located in the cone-shaped shielding zone hence the upfield shift.

Magnetic properties of most common nuclei

H and C aren't the only nuclei susceptible to NMR experiments. A number of different nuclei can also be
detected, although the use of such techniques is generally rare due to small relative sensitivities in NMR experiments
(compared to H) of the nuclei in question, the other factor for rare use being their slender representation in
nature/organic compounds.

Isotope

Occurrence

Magnetic

Electric quadrupole

Operating frequency Relative

at 7 T sensitivity

(MHz)

1.1

1/2

0.70220

99.64

0.40358

7.1 xlO" 2

0.37

1/2

-0.28304

99.76

0.0317

5/2

-1.8930

-4.0 x 10"

100

1/2

2.6273

10.69
282.40

0.0159
0.00101
0.00104

0.0291
0.834

Chemical Shift

31p

100

1/2

1.1205

121.49

0.0664

35 CI

75.4

3/2

0.92091

-7.9

tlO' 2

29.41

0.0047

37 CI

24.6

3/2

0.68330

-6.2

tlO' 2

24.48

0.0027

Ma,,K

tic properties of
3 n„uclei [5]

H, C, N, F and P are the five nuclei that have the greatest importance in NMR experiments:

• H because of high sensitivity and vast occurrence in organic compounds

• C because of being the key component of all organic compounds despite occurring at a low abundance (1.1
compared to the major isotope of carbon C, which has a spin of and therefore is NMR inactive.

• N because of being a key component of important biomolecules such as proteins and DNA

• F because of high relative sensitivity

• P because of frequent occurrence in organic compounds and moderate relative sensitivity

Other chemical shifts

The related Knight shift (first reported in 1949) is observed with pure metals. The NMR chemical shift in its present
day meaning first appeared in journals in 1950. Chemical shifts with a different meaning appear in X-ray
photoelectron spectroscopy as the shift in atomic core-level energy due to a specific chemical environment. The term
is also used in Mossbauer spectroscopy, where similarly to NMR it refers to a shift in peak position due to the local
chemical bonding environment. As is the case for NMR the chemical shift reflects the electron density at the atomic
nucleus.

See also

• Carbon-13NMR

• MRI

• NMR spectroscopy

• 2D-FT NMRI and Spectroscopy

• Nuclear magnetic resonance

• Protein NMR

• Proton NMR

• Solid-state NMR

• Zeeman effect

External links

I VI

www.chem.wisc.edu
BioMagResBank [8]
wwwchem.csustan.edu
Proton chemical shifts
Carbon chemical shifts

|I0|

• Online tutorials (these generally involve combined use of IR, H NMR, C NMR and mass spectrometry)

• Problem set 1, advanced (see also this link for more background information on spin-spin coupling)

• Problem set 2, moderate [13]

• Problem set 4, moderate, German language (don't let that scare you away!)

• Problem set 5, the best!

Chemical Shift

• Combined solutions to problem set 5 (Problems 1-32) L1DJ and (Problems 33-64) L1 ' J

References

ectrometric Identification of organic Compounds Silverstein, Bassler, Morrill 4th Ed. ISBN 047109700
2] Organic Spectroscopy William Kemp 3rd Ed. ISBN 0333417674
3] Basic 'H - U C-NMR spectroscopy Metin Balei ISBN 04445 181 18
[4] In units of the nuclear magneton
5] CRC Handbook of Chemistry and Physics 65Th Ed

Short History of Three Chemical Shifts Shin-ichi Nagaoka Vol. 84 No. 5 May 2007 Journal of Chemical Education 8(
7] http://www.chem.wisc.edu/areas/reich/handouls/nmr-h/hdata.htm
8] http://www.bmrb.wisc.edu

[9] http://wwwchem.csustan.edu/Tutorials/NMRTABLE.HTM
iii| I I hem.tt li i i/hand mi ila.hlm

1] http://www.chem.ucla.edu/~webspectra/

.' I Imp li li I i i li. I I I

3] http://orgchem.colorado.edu/hndbksupporl/speelprob/problems.html

4] http://www chen nn p I d an de/lools/kombil.htm

5] http://www.nd.edu/~smithgrp/structure/workbook html
16] http://ww\\ n i u mill q Iru lui mswersl ^ dlh

7] http://www.nd.edu/~smil ui mswers33-64.GIF

Knight shift

The Knight shift is a shift in the nuclear magnetic resonance frequency of a paramagnetic substance first published

in 1949 by the American physicist Walter David Knight.

The Knight shift is due to the conduction electrons in metals. They introduce an "extra" effective field at the nuclear

site, due to the spin orientations of the conduction electrons in the presence of an external field. This is responsible

for the shift observed in the nuclear magnetic resonance. The shift comes from two sources, one is the Pauli

paramagnetic spin susceptibility, the other is the s-component wavefunctions at the nucleus.

Depending on the electronic structure, Knight shift may be temperature dependent. However, in metals which

normally have a broad featureless electronic density of states, Knight shifts are temperature independent.

Robinson oscillator

The Robinson oscillator (or Robinson marginal oscillator) is an electronic circuit used in the field of Nuclear
Magnetic Resonance (NMR). The oscillator forms the underlying basis of Magnetic Resonance Imaging (MRI)
systems used in many hospitals. It was invented by the British physicist Neville Robinson.

References

• Deschamps, P., Vaissiere, J. and Sullivan, N. S., Integrated circuit Robinson oscillator for NMR detection ^ ,
Review of Scientific Instruments, 48(6):664-668, June 1977. DOI 10.1063/1.1135103

• Wilson, K. J. and Vallabhan, C. P. G., An improved MOSFET-based Robinson oscillator for NMR detection [2] ,
Meas. Sci. Technol, l(5):458-460, May 1990. DOI 10.1088/0957-0233/1/5/015

References

ent.aip.org/RSINAK/v48/i6/664_l.html
v.iop.org/EJ/abstract/0957-0233/1/5/015

Relaxation

Relaxation stands quite generally for a release of tension, a return to equilibrium.
In the sciences, the term is used in the following ways:

• Relaxation (physics), and more in particular:

• Relaxation (NMR), processes by which nuclear magnetization returns to the equilibrium distribution

• Dielectric relaxation, the delay in the dielectric constant of a material

• Structural relaxation, responsible for the glass transition

• In mathematics:

• Relaxation technique (mathematics), a technique for transforming hard constraints into easier ones

• Relaxation method, for numerically solving elliptic partial differential equations

• In computer science:

• Relaxation, the act of substituting alternative program code during linking
In Physiology, Hypnotism, Meditation, Recreation:

• Relaxation technique, an activity that helps a person to relax

• Relaxed in Flow (psychology), a state of arousal, flow, over-learned self-control and relaxation

• Relaxation (psychology), the emotional state of low tension
See also:

• Tension (music)

Chemical Shift

techniques such as proton NMR and carbon- 13 NMR.

Operating frequency

The operating frequency C^oof a magnet is calculated from the Larmor equation

Thus, the proton operating frequency for a 1 T magnet is calculated as:
2.79 x 5.05 x 10- 27 J/T
"■ = ^ = 6.62 xlO^J.* (1/2) X ' T = 425MHZ

Chemical shift referencing

Chemical shift 5 is usually expressed in parts per million (ppm) by frequency, because it is calculated from:

difference in precession frequency between two nuclei

= :

operating frequency of the magnet

Thus, an NMR signal at 300 Hz from TMS at an applied frequency of 300MHz has a chemical shift of:
300Hz ,..„„.,

300X10° Hz =1Xl ° =lppm

Chemical Shift

The induced magnetic field

Nuclei are found to resonate in a wide range to the left (or more rare to the right) of the internal standard. When a
signal is found with a higher chemical shift:

• the applied effective magnetic field is lower, if the resonance frequency is fixed, (as in old traditional CW
spectrometers)

• the frequency is higher, when the applied magnetic field is static, (normal case in FT spectrometers)

• the nucleus is more deshielded

• the signal or shift is downfield or at low field or paramagnetic

Conversely a lower chemical shift is called a diamagnetic shift, and is upfield and more shielded.

Diamagnetic shielding

Factors causing chemical shifts

Important factors influencing chemical shift are electron density, electronegativity of neighboring groups and
anisotropic induced magnetic field effects.

Chemical Shift

The protons in aromatic compounds are shifted downfield even further with a signal for benzene at 7.73 ppm as a
consequence of a diamagnetic ring current.

Magnetic properties of most common nuclei

Isotope

Occurrence

Magnetic

Electric quadrupole

Operating frequency Relative

at 7 T sensitivity

(MHz)

1.1

1/2

0.70220

99.64

0.40358

7.1 xlO" 2

0.37

1/2

-0.28304

99.76

0.0317

5/2

-1.8930

-4.0 x 10"

100

1/2

2.6273

10.69
282.40

0.0159
0.00101
0.00104

0.0291
0.834

Chemical Shift

31p

100

1/2

1.1205

121.49

0.0664

35 CI

75.4

3/2

0.92091

-7.9

tlO' 2

29.41

0.0047

37 CI

24.6

3/2

0.68330

-6.2

tlO' 2

24.48

0.0027

Ma,,K

tic properties of
3 n„uclei [5]

H, C, N, F and P are the five nuclei that have the greatest importance in NMR experiments:

• H because of high sensitivity and vast occurrence in organic compounds

• N because of being a key component of important biomolecules such as proteins and DNA

• F because of high relative sensitivity

• P because of frequent occurrence in organic compounds and moderate relative sensitivity

Other chemical shifts

See also

• Carbon-13NMR

• MRI

• NMR spectroscopy

• 2D-FT NMRI and Spectroscopy

• Nuclear magnetic resonance

• Protein NMR

• Proton NMR

• Solid-state NMR

• Zeeman effect

External links

I VI

www.chem.wisc.edu
BioMagResBank [8]
wwwchem.csustan.edu
Proton chemical shifts
Carbon chemical shifts

|I0|

• Online tutorials (these generally involve combined use of IR, H NMR, C NMR and mass spectrometry)

• Problem set 1, advanced (see also this link for more background information on spin-spin coupling)

• Problem set 2, moderate [13]

• Problem set 4, moderate, German language (don't let that scare you away!)

• Problem set 5, the best!

Chemical Shift

• Combined solutions to problem set 5 (Problems 1-32) L J and (Problems 33-64) L J

References

[1] Spectrometric Identification of organic Compounds Silverstein, Bassler, Morrill 4th Ed. ISBN 047109700

[2] Organic Spectroscopy William Kemp 3rd Ed. ISBN 0333417674

[3] Basic 'H - Li C-NMR spectroscopy Metin Balei ISBN 04445 181 18

[4] In units of the nuclear magneton

[5] CRC Handbook of Chemistry and Physics 65Th Ed

[6] A Short History of Three Chemical Shifts Shin-ichi Nagaoka Vol. 84 No. 5 May 2007 Journal of Chemical Education 8(

Fourier transform

In mathematics, the Fourier transform (often abbreviated FT) is an operation that transforms one complex-valued
function of a real variable into another. In such applications as signal processing, the domain of the original function
is typically time and is accordingly called the time domain. The domain of the new function is typically called the
frequency domain, and the new function itself is called the frequency domain representation of the original function.
It describes which frequencies are present in the original function. This is analogous to describing a musical chord in
terms of the notes being played. In effect, the Fourier transform decomposes a function into oscillatory functions.
The term Fourier transform refers both to the frequency domain representation of a function, and to the process or
formula that "transforms" one function into the other.

The Fourier transform and its generalizations are the subject of Fourier analysis. In this specific case, both the time
and frequency domains are unbounded linear continua. It is possible to define the Fourier transform of a function of
several variables, which is important for instance in the physical study of wave motion and optics. It is also possible
to generalize the Fourier transform on discrete structures such as finite groups, efficient computation of which
through a fast Fourier transform is essential for high-speed computing.

is Fourier transform

Fourier series

Discrete Fourier transform

Discrete-time Fourier
transform

Related transforms

Definition

There are several common conventions for defining the Fourier transform of an integrable function / :
(Kaiser 1994). This article will use the definition:

/(£) = / f{x)e 2mx ^ dx, for every real number g.

When the independent variable x represents time (with SI unit of seconds), the transform variable g represents
frequency (in hertz). Under suitable conditions, /can be reconstructed from f by the inverse transform:

f(x)= I /(£) e 27ria *df, for every real number x.

For other common conventions and notations, including using the angular frequency co instead of the frequency g,
see Other conventions and Other notations below. The Fourier transform on Euclidean space is treated separately, in

Fourier transform

which the variable x often represents position and g

Introduction

The motivation for the Fourier transform comes from the study of Fourier series. In the study of Fourier series,
complicated periodic functions are written as the sum of simple waves mathematically represented by sines and
cosines. Due to the properties of sine and cosine it is possible to recover the amount of each wave in the sum by an
integral. In many cases it is desirable to use Euler's formula, which states that e m = cos 2jz6 + i sin 2jz0, to write
Fourier series in terms of the basic waves e ' . This has the advantage of simplifying many of the formulas involved
and providing a formulation for Fourier series that more closely resembles the definition followed in this article. This
passage from sines and cosines to complex exponentials makes it necessary for the Fourier coefficients to be
complex valued. The usual interpretation of this complex number is that it gives you both the amplitude (or size) of
the wave present in the function and the phase (or the initial angle) of the wave. This passage also introduces the
need for negative "frequencies". If 6 were measured in seconds then the waves e m and e~ m would both complete
one cycle per second, but they represent different frequencies in the Fourier transform. Hence, frequency no longer
measures the number of cycles per unit time, but is closely related.

We may use Fourier series to motivate the Fourier transform as follows. Suppose that /is a function which is zero
outside of some interval [-L/2, L/2]. Then for any 7>Lwe may expand / in a Fourier series on the interval
[-772,772], where the "amount" (denoted by c ) of the wave e mn in the Fourier series of /is given by

/r/2
e-" mnx/T f(x)dx
-772

and/ should be given by the formula

m = 7£ £ />/T)e 2 ^/T
If we let I = n/T, and we let Ag = (n + \)IT - nIT = \IT, then this last sum becomes the Riemann sum

By letting T — > °° this Riemann sum converges to the integral for the inverse Fourier transform given in the
Definition section. Under suitable conditions this argument may be made precise (Stein & Shakarchi 2003). Hence,
as in the case of Fourier series, the Fourier transform can be thought of as a function that measures how much of
each individual frequency is present in our function, and we can recombine these waves by using an integral (or
"continuous sum") to reproduce the original function.

The following images provide a visual illustration of how the Fourier transform measures whether a frequency is
present in a particular function. The function depicted f(f\ = cos(67r£)e~ ,ri oscillates at 3 hertz (if t measures
seconds) and tends quickly to 0. This function was specially chosen to have a real Fourier transform which can easily
be plotted. The first image contains its graph. In order to calculate f(3)we must integrate e _2jr,(3r) /(f). The second
image shows the plot of the real and imaginary parts of this function. The real part of the integrand is almost always
positive, this is because when /(f) is negative, then the real part of e~ '' is negative as well. Because they oscillate
at the same rate, when /(f) is positive, so is the real part of e~ . The result is that when you integrate the real part

of the integrand you get a relatively large number (in this case 0.5). On the other hand, when you try to measure a
frequency that is not present, as in the case when we look at f (5)> tne integrand oscillates enough so that the
integral is very small. The general situation may be a bit more complicated than this, but this in spirit is how the
Fourier transform measures how much of an individual frequency is present in a function f(t).

Fourier transform

< (riginal function showing Real and imaginary parts of Real and imaginary parts of Fourier transform with 3 and 5

oscillation 3 hertz. integrand for Fourier transform integrand for Fourier transform hertz labeled,

at 3 hertz at 5 hertz

Properties of the Fourier transform

An integrable function is a function/on the real line that is Lebesgue-measurable and satisfies
/ \f(x)\dx <oo.

Basic properties

Given integrable functions f{x), g(x), and h(x) denote their Fourier transforms by f(f), g{£) > an d h(f)

respectively. The Fourier transform has the following basic properties (Pinsky 2002).

Linearity

For any complex numbers a and b, if h(x) = af{x) + bg(x), then h(£) = a ■ /(£) + b ■ §(£)■
Translation

For any real number* if h(x) =f(x - x Q ), then ^(£) = e~ 2lxixoi f {£) .
Modulation

For any real number g if h(x) = e 2m %f(x), then U^\ = fU — £o)-

For a non-zero real number a, if h(x) =f(ax), then /l(£) = -j — :j I — ) ■ The case a = -1 leads to the

\a\ \aj

time-reversal property , which states: if h(x) =f(-x), then h(£) = / ( — £)■
Conjugation

If h(x) =J(xJAhen ~ m = J{Z{y

In particular, if/is real, then one has the reality condition f(—f) = f(f).
And if/is purely imaginary, then f(—f) = —?(£).
Convolution

Uk{x) = (/*<?}(», then h(0 = fa)-m-

Fourier transform

Uniform continuity and the Riemann-Lebesgue lemma

/(£) -► as |e| -> oo.
The Fourier transform f of an integrable function / is bounded and continuous, but need not be integrable - for
example, the Fourier transform of the rectangular function, which is a step function (and hence integrable) is the sine
function, which is not Lebesgue integrable, though it does have an improper integral: one has an analog to the
alternating harmonic series, which is a convergent sum but not absolutely convergent.

It is not possible in general to write the inverse transform as a Lebesgue integral. However, when both/and f are
integrable, the following inverse equality holds true for almost every x:

fw=£lf ($***<%■

Almost everywhere, / is equal to the continuous function given by the right-hand side. Iff is given as continuous

function on the line, then equality holds for every x.

A consequence of the preceding result is that the Fourier transform is injective on L (R).

The Plancherel theorem and Parseval's theorem

Let f(x) and g(x) be integrable, and let f(£\and (HO be their Fourier transforms. If f(x) and g(x) are also
square-integrable, then we have Parseval's theorem (Rudin 1987, p. 187):

where the bar denotes complex conjugation.

The Plancherel theorem, which is equivalent to Parseval's theorem, states (Rudin 1987, p. 186):

/_J/(x)| 2 ^ = /_^|/(0| 2 ^
The Plancherel theorem makes it possible to define the Fourier transform for functions in L (R), as described in
Generalizations below. The Plancherel theorem has the interpretation in the sciences that the Fourier transform
preserves the energy of the original quantity. It should be noted that depending on the author either of these theorems
might be referred to as the Plancherel theorem or as Parseval's theorem.
See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.

Fourier transform

Poisson summation formula

The Poisson summation formula provides a link between the study of Fourier transforms and Fourier Series. Given
an integrable function/we can consider the periodization off given by:

f{x) = ^2f(x + k),

kez
where the summation is taken over the set of all integers k. The Poisson summation formula relates the Fourier series
of J to the Fourier transform of/. Specifically it states that the Fourier series of J is given by:

Convolution theorem

The Fourier transform translates between convolution and multiplication of functions. If fix) and g(x) are integrable
functions with Fourier transforms ff£)and ^(^respectively, then the Fourier transform of the convolution is
given by the product of the Fourier transforms f(£\and g(£) (under other conventions for the definition of the
Fourier transform a constant factor may appear).
This means that if:

K X ) = U *g)( x ) = J f(y)g{x-y)dy,

where * denotes the convolution operation, then:

In linear time invariant (LTI) system theory, it is common to interpret g(x) as the impulse response of an LTI system
with input fix) and output h(x), since substituting the unit impulse for/(;t) yields h(x) = g(x). In this case, g(£)
represents the frequency response of the system.

Conversely, if fix) can be decomposed as the product of two square integrable functions p(x) and q(x), then the
Fourier transform of fix) is given by the convolution of the respective Fourier transforms p(£) and q(£) ■

Cross-correlation theorem

In an analogous manner, it can be shown that if h(x) is the cross-correlation of/(x) and g(x):

H x ) = if * 9){z) = j_ f(y)g(x + y)dy

then the Fourier transform of h(x) is:

As a special case, the autocorrelation of function fix) is:

Kx) = (/ * f)(x) = jT f(y)f(x + y) dy
,'hicli

Ho =7(fl/(o = i/(oi 2 -

Fourier transform

Eigenfunctions

One important choice of an orthonormal basis for L (R) is given by the Hermite functions

2 1 / 4 2

%j) n {x) = -^e^ x H n {2x^/Tx),

Vn!
where H n (x) are the "probabilist's" Hermite polynomials, defined by H (x) = (-l)"exp(x 12) D" exp(-x 12). Under
this convention for the Fourier transform, we have that

MO = HTMO-

In other words, the Hermite functions form a complete orthonormal system of eigenfunctions for the Fourier
transform on L (R) (Pinsky 2002). However, this choice of eigenfunctions is not unique. There are only four
different eigenvalues of the Fourier transform (±1 and ±i) and any linear combination of eigenfunctions with the
same eigenvalue gives another eigenfunction. As a consequence of this, it is possible to decompose L (R) as a direct
sum of four spaces H H H and H where the Fourier transform acts on H simply by multiplication by i . This
approach to define the Fourier transform is due to N. Wiener (Duoandikoetxea 2001). The choice of Hermite
functions is convenient because they are exponentially localized in both frequency and time domains, and thus give
rise to the fractional Fourier transform used in time-frequency analysis (Boashash 2003).

Fourier transform on Euclidean space

The Fourier transform can be in any arbitrary number of dimensions n. As with the one-dimensional case there are
many conventions, for an integrable function/(x) this article takes the definition:

f(0=Hf)(0= [ f(x)e- 2 ™<dx
where x and g are n-dimensional vectors, and x ■ g is the dot product of the vectors. The dot product is sometimes
written as ( x ,£) .

All of the basic properties listed above hold for the n-dimensional Fourier transform, as do Plancherel's and
Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and the
Riemann-Lebesgue lemma holds. (Stein & Weiss 1971)

Uncertainty principle

Generally speaking, the more concentrated fix) is, the more spread out its Fourier transform f(f) must be. In

particular, the scaling property of the Fourier transform may be seen as saying: if we "squeeze" a function in x, its

Fourier transform "stretches out" in g. It is not possible to arbitrarily concentrate both a function and its Fourier

transform.

The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an

Uncertainty Principle, and is formalized by viewing a function and its Fourier transform as conjugate variables

with respect to the symplectic form on the time-frequency domain: from the point of view of the linear canonical

transformation, the Fourier transform is rotation by 90° in the time-frequency domain, and preserves the symplectic

form.

Suppose fix) is an integrable and square-integrable function. Without loss of generality, assume that fix) is

normalized:

J~Jf{x)\ 2 dx = l.

It follows from the Plancherel theorem that f(f) is also normalized.

The spread around x = may be measured by the dispersion about zero (Pinsky 2002) defined by

Fourier transform

D (f) = J°^x 2 \f(x)\ 2 dx.

In probability terms, this is the second moment of \f(x) | 2 about zero.

The Uncertainty principle states that, if/(x) is absolutely continuous and the functions x-f(x) and/(x) are square

integrable, then

Do(f)D (f) > ^- 2 (Pinsky2002).
The equality is attained only in the case f( x \ = d e ~ 7rx2 / cr2 (hence f(f\ = a Ci e~ w<j2 ^ ) where a>
is arbitrary and C is such that /is L -normalized (Pinsky 2002). In other words, where /is a (normalized) Gaussian
function, centered at zero.
In fact, this inequality implies that:

(|jx-x ) 2 i/(x)i 2 ^) (/je-^o) 2 i/(0l 2 ^) > ^

for any x , £ in R (Stein & Shakarchi 2003).

In quantum mechanics, the momentum and position wave functions are Fourier transform pairs, to within a factor of
Planck's constant. With this constant properly taken into account, the inequality above becomes the statement of the
Heisenberg uncertainty principle (Stein & Shakarchi 2003).

Spherical harmonics

Let the set of homogeneous harmonic polynomials of degree k on R" be denoted by A The set A consists of the
solid spherical harmonics of degree k. The solid spherical harmonics play a similar role in higher dimensions to the
Hermite polynomials in dimension one. Specifically, if f(x) = e~ n x 2P(x) for some P(x) in A then
/"(£) = i~ k /(£)■ Let the set H be the closure in L (R") of linear combinations of functions of the form/(lxl)P(x)
where P(x) is in A The space L (R n ) is then a direct sum of the spaces H and the Fourier transform maps each
space H to itself and is possible to characterize the action of the Fourier transform on each space H (Stein & Weiss
1971). Let/(x) =f (\x\)P(x) (with P(x) in Ap, then /(f) = F (|f |)P(£) where

F (r) = 2^-^"^ j™ Us)J {n+2k - 2)l2 {2vrs)s^ k ^ds.

Here J denotes the Bessel function of the first kind with order (n + 2k- 2)12. When k = this gives a

useful formula for the Fourier transform of a radial function (Grafakos 2004).

Restriction problems

In higher dimensions it becomes interesting to study restriction problems for the Fourier transform. The Fourier
transform of an integrable function is continuous and the restriction of this function to any set is defined. But for a
square-integrable function the Fourier transform could be a general class of square integrable functions. As such, the
restriction of the Fourier transform of an L (R") function cannot be defined on sets of measure 0. It is still an active
area of study to understand restriction problems in if for 1 < p < 2. Surprisingly, it is possible in some cases to
define the restriction of a Fourier transform to a set S, provided S has non-zero curvature. The case when S is the unit
sphere in R" is of particular interest. In this case the Tomas-Stein restriction theorem states that the restriction of the
Fourier transform to the unit sphere in R" is a bounded operator on if provided 1 < p < (2n + 2) / (« + 3).
One notable difference between the Fourier transform in 1 dimension versus higher dimensions concerns the partial
sum operator. Consider an increasing collection of measurable sets E indexed by R G (0,°°): such as balls of radius
R centered at the origin, or cubes of side 2R. For a given integrable function/ consider the function/ defined by:

Jer

Fourier transform

Suppose in addition that /is in L p (R n ). For n = 1 and 1 < p < °o, if one takes E = (-R, R), then/ converges to /in
L p as R tends to infinity, by the boundedness of the Hilbert transform. Naively one may hope the same holds true for
n > 1. In the case that E is taken to be a cube with side length R, then convergence still holds. Another natural
candidate is the Euclidean ball E = {§ : 1^1 < R}. In order for this partial sum operator to converge, it is necessary
that the multiplier for the unit ball be bounded in L p (R n ). For n> 2 it is a celebrated theorem of Charles Fefferman
that the multiplier for the unit ball is never bounded unless p = 2 (Duoandikoetxea 2001). In fact, when p * 2, this
shows that not only may/ fail to converge to /in L p , but for some functions /€ L p (R n ),f is not even an element of
L p .

Generalizations

Fourier transform on other function spaces

It is possible to extend the definition of the Fourier transform to other spaces of functions. Since compactly
supported smooth functions are integrable and dense in L (R), the Plancherel theorem allows us to extend the
definition of the Fourier transform to general functions in L (R) by continuity arguments. Further J^: L (R) — >
L (R) is a unitary operator (Stein & Weiss 1971, Thm. 2.3). Many of the properties remain the same for the Fourier
transform. The Hausdorff- Young inequality can be used to extend the definition of the Fourier transform to include
functions in L P (R) for 1 < p < 2. Unfortunately, further extensions become more technical. The Fourier transform of
functions in L p for the range 2 < p < °° requires the study of distributions (Katznelson 1976). In fact, it can be shown
that there are functions in L p with p>2 so that the Fourier transform is not defined as a function (Stein & Weiss
1971).

Fourier-Stieltjes transform

The Fourier transform of a finite Borel measure fi on R" is given by (Pinsky 2002):

•J IV

m= / e- 2 ™-^

This transform continues to enjoy many of the properties of the Fourier transform of integrable functions. One
notable difference is that the Riemann-Lebesgue lemma fails for measures (Katznelson 1976). In the case that
d/i =J{x) dx, then the formula above reduces to the usual definition for the Fourier transform of/ In the case that \i is
the probability distribution associated to a random variable X, the Fourier-Stieltjes transform is closely related to the
characteristic function, but the typical conventions in probability theory take e' 1 ' 5 instead of e~ mx% (Pinsky 2002). In
the case when the distribution has a probability density function this definition reduces to the Fourier transform
applied to the probability density function, again with a different choice of constants.

The Fourier transform may be used to give a characterization of continuous measures. Bochner's theorem
characterizes which functions may arise as the Fourier-Stieltjes transform of a measure (Katznelson 1976).
Furthermore, the Dirac delta function is not a function but it is a finite Borel measure. Its Fourier transform is a
constant function (whose specific value depends upon the form of the Fourier transform used).

Fourier transform

Tempered distributions

The Fourier transform maps the space of Schwartz functions to itself, and gives a homeomorphism of the space to
itself (Stein & Weiss 1971). Because of this it is possible to define the Fourier transform of tempered distributions.
These include all the integrable functions mentioned above and have the added advantage that the Fourier transform
of any tempered distribution is again a tempered distribution.

The following two facts provide some motivation for the definition of the Fourier transform of a distribution. First let
/and g be integrable functions, and let f and g be their Fourier transforms respectively. Then the Fourier transform
obeys the following multiplication formula (Stein & Weiss 1971),

/ f{x)g{x)dx= I f(x)g(x)dx.

Secondly, every integrable function /defines a distribution T by the relation

Tf(ip) = / f(x)(f(x)dx for all Schwartz functions q>.
In fact, given a distribution T, we define the Fourier transform by the relation

T(ip) = T(ip) for all Schwartz functions 99.
It follows that

Tf = T f .

Distributions can be differentiated and the above mentioned compatibility of the Fourier transform with
differentiation and convolution remains true for tempered distributions.

Locally compact abelian groups

The Fourier transform may be generalized to any locally compact abelian group. A locally compact abelian group is
an abelian group which is at the same time a locally compact Hausdorff topological space so that the group
operations are continuous. If G is a locally compact abelian group, it has a translation invariant measure \i, called
Haar measure. For a locally compact abelian group G it is possible to place a topology on the set of characters q so
that q is also a locally compact abelian group. For a function / in L (G) it is possible to define the Fourier
transform by (Katznelson 1976):

f{Z) = Jt{x)f(x)dii for any (eG.
Locally compact Hausdorff space

The Fourier transform may be generalized to any locally compact Hausdorff space, which recovers the topology but
loses the group structure.

Given a locally compact Hausdorff topological space X, the space A=C (X) of continuous complex-valued functions
on X which vanish at infinity is in a natural way a commutative C*-algebra, via pointwise addition, multiplication,
complex conjugation, and with norm as the uniform norm. Conversely, the characters of this algebra A, denoted
$^4 , are naturally a topological space, and can be identified with evaluation at a point of x, and one has an is
isomorphism Cq(X) — > Co(3?a)- ^ n tne case wnere X=R is the real line, this is exactly the Fourier transform.

Fourier transform

Non-abelian groups

The Fourier transform can also be defined for functions on a non-abelian group, provided that the group is compact.
Unlike the Fourier transform on an abelian group, which is scalar-valued, the Fourier transform on a non-abelian
group is operator-valued (Hewitt & Ross 1971, Chapter 8). The Fourier transform on compact groups is a major tool
in representation theory (Knapp 2001) and non-commutative harmonic analysis.

Let G be a compact Hausdorff topological group. Let 2 denote the collection of all isomorphism classes of
finite-dimensional irreducible unitary representations, along with a definite choice of representation lr a ' on the
Hilbert space H of finite dimension d for each a € 2. If [x is a finite Borel measure on G, then the Fourier-Stieltjes
transform of \y is the operator on H defined by

where JJ^is the complex-conjugate representation of U acting on H . As in the abelian case, if \i is absolutely
continuous with respect to the left-invariant probability measure X on G, then it is represented as

dfi = fdX
for some/G L (X). In this case, one identifies the Fourier transform off with the Fourier-Stieltjes transform of \y.
The mapping fj,h^ p, defines an isomorphism between the Banach space M(G) of finite Borel measures (see rca
space) and a closed subspace of the Banach space C (2) consisting of all sequences E=(E ) indexed by 2 of
(bounded) linear operators E : H — > H for which the norm

||£|| = S up||£ ff ||°

<xG£

is finite. The "convolution theorem" asserts that, furthermore, this isomorphism of Banach spaces is in fact an
isomorphism of C algebras into a subspace of C^(2), in which M(G) is equipped with the product given by
convolution of measures and C (2) the product given by multiplication of operators in each index o.
The Peter-Weyl theorem holds, and a version of the Fourier inversion formula (Plancherel's theorem) follows: if
/€L 2 (G), then

f(g) = Y,dMf(<r)U^)

where the summation is understood as convergent in the L sense.

The generalization of the Fourier transform to the noncommutative situation has also in part contributed to the
development of noncommutative geometry. In this context, a categorical generalization of the Fourier transform to
noncommutative groups is Tannaka-Krein duality, which replaces the group of characters with the category of
representations. However, this loses the connection with harmonic functions.

Alternatives

In signal processing terms, a function (of time) is a representation of a signal with perfect time resolution, but no
frequency information, while the Fourier transform has perfect frequency resolution, but no time information: the
magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by
phase (argument of the Fourier transform at a point), and standing waves are not localized in time - a sine wave
continues out to infinity, without decaying. This limits the usefulness of the Fourier transform for analyzing signals
that are localized in time, notably transients, or any signal of finite extent.

As alternatives to the Fourier transform, in time-frequency analysis, one uses time-frequency transforms or
time-frequency distributions to represent signals in a form that has some time information and some frequency
information - by the uncertainty principle, there is a trade-off between these. These can be generalizations of the
Fourier transform, such as the short-time Fourier transform or fractional Fourier transform, or can use different
functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the

Fourier transform

(continuous) Fourier transform being the continuous wavelet transform. (Boashash 2003).

Applications

Analysis of differential equations

Fourier transforms and the closely related Laplace transforms are widely used in solving differential equations. The
Fourier transform is compatible with differentiation in the following sense: if fix) is a differentiable function with
Fourier transform /(£), then the Fourier transform of its derivative is given by 27ri£/(£)- This can be used to
transform differential equations into algebraic equations. Note that this technique only applies to problems whose
domain is the whole set of real numbers. By extending the Fourier transform to functions of several variables partial
differential equations with domain R n can also be translated into algebraic equations.

FT-NMR, FT-IR, FT-NIR and MRI

The Fourier transform is also used in nuclear magnetic resonance (NMR) and in other kinds of spectroscopy, e.g.
infrared (FT-IR). In NMR an exponentially-shaped free induction decay (FID) signal is acquired in the time domain
and Fourier-transformed to a Lorentzian line-shape in the frequency domain. The Fourier transform is also used in
magnetic resonance imaging (MRI) and mass spectrometry.

Domain and range of the Fourier transform

It is often desirable to have the most general domain for the Fourier transform as possible. The definition of Fourier
transform as an integral naturally restricts the domain to the space of integrable functions. Unfortunately, there is no
simple characterizations of which functions are Fourier transforms of integrable functions (Stein & Weiss 1971). It is
possible to extend the domain of the Fourier transform in various ways, as discussed in generalizations above. The
following list details some of the more common domains and ranges on which the Fourier transform is defined.

• The space of Schwartz functions is closed under the Fourier transform. Schwartz functions are rapidly decaying
functions and do not include all functions which are relevant for the Fourier transform. More details may be found
in (Stein & Weiss 1971).

• The space If maps into the space L q , where lip + \lq=\ and 1 < p < 2 (Hausdorff- Young inequality).

• In particular, the space L is closed under the Fourier transform, but here the Fourier transform is no longer
defined by integration.

• The space L of Lebesgue integrable functions maps into C , the space of continuous functions that tend to zero at
infinity - not just into the space lf° of bounded functions (the Riemann-Lebesgue lemma).

• The set of tempered distributions is closed under the Fourier transform. Tempered distributions are also a form of
generalization of functions. It is in this generality that one can define the Fourier transform of objects like the
Dirac comb.

Other notations

Other common notations for /(£) are: /(f), F(£), T (/)(£), (Tf) (£), T(f), F{u), F(jw),
J-\f\ and J 7 (fit)) .Though less commonly other notations are used. Denoting the Fourier transform by a
capital letter corresponding to the letter of function being transformed (such as fix) and F(g)) is especially common
in the sciences and engineering. In electronics, the omega (<w) is often used instead of g due to its interpretation as
angular frequency, sometimes it is written as F(jco), where j is the imaginary unit, to indicate its relationship with the
Laplace transform, and sometimes it is written informally as F(2nf) in order to use ordinary frequency.

Fourier transform

The interpretation of the complex function f(f\may be aided by expressing it in polar coordinate form:
/(£) = A(^)e %v ^ m terms of the two real functions A(|) and cp(g) where:

^(0 = 1/(01.

is the amplitude and

is the phase (see arg function).

Then the inverse transform can be written:

f(x) = f°° A{£) e'^+rtO) df ,

which is a recombination of all the frequency components of fix). Each component is a complex sinusoid of the
form e m whose amplitude is A(|) and whose initial phase angle (at x = 0) is 99(g).

The Fourier transform may be thought of as a mapping on function spaces. This mapping is here denoted Jfand
■?"(/) is used to denote the Fourier transform of the function/. This mapping is linear, which means that J 7 can
also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra
of applying a linear transformation to a vector (here the function/) can be used to write J-"f instead of J-(f\
Since the result of applying the Fourier transform is again a function, we can be interested in the value of this
function evaluated at the value g for its variable, and this is denoted either as J-(f) (£) or as (•?""/) (£)• Notice that
in the former case, it is implicitly understood that J^is applied first to /and then the resulting function is evaluated
at g, not the other way around.

In mathematics and various applied sciences it is often necessary to distinguish between a function /and the value of
/when its variable equals x, denoted f(x). This means that a notation like JT(/(x)) formally can be interpreted as
the Fourier transform of the values of/ at x. Despite this flaw, the previous notation appears frequently, often when a
particular function or a function of a particular variable is to be transformed. For example, jF(rect(s) ) = sinc(^)
is sometimes used to express that the Fourier transform of a rectangular function is a sine function, or
J?(f( x _|_ xq)) = J-(f(x))e 27T ^ X0 is used to express the shift property of the Fourier transform. Notice, that the
last example is only correct under the assumption that the transformed function is a function of x, not of x .

Other conventions

There are three common conventions for defining the Fourier transform. The Fourier transform is often written in
terms of angular frequency: w = 2n'£, whose units are radians per second.
The substitution g = <y/(2jt) into the formulas above produces this convention:

/>)= f f(x)e-^dx.
Under this convention, the inverse transform becomes:

/(*) = 7^w f h^y^du.

Unlike the convention followed in this article, when the Fourier transform is defined this way, it is no longer a

unitary transformation on L (R n ). There is also less symmetry between the formulas for the Fourier transform and its

inverse.

Another popular convention is to split the factor of Clxf evenly between the Fourier transform and its inverse, which

leads to definitions:

Fourier transform

fir)

"(2:

'^ 2 L

fi^e^du.

Under this convention, the Fourier transform is again a unitary transformation on L (R ). It also i
symmetry between the Fourier transform and its inverse.

Variations of all three conventions can be created by conjugating the complex-exponential kernel of both the forward
and the reverse transform. The signs must be opposites. Other than that, the choice is (again) a matte

Summary of popular forms of the Fourier transform

ordinary frequency g (hertz)

angular frequency to (rad/s)

/i(0 = J n f(x)e-^<dx = / 2 (2<) = (2^)"/ 2 / 3 (2<)

j,(.)g J j f /(*- J ,=/ 1 g)=(^i(„)

The ordinary-frequency convention (which is used in this article) is the one most often found in the mathematics
literature. In the physics literature, the two angular-frequency conventions are more commonly used.
As discussed above, the characteristic function of a random variable is the same as the Fourier-Stieltjes transform of
its distribution measure, but in this context it is typical to take a different convention for the constants. Typically
characteristic function is defined E(e lt ' X ) = / e^'^dfix^)- ^ s m tne case °f tne "non-unitary angular

frequency" convention above, there is no factor of 2jz appearing in either of the integral, or in the exponential.
Unlike any of the conventions appearing above, this convention takes the opposite sign in the exponential.

Tables of important Fourier transforms

The following tables record some closed form Fourier transforms. For functions f(x) , g(x) and h(x) denote their
Fourier transforms by f , g, and ^ respectively. Only the three most common conventions are included. It is
sometimes useful to notice that entry 105 gives a relationship between the Fourier transform of a function and the
original function, which can be seen as relating the Fourier transform and its inverse.

Functional relationships

The Fourier transforms in this table may be found in (Erdelyi 1954) or the appendix of (Kammler 2000).

Fourier transform

Function

Fourier transform

Remarks

unitary, ordinary frequency

unitary, angular frequency

non-unitary, angular
frequency

fc) = £ o me-***dx

K ^ = wJ x J {x)e ^ dx

f{ V )= j^J(x)e—dx

Definition

101

a-f{x)+b-g(x)

a-ho+b-m

a-/H + 6-ffH

a • />) + 6 • $(«/)

Linearity

102

f(x - a)

e- 2 ™ ? /(0

e--/(c)

e— /»

Shift in
domain

103

e 2 — f(x)

/(?-«)

/> - 2™)

/> - 27m)

Shift in
frequency
domain,
dual of
102

104

/(ax)

£'(:)

^(:)

Scaling in
the time
domain. If

large, then

/Mis

concentratec
around

ii?

flattens.

105

/»

f(-0

/(-")

2tt/(- I /)

Duality.
Here f

calculated
using the

method as

Fourier

transform

column.

Results

swapping
"dummy"
variables

^or^or

106

d"f(x)
dx"

(27TiOV(0

MVM

H"/»

107

x-f(x)

/ 1 \"«f/K)

V2tJ df»

.„g/H

1 dv n

This is the
dual of
106

Fourier transform

108

(/*<?)(*)

Horn

V^fHaH

/»$(")

The

f*9

denotes the
convolution
of /and
g — this

theorem

109

f{x)g[x)

(f*9)(0

(/*s)H

^(/*0M

This is the
dual of
108

110

For /(x) a purely

/ (-0 = W)

/(- W ) = 7h

j(- v ) = W)

symmetry.

indicates
the

conjugate.

111

For /(x) a purely

f(oj\ f(£,) an ^ i /(V)are purely real even functions.

real even function

112

For /(x) a purely

/(a;), /(O anc ' /(^)are purely imaginary odd functions.

real odd function

Square-integrable functions

The Fourier transforms in this table may be found in (Campbell & Foster 1948), (Erdelyi 1954), or the appendix of
(Kammler 2000).

Fourier transform
unitary, ordinary frequency

Fourier transform
unitary, angular frequency

Fourier transform
non-unitary, angular

frequency

/(e) = f° f{x)e-™°*dx / H = -L r J{x)e—dx /» = (^ f(x),

■J— oo yZTT J —co -/— oo

<£)

'(£;)

°(£)

The rectangular
pulse and the

noruutUz.ed sine-
function, here

defined as

sill(.T.V)A.T.V)

Dual of rule
201. The

function is an
ideal low-pass
filter, and the
sine function is
the non-causal

response of such
a filter.

Fourier transform

203

sine 2 (ax)

*-(9

^r*(£)

R' tri (i)

The function
tri(*) is the
triangular
function

204

tri(ax)

h--(!)

\/W V2^J

R- Sinc2 (i)

Dual of rule

203.

20:-

e— «(x)

a + ii/

The function
u(x) is the
Heaviside unit

V^(a + l cv)

step function

and fl >0.

21 th

1 _^

This shows that,
for the unitary
Fourier

transforms, the
Gaussian
function
exp(-a:r 2 ) is its
own Fourier
transform for
some choice of
a. For this to be

Re(a)>0.

207

e -l-l

vf-^

2a
a 2 + z/ 2

For a>0. That
is, the Fourier
transform of a

a 2 + 47r 2^2

decaying

exponential

function is a

Lorentzian

function.

208

J_M

^(-zr-f/„_ 1 (2<)

■vf^ectQ

The functions J
(x) are the n-th

functions of the

• \/l — 47r 2 ^ 2 rect(7rf )

first kind. The

functions U (x)

are the

Chebyshev

polynomial of

the second kind.

See 315 and 316

below.

21 W

sech(ax)

^ d '(^)

^ sech S")

a S6Ch (2^")

Hyperbolic
Fourier

Fourier transform 28

Distributions

The Fourier transforms in this table may be found in (Erdelyi 1954) or the appendix of (Kammler 2000).

Function Fourier transform Fourier transform Fourier transform Remarks

unitary, ordinary frequency unitary, angular frequency non-unitary, angular frequency

/(X) /(£) = /_" /(-)e- 2 ^ dx / H = _L |_~ f{x)e -^ dx /(,) = |_" f( x)e — dx

8{£) V2^-5(u) 2k5(l>) The distribution 5(g) denotes the Dirac delta f

1 11 Dual of rule 301.

2n5(v - a)

This follows from 103 and 301.

.(ax?)

U"^ + ^ + 9

5{{jJ — a) + 5{uj + a) n (8(v — a) + 5(v + a)) This follows from ru l es Wl and 303 using Eu

Max) = {e- + e— )/2

6(u> + a)- S(cj - a) itt {5{v + a) - 5{v - a)) ™ s foll ™ s fr om 101 ™* ™ "sing

2 sin(ax) = (e*» - e"^)/(2i)

v^ V 4a 4/

-/f sin (V-i) ^^(^"i) "yf sm (^"ij

" S (n) (e) »"^ W (w) 27ri"<jM (!/) Here, » is a natural number and tfW (f )is th

distribution dcn\ali\c ol the Dirac delta fund
This rule follows from rules 107 and 30 1 . Coi
this rule with 101. wc can transform all polyni

y n (£) Hjf —ins°"a(v) Here sgn(g) is the sign function. Note that 1/x

~~ V ~0 S S n V u; / distribution. It is necessary to use the Cauchy

principal vain hen 1 tin iaain-,1 li it I
functions. This rule is useful in stud; inc. die I
transform.

■g$&™ -^if-«) -^■^-m -7^9^:

l/x" is the homogeneous dislribulion defined 1
regularizing the singularity via

in(7ra/2)r(a+l) -2 sin(7ra/2)l> + 1) 2 sin( TO /2)r(a + 1) If Rea>-l,then |x|" is a locally integrable

|27rf|»+ 1 v^ M" +1 ' Ic^l"-^ 1 function, and so a tempered distribution. The:

i,in plane l he s] i ni| d

distributions. It admits a unique meromorphie
extension to a tempered distribution, also dene
|x| Q for a * -2, -4, ... (See homogeneous

Fourier transform

£ S(x-nT)

E'- f

The

dualofrul

309. This time the

3 ourier

Cauchy

value.

Ilk

function k

x) is the Heaviside

nit step f

this

follows fro

m rules 101, 301, an

d 312.

Tin

function i

known as the Dirac

comb lu

Ihi

result can

be derived from 302

and 102

£ e™ = 2n £ 6(x + 27vk).,

2rect«)

2(-i)"T„(2Qrect«)

V 7 ! - 4tt 2 C 2

2 (-*)"T»rect (|) 2(-i)"T n (i/)rect (0

The function ./ (\

1 His in a generalization of 3 1 5. The func
the n-th order Bessel function of first kin
function T (,v) is the Chebyshev polynon

Two-dimensional functions

ordinary frequency

;.,{,) = J[f(x,y)e-" <s "+ e «<d

„,^ = lJJf<,.y)e-"^>*

angular frequency

: (^^ y ) = JJf(x,y)e-^+^d

\ab\

1 -("1/

2v\ab\ e

|afe| e

Fourier transform

Formulas for general n -dimensional functions

Function

Fourier transform
unitary, ordinary frequency

Fourier transform
unitary, angular frequency

Fourier transform

non-unitary, angular

frequency

Remarks

/(*)

fc) = j^me-^rx

^=^L f{x)e ^ dnx

/( " )= L f{x)e ~ ixvirx

Fourier transform

xp.ntMXi-MY

■- s r(6 + i)\t\-wv- s
■j r „ /2+4 (27riei)

^r(5 + i)|-|
■■WCM)

fund

on of

the ii

terval

[0,1]

The

lunclion

r(x)

sthe

Bessel
function of
the first
kind with

n/2 + d.
Taking
n = 2 and
6 =

102. (Stein
& Weiss
1971, Thm.

4.13)

See Riesz

potential.
The

formula
also holds

but then the

function

Fourier

transform

understood
as suitably

regularized

tempered

distributions.

See

homogeneou

distribution.

Fourier transform

See also

Fourier series
Fast Fourier transform
Laplace transform
Discrete Fourier transform

• DFT matrix

Discrete-time Fourier transform
Fourier-Deligne transform
Fractional Fourier transform
Linear canonical transform
Fourier sine transform
Short-time Fourier transform
Fourier inversion theorem
Analog signal processing
Transform (mathematics)
Integral transform

• Hartley transform

• Hankel transform

References

• Boashash, B., ed. (2003), Time-Frequency Signal Analysis and Processing: A Comprehensive Reference, Oxford:
Elsevier Science

• Bochner S., Chandrasekharan K. (1949), Fourier Transforms, Princeton University Press

• Bracewell, R. N. (2000), The Fourier Transform and Its Applications (3rd ed.), Boston: McGraw-Hill.

• Campbell, George; Foster, Ronald (1948), Fourier Integrals for Practical Applications, New York: D. Van
Nostrand Company, Inc..

• Duoandikoetxea, Javier (2001), Fourier Analysis, American Mathematical Society, ISBN 0-8218-2172-5.

• Dym, H; McKean, H (1985), Fourier Series and Integrals, Academic Press, ISBN 978-0122264511.

• Erdelyi, Arthur, ed. (1954), Tables of Integral Transforms, 1, New Your: McGraw-Hill

• Fourier, J. B. Joseph (1822), The'orie Analytique de la Chaleur [1] , Paris

• Grafakos, Loukas (2004), Classical and Modern Fourier Analysis, Prentice-Hall, ISBN 0- 1 3-035399-X.

• Hewitt, Edwin; Ross, Kenneth A. (1970), Abstract harmonic analysis. Vol. II: Structure and analysis for compact
groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band
152, Berlin, New York: Springer- Verlag, MR0262773.

• Hormander, L. (1976), Linear Partial Differential Operators, Volume 1, Springer- Verlag, ISBN 978-3540006626.

• James, J.F. (2002), A Student's Guide to Fourier Transforms (2nd ed.), New York: Cambridge University Press,
ISBN 0-521-00428-4.

• Kaiser, Gerald (1994), A Friendly Guide to Wavelets, Birkhauser, ISBN 0-8176-371 1-7

• Kammler, David (2000), A First Course in Fourier Analysis, Prentice Hall, ISBN 0-13-578782-3

• Katznelson, Yitzhak (1976), An introduction to Harmonic Analysis, Dover, ISBN 0-486-63331-4

• Knapp, Anthony W. (2001), Representation Theory of Semisimple Groups: An Overview Based on Examples ,
Princeton University Press, ISBN 978-0-691-09089-4

• Pinsky, Mark (2002), Introduction to Fourier Analysis and Wavelets, Brooks/Cole, ISBN 0-534-37660-6

• Polyanin, A. D.; Manzhirov, A. V. (1998), Han, a wkoflnl ralEqi alums, Boca Raton: CRC Press,
ISBN 0-8493-2876-4.

• Rudin, Walter (1987), Real and Complex Analysis (Third ed.), Singapore: McGraw Hill, ISBN 0-07-100276-6.

Fourier transform

• Stein, Elias; Shakarchi, Rami (2003), Fourier Analysis: An introduction, Princeton University Press,
ISBN 0-691-1 1384-X.

• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.:
Princeton University Press, ISBN 978-0-691-08078-9.

• Wilson, R. G. (1995), Fourier Series and Optical Transform Techniques in Contemporary Optics, New York:
Wiley, ISBN 0471303577.

• Yosida, K. (1968), Functional Analysis, Springer- Verlag, ISBN 3-540-58654-7.

External links

• Fourier Series Applet (Tip: drag magnitude or phase dots up or down to change the wave form).

• Stephan Bernsee's FFTlab [4] (Java Applet)

• Tables of Integral Transforms at Eq World: The World of Mathematical Equations.

• Weisstein, Eric W., "Fourier Transform J " from MathWorld.

• Fourier Transform Module by John H. Mathews

• The DFT "a Pied": Mastering The Fourier Transform in One Day [8] at The DSP Dimension

• An Interactive Flash Tutorial for the Fourier Transform

References

[1] http://books.google.com/?id=TDQJAAAAIAAJ&printsec=frontcover&dq=Th%C3%A9orie+analytique+de+la+chaleur&q

[2] http://books.google.com/?id=QCcWlh835pwC

[3] http://www.weslga.edu/~jhasbun/osp/Fourier.htm

[4] http://www.dspdimension.com/fftlab/

[5] http://eqworld.ipmnet.ru/en/auxi liarv/aux intlrans.htm

|'i| Imp in 'ill ■■ill ■■Hi in I mil i i I i in l"im I I

[7] http://math.fullerton.edu/nialliew s/c2003/K>uriei TransformMod.html

[8] http://www.dspdiinension.eoin/admin/dft-a-pied/

[9] http://www.fourier-series.eom/f transform/index.html

Discrete Fourier transform

Fourier transforms

Continuous Fourier transform

Fourier series

Discrete Fourier transform

Discrete-time Fourier

I ran-- form

Related transforms

In mathematics, the discrete Fourier transform (DFT) is a specific kind of Fourier transform, used in Fourier
analysis. It transforms one function into another, which is called the frequency domain representation, or simply the
DFT, of the original function (which is often a function in the time domain). But the DFT requires an input function
that is discrete and whose non-zero values have a limited (finite) duration. Such inputs are often created by sampling
a continuous function, like a person's voice. Unlike the discrete-time Fourier transform (DTFT), it only evaluates
enough frequency components to reconstruct the finite segment that was analyzed. Using the DFT implies that the
finite segment that is analyzed is one period of an infinitely extended periodic signal; if this is not actually true, a
window function has to be used to reduce the artifacts in the spectrum. For the same reason, the inverse DFT cannot
reproduce the entire time domain, unless the input happens to be periodic (forever). Therefore it is often said that the
DFT is a transform for Fourier analysis of finite-domain discrete-time functions. The sinusoidal basis functions of
the decomposition have the same properties.

The input to the DFT is a finite sequence of real or complex numbers (with more abstract generalizations discussed
below), making the DFT ideal for processing information stored in computers. In particular, the DFT is widely
employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, to solve
partial differential equations, and to perform other operations such as convolutions or multiplying large integers. A
key enabling factor for these applications is the fact that the DFT can be computed efficiently in practice using a fast
Fourier transform (FFT) algorithm.

FFT algorithms are so commonly employed to compute DFTs that the term "FFT" is often used to mean "DFT" in
colloquial settings. Formally, there is a clear distinction: "DFT" refers to a mathematical transformation or function,
regardless of how it is computed, whereas "FFT" refers to a specific family of algorithms for computing DFTs. The
terminology is further blurred by the (now rare) synonym finite Fourier transform for the DFT, which apparently
predates the term "fast Fourier transform" (Cooley et al., 1969) but has the same initialism.

Definition

The sequence of N complex numbers x ..., x is transformed into the sequence of N complex numbers X ...,
X by the DFT according to the formula:

AT-1

X k = Y, x n e-'^ kn fc = 3 .

where i is the imaginary unit and e ^pis a primitive N'th root of unity. (This expression can also be written in ten

of a DFT matrix; when scaled appropriately it becomes a unitary matrix and the X can thus be viewed as coefficie

of x in an orthonormal basis.)

The transform is sometimes denoted by the symbol J? , as in X = J- {x} or J 7 (x) or J^x •

The inverse discrete Fourier transform (IDFT) is given by

Discrete Fourier transform

-V£ x * Vhn

x n = ^Y. X ^ n = 0,...,N-l.

A simple description of these equations is that the complex numbers X k represent the amplitude and phase of the
different sinusoidal components of the input "signal" X n . The DFT computes the X k fr° m the X n , while the
IDFT shows how to compute the £ n as a sum of sinusoidal components (\ /N)X k e^ Lkn w i tri frequency k/N
cycles per sample. By writing the equations in this form, we are making extensive use of Euler's formula to express
sinusoids in terms of complex exponentials, which are much easier to manipulate. In the same way, by writing X k
in polar form, we obtain the sinusoid amplitude A k /N&nd phase ^> fe from the complex modulus and argument of
X k , respectively:

A k = \X k \ = y/Re(X k f + Jm{X k y,

<p k = arg(X fc ) = atan2(lm(X fe ),Re(X fc )),
where atan2 is the two-argument form of the arctan function. Note that the normalization factor multiplying the DFT
and IDFT (here 1 and 1/N) and the signs of the exponents are merely conventions, and differ in some treatments. The
only requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that the
product of their normalization factors be UN. A normalization of 1 /\/jVfor both the DFT and IDFT makes the
transforms unitary, which has some theoretical advantages, but it is often more practical in numerical computation to
perform the scaling all at once as above (and a unit scaling can be convenient in other ways).

(The convention of a negative sign in the exponent is often convenient because it means that X k is the amplitude of
a "positive frequency" 2-Trfc/iV ■ Equivalently, the DFT is often thought of as a matched filter: when looking for a
frequency of +1, one correlates the incoming signal with a frequency of -1.)
In the following discussion the terms "sequence" and "vector" will be considered interchangeable.

Properties
Completeness

The discrete Fourier transform is an invertible, linear transformation

with C denoting the set of complex numbers. In other words, for any N>0, an iV-dimensional complex vector has a
DFT and an IDFT which are in turn iV-dimensional complex vectors.

Orthogonality

The vectors e ^kn form an orthogonal basis over the set of iV-dimensional complex vectors:

t(^ kn ){e-^ n ) = NS M

where S kk , is the Kronecker delta. This orthogonality condition can be used to derive the formula for the IDFT from
the definition of the DFT, and is equivalent to the unitarity property below.

Discrete Fourier transform

The Plancherel theorem and Parseval's theorem

If X .and Y are the DFTs of x and y respectively then the Plancherel theorem states:
JV-1 i JV-1

n =0 VV fc=0

where the star denotes complex conjugation. Parseval's theorem is a special case of the Plancherel theorem and

stales:

JV-1 i JV-1

These theorems are also equivalent to the unitary condition below.

Periodicity

If the expression that defines the DFT is evaluated for all integers k instead of just for k = 0, . . . , N — 1, then
the resulting infinite sequence is a periodic extension of the DFT, periodic with period N.
The periodicity can be shown directly from the definition:

JV-1 JV-1 JV-1

X k+N d = f Y, x n e-^ k+N > = J2 x n e~^ kn e^= ^ x n e~^ kn = X k .

n=0 n=0 1 n=0

Similarly, it can be shown that the IDFT formula leads to a periodic extension.

The shift theorem

Multiplying £ n by a linear phase e ^-nmfoi some integer m corresponds to a circular shift of the output X k '■
Xfcis replaced by X^_ m , where the subscript is interpreted modulo N (i.e., periodically). Similarly, a circular
shift of the input ^corresponds to multiplying the output X^by a linear phase. Mathematically, if {^J
represents the vector x then
if-F(K» fc =X fc

then p({ Xn . e ¥— }) fc = Xk _ m

and H{Xn-m})k = X k ■ e~'^ km

Circular convolution theorem and cross-correlation theorem

The convolution theorem for the continuous and discrete time Fourier transforms indicates that a convolution of two
infinite sequences can be obtained as the inverse transform of the product of the individual transforms. With
sequences and transforms of length N, a circularity arises:

N '

k=0

fc=0 \l=0 / \m=0

JV-1 JV-1 / 1 JV-1

■•)

The quantity in parentheses is for all values of m except those of the form n — I — pN , where p is any integer.
At those values, it is 1 . It can therefore be replaced by an infinite sum of Kronecker delta functions, and we continue
accordingly. Note that we can also extend the limits of m to infinity, with the understanding that the x and y
sequences are defined as outside [0,N-1]:

Discrete Fourier transform

t-' {X ■ Y} n = £ Xl £ y™ I E *".(«-'-**

iV-1 oo / oo

E X M E »»-«-pJV = ( X *yN)n

1=0 \p=-oo

which is the convolution of the x sequence with a periodically extended y sequence defined by:
(Yn)™ = E »("-*")■

p=-oo

Similarly, it can be shown that:

iV-1

•7 7-1 {X* ■ Y} n = E x *\ ■ (VN) n +i = f (x*y N ) n ,
1=0
which is the cross-correlation of x and yN •

A direct evaluation of the convolution or correlation summation (above) requires O (A/ 2 ) operations for an output
sequence of length N. An indirect method, using transforms, can take advantage of the 0(iVlog N) efficiency of
the fast Fourier transform (FFT) to achieve much better performance. Furthermore, convolutions can be used to
efficiently compute DFTs via Rader's FFT algorithm and Bluestein's FFT algorithm.

Methods have also been developed to use circular convolution as part of an efficient process that achieves normal
(non-circular) convolution with an xor ysequence potentially much longer than the practical transform size (N).
Two such methods are called overlap-save and overlap-add .

Convolution theorem duality

It can also be shown that:

iv-i

f (x • y} k = Y, x ™-y™- e ~^ kn

= — (X * Y]>j)fe, which is the circular convolution of X an d Y-

Trigonometric interpolation polynomial

The trigonometric interpolation polynomial

p(t) = ^ [X + X ie lt + ■■■ + X N/2 _ ie (N ^ lt + X N/2 cos(Nt/2) + X N/2+1 e ( - N ^ +1 ^ + ■ • ■ + X N _

N even ,

P(t) = ^ [ X ° + X ^ + ■■■ + X VN/2\e VN/m + ^ W 2j+l^ LAf/2Jit + ■ ■ ■ + X*_ie- tt ]far N

odd,
where the coefficients X are given by the DFT of x above, satisfies the interpolation property p(27rn/N) — X n
forn = 0,...,N -1.
For even N, notice that the Nyquist component — ' cos(Nt/2) i s handled specially.

Discrete Fourier transform

This interpolation is not unique: aliasing implies that one could add N to any of the complex-sinusoid frequencies
(e.g. changing e _i *to gi(JV-l)t) without changing the interpolation property, but giving different values in
between the X n points. The choice above, however, is typical because it has two useful properties. First, it consists
of sinusoids whose frequencies have the smallest possible magnitudes, and therefore minimizes the mean-square
slope / \p'(t) \ 2 dt of the interpolating function. Second, if the X n are real numbers, then p(f ) is real as well.

In contrast, the most obvious trigonometric interpolation polynomial is the one in which the frequencies range from
to ]V — 1 (instead of roughly — TV/ 2 to -\-N/1s& above), similar to the inverse DFT formula. This
interpolation does not minimize the slope, and is not generally real-valued for real X n ; its use is a common mistake.

The unitary DFT

Another way of looking at the DFT i
Vandermonde matrix:

where

: that in the above discussion, the DFT can be expressed as £

0-0

,01

O-(JV-l)

iV N

LO N

1-0

, ,1-1

. i-(JV-i)

(iV-l)-O (JV-l)-l

-27TI/.Y

u N = e ■
is a primitive Nth root of unity. The inverse transform is then given by the inverse of the above matrix:

With unitary normalization constants 1/yJV, the DFT becomes a unitary transformation, defined by a unitary

U = F/v / iV

IT 1 = U*

|det(U)| = l
where det() is the determinant function. The determinant is the product of the eigenvalues, which are always ±1 or
±i as described below. In a real vector space, a unitary transformation can be thought of as simply a rigid rotation
of the coordinate system, and all of the properties of a rigid rotation can be found in the unitary DFT.
The orthogonality of the DFT is now expressed as an orthonormality condition (which arises in many areas of
mathematics as described in root of unity):

,„=o

If Xi s defined as the unitary DFT of the vector xthen
JV-1

n=0

and the Plancherel theorem is expressed as:

JV-1 JV-1

E *»»; = E x ^:

71=0 fc=0

If we view the DFT as just a coordinate transformation which simply specifies the components of a vector in a new
coordinate system, then the above is just the statement that the dot product of two vectors is preserved under a
unitary DFT transformation. For the special case X = y, this implies that the length of a vector is preserved as

Discrete Fourier transform

well — this is just Parseval's theorem:

JV-1 JV-1

E ix.i 2 = x; i^i 2

n,=0 fc=0

Expressing the inverse DFT in terms of the DFT

A useful property of the DFT is that the inverse DFT can be easily expressed in terms of the (forward) DFT, via
several well-known "tricks". (For example, in computations, it is often convenient to only implement a fast Fourier
transform corresponding to one transform direction and then to get the other transform direction from the first.)
First, we can compute the inverse DFT by reversing the inputs:

T-\{x n }) = T({x N _ n })/N

(As usual, the subscripts are interpreted modulo N; thus, for n = Q, we have £jv-0 = Xq.)
Second, one can also conjugate the inputs and outputs:

JP'- 1 (x)=^(x*)7iV
Third, a variant of this conjugation trick, which is sometimes preferable because it requires no modification of the
data values, involves swapping real and imaginary parts (which can be done on a computer simply by modifying
pointers). Define swap( X n ) as X n with its real and imaginary parts swapped — that is, if x n = a + bi then
swap( X n ) is fa -\- ai ■ Equivalently, swap( X n ) equals ix* ■ Then

^ _1 (x) = swap(T(swap(x)})/7V
That is, the inverse transform is the same as the forward transform with the real and imaginary parts swapped for
both input and output, up to a normalization (Duhamel et ai, 1988).

The conjugation trick can also be used to define a new transform, closely related to the DFT, that is involutary — that
is, which is its own inverse. In particular, T(x) = J-(x*)/vN^ s clearly its own inverse: T(T(li.)) — X- A
closely related involutary transformation (by a factor of (1+0 Nl) is 77(x) = J-"((l + i)n*)/ \/2N > since the
(1 + i) factors in H ( H (x)) cancel the 2. For real inputs X, the real part of 7/(x)is none other than the
discrete Hartley transform, which is also involutary.

Eigenvalues and eigenvectors

The eigenvalues of the DFT matrix are simple and well-known, whereas the eigenvectors are complicated, not

unique, and are the subject of ongoing research.

Consider the unitary form TJ defined above for the DFT of length N, where

m,n Vn n Vn

This matrix satisfies the equation:

U 4 = I.

This can be seen from the inverse properties above: operating TJ twice gives the original data in reverse order, so
operating TJfour times gives back the original data and is thus the identity matrix. This means that the eigenvalues
\ satisfy a characteristic equation:

A 4 = l.

Therefore, the eigenvalues of TJare the fourth roots of unity: \ is +1, -1, +i, or -i.

Since there are only four distinct eigenvalues for this JV X N msA nx, they have some multiplicity. The multiplicity
gives the number of linearly independent eigenvectors corresponding to each eigenvalue. (Note that there are N
independent eigenvectors; a unitary matrix is never defective.)

Discrete Fourier transform

The problem of their multiplicity was solved by McClellan and Parks (1972), although it was later shown to have
been equivalent to a problem solved by Gauss (Dickinson and Steiglitz, 1982). The multiplicity depends on the value
of N modulo 4, and is given by the following table:

Multiplicities of the eigenvalues X of the unitary DFT matrix U as a function of the
transform size N (in terms of an integer m).

sizeN

l = +l

l = -l

/. = -I

l = +i

m+1

m- 1

4m +1

m+1

Am + 2

m+1

Am + 3

m+1

m + l

No simple analytical formula for general eigenvectors is known. Moreover, the eigenvectors are not unique because
any linear combination of eigenvectors for the same eigenvalue is also an eigenvector for that eigenvalue. Various
researchers have proposed different choices of eigenvectors, selected to satisfy useful properties like orthogonality
and to have "simple" forms (e.g., McClellan and Parks, 1972; Dickinson and Steiglitz, 1982; Grlinbaum, 1982;
Atakishiyev and Wolf, 1997; Candan et al, 2000; Hanna et al., 2004; Gurevich and Hadani, 2008). However two
simple closed-form analytical eigenvectors for special DFT period N were found (Kong, 2008):
For DFT period N=2L+ 1 =4^+1, where K is an integer, the following is an eigenvector of DFT:

F(rn) = JJ |cos (^mj - cos (-^sjj

For DFT period N=2L = AK, where K is an ii

r, the follow

ing !•

of DFT:

F(m) — sin I

/2tt

,5"

II l cos ( ~ m I — cos (
s=K+l

The choice of eigenvectors of the DFT matrix has become important in recent years in order to define a discrete
analogue of the fractional Fourier transform — the DFT matrix can be taken to fractional powers by exponentiating
the eigenvalues (e.g., Rubio and Santhanam, 2005). For the continuous Fourier transform, the natural orthogonal
eigenfunctions are the Hermite functions, so various discrete analogues of these have been employed as the
eigenvectors of the DFT, such as the Kravchuk polynomials (Atakishiyev and Wolf, 1997). The "best" choice of
eigenvectors to define a fractional discrete Fourier transform remains an open question, however.

The real-input DFT

If Xq, . . . , Xjv-iare real numbers, as they often are in practical applications, then the DFT obeys the symmetry:

x k = x* N _ k .

The star denotes complex conjugation. The subscripts are interpreted modulo N.

Therefore, the DFT output for real inputs is half redundant, and one obtains the complete information by only

looking at roughly half of the outputs Xq, . . . , -X"iv-i- In this case, the "DC" element XqIs purely real, and for

even N the "Nyquist" element X/yrtis a ls° rea k so there are exactly N non-redundant real numbers in the first half

+ Nyquist element of the complex output X.

Using Euler's formula, the interpolating trigonometric polynomial can then be interpreted as a sum of sine and cosine

functions.

Discrete Fourier transform

Generalized/shifted DFT

It is possible to shift the transform sampling in time and/or frequency domain by some real shifts a and b,
respectively. This is sometimes known as a generalized DFT (or GDFT), also called the shifted DFT or offset
DFT, and has analogous properties to the ordinary DFT:

X k = Y, x n e-'^ k+h ^ n+ ^ k = 0, . . . , N - 1.

Most often, shifts of 1/2 (half a sample) are used. While the ordinary DFT corresponds to a periodic signal in both
time and frequency domains, a = l/2produces a signal that is anti-periodic in frequency domain (
Xk+N = —X^) and vice-versa for b — 1/2- Thus, the specific case of a — b — l/2is known as an odd-time
odd-frequency discrete Fourier transform (or O DFT). Such shifted transforms are most often used for symmetric
data, to represent different boundary symmetries, and for real-symmetric data they correspond to different forms of
the discrete cosine and sine transforms.
Another interesting choice is a — b — —(N — l)/2, which is called the centered DFT (or CDFT). The

centered DFT has the useful property that, when N is a multiple of four, all four of its eigenvalues (see above) have
equal multiplicities (Rubio and Santhanam, 2005)

The discrete Fourier transform can be viewed as a special case of the z-transform, evaluated on the unit circle in the
complex plane; more general z-transforms correspond to complex shifts a and b above.

Multidimensional DFT

The ordinary DFT transforms a one-dimensional sequence or array X n that is a function of exactly one discrete
variable n. The multidimensional DFT of a multidimensional array 2 ; n 1 ,n 2 ,...,n d that is a function of d discrete
variables ri£ = 0, 1, . . . , Nj> — lfor £ in 1, 2, . . . , d is defined by:
iVi-1 / N 2 -l ( N d -1

Y, , , — V L,> kini V ,,, k2U2 ... V ,,) kdnd ■ t

ni =0 \ n 2 =0 \ n d =0

where uj n = exp(— 27rz/A^)as above and the d output indices run from hi = 0, 1, . . . , N^ — 1. This is
more compactly expressed in vector notation, where we define n = {n\, ^2? • • • j n d) anc *
k= (hi, &2, . . • , &d) as ^-dimensional vectors of indices from to N — 1, which we define as

N - 1 = (iVj_- 1, N 2 - 1, . . . , N d - 1) :

X k = ^e- 2 ^<^x n ,
•i i.i
where the division n/Nis defined as n/N = (ni/Ni, . . . , n^/N^to be performed element-wise, and the
sum denotes the set of nested summations above.
The inverse of the multi-dimensional DFT is, analogous to the one-dimensional case, given by:

: 1 y e 2^n.(k/N) x

As the one-dimensional DFT expresses the input X n as a superposition of sinusoids, the multidimensional DFT
expresses the input as a superposition of plane waves, or sinusoids. The direction of oscillation in space is k/N-
The amplitudes are X\^- This decomposition is of great importance for everything from digital image processing
(two-dimensional) to solving partial differential equations. The solution is broken up into plane waves.
The multidimensional DFT can be computed by the composition of a sequence of one-dimensional DFTs along each
dimension. In the two-dimensional case ^ m ,n 2 the ^independent DFTs of the rows (i.e., along Tl^) are
computed first to form a new array Vmfa- Then the JV2 independent DFTs of y along the columns (along Tli) are

Discrete Fourier transform 42

computed to form the final result X/ Cl ^ 2 . Alternatively the columns can be computed first and then the rows. The order is
immaterial because the nested summations above commute.

An algorithm to compute a one-dimensional DFT is thus sufficient to efficiently compute a multidimensional DFT.
This approach is known as the row-column algorithm. There are also intrinsically multidimensional FFT algorithms.

The real-input multidimensional DFT

For input data x ni,n 2 ,...,n d consisting of real numbers, the DFT outputs have a conjugate symmetry similar to the
one-dimensional case above:

-Xfcl,fc 2 ,...,fc d = ^■N 1 -k 1 ,N 2 -k 2 ,...,N d -k d ^

where the star again denotes complex conjugation and the g -th subscript is again interpreted modulo Ni (for

£= 1,2,. ..,d).
Applications

The DFT has seen wide usage across a large number of fields; we only sketch a few examples below (see also the
references at the end). All applications of the DFT depend crucially on the availability of a fast algorithm to compute
discrete Fourier transforms and their inverses, a fast Fourier transform.

Spectral analysis

When the DFT is used for spectral analysis, the {2^} sequence usually represents a finite set of uniformly-spaced
time-samples of some signal x{t), where t represents time. The conversion from continuous time to samples
(discrete-time) changes the underlying Fourier transform of x(t) into a discrete-time Fourier transform (DTFT),
which generally entails a type of distortion called aliasing. Choice of an appropriate sample-rate (see Nyquist
frequency) is the key to minimizing that distortion. Similarly, the conversion from a very long (or infinite) sequence
to a manageable size entails a type of distortion called leakage, which is manifested as a loss of detail (aka
resolution) in the DTFT. Choice of an appropriate sub-sequence length is the primary key to minimizing that effect.
When the available data (and time to process it) is more than the amount needed to attain the desired frequency
resolution, a standard technique is to perform multiple DFTs, for example to create a spectrogram. If the desired
result is a power spectrum and noise or randomness is present in the data, averaging the magnitude components of
the multiple DFTs is a useful procedure to reduce the variance of the spectrum (also called a periodogram in this
context); two examples of such techniques are the Welch method and the Bartlett method; the general subject of
estimating the power spectrum of a noisy signal is called spectral estimation.

A final source of distortion (or perhaps illusion) is the DFT itself, because it is just a discrete sampling of the DTFT,
which is a function of a continuous frequency domain. That can be mitigated by increasing the resolution of the
DFT. That procedure is illustrated in the discrete-time Fourier transform article.

• The procedure is sometimes referred to as zero-padding, \\ hich is a particular implementation used in conjui
with the fast Fourier transform (FFT) algorithm. The inefficiency of performing multiplications and additions
with zero-valued "samples" is more than offset by the inherent efficiency of the FFT.

• As already noted, leakage imposes a limit on the inherent resolution of the DTFT. So there is a practical limit tc
the benefit that can be obtained from a fine-grained DFT.

Discrete Fourier transform

Data compression

The field of digital signal processing relies heavily on operations in the frequency domain (i.e. on the Fourier
transform). For example, several lossy image and sound compression methods employ the discrete Fourier
transform: the signal is cut into short segments, each is transformed, and then the Fourier coefficients of high
frequencies, which are assumed to be unnoticeable, are discarded. The decompressor computes the inverse transform
based on this reduced number of Fourier coefficients. (Compression applications often use a specialized form of the
DFT, the discrete cosine transform or sometimes the modified discrete cosine transform.)

Partial differential equations

Discrete Fourier transforms are often used to solve partial differential equations, where again the DFT is used as an
approximation for the Fourier series (which is recovered in the limit of infinite N). The advantage of this approach is
that it expands the signal in complex exponentials e lnx , which are eigenfunctions of differentiation: dldx e mx = in e mx .
Thus, in the Fourier representation, differentiation is simple — we just multiply by i n. (Note, however, that the
choice of n is not unique due to aliasing; for the method to be convergent, a choice similar to that in the
trigonometric interpolation section above should be used.) A linear differential equation with constant coefficients is
transformed into an easily solvable algebraic equation. One then uses the inverse DFT to transform the result back
into the ordinary spatial representation. Such an approach is called a spectral method.

Polynomial multiplication

Suppose we wish to compute the polynomial product c(x) = a(x) ■ b(x). The ordinary product expression for the
coefficients of c involves a linear (acyclic) convolution, where indices do not "wrap around." This can be rewritten
as a cyclic convolution by taking the coefficient vectors for a(x) and b(x) with constant term first, then appending
zeros so that the resultant coefficient vectors a and b have dimension d > deg(a(x)) + deg(b(x)). Then,

c = a* b

Where c is the vector of coefficients for c(x), and the convolution operator * is defined so

: J^ a m b n _ m mod d n = 0,l...,d-l

But convolution becomes multiplication under the DFT:

T{c) = :F(a).F(b)

Here the vector product is taken elementwise. Thus the coefficients of the product polynomial c(x) are just the terms
0, ..., deg(a(x)) + deg(b(x)) of the coefficient vector

c = ^- 1 (^(a)^(b)).
With a fast Fourier transform, the resulting algorithm takes O (NlogN) arithmetic operations. Due to its simplicity
and speed, the Cooley-Tukey FFT algorithm, which is limited to composite sizes, is often chosen for the transform
operation. In this case, d should be chosen as the smallest integer greater than the sum of the input polynomial
degrees that is factorizable into small prime factors (e.g. 2, 3, and 5, depending upon the FFT implementation).

Discrete Fourier transform

Multiplication of large integers

The fastest known algorithms for the multiplication of very large integers use the polynomial multiplication method
outlined above. Integers can be treated as the value of a polynomial evaluated specifically at the number base, with
the coefficients of the polynomial corresponding to the digits in that base. After polynomial multiplication, a
relatively low-complexity carry-propagation step completes the multiplication.

Some discrete Fourier transform pairs

Some DFT pairs

JV fc=0

Xk = y- Xne -i2wk n /N

n=0

Note

x n e' 2 ^ N

X k - e

Shift theorem

x n -i

X k e- aM ' N

x„€R

x k = X* N _ k

Real DFT

J N if a = ^ k l N

from the geometric progression formula

ll-ae— 2 **/" ° CrVV1 " C

(V)

(l + e-™'*)"- 1

from the binomial theorem

(± if 2n < W or 2(N - n) < W
y otherwise

( 1 if k =

1 — . /t t \ otherwise

x n is a rectangular window function of W points centered on
Xq, where Wis an odd integer, and X^is a sine-like function

Derivation as Fourier series

The DFT can be derived as a truncation of the Fourier series of a periodic sequence of impulses.

Generalizations
Representation theory

The DFT can be interpreted as the complex-valued representation theory of the finite cyclic group. In other words, a

sequence of n complex numbers can be thought of as an element of «-dimensional complex space C n ,or

equivalently a function from the finite cyclic group of order n to the complex numbers, Z/jlZ — > C.This latter

may be suggestively written ^Z/n.Zto emphasize that this is a complex vector space whose coordinates are indexed

by the «-element set Z/nZ.

From this point of view, one may generalize the DFT to representation theory generally, or more narrowly to the

representation theory of finite groups.

More narrowly still, one may generalize the DFT by either changing the target (taking values in a field other than the

complex numbers), or the domain (a group other than a finite cyclic group), as detailed in the sequel.

Discrete Fourier transform

Other fields

Many of the properties of the DFT only depend on the fact that g -^pis a primitive root of unity, sometimes
denoted Ci'jvor W N (so that cjjY = 1)- Such properties include the completeness, orthogonality,
Plancherel/Parseval, periodicity, shift, convolution, and unitarity properties above, as well as many FFT algorithms.
For this reason, the discrete Fourier transform can be defined by using roots of unity in fields other than the complex
numbers, and such generalizations are commonly called number-theoretic transforms (NTTs) in the case of finite
fields. For more information, see number-theoretic transform and discrete Fourier transform (general).

Other finite groups

The standard DFT acts on a sequence x x , ■ ■-,*„ of complex numbers, which can be viewed as a function { 0, 1 ,
...,N- 1 } — > C. The multidimensional DFT acts on multidimensional sequences, which can be viewed as functions

{0, 1, . . . , N x - 1} x ■ • ■ x {0, 1, . . . , N d - 1} -> C.

This suggests the generalization to Fourier transforms on arbitrary finite groups, which act on functions G — > C
where G is a finite group. In this framework, the standard DFT is seen as the Fourier transform on a cyclic group,
while the multidimensional DFT is a Fourier transform on a direct sum of cyclic groups.

Alternatives

As with other Fourier transforms, there are various alternatives to the DFT for various applications, prominent
among which are wavelets. The analog of the DFT is the discrete wavelet transform (DWT). From the point of view
of time-frequency analysis, a key limitation of the Fourier transform is that it does not include location information,
only frequency information, and thus has difficulty in representing transients. As wavelets have location as well as
frequency, they are better able to represent location, at the expense of greater difficulty representing frequency. For
details, see comparison of the discrete wavelet transform with the discrete Fourier transform.

See also

• DFT matrix

• Fast Fourier transform

• List of Fourier-related transforms
. fftw

References

• Brigham, E. Oran (1988). The fast Fourier transform and its applications. Englewood Cliffs, N.J.: Prentice Hall.
ISBN 0-13-307505-2.

• Oppenheim, Alan V.; Schafer, R. W.; and Buck, J. R. (1999). Discrete-time signal processing. Upper Saddle
River, N.J.: Prentice Hall. ISBN 0-13-754920-2.

• Smith, Steven W. (1999). "Chapter 8: The Discrete Fourier Transform" [3] . The Scientist and Engineer's Guide to
Digital Signal Processing (Second ed.). San Diego, Calif.: California Technical Publishing. ISBN 0-9660176-3-3.

• Cormen, Thomas H.; Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein (2001). "Chapter 30:
Polynomials and the FFT". Introduction to Algorithms (Second ed.). MIT Press and McGraw-Hill. pp. 822-848.
ISBN 0-262-03293-7. esp. section 30.2: The DFT and FFT, pp. 830-838.

• P. Duhamel, B. Piron, and J. M. Etcheto (1988). "On computing the inverse DFT". IEEE Trans. Acoust, Speech
andSig. Processing 36 (2): 285-286. doi: 10. 1109/29. 15 19.

• J. H. McClellan and T. W. Parks (1972). "Eigenvalues and eigenvectors of the discrete Fourier transformation".
IEEE Trans. Audio Electroacoust. 20 (1): 66-74. doi: 10. 1109/TAU. 1972. 1162342.

Discrete Fourier transform

• Bradley W. Dickinson and Kenneth Steiglitz (1982). "Eigenvectors and functions of the discrete Fourier
transform". IEEE Trans. Acoust., Speech and Sig. Processing 30 (1): 25-31. doi: 10.1 109/TASSP. 1982.1 163843.
(Note that this paper has an apparent typo in its table of the eigenvalue multiplicities: the +il-i columns are
interchanged. The correct table can be found in McClellan and Parks, 1972, and is easily confirmed numerically.)

• F. A. Griinbaum (1982). "The eigenvectors of the discrete Fourier transform". /. Math. Anal. Appl. 88 (2):
355-363. doi: 10. 1016/0022-247X(82)90199-8.

• Natig M. Atakishiyev and Kurt Bernardo Wolf (1997). "Fractional Fourier-Kravchuk transform". J. Opt. Soc. Am.
A 14 (7): 1467-1477. doi: 10. 1364/JOSAA. 14.001467.

• C. Candan, M. A. Kutay and H. M.Ozaktas (2000). "The discrete fractional Fourier transform". IEEE Trans, on
Signal Processing 48 (5): 1329-1337. doi: 10. 1109/78.839980.

• Magdy Tawfik Hanna, Nabila Philip Attalla Seif, and Waleed Abd El Maguid Ahmed (2004).
"Hermite-Gaussian-like eigenvectors of the discrete Fourier transform matrix based on the singular-value
decomposition of its orthogonal projection matrices". IEEE Trans. Circ. Syst. 1 51 (11): 2245-2254.
doi: 10. 1 109/TCSI.2004.836850.

• Shamgar Gurevich and Ronny Hadani (2009). "On the diagonalization of the discrete Fourier transform". Applied
and Computational Harmonic Analysis 27 (1): 87-99. doi: 10.1016/j.acha.2008. 11.003. preprint at
arXiv:0808.3281.

• Shamgar Gurevich, Ronny Hadani, and Nir Sochen (2008). "The finite harmonic oscillator and its applications to
sequences, communication and radar". IEEE Transactions on Information Theory 54 (9): 4239-4253.
doi:10.1109/TIT.2008.926440. preprint at arXiv:0808.1495.

• Juan G Vargas-Rubio and Balu Santhanam (2005). "On the multiangle centered discrete fractional Fourier
transform". IEEE Sig. Proc. Lett. 12 (4): 273-276. doi:10.1109/LSP.2005.843762.

• J. Cooley, P. Lewis, and P. Welch (1969). "The finite Fourier transform". IEEE Trans. Audio Electroacoustics 17
(2): 77-85. doi: 10. 1109/TAU. 1969. 1162036.

• F.N. Kong (2008). "Analytic Expressions of Two Discrete Hermite-Gaussian Signals". IEEE Trans. Circuits and
Systems -II: Express Briefs. 55 (1): 56-60. doi:10.1109/TCSII.2007.909865.

r|4|

External links

• Interactive flash tutorial on the DFT L '

• Mathematics of the Discrete Fourier Transform by Julius O. Smith III

• Fast implementation of the DFT - coded in C and under General Public License (GPL) L J

• Example of how DFT spectral analysis is used in engineering studies of the Otto Struve 2. lm telescope

• The DFT "a Pied": Mastering The Fourier Transform in One Day

References

[1] T. G. Stockham, Jr., "High-speed convolution and correlation," in 1966 Proc. AFIPS Spring Joint Computing Conf. Reprinted in Digital
Signal Processing, L. R. Rabiner and C. M. Rader, editors, New York: IEEE Press, 1972.

|2| Santhanam. Bah': Santhanam. Thalana\ar S. "Di screw (uiuss He rmile turn lions and eigenvc < tors oj the cam red di screw Fourier transfon
(http://thamakau.usc.edu/Proceedings/ICASSP 2007/pdfs/0301385.pdf), Proceedings of the 32nd IEEE International Conference on
Acoustics, Speech, and Signal Processing (ICASSP 2007, SPTM-P12.4), vol. Ill, pp. 1385-1388.

[3] http://www.dspguide.eom/ch8/l.htm

| ! i hllp://\\ \\ u Courier series.com/Courierseries2/OFr_tutorial.html

[5] http://ccrma.stanford.edu/~jos/mdft/mdft.html

[6] http://www.fftw.org

[7] http://nexus.as.utexas.edu/kuehne/3_4%22.html

Fast Fourier transform

A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its
inverse. There are many distinct FFT algorithms involving a wide range of mathematics, from simple
complex-number arithmetic to group theory and number theory; this article gives an overview of the available
techniques and some of their general properties, while the specific algorithms are described in subsidiary articles
linked below.

A DFT decomposes a sequence of values into components of different frequencies. This operation is useful in many
fields (see discrete Fourier transform for properties and applications of the transform) but computing it directly from
the definition is often too slow to be practical. An FFT is a way to compute the same result more quickly: computing
a DFT of N points in the naive way, using the definition, takes 0(N ) arithmetical operations, while an FFT can
compute the same result in only 0(N log N) operations. The difference in speed can be substantial, especially for
long data sets where N may be in the thousands or millions — in practice, the computation time can be reduced by
several orders of magnitude in such cases, and the improvement is roughly proportional to N/log(N). This huge
improvement made many DFT -based algorithms practical; FFTs are of great importance to a wide variety of
applications, from digital signal processing and solving partial differential equations to algorithms for quick
multiplication of large integers.

The most well known FFT algorithms depend upon the factorization of N, but (contrary to popular misconception)
there are FFTs with 0(N log AO complexity for all N, even for prime N. Many FFT algorithms only depend on the
fact that -^jpis an JV m primitive root of unity, and thus can be applied to analogous transforms over any finite
field, such as number- theoretic transforms.

Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a UN factor, any FFT
algorithm can easily be adapted for it.

Definition and speed

An FFT computes the DFT and produces exactly the same result as evaluating the DFT definition directly; the only
difference is that an FFT is much faster. (In the presence of round-off error, many FFT algorithms are also much
more accurate than evaluating the DFT definition directly, as discussed below.)
Let x ...., x be complex numbers. The DFT is defined by the formula

JV-1

X k = J2 x n e- i2wk % k = 0,...,N -1.

n=0

Evaluating this definition directly requires 0(N ) operations: there are N outputs X and each output requires a sum
of N terms. An FFT is any method to compute the same results in 0(AHog AO operations. More precisely, all known
FFT algorithms require @(N log AO operations (technically, O only denotes an upper bound), although there is no
proof that better complexity is impossible.

To illustrate the savings of an FFT, consider the count of complex multiplications and additions. Evaluating the
DFT's sums directly involves N complex multiplications and N(N- 1) complex additions [of which 0(AO operations
can be saved by eliminating trivial operations such as multiplications by 1]. The well-known radix-2 Cooley-Tukey
algorithm, for N a power of 2, can compute the same result with only (A/2) log N complex multiplies (again,
ignoring simplifications of multiplications by 1 and similar) and Nlog N complex additions. In practice, actual
performance on modern computers is usually dominated by factors other than arithmetic and is a complicated subject
(see, e.g., Frigo & Johnson, 2005), but the overall improvement from &(N ) to ©(A^ log AO remains.

Fast Fourier transform

Computational issues

Bounds on complexity and operation counts

A fundamental question of longstanding theoretical interest is to prove lower bounds on the complexity and exact
operation counts of fast Fourier transforms, and many open problems remain. It is not even rigorously proved
whether DFTs truly require Q(N log JV)(i-e., order N log A^ or greater) operations, even for the simple case of
power of two sizes, although no algorithms with lower complexity are known. In particular, the count of arithmetic
operations is usually the focus of such questions, although actual performance on modern-day computers is
determined by many other factors such as cache or CPU pipeline optimization.

Following pioneering work by Winograd (1978), a tight 0(JV) lower bound is known for the number of real
multiplications required by an FFT. It can be shown that only 4N — 2 logo N — 2 log 2 N — 4 irrational real
multiplications are required to compute a DFT of power-of-two length _/y = 2 m • Moreover, explicit algorithms
that achieve this count are known (Heideman & Burrus, 1986; Duhamel, 1990). Unfortunately, these algorithms
require too many additions to be practical, at least on modern computers with hardware multipliers.
A tight lower bound is not known on the number of required additions, although lower bounds have been proved
under some restrictive assumptions on the algorithms. In 1973, Morgenstern proved an Q(N log N) lower bound
on the addition count for algorithms where the multiplicative constants have bounded magnitudes (which is true for
most but not all FFT algorithms). Pan (1986) proved an Q(N log iV)lower bound assuming a bound on a measure
of the FFT algorithm's "asynchronicity", but the generality of this assumption is unclear. For the case of
power-of-two ]\T, Papadimitriou (1979) argued that the number 7Vlog 2 N of complex-number additions achieved
by Cooley-Tukey algorithms is optimal under certain assumptions on the graph of the algorithm (his assumptions
imply, among other things, that no additive identities in the roots of unity are exploited). (This argument would
imply that at least 2 N log 2 Aureal additions are required, although this is not a tight bound because extra additions
are required as part of complex-number multiplications.) Thus far, no published FFT algorithm has achieved fewer

than TV logo N complex-number additions (or their equivalent) for power-of-two /y ■

A third problem is to minimize the total number of real multiplications and additions, sometimes called the
"arithmetic complexity" (although in this context it is the exact count and not the asymptotic complexity that is being
considered). Again, no tight lower bound has been proven. Since 1968, however, the lowest published count for
power-of-two yy was long achieved by the split-radix FFT algorithm, which requires 4N log 2 N — 6iV + 8real

multiplications and additions for JV > 1 ■ This was recently reduced to ~ — N log 2 N (Johnson and Frigo,

2007; Lundy and Van Buskirk, 2007).

Most of the attempts to lower or prove the complexity of FFT algorithms have focused on the ordinary complex-data
case, because it is the simplest. However, complex-data FFTs are so closely related to algorithms for related
problems such as real-data FFTs, discrete cosine transforms, discrete Hartley transforms, and so on, that any
improvement in one of these would immediately lead to improvements in the others (Duhamel & Vetterli, 1990).

Accuracy and approximations

All of the FFT algorithms discussed below compute the DFT exactly (in exact arithmetic, i.e. neglecting
floating-point errors). A few "FFT" algorithms have been proposed, however, that compute the DFT approximately,
with an error that can be made arbitrarily small at the expense of increased computations. Such algorithms trade the
approximation error for increased speed or other properties. For example, an approximate FFT algorithm by
Edelman et al. (1999) achieves lower communication requirements for parallel computing with the help of a fast
multipole method. A wavelet-based approximate FFT by Guo and Burrus (1996) takes sparse inputs/outputs
(time/frequency localization) into account more efficiently than is possible with an exact FFT. Another algorithm for
approximate computation of a subset of the DFT outputs is due to Shentov et al. (1995). Only the Edelman algorithm

Fast Fourier transform

works equally well for sparse and non-sparse data, however, since it is based on the compressibility (rank deficiency)
of the Fourier matrix itself rather than the compressibility (sparsity) of the data.

Even the "exact" FFT algorithms have errors when finite-precision floating-point arithmetic is used, but these errors
are typically quite small; most FFT algorithms, e.g. Cooley-Tukey, have excellent numerical properties. The upper
bound on the relative error for the Cooley-Tukey algorithm is 0(e log N), compared to 0(eN ) for the naive DFT
formula (Gentleman and Sande, 1966), where e is the machine floating-point relative precision. In fact, the root
mean square (rms) errors are much better than these upper bounds, being only 0(e Vlog N) for Cooley-Tukey and
0(e VAO for the naive DFT (Schatzman, 1996). These results, however, are very sensitive to the accuracy of the
twiddle factors used in the FFT (i.e. the trigonometric function values), and it is not unusual for incautious FFT
implementations to have much worse accuracy, e.g. if they use inaccurate trigonometric recurrence formulas. Some
FFTs other than Cooley-Tukey, such as the Rader-Brenner algorithm, are intrinsically less stable.
In fixed-point arithmetic, the finite-precision errors accumulated by FFT algorithms are worse, with rms errors
growing as O(VA0 for the Cooley-Tukey algorithm (Welch, 1969). Moreover, even achieving this accuracy requires
careful attention to scaling in order to minimize the loss of precision, and fixed-point FFT algorithms involve
rescaling at each intermediate stage of decompositions like Cooley-Tukey.

To verify the correctness of an FFT implementation, rigorous guarantees can be obtained in 0(N log AO time by a
simple procedure checking the linearity, impulse-response, and time-shift properties of the transform on random
inputs (Ergiin, 1995).

Algorithms
Cooley-Tukey algorithm

By far the most common FFT is the Cooley-Tukey algorithm. This is a divide and conquer algorithm that

recursively breaks down a DFT of any composite size N = N N into many smaller DFTs of sizes N and N , along

with O(A0 multiplications by complex roots of unity traditionally called twiddle factors (after Gentleman and Sande,

1966).

This method (and the general idea of an FFT) was popularized by a publication of J. W. Cooley and J. W. Tukey in

1965, but it was later discovered (Heideman & Burrus, 1984) that those two authors had independently re-invented

an algorithm known to Carl Friedrich Gauss around 1805 (and subsequently rediscovered several times in limited

forms).

The most well-known use of the Cooley-Tukey algorithm is to divide the transform into two pieces of size TV/ 2 at

each step, and is therefore limited to power-of-two sizes, but any factorization can be used in general (as was known

to both Gauss and Cooley/Tukey). These are called the radix-2 and mixed-radix cases, respectively (and other

variants such as the split-radix FFT have their own names as well). Although the basic idea is recursive, most

traditional implementations rearrange the algorithm to avoid explicit recursion. Also, because the Cooley-Tukey

algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT,

such as those described below.

Other FFT algorithms

There are other FFT algorithms distinct from Cooley-Tukey. For TV = TVi A^with coprime TVi and TV2, one can
use the Prime-Factor (Good-Thomas) algorithm (PFA), based on the Chinese Remainder Theorem, to factorize the
DFT similarly to Cooley-Tukey but without the twiddle factors. The Rader-Brenner algorithm (1976) is a
Cooley-Tukey-like factorization but with purely imaginary twiddle factors, reducing multiplications at the cost of
increased additions and reduced numerical stability; it was later superseded by the split-radix variant of
Cooley-Tukey (which achieves the same multiplication count but with fewer additions and without sacrificing
accuracy). Algorithms that recursively factorize the DFT into smaller operations other than DFTs include the Bruun

Fast Fourier transform 5(

and QFT algorithms. (The Rader-Brenner and QFT algorithms were proposed for power-of-two sizes, but it is
possible that they could be adapted to general composite n . Bruun's algorithm applies to arbitrary even composite
sizes.) Bruun's algorithm, in particular, is based on interpreting the FFT as a recursive factorization of the
polynomial Z N _ \, here into real-coefficient polynomials of the form Z M _ land Z 2M -\- az M + 1 ■
Another polynomial viewpoint is exploited by the Winograd algorithm, which factorizes Z N _ ^into cyclotomic
polynomials — these often have coefficients of 1, 0, or -1, and therefore require few (if any) multiplications, so
Winograd can be used to obtain minimal-multiplication FFTs and is often used to find efficient algorithms for small
factors. Indeed, Winograd showed that the DFT can be computed with only O(N) irrational multiplications,
leading to a proven achievable lower bound on the number of multiplications for power-of-two sizes; unfortunately,
this comes at the cost of many more additions, a tradeoff no longer favorable on modern processors with hardware
multipliers. In particular, Winograd also makes use of the PFA as well as an algorithm by Rader for FFTs of prime
size.

Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime JV , expresses a
DFT of prime size n as a cyclic convolution of (composite) size j\T _ 1 , which can then be computed by a pair of
ordinary FFTs via the convolution theorem (although Winograd uses other convolution methods). Another
prime-size FFT is due to L. I. Bluestein, and is sometimes called the chirp-z algorithm; it also re-expresses a DFT as
a convolution, but this time of the same size (which can be zero-padded to a power of two and evaluated by radix-2
Cooley-Tukey FFTs, for example), via the identity n k = -{k - nf/2 + n 2 /2 + k 2 /2-

FFT algorithms specialized for real and/or symmetric data

In many applications, the input data for the DFT are purely real, in which case the outputs satisfy the symmetry

and efficient FFT algorithms have been designed for this situation (see e.g. Sorensen, 1987). One approach consists
of taking an ordinary algorithm (e.g. Cooley-Tukey) and removing the redundant parts of the computation, saving
roughly a factor of two in time and memory. Alternatively, it is possible to express an even-length real-input DFT as
a complex DFT of half the length (whose real and imaginary parts are the even/odd elements of the original real
data), followed by O(A0 post-processing operations.

It was once believed that real-input DFTs could be more efficiently computed by means of the discrete Hartley
transform (DHT), but it was subsequently argued that a specialized real-input DFT algorithm (FFT) can typically be
found that requires fewer operations than the corresponding DHT algorithm (FHT) for the same number of inputs.
Bruun's algorithm (above) is another method that was initially proposed to take advantage of real inputs, but it has
not proved popular.

There are further FFT specializations for the cases of real data that have even/odd symmetry, in which case one can
gain another factor of (roughly) two in time and memory and the DFT becomes the discrete cosine/sine transform(s)
(DCT/DST). Instead of directly modifying an FFT algorithm for these cases, DCTs/DSTs can also be computed via
FFTs of real data combined with O(A0 pre/post processing.

Fast Fourier transform

Multidimensional FFTs

As defined in the multidimensional DFT article, the multidimensional DFT

x k = J2 € ~ 2wik ' HN)x "

transforms an array X n with a ^-dimensional vector of indices n = (rii, n^-, ■ ■ ■ , Tl^by a set of d nested

summations (over rhj — . . . Nj — lfor each j), where the division n/N, defined as

n/N = ijlijNi, ■ • ■ jTld/Nd), is performed element-wise. Equivalently, it is simply the composition of a

sequence of d sets of one-dimensional DFTs, performed along one dimension at a time (in any order).

This compositional viewpoint immediately provides the simplest and most common multidimensional DFT

algorithm, known as the row-column algorithm (after the two-dimensional case, below). That is, one simply

performs a sequence of d one-dimensional FFTs (by any of the above algorithms): first you transform along the Tl\

dimension, then along the ^dimension, and so on (or actually, any ordering will work). This method is easily

shown to have the usual 0(iVlog N) complexity, where N = N1N2 • • • Njis the total number of data points

transformed. In particular, there are N/Ni transforms of size N±, etcetera, so the complexity of the sequence of

FFTs is:

N N

—0(N 1 \ 0g N 1 ) + -

= O (N [log JVi + ■ ■ ■ + log N d }) = 0(N log N).
In two dimensions, the X^can be viewed as an ri\ X n^ matrix, and this algorithm corresponds to first performing
the FFT of all the rows and then of all the columns (or vice versa), hence the name.

In more than two dimensions, it is often advantageous for cache locality to group the dimensions recursively. For
example, a three-dimensional FFT might first perform two-dimensional FFTs of each planar "slice" for each fixed
ri\, and then perform the one-dimensional FFTs along the ^direction. More generally, an asymptotically optimal
cache-oblivious algorithm consists of recursively dividing the dimensions into two groups {rii 1 ■ • ■ , 7^/2) and
( n d/2+l 1 ' ' ' 5 ^d) tnat are transformed recursively (rounding if d is not even) (see Frigo and Johnson, 2005). Still,
this remains a straightforward variation of the row-column algorithm that ultimately requires only a one-dimensional
FFT algorithm as the base case, and still has OiNXog N) complexity. Yet another variation is to perform matrix
transpositions in between transforming subsequent dimensions, so that the transforms operate on contiguous data;
this is especially important for out-of-core and distributed memory situations where accessing non-contiguous data is

t from the row-column algorithm, although all of
them have OiNXog N) complexity. Perhaps the simplest non-row-column FFT is the vector-radix FFT algorithm,
which is a generalization of the ordinary Cooley-Tukey algorithm where one divides the transform dimensions by a
vector r = (fi, T2, ■ ■ ■ , V"d)°f radices at each step. (This may also have cache benefits.) The simplest case of
vector-radix is where all of the radices are equal (e.g. vector-radix-2 divides all of the dimensions by two), but this is
not necessary. Vector radix with only a single non-unit radix at a time, i.e. r = (1, ■ ■ ■ , 1, r, 1, • ■ ■ , 1), is
essentially a row-column algorithm. Other, more complicated, methods include polynomial transform algorithms due
to Nussbaumer (1977), which view the transform in terms of convolutions and polynomial products. See Duhamel
and Vetterli (1990) for more information and references.

Fast Fourier transform

Other generalizations

An 0(N logAO generalization to spherical harmonics on the sphere S with N nodes was described by
Mohlenkamp (1999), along with an algorithm conjectured (but not proven) to have 0(N log N) complexity;
Mohlenkamp also provides an implementation in the libftsh library .A spherical-harmonic algorithm with 0(N
log N) complexity is described by Rokhlin and Tygert (2006).

Various groups have also published "FFT" algorithms for non-equispaced data, as reviewed in Potts et al. (2001).
Such algorithms do not strictly compute the DFT (which is only defined for equispaced data), but rather some
approximation thereof (a non-uniform discrete Fourier transform, or NDFT, which itself is often computed only
approximately).

See also

Split-radix FFT algorithm

Prime-factor FFT algorithm

Bruun's FFT algorithm

Rader's FFT algorithm

Bluestein's FFT algorithm

Butterfly diagram - a diagram used to describe FFTs.

Odlyzko-Schonhage algorithm applies the FFT to finite Dirichlet series.

Overlap add/Overlap save - efficient convolution methods using FFT for long signals

Spectral music (involves application of FFT analysis to musical composition)

Spectrum analyzers - Devices that perform an FFT

FFTW "Fastest Fourier Transform in the West" - 'C library for the discrete Fourier transform (DFT) in one or

more dimensions.

Time Series

Math Kernel Library

„i-l

References

• N. Brenner and C. Rader, 1976, A New Principle for Fast Fourier Transformation LZJ , IEEE Acoustics, Speech 6
Signal Processing 24: 264-266.

• Brigham, E.O. (2002), The Fast Fourier Transform, New York: Prentice-Hall

• Cooley, James W., and John W. Tukey, 1965, "An algorithm for the machine calculation of complex Fourier
series," Math. Comput. 19: 297-301.

• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein, 2001. Introduction to Algorithms,
2nd. ed. MIT Press and McGraw-Hill. ISBN 0-262-03293-7. Especially chapter 30, "Polynomials and the FFT."

• Pierre Duhamel, 1990, Algorithms meeting the lower bounds on the multiplicative complexity of length- 2"
DFTs and their connection with practical algorithms (doi: 10. 1109/29.60070), IEEE Trans. Acoust. Speech. Sig.
Proc. 38: 1504-151.

• P. Duhamel and M. Vetterli, 1990, Fast Fourier transforms: a tutorial review and a state of the art
(doi:10.1016/0165-1684(90)90158-U), Signal Processing 19: 259-299.

• A. Edelman, P. McCorquodale, and S. Toledo, 1999, The Future Fast Fourier Transform?
(doi:10.1137/S1064827597316266), SI AM J. Sci. Computing 20: 1094-1114.

• Funda Ergtin, 1995, Testing multivariate linear functions: Overcoming the generator bottleneck
(doi:10.1145/225058.225167), Proc. 27th ACM Symposium on the Theory of Computing: 407-416.

• M. Frigo and S. G. Johnson, 2005, "The Design and Implementation of FFTW3 [3] ," Proceedings of the IEEE 93:
216-231.

Fast Fourier transform

• Carl Friedrich Gauss, 1866. "Nachlass: Theoria interpolationis methodo nova tractata," Werke band 3, 265-327.
Gottingen: Konigliche Gesellschaft der Wissenschaften.

W. M. Gentleman and G. Sande, 1966, "Fast Fourier transforms— for fun and profit," Proc. AFIPS 29: 563-578.
H. Guo and C. S. Burrus, 1996, Fast approximate Fourier transform via wavelets transform
(doi: 10. 11 17/12.255236), Proc. SPIE Intl. Soc. Opt. Eng. 2825: 250-259.
H. Guo, G. A. Sitton, C. S. Burrus, 1994, The Quick Discrete Fourier Transform

(doi: 10. 1109/ICASSP. 1994.389994), Proc. IEEE Conf. Acoust. Speech and Sig. Processing (ICASSP) 3:
445-448.

Heideman, M. T., D. H. Johnson, and C. S. Burrus, "Gauss and the history of the fast Fourier transform ," IEEE
ASSP Magazine, 1, (4), 14-21 (1984).

Michael T. Heideman and C. Sidney Burrus, 1986, On the number of multiplications necessary to compute a
length- 2"DFT [5] , IEEE Trans. Acoust. Speech. Sig. Proc. 34: 91-95.

S. G. Johnson and M. Frigo, 2007. "A modified split-radix FFT with fewer arithmetic operations ," IEEE Trans.
Signal Processing 55 (1): 111-119.

T. Lundy and J. Van Buskirk, 2007. "A new matrix approach to real FFTs and convolutions of length 2 ,"
Computing 80 (1): 23-45.

Jacques Morgenstern, 1973, Note on a lower bound of the linear complexity of the fast Fourier transform
(doi:10.1145/321752.321761),/. ACM 20: 305-306.

M. J. Mohlenkamp, 1999, "A fast transform for spherical harmonics", J. Fourier Anal. Appl. 5, 159-184.
(preprint [8] )

H. J. Nussbaumer, 1977, Digital filtering using polynomial transforms (doi: 10. 1049/el: 19770280), Electronics
Lett. 13: 386-387.

V. Pan, 1986, The trade-off between the additive complexity and the asyncronicity of linear and bilinear
algorithms (doi: 10. 1016/0020-0190(86)90035-9), Information Proc. Lett. 22: 11-14.

Christos H. Papadimitriou, 1979, Optimality of the fast Fourier transform (doi:10.1 145/322108.3221 18), /. ACM
26: 95-102.

D. Potts, G. Steidl, and M. Tasche, 2001. "Fast Fourier transforms for nonequispaced data: A tutorial , in: J.J.
Benedetto and P. Ferreira (Eds.), Modan plin heory: Mathei t and ' plic ations (Birkhauser).
Vladimir Rokhlin and Mark Tygert, 2006, "Fast algorithms for spherical harmonic expansions ," SI AM J. Sci.
Computing 27 (6): 1903-1928.

James C. Schatzman, 1996, Accuracy of the discrete Fourier transform and the fast Fourier transform , SIAM
J. Sci. Comput. 17: 1150-1166.

O. V. Shentov, S. K. Mitra, U. Heute, and A. N. Hossen, 1995, Subband DFT. I. Definition, interpretations and
extensions (doi: 10. 1016/0165-1684(94)00103-7), Signal Processing 41: 261-277.

H. V. Sorensen, D. L. Jones, M. T. Heideman, and C. S. Burrus, 1987, Real-valued fast Fourier transform
algorithms [12] , IEEE Trans. Acoust. Speech Sig. Processing ASSP-35: 849-863. See also Corrections to
"Real-valued fast Fourier transform algorithms"

Peter D. Welch, 1969, A fixed-point fast Fourier transform error analysis , IEEE Trans. Audio
Electroacoustics 17: 151-157.
S. Winograd, 1978, On computing the discrete Fourier transform [15] , Math. Computation 32: 175-199.

Fast Fourier transform

External links

.[16] ,

- Fast Fourier Transforms , Connexions online book edited by C. Sidney Burrus, with chapters by C. Sidney
Burrus, Ivan Selesnick, Markus Pueschel, Matteo Frigo, and Steven G. Johnson (2008).

• Links to FFT code and information online.

• National Taiwan University - FFT

• FFT programming in C++ — Cooley-Tukey algorithm.

• Online documentation, links, book, and code.

• Using FFT to construct aggregate probability distributions

• Sri Welaratna, "30 years of FFT Analyzers , Sound and Vibration (January 1997, 30th anniversary issue). A
historical review of hardware FFT devices.

• FFT Basics and Case Study Using Multi-Instrument

• FFT Textbook notes, PPTs at Holistic Numerical Methods Institute.

• ALGLIB FFT Code GPL Licensed multilanguage (VBA, C++, Pascal, etc.) numerical analysis and data
processing library.

References

http://www.math.ohiou.edu/~mjm/research/lihftsh.html

http://ieeexplore. ieee. org/search/ wrapper .j sp?arnumber= 1 1 62805

http://fftw.org/fftw-paper-ieee.pdf

http://ieeexplore.icce. org/xp]s/abs_a]].jsp?annimher=l 162257

1 i M II i i u numher=l 164785

http://www.fftw.org/newsplit.pdf

http://dx.doi.org/10.1007/BF01261607

http://www.math.ohiou.edu/~mjm/research/MOHLEN1999P.pdf

tz.de/~potts/paper/ndft.pdf
<n. yale.edu/~mwt7/sph2. pdf

g/citation.cfm?id=240432

( i ,i I ipp |p i nil ml - I '
■c.ort'/scarch/w rapper.jsp'.'arnumber-l ICo28 I
i a I' ,| i i i ii mil (

org/view/00255718/di970565/97p0015m/0

http://pantheo
http://portal.aci
http://ieeexplor
http://ieeexplor
http://ieeexplor
http://www.jstor.oi
http://cnx. org/content/col 1 0550/
http://www.fftw.org/links.html
http://www.cmlab.csie.ntu.edu.tw/cml/dsp/ln
http://www.librow.com/articles/article-10
http ://www.jjj . de/fxt/

http://www.vosesoftware.eom/ModelRiskHelp/index.htm#Aggregate_dislii I hi linn-,,'
\ggregate_modeliiiL'__ _Fasl_Fourier_Transform_FFT_method.htm
http://www.dataphysics.com/support/library/downloads/articles/DP-30%20Years%20of%20FFT.pdf
http://www.multi-instrument.com/doc/D1002/FFT_Basics_and_Case_Study_usinj' __ Mull! [nstrument_D1002.pdf

iii| M In I.- n. ii i In i..|i in I I

http://www.alglib.net/fasttransforms/fft.php

ing/coding/transform/fft.html

Fourier transform spectroscopy

Fourier transform spectroscopy is a measurement technique whereby spectra are collected based on measurements
of the coherence of a radiative source, using time-domain or space-domain measurements of the electromagnetic
radiation or other type of radiation. It can be applied to a variety of types of spectroscopy including optical
spectroscopy, infrared spectroscopy (FT IR, FT-NIRS), Nuclear Magnetic Resonance (NMR) and Magnetic
Resonance Spectroscopic Imaging (MRSI) , mass spectrometry and electron spin resonance spectroscopy. There
are several methods for measuring the temporal coherence of the light, including the continuous wave Michelson or
Fourier transform spectrometer and the pulsed Fourier transform spectrograph (which is more sensitive and has a
much shorter sampling time than conventional spectroscopic techniques, but is only applicable in a laboratory
environment).

The term "Fourier transform spectroscopy" reflects the fact that in all these techniques, a Fourier transform is
required to turn the raw data into the actual spectrum.

Conceptual introduction
Measuring an emission spectrum

One of the most basic tasks in spectroscopy is to characterize the
spectrum of a light source: How much light is emitted at each different
wavelength. The most straightforward way to measure a spectrum is to
pass the light through a monochromator, an instrument that blocks all
of the light except the light at a certain wavelength (the un-blocked
wavelength is set by a knob on the monochromator). Then the intensity
of this remaining (single-wavelength) light is measured. The measured
intensity directly indicates how much light is emitted at that
wavelength. By varying the monochromator' s wavelength setting, the
full spectrum can be measured. This simple scheme in fact describes
how some spectrometers work.

xX>

An example of a spectrum: The spectrum of light
emitted by the blue flame of a butane torch. The
horizontal axis is the wavelength of light, and the
vertical axis represents how much light is emitted
by the torch at that wavelength.

Fourier transform spectroscopy is a less intuitive way to get the same

information. Rather than allowing only one wavelength at a time to

pass through to the detector, this technique lets through a beam

containing many different wavelengths of light at once, and measures the total beam intensity. Next, the beam is

modified to contain a different combination of wavelengths, giving a second data point. This process is repeated

many times. Afterwards, a computer takes all this data and works backwards to infer how much light there is at each

wavelength.

To be more specific, between the light source and the detector, there is a certain configuration of mirrors that allows
some wavelengths to pass through but blocks others (due to wave interference). The beam is modified for each new
data point by moving one of the mirrors; this changes the set of wavelengths that can pass through.
As mentioned, computer processing is required to turn the raw data (light intensity for each mirror position) into the
desired result (light intensity for each wavelength). The processing required turns out to be a common algorithm
called the Fourier transform (hence the name, "Fourier transform spectroscopy"). The raw data is sometimes called
an "interferogram".

Fourier transform spectroscopy

Measuring an absorption spectrum

The method of Fourier transform spectroscopy can also be used for

absorption spectroscopy. The primary example is "FTIR

Spectroscopy", a common technique in chemistry.

In general, the goal of absorption spectroscopy is to measure how well

a sample absorbs or transmits light at each different wavelength.

Although absorption spectroscopy and emission spectroscopy are

different in principle, they are closely related in practice; any technique

for emission spectroscopy can also be used for absorption

spectroscopy. First, the emission spectrum of a broadband lamp is

measured (this is called the "background spectrum"). Second, the

emission spectrum of the same lamp shining tin cuyji the sample is

measured (this is called the "sample spectrum"). The sample will

absorb some of the light, causing the spectra to be different. The ratio

of the "sample spectrum" to the "background spectrum" is directly related to the sample's absorption spectrum.

Accordingly, the technique of "Fourier transform spectroscopy" can be used both for measuring emission spectra (for
example, the emission spectrum of a star), and absorption spectra (for example, the absorption spectrum of a glass of

liquid).

An "interferogram" from a Fourier transform
spectrometer. The horizontal axis is the position
of the mirror, and the vertical axis is the amount
of light detected. This is the "raw data" which can
be Fourier transformed into an actual spectrum.

Continuous wave Michelson or Fourier transform spectrograph

The Michelson spectrograph is similar to the
instrument used in the Michelson-Morley experiment.
Light from the source is split into two beams by a
half-silvered mirror, one is reflected off a fixed mirror
and one off a moving mirror which introduces a time
delay — the Fourier transform spectrometer is just a
Michelson interferometer with a movable mirror. The
beams interfere, allowing the temporal coherence of the
light to be measured at each different time delay
setting, effectively converting the time domain into a
spatial coordinate. By making measurements of the
signal at many discrete positions of the moving mirror,
the spectrum can be reconstructed using a Fourier
transform of the temporal coherence of the light.
Michelson spectrographs are capable of very high
spectral resolution observations of very bright sources.
The Michelson or Fourier transform spectrograph was
popular for infra-red applications at a time when
infra-red astronomy only had single pixel detectors.
Imaging Michelson spectrometers are a possibility, but i
instruments which are easier to construct.

coherent
light source

CZr-

The Fourier transform --pectronieter is just a Micl
interferometer but one of the two fully-reflecting mirror
allowing a variable delay (in the travel-time of the li
included in one of the beams.

general have been supplanted by imaging Fabry-Perot

Fourier transform spectroscopy

Extracting the spectrum

The intensity as a function of the path length difference in the interferometer pand wavenumber y = \j\ is

I(p t y) = I{v) [1 + cos(2tti>p)] ,
where I{y) is the spectrum to be determined. Note that it is not necessary for I{y) to be modulated by the sample
before the interferometer. In fact, most FTIR spectrometers place the sample after the interferometer in the optical
path. The total intensity at the detector is

I(p)= f I(p,v)dv=l I{v)[l + co$(2itvp)]di>.
Jo Jo

This is just a Fourier cosine transform. The inverse gives us our desired result in terms of the measured quantity

HP)'-

I(v) = 4 / [Up) - \l(p = 0)] cos(27ri>p)dp.
Jo

Pulsed Fourier transform spectrometer

A pulsed Fourier transform spectrometer does not employ transmittance techniques. In the most general description
of pulsed FT spectrometry, a sample is exposed to an energizing event which causes a periodic response. The
frequency of the periodic response, as governed by the field conditions in the spectrometer, is indicative of the
measured properties of the analyte.

Examples of Pulsed Fourier transform spectrometry

In magnetic spectroscopy (EPR, NMR), an RF pulse in a strong ambient magnetic field is used as the energizing
event. This turns the magnetic particles at an angle to the ambient field, resulting in gyration. The gyrating spins then
induce a periodic current in a detector coil. Each spin exhibits a characteristic frequency of gyration (relative to the
field strength) which reveals information about the analyte.

In Fourier transform mass spectrometry, the energizing event is the injection of the charged sample into the strong
electromagnetic field of a cyclotron. These particles travel in circles, inducing a current in a fixed coil on one point
in their circle. Each traveling particle exhibits a characteristic cyclotron frequency-field ratio revealing the masses in
the sample.

The Free Induction Decay

Pulsed FT spectrometry gives the advantage of requiring a single, time-dependent measurement which can easily
deconvolute a set of similar but distinct signals. The resulting composite signal, is called a free induction decay,
because typically the signal will decay due to inhomogeneities in sample frequency, or simply unrecoverable loss of
signal due to entropic loss of the property being measured.

Stationary Forms of Fourier Transform Spectrometers

In addition to the scanning forms of Fourier transform spectrometers, there are a number of stationary or
self-scanned forms. While the analysis of the interferometric output is similar to that of the typical scanning
interferometer, significant differences apply, as shown in the published analyses. Some stationary forms retain the
Fellgett multiplex advantage, and their use in the spectral region where detector noise limits apply is similar to the
scanning forms of the FTS. In the photon-noise limited region, the application of stationary interferometers is
dictated by specific consideration for the spectral region and the application.

Fourier transform spectroscopy

Fellgett Advantage

One of the most important advantages of Fourier transform spectroscopy was shown by P.B. Fellgett, an early
advocate of the method. The Fellgett advantage, also known as the multiplex principle, states that a multiplex
spectrometer such as the Fourier transform spectroscopy will produce a gain of the order of the square root of m in
the signal-to-noise ratio of the resulting spectrum, when compared with an equivalent scanning monochromator,
where m is the number of elements comprising the resulting spectrum when the measurement noise is dominated by
detector noise.

Converting spectra from time domain to frequency domain

S(t) = ^ /(i^e-^ du
The sum is performed over all contributing frequencies to give a signal S(t) in the time domain.
l(v) = r S(t)e il/2nt dt

gives non-zero value when S(t) contains a component that matches the oscillating function.
Remember that

See also

• Applied spectroscopy

• 2D-FT NMRI and Spectroscopy

• Forensic chemistry

• Forensic polymer engineering

• nuclear magnetic resonance

• Infrared spectroscopy

External links

• Description of how a Fourier transform spectrometer works

• The Michelson or Fourier transform spectrograph

• Internet Journal of Vibrational Spectroscopy - How FTIR works

• Fourier Transform Spectroscopy Topical Meeting and Tabletop Exhibit

References

[1] Antoine Abn_ in w 1 i i i in I i \ Pi ess: Cambridge,

[2] Peter Atkins, Julio De Paula. 2006. Physical Chemistry., 8th ed. Oxford University Press: Oxford, UK.

[3] US Patent No. 4,976,542 Digital Array Scanned Interferometer, issued Dec. 1 1 , 1990

| i] luip://scienceworld. wolfram.com/phy sics/FourierTransfoniiSpcclronictcr. html

[5] http://www. astro h\ ini ,i ul com ph nol

[6] http://www.ijvs.comA olunic.Vedition.Vsectionl.html#Feature

[7] http://www.osa.org/meetin i p I niis/rts/default.aspx

NMR Spectroscopy

Nuclear magnetic resonance spectroscopy, most
commonly known as NMR spectroscopy, is the
name given to a technique which exploits the
magnetic properties of certain nuclei. For details
regarding this phenomenon and its origins, refer to
the nuclear magnetic resonance article. The most
important applications for the organic chemist are
proton NMR and carbon- 13 NMR spectroscopy. In
principle, NMR is applicable to any nucleus
possessing spin.

Many types of information can be obtained from an
NMR spectrum. Much like using infrared
spectroscopy (IR) to identify functional groups,
analysis of a NMR spectrum provides information
on the number and type of chemical entities in a
molecule. However, NMR provides much more
information than IR.

The impact of NMR spectroscopy on the natural

sciences has been substantial. It can, among other

things, be used to study mixtures of analytes, to

understand dynamic effects such as change in

temperature and reaction mechanisms, and is an

invaluable tool in understanding protein and

nucleic acid structure and function. It can be applied to a wide variety of samples, both ii

si. He.

the solution and the solid

NMR Spectroscopy

Basic NMR techniques

The NMR sample is prepared in a thin-walled glass
tube - an NMR tube.

When placed in a magnetic field, NMR active nuclei (such as H
or "C) absorb at a frequency characteristic of the isotope. The
resonant frequency, energy of the absorption and the intensity of
the signal are proportional to the strength of the magnetic field.
For example, in a 21 tesla magnetic field, protons resonate at
900 MHz. It is common to refer to a 21 T magnet as a 900 MHz
magnet, although different nuclei resonate at a different frequency
at this field strength.

In the Earth's magnetic field the same nuclei resonate at audio
frequencies. This effect is used in Earth's field NMR spectrometers
and other instruments. Because these instruments are portable and
inexpensive, they are often used for teaching and field work.

Chemical shift

Depending on the local chemical environment, different protons in
a molecule resonate at slightly different frequencies. Since both
this frequency shift and the fundamental resonant frequency are
directly proportional to the strength of the magnetic field, the shift

is converted into afield-independent dimensionless value known as the chemical shift. The chemical shift is reported
as a relative measure from some reference resonance frequency. (For the nuclei H, C, and Si, TMS
(tetramethylsilane) is commonly used as a reference.) This difference between the frequency of the signal and the
frequency of the reference is divided by frequency of the reference signal to give the chemical shift. The frequency
shifts are extremely small in comparison to the fundamental NMR frequency. A typical frequency shift might be 100
Hz, compared to a fundamental NMR frequency of 100 MHz, so the chemical shift is generally expressed in parts
per million (ppm). To be able to detect such small frequency differences it is necessary, that the external magnetic
field varies much less throughout the sample volume. High resolution NMR spectrometers use shims to adjust the
homogeneity of the magnetic field to parts per billion (ppb) in a volume of a few cubic centimeters.

By understanding different chemical environments, the chemical shift can be used to obtain some structural
information about the molecule in a sample. The conversion of the raw data to this information is called assigning
the spectrum. For example, for the H-NMR spectrum for ethanol (CH CH OH), one would expect three specific
signals at three specific chemical shifts: one for the CH group, one for the CH group and one for the OH group. A
typical CH group has a shift around 1 ppm, a CH attached to an OH has a shift of around 4 ppm and an OH has a
shift around 2-3 ppm depending on the solvent used.

Because of molecular motion at room temperature, the three methyl protons average out during the course of the
NMR experiment (which typically requires a few ms). These protons become degenerate and form a peak at the
same chemical shift.

The shape and size of peaks are indicators of chemical structure too. In the example above — the proton spectrum of
ethanol — the CH peak would be three times as large as the OH. Similarly the CH peak would be twice the size of
the OH peak but only 2/3 the size of the CH peak.

Modern analysis software allows analysis of the size of peaks to understand how many protons give rise to the peak.
This is known as integration — a mathematical process which calculates the area under a graph (essentially what a
spectrum is). The analyst must integrate the peak and not measure its height because the peaks also have width — and
thus its size is dependent on its area not its height. However, it should be mentioned that the number of protons, or

NMR Spectroscopy

any other observed nucleus, is only proportional to the intensity, or the integral, of the NMR signal, in the very
simplest one-dimensional NMR experiments. In more elaborate experiments, for instance, experiments typically
used to obtain carbon- 13 NMR spectra, the integral of the signals depends on the relaxation rate of the nucleus, and
its scalar and dipolar coupling constants. Very often these factors are poorly known - therefore, the integral of the
NMR signal is very difficult to interpret in more complicated NMR experiments.

J-coupling

Multiplicity

Intensity Ratio

Singlet (s)

Doublet (d)

1:1

Triplet (t)

1:2:1

Quartet (q)

1:3:3:1

Quintet

1:4:6:4:1

Sextet

1:5:10:10:5:1

Septet

1:6:15:20:15:6:1

Some of the most useful information for structure determination in a one-dimensional NMR spectrum comes from
J-coupling or scalar coupling (a special case of spin-spin coupling) between NMR active nuclei. This coupling
arises from the interaction of different spin states through the chemical bonds of a molecule and results in the
splitting of NMR signals. These splitting patterns can be complex or simple and, likewise, can be straightforwardly
interpretable or deceptive. This coupling provides detailed insight into the connectivity of atoms in a molecule.
Coupling to n equivalent (spin Vi) nuclei splits the signal into a «+l multiplet with intensity ratios following Pascal's
triangle as described on the right. Coupling to additional spins will lead to further splittings of each component of the
multiplet e.g. coupling to two different spin Vi nuclei with significantly different coupling constants will lead to a
doublet of doublets (abbreviation: dd). Note that coupling between nuclei that are chemically equivalent (that is,
have the same chemical shift) has no effect of the NMR spectra and couplings between nuclei that are distant
(usually more than 3 bonds apart for protons in flexible molecules) are usually too small to cause observable
splittings. Long-range couplings over more than three bonds can often be observed in cyclic and aromatic
compounds, leading to more complex splitting patterns.

For example, in the proton spectrum for ethanol described above, the CH group is split into a triplet with an
intensity ratio of 1:2:1 by the two neighboring CH protons. Similarly, the CH is split into a quartet with an
intensity ratio of 1:3:3:1 by the three neighboring CH protons. In principle, the two CH protons would also be split
again into a doublet to form a double! of quartets by the hydroxyl proton, but intermolecular exchange of the acidic
hydroxyl proton often results in a loss of coupling information.

Coupling to any spin Yi nuclei such as phosphorus-31 or fluorine-19 works in this fashion (although the magnitudes
of the coupling constants may be very different). But the splitting patterns differ from those described above for
nuclei with spin greater than Vi because the spin quantum number has more than two possible values. For instance,
coupling to deuterium (a spin 1 nucleus) splits the signal into a 1:1:1 triplet because the spin 1 has three spin states.
Similarly, a spin 3/2 nucleus splits a signal into a 1:1:1:1 quartet and so on.

Coupling combined with the chemical shift (and the integration for protons) tells us not only about the chemical
environment of the nuclei, but also the number of neighboring NMR active nuclei within the molecule. In more
complex spectra with multiple peaks at similar chemical shifts or in spectra of nuclei other than hydrogen, coupling
is often the only way to distinguish different nuclei.

NMR Spectroscopy

Second-order (or strong) coupling

The above description assumes that the coupling constant is small in comparison with the difference in NMR
frequencies between the inequivalent spins. If the shift separation decreases (or the coupling strength increases), the
multiplet intensity patterns are first distorted, and then become more complex and less easily analyzed (especially if
more than two spins are involved). Intensification of some peaks in a multiplet is achieved at the expense of the
remainder, which sometimes almost disappear in the background noise, although the integrated area under the peaks
remains constant. In most high-field NMR, however, the distortions are usually modest and the characteristic
distortions (roofing) can in fact help to identify related peaks.

Second-order effects decrease as the frequency difference between multiplets increases, so that high-field (i.e.
high-frequency) NMR spectra display less distortion than lower frequency spectra. Early spectra at 60 MHz were
more prone to distortion than spectra from later machines typically operating at frequencies at 200 MHz or above.

Magnetic inequivalence

More subtle effects can occur if chemically equivalent spins (i.e. nuclei related by symmetry and so having the same
NMR frequency) have different coupling relationships to external spins. Spins that are chemically equivalent but are
not indistinguishable (based on their coupling relationships) are termed magnetically inequivalent. For example, the
4 H sites of 1,2-dichlorobenzene divide into two chemically equivalent pairs by symmetry, but an individual member
of one of the pairs has different couplings to the spins making up the other pair. Magnetic inequivalence can lead to
highly complex spectra which can only be analyzed by computational modeling. Such effects are more common in
NMR spectra of aromatic and other non-flexible systems, while conformational averaging about C-C bonds in
flexible molecules tends to equalize the couplings between protons on adjacent carbons, reducing problems with
magnetic inequivalence.

Correlation spectroscopy

Correlation spectroscopy is one of several types of two-dimensional nuclear magnetic resonance (NMR)
spectroscopy. This type of NMR experiment is best known by its acronym, COSY. Other types of two-dimensional
NMR include J-spectroscopy, exchange spectroscopy (EXSY), Nuclear Overhauser effect spectroscopy (NOESY),
total correlation spectroscopy (TOCSY) and heteronuclear correlation experiments, such as HSQC, HMQC, and
HMBC Two-dimensional NMR spectra provide more information about a molecule than one-dimensional NMR
spectra and are especially useful in determining the structure of a molecule, particularly for molecules that are too
complicated to work with using one-dimensional NMR. The first two-dimensional experiment, COSY, was proposed
by Jean Jeener, a professor at Universite Libre de Bruxelles, in 1971. This experiment was later implemented by
Walter P. Aue, Enrico Bartholdi and Richard R. Ernst, who published their work in 1976. [2]

Solid-state nuclear magnetic resonance

A variety of physical circumstances does not allow molecules to be studied in solution, and at the same time not by
other spectroscopic techniques to an atomic level, either. In solid-phase media, such as crystals, microcrystalline
powders, gels, anisotropic solutions, etc., it is in particular the dipolar coupling and chemical shift anisotropy that
become dominant to the behaviour of the nuclear spin systems. In conventional solution-state NMR spectroscopy,
these additional interactions would lead to a significant broadening of spectral lines. A variety of techniques allows
to establish high-resolution conditions, that can, at least for C spectra, be comparable to solution-state NMR
spectra.

Two important concepts for high-resolution solid-state NMR spectroscopy are the limitation of possible molecular
orientation by sample orientation, and the reduction of anisotropic nuclear magnetic interactions by sample spinning.
Of the latter approach, fast spinning around the magic angle is a very prominent method, when the system comprises
spin 1/2 nuclei. A number of intermediate techniques, with samples of partial alignment or reduced mobility, is

NMR Spectroscopy

currently being used in NMR spectroscopy.

Applications in which solid-state NMR effects occur are often related to structure investigations on membrane
proteins, protein fibrils or all kinds of polymers, and chemical analysis in inorganic chemistry, but also include
"exotic" applications like the plant leaves and fuel cells.

NMR spectroscopy applied to proteins

Much of the recent innovation within NMR spectroscopy has been within the field of protein NMR, which has
become a very important technique in structural biology. One common goal of these investigations is to obtain high
resolution 3-dimensional structures of the protein, similar to what can be achieved by X-ray crystallography. In
contrast to X-ray crystallography, NMR is primarily limited to relatively small proteins, usually smaller than 35 kDa,
though technical advances allow ever larger structures to be solved. NMR spectroscopy is often the only way to
obtain high resolution information on partially or wholly intrinsically unstructured proteins. It is now a common tool
for the determination of Conformation Activity Relationships where the structure before and after interaction with,
for example, a drug candidate is compared to its known biochemical activity.

Proteins are orders of magnitude larger than the small organic molecules discussed earlier in this article, but the same
NMR theory applies. Because of the increased number of each element present in the molecule, the basic ID spectra
become crowded with overlapping signals to an extent where analysis is impossible. Therefore, multidimensional (2,
3 or 4D) experiments have been devised to deal with this problem. To facilitate these experiments, it is desirable to
isotopically label the protein with C and N because the predominant naturally occurring isotope C is not
NMR-active, whereas the nuclear quadrupole moment of the predominant naturally occurring N isotope prevents
high resolution information to be obtained from this nitrogen isotope. The most important method used for structure
determination of proteins utilizes NOE experiments to measure distances between pairs of atoms within the
molecule. Subsequently, the obtained distances are used to generate a 3D structure of the molecule by solving a
distance geometry problem.

See also

• distance geometry

• In vivo magnetic resonance spectroscopy

• Low field NMR

• Magnetic Resonance Imaging

• Nuclear Magnetic Resonance

• NMR spectra database

• NMR tube - includes a section on sample preparation

• Protein nuclear magnetic resonance spectroscopy

• NMR spectroscopy of stereoisomers

External links

• Protein NMR- A Practical Guide Practical guide to NMR, in particular protein NMR assignment

• James Keeler. "Understanding NMR Spectroscopy" (reprinted at University of Cambridge). University of
California, Irvine. Retrieved 2007-05-11.

• The Basics of NMR - A non-technical overview of NMR theory, equipment, and techniques by Dr. Joseph
Hornak, Professor of Chemistry at RIT

• NMRWiki.ORG [6] project, a Wiki dedicated to NMR, MRI, and EPR.

• NMR spectroscopy for organic chemistry

• The Spectral Game NMR spectroscopy game.

NMR Spectroscopy

Spectra libray NMR spectroscopy library

y [9] N

• Obtaining dihedral angles from J coupling constants

• Another Javascript-like NMR coupling constant to dihedral

• NMR Spectroscopy Citizendium article on NMR Spectroscopy
Free NMR processing, analysis and simulation software

• WINDNMR-Pro - simulation software for interactive calculation of first and second-order spin-coupled
multiplets and a variety of DNMR lineshapes.

• CARA - resonance assignment software developed at the Wuthrich group

• NMRShiftDB - open database and NMR prediction website

• Spinworks [16]

• SPINUS website that uses neural networks to predict NMR spectra from chemical structures

• MD-jeep : free software for solving distance geometry problems related to NMR data

References

1 1 1 Jai i i la I ii i i l II \ w I I i li in ul I i i li 11 h i I i | li i i i i i

University ot C mi i id W In f California. lr\ inc. . Rclric\cd 2007 I

[2] Martin, G.E; Zekter, A.S., Two-Dimensional NMR Methods for Establishing Molecular Connectivity; VCH Publishers, Inc: New York, 1
(p.59)

|3| litlp://\\ v u .protein nmr.ortMik

| ! ] htlp://\\ \\ \\ kcclcr.ch.caiii.ac.uk/leclurcs/lr\ inc/

[5] http://www.cis.rit.edu/htbooks/nmr/

[6] http://nmr\\ iki.ory

[7] http : // w w w . organic world wide, net/nmr . html

[8] http://spectralgame.com

[9] http://nmr.chinanmr.cn/guide/eNMR/eNMRindex.html

[10] http://www.jonathanpmiller.com/Karplus.html

[11] http://www.spectroscopynow.com/FCKeditor/UserFiles/File/specNOW/HTML%20files/General_Karplus_Calculator.htm

[12] http://en.citizendium.org/\\ ;ki/NMR_spectroscopy

[13] http://www.chem.wisc.edu/areas/reich/plt/windnmr.htm

[14] http://cara.nmr.ch

[15] http://www.nmrshiftdb.org

[16] http://www.umanitoba.ca/chemisii > /nmr/spinw orks/

1 17 1 hUp://uH\\2.chcmic.uni erlangen.de/sen ices/spinus/

[18] http ://www. antoniomucherino.it/en/mdjeep. php

2D-NMR

Correlation spectroscopy (COSY) is one of several types of two-dimensional nuclear magnetic resonance (NMR)
spectroscopy. Other types of two-dimensional NMR include J-spectroscopy, exchange spectroscopy (EXSY), and
Nuclear Overhauser effect spectroscopy (NOESY). Two-dimensional NMR spectra provide more information about
a molecule than one-dimensional NMR spectra and are especially useful in determining the structure of a molecule,
particularly for molecules that are too complicated to work with using one-dimensional NMR. The first
two-dimensional experiment, COSY, was proposed by Jean Jeener, a professor at the Universite Libre de Bruxelles,
in 1971. This experiment was later implemented by Walter P. Aue, Enrico Bartholdi and Richard R. Ernst, who
published their work in 1976.

Principles

A two-dimensional NMR experiment involves a series of one-dimensional experiments. Each experiment consists of
a sequence of radio frequency pulses with delay periods in between them. It is the timing, frequencies, and intensities
of these pulses that distinguish different NMR experiments from one another. During some of the delays, the nuclear
spins are allowed to freely precess (rotate) for a determined length of time known as the evolution time. The
frequencies of the nuclei are detected after the final pulse. By incrementing the evolution time in successive
experiments, a two-dimensional data set is generated from a series of one-dimensional experiments.
An example of a two-dimension NMR experiment is the homonuclear correlation spectroscopy (COSY) sequence,
which consists of a pulse (pi) followed by an evolution time (tl) followed by a second pulse (p2) followed by a
measurement time (t2). A computer is used to compile the spectra as a function of the evolution time (tl). Finally,
the Fourier transform is used to convert the time-dependent signals into a two-dimensional spectrum.
The two-dimensional spectrum that results from the COSY experiment shows the frequencies for a single isotope
(usually hydrogen, H) along both axes. (Techniques have also been devised for generating heteronuclear correlation
spectra, in which the two axes correspond to different isotopes, such as C and H.) The intensities of the peaks in
the spectrum can be represented using a third dimension. More commonly, intensity is indicated using contours or
different colors. The spectrum is interpreted starting from the diagonal, which consists of a series of peaks. The
peaks that appear off of the diagonal are called cross-peaks. The cross-peaks are symmetrical (both above and below)
along the diagonal and indicate which hydrogen atoms are spin-spin coupled to each other. One can determine which
atoms are connected to one another by only a few chemical bonds by matching the center of a cross-peak with the
center of each of two corresponding diagonal peaks. The peaks on the diagonal when matched with cross-peaks are
coupled to each other.

For example: a CH CH COCH molecule 2-butanone would show three peaks on the diagonal, due to the three
distinct hydrogen groups. By drawing a line straight down from a cross-peak to the point on the diagonal directly
above or below it, and then drawing a line from the cross-peak directly across to another peak on the diagonal, one
can determine which peaks are coupled. This is done in such a way that the lines from the cross-peak form a 90°
angle between the two peaks on the diagonal. The matching peaks, as determined by using the cross-peaks, indicate
which hydrogens are coupled, giving a clearer understanding of the structure of the molecule under examination.

1 , 1.

u .

fcj

Ill [,„

i:l?

■ f h

3roton-COSY experiment o

i Pn\

tM s ;

To the right is an example of a COSY NMR
spectrum of progesterone in DMSO-d6. The
spectrum that appears along both the x -
and y -axes is a regular one dimensional H
NMR spectrum. The COSY is read along
the diagonal - where the bulk of the peaks
appear. Cross-peaks appear symmetrically
above and below the diagonal.

COSY NMR

COSY-90 is the most common COSY

experiment. In COSY-90, the sample is

irradiated with a radio frequency pulse, pi,

which tilts the nuclear spin by 90°. After pi,

the sample is allowed to freely precess

during an evolution period (tl). A second

90° pulse, p2, is then applied, after which the experimental data are acquired. This is done repeatedly using a series

of different evolution periods (tl). At the conclusion of data acquisition the data is Fourier transformed in each

dimension to generate the two dimensional spectrum. It is only because the evolution period is varied that

cross-peaks appear in the spectrum.

Cross-peaks result from a phenomenon

called magnetization transfer. In

COSY, magnetization transfer occurs

through the chemical bonds rather than

through space.

Another member of the COSY family
is COSY-45. In COSY-45 a 45° pulse
is used instead of a 90° pulse for the
first pulse, pi. The advantage of a
COSY-45 is that the diagonal-peaks are less pronounced, making it simpl<

In COSY NMR, the resonance signal from the sample is read in period t2 follow In; 1 h<
experimental magnetic pulses pi and p2 separated by a variable period tl.

3 match cross-peaks near the diagonal
s can be elucidated from a COSY-45
spectrum. This is not possible using COSY-90. ^ uo I ' tJ - V5 - , - w ' :, lyU Overall, the COSY-45 offers a cleaner spectrum
while the COSY-90 is more sensitive. Related COSY techniques include double quantum filtered COSY and
multiple quantum filtered COSY.

COSY NMR has useful applications. Organic chemists often use COSY to elucidate structural data on molecules that
are not satisfactorily represented in a one-dimensional NMR spectrum. Using cross-peaks, along with the diagonal
spectrum, one can often discover much about the structure of an unknown molecule.

NOESY

In NOESY, the Nuclear Overhauser effect (NOE) between nuclear spins is used to establish the correlations. Hence
the cross-peaks in the resulting two-dimensional spectrum connect resonances from spins that are spatially close.
NOESY spectra from large biomolecules can often be assigned using Sequential Walking.

The NOESY experiment can also be performed in a one-dimensional fashion by pre-selecting individual resonances.
The spectra are read with the pre-selected nuclei giving a large, negative signal while neighboring nuclei are
identified by weaker, positive signals. This only reveals which peaks have measurable NOEs to the resonance of
interest but obviously takes much less time than the full 2D experiment. In addition, if a pre-selected nucleus
changes environment within the time scale of the experiment, multiple negative signals may be observed. This offers
exchange information similar to EXSY (i.e. exchange spectroscopy) NMR spectroscopy.

HMBC & HMQC

HMQC (Heteronuclear Multiple Quantum Coherence) and HMBC (Heteronuclear Multiple Bond Coherence) are 2D
inverse correlation techniques that allow for the determination of connectivity between two different nuclear species.
HMQC is selective for direct coupling and HMBC gives longer range couplings (2-4 bond coupling).

See also

• Exclusive correlation spectroscopy

• Two dimensional correlation analysis

User:Bci2/2D-FT NMRI and Spectroscopy

2D-FT Nuclear Magnetic resonance imaging (2D-FT NMRI), or Two-dimensional Fourier transform magnetic
resonance imaging (NMRI), is primarily a non— invasive imaging technique most commonly used in biomedical
research and medical radiology/nuclear medicine/MRI to visualize structures and functions of the living systems and
single cells. For example it can provides fairly detailed images of a human body in any selected cross-sectional
plane, such as longitudinal, transversal, sagital, etc. NMRI provides much greater contrast especially for the different
soft tissues of the body than computed tomography (CT) as its most sensitive option observes the nuclear spin
distribution and dynamics of highly mobile molecules that contain the naturally abundant, stable hydrogen isotope
H as in plasma water molecules, blood, disolved metabolites and fats. This approach makes it most useful in
cardiovascular, oncological (cancer), neurological (brain), musculoskeletal, and cartilage imaging. Unlike CT, it uses
no ionizing radiation, and also unlike nuclear imaging it does not employ any radioactive isotopes. Some of the first
MRI images reported were published in 1973 and the first study performed on a human took place on July 3,
1977. Earlier papers were also published by Peter Mansfield in UK (Nobel Laureate in 2003), and R. Damadian
in the USA, (together with an approved patent for magnetic imaging). Unpublished "high-resolution' (50 micron
resolution) images of other living systems, such as hydrated wheat grains, were obtained and communicated in UK
in 1977-1979, and were subsequently confirmed by articles published in Nature.

User:Bci2/2D-FT NMRI and Spectroscopy

NMRI Principle

Certain nuclei such as H nuclei, or
'fermions' have spin-1/2, because there
are two spin states, referred to as "up"
and "down" states. The nuclear
magnetic resonance absorption
phenomenon occurs when samples
containing such nuclear spins are
placed in a static magnetic field and a
very short radiofrequency pulse is
applied with a center, or carrier,
frequency matching that of the
transition between the up and down
states of the spin-1/2 H nuclei that
were polarized by the static magnetic
field. Very low field schemes have
also been recently reported.

linical dunjnoxtio am! biomedical re

Chemical Shifts

NMR is a very useful family of techniques for chemical and biochemical research because of the chemical shift; this
effect consists in a frequency shift of the nuclear magnetic resonance for specific chemical groups or atoms as a
result of the partial shielding of the corresponding nuclei from the applied, static external magnetic field by the
electron orbitals (or molecular orbitals) surrounding such nuclei present in the chemical groups. Thus, the higher the
electron density surounding a specific nucleus the larger the chemical shift will be. The resulting magnetic field at
the nucleus is thus lower than the applied external magnetic field and the resonance frequencies observed as a result
of such shielding are lower than the value that would be observed in the absence of any electronic orbital shielding.
Furthermore, in order to obtain a chemical shift value independent of the strength of the applied magnetic field and
allow for the direct comparison of spectra obtained at different magnetic field values, the chemical shift is defined by
the ratio of the strength of the local magnetic field value at the observed (electron orbital-shielded) nucleus by the
external magnetic field strength, H / H The first NMR observations of the chemical shift, with the correct
physical chemistry interpretation, were reported for F containing compounds in the early 1950's by Herbert S.
Gutowsky and Charles P. Slichter from the University of Illinois at Urbana (USA).

NMR Imaging Principles

A number of methods have been devised for combining magnetic field gradients and radiofrequency pulsed
excitation to obtain an image. Two major maethods involve either 2D -FT or 3D-FT reconstruction from
projections, somewhat similar to Computed Tomography, with the exception of the image interpretation that in the
former case must include dynamic and relaxation/contrast enhancement information as well. Other schemes involve
building the NMR image either point-by-point or line-by-line. Some schemes use instead gradients in the rf field
rather than in the static magnetic field. The majority of NMR images routinely obtained are either by the
Two-Dimensional Fourier Transform (2D-FT) technique (with slice selection), or by the Three-Dimensional
Fourier Transform (3D— FT) techniques that are however much more time consuming at present. 2D-FT NMRI is
sometime called in common parlance a "spin-warp". An NMR image corresponds to a spectrum consisting of a
number of ^spatial frequencies' at different locations in the sample investigated, or in a patient. A two-dimensional

User:Bci2/2D-FT NMRI and Spectroscopy

Fourier transformation of such a "real" image may be considered as a representation of such "real waves" by a matrix
of spatial frequencies known as the k-space. We shall see next in some mathematical detail how the 2D-FT
computation works to obtain 2D-FT NMR images.

Two-dimensional Fourier transform imaging and spectroscopy

A two-dimensional Fourier transform (2D-FT) is computed numerically or carried out in two stages, 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 both NMRI and two-dimensional nuclear magnetic
resonance spectroscopy (2D-FT NMRS) that allows the three-dimensional reconstruction of polymer and
biopolymer structures at atomic resolution]]. for molecular weights (Mw) of dissolved biopolymers in aqueous
solutions (for example) up to about 50,000 Mw. For larger biopolymers or polymers, more complex methods have
been developed to obtain limited structural resolution needed for partial 3D-reconstructions of higher molecular
structures, e.g. for up 900,000 Mw or even oriented microcrystals in aqueous suspensions or single crystals; such
methods have also been reported for in vivo 2D-FT NMR spectroscopic studies of algae, bacteria, yeast and certain
mammalian cells, including human ones. The 2D-FT method is also widely utilized in optical spectroscopy, such as
2D-FT MR hyperspectral imaging (2D-FT NIR-HS), or in MRI imaging for research and clinical, diagnostic
applications in Medicine. In the latter case, 2D-FT NIR-HS has recently allowed the identification of single,
malignant cancer cells surrounded by healthy human breast tissue at about 1 micron resolution, well-beyond the
resolution obtainable 2D-FT NMRI for such systems in the limited time available for such diagnostic investigations
(and also in magnetic fields up to the FDA approved magnetic field strength H of 4.7 T, as shown in the top image
of the state-of-the-art NMRI instrument). A more precise mathematical definition of the "double' (2D) Fourier
transform involved in both 2D NMRI and 2D-FT NMRS is specified next, and a precise example follows this
generally accepted definition.

2D-FT Definition

A 2D-FT, or two-dimensional Fourier transform, is a standard Fourier transformation of a function of two variables,
f( x li x 2)> carr i e d fi rst in the first variable X\, followed by the Fourier transform in the second variable a^of the
resulting function F(si, X2) ■ Note that in the case of both 2D-FT NMRI and 2D-FT NMRS the two independent
variables in this definition are in the time domain, whereas the results of the two successive Fourier transforms have,
of course, frequencies as the independent variable in the NMRS, and ultimately spatial coordinates for both 2D
NMRI and 2D-FT NMRS following computer structural recontructions based on special algorithms that are different
from FT or 2D-FT. Moreover, such structural algorithms are different for 2D NMRI and 2D-FT NMRS: in the
former case they involve macroscopic, or anatomical structure detrmination, whereas in the latter case of 2D-FT
NMRS the atomic structure reconstruction algorithms are based on the quantum theory of a microphysical (quantum)
process such as nuclear Overhauser enhancement NOE, or specific magnetic dipole-dipole interactions between
neighbor nuclei.

User:Bci2/2D-FT NMRI and Spectroscopy

Example 1

A 2D Fourier transformation and phase correction is applied to a set of 2D NMR (FID) signals : s(ti , i 2 )yi e lding a
real 2D-FT NMR "spectrum' (collection of ID FT-NMR spectra) represented by a matrix S whose elements are

S{y\,U2) = Re / / cos(viti)exp(~ tL " 2i2 's(ti,t2)dtidt2
where : fiand : ^denote the discrete indirect double-quantum and single-quantum(detection) axes, respectively,
in the 2D NMR experiments. Next, the \emph{covariance matrix} is calculated in the frequency domain according to
the following equation

C(i/ 2 ,h> 2 ) = S T S = ^2[S(i/ u v' 2 )S{v u v 2 )\, W ith : I / 2) ^taking all possible single-quantum
frequency values and with the summation carried out over all discrete, double quantum frequencies : V\ .

Example 2

Atomic Structure from 2D-FT STEM Images 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 (obtained at Cornell
University). 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 Chemistry, Physiology and Medicine received very significant recognition.

Brief explanation of NMRI diagnostic uses in Pathology

As an example, a diseased tissue such as a malign tumor, can be detected by 2D-FT NMRI because the hydrogen
nuclei of molecules in different tissues return to their equilibrium spin state at different relaxation rates, and also
because of the manner in which a malign tumor spreads and grows rapidly along the blood vessels adjacent to the
tumor, also inducing further vascularization to occur. By changing the pulse delays in the RF pulse sequence
employed, and/or the RF pulse sequence itself, one may obtain a "relaxation— based contrast', or contrast
enhancement between different types of body tissue, such as normal vs. diseased tissue cells for example. Excluded
from such diagnostic observations by NMRI are all patients with ferromagnetic metal implants, (e.g., cochlear
implants), and all cardiac pacemaker patients who cannot undergo any NMRI scan because of the very intense
magnetic and RF fields employed in NMRI which would strongly interfere with the correct functioning of such
pacemakers. It is, however, conceivable that future developments may also include along with the NMRI diagnostic
treatments with special techniques involving applied magnetic fields and very high frequency RF. Already, surgery
with special tools is being experimented on in the presence of NMR imaging of subjects. Thus, NMRI is used to
image almost every part of the body, and is especially useful for diagnosis in neurological conditions, disorders of
the muscles and joints, for evaluating tumors, such as in lung or skin cancers, abnormalities in the heart (especially
in children with hereditary disorders), blood vessels, CAD, atherosclerosis and cardiac infarcts (courtesy of Dr.
Robert R. Edelman)

See also

User:Bci2/2D-FT NMRI and Spectroscopy

Nuclear magnetic resonance (NMR)
Medical imaging

Protein nuclear magnetic resonance spe
Kurt Wuthrich

Chemical shifl

Computed tomography (CT)

Fourier transform spectroscopy FTS)

il I IglK li

Solid-state NMR
Herbert S. Gutowsky
John S. Waugh
Charles P. Slichter
FT-NIRS (NIR)
Magnetic re
elaslograpln

Earth's field NMR (EFNMR)
Robinson oscillator

Reference list

[I] Lauterbur, PC. Nobel Laureate in 2(1(13 I 1973). "Image Formation b_\ Induced Local Interactions: Examples of Fmplowng Nuclear Magnetic
Resonance". Nature 242: 190-1. doi:10.1038/242190a0.

[2] [http://www.howstuffworks.com/mri.htm/printable Howstuffworks "How MRI Works"

[3] Peter Mansfield. 2003. Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI (http://www.parteqinnovations.com/pdf-doc/

fandr-Gazl006.pdf)
[4] Antoine Ait igam / u ' ' / pp.. Cambridge Uni\ei i i n I 1

[5] Raftery D (August 2006). "MRI without the magnet" (http://www.pubmedcenlral.nih. go\/articlerender. lcgi?tool=pmcentrez&

artid=1568902). Proc Natl Acad Sci USA. 103 (34): 12657-8. doi:10.1073/pnas.0605625103. PMID 16912110. PMC 1568902.
[6] Wu Y, Chesler DA. Glimchei MJ. ct al (Februars 1999). "Multinuclear -.i -■ i i<.S state three dimensional MRI of bone and synthetic calcium

phosphates" (http://www.pnas.org/cgi/pmidlookup?view=long&pmid=9990066). Proc. Natl. Acad. Sci. U.S.A. 96 (4): 1574-8.

doi:10.1073/pnas.96.4.1574. PMID 9990066. PMC 15521. .
[7] http: // w w w . math. cuhk. edu. hk/course/mat207 1 a/lec 1 _08 .ppt
[8] *Haacke, L Mark: Brown. Robert F: Thompson. Michael: Venkatesan. Ramesh ( 1999), Magm lit resonance i mug in;.;: physical principles and

sequence design. New York: J. Wiley & Sons. ISBN 0-471-35128-8.
[9] Richard R. Ernst. 1992 Nu lear Magnetii R< onance Fouriet transform (2D-FT) Spectroscopy. Nobel Lecture (http://nobelprize.org/

iiobeLpri/ l» it ii laureate 1 99 1 /crust lectin pdl i n December 9 1992
[10] http://en.wikipedia.Org/wiki/Nuclear_magnetic_resonance#Nuclear_spin_and_magnets Kurt Wutrich in 1982-1986 : 2D-FT NMR of

[II] Charles P. Slichter. 1996. Principles of Magnetic Resonance. Springer: Berlin and New York, Third Edition., 65 1pp. ISBN 0-387-50157-6.
[12] http://www.physorg.com/newsl29395045.html

1 13 1 1 il] 1 I 1 1 ii I L 11 I in 1 I 1 ill 11 1 I

I i!| Protein structure determination in solution b\ NMR spcclrosa>p\ (htlp://««u.ncbi.nlm.nih.go\/entrez/i]uer\ .lcgi , . , cmd=Retrie\e&

db=pubmed&dopt=Abstract&list_uids=2266107&query_hl=33&itool=pubmed_docsum) Wuthrich K. J Biol Chem. 1990 December

25;265(36):22059-62.
[15] http://www.mr-tip.com/servl.php?type=img&img=Cardiac' < 20Infarcl ' i 2()Short%20Axis%20Cine%204

References

• Antoine Abragam. 1968. Principles ofNucleai M.r m tic Resonance., 895 pp., Cambridge University Press:
Cambridge, UK.

• Charles P. Slichter. 1996. Principles of Magnetic Resonance. Springer: Berlin and New York, Third Edition.,
651pp. ISBN 0-387-50157-6.

• Kurt Wuthrich. 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 (http://nobelprize.org/nobel_prizes/chemistry/laureates/2002/wutrich-lecture.pdf)

• Protein structure determination in solution by NMR spectroscopy (http://www.ncbi.nlm.nih.gov/entrez/
query. fcgi?cmd=Retrieve&db=pubmed&dopt=Abstract&list_uids=2266107&query_hl=33&
itool=pubmed_docsum) Wuthrich K. J Biol Chem. 1990 December 25;265(36):22059-62

• 2D-FT NMRI Instrument image: A JPG color image of a 2D-FT NMRI "monster' Instrument (http://upload.
wikimedia.org/wikipedia/en/b/bf/HWB-NMRv900.jpg).

User:Bci2/2D-FT NMRI and Spectroscopy

• Richard R. Ernst. 1992. Nuclear Magnetic Resonance Fourier Transform (2D-FT) Spectroscopy. Nobel Lecture
(http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernst-lecture.pdf), on December 9, 1992.

• Peter Mansfield. 2003. Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI (http://www.
parteqinnovations.com/pdf-doc/fandr-Gazl006.pdf)

• D. Benett. 2007. PhD Thesis. Worcester Polytechnic Institute. PDF of 2D-FT Imaging Applications to NMRI in
Medical Research. (http://www.wpi.edu/Pubs/ETD/Available/etd-081707-080430/unrestricted/dbennett.

pdf) Worcester Polytechnic Institute. (Includes many 2D-FT NMR images of human brains.)

• Paul Lauterbur. 2003. Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI. (http://nobelprize.org/
nobel_prizes/medicine/laureates/2003/)

• Jean Jeener. 1971. Two-dimensional Fourier Transform NMR, presented at an Ampere International Summer
School, Basko Polje, unpublished. A verbatim quote follows from Richard R. Ernst's Nobel Laureate Lecture
delivered on December 2, 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 (Jean Jeneer,1971 }). He
suggested a 2D Fourier transform experiment consisting of two $\pi/2$ pulses with a variable time $ t_l$ 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 $s(t_l,t_2)$ of the two-pulse sequence and
Fourier-transformation with respect to both time variables produces a two-dimensional spectrum $S(0_1,0_2)$
of the desired form. This two-pulse experiment by Jean Jeener is the forefather of a whole class of $2D$
experiments that can also easily be expanded to multidimensional spectroscopy.

• Dudley, Robert, L (1993). "High-Field NMR Instrumentation". Ch. 10 in Physical Chemistry of Food Processes
(New York: Van Nostrand-Reinhold) 2: 421-30. ISBN 0-442-00582-2.

• Baianu, I.C.; Kumosinski, Thomas (August 1993). "NMR Principles and Applications to Structure and
Hydration,". Ch.9 in Physical Chemistry of Food Processes (New York: Van Nostrand-Reinhold) 2: 338-420.
ISBN 0-442-00582-2.

• Haacke, E Mark; Brown, Robert F; Thompson, Michael; Venkatesan, Ramesh (1999). Magnetic resonance
imaging: physical pi inciples and sequence design. New York: J. Wiley & Sons. ISBN 0-471-35128-8.

• Raftery D (August 2006). "MRI without the magnet" (http://www.pubmedcentral.nih.gov/articlerender.
fcgi?tool=pmcentrez&artid=1568902). Proc Natl Acad Sci USA. 103 (34): 12657-8.
doi:10.1073/pnas.0605625103. PMID 16912110. PMC 1568902.

• Wu Y, Chesler DA, Glimcher MJ, et al (February 1999). "Multinuclear solid-state three-dimensional MRI of
bone and synthetic calcium phosphates" (http://www.pnas. org/cgi/pmidlookup?view=long&pmid=9990066).
Proc. Natl. Acad. Sci. U.S.A. 96 (4): 1574-8. doi:10.1073/pnas.96.4.1574. PMID 9990066. PMC 15521.

External links

• Cardiac Infarct or "heart attack" Imaged in Real Time by 2D-FT NMRI (http://www.mr-tip.com/exam_gifs/
cardiac_infarct_short_axis_cine_6.gif)

• 3D Animation Movie about MRI Exam (http://www.patiencys.com/MRI/)

• Interactive Flash Animation on MRI (http://www.e-mri.org) - Online Magnetic Resonance Imaging physics and
technique course

• International Society for Magnetic Resonance in Medicine (http://www.ismrm.org)

• Danger of objects flying into the scanner (http://www.simplyphysics.com/flying_objects.html)

User:Bci2/2D-FT NMRI and Spectroscopy

Related Wikipedia websites

Medical imaging

Computed tomography

Magnetic resonance microscopy

Fourier transform spectroscopy

FT-NIRS

Magnetic resonance elastography

Nuclear magnetic resonance (NMR)

Chemical shift

Relaxation

Robinson oscillator

Earth's field NMR (EFNMR)

Rabi cycle
This article incorporates material by the original author from 2D -FT MR- Imaging and related Nobel awards (http:/
/planetphysics. org/ encyclopedia/ 2DFTImaging.html) on PlanetPhysics (http://planetphysics.org/), which is
licensed under the GFDL.

Solid-state nuclear magnetic resonance

Solid-state NMR (SSNMR) spectroscopy is
a kind of nuclear magnetic resonance
(NMR) spectroscopy, characterized by the
presence of anisotropic (directionally
dependent) interactions.

Introduction

Basic concepts A spin interacts with a

magnetic or an electric field. Spatial

proximity and/or a chemical bond between

two atoms can give rise to interactions

between nuclei. In general, these

interactions are orientation dependent. In

media with no or little mobility (e.g.

crystals, powders, large membrane vesicles,

molecular aggregates), anisotropic

interactions have a substantial influence on

the behaviour of a system of nuclear spins.

In contrast, in a classical liquid-state NMR experiment, Brownian motion leads to an averaging of anisotropic

interactions. In such cases, these interactions can be neglected on the time-scale of the NMR experiment.

Examples of anisotropic nuclear interactions Two directionally dependent interactions commonly found in
solid-state NMR are the chemical shift anisotropy (CSA) and the internuclear dipolar coupling. Many more such
interactions exist, such as the anisotropic J-coupling in NMR, or in related fields, such as the g-tensor in electron
spin resonance. In mathematical terms, all these interactions can be described using the same formalism.

te 900 MHz (21 .1 T ) NMR spectrometer at the Canadian National
Ultrahigh-field NMR Facility for Solids.

Solid-state nuclear magnetic resonance

Experimental background Anisotropic interactions modify the nuclear spin energy levels (and hence the resonance

frequency) of all sites in a molecule, and often contribute to a line-broadening effect in NMR spectra. However,

there is a range of situations when their presence can either not be avoided, or is even particularly desired, as they

encode structural parameters, such as orientation information, on the molecule of interest.

High-resolution conditions in solids (in a

wider sense) can be established using magic

angle spinning (MAS), macroscopic sample

orientation, combinations of both of these

techniques, enhancement of mobility by

highly viscous sample conditions, and a

variety of radio frequency (RF) irradiation

patterns. While the latter allows decoupling

of interactions in spin space, the others

facilitate averaging of interactions in real

space. In addition, line-broadening effects

from microscopic inhomogeneities can be

reduced by appropriate methods of sample

preparation.

mia MAS re
8 kHz, 3.2 n

rs (left to right), 7 it
i for 23 kHz, 2.5 mi

n diameter for MAS up to 8 kHz, <
for 35 kHz, 1.3 mm for 70 kHz.

Under decoupling conditions, isotropic
interactions can report on the local structure,
e.g. by the isotropic chemical shift. In
addition, decoupled interactions can be

selectively re-introduced (recoupling"), and used, for example, for controlled de-phasing or transfer of polarization,
which allows to derive a number of structural parameters.

Solid-state NMR line widths The residual line width (full width at half max) of " C nuclei under MAS conditions at
5-15 kHz spinning rate is typically in the order of 0.5-2 ppm, and may be comparable to solution-state NMR
conditions. Even at MAS rates of 20 kHz and above, however, non linear groups (not a straight line) of the same
nuclei linked via the homonuclear dipolar interactions can only be suppressed partially, leading to line widths of 0.5
ppm and above, which is considerably more than in optimal solution state NMR conditions. Other interactions such
as the quadrupolar interaction can lead to line widths of 1000's of ppm due to the strength of the interaction. The
first-order quadrupolar broadening is largely suppressed by sufficiently fast MAS, but the second-order quadrupolar
broadening has a different angular dependence and cannot be removed by spinning at one angle alone. Ways to
achieve isotropic lineshapes for quadrupolar nuclei include spinning at two angles simultaneously (DOR),
sequentially (DAS), or through refocusing the second-order quadrupolar interaction with a two-dimensional
experiment such as MQMAS or STMAS.

Anisotropic interactions in solution-state NMR From the perspective of solution-state NMR, it can be desirable to
reduce motional averaging of dipolar interactions by alignment media. The order of magnitude of these residual
dipolar couplings (RDCs) are typically of only a few rad/Hz, but do not destroy high-resolution conditions, and
provide a pool of information, in particular on the orientation of molecular domains with respect to each other.
Dipolar truncation The dipolar coupling between two nuclei is inversely proportional to the cube of their distance.
This has the effect that the polarization transfer mediated by the dipolar interaction is cut off in the presence of a
third nucleus (all of the same kind, e.g. C) close to one of these nuclei. This effect is commonly referred to as
dipolar truncation. It has been one of the major obstacles in efficient extraction of internuclear distances, which are
crucial in the structural analysis of biomolecular structure. By means of labeling schemes or pulse sequences,
however, it has become possible to circumvent this problem in a number of ways.

Solid-state nuclear magnetic resonance

Nuclear spin interactions in the solid phase
Chemical shielding

The chemical shielding is a local property of each nucleus, and depends on the external magnetic field.

Specifically, the external magnetic field induces currents of the electrons in molecular orbitals. These induced

currents create local magnetic fields that often vary across the entire molecular framework such that nuclei in distinct

molecular environments usually experience unique local fields from this effect.

Under sufficiently fast magic angle spinning, or in solution-state NMR, the directionally dependent character of the

chemical shielding is removed, leaving the isotropic chemical shift.

J-coupling

The J-coupling or indirect nuclear spin-spin coupling (sometimes also called "scalar" coupling despite the fact that J
is a tensor quantity) describes the interaction of nuclear spins through chemical bonds.

Dipolar coupling

Main article: Dipolar coupling (NMR)

Nuclear spins exhibit a dipole moment, which interacts with the dipole
moment of other nuclei (dipolar coupling). The magnitude of the interaction
is dependent on the spin species, the internuclear distance, and the orientation
of the vector connecting the two nuclear spins with respect to the external
magnetic field B (see figure). The maximum dipolar coupling is given by the
dipolar coupling constant d.

fi/A)7i72

47T r 3
where r is the distance between the nuclei, and y and y are the gyromagnetic ratios of the nuclei. In a strong
magnetic field, the dipolar coupling depends on the orientation of the internuclear vector with the external magnetic
field by

Doc3cos 2 #-l-
Consequently, two nuclei with a dipolar coupling vector at an angle of 6 =54.7° to a strong external magnetic field,
which is the angle where D becomes zero, have zero dipolar coupling. 8 is called the magic angle. One technique
for removing dipolar couplings, at least to some extent, is magic angle spinning.

Quadrupolar interaction

Nuclei with a spin greater than one-half have a non spherical charge distribution. This is known as a quadrupolar
nucleus. A non spherical charge distribution can interact with an electric field gradient caused by some form of
non-symmetry (e.g. in a trigonal bonding atom there are electrons around it in a plane, but not above or below it) to
produce a change in the energy level in addition to the Zeeman effect. The quadrupolar interaction is the largest
interaction in NMR apart from the Zeeman interaction and they can even become comparable in size. Due to the
interaction being so large it can not be treated to just the first order, like most of the other interactions. This means
you have a first and second order interaction, which can be treated separately. The first order interaction has an
angular dependency with respect to the magnetic field of (3 COS 2 6 — 1) (the P2 Legendre polynomial), this means

Solid-state nuclear magnetic resonance

that if you spin the sample at q _ arc tan \^(~^^-^^°) Y ou can average out the first order interaction over one rotor period (all otl

interactions apart from Zeeman, Chemical shift, paramagnetic and J coupling also have this angular dependency).

However, the second order interaction depends on the P4 Legendre polynomial which has zero points at 30.6° and

70.1°. These can be taken advantage of by either using DOR (DOuble angle Rotation) where you spin at two angles

at the same time, or DAS (Double Angle Spinning) where you switch quickly between the two angles. But these

techniques suffer from the fact that they require special hardware (probe). A revolutionary advance is Lucio

Frydman's multiple quantum magic angle spinning (MQMAS) NMR in 1995 and it has become a routine method for

obtaining high resolution solid-state NMR spectra of quadrupolar nuclei . A similar method to MQMAS is

satellite transisition magic angle spinning (STMAS) NMR proposed by Zhehong Gan in 2000.

Other interactions

Paramagnetic substances are subjec

History

See also: nuclear magnetic resonance or NMR spectroscopy articles for an account on discoveries in NMR and NMR

spectroscopy in general.

History of discoveries of NMR phenomena, and the development of solid-state NMR spectroscopy:

Purcell, Torrey and Pound: "nuclear induction" on H in paraffin 1945, at about the same time Bloch et al. on H in

water.

Modern solid-state NMR spectroscopy

Methods and techniques

(Frequency channel 1,
e.g. >H)

Basic example

A fundamental RF pulse sequence and building-block in most
solid-state NMR experiments is cross-polarization (CP) [Waugh et
al.]. It can be used to enhance the signal of nuclei with a low
gyromagnetic ratio (e.g. C, N) by magnetization transfer from
nuclei with a high gyromagnetic ratio (e.g. H), or as spectral
editing method (e.g. directed 15 N-> 13 C CP in protein
spectroscopy). In order to establish magnetization transfer, the RF
pulses applied on the two frequency channels must fulfill the
Hartmann-Hahn condition [Hartmann, 1962]. Under MAS, this
condition defines a relationship between the voltage through the
RF coil and the rate of sample rotation. Experimental optimization
of such conditions is one of the routine tasks in performing a
(solid-state) NMR experiment.

CP is a basic building block of most pulse sequences in solid-state

NMR spectroscopy. Given its importance, a pulse sequence

employing direct excitation of H spin polarization, followed by CP transfer to and signal detection of C, N) or

similar nuclei, is itself often referred to as CP experiment, or, in conjunction with MAS, as CP-MAS [Schaefer and

Stejskal, 1976]. It is the typical starting point of an investigation using solid-state NMR spectroscopy.

Tcp

^ l^

Signal
acquisition

■>

(Frequency channel 2
e.g. ,3 Q

CP pulse seque

J \J time

Solid-state nuclear magnetic resonance

Decoupling

Nuclear spin interactions need to be removed (decoupled) in order to increase the resolution of NMR spectra, and to

isolate spin systems.

A technique that can substantially reduce or remove the chemical shift anisotropy, the dipolar coupling is sample

rotation (most commonly magic angle spinning, but also off-magic angle spinning).

Homonuclear RF decoupling decouples spin interactions of nuclei which are the same as those which are being

detected. Heteronuclear RF decoupling decouples spin interactions of other nuclei.

Recoupling

Although the broadened lines are often not desired, dipolar couplings between atoms in the crystal lattice can also

provide very useful information. Dipolar coupling are distance dependent, and so they may be used to calculate

interatomic distances in isotopically labeled molecules.

Because most dipolar interactions are removed by sample spinning, recoupling experiments are needed to

re-introduce desired dipolar couplings so they can be measured.

An example of a recoupling experiment is the Rotational Echo DOuble Resonance (REDOR) experiment [Gullion

andSchaefer, 1989].

Applications
Biology

Membrane proteins and amyloid fibrils, the latter related to Alzheimer's disease and Parkinson's disease, are two
examples of application where solid-state NMR spectroscopy complements solution-state NMR spectroscopy and
beam diffraction methods (e.g. X-ray crystallography, electron microscopy).

Chemistry

Solid-state NMR spectroscopy serves as an analysis tool in organic and inorganic chemistry. SSNMR is also a
valuable tool to study local dynamics, kinetics, and thermodynamics of a variety of systems.

References

[1] http://nmr900.c;

|2| Isotropic Special of Half Integer Quadrupolar Spins from Bidimensional Magic-Angle Spinning NMR Lucio Frydman and John S.

Hardwood, J. Am. Chem. Soc, 1995, J 17, 5367—5368, (1995)
[3] Two-dimensional Magic Angle Spinning Isotropic Reconstruction Sequence-, lor Quadru polar Nuclei . f). IS lassiol. B. Touzo. 1). Trumeau. J.

P. Coutures, J. Virlet, P. Florian and P. J. Grandinetti , Solid-State NMR , 6, 73 (1996)

See also

Preclinical imaging
Cardiac PET

Biomedical informatics

i % jit I lin in in i ( n iiuiiii ition hi

Medicine

Digital Mammography and PACS

EMMI European Master in Molecular Imaging

Fotofinder

Full-body scan

Magnetic field imaging • Pneumoencephalogram

Medical examination • Radiology information s\ stem

Medical radiography • Segmentation (image processing)

Medical test • Signal-to-noise ratio

Neuroimaging • Society for Imaging Science and Technology

Non invasive (medical) • Tomogram

PACS • Virtopsy

JPEG 2000

compression

JPIP streaming

See also

Earth's field NMR (EFNMR)

Electron spin resonance (spin plnsicsi
History of brain imaging
Medical imaging

\i i ii I ininiun •

Jemris (open source MRI simulator)

Magnetic Resonance Imcivjng Ijoiirm
Magnetic resonance microscopy
Magnetic Particle Imaging (MPI)
Magnetic resonance elastography
Neuroiniaging software
Nephrogenic fibrosing dermopathy
Nobel Prize cc

Nuclear magnetic resonance (NMR)
2D-FT NMRI and Spectroscopy
Relaxation
Robinson oscillator
Rabi cycle

VillOpsN

See also

• Ferromagnetic resonance

• Spin labels

• Site-directed spin labeling

• Spin trapping

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