Magnetic Resonance
Spectroscopy and
lmaging:NMR, MRI and ESR
pen source mwlib toolkit. See hUp://code. pediapress.com/ for more informatioi
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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
2DNMR 65
User:Bci2/2DFT NMRI and Spectroscopy 67
Solidstate 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, overlearned selfcontrol and relaxation
• Relaxation (psychology), the emotional state of low tension
See also:
• Tension (music)
Chemical Shift
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. Electrondonating alkyl groups, for example, lead to
increased shielding while electronwithdrawing 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 electronpoor tropylium
ion has its protons downfield at 9.17 ppm, those of the electronrich 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 threedimensional space where a nucleus experiences diamagnetic shift is called the shielding zone with a
conelike 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 23 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 coneshaped 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
1
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 Xray
photoelectron spectroscopy as the shift in atomic corelevel 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
• Carbon13NMR
• MRI
• NMR spectroscopy
• 2DFT NMRI and Spectroscopy
• Nuclear magnetic resonance
• Protein NMR
• Proton NMR
• Solidstate 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 spinspin 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 132) L1DJ and (Problems 3364) 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 CNMR 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 Shinichi Nagaoka Vol. 84 No. 5 May 2007 Journal of Chemical Education 8(
7] http://www.chem.wisc.edu/areas/reich/handouls/nmrh/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 mswers3364.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 scomponent 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
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):664668, June 1977. DOI 10.1063/1.1135103
• Wilson, K. J. and Vallabhan, C. P. G., An improved MOSFETbased Robinson oscillator for NMR detection [2] ,
Meas. Sci. Technol, l(5):458460, May 1990. DOI 10.1088/09570233/1/5/015
References
ent.aip.org/RSINAK/v48/i6/664_l.html
v.iop.org/EJ/abstract/09570233/1/5/015
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, overlearned selfcontrol and relaxation
• Relaxation (psychology), the emotional state of low tension
See also:
• Tension (music)
Chemical Shift
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. Electrondonating alkyl groups, for example, lead to
increased shielding while electronwithdrawing 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 electronpoor tropylium
ion has its protons downfield at 9.17 ppm, those of the electronrich 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 threedimensional space where a nucleus experiences diamagnetic shift is called the shielding zone with a
conelike 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 23 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 coneshaped 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
1
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 Xray
photoelectron spectroscopy as the shift in atomic corelevel 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
• Carbon13NMR
• MRI
• NMR spectroscopy
• 2DFT NMRI and Spectroscopy
• Nuclear magnetic resonance
• Protein NMR
• Proton NMR
• Solidstate 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 spinspin 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 132) L J and (Problems 3364) 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 CNMR 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 Shinichi 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 complexvalued
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 highspeed computing.
is Fourier transform
Fourier series
Discrete Fourier transform
Discretetime 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
n4
w
< (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 Lebesguemeasurable 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 nonzero real number a, if h(x) =f(ax), then /l(£) = j — :j I — ) ■ The case a = 1 leads to the
\a\ \aj
timereversal 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 RiemannLebesgue 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 righthand 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
squareintegrable, 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{xy)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(£) ■
Crosscorrelation theorem
In an analogous manner, it can be shown that if h(x) is the crosscorrelation 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 timefrequency analysis (Boashash 2003).
Fourier transform on Euclidean space
The Fourier transform can be in any arbitrary number of dimensions n. As with the onedimensional 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 ndimensional 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 ndimensional Fourier transform, as do Plancherel's and
Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and the
RiemannLebesgue 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 tradeoff 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 timefrequency domain: from the point of view of the linear canonical
transformation, the Fourier transform is rotation by 90° in the timefrequency domain, and preserves the symplectic
form.
Suppose fix) is an integrable and squareintegrable 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 xf(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:
(jxx ) 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
squareintegrable 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 nonzero curvature. The case when S is the unit
sphere in R" is of particular interest. In this case the TomasStein 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).
FourierStieltjes 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 RiemannLebesgue 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 FourierStieltjes 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 FourierStieltjes 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 complexvalued 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
Nonabelian groups
The Fourier transform can also be defined for functions on a nonabelian group, provided that the group is compact.
Unlike the Fourier transform on an abelian group, which is scalarvalued, the Fourier transform on a nonabelian
group is operatorvalued (Hewitt & Ross 1971, Chapter 8). The Fourier transform on compact groups is a major tool
in representation theory (Knapp 2001) and noncommutative harmonic analysis.
Let G be a compact Hausdorff topological group. Let 2 denote the collection of all isomorphism classes of
finitedimensional 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 FourierStieltjes
transform of \y is the operator on H defined by
where JJ^is the complexconjugate representation of U acting on H . As in the abelian case, if \i is absolutely
continuous with respect to the leftinvariant 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 FourierStieltjes 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 PeterWeyl 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 TannakaKrein 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 timefrequency analysis, one uses timefrequency transforms or
timefrequency distributions to represent signals in a form that has some time information and some frequency
information  by the uncertainty principle, there is a tradeoff between these. These can be generalizations of the
Fourier transform, such as the shorttime 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.
FTNMR, FTIR, FTNIR and MRI
The Fourier transform is also used in nuclear magnetic resonance (NMR) and in other kinds of spectroscopy, e.g.
infrared (FTIR). In NMR an exponentiallyshaped free induction decay (FID) signal is acquired in the time domain
and Fouriertransformed to a Lorentzian lineshape 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 RiemannLebesgue 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 complexexponential 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 ordinaryfrequency convention (which is used in this article) is the one most often found in the mathematics
literature. In the physics literature, the two angularfrequency conventions are more commonly used.
As discussed above, the characteristic function of a random variable is the same as the FourierStieltjes 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 "nonunitary 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
Fourier transform
Fourier transform
Remarks
unitary, ordinary frequency
unitary, angular frequency
nonunitary, angular
frequency
m
fc) = £ o me***dx
K ^ = wJ x J {x)e ^ dx
f{ V )= j^J(x)e—dx
Definition
101
af{x)+bg(x)
aho+bm
a/H + 6ffH
a • />) + 6 • $(«/)
Linearity
102
f(x  a)
e 2 ™ ? /(0
e/(c)
e— /»
Shift in
domain
103
e 2 — f(x)
/(?«)
/>  2™)
/>  27m)
Shift in
frequency
domain,
dual of
102
104
/(ax)
m
£'(:)
^(:)
Scaling in
the time
domain. If
large, then
/Mis
concentratec
around
ii?
flattens.
105
/»
f(0
/(")
2tt/( I /)
Duality.
Here f
calculated
using the
method as
Fourier
transform
column.
Results
swapping
"dummy"
variables
^or^or
106
d"f(x)
dx"
(27TiOV(0
MVM
H"/»
107
xf(x)
/ 1 \"«f/K)
V2tJ df»
.„g/H
1 dv n
This is the
dual of
106
Fourier transform
108
(/*<?)(*)
Horn
V^fHaH
/»$(")
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
Squareintegrable 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
nonunitary, 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 lowpass
filter, and the
sine function is
the noncausal
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)
1
1
1
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 ll
2a
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
^(zrf/„_ 1 (2<)
■vf^ectQ
The functions J
(x) are the nth
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 nonunitary, 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 (Vi) ^^(^"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(xnT)
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 nth order Bessel function of first kin
function T (,v) is the Chebyshev polynon
Twodimensional 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
nonunitary, angular
frequency
Remarks
/(*)
fc) = j^me^rx
^=^L f{x)e ^ dnx
/( " )= L f{x)e ~ ixvirx
Fourier transform
xp.ntMXiMY
■ 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
Discretetime Fourier transform
FourierDeligne transform
Fractional Fourier transform
Linear canonical transform
Fourier sine transform
Shorttime Fourier transform
Fourier inversion theorem
Analog signal processing
Transform (mathematics)
Integral transform
• Hartley transform
• Hankel transform
References
• Boashash, B., ed. (2003), TimeFrequency 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: McGrawHill.
• 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 0821821725.
• Dym, H; McKean, H (1985), Fourier Series and Integrals, Academic Press, ISBN 9780122264511.
• Erdelyi, Arthur, ed. (1954), Tables of Integral Transforms, 1, New Your: McGrawHill
• Fourier, J. B. Joseph (1822), The'orie Analytique de la Chaleur [1] , Paris
• Grafakos, Loukas (2004), Classical and Modern Fourier Analysis, PrenticeHall, ISBN 0 1 3035399X.
• 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 9783540006626.
• James, J.F. (2002), A Student's Guide to Fourier Transforms (2nd ed.), New York: Cambridge University Press,
ISBN 0521004284.
• Kaiser, Gerald (1994), A Friendly Guide to Wavelets, Birkhauser, ISBN 08176371 17
• Kammler, David (2000), A First Course in Fourier Analysis, Prentice Hall, ISBN 0135787823
• Katznelson, Yitzhak (1976), An introduction to Harmonic Analysis, Dover, ISBN 0486633314
• Knapp, Anthony W. (2001), Representation Theory of Semisimple Groups: An Overview Based on Examples ,
Princeton University Press, ISBN 9780691090894
• Pinsky, Mark (2002), Introduction to Fourier Analysis and Wavelets, Brooks/Cole, ISBN 0534376606
• Polyanin, A. D.; Manzhirov, A. V. (1998), Han, a wkoflnl ralEqi alums, Boca Raton: CRC Press,
ISBN 0849328764.
• Rudin, Walter (1987), Real and Complex Analysis (Third ed.), Singapore: McGraw Hill, ISBN 0071002766.
Fourier transform
• Stein, Elias; Shakarchi, Rami (2003), Fourier Analysis: An introduction, Princeton University Press,
ISBN 06911 1384X.
• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.:
Princeton University Press, ISBN 9780691080789.
• 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 3540586547.
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/dftapied/
[9] http://www.fourierseries.eom/f transform/index.html
Discrete Fourier transform
Discrete Fourier transform
Fourier transforms
Continuous Fourier transform
Fourier series
Discrete Fourier transform
Discretetime 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 nonzero values have a limited (finite) duration. Such inputs are often created by sampling
a continuous function, like a person's voice. Unlike the discretetime 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 finitedomain discretetime 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:
AT1
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,...,Nl.
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 twoargument 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 oppositesign 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" 2Trfc/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 iVdimensional complex vector has a
DFT and an IDFT which are in turn iVdimensional complex vectors.
Orthogonality
The vectors e ^kn form an orthogonal basis over the set of iVdimensional 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:
JV1 i JV1
n =0 VV fc=0
where the star denotes complex conjugation. Parseval's theorem is a special case of the Plancherel theorem and
stales:
JV1 i JV1
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:
JV1 JV1 JV1
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
ifF(K» fc =X fc
then p({ Xn . e ¥— }) fc = Xk _ m
and H{Xnm})k = X k ■ e~'^ km
Circular convolution theorem and crosscorrelation 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
JV1 JV1 / 1 JV1
■•)
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,N1]:
Discrete Fourier transform
t' {X ■ Y} n = £ Xl £ y™ I E *".(«'**
iV1 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:
iV1
•7 71 {X* ■ Y} n = E x *\ ■ (VN) n +i = f (x*y N ) n ,
1=0
which is the crosscorrelation 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
(noncircular) convolution with an xor ysequence potentially much longer than the practical transform size (N).
Two such methods are called overlapsave and overlapadd .
Convolution theorem duality
It can also be shown that:
ivi
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 complexsinusoid frequencies
(e.g. changing e _i *to gi(JVl)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 meansquare
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 realvalued 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 £
00
,01
O(JVl)
N
iV N
LO N
10
, ,11
. i(JVi)
(iVl)O (JVl)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
JV1
n=0
and the Plancherel theorem is expressed as:
JV1 JV1
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:
JV1 JV1
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 wellknown "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 £jv0 = 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 wellknown, 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
Am
m+1
m
m
m 1
4m +1
m+1
m
m
m
Am + 2
m+1
m+1
m
m
Am + 3
m+1
m+1
m + l
m
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 closedform 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"
fr
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 realinput DFT
If Xq, . . . , Xjviare 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"ivi 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 nonredundant 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 antiperiodic in frequency domain (
Xk+N = —X^) and viceversa for b — 1/2 Thus, the specific case of a — b — l/2is known as an oddtime
oddfrequency discrete Fourier transform (or O DFT). Such shifted transforms are most often used for symmetric
data, to represent different boundary symmetries, and for realsymmetric 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 ztransform, evaluated on the unit circle in the
complex plane; more general ztransforms correspond to complex shifts a and b above.
Multidimensional DFT
The ordinary DFT transforms a onedimensional 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:
iVi1 / 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 elementwise, and the
sum denotes the set of nested summations above.
The inverse of the multidimensional DFT is, analogous to the onedimensional case, given by:
: 1 y e 2^n.(k/N) x
As the onedimensional 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
(twodimensional) 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 onedimensional DFTs along each
dimension. In the twodimensional 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 onedimensional DFT is thus sufficient to efficiently compute a multidimensional DFT.
This approach is known as the rowcolumn algorithm. There are also intrinsically multidimensional FFT algorithms.
The realinput 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
onedimensional 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 uniformlyspaced
timesamples of some signal x{t), where t represents time. The conversion from continuous time to samples
(discretetime) changes the underlying Fourier transform of x(t) into a discretetime Fourier transform (DTFT),
which generally entails a type of distortion called aliasing. Choice of an appropriate samplerate (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 subsequence 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 discretetime Fourier transform article.
• The procedure is sometimes referred to as zeropadding, \\ hich is a particular implementation used in conjui
with the fast Fourier transform (FFT) algorithm. The inefficiency of performing multiplications and additions
with zerovalued "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 finegrained 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...,dl
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 CooleyTukey 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 lowcomplexity carrypropagation 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
a"
J N if a = ^ k l N
from the geometric progression formula
llae— 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 sinelike 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 complexvalued 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 numbertheoretic transforms (NTTs) in the case of finite
fields. For more information, see numbertheoretic 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 timefrequency 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 Fourierrelated transforms
. fftw
References
• Brigham, E. Oran (1988). The fast Fourier transform and its applications. Englewood Cliffs, N.J.: Prentice Hall.
ISBN 0133075052.
• Oppenheim, Alan V.; Schafer, R. W.; and Buck, J. R. (1999). Discretetime signal processing. Upper Saddle
River, N.J.: Prentice Hall. ISBN 0137549202.
• 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 0966017633.
• 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 McGrawHill. pp. 822848.
ISBN 0262032937. esp. section 30.2: The DFT and FFT, pp. 830838.
• P. Duhamel, B. Piron, and J. M. Etcheto (1988). "On computing the inverse DFT". IEEE Trans. Acoust, Speech
andSig. Processing 36 (2): 285286. 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): 6674. 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): 2531. doi: 10.1 109/TASSP. 1982.1 163843.
(Note that this paper has an apparent typo in its table of the eigenvalue multiplicities: the +ili 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):
355363. doi: 10. 1016/0022247X(82)901998.
• Natig M. Atakishiyev and Kurt Bernardo Wolf (1997). "Fractional FourierKravchuk transform". J. Opt. Soc. Am.
A 14 (7): 14671477. 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): 13291337. doi: 10. 1109/78.839980.
• Magdy Tawfik Hanna, Nabila Philip Attalla Seif, and Waleed Abd El Maguid Ahmed (2004).
"HermiteGaussianlike eigenvectors of the discrete Fourier transform matrix based on the singularvalue
decomposition of its orthogonal projection matrices". IEEE Trans. Circ. Syst. 1 51 (11): 22452254.
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): 8799. 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): 42394253.
doi:10.1109/TIT.2008.926440. preprint at arXiv:0808.1495.
• Juan G VargasRubio and Balu Santhanam (2005). "On the multiangle centered discrete fractional Fourier
transform". IEEE Sig. Proc. Lett. 12 (4): 273276. doi:10.1109/LSP.2005.843762.
• J. Cooley, P. Lewis, and P. Welch (1969). "The finite Fourier transform". IEEE Trans. Audio Electroacoustics 17
(2): 7785. doi: 10. 1109/TAU. 1969. 1162036.
• F.N. Kong (2008). "Analytic Expressions of Two Discrete HermiteGaussian Signals". IEEE Trans. Circuits and
Systems II: Express Briefs. 55 (1): 5660. doi:10.1109/TCSII.2007.909865.
r4
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., "Highspeed 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, SPTMP12.4), vol. Ill, pp. 13851388.
[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
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
complexnumber 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 roundoff 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
JV1
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 wellknown radix2 CooleyTukey
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)(ie., 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 modernday 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 poweroftwo 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
poweroftwo ]\T, Papadimitriou (1979) argued that the number 7Vlog 2 N of complexnumber additions achieved
by CooleyTukey 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 complexnumber multiplications.) Thus far, no published FFT algorithm has achieved fewer
than TV logo N complexnumber additions (or their equivalent) for poweroftwo /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
poweroftwo yy was long achieved by the splitradix FFT algorithm, which requires 4N log 2 N — 6iV + 8real
34
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 complexdata
case, because it is the simplest. However, complexdata FFTs are so closely related to algorithms for related
problems such as realdata 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
floatingpoint 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 waveletbased 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 nonsparse 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 finiteprecision floatingpoint arithmetic is used, but these errors
are typically quite small; most FFT algorithms, e.g. CooleyTukey, have excellent numerical properties. The upper
bound on the relative error for the CooleyTukey algorithm is 0(e log N), compared to 0(eN ) for the naive DFT
formula (Gentleman and Sande, 1966), where e is the machine floatingpoint relative precision. In fact, the root
mean square (rms) errors are much better than these upper bounds, being only 0(e Vlog N) for CooleyTukey 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 CooleyTukey, such as the RaderBrenner algorithm, are intrinsically less stable.
In fixedpoint arithmetic, the finiteprecision errors accumulated by FFT algorithms are worse, with rms errors
growing as O(VA0 for the CooleyTukey algorithm (Welch, 1969). Moreover, even achieving this accuracy requires
careful attention to scaling in order to minimize the loss of precision, and fixedpoint FFT algorithms involve
rescaling at each intermediate stage of decompositions like CooleyTukey.
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, impulseresponse, and timeshift properties of the transform on random
inputs (Ergiin, 1995).
Algorithms
CooleyTukey algorithm
By far the most common FFT is the CooleyTukey 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 reinvented
an algorithm known to Carl Friedrich Gauss around 1805 (and subsequently rediscovered several times in limited
forms).
The most wellknown use of the CooleyTukey algorithm is to divide the transform into two pieces of size TV/ 2 at
each step, and is therefore limited to poweroftwo sizes, but any factorization can be used in general (as was known
to both Gauss and Cooley/Tukey). These are called the radix2 and mixedradix cases, respectively (and other
variants such as the splitradix 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 CooleyTukey
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 CooleyTukey. For TV = TVi A^with coprime TVi and TV2, one can
use the PrimeFactor (GoodThomas) algorithm (PFA), based on the Chinese Remainder Theorem, to factorize the
DFT similarly to CooleyTukey but without the twiddle factors. The RaderBrenner algorithm (1976) is a
CooleyTukeylike 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 splitradix variant of
CooleyTukey (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 RaderBrenner and QFT algorithms were proposed for poweroftwo 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 realcoefficient 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 minimalmultiplication 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 poweroftwo 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
primesize FFT is due to L. I. Bluestein, and is sometimes called the chirpz algorithm; it also reexpresses a DFT as
a convolution, but this time of the same size (which can be zeropadded to a power of two and evaluated by radix2
CooleyTukey 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. CooleyTukey) 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 evenlength realinput 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 postprocessing operations.
It was once believed that realinput DFTs could be more efficiently computed by means of the discrete Hartley
transform (DHT), but it was subsequently argued that a specialized realinput 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 elementwise. Equivalently, it is simply the composition of a
sequence of d sets of onedimensional 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 rowcolumn algorithm (after the twodimensional case, below). That is, one simply
performs a sequence of d onedimensional 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 threedimensional FFT might first perform twodimensional FFTs of each planar "slice" for each fixed
ri\, and then perform the onedimensional FFTs along the ^direction. More generally, an asymptotically optimal
cacheoblivious 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 rowcolumn algorithm that ultimately requires only a onedimensional
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 outofcore and distributed memory situations where accessing noncontiguous data is
t from the rowcolumn algorithm, although all of
them have OiNXog N) complexity. Perhaps the simplest nonrowcolumn FFT is the vectorradix FFT algorithm,
which is a generalization of the ordinary CooleyTukey 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
vectorradix is where all of the radices are equal (e.g. vectorradix2 divides all of the dimensions by two), but this is
not necessary. Vector radix with only a single nonunit radix at a time, i.e. r = (1, ■ ■ ■ , 1, r, 1, • ■ ■ , 1), is
essentially a rowcolumn 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 sphericalharmonic algorithm with 0(N
log N) complexity is described by Rokhlin and Tygert (2006).
Various groups have also published "FFT" algorithms for nonequispaced 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 nonuniform discrete Fourier transform, or NDFT, which itself is often computed only
approximately).
See also
Splitradix FFT algorithm
Primefactor FFT algorithm
Bruun's FFT algorithm
Rader's FFT algorithm
Bluestein's FFT algorithm
Butterfly diagram  a diagram used to describe FFTs.
OdlyzkoSchonhage 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
„il
References
• N. Brenner and C. Rader, 1976, A New Principle for Fast Fourier Transformation LZJ , IEEE Acoustics, Speech 6
Signal Processing 24: 264266.
• Brigham, E.O. (2002), The Fast Fourier Transform, New York: PrenticeHall
• Cooley, James W., and John W. Tukey, 1965, "An algorithm for the machine calculation of complex Fourier
series," Math. Comput. 19: 297301.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein, 2001. Introduction to Algorithms,
2nd. ed. MIT Press and McGrawHill. ISBN 0262032937. 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: 1504151.
• P. Duhamel and M. Vetterli, 1990, Fast Fourier transforms: a tutorial review and a state of the art
(doi:10.1016/01651684(90)90158U), Signal Processing 19: 259299.
• A. Edelman, P. McCorquodale, and S. Toledo, 1999, The Future Fast Fourier Transform?
(doi:10.1137/S1064827597316266), SI AM J. Sci. Computing 20: 10941114.
• 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: 407416.
• M. Frigo and S. G. Johnson, 2005, "The Design and Implementation of FFTW3 [3] ," Proceedings of the IEEE 93:
216231.
Fast Fourier transform
• Carl Friedrich Gauss, 1866. "Nachlass: Theoria interpolationis methodo nova tractata," Werke band 3, 265327.
Gottingen: Konigliche Gesellschaft der Wissenschaften.
W. M. Gentleman and G. Sande, 1966, "Fast Fourier transforms— for fun and profit," Proc. AFIPS 29: 563578.
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: 250259.
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:
445448.
Heideman, M. T., D. H. Johnson, and C. S. Burrus, "Gauss and the history of the fast Fourier transform ," IEEE
ASSP Magazine, 1, (4), 1421 (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: 9195.
S. G. Johnson and M. Frigo, 2007. "A modified splitradix FFT with fewer arithmetic operations ," IEEE Trans.
Signal Processing 55 (1): 111119.
T. Lundy and J. Van Buskirk, 2007. "A new matrix approach to real FFTs and convolutions of length 2 ,"
Computing 80 (1): 2345.
Jacques Morgenstern, 1973, Note on a lower bound of the linear complexity of the fast Fourier transform
(doi:10.1145/321752.321761),/. ACM 20: 305306.
M. J. Mohlenkamp, 1999, "A fast transform for spherical harmonics", J. Fourier Anal. Appl. 5, 159184.
(preprint [8] )
H. J. Nussbaumer, 1977, Digital filtering using polynomial transforms (doi: 10. 1049/el: 19770280), Electronics
Lett. 13: 386387.
V. Pan, 1986, The tradeoff between the additive complexity and the asyncronicity of linear and bilinear
algorithms (doi: 10. 1016/00200190(86)900359), Information Proc. Lett. 22: 1114.
Christos H. Papadimitriou, 1979, Optimality of the fast Fourier transform (doi:10.1 145/322108.3221 18), /. ACM
26: 95102.
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): 19031928.
James C. Schatzman, 1996, Accuracy of the discrete Fourier transform and the fast Fourier transform , SIAM
J. Sci. Comput. 17: 11501166.
O. V. Shentov, S. K. Mitra, U. Heute, and A. N. Hossen, 1995, Subband DFT. I. Definition, interpretations and
extensions (doi: 10. 1016/01651684(94)001037), Signal Processing 41: 261277.
H. V. Sorensen, D. L. Jones, M. T. Heideman, and C. S. Burrus, 1987, Realvalued fast Fourier transform
algorithms [12] , IEEE Trans. Acoust. Speech Sig. Processing ASSP35: 849863. See also Corrections to
"Realvalued fast Fourier transform algorithms"
Peter D. Welch, 1969, A fixedpoint fast Fourier transform error analysis , IEEE Trans. Audio
Electroacoustics 17: 151157.
S. Winograd, 1978, On computing the discrete Fourier transform [15] , Math. Computation 32: 175199.
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++ — CooleyTukey 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 MultiInstrument
• 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/fftwpaperieee.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'.'arnumberl 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/article10
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/DP30%20Years%20of%20FFT.pdf
http://www.multiinstrument.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
Fourier transform spectroscopy is a measurement technique whereby spectra are collected based on measurements
of the coherence of a radiative source, using timedomain or spacedomain 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, FTNIRS), 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 unblocked
wavelength is set by a knob on the monochromator). Then the intensity
of this remaining (singlewavelength) 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>
.
M
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 MichelsonMorley experiment.
Light from the source is split into two beams by a
halfsilvered 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 infrared applications at a time when
infrared 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 fullyreflecting mirror
allowing a variable delay (in the traveltime of the li
included in one of the beams.
general have been supplanted by imaging FabryPerot
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 frequencyfield ratio revealing the masses in
the sample.
The Free Induction Decay
Pulsed FT spectrometry gives the advantage of requiring a single, timedependent 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
selfscanned 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 photonnoise 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 signaltonoise 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 nonzero value when S(t) contains a component that matches the oscillating function.
Remember that
See also
• Applied spectroscopy
• 2DFT NMRI and Spectroscopy
• Forensic chemistry
• Forensic polymer engineering
• nuclear magnetic resonance
• Infrared spectroscopy
External links
in
• 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
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 thinwalled 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 afieldindependent 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 HNMR 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 23 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 onedimensional 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.
Jcoupling
Multiplicity
Intensity Ratio
Singlet (s)
1
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 onedimensional NMR spectrum comes from
Jcoupling or scalar coupling (a special case of spinspin 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. Longrange 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 phosphorus31 or fluorine19 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
Secondorder (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 highfield NMR, however, the distortions are usually modest and the characteristic
distortions (roofing) can in fact help to identify related peaks.
Secondorder effects decrease as the frequency difference between multiplets increases, so that highfield (i.e.
highfrequency) 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 1,2dichlorobenzene 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 nonflexible systems, while conformational averaging about CC 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 twodimensional nuclear magnetic resonance (NMR)
spectroscopy. This type of NMR experiment is best known by its acronym, COSY. Other types of twodimensional
NMR include Jspectroscopy, exchange spectroscopy (EXSY), Nuclear Overhauser effect spectroscopy (NOESY),
total correlation spectroscopy (TOCSY) and heteronuclear correlation experiments, such as HSQC, HMQC, and
HMBC Twodimensional NMR spectra provide more information about a molecule than onedimensional NMR
spectra and are especially useful in determining the structure of a molecule, particularly for molecules that are too
complicated to work with using onedimensional NMR. The first twodimensional 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]
Solidstate 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 solidphase 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 solutionstate NMR spectroscopy,
these additional interactions would lead to a significant broadening of spectral lines. A variety of techniques allows
to establish highresolution conditions, that can, at least for C spectra, be comparable to solutionstate NMR
spectra.
Two important concepts for highresolution solidstate 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 solidstate 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 3dimensional structures of the protein, similar to what can be achieved by Xray crystallography. In
contrast to Xray 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
NMRactive, 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 20070511.
• The Basics of NMR  A nontechnical 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 Javascriptlike NMR coupling constant to dihedral
• NMR Spectroscopy Citizendium article on NMR Spectroscopy
Free NMR processing, analysis and simulation software
• WINDNMRPro  simulation software for interactive calculation of first and secondorder spincoupled
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
• MDjeep : 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., TwoDimensional 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
2DNMR
Correlation spectroscopy (COSY) is one of several types of twodimensional nuclear magnetic resonance (NMR)
spectroscopy. Other types of twodimensional NMR include Jspectroscopy, exchange spectroscopy (EXSY), and
Nuclear Overhauser effect spectroscopy (NOESY). Twodimensional NMR spectra provide more information about
a molecule than onedimensional NMR spectra and are especially useful in determining the structure of a molecule,
particularly for molecules that are too complicated to work with using onedimensional NMR. The first
twodimensional 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 twodimensional NMR experiment involves a series of onedimensional 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 twodimensional data set is generated from a series of onedimensional experiments.
An example of a twodimension 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 timedependent signals into a twodimensional spectrum.
The twodimensional 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 crosspeaks. The crosspeaks are symmetrical (both above and below)
along the diagonal and indicate which hydrogen atoms are spinspin 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 crosspeak with the
center of each of two corresponding diagonal peaks. The peaks on the diagonal when matched with crosspeaks are
coupled to each other.
For example: a CH CH COCH molecule 2butanone would show three peaks on the diagonal, due to the three
distinct hydrogen groups. By drawing a line straight down from a crosspeak to the point on the diagonal directly
above or below it, and then drawing a line from the crosspeak 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 crosspeak form a 90°
angle between the two peaks on the diagonal. The matching peaks, as determined by using the crosspeaks, indicate
which hydrogens are coupled, giving a clearer understanding of the structure of the molecule under examination.
1 , 1.
u .
2D
fcj
Ill [,„
i:l?
■ f h
3rotonCOSY experiment o
i Pn\
tM s ;
To the right is an example of a COSY NMR
spectrum of progesterone in DMSOd6. 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. Crosspeaks appear symmetrically
above and below the diagonal.
COSY NMR
COSY90 is the most common COSY
experiment. In COSY90, 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
crosspeaks appear in the spectrum.
Crosspeaks 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 COSY45. In COSY45 a 45° pulse
is used instead of a 90° pulse for the
first pulse, pi. The advantage of a
COSY45 is that the diagonalpeaks 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 crosspeaks near the diagonal
s can be elucidated from a COSY45
spectrum. This is not possible using COSY90. ^ uo I ' tJ  V5  ,  w ' :, lyU Overall, the COSY45 offers a cleaner spectrum
while the COSY90 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 onedimensional NMR spectrum. Using crosspeaks, 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 crosspeaks in the resulting twodimensional 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 onedimensional fashion by preselecting individual resonances.
The spectra are read with the preselected 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 preselected 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 (24 bond coupling).
See also
• Exclusive correlation spectroscopy
• Two dimensional correlation analysis
User:Bci2/2DFT NMRI and Spectroscopy
2DFT Nuclear Magnetic resonance imaging (2DFT NMRI), or Twodimensional 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 crosssectional
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 "highresolution' (50 micron
resolution) images of other living systems, such as hydrated wheat grains, were obtained and communicated in UK
in 19771979, and were subsequently confirmed by articles published in Nature.
User:Bci2/2DFT NMRI and Spectroscopy
NMRI Principle
Certain nuclei such as H nuclei, or
'fermions' have spin1/2, 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 spin1/2 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 orbitalshielded) 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 3DFT 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 pointbypoint or linebyline. 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
TwoDimensional Fourier Transform (2DFT) technique (with slice selection), or by the ThreeDimensional
Fourier Transform (3D— FT) techniques that are however much more time consuming at present. 2DFT NMRI is
sometime called in common parlance a "spinwarp". 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 twodimensional
User:Bci2/2DFT 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 kspace. We shall see next in some mathematical detail how the 2DFT
computation works to obtain 2DFT NMR images.
Twodimensional Fourier transform imaging and spectroscopy
A twodimensional Fourier transform (2DFT) is computed numerically or carried out in two stages, both involving
"standard', onedimensional 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 2DFT analysis is a very powerful method for both NMRI and twodimensional nuclear magnetic
resonance spectroscopy (2DFT NMRS) that allows the threedimensional 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 3Dreconstructions 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 2DFT NMR spectroscopic studies of algae, bacteria, yeast and certain
mammalian cells, including human ones. The 2DFT method is also widely utilized in optical spectroscopy, such as
2DFT MR hyperspectral imaging (2DFT NIRHS), or in MRI imaging for research and clinical, diagnostic
applications in Medicine. In the latter case, 2DFT NIRHS has recently allowed the identification of single,
malignant cancer cells surrounded by healthy human breast tissue at about 1 micron resolution, wellbeyond the
resolution obtainable 2DFT 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 stateoftheart NMRI instrument). A more precise mathematical definition of the "double' (2D) Fourier
transform involved in both 2D NMRI and 2DFT NMRS is specified next, and a precise example follows this
generally accepted definition.
2DFT Definition
A 2DFT, or twodimensional 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 2DFT NMRI and 2DFT 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 2DFT NMRS following computer structural recontructions based on special algorithms that are different
from FT or 2DFT. Moreover, such structural algorithms are different for 2D NMRI and 2DFT NMRS: in the
former case they involve macroscopic, or anatomical structure detrmination, whereas in the latter case of 2DFT
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 dipoledipole interactions between
neighbor nuclei.
User:Bci2/2DFT 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 2DFT NMR "spectrum' (collection of ID FTNMR 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 doublequantum and singlequantum(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 singlequantum
frequency values and with the summation carried out over all discrete, double quantum frequencies : V\ .
Example 2
Atomic Structure from 2DFT STEM Images of electron distributions in a hightemperature cuprate
superconductor "paracrystal' reveal both the domains (or "location') and the local symmetry of the 'pseudogap' in the
electronpair correlation band responsible for the high— temperature superconductivity effect (obtained at Cornell
University). So far there have been three Nobel prizes awarded for 2DFT NMR/MRI during 19922003, and an
additional, earlier Nobel prize for 2DFT of Xray data ("CAT scans'); recently the advanced possibilities of 2DFT
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 2DFT 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:Bci2/2DFT 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
Solidstate NMR
Herbert S. Gutowsky
John S. Waugh
Charles P. Slichter
FTNIRS (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: 1901. 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/pdfdoc/
fandrGazl006.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): 126578. 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): 15748.
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 0471351288.
[9] Richard R. Ernst. 1992 Nu lear Magnetii R< onance Fouriet transform (2DFT) 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 19821986 : 2DFT NMR of
[II] Charles P. Slichter. 1996. Principles of Magnetic Resonance. Springer: Berlin and New York, Third Edition., 65 1pp. ISBN 0387501576.
[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):2205962.
[15] http://www.mrtip.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 0387501576.
• Kurt Wuthrich. 1986, NMR of Proteins and Nucleic Acids., J. Wiley and Sons: New York, Chichester, Brisbane,
Toronto, Singapore. ( Nobel Laureate in 2002 for 2DFT NMR Studies of Structure and Function of Biological
Macromolecules (http://nobelprize.org/nobel_prizes/chemistry/laureates/2002/wutrichlecture.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):2205962
• 2DFT NMRI Instrument image: A JPG color image of a 2DFT NMRI "monster' Instrument (http://upload.
wikimedia.org/wikipedia/en/b/bf/HWBNMRv900.jpg).
User:Bci2/2DFT NMRI and Spectroscopy
• Richard R. Ernst. 1992. Nuclear Magnetic Resonance Fourier Transform (2DFT) Spectroscopy. Nobel Lecture
(http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/ernstlecture.pdf), on December 9, 1992.
• Peter Mansfield. 2003. Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI (http://www.
parteqinnovations.com/pdfdoc/fandrGazl006.pdf)
• D. Benett. 2007. PhD Thesis. Worcester Polytechnic Institute. PDF of 2DFT Imaging Applications to NMRI in
Medical Research. (http://www.wpi.edu/Pubs/ETD/Available/etd081707080430/unrestricted/dbennett.
pdf) Worcester Polytechnic Institute. (Includes many 2DFT 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. Twodimensional 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 twodimensional (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 twopulse sequence and
Fouriertransformation with respect to both time variables produces a twodimensional spectrum $S(0_1,0_2)$
of the desired form. This twopulse 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). "HighField NMR Instrumentation". Ch. 10 in Physical Chemistry of Food Processes
(New York: Van NostrandReinhold) 2: 42130. ISBN 0442005822.
• 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 NostrandReinhold) 2: 338420.
ISBN 0442005822.
• 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 0471351288.
• 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): 126578.
doi:10.1073/pnas.0605625103. PMID 16912110. PMC 1568902.
• Wu Y, Chesler DA, Glimcher MJ, et al (February 1999). "Multinuclear solidstate threedimensional 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): 15748. doi:10.1073/pnas.96.4.1574. PMID 9990066. PMC 15521.
External links
• Cardiac Infarct or "heart attack" Imaged in Real Time by 2DFT NMRI (http://www.mrtip.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.emri.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:Bci2/2DFT NMRI and Spectroscopy
Related Wikipedia websites
Medical imaging
Computed tomography
Magnetic resonance microscopy
Fourier transform spectroscopy
FTNIRS
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.
Solidstate nuclear magnetic resonance
Solidstate 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 liquidstate NMR experiment, Brownian motion leads to an averaging of anisotropic
interactions. In such cases, these interactions can be neglected on the timescale of the NMR experiment.
Examples of anisotropic nuclear interactions Two directionally dependent interactions commonly found in
solidstate NMR are the chemical shift anisotropy (CSA) and the internuclear dipolar coupling. Many more such
interactions exist, such as the anisotropic Jcoupling in NMR, or in related fields, such as the gtensor 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
Ultrahighfield NMR Facility for Solids.
Solidstate 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 linebroadening 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.
Highresolution 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, linebroadening 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 reintroduced (recoupling"), and used, for example, for controlled dephasing or transfer of polarization,
which allows to derive a number of structural parameters.
Solidstate NMR line widths The residual line width (full width at half max) of " C nuclei under MAS conditions at
515 kHz spinning rate is typically in the order of 0.52 ppm, and may be comparable to solutionstate 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
firstorder quadrupolar broadening is largely suppressed by sufficiently fast MAS, but the secondorder 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 secondorder quadrupolar interaction with a twodimensional
experiment such as MQMAS or STMAS.
Anisotropic interactions in solutionstate NMR From the perspective of solutionstate 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 highresolution 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.
Solidstate 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 solutionstate NMR, the directionally dependent character of the
chemical shielding is removed, leaving the isotropic chemical shift.
Jcoupling
The Jcoupling or indirect nuclear spinspin 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 onehalf 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
nonsymmetry (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
Solidstate 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 solidstate 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 solidstate 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 solidstate NMR spectroscopy
Methods and techniques
©
(Frequency channel 1,
e.g. >H)
Basic example
A fundamental RF pulse sequence and buildingblock in most
solidstate NMR experiments is crosspolarization (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
HartmannHahn 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
(solidstate) NMR experiment.
CP is a basic building block of most pulse sequences in solidstate
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 CPMAS [Schaefer and
Stejskal, 1976]. It is the typical starting point of an investigation using solidstate NMR spectroscopy.
Tcp
en
2
1
^ l^
Signal
acquisition
■>
(Frequency channel 2
e.g. ,3 Q
CP pulse seque
J \J time
Solidstate 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 offmagic 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
reintroduce 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 solidstate NMR spectroscopy complements solutionstate NMR spectroscopy and
beam diffraction methods (e.g. Xray crystallography, electron microscopy).
Chemistry
Solidstate 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 MagicAngle Spinning NMR Lucio Frydman and John S.
Hardwood, J. Am. Chem. Soc, 1995, J 17, 5367—5368, (1995)
[3] Twodimensional 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 , SolidState NMR , 6, 73 (1996)
Suggested readings for beginners
• High Resolution SolidState NMR of Quadrupolar Nuclei (http://www.grandinetti.org/assets/
GrandinettiRMCBruker2007.pdf) Grandinetti ENC Tutorial
• David D. Laws, HansMarcus L. Bitter, and Alexej Jerschow, "SolidState NMR Spectroscopic Methods in
Chemistry", Angewandte Chemie International Edition (engl.), Vol. 41, pp. 3096 (2002)
doi:10.1002/15213773(20020902)41:17<3096::AIDANIE3096>3.0.CO;2X
• Levitt, Malcolm H, Spin Dynamics: Basics of Nuclear Magnetic Resonance, Wiley, Chichester, United Kingdom,
2001. (NMR basics, including solids)
• Duer, Melinda J., Introduction to SolidState NMR Spectroscopy, Blackwell, Oxford, 2004. (Some detailed
examples of SSNMR spectroscopy)
Solidstate nuclear magnetic resonance
Advanced readings
Books and major review articles
• McDermott, A, Structure and Dynamics of Membrane Proteins by Magic Angle Spinning SolidState NMR (http:/
/arjourn als.annualreviews.org/doi/abs/ 10. 1146/ annurev.biophys. 050708. 1337 19) Annual Review of
Biophysics, v. 38, 2009.
• Mehring, M, Principles of High Resolution NMR in Solids, 2nd ed., Springer, Heidelberg, 1983.
• Slichter, C. P., Principles of Magnetic Resonance, 3rd ed., Springer, Heidelberg, 1990.
• Gerstein, B. C. and Dybowski, C, Transient Techniques in NMR of Solids, Academic Press, San Diego, 1985.
• SchmidtRohr, K. and Spiess, H.W I dtidin n n I did \tatt <" < </ Polymers, Academic Press, San
Diego, 1994.
• Dybowski, C. and Lichter, R. L., NMR Spectroscopy Techniques, Marcel Dekker, New York, 1987.
• Ramamoorthy, A., NMR Spectroscopy of Biological Solids. Taylor & Francis, New York, 2006.
General
References to books and research articles
• Andrew, E. R., Bradbury, A. and Eades, R. G., "Removal of Dipolar Broadening of Nuclear Magnetic Resonance
Spectra of Solids by Specimen Rotation," Nature 183, 1802, (1959)
• Ernst, Bodenhausen, Wokaun: Principl u I <i, ti > mce in One and Two Dimensions
• Hartmann S.R., Hahn E.L., "Nuclear Double Resonance in the Rotating Frame" Phys. Rev. 128 (1962) 2042.
• Pines A., Gibby M.G, Waugh J.S., "Protonenhanced NMR of dilute spins in solids" J. Chem. Phys. 59, 56990,
(1973)
• Purcell, Torrey and Pound (1945).
• Schaefer, J. and Stejskal, E. O., "Carbon13 Nuclear Magnetic Resonance of Polymers Spinning at the Magic
Angle," Journal of the American Chemical Society 98, 1031 (1976).
• Gullion, T. and Schaefer, J., "RotationalEcho, DoubleResonance NMR," J. Magn. Reson., 81, 196 (1989).
External links
• NMRWiki.ORG (http://www.nmrwiki.org) NMR resource you can edit
• SSNMRBLOG (http://ssnmr.blogspot.com/) SolidState NMR Literature Blog by Prof. Rob Schurko's
SolidState NMR group at the University of Windsor
• www.ssnmr.org (http://www.ssnmr.org) Rocky Mountain Conference on SolidState NMR
• Varian Inc (http://www.varianinc.com) NMR system/product manufacturer
Magnetic resonance microscopy
Magnetic resonance microscopy
Magnetic Resonance Microscopy (MRM, pMRI) is Magnetic Resonance Imaging (MRI) at a microscopic level. A
strict definition is MRI having voxel resolutions of better than 100 am 3
Nomenclature
Many scientist in the field consider the name Magnetic Resonance Microscopy to be a misnomer, since the images
produced are much worse than those produced by even a marginal optical or electron microscope. As such, the name
High Resolution Magnetic Resonance Imaging is often preferred in scientific literature on the subject. In fact, the
term is most widely used by the High Resolution Magnetic Resonance Imaging group from Duke University, headed
by Allan Johnson.
Differences between MRI and MRM
• Resolution: Typical medical MRI resolution is about 1 mm 3 ; the desired resolution of MRM is 100 pm 3 or
smaller.
• Specimen size: Medical MRI machines are designed so that a patient may fit inside. MRM chambers are usually
small, typically less than 1 cm 3 .
Current status of MRM
Although MRI is very common for medical applications, MRM is still developed in laboratories. The major barriers
for practical MRM include:
• Magnetic field gradient: High gradient focus the magnetic resonance in a smaller volume (smaller point spread
function), results in a better spatial resolution. The gradients for MRM are typically 50 to 100 times those of
clinical systems. However, the construction of radio frequency (RF) coil used in MRM does not allow ultrahigh
gradient.
• Sensitivity: Because the voxels for MRM can be 1/100,000 of those in MRI, the signal will be proportionately
weaker
Alternative MRM
Magnetic Resonance Force Microscopy (MRFM) is claimed to have nm 3 scale resolutions. It improves the
sensitivity issue by introducing microfabricated cantilever to measure tiny signals. The magnetic gradient is
generated by a micrometrescale magnetic tip, yielding a typical gradient 10 million times larger than those of
clinical systems. This technique is still in the beginning stage. Because the specimen need to be in high vacuum at
cryogenic temperatures, MRFM can be only used for solid state rr
Magnetic resonance microscopy
External links
• Introduction to Magnetic Resonance Microscopy Auditory Research Laboratory at the Univ. of North Carolina.
References
[1] P. Glover and P. Mansfield, Limits to magnetic resonance microscopy, Rep. Prog. Phys. 65 148915 1 ! . 2002
[2] R. Maronpot Applications of Magnetic Resonance Microscop>. Toxicologic Palholog). 32( Suppl. 2):42^8, 2004
[3] http://chaweb2.ined.unc.cdu/henson_mnn/pages/inrmfaq.html#MRMAnchor
Imaging
Medical imaging
Medical imaging is the technique and process used to create images of the human body (or parts and function
thereof) for clinical purposes (medical procedures seeking to reveal, diagnose or examine disease) or medical science
(including the study of normal anatomy and physiology). Although imaging of removed organs and tissues can be
performed for medical reasons, such procedures are not usually referred to as medical imaging, but rather are a part
of pathology.
As a discipline and in its widest sense, it is part of biological imaging and incorporates radiology (in the wider
sense), nuclear medicine, investigative radiological sciences, endoscopy, (medical) thermography, medical
photography and microscopy (e.g. for human pathological investigations).
Measurement and recording techniques which are not primarily designed to produce images, such as
electroencephalography (EEG), magnetoencephalography (MEG), Electrocardiography (EKG) and others, but which
produce data susceptible to be represented as maps (i.e. containing positional information), can be seen as forms of
medical imaging.
Overview
In the clinical context, medical imaging is generally equated to radiology or "clinical imaging" and the medical
practitioner responsible for interpreting (and sometimes acquiring) the images is a radiologist. Diagnostic
radiography designates the technical aspects of medical imaging and in particular the acquisition of medical images.
The radiographer or radiologic technologist is usually responsible for acquiring medical images of diagnostic
quality, although some radiological interventions are performed by radiologists. While radiology is an evaluation of
anatomy, nuclear medicine provides functional assessment.
As a field of scientific investigation, medical imaging constitutes a subdiscipline of biomedical engineering,
medical physics or medicine depending on the context: Research and development in the area of instrumentation,
image acquisition (e.g. radiography), modelling and quantification are usually the preserve of biomedical
engineering, medical physics and computer science; Research into the application and interpretation of medical
images is usually the preserve of radiology and the medical subdiscipline relevant to medical condition or area of
medical science (neuroscience, cardiology, psychiatry, psychology, etc.) under investigation. Many of the techniques
developed for medical imaging also have scientific and industrial applications.
Medical imaging is often perceived to designate the set of techniques that noninvasively produce images of the
internal aspect of the body. In this restricted sense, medical imaging can be seen as the solution of mathematical
inverse problems. This means that cause (the properties of living tissue) is inferred from effect (the observed signal).
In the case of ultrasonography the probe consists of ultrasonic pressure waves and echoes inside the tissue show the
internal structure. In the case of projection radiography, the probe is Xray radiation which is absorbed at different
rates in different tissue types such as bone, muscle and fat.
The term noninvasive is a term based off of the fact that following medical imaging modalities do not penetrate the
skin physically. But on the electromagnetic and radiation level, they are quite invasive. From the high energy
photons in XRay Computed Tomography, to the 2+ Tesla coils of an MRI device, these modalities alter the physical
and chemical reactions of the body in order to obtain data.
Imaging technology
Radiography
Two forms of radiographic images are in use in medical imaging; projection radiography and fluoroscopy, with the
latter being useful for intraoperative and catheter guidance. These 2D techniques are still in wide use despite the
advance of 3D tomography due to the low cost, high resolution, and depending on application, lower radiation
dosages. This imaging modality utilizes a wide beam of x rays for image acquisition and is the first imaging
technique available in modern medicine.
• Fluoroscopy produces realtime images of internal structures of the body in a similar fashion to radiography, but
employs a constant input of xrays, at a lower dose rate. Contrast media, such as barium, iodine, and air are used
to visualize internal organs as they work. Fluoroscopy is also used in imageguided procedures when constant
feedback during a procedure is required. An image receptor is required to convert the radiation into an image after
it has passed through the area of interest. Early on this was a fluorescing screen, which gave way to an Image
Amplifier (IA) which was a large vacuum tube that had the receiving end coated with cesium iodide, and a mirror
at the opposite end. Eventually the mirror was replaced with a TV camera.
• Projectional radiographs, more commonly known as xrays, are often used to determine the type and extent of a
fracture as well as for detecting pathological changes in the lungs. With the use of radioopaque contrast media,
such as barium, they can also be used to visualize the structure of the stomach and intestines  this can help
diagnose ulcers or certain types of colon cancer.
Magnetic resonance imaging (MRI)
A magnetic resonance imaging instrument (MRI scanner), or "nuclear
magnetic resonance (NMR) imaging" scanner as it was originally
known, uses powerful magnets to polarise and excite hydrogen nuclei
(single proton) in water molecules in human tissue, producing a
detectable signal which is spatially encoded, resulting in images of the
body. MRI uses three electromagnetic fields: a very strong (on the
order of units of teslas) static magnetic field to polarize the hydrogen
nuclei, called the static field; a weaker timevarying (on the order of
1 kHz) field(s) for spatial encoding, called the gradient field(s); and a
weak radiofrequency (RF) field for manipulation of the hydrogen
nuclei to produce measurable signals, collected through an RF antenna.
Like CT, MRI traditionally creates a two dimensional image of a thin
"slice" of the body and is therefore considered a tomographic imaging
technique. Modern MRI instruments are capable of producing images
in the form of 3D blocks, which may be considered a generalisation of
the singleslice, tomographic, concept. Unlike CT, MRI does not
involve the use of ionizing radiation and is therefore not associated
with the same health hazards. For example, because MRI has only been in use since the early 1980s, there are no
known longterm effects of exposure to strong static fields (this is the subject of some debate; see 'Safety' in MRI)
and therefore there is no limit to the number of scans to which an individual can be subjected, in contrast with Xray
and CT. However, there are wellidentified health risks associated with tissue heating from exposure to the RF field
and the presence of implanted devices in the body, such as pace makers. These risks are strictly controlled as part of
the design of the instrument and the scanning protocols used.
Because CT and MRI are sensitive to different tissue properties, the appearance of the images obtained with the two
techniques differ markedly. In CT, Xrays must be blocked by some form of dense tissue to create an image, so the
image quality when looking at soft tissues will be poor. In MRI, while any nucleus with a net nuclear spin can be
used, the proton of the hydrogen atom remains the most widely used, especially in the clinical setting, because it is
so ubiquitous and returns a large signal. This nucleus, present in water molecules, allows the excellent softtissue
contrast achievable with MRI.
Nuclear medicine
Nuclear medicine encompasses both diagnostic imaging and treatment of disease, and may also be referred to as
molecular medicine or molecular imaging & therapeutics . Nuclear medicine uses certain properties of isotopes
and the energetic particles emitted from radioactive material to diagnose or treat various pathology. Different from
the typical concept of anatomic radiology, nuclear medicine enables assessment of physiology. This functionbased
approach to medical evaluation has useful applications in most subspecialties, notably oncology, neurology, and
cardiology. Gamma cameras are used in e.g. scintigraphy, SPECT and PET to detect regions of biologic activity that
may be associated with disease. Relatively short lived isotope, such as I is administered to the patient. Isotopes
are often preferentially absorbed by biologically active tissue in the body, and can be used to identify tumors or
fracture points in bone. Images are acquired after collimated photons are detected by a crystal that gives off a light
signal, which is in turn amplified and converted into count data.
• Scintigraphy ("scint") is a form of diagnostic test wherein radioisotopes are taken internally, for example
intravenously or orally. Then, gamma camera capture and form twodimensional images from the radiation
emitted by the radiopharmaceuticals. For example, technetiumlabeled isoniazid (INH) and ethambutol (EMB)
has been used for tubercular imaging for early diagnosis of tuberculosis
• SPECT is a 3D tomographic technique that uses gamma camera data from many projections and can be
reconstructed in different planes. A dual detector head gamma camera combined with a CT scanner, which
provides localization of functional SPECT data, is termed a SPECT/CT camera, and has shown utility in
advancing the field of molecular imaging. In most other medical imaging modalities, energy is passed through the
body and the reaction or result is read by detectors. In SPECT imaging, the patient is injected with a radioisotope,
most commonly Thallium 201TI, Technetium 99mTC, Iodine 1231, and Gallium 68Ga
. The radioactive gamma rays are emitted through the body as the natural decaying process of these isotopes takes
place. The emissions of the gamma rays are captured by detectors that surround the body. This essentially means that
the human is now the source of the radioactivity, rather than the medical imaging devices such as XRay, CT, or
Ultrasound.
• Positron emission tomography (PET) uses coincidence detection to image functional processes. Shortlived
positron emitting isotope, such as F, is incorporated with an organic substance such as glucose, creating
F18fluorodeoxyglucose, which can be used as a marker of metabolic utilization. Images of activity distribution
throughout the body can show rapidly growing tissue, like tumor, metastasis, or infection. PET images can be
viewed in comparison to computed tomography scans to determine an anatomic correlate. Modern scanners
combine PET with a CT, or even MRI, to optimize the image reconstruction involved with positron imaging. This
is performed on the same equipment without physically moving the patient off of the gantry. The resultant hybrid
of functional and anatomic imaging information is a useful tool in noninvasive diagnosis and patient
management.
Photoacoustic imaging
Photoacoustic imaging is a recently developed hybrid biomedical imaging modality based on the photoacoustic
effect. It combines the advantages of optical absorption contrast with ultrasonic spatial resolution for deep imaging
in (optical) diffusive or quasidiffusive regime. Recent studies have shown that photoacoustic imaging can be used in
vivo for tumor angiogenesis monitoring, blood oxygenation mapping, functional brain imaging, and skin melanoma
detection, etc.
Breast Thermography
Digital infrared imaging thermography is based on the principle that metabolic activity and vascular circulation in
both precancerous tissue and the area surrounding a developing breast cancer is almost always higher than in normal
breast tissue. Cancerous tumors require an everincreasing supply of nutrients and therefore increase circulation to
their cells by holding open existing blood vessels, opening dormant vessels, and creating new ones
(neoangiogenesis). This process frequently results in an increase in regional surface temperatures of the breast.
Digital infrared imaging uses extremely sensitive medical infrared cameras and sophisticated computers to detect,
analyze, and produce highresolution diagnostic images of these temperature variations. Because of DII's sensitivity,
these temperature variations may be among the earliest signs of breast cancer and/or a precancerous state of the
breast [5] .
Tomography
Tomography is the method of imaging a single plane, or slice, of an object resulting in a tomogram. There are
several forms of tomography:
• Linear tomography: This is the most basic form of tomography. The Xray tube moved from point "A" to point
"B" above the patient, while the cassette holder (or "bucky") moves simultaneously under the patient from point
"B" to point "A." The fulcrum, or pivot point, is set to the area of interest. In this manner, the points above and
below the focal plane are blurred out, just as the background is blurred when panning a camera during exposure.
No longer carried out and replaced by computed tomography.
• Poly tomography: This was a complex form of tomography. With this technique, a number of geometrical
movements were programmed, such as hypocycloidic, circular, figure 8, and elliptical. Philips Medical Systems
[6] produced one such device called the 'Poly tome.' This unit was still in use into the 1990s, as its resulting
images for small or difficult physiology, such as the inner ear, was still difficult to image with CTs at that time.
As the resolution of CTs got better, this procedure was taken over by the CT.
• Zonography: This is a variant of linear tomography, where a limited arc of movement is used. It is still used in
some centres for visualising the kidney during an intravenous urogram (IVU).
• Orthopantomography (OPT or OPG): The only common tomographic examination in use. This makes use of a
complex movement to allow the radiographic examination of the mandible, as if it were a flat bone. It is often
referred to as a "Panorex", but this is incorrect, as it is a trademark of a specific company.
• Computed Tomography (CT), or Computed Axial Tomography (CAT: A CT scan, also known as a CAT scan), is
a helical tomography (latest generation), which traditionally produces a 2D image of the structures in a thin
section of the body. It uses Xrays. It has a greater ionizing radiation dose burden than projection radiography;
repeated scans must be limited to avoid health effects. CT is based off of the same principles as XRay projections
but in this case, the patient is enclosed in a surrounding ring of detectors assigned with 5001000 scintillation
detectors
. This being the fourthgeneration XRay CT scanner geometry. Previously in older generation scanners, the XRay
beam was paired by a translating source and detector.
Ultrasound
Medical ultrasonography uses high frequency broadband sound waves in the megahertz range that are reflected by
tissue to varying degrees to produce (up to 3D) images. This is commonly associated with imaging the fetus in
pregnant women. Uses of ultrasound are much broader, however. Other important uses include imaging the
abdominal organs, heart, breast, muscles, tendons, arteries and veins. While it may provide less anatomical detail
than techniques such as CT or MRI, it has several advantages which make it ideal in numerous situations, in
particular that it studies the function of moving structures in realtime, emits no ionizing radiation, and contains
speckle that can be used in elastography. Ultrasound is also used as a popular research tool for capturing raw data,
that can be made available through an Ultrasound research interface, for the purpose of tissue characterization and
implementation of new image processing techniques. The concepts of ultrasound differ from other medical imaging
modalities in the fact that it is operated by the transmission and receipt of sound waves. The high frequency sound
waves are sent into the tissue and depending on the composition of the different tissues; the signal will be attenuated
and returned at separate intervals. A path of reflected sound waves in a multilayered structure can be defined by an
input acoustic impedance( Ultrasound sound wave) and the Reflection and transmission coefficients of the relative
structures . It is very safe to use and does not appear to cause any adverse effects, although information on this is
not well documented. It is also relatively inexpensive and quick to perform. Ultrasound scanners can be taken to
critically ill patients in intensive care units, avoiding the danger caused while moving the patient to the radiology
department. The real time moving image obtained can be used to guide drainage and biopsy procedures. Doppler
capabilities on modern scanners allow the blood flow in arteries and veins to be assessed.
Medical imaging topics
Maximizing imaging procedure use
The amount of data obtained in a single MR or CT scan is very extensive. Some of the data that radiologists discard
could save patients time and money, while reducing their exposure to radiation and risk of complications from
invasive procedures. J
Creation of threedimensional images
Recently, techniques have been developed to enable CT, MRI and ultrasound scanning software to produce 3D
images for the physician. Traditionally CT and MRI scans produced 2D static output on film. To produce 3D
images, many scans are made, then combined by computers to produce a 3D model, which can then be manipulated
by the physician. 3D ultrasounds are produced using a somewhat similar technique. In diagnosing disease of the
viscera of abdomen,ultrasound is particularly sensitive on imaging of biliary tract,urinary tract and female
reproductive organs(ovary,fallopian tubes). As for example,diagnosis of gall stone by dilatation of common bile duct
and stone in common bile duct . With the ability to visualize important structures in great detail, 3D visualization
methods are a valuable resource for the diagnosis and surgical treatment of many pathologies. It was a key resource
for the famous, but ultimately unsuccessful attempt by Singaporean surgeons to separate Iranian twins Ladan and
Laleh Bijani in 2003. The 3D equipment was used previously for similar operations with great success.
Other proposed or developed techniques include:
• Diffuse optical tomography
• Elastography
• Electrical impedance tomography
• Optoacoustic imaging
• Ophthalmology
• Ascan
• Bscan
• Corneal topography
• Optical coherence tomography
• Scanning laser ophthalmoscopy
Some of these techniques are still at a research stage and not yet used in clinical routines.
Compression of medical images
Medical imaging techniques produce very large amounts of data, especially from CT, MRI and PET modalities. As a
result, storage and communications of electronic image data are prohibitive without the use of compression. JPEG
2000 is the stateoftheart image compression DICOM standard for storage and transmission of medical images.
The cost and feasibility of accessing large image data sets over low or various bandwidths are further addressed by
use of another DICOM standard, called JPIP, to enable efficient streaming of the JPEG 2000 compressed image data.
Nondiagnostic imaging
Neuroimaging has also been used in experimental circumstances to allow people (especially disabled persons) to
control outside devices, acting as a brain computer interface.
Archiving and recording
Used primarily in ultrasound imaging, capturing the image a medical imaging device is required for archiving and
telemedicine applications. In most scenarios, a frame grabber is used in order to capture the video signal from the
medical device and relay it to a computer for further processing and operations.
Open source software for medical image analysis
Several open source software packages are available for performing analysis of medical images:
• ImageJ
• 3D Sheer
• ITK
• OsiriX
• Gemldent
• MicroDicom
• Free Surfer
Use in pharmaceutical clinical trials
Medical imaging has become a major tool in clinical trials since it enables rapid diagnosis with visualization and
quantitative assessment.
A typical clinical trial goes through multiple phases and can take up to eight years. Clinical endpoints or outcomes
are used to determine whether the therapy is safe and effective. Once a patient reaches the endpoint, he/she is
generally excluded from further experimental interaction. Trials that rely solely on clinical endpoints are very costly
as they have long durations and tend to need large number of patients.
In contrast to clinical endpoints, surrogate endpoints have been shown to cut down the time required to confirm
whether a drug has clinical benefits. Imaging biomarkers (a characteristic that is objectively measured by an imaging
technique, which is used as an indicator of pharmacological response to a therapy) and surrogate endpoints have
shown to facilitate the use of small group sizes, obtaining quick results with good statistical power.
Imaging is able to reveal subtle change that is indicative of the progression of therapy that may be missed out by
more subjective, traditional approaches. Statistical bias is reduced as the findings are evaluated without any direct
patient contact.
For example, measurement of tumour shrinkage is a commonly used surrogate endpoint in solid tumour response
evaluation. This allows for faster and more objective assessment of the effects of anticancer drugs. In evaluating the
extent of Alzheimer's disease, it is still prevalent to use behavioural and cognitive tests. MRI scans on the entire
brain can accurately pinpoint hippocampal atrophy rate while PET scans is able to measure the brain's metabolic
activity by measuring regional glucose metabolism.
An imagingbased trial will usually be made up of three components:
1. A realistic imaging protocol. The protocol is an outline that standardizes (as far as practically possible) the way in
which the images are acquired using the various modalities (PET, SPECT, CT, MRI). It covers the specifics in
which images are to be stored, processed and evaluated.
2. An imaging centre that is responsible for collecting the images, perform quality control and provide tools for data
storage, distribution and analysis. It is important for images acquired at different time points are displayed in a
standardised format to maintain the reliability of the evaluation. Certain specialised imaging contract research
organizations provide to end medical imaging services, from protocol design and site management through to data
quality assurance and image analysis.
3. Clinical sites that recruit patients to generate the images to send back to the imaging centre.
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
Fullbody scan
Magnetic field imaging • Pneumoencephalogram
Medical examination • Radiology information s\ stem
Medical radiography • Segmentation (image processing)
Medical test • Signaltonoise ratio
Neuroimaging • Society for Imaging Science and Technology
Non invasive (medical) • Tomogram
PACS • Virtopsy
JPEG 2000
compression
JPIP streaming
Further reading
• Burger, Wilhelm; Burge, Mark James, eds (2008). Digital Image Processing: An Algorithmic Introduction using
Java. Texts in Computer Science series. New York: Springer Science+Business Media.
doi: 10. 1007/9781846289682. ISBN 9781846283796.
• Baert, Albert L., ed (2008). Encyclop i >fDu nt I maging.I thn: Springer Verlag.
doi: 10. 1007/9783540352808. ISBN 9783540352785.
• Tony F. Chan and Jackie Shen (2005). Image Processing and Analysis  Variational, PDE, Wavelet, and
Stochastic Methods [13] . SIAM
• Terry Yoo(Editor) (2004), Insight into Images.
• Robb, RA (1999). Biomedical Imaging, Visualization, and Analysis. John Wiley & Sons, Inc. ISBN 0471283533.
• Journal of Digital Imaging (New York: Springer Science+Business Media). ISSN 08971889.
• Using JPIP for StandardCompliant Sharing of Medical Image Data a white paper by Aware Inc.
External links
• Medical imaging at the Open Directory Project
• Medical Image Database Free Indexed Online Images
• http://www.aware.com/imaging/accuradjpip.htm What is JPIP?
References
[I] Society of Nuclear Medicine (http://www.snm.org)
2 I 1 li 'i n i] i in iii ph lulu in i 1 diclionai ih li i nan m mi i ph Citin Dor] ind \1 li il D u n i
for Health Consumers, 2007 by Saunders: Saunders Coinprehensne Yeterinan, Dictionary, 3 ed. 2007; McGrawHill Concise Dictionary of
Modern Medicine, 2002 by The McGrawHill Companies
[3] Singh, Namrala Singh. Clinical K\ a] nation of Radiolabeled Drugs for Tubercular Imaging. LAP Lambert Academic Publishing (2010).
ISBN13: 9783838378381
[4] Dhawan P, A. (2003). Medical Imaging Analysis. Hoboken, NJ: WileyInterscience Publication
[5] http://www.breastthermography.com/breast_thermography_mf.htm
[6] http://www.mcdical.philips.coni/iiiaiii/iiKlcx.asp
[7] Dhawan P, A. (2003). Medical Imaging Analysis. Hoboken, NJ: WileyInterscience Publication
[8] Dhawan P, A. (2003). Medical Imaging Analysis. Hoboken, NJ: WileyInterscience Publication
 1 )] 1 . heir G. Wast no nt not id n Ih in , n in 1 1 in p < lu hup \\\ liatini ia ns; i in n lispl irl K
113619/1541872). Diagnostic Imaging. March 19, 2010.
[10] Udupa, J.K. and Herman, G. T, 3D Imaging in Medicine, 2nd Edition, CRC Press, 2000
[II] Treating Medical Ailments in Real Time I htlp://\\ \\ \\ .epiphan.com/solulii>ns_new/?arid=16)
[12] Hajnal, J. V., Hawkes, D. J., & Hill, D. L. (2001). Medical Image Registration. CRC Press.
MM hup ■' i.k I leu. ii I .ii .■ .1 p.iL'i ■ 1 1 . I ■ i . . • i . . •  l ill l
1 1 1 1 hit] a i in i n i n i in i \ i p ii ill tin
[15] http://www.aw arc. cum/iniaging/w hitepapcrs.hlni
[16] http://www.dmoz.Org/Health/Medicine/Imaging//
I i l\ http://rad.usuhs.edu/mcdpix/indcx.html?
MRI
Magnetic resonance imaging (MRI), or nuclear magnetic resonance
imaging (NMRI), is primarily a noninvasive medical imaging
technique used in radiology to visualize detailed internal structure and
limited function of the body. MRI provides much greater contrast
between the different soft tissues of the body than computed
tomography (CT) does, making it especially useful in neurological
(brain), musculoskeletal, cardiovascular, and oncological (cancer)
imaging.
Unlike CT, MRI uses no ionizing radiation. Rather, it uses a powerful
magnetic field to align the nuclear magnetization of (usually) hydrogen
atoms in water in the body. Radio frequency (RF) fields are used to
systematically alter the alignment of this magnetization. This causes
the hydrogen nuclei to produce a rotating magnetic field detectable by
the scanner. This signal can be manipulated by additional magnetic
fields to build up enough information to construct an image of the
body. [1] :36
Magnetic resonance imaging is a relatively new technology. The first
MR image was published in 1973 and the first crosssectional
image of a living mouse was published in January 1974. The first
studies performed on humans were published in 1977. By
comparison, the first human Xray image was taken in 1895.
Magnetic resonance imaging is a development of nuclear magnetic
resonance. Originally, the technique was referred to as nuclear
magnetic resonance imaging (NMRI). However, because the word
nuclear was associated in the public mind with ionizing radiation
exposure, it is generally now referred to simply as MRI. Scientists still
use the term NMRI when discussing nonmedical devices operating on
the same principles. The term magnetic resonance tomography (MRT)
is also sometimes used.
Sagittal MR image of the knee
Parasagittal MRI of the head, with aliasing
ariilactN (ikinc and forehead appear at the back of
the head)
How MRI works
The body is largely composed of water molecules. Each water molecule has two hydrogen nuclei or protons. When a
person goes inside the powerful magnetic field of the scanner, the magnetic moments of some of these protons
changes, and aligns with the direction of the field.
Photons are the carriers of electromagnetic radiation such as light, and electromagnetism. All photons travel at the
speed of light in a vacuum and therefore cannot carry differing amounts of energy due to their velocity. Their energy
is manifested by frequency — higher energy, higher frequency.
In an MRI machine a radio frequency transmitter is briefly turned on, producing an electromagnetic field. The
photons of this field have just the right energy, known as the resonance frequency, to flip the spin of the aligned
protons in the body. As the intensity and duration of the field increases, more aligned spins are affected. After the
field is turned off, the protons decay to the original spindown state and the difference in energy between the two
states is released as a photon. It is these photons that produce the electromagnetic signal that the scanner detects. The
frequency the protons resonate at depends on the strength of the magnetic field. As a result of conservation of
energy, this also dictates the frequency of the released photons. The photons released when the field is removed have
an energy — and therefore a frequency — due to the amount of energy the protons absorbed while the field was
It is this relationship between fieldstrength and frequency that allows the use of nuclear magnetic resonance for
imaging. Additional magnetic fields are applied during the scan to make the magnetic field strength depend on the
position within the patient, providing a straightforward method to control where the protons are excited by the radio
photons. These fields are created by passing electric currents through solenoids, known as gradient coils. Since these
coils are within the bore of the scanner, there are large forces between them and the main field coils, producing most
of the noise that is heard during operation. Without efforts to dampen this noise, it can approach 130 decibels (dB)
with strong fields (see also #Acoustic_noise).
An image can be constructed because the protons in different tissues return to their equilibrium state at different
rates, which is a difference that can be detected. By changing the parameters on the scanner, this effect is used to
create contrast between different types of body tissue or between other properties, as in fMRI and diffusion MRI.
Contrast agents may be injected intravenously to enhance the appearance of blood vessels, tumors or inflammation.
Contrast agents may also be directly injected into a joint in the case of arthrograms, MRI images of joints. Unlike
CT, MRI uses no ionizing radiation and is generally a very safe procedure. Nonetheless the strong magnetic fields
and radio pulses can affect metal implants, including cochlear implants and cardiac pacemakers. In the case of
cardiac pacemakers, the results can sometimes be lethal , so patients with such implants are generally not eligible
for MRI.
MRI is used to image every part of the body, and is particularly useful for tissues with many hydrogen nuclei and
little density contrast, such as the brain, muscle, connective tissue and most tumors.
Applications
In clinical practice, MRI is used to distinguish pathologic tissue (such as a brain tumor) from normal tissue. One
advantage of an MRI scan is that it is believed to be harmless to the patient. It uses strong magnetic fields and
nonionizing radiation in the radio frequency range, unlike CT scans and traditional Xrays, which both use ionizing
radiation.
While CT provides good spatial resolution (the ability to distinguish two separate structures an arbitrarily small
distance from each other), MRI provides comparable resolution with far better contrast resolution (the ability to
distinguish the differences between two arbitrarily similar but not identical tissues). The basis of this ability is the
complex library of pulse sequences that the modern medical MRI scanner includes, each of which is optimized to
provide image contrast based on the chemical sensitivity of MRI.
For example, with particular values of the echo time (T ' ) and the repetition time (T ' ), which are basic parameters of
image acquisition, a sequence takes on the property of T weighting. On a T weighted scan, water and
fluidcontaining tissues are bright (most modern T sequences are actually fast T sequences) and fatcontaining
tissues are dark. The reverse is true for T weighted images. Damaged tissue tends to develop edema, which makes a
T weighted sequence sensitive for pathology, and generally able to distinguish pathologic tissue from normal tissue.
With the addition of an additional radio frequency pulse and additional manipulation of the magnetic gradients, a
T weighted sequence can be converted to a FLAIR sequence, in which free water is now dark, but edematous
tissues remain bright. This sequence in particular is currently the most sensitive way to evaluate the brain for
demyelinating diseases, such as multiple sclerosis.
The typical MRI examination consists of 520 sequences, each of which are chosen to provide a particular type of
information about the subject tissues. This information is then synthesized by the interpreting physician.
Basic MRI scans
^weighted MRI
T weighted scans use a gradient echo (GRE) sequence, with short T and short T . This is one of the basic types of
MR contrast and is a commonly run clinical scan. The T weighting can be increased (improving contrast) with the
use of an inversion pulse as in an MPRAGE sequence. Due to the short repetition time (T ) this scan can be run
very fast allowing the collection of high resolution 3D datasets. A T reducing gadolinium contrast agent is also
commonly used, with a T scan being collected before and after administration of contrast agent to compare the
difference. In the brain T weighted scans provide good gray matter/white matter contrast, in other words put
SIMPLY, Tl Weighted Images highlights fat deposition.
r 2 weighted MRI
7 weighted scans use a spin echo (SE) sequence, with long T and long T . They have long been the clinical
workhorse as the spin echo sequence is less susceptible to inhomogeneities in the magnetic field. They are
particularly well suited to edema as they are sensitive to water content (edema is characterized by increased water
content). In other words, put more simply, T2 weighted images light up liquid on the images being visualized.
T* 2 weighted MRI
T (pronounced "T 2 star") weighted scans use a gradient echo (GRE) sequence, with long T and long T . The
gradient echo sequence used does not have the extra refocusing pulse used in spin echo so it is subject to additional
losses above the normal T decay (referred to as T '), these taken together are called T . This also makes it more
prone to susceptibility losses at air/tissue boundaries, but can increase contrast for certain types of tissue, such as
venous blood.
Spin density weighted MRI
Spin density, also called proton density, weighted scans try to have no contrast from either T or T decay, the only
signal change coming from differences in the amount of available spins (hydrogen nuclei in water). It uses a spin
echo or sometimes a gradient echo sequence, with short T and long T .
Specialized MRI scans
Diffusion MRI
Diffusion MRI measures the diffusion of water molecules in biological
tissues. In an isotropic medium (inside a glass of water for example) water
molecules naturally move randomly according to turbulence and Brownian
motion. In biological tissues however, where the Reynold's number is low
enough for flows to be laminar, the diffusion may be anisotropic. For
example a molecule inside the axon of a neuron has a low probability of
crossing the myelin membrane. Therefore the molecule moves principally
along the axis of the neural fiber. If we know that molecules in a particular
voxel diffuse principally in one direction we can make the assumption that
the majority of the fibers in this area are going parallel to that direction.
The recent development of diffusion tensor imaging (DTI) enables
diffusion to be measured in multiple directions and the fractional anisotropy
in each direction to be calculated for each voxel. This enables researchers to
make brain maps of fiber directions to examine the connectivity of different regions in the brain (using tractography)
or to examine areas of neural degeneration and demyelination in diseases like Multiple Sclerosis.
Another application of diffusion MRI is diffusionweighted imaging (DWI). Following an ischemic stroke, DWI is
highly sensitive to the changes occurring in the lesion. It is speculated that increases in restriction (barriers) to
water diffusion, as a result of cytotoxic edema (cellular swelling), is responsible for the increase in signal on a DWI
scan. The DWI enhancement appears within 510 minutes of the onset of stroke symptoms (as compared with
computed tomography, which often does not detect changes of acute infarct for up to 46 hours) and remains for up
to two weeks. Coupled with imaging of cerebral perfusion, researchers can highlight regions of "perfusion/diffusion
mismatch" that may indicate regions capable of salvage by reperfusion therapy.
Like many other specialized applications, this technique is usually coupled with a fast image acquisition sequence,
such as echo planar imaging sequence.
Magnetization Transfer MRI
Magnetization transfer (MT) refers to the transfer of longitudinal magnetization from free water protons to hydration
water protons in NMR and MRI.
In magnetic resonance imaging of molecular solutions, such as protein solutions, two types of water molecules, free
(bulk) and hydration (bound), are found. Free water protons have faster average rotational frequency and hence less
fixed water molecules that may cause local field inhomogeneity. Because of this uniformity, most free water protons
have resonance frequency lying narrowly around the normal proton resonance frequency of 63 MHz (at 1.5 teslas).
This also results in slower transverse magnetization dephasing and hence longer T Conversely, hydration water
molecules are slowed down by interaction with solute molecules and hence create field inhomogeneities that lead to
wider resonance frequency spectrum.
Fluid attenuated inversion recovery (FLAIR)
Fluid Attenuated Inversion Recovery (FLAIR) is an inversionrecovery pulse sequence used to null signal from
fluids. For example, it can be used in brain imaging to suppress cerebrospinal fluid (CSF) so as to bring out the
periventricular hyperintense lesions, such as multiple sclerosis (MS) plaques. By carefully choosing the inversion
time TI (the time between the inversion and excitation pulses), the signal from any particular tissue can be
suppressed.
Magnetic resonance angiography
Magnetic resonance angiography (MRA) generates pictures of the
arteries to evaluate them for stenosis (abnormal narrowing) or
aneurysms (vessel wall dilatations, at risk of rupture). MRA is
often used to evaluate the arteries of the neck and brain, the
thoracic and abdominal aorta, the renal arteries, and the legs
(called a "runoff"). A variety of techniques can be used to
generate the pictures, such as administration of a paramagnetic
contrast agent (gadolinium) or using a technique known as
"flowrelated enhancement" (e.g. 2D and 3D timeofflight
sequences), where most of the signal on an image is due to blood
that recently moved into that plane, see also FLASH MRI.
Techniques involving phase accumulation (known as phase
contrast angiography) can also be used to generate flow velocity
maps easily and accurately. Magnetic resonance venography (MRV) is a similar procedure that is used to image
veins. In this method, the tissue is now excited inferiorly, while signal is gathered in the plane immediately superior
to the excitation plane — thus imaging the venous blood that recently moved from the excited plane
Magnetic resonance gated intracranial CSF dynamics (MRGILD)
Magnetic resonance gated intracranial cerebrospinal fluid (CSF) or liquor dynamics (MRGILD) technique is an MR
sequence based on bipolar gradient pulse used to demonstrate CSF pulsatile flow in ventricles, cisterns, aqueduct of
Sylvius and entire intracranial CSF pathway. It is a method for analyzing CSF circulatory system dynamics in
patients with CSF obstructive lesions such as normal pressure hydrocephalus. It also allows visualization of both
arterial and venous pulsatile blood flow in vessels without use of contrast agents.
„ [13] [14]
Diastolic time data acquisition (I)I'DA). Systolic time data acquisition (STDA).
Magnetic resonance spectroscopy
Magnetic resonance spectroscopy (MRS) is used to measure the levels of different metabolites in body tissues. The
MR signal produces a spectrum of resonances that correspond to different molecular arrangements of the isotope
being "excited". This signature is used to diagnose certain metabolic disorders, especially those affecting the
brain, ' and to provide information on tumor metabolism. J
Magnetic resonance spectroscopic imaging (MRSI) combines both spectroscopic and imaging methods to produce
spatially localized spectra from within the sample or patient. The spatial resolution is much lower (limited by the
available SNR), but the spectra in each voxel contains information about many metabolites. Because the available
signal is used to encode spatial and spectral information, MRSI requires high SNR achievable only at higher field
strengths (3 T and above).
Functional MRI
Functional MRI (fMRI) measures signal changes in the brain that are
due to changing neural activity. The brain is scanned at low resolution
but at a rapid rate (typically once every 23 seconds). Increases in
neural activity cause changes in the MR signal via T changes; this
mechanism is referred to as the BOLD (bloodoxygenlevel dependent)
effect. Increased neural activity causes an increased demand for
oxygen, and the vascular system actually overcompensates for this,
increasing the amount of oxygenated hemoglobin relative to
deoxygenated hemoglobin. Because deoxygenated hemoglobin
attenuates the MR signal, the vascular response leads to a signal
increase that is related to the neural activity. The precise nature of the
relationship between neural activity and the BOLD signal is a subject
of current research. The BOLD effect also allows for the generation of
high resolution 3D maps of the venous vasculature within neural tissue.
While BOLD signal is the most common method employed for neuroscience studies in human subjects, the flexible
nature of MR imaging provides means to sensitize the signal to other aspects of the blood supply. Alternative
techniques employ arterial spin labeling (ASL) or weight the MRI signal by cerebral blood flow (CBF) and cerebral
blood volume (CBV). The CBV method requires injection of a class of MRI contrast agents that are now in human
clinical trials. Because this method has been shown to be far more sensitive than the BOLD technique in preclinical
studies, it may potentially expand the role of fMRI in clinical applications. The CBF method provides more
quantitative information than the BOLD signal, albeit at a significant loss of detection sensitivity.
Interventional MRI
The lack of harmful effects on the patient and the operator make MRI wellsuited for "interventional radiology",
where the images produced by a MRI scanner are used to guide minimally invasive procedures. Of course, such
procedures must be done without any ferromagnetic instruments.
A specialized growing subset of interventional MRI is that of intraoperative MRI in which the MRI is used in the
surgical process. Some specialized MRI systems have been developed that allow imaging concurrent with the
surgical procedure. More typical, however, is that the surgical procedure is temporarily interrupted so that MR
images can be acquired to verify the success of the procedure or guide subsequent surgical work.
Radiation therapy simulation
Because of MRI's superior imaging of soft tissues, it is now being utilized to specifically locate tumors within the
body in preparation for radiation therapy treatments. For therapy simulation, a patient is placed in specific,
reproducible, body position and scanned. The MRI system then computes the precise location, shape and orientation
of the tumor mass, correcting for any spatial distortion inherent in the system. The patient is then marked or tattooed
with points that, when combined with the specific body position, permits precise triangulation for radiation therapy.
Current density imaging
Current density imaging (CDI) endeavors to use the phase information from images to reconstruct current densities
within a subject. Current density imaging works because electrical currents generate magnetic fields, which in turn
affect the phase of the magnetic dipoles during an imaging sequence. To date no successful CDI has been performed
using biological currents, but several studies have been published that involve currents applied through a pair of
electrodes.
Magnetic resonance guided focused ultrasound
In MRgFUS therapy, ultrasound beams are focused on a tissue — guided and controlled using MR thermal
imaging — and due to the significant energy deposition at the focus, temperature within the tissue rises to more than
65 °C (150 °F), completely destroying it. This technology can achieve precise "ablation" of diseased tissue. MR
imaging provides a threedimensional view of the target tissue, allowing for precise focusing of ultrasound energy.
The MR imaging provides quantitative, realtime, thermal images of the treated area. This allows the physician to
ensure that the temperature generated during each cycle of ultrasound energy is sufficient to cause thermal ablation
within the desired tissue and if not, to adapt the parameters to ensure effective treatment.
Multinuclear imaging
Hydrogen is the most frequently imaged nucleus in MRI because it is present in biological tissues in great
abundance. However, any nucleus with a net nuclear spin could potentially be imaged with MRI. Such nuclei include
helium3, carbon13, fluorine19, oxygen17, sodium23, phosphorus31 and xenon129. Na, P and O are
naturally abundant in the body, so can be imaged directly. Gaseous isotopes such as He or Xe must be
hyperpolarized and then inhaled as their nuclear density is too low to yield a useful signal under normal conditions.
O, C and F can be administered in sufficient quantities in liquid form (e.g. Owater, Cglucose solutions or
perfluorocarbons) that hyperpolarization is not a necessity.
Multinuclear imaging is primarily a research technique at present. However, potential applications include functional
imaging and imaging of organs poorly seen on H MRI (e.g. lungs and bones) or as alternative contrast agents.
Inhaled hyperpolarized He can be used to image the distribution of air spaces within the lungs. Injectable solutions
containing C or stabilized bubbles of hyperpolarized Xe have been studied as contrast agents for angiography
and perfusion imaging. P can potentially provide information on bone density and structure, as well as functional
imaging of the brain.
Susceptibility weighted imaging (SWI)
Susceptibility weighted imaging (SWI), is a new type of contrast in MRI different from spin density, T or T
imaging. This method exploits the susceptibility differences between tissues and uses a fully velocity compensated,
three dimensional, RF spoiled, highresolution, 3D gradient echo scan. This special data acquisition and image
processing produces an enhanced contrast magnitude image very sensitive to venous blood, hemorrhage and iron
storage. It is used to enhance the detection and diagnosis of tumors, vascular and neurovascular diseases (stroke and
hemorrhage, multiple sclerosis, Alzheimer's), and also detects traumatic brain injuries that may not be diagnosed
using other methods. [18][19]
Other specialized MRI techniques
MRI is a new and active field of research and new methods and variants are often published when they are able to
get better results in specific fields. Examples of these recent improvements are T weighted turbo spinecho (T
TSE MRI), double inversion recovery MRI (DIRMRI) or phasesensitive inversion recovery MRI (PSIRMRI), all
of them able to improve imaging of the brain lesions . Another example is MPRAGE
(magnetizationprepared rapid acquisition with gradient echo) , which improves images of multiple sclerosis
cortical lesions
Portable instruments
Portable magnetic resonance instruments are available for use in education and field research. Using the principles of
Earth's field NMR, they have no powerful polarizing magnet, so that such instruments can be small and inexpensive.
Some can be used for both EFNMR spectroscopy and MRI imaging . The low strength of the Earth's field results
in poor signal to noise ratios, requiring long scan times to capture spectroscopic data or build up MRI images.
Research with atomic magnetometers have discussed the possibility for cheap and portable MRI instruments without
the large magnet.
2S 2fr
MRI versus CT
A computed tomography (CT) scanner uses Xrays, a type of ionizing radiation, to acquire its images, making it a
good tool for examining tissue composed of elements of a higher atomic number than the tissue surrounding them,
such as bone and calcifications (calcium based) within the body (carbon based flesh), or of structures (vessels,
bowel). MRI, on the other hand, uses nonionizing radio frequency (RF) signals to acquire its images and is best
suited for noncalcified tissue, though MR images can also be acquired from bones and teetlr as well as fossils.
CT may be enhanced by use of contrast agents containing elements of a higher atomic number than the surrounding
flesh such as iodine or barium. Contrast agents for MRI have paramagnetic properties, e.g., gadolinium and
manganese.
Both CT and MRI scanners are able to generate multiple twodimensional crosssections (slices) of tissue and
threedimensional reconstructions. Unlike CT, which uses only Xray attenuation to generate image contrast, MRI
has a long list of properties that may be used to generate image contrast. By variation of scanning parameters, tissue
contrast can be altered and enhanced in various ways to detect different features. (See Applications above.)
MRI can generate crosssectional images in any plane (including oblique planes). In the past, CT was limited to
acquiring images in the axial (or near axial) plane. The scans used to be called Computed Axial Tomography scans
(CAT scans). However, the development of multidetector CT scanners with nearisotropic resolution, allows the CT
scanner to produce data that can be retrospectively reconstructed in any plane with minimal loss of image quality.
For purposes of tumor detection and identification in the brain, MRI is generally superior. However, in the
case of solid tumors of the abdomen and chest, CT is often preferred due to less motion artifact. Furthermore, CT
usually is more widely available, faster, less expensive, and may be less likely to require the person to be sedated or
anesthetized.
MRI is also best suited for cases when a patient is to undergo the exam several times successively in the short term,
because, unlike CT, it does not expose the patient to the hazards of ionizing radiation.
llion USD. 3.0 tesla
i cost up to $500,000
Economics of MRI
MRI equipment is expensive. 1.5 tesla scanners often cost between $1 million and $1.5
scanners often cost between $2 million and $2.3 million USD. Construction of MRI suites
USD, or more, depending on project scope.
MRI scanners have been significant sources of revenue for healthcare
providers in the US. This is because of favorable reimbursement rates
from insurers and federal government programs. Insurance
reimbursement is provided in two components, an equipment charge
for the actual performance of the MRI scan and professional charge for
the radiologist's review of the images and/or data. In the US Northeast,
an equipment charge might be $3,500 and a professional charge might
be $350 although the actual fees received by the equipment owner
and interpreting physician are often quite less and depend on the rates
negotiated with insurance companies or determined by governmental
action as in the Medicare Fee Schedule. For example, an orthopedic
surgery group in Illinois billed a charge of $1,116 for a knee MRI in 2007 but the Medicare reimbursement in 2007
was only $470.91 . Many insurance companies require preapproval of an MRI procedure as a condition for
coverage.
In the US, the 2007 Deficit Reduction Act (DRA) significantly reduced reimbursement rates paid by federal
insurance programs for the equipment component of many scans, shifting the economic landscape. Many private
insurers have followed suit.
Looking througl
Installation of the MRI unit
r*%
P
K *
—
Heavy lifting equipment is used to install the
MRI unit.
An MRI unit is a rather large item, typically requiring heavy
equipment (such as cranes) to move the unit to its final location. Once
the MRI unit is in place, the room that houses it is usually "built up"
around the unit itself. See this page for an example of the
complexity involved in installing an MRI unit in a clinical setting.
Safety
Death and injuries have occurred from projectiles created by the
magnetic field, although few compared to the millions of examinations
administered. MRI makes use of powerful magnetic fields that,
though not known to cause direct biological damage, can interfere with
metallic and electromechanical devices. Additional (small) risks are
presented by the radio frequency systems, components or elements of
the MRI system's operation, elements of the scanning procedure and
medications that may be administered to facilitate MRI imaging.
Of great concern is the dramatic increase in the number of reported MRI accidents to the U.S. Food and Drug
Administration (FDA). Since 2004, the last year in which a decline in the number of MRI accidents was reported, the
full spectrum of MRI accidents has increased significantly in the following years. The 2008 FDA accident report
data culminates in a 277% increase over the 2004 rate.
There are many steps that the MRI patient and referring physician can take to help reduce the remaining risks,
including providing a full, accurate and thorough medical history to the MRI provider.
Several of the specific MRI safety considerations are identified below:
Implants and foreign bodies
Pacemakers are generally considered an absolute contraindication towards MRI scanning, though highly specialized
protocols have been developed to permit scanning of select pacing devices. Several cases of arrhythmia or death
have been reported in patients with pacemakers who have undergone MRI scanning without appropriate precautions.
Other electronic implants have varying contraindications, depending upon scanner technology, and implant
properties, scanning protocols and anatomy being imaged.
Many other forms of medical or biostimulation implants may be contraindicated for MRI scans. These may include
vagus nerve stimulators, implantable cardioverterdefibrillators, loop recorders, insulin pumps, cochlear implants,
deep brain stimulators, and many others. Medical device patients should always present complete information
(manufacturer, model, serial number and date of implantation) about all implants to both the referring physician and
to the radiologist or technologist before entering the room for the MRI scan.
While these implants pose a current problem, scientists and manufacturers are working on improved designs that
reduce risks to medical device operations. One such development in the works is a nanocoating for implants
intended to screen them from the radio frequency waves, helping to make MRI exams available to patients currently
prohibited from receiving them. The current article for this is from New Scientist.
Ferromagnetic foreign bodies (e.g. shell fragments), or metallic implants (e.g. surgical prostheses, aneurysm clips)
are also potential risks, and safety aspects need to be considered on an individual basis. Interaction of the magnetic
and radio frequency fields with such objects can lead to trauma due to movement of the object in the magnetic field,
thermal injury from radiofrequency induction heating of the object, or failure of an implanted device. These issues
are especially problematic when dealing with the eye. Most MRI centers require an orbital xray to be performed on
anyone suspected of having metal fragments in their eyes, something not uncommon in metalworking.
Because of its nonferromagnetic nature and poor electrical conductivity, titanium and its alloys are useful for long
term implants and surgical instruments intended for use in imageguided surgery. In particular, not only is titanium
safe from movement from the magnetic field, but artifacts around the implant are less frequent and less severe than
with more ferromagnetic materials e.g. stainless steel. Artifacts from metal frequently appear as regions of empty
space around the implant — frequently called 'blackhole artifact'. E.g. a 3 mm titanium alloy coronary stent may
appear as a 5 mm diameter region of empty space on MRI, whereas around a stainless steel stent, the artifact may
extend for 1020 mm or more.
In 2006, a new classification system for implants and ancillary clinical devices has been developed by ASTM
International and is now the standard supported by the US Food and Drug Administration:
MRSafe — The device or implant is completely nonmagnetic,
nonelectric ally conductive, and nonRF reactive, eliminating all of the
primary potential threats during an MRI procedure.
MR
MRConditional — A device or implant that may contain magnetic,
electrically conductive or RFreactive components that is safe for
operations in proximity to the MRI, provided the conditions for safe
operation are defined and observed (such as 'tested safe to 1.5 teslas' or
'safe in magnetic fields below 500 gauss in strength').
MRUnsafe — Nearly selfexplanatory, this category is reserved for
objects that are significantly ferromagnetic and pose a clear and direct
threat to persons and equipment within the magnet room.
Though the current classification system was originally developed for
regulatoryapproved medical devices, it is being applied to all manner
of items, appliances and equipment intended for use in the MR
environment.
In the case of pacemakers, the risk is thought to be primarily RF
induction in the pacing electrodes/wires causing inappropriate pacing
of the heart, rather than the magnetic field affecting the pacemaker
itself. Much research and development is being undertaken, and many
tools are being developed to predict RF field effects inside the body.
Patients who have been prescribed MRI exams who are concerned
about safety may be interested in the 10 Questions To Ask Your MRI Provider
MRI providers who wish to measure the degree to which they have effectively addressed the safety issues for
patients and staff may be interested in the MRI Suite Safety Calculator provided through a radiology website.
Projectile or missile effect
As a result of the very high strength of the magnetic field needed to produce scans (frequently up to 60,000 times the
Earth's own magnetic field effects), there are several incidental safety issues addressed in MRI facilities.
Missileeffect accidents, where ferromagnetic objects are attracted to the center of the magnet, have resulted in injury
and death. A video simulation of a fatal projectile effect accident illustrates the extreme power that
contemporary MRI equipment can exert on ferromagnetic objects.
To reduce the risks of projectile accidents, ferromagnetic objects and devices are typically prohibited in proximity to
the MRI scanner, with nonferromagnetic versions of many tools and devices typically retained by the scanning
facility. Patients undergoing MRI examinations are required to remove all metallic objects, often by changing into a
gown or scrubs.
Ferromagnetic detection devices are used by some sites as a supplement conventional screening techniques, and are
now recommended by the American College of Radiology's Guidance Document for Safe MR Practices: 2007
and the United States' Veterans Administration's Design Guide
The magnetic field and the associated risk of missileeffect accidents remains a permanent hazard, as
superconductive MRI magnets are kept permanently energized and so retain their magnetic field in the event of a
power outage.
Radio frequency energy
A powerful radio transmitter is needed for excitation of proton spins. This can heat the body to the point of risk of
hyperthermia in patients, particularly in obese patients or those with thermoregulation disorders. Several countries
have issued restrictions on the maximum specific absorption rate that a scanner may produce.
Peripheral nerve stimulation (PNS)
The rapid switching on and off of the magnetic field gradients is capable of causing nerve stimulation. Volunteers
report a twitching sensation when exposed to rapidly switched fields, particularly in their extremities. The reason the
peripheral nerves are stimulated is that the changing field increases with distance from the center of the gradient
coils (which more or less coincides with the center of the magnet). Note however that when imaging the head, the
heart is far offcenter and induction of even a tiny current into the heart must be avoided at all costs. Although PNS
was not a problem for the slow, weak gradients used in the early days of MRI, the strong, rapidly switched gradients
used in techniques such as EPI, fMRI, diffusion MRI, etc. are indeed capable of inducing PNS. American and
European regulatory agencies insist that manufacturers stay below specified dB/dt limits {dRIdt is the change in field
per unit time) or else prove that no PNS is induced for any imaging sequence. As a result of dB/dt limitation,
commercial MRI systems cannot use the full rated power of their gradient amplifiers.
Acoustic noise
Switching of field gradients causes a change in the Lorentz force experienced by the gradient coils, producing
minute expansions and contractions of the coil itself. As the switching is typically in the audible frequency range, the
resulting vibration produces loud noises (clicking or beeping). This is most marked with highfield machines and
rapidimaging techniques in which sound intensity can reach 120 dB(A) (equivalent to a jet engine at takeoff)
As a reference, 120 dB is the threshold of loudness causing sensation in the human ear canal — tickling, and 140 dB
is the threshold of ear pain. Since decibel is a logarithmic measurement, a 10 dB increase equates to a 10fold
increase in intensity — which, in acoustics, is roughly equal to a doubling of loudness.
Appropriate use of ear protection is essential for anyone inside the MRI scanner room during the examination.
Cryogens
As described above in #Scanner construction and operation, many MRI scanners rely on cryogenic liquids to enable
superconducting capabilities of the electromagnetic coils within. Though the cryogenic liquids used are nontoxic,
their physical properties present specific hazards.
An unintentional shutdown of a superconducting electromagnet, an event known as "quench", involves the rapid
boiling of liquid helium from the device. If the rapidly expanding helium cannot be dissipated through an external
vent, sometimes referred to as 'quench pipe', it may be released into the scanner room where it may cause
displacement of the oxygen and present a risk of asphyxiation.
Liquid helium, the most commonly used cryogen in MRI, undergoes near explosive expansion as it changes from
liquid to a gaseous state. Rooms built in support of superconducting MRI equipment should be equipped with
pressure relief mechanisms and an exhaust fan, in addition to the required quench pipe.
Since a quench results in rapid loss of all cryogens in the magnet, recommissioning the magnet is expensive and
timeconsuming. Spontaneous quenches are uncommon, but may also be triggered by equipment malfunction,
improper cryogen fill technique, contaminants inside the cryostat, or extreme magnetic or vibrational disturbances.
Contrast agents
The most commonly used intravenous contrast agents are based on chelates of gadolinium. In general, these agents
have proved safer than the iodinated contrast agents used in Xray radiography or CT. Anaphylactoid reactions are
rare, occurring in approx. 0.030.1%. Of particular interest is the lower incidence of nephrotoxicity, compared
with iodinated agents, when given at usual doses — this has made contrastenhanced MRI scanning an option for
patients with renal impairment, who would otherwise not be able to undergo contrastenhanced CT.
Although gadolinium agents have proved useful for patients with renal impairment, in patients with severe renal
failure requiring dialysis there is a risk of a rare but serious illness, nephrogenic systemic fibrosis, that may be linked
to the use of certain gadoliniumcontaining agents. The most frequently linked is gadodiamide, but other agents have
been linked too. Although a causal link has not been definitively established, current guidelines in the United
States are that dialysis patients should only receive gadolinium agents where essential, and that dialysis should be
performed as soon as possible after the scan to remove the agent from the body promptly. In Europe, where more
gadoliniumcontaining agents are available, a classification of agents according to potential risks has been
released. Recently a new contrast agent named gadoxetate, brand name Eovist (US) or Primovist (EU), was
approved for diagnostic use: this has the theoretical benefit of a dual excretion path.
Pregnancy
No effects of MRI on the fetus have been demonstrated. In particular, MRI avoids the use of ionizing radiation, to
which the fetus is particularly sensitive. However, as a precaution, current guidelines recommend that pregnant
women undergo MRI only when essential. This is particularly the case during the first trimester of pregnancy, as
organogenesis takes place during this period. The concerns in pregnancy are the same as for MRI in general, but the
fetus may be more sensitive to the effects — particularly to heating and to noise. However, one additional concern is
the use of contrast agents; gadolinium compounds are known to cross the placenta and enter the fetal bloodstream,
and it is recommended that their use be avoided.
Despite these concerns, MRI is rapidly growing in importance as a way of diagnosing and monitoring congenital
defects of the fetus because it can provide more diagnostic information than ultrasound and it lacks the ionizing
radiation of CT. MRI without contrast agents is the imaging mode of choice for presurgical, inutero diagnosis and
evaluation of fetal tumors, primarily teratomas, facilitating open fetal surgery, other fetal interventions, and planning
for procedures (such as the EXIT procedure) to safely deliver and treat babies whose defects would otherwise be
fatal.
Claustrophobia and discomfort
Due to the construction of some MRI scanners, they can be potentially unpleasant to lie in. Older models of closed
bore MRI systems feature a fairly long tube or tunnel. The part of the body being imaged must lie at the center of the
magnet, which is at the absolute center of the tunnel. Because scan times on these older scanners may be long
(occasionally up to 40 minutes for the entire procedure), people with even mild claustrophobia are sometimes unable
to tolerate an MRI scan without management. Modern scanners may have larger bores (up to 70 cm) and scan times
are shorter. This means that claustrophobia is less of an issue, and many patients now find MRI an innocuous and
easily tolerated procedure.
Nervous patients may still find the following strategies helpful:
• Advance preparation
• visiting the scanner to see the room and practice lying on the table
• visualization techniques
• chemical sedation
• general anesthesia
• Coping while inside the scanner
• holding a "panic button"
• closing eyes as well as covering them (e.g. washcloth, eye mask)
• listening to music on headphones or watching a movie with a Headmounted display while in the machine
Alternative scanner designs, such as open or upright systems, can also be helpful where these are available. Though
open scanners have increased in popularity, they produce inferior scan quality because they operate at lower
magnetic fields than closed scanners. However, commercial 1.5 tesla open systems have recently become available,
providing much better image quality than previous lower field strength open models
For babies and young children chemical sedation or general anesthesia are the norm, as these subjects cannot be
instructed to hold still during the scanning session. Obese patients and pregnant women may find the MRI machine
to be a tight fit. Pregnant women may also have difficulty lying on their backs for an hour or more without moving.
Guidance
Safety issues, including the potential for biostimulation device interference, movement of ferromagnetic bodies, and
incidental localized heating, have been addressed in the American College of Radiology's White Paper on MR
Safety, which was originally published in 2002 and expanded in 2004. The ACR White Paper on MR Safety has been
rewritten and was released early in 2007 under the new title ACR Guidance Document for Safe MR Practices
In December 2007, the Medicines in Healthcare product Regulation Agency (MHRA), a UK healthcare regulatory
body, issued [heir Safely Guidelines for Magnetic Resonance Imaging Equipment in Clinical Use
In February 2008, the Joint Commission, a US healthcare accrediting organization, issued a Sentinel Event Alert #38
, their highest patient safety advisory, on MRI safety issues.
In July 2008, the United States Veterans Administration, a federal governmental agency serving the healthcare needs
of former military personnel, issued a substantial revision to their MRI Design Guide , which includes physical or
facility safety considerations.
The European Physical Agents Directive
The European Physical Agents (Electromagnetic Fields) Directive is legislation adopted in European legislature.
Originally scheduled to be required by the end of 2008, each individual state within the European Union must
include this directive in its own law by the end of 2012. Some member nations passed complying legislation and are
now attempting to repeal their state laws in expectation that the final version of the EU Physical Agents Directive
will be substantially revised prior to the revised adoption date.
The directive applies to occupational exposure to electromagnetic fields (not medical exposure) and was intended to
limit workers' acute exposure to strong electromagnetic fields, as may be found near electricity substations, radio or
television transmitters or industrial equipment. However, the regulations impact significantly on MRI, with separate
sections of the regulations limiting exposure to static magnetic fields, changing magnetic fields and radio frequency
energy. Field strength limits are given, which may not be exceeded. An employer may commit a criminal offense by
allowing a worker to exceed an exposure limit, if that is how the Directive is implemented in a particular member
state.
The Directive is based on the international consensus of established effects of exposure to electromagnetic fields,
and in particular the advice of the European Commissions's advisor, the International Commission on NonIonizing
Radiation Protection (ICNIRP). The aims of the Directive, and the ICNIRP guidelines it is based on, are to prevent
exposure to potentially harmful fields. The actual limits in the Directive are very similar to the limits advised by the
Institute of Electrical and Electronics Engineers, with the exception of the frequencies produced by the gradient
coils, where the IEEE limits are significantly higher.
Many Member States of the EU already have either specific EMF regulations or (as in the UK) a general requirement
under workplace health and safety legislation to protect workers against electromagnetic fields. In almost all cases
the existing regulations are aligned with the ICNIRP limits so that the Directive should, in theory, have little impact
on any employer already meeting their legal responsibilities.
The introduction of the Directive has brought to light an existing potential issue with occupational exposures to MRI
fields. There are at present very few data on the number or types of MRI practice that might lead to exposures in
excess of the levels of the Directive. There is a justifiable concern amongst MRI practitioners that if the
Directive were to be enforced more vigorously than existing legislation, the use of MRI might be restricted, or
working practices of MRI personnel might have to change.
In the initial draft a limit of static field strength to 2 T was given. This has since been removed from the regulations,
and whilst it is unlikely to be restored as it was without a strong justification, some restriction on static fields may be
reintroduced after the matter has been considered more fully by ICNIRP. The effect of such a limit might be to
restrict the installation, operation and maintenance of MRI scanners with magnets of 2 T and stronger. As the
increase in field strength has been instrumental in developing higher resolution and higher performance scanners,
this would be a significant step back. This is why it is unlikely to happen without strong justification.
Individual government agencies and the European Commission have now formed a working group to examine the
implications on MRI and to try to address the issue of occupational exposures to electromagnetic fields from MRI.
Threedimensional (3D) image reconstruction
The principle
Because contemporary MRI scanners offer isotropic, or near isotropic, resolution, display of images does not need to
be restricted to the conventional axial images. Instead, it is possible for a software program to build a volume by
'stacking' the individual slices one on top of the other. The program may then display the volume in an alternative
manner.
3D rendering techniques
Surface rendering
A threshold value of greyscale density is chosen by the operator (e.g. a level that corresponds to fat). A
threshold level is set, using edge detection image processing algorithms. From this, a 3dimensional model can
be constructed and displayed on screen. Multiple models can be constructed from various different thresholds,
allowing different colors to represent each anatomical component such as bone, muscle, and cartilage.
However, the interior structure of each element is not visible in this mode of operation.
Volume rendering
Surface rendering is limited in that it only displays surfaces that meet a threshold density, and only displays
the surface closest to the imaginary viewer. In volume rendering, transparency and colors are used to allow a
better representation of the volume to be shown in a single image  e.g. the bones of the pelvis could be
displayed as semitransparent, so that even at an oblique angle, one part of the image does not conceal another.
Image segmentation
Where different structures have similar threshold density, it can become impossible to separate them simply by
adjusting volume rendering parameters. The solution is called segmentation, a manual or automatic procedure that
can remove the unwanted structures from the image.
2003 Nobel Prize
Reflecting the fundamental importance and applicability of MRI in medicine, Paul Lauterbur of the University of
Illinois at UrbanaChampaign and Sir Peter Mansfield of the University of Nottingham were awarded the 2003
Nobel Prize in Physiology or Medicine for their "discoveries concerning magnetic resonance imaging". The Nobel
citation acknowledged Lauterbur's insight of using magnetic field gradients to determine spatial localization, a
discovery that allowed rapid acquisition of 2D images. Mansfield was credited with introducing the mathematical
formalism and developing techniques for efficient gradient utilization and fast imaging. The actual research that won
the prize was done almost 30 years before, while Paul Lauterbur was at Stony Brook University in New York.
The award was vigorously protested by Raymond Vahan Damadian, founder of FONAR Corporation, who claimed
that he invented the MRI, and that Lauterbur and Mansfield had merely refined the technology. An ad hoc
group, called "The Friends of Raymond Damadian", took out fullpage advertisements in the New York Times and
The Washington Post entitled "The Shameful Wrong That Must Be Righted", demanding that he be awarded at least
a share of the Nobel Prize. Also, even earlier, in the Soviet Union, Vladislav Ivanov filed (in 1960) a document
with the USSR State Committee for Inventions and Discovery at Leningrad for a Magnetic Resonance Imaging
device , although this was not approved until the 1970s. In a letter to Physics Today, Herman Carr pointed out
n earlier use of field gradients for onedimensional MR imaging.
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)
2DFT NMRI and Spectroscopy
Relaxation
Robinson oscillator
Rabi cycle
VillOpsN
Further reading
• Simon, Merrill; Mattson, James S (1996). The pioneers of NMR and magnetic resonance in medicine: The story of
MRI. Ramat Gan, Israel: BarIlan University Press. ISBN 0961924314.
• Haacke, E Mark; Brown, Robert F; Thompson, Michael; Venkatesan, Ramesh (1999). Magnetic r
imaging. Ph) rical pi inciples cud ■sequence design. New York: J. Wiley & Sons. ISBN 0471351288.
• Lee, S. C. et al., (2001). One Micrometer Resolution NMR Microscopy. J. Magn. Res., 150: 207213.
External links
Listing of MRI Types [68]  All Inclusive Listing of MRI Types
www.mritutorial.com [ 9]  MRITUTORIAL.COM I A free learning repository about neuroimaging
http://www.imaios.com/en/eCourses/eMRI L J MRI stepbystep, interactive course on magnetic resonance
imaging
MDCT  Free Radiology Resource For Radiographers, Radiologists and Technical Assistants
A Guided Tour of MRI: An introduction for laypeople National High Magnetic Field Laboratory
Joseph P. Hornak, Ph.D. The Basics of MRI L . Underlying physics and technical aspects.
Video: What to Expect During Your MRI Exam from the Institute for Magnetic Resonance Safety, Education,
and Research (IMRSER)
Interactive Flash Animation on MRI  Online Magnetic Resonance Imaging physics ami technique course
International Society for Magnetic Resonance in Medicine L J
Article on helium scarcity and potential effects on NMR and MRI communities
Danger of objects flying into the scanner
Video compiled of MRI scans showing arachnoid cyst
JEMRIS [80]  Parallel and singlecore general MRI Simulator
Professor Laurance Hall  Daily Telegraph obituary
mriphysics.com  Online MRI physics textbook.
Animations and Simulations
• The animations show spin, modification of spin with magnetic fields and radio frequency pulses, relaxation and
pressesion, spin echo sequence, inversion recovery sequence, gradient echo sequence made by BIGS
References
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ESR Spectroscopy and Microspectroscopy
ESR
Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a technique for
studying chemical species that have one or more unpaired electrons, such as organic and inorganic free radicals or
inorganic complexes possessing a transition metal ion. The basic physical concepts of EPR are analogous to those of
nuclear magnetic resonance (NMR), but it is electron spins that are excited instead of spins of atomic nuclei.
Because most stable molecules have all their electrons paired, the EPR technique is less widely used than NMR.
However, this limitation to paramagnetic species also means that the EPR technique is one of great specificity, since
ordinary chemical solvents and matrices do not give rise to EPR spectra.
EPR was first observed in Kazan State University by Soviet physicist Yevgeny Zavoisky in 1944, and was
developed independently at the same time by Brebis Bleaney at Oxford University.
Theory
Origin of an EPR signal
Every electron has a magnetic moment and
spin quantum number s = 1/2, with magnetic
components m = +1/2 and m = 1/2. In the
presence of an external magnetic field with
strength B the electron's magnetic moment
aligns itself either parallel (m = 1/2) or
antiparallel (m = +1/2) to the field, each
alignment having a specific energy (see the
Zeeman effect). The parallel alignment
corresponds to the lower energy state, and
the separation between it and the upper state
is /\E = guB where g is the electron's
socalled gfactor (see also the Lande gfactor) and \i
of the energy levels is directly proportional to the mag
s the Bohr magneton. This equation implies that the splitl
itic field's strength, as shown in the diagram below.
An unpaired electron can move between the two energy levels by either absorbing or emitting electromagnetic
radiation of energy e = h j^such that the resonance condition, e = /\E, is obeyed. Substituting in e = h v&nA /\
E = gJi„B leads to the fundamental equation of EPR spectroscopy: h v= guB Experimentally, this equation
permits a large combination of frequency and magnetic field values, but the great majority of EPR measurements are
made with microwaves in the 9000  10000 MHz (9  10 GHz) region, with fields corresponding to about 3500 G
(0.35 T). See below for other fieldfrequency combinations.
In principle, EPR spectra can be generated by either varying the photon frequency incident on a sample while
holding the magnetic field constant, or doing the reverse. In practice, it is usually the frequency which is kept fixed.
A collection of paramagnetic centers, such as free radicals, is exposed to microwaves at a fixed frequency. By
increasing an external magnetic field, the gap between the m = +1/2 and m = 111 energy states is widened until it
matches the energy of the microwaves, as represented by the doublearrow in the diagram above. At this point the
unpaired electrons can move between their two spin states. Since there typically are more electrons in the lower
state, due to the MaxwellBoltzmann distribution (see below), there is a net absorption of energy, and it is this
absorption which is monitored and converted into a spectrum.
As an example of how hv = S} l R B can be
used, consider the case of a free electron,
which has g = 2.0023, [1] and the simulated
spectrum shown at the right in two different
forms. For the microwave frequency of
9388.2 MHz, the predicted resonance
position is a magnetic field of about B = h
ul gji B = 0.3350 tesla = 3350 gauss, as
shown. Note that while two forms of the
same spectrum are presented in the figure,
most EPR spectra are recorded and
published only as first derivatives.
3346
3348 3350 3352 3354
Magnetic Field Strength (G)
Because of electronnuclear mass
differences, the magnetic moment of an
electron is substantially larger than the
corresponding quantity for any nucleus, so
that a much higher electromagnetic frequency is needed to bring about a spin resonance with an electron than with a
nucleus, at identical magnetic field strengths. For example, for the field of 3350 G shown at the right, spin resonance
occurs near 9388.2 MHz for an electron compared to only about 14.3 MHz for H nuclei. (For NMR spectroscopy,
the corresponding resonance equation is hu = S^ffir, where g^and fj, depend on the nucleus under study.)
MaxwellBoltzmann distribution
In practice, EPR samples consist of collections of many paramagnetic species, and not single isolated paramagnetic
centers. If the population of radicals is in thermodynamic equilibrium, its statistical distribution is described by the
MaxwellBoltzmann equation
"upper / Supper" Slower \ f AE \ ( € \ ( kv \
^ = exp { — *r — ) = exp r ifeT J = exp {^) = exp {if)
where "upperis the number of paramagnetic centers occupying the upper energy state, k is the Boltzmann
constant, and J 1 is the temperature in kelvins. At 298 K, Xband microwave frequencies ( v ~ 9.75 GHz) give
"upper/slower ~ 0.998, meaning that the upper energy level has a smaller population than the lower one. Therefore,
transitions from the lower to the higher level are more probable than the reverse, which is why there is a net
absorption of energy.
The sensitivity of the EPR method (i.e., the minimum number of detectable spins iV m i n ) depends on the photon
frequency v according to
N = hV
where A;iis a constant, V^ s tne sample's volume, QqIS the unloaded quality factor of the microwave cavity
(sample chamber), kf is the cavity filling coefficient, and pis the microwave power in the spectrometer cavity.
With kfcind pbeing constants, _/V m i n ~ (Qo^ 2 ) ^ * e ' ^min~ v~ a ' wnere a ~ 15 ln practice, a can
change varying from 0.5 to 4.5 depending on spectrometer characteristics, resonance conditions, and sample size. In
other words, the higher the spectrometer frequency the lower the detection limit ( A^ m ; n ), meaning greater
sensitivity.
Spectral parameters
In real systems, electrons are normally not solitary, but are associated with one or more atoms. There are several
important consequences of this:
1 . An unpaired electron can gain or lose angular momentum, which can change the value of its gfactor, causing it
to differ from g . This is especially significant for chemical systems with transitionmetal ions.
2. If an atom with which an unpaired electron is associated has a nonzero nuclear spin, then its magnetic moment
will affect the electron. This leads to the phenomenon of hyperfine coupling, analogous to Jcoupling in NMR,
splitting the EPR resonance signal into doublets, triplets and so forth.
3. Interactions of an unpaired electron with its environment influence the shape of an EPR spectral line. Line shapes
can yield information about, for example, rates of chemical reactions.
4. The gfactor and hyperfine coupling in an atom or molecule may not be the same for all orientations of an
unpaired electron in an external magnetic field. This anisotropy depends upon the electronic structure of the atom
or molecule (e.g., free radical) in question, and so can provide information about the atomic or molecular orbital
containing the unpaired electron.
The g factor
Knowledge of the gfactor can give information about a paramagnetic center's electronic structure. An unpaired
electron responds not only to a spectrometer's applied magnetic field B but also to any local magnetic fields of
atoms or molecules. The effective field B experienced by an electron is thus written
B eS = B (l  a)
where a includes the effects of local fields ( crcan be positive or negative). Therefore, the h v= g/i„B
resonance condition (above) is rewritten as follows:
hu = g e /J, B B eS = g e ^ B B (l  a)
The quantity g (1  o) is denoted g and called simply the gfactor, so that the final resonance equation becomes
hv = gfi B B
This last equation is used to determine g in an EPR experiment by measuring the field and the frequency at which
resonance occurs. If g does not equal g the implication is that the ratio of the unpaired electron's spin magnetic
moment to its angular momentum differs from the free electron value. Since an electron's spin magnetic moment is
constant (approximately the Bohr magneton), then the electron must have gained or lost angular momentum through
spinorbit coupling. Because the mechanisms of spinorbit coupling are well understood, the magnitude of the
change gives information about the nature of the atomic or molecular orbital containing the unpaired electron.
Hyperfine coupling
Since the source of an EPR spectrum is a change in an electron's spin state, it might be thought that all EPR spectra
would consist of a single line. However, the interaction of an unpaired electron, by way of its magnetic moment,
with nearby nuclear spins, results in additional allowed energy states and, in turn, multilined spectra. In such cases,
the spacing between the EPR spectral lines indicates the degree of interaction between the unpaired electron and the
perturbing nuclei. The hyperfine coupling constant of a nucleus is directly related to the spectral line spacing and, in
the simplest cases, is essentially the spacing itself.
Two common mechanisms by which electrons and nuclei interact are the Fermi contact interaction and by dipolar
interaction. The former applies largely to the case of isotropic interactions (independent of sample orientation in a
magnetic field) and the latter to the case of anisotropic interactions (spectra dependent on sample orientation in a
magnetic field). Spin polarization is a third mechanism for interactions between an unpaired electron and a nuclear
spin, being especially important for n electron organic radicals, such as the benzene radical anion. The symbols "a"
or "A" are used for isotropic hyperfine coupling constants while "B" is usually employed for anisotropic hyperfine
coupling constants.
In many cases, the isotropic hyperfine splitting pattern for a radical freely tumbling in a solution (isotropic system)
can be predicted.
• For a radical having M equivalent nuclei, each with a spin of /, the number of EPR lines expected is 2MI + 1 . As
an example, the methyl radical, CH has three H nuclei each with / = 1/2, and so the number of lines expected is
2MI + 1 = 2(3)(l/2) + 1=4, which is as observed.
• For a radical having M equivalent nuclei, each with a spin of / , and a group of M equivalent nuclei, each with a
spin of / the number of lines expected is (2M I +1) (2M I + 1). As an example, the methoxymethyl radical,
H C(OCH ), has two equivalent H nuclei each with / = 1/2 and three equivalent H nuclei each with / = 1/2, and
so the number of lines expected is (2M / + 1) (ZM^ + 1) = [2(2)(l/2) + l][2(3)(l/2) + 1] = [3] [4] = 12, again as
observed.
H
1
1
f+
HO SB «• WO *
Simulated EPR spectrum of the CH radical
• The above can be extended to predict the
number of lines for any number of nuclei.
While it is easy to predict the number of
lines a radical's EPR spectrum should show,
the reverse problem, unraveling a complex
multiline EPR spectrum and assigning the
various spacings to specific nuclei, is more
difficult.
In the oftencountered case of / = 1/2 nuclei
(e.g., 1 H, 19 F, 31 P), the line intensities
produced by a population of radicals, each
possessing M equivalent nuclei, will follow Pascal's triangle. For example, the spectrum at the right shows that the
three l U nuclei of the CH radical give rise to 2MI + 1 = 2(3)(l/2) +1=4 lines with a 1:3:3:1 ratio. The line spacing
gives a hyperfine coupling constant of a = 23 G for each of the three H nuclei. Note again that the lines in this
spectrum are first derivatives of absoiptions.
As a second example, consider the
methoxymethyl radical, H 2 C(OCH 3 ). The
two equivalent methyl hydrogens will give
an overall 1:2:1 EPR pattern, each
component of which is further split by the
three methoxy hydrogens into a 1:3:3:1
pattern to give a total of 3x4 = 12 lines, a
triplet of quartets. A simulation of the
observed EPR spectrum is shown at the
right, and agrees with the 12line prediction
and the expected line intensities. Note that
the smaller coupling constant (smaller line
spacing) is due to the three methoxy hydrogens, while the larger coupling constant (line spacing) is from the two
hydrogens bonded directly to the carbon atom bearing the unpaired electron. It is often the case that coupling
constants decrease in size with distance from a radical's unpaired electron, but there are some notable exceptions,
such as the ethyl radical (CH CH ).
Resonance linewidth definition
Resonance linewidths are defined in terms of the magnetic induction B, and its corresponding units, and are
measured along the x axis of an EPR spectrum, from a line's center to a chosen reference point of the line. These
defined widths are called halfwidths and possess some advantages: for asymmetric lines values of left and right
halfwidth can be given. The halfwidth ABh is the distance measured from the line's center to the point in which
absorption value has half of maximal absorption value in the center of resonance line. First inclination width
ABi MS a distance from center of the line to the point of maximal absorption curve inclination. In practice, a full
definition of linewidth is used. For symmetric lines, halfwidth ABi i^ — 2AB^ , and full inclination width
AB max = 2AB ls
Applications
EPR spectroscopy is used in various branches of science, such as chemistry and physics, for the detection and
identification of free radicals and paramagnetic centers such as F centers. EPR is a sensitive, specific method for
studying both radicals formed in chemical reactions and the reactions themselves. For example, when frozen water
(solid HO) is decomposed by exposure to highenergy radiation, radicals such as H, OH, and HO are produced.
Such radicals can be identified and studied by EPR. Organic and inorganic radicals can be detected in
electrochemical systems and in materials exposed to UV light. In many cases, the reactions to make the radicals and
the subsequent reactions of the radicals are of interest, while in other cases EPR is used to provide information on a
radical's geometry and the orbital of the unpaired electron.
Medical and biological applications of EPR also exist. Although radicals are very reactive, and so do not normally
occur in high concentrations in biology, special reagents have been developed to spinlabel molecules of interest.
These reagents are particularly useful in biological systems. Speciallydesigned nonreactive radical molecules can
attach to specific sites in a biological cell, and EPR spectra can then give information on the environment of these
socalled spinlabel or spinprobes.
A type of dosimetry system has been designed for reference standards and routine use in medicine, based on EPR
signals of radicals from irradiated polycrystalline aalanine(the alanine deamination radical, the hydrogen
abstraction radical, and the (CO"(OH))=C(CH )NH radical) . This method is suitable for measuring gamma and
xrays, electrons, protons, and highlinear energy transfer (LET) radiation of doses in the 1 Gy to 100 kGy range.
EPR spectroscopy can only be applied to systems in which the balance between radical decay and radical formation
keeps the freeradicals concentration above the detection limit of the spectrometer used. This can be a particularly
severe problem in studying reactions in liquids. An alternative approach is to slow down reactions by studying
samples held at cryogenic temperatures, such as 77 K (liquid nitrogen) or 4.2 K (liquid helium). An example of this
work is the study of radical reactions in single crystals of amino acids exposed to xrays, work that sometimes leads
to activation energies and rate constants for radical reactions.
The study of radiationinduced free radicals in biological substances (for cancer research) poses the additional
problem that tissue contains water, and water (due to its electric dipole moment) has a strong absorption band in the
microwave region used in EPR spectrometers.
EPR also has been used by archaeologists for the dating of teeth. Radiation damage over long periods of time creates
free radicals in tooth enamel, which can then be examined by EPR and, after proper calibration, dated. Alternatively,
material extracted from the teeth of people during dental procedures can be used to quantify their cumulative
exposure to ionizing radiation. People exposed to radiation from the Chernobyl disaster have been examined by this
method.
, 141 5
Radiationsterilized foods have been examined with EPR spectroscopy, the aim being to develop methods to
determine if a particular food sample has been irradiated and to what dose.
Because of its high sensitivity, EPR was used recently to measure the quantity of energy used locally during a
mechanochemical milling process.
Highfield highfrequency measurements
Highfieldhighfrequency EPR measurements are sometimes needed to detect subtle spectroscopic details.
However, for many years the use of electromagnets to produce the needed fields above 1.5 T was impossible, due
principally to limitations of traditional magnet materials. The first multifunctional millimeter EPR spectrometer with
a superconducting solenoid was described in the early 1970s by Prof. Y. S. Lebedev's group (Russian Institute of
Chemical Physics, Moscow) in collaboration with L. G. Oranski's group (Ukrainian Physics and Technics Institute,
Donetsk) which began working in the Institute of Problems of Chemical Physics, Chernogolovka around 1975.
Two decades later, a Wband EPR spectrometer was produced as a small commercial line by the German Bruker
Company, initiating the expansion of Wband EPR techniques into mediumsized academic laboratories. Today there
still are only a few scientific centers in the world capable of highfieldhighfrequency EPR, among them are the
Grenoble High Magnetic Field Laboratory in Grenoble, France, the Physics Department in Freie Universitat Berlin,
the National High Magnetic Field Laboratory in Tallahassee, US, the National Center for Advanced ESR
Technology (ACERT) at Cornell University in Ithaca, US, the Department of Physiology and Biophysics at Albert
Einstein College of Medicine, Bronx, NY, the IFW in Dresden, Germany, the Institute of Physics of Complex Matter
in Lausanne in Switzerland, and the Institute of Physics of the Leiden University, Netherlands.
9 GHz i 35 GHz 95 GHz 140 GHi
3
le radical as a function of frequency. Note the improvement in
resolution from left to right. [7]
Waveband
L
S
c
X
P
K
Q
U
V
E
W
F
D

J

A/mm
300
100
75
30
20
12.5
8.5
6
4.6
4
3.2
2.7
2.1
1.6
1.1
0.83
zz/GHz
1
3
4
10
15
24
35
50
65
75
95
111
140
190
285
360
So/T
0.03
0.11
0.14
0.33
0.54
0.86
1.25
1.8
2.3
2.7
3.5
3.9
4.9
6.8
10.2
,2.8
The EPR waveband is stipulated by the frequency or wavelength of a spectrometer's microwave source (see Table).
EPR experiments often are conducted at X and, less commonly, Q bands, mainly due to the ready availability of the
necessary microwave components (which originally were developed for radar applications). A second reason for
widespread X and Q band measurements is that electromagnets can reliably generate fields up to about 1 tesla.
However, the low spectral resolution over gfactor at these wavebands limits the study of paramagnetic centers with
comparatively low anisotropic magnetic parameters. Measurements at v > 40 GHz, in the millimeter wavelength
region, offer the following advantages:
1. EPR spectra are simplified due to the reduction of secondorder effects at high fields.
2. Increase in orientation selectivity and sensitivity in the investigation of disordered systems.
3. The informativity and precision of pulse methods, e.g., ENDOR also increase at high magnetic fields.
4. Accessibility of spin systems with larger zerofield splitting due to the larger microwave quantum energy h v .
5. The higher spectral resolution over gfactor, which increases with irradiation frequency v and external magnetic
field B This is used to investigate the structure, polarity, and dynamics of radical microenvironments in
spinmodified organic and biological systems through the spin label and probe method. The figure shows how
spectral resolution improves with increasing frequency.
6. Saturation of paramagnetic centers occurs at a comparatively low microwave polarizing field B due to the
exponential dependence of the number of excited spins on the radiation frequency v . This effect can be
successfully used to study the relaxation and dynamics of paramagnetic centers as well as of superslow motion in
the systems under study.
7. The crossrelaxation of paramagnetic centers decreases dramatically at high magnetic fields, making it easier to
obtain moreprecise and morecomplete information about the system under study.
See also
• Ferromagnetic resonance
• Spin labels
• Sitedirected spin labeling
• Spin trapping
Further reading
Many good books and papers are available on the subject of EPR spectroscopy, including those listed here.
Essentially all details in this article can be found in these.
• Altshuler, S. A.; Kozirev, B. M. (1964). Electron Paramagnetic Resonance. New York: Academic Press.
• Carrington, A.; McLachlan A. (1967). Introduction to Magnetic Resonance. London: Harper and Row.
ISBN 0470265728.
• Galkin, A. A.; Grinberg, O. Y., Dubinskii, A. A., Kabdin, N. N., Krymov, V. N., Kurochkin, V. I., Lebedev, Y.
S., Oransky, L. G., Shuvalov, V. F. (1977). "EPR Spectrometer in 2mm Range for Chemical Research". Instrum.
Experim. Techn. 20 (4): 1229.
• Krinichnyi, V. I. (1995). 2mm Wave Band EPR Spectroscopy of Condensed Systems. Boca Raton, Florida: CRC
Press.
• Lebedev, Y. S. (1994). "2". HighField ESR in Electron Spin Resonance. 14. Cambridge: Royal Society of
Chemistry, p. 63.
• Rhodes, C. J. (2000). Toxicology of the Human Environment  The Critical Role of Free Radicals. Taylor and
Francis. ISBN 0748409165.  Provides an overview of the role of free radicals in biology and of the use of
electron spin resonance in their detection.
• Symons, M. (1978). Chemical and Biochemical Aspects of ElectronSpin Resonance Spectroscopy. New York:
Wiley. ISBN 0442302290.
• Weil, J. A.; Bolton, J. R., Wertz, J. E. (2001). Electron Paramagnetic Resonance: Elementary Theory and
Practical Applications. New York: WileyInterscience. ISBN 0471572349.
• Weltner, W. (1983). Magnetic Atoms and Molecules. New York: Van Nostrand Reinhold. ISBN 0442292066.
• Wertz, J. E.; Bolton, J. R. (1972). Electron Spin Resonance: Elementary Theory and Practical Applications. New
York: McGrawHill. ISBN 0070694540.
• Protein structure elucidation by EPR: Steinhoff, H.J. (2002). "Methods for study of protein dynamics and
proteinprotein interaction in proteinubiquitination by electron paramagnetic resonance spectroscopy". Frontiers
in Bioscience 7: 97110. doi:10.2741/stein.
External links
• NMRWiki.ORG [6] project
• Electron Magnetic Resonance Program National High Magnetic Field Laboratory
References
[1] Odom, B.; Hanneke, D.; D'Urso, B.; and Gabrielse, G. (2006). "New Measurement of the Electron Magnetic Moment Using a OneElectron
Quantum Cyclotron". Phyical Review Letters 97: 030801. doi:10.1103/PhysRevLett.97.030801.
1 2\ Strict I \ speaking. "a" refers to the Inperfine splitting constant, a line spacing measured in magnetic field units, while .1 ami />' refer Id
hyperfine coupling constants measured in frequenc\ units. Splitting and coupling constants are proportional, but not identical. The book by
Wertz and Bolton has more information (pp. 16 :imi I 12).
[3] "Dosimetry Systems". Journal of the ICRU 8 (5). 2008. doi:10.1093/jicru/ndn027.
[4] Gualtieri, G.; Colacicchia, S, Sgattonic, R., Giannonic, M. (2001). "The Chernobyl Accident: EPR Dosimetry on Dental Enamel of Children".
Applied Radiation and Isotopes 55 (1): 71  79. doi:10.1016/S09698043(00)00351l. PMID 11339534.
[5] Chumak, V.; Sholom, S.; Pasalskaya, L. (1999). "Application of High Precision EPR Dosimetry with Teeth for Reconstruction of Doses to
Chernobyl Populations" (http://rpd.oxfordjournals.org/cgi/content/abstract/84/l4/515). Radiation Protection Dosimetry 84: 515520. .
[6] Baron, M., Chamayou, A., Marchioro, L., Raffi. J. (200s i. "Radicalar probes to measure the action of energ_\ on aanular materials". Adv.
Powder Technol 16 (3): 199212. doi:10.1163/1568552053750242.
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