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Magnetic Resonance 
Spectroscopy and 
lmaging:NMR, MRI and ESR 

pen source mwlib toolkit. See hUp://code. for more informatioi 
PDF generated at: Sat, 31 Jul 2010 23:39:49 UTC 



Nuclear Magnetic Resonance x 

Relaxation 1 

Chemical Shift 2 

Knight shift 6 

Robinson oscillator 7 

Relaxation 7 

Chemical Shift 8 

Fourier transform 12 

Discrete Fourier transform 34 

Fast Fourier transform 47 

Fourier transform spectroscopy 55 

NMR Spectroscopy 59 

2D-NMR 65 

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

Solid-state nuclear magnetic resonance 73 

Magnetic resonance microscopy 79 

Imaging 81 

Medical imaging 8 1 

MRI 89 

ESR Spectroscopy and Microspectroscopy 10 8 

ESR 108 


Article Sources and Contributors 

Image Sources, Licenses and Contributors 

Article Licenses 

Nuclear Magnetic Resonance 


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

• Relaxation (physics), and more in particular: 

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

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

• Structural relaxation, responsible for the glass transition 

• In mathematics: 

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

• Relaxation method, for numerically solving elliptic partial differential equations 

• In computer science: 

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

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

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

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

• Tension (music) 

Chemical Shift 

Chemical Shift 

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

techniques such as proton NMR and carbon- 13 NMR. 

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

Operating frequency 

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

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

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

Chemical shift referencing 

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

difference in precession frequency between two nuclei 

= : 

operating frequency of the magnet 

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

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

300X10° Hz =1Xl ° =lppm 

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

Chemical Shift 

The induced magnetic field 

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

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

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

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

• the nucleus is more deshielded 

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

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

Diamagnetic shielding 

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

Factors causing chemical shifts 

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

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

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

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

Chemical Shift 

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

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

Magnetic properties of most common nuclei 

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




Electric quadrupole 

Operating frequency Relative 

at 7 T sensitivity 








7.1 xlO" 2 








-4.0 x 10" 







Chemical Shift 







35 CI 





tlO' 2 



37 CI 





tlO' 2 




tic properties of 
3 n„uclei [5] 

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

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

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

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

• F because of high relative sensitivity 

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

Other chemical shifts 

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

See also 

• Carbon-13NMR 

• MRI 

• NMR spectroscopy 

• 2D-FT NMRI and Spectroscopy 

• Nuclear magnetic resonance 

• Protein NMR 

• Proton NMR 

• Solid-state NMR 

• Zeeman effect 

External links 

BioMagResBank [8] 
Proton chemical shifts 
Carbon chemical shifts 


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

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

• Problem set 2, moderate [13] 

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

• Problem set 5, the best! 

Chemical Shift 

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


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

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

iii| I I li i i/hand mi ila.hlm 


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


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

5] html 
16] http://ww\\ n i u mill q Iru lui mswersl ^ dlh 

7] ui mswers33-64.GIF 

Knight shift 

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

in 1949 by the American physicist Walter David Knight. 

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

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

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

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

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

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

Robinson oscillator 

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. 


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

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



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

• Relaxation (physics), and more in particular: 

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

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

• Structural relaxation, responsible for the glass transition 

• In mathematics: 

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

• Relaxation method, for numerically solving elliptic partial differential equations 

• In computer science: 

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

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

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

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

• Tension (music) 

Chemical Shift 

Chemical Shift 

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

techniques such as proton NMR and carbon- 13 NMR. 

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

Operating frequency 

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

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

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

Chemical shift referencing 

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

difference in precession frequency between two nuclei 

= : 

operating frequency of the magnet 

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

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

300X10° Hz =1Xl ° =lppm 

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

Chemical Shift 

The induced magnetic field 

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

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

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

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

• the nucleus is more deshielded 

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

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

Diamagnetic shielding 

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

Factors causing chemical shifts 

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

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

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

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

Chemical Shift 

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

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

Magnetic properties of most common nuclei 

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




Electric quadrupole 

Operating frequency Relative 

at 7 T sensitivity 








7.1 xlO" 2 








-4.0 x 10" 







Chemical Shift 







35 CI 





tlO' 2 



37 CI 





tlO' 2 




tic properties of 
3 n„uclei [5] 

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

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

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

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

• F because of high relative sensitivity 

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

Other chemical shifts 

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

See also 

• Carbon-13NMR 

• MRI 

• NMR spectroscopy 

• 2D-FT NMRI and Spectroscopy 

• Nuclear magnetic resonance 

• Protein NMR 

• Proton NMR 

• Solid-state NMR 

• Zeeman effect 

External links 

BioMagResBank [8] 
Proton chemical shifts 
Carbon chemical shifts 


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

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

• Problem set 2, moderate [13] 

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

• Problem set 5, the best! 

Chemical Shift 

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


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

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

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

[4] In units of the nuclear magneton 

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

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

Fourier transform 

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

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

is Fourier transform 

Fourier series 

Discrete Fourier transform 

Discrete-time Fourier 

Related transforms 


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 


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 

e-" mnx/T f(x)dx 

and/ should be given by the formula 

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

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

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

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

Fourier transform 



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

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

at 3 hertz at 5 hertz 

Properties of the Fourier transform 

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

Basic properties 

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

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


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

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

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

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

\a\ \aj 

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

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) = —?(£). 

Uk{x) = (/*<?}(», then h(0 = fa)-m- 

Fourier transform 

Uniform continuity and the Riemann-Lebesgue lemma 

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

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

fw=£lf ($***<%■ 

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

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

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

The Plancherel theorem and Parseval's theorem 

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

where the bar denotes complex conjugation. 

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

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

Fourier transform 

Poisson summation formula 

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

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

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

Convolution theorem 

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

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

where * denotes the convolution operation, then: 

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

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

Cross-correlation theorem 

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

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

then the Fourier transform of h(x) is: 

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

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

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

Fourier transform 


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), 

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 


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

Fourier transform on Euclidean space 

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

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

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

Uncertainty principle 

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

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

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


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

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

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

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


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


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

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

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

Fourier transform 

D (f) = J°^x 2 \f(x)\ 2 dx. 

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

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

integrable, then 

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

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

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

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

Spherical harmonics 

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

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

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

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

Restriction problems 

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


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 . 


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 

Fourier-Stieltjes transform 

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

•J IV 

m= / e- 2 ™-^ 

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

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

Fourier transform 

Tempered distributions 

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

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

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

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

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

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

Tf = T f . 

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

Locally compact abelian groups 

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

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

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

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

Fourier transform 

Non-abelian groups 

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

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

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

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

||£|| = S up||£ ff ||° 


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

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

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

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


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

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

Fourier transform 

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


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. 


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

Domain and range of the Fourier transform 

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

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

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

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

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

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

Other notations 

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

Fourier transform 

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

^(0 = 1/(01. 

is the amplitude and 

is the phase (see arg function). 

Then the inverse transform can be written: 

f(x) = f°° A{£) e'^+rtO) df , 

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

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

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

Other conventions 

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

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

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

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

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


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

leads to definitions: 

Fourier transform 



'^ 2 L 


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

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

Summary of popular forms of the Fourier transform 

ordinary frequency g (hertz) 

angular frequency to (rad/s) 

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

j,(.)g J j f /(*- J ,=/ 1 g)=(^i(„) 

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

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

Tables of important Fourier transforms 

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

Functional relationships 

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

Fourier transform 


Fourier transform 

Fourier transform 

Fourier transform 


unitary, ordinary frequency 

unitary, angular frequency 

non-unitary, angular 


fc) = £ o me-***dx 

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

f{ V )= j^J(x)e—dx 





a-/H + 6-ffH 

a • />) + 6 • $(«/) 



f(x - a) 

e- 2 ™ ? /(0 


e— /» 

Shift in 


e 2 — f(x) 


/> - 2™) 

/> - 27m) 

Shift in 
dual of 






Scaling in 
the time 
domain. If 

large, then 









2tt/(- I /) 

Here f 

using the 

method as 














/ 1 \"«f/K) 

V2tJ df» 


1 dv n 

This is the 
dual of 

Fourier transform 








denotes the 
of /and 
g — this 







This is the 
dual of 


For /(x) a purely 

/ (-0 = W) 

/(- W ) = 7h 

j(- v ) = W) 





For /(x) a purely 

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

real even function 


For /(x) a purely 

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

real odd function 

Square-integrable functions 

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

Fourier transform 
unitary, ordinary frequency 

Fourier transform 
unitary, angular frequency 

Fourier transform 
non-unitary, angular 


/(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 


Dual of rule 
201. The 

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

response of such 
a filter. 

Fourier transform 


sine 2 (ax) 



R' tri (i) 

The function 
tri(*) is the 




\/W V2^J 

R- Sinc2 (i) 

Dual of rule 



e— «(x) 




a + ii/ 

The function 
u(x) is the 
Heaviside unit 

V^(a + l cv) 

step function 

and fl >0. 

21 th 

1 _^ 

This shows that, 
for the unitary 

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



e -l-l 



a 2 + z/ 2 

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

a 2 + 47r 2^2 



function is a 





^(-zr-f/„_ 1 (2<) 


The functions J 
(x) are the n-th 

functions of the 

• \/l — 47r 2 ^ 2 rect(7rf ) 

first kind. The 

functions U (x) 

are the 


polynomial of 

the second kind. 

See 315 and 316 


21 W 


^ d '(^) 

^ sech S") 

a S6Ch (2^") 


Fourier transform 28 


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

Function Fourier transform Fourier transform Fourier transform Remarks 

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

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

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

1 11 Dual of rule 301. 

2n5(v - a) 

This follows from 103 and 301. 


U"^ + ^ + 9 

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

Max) = {e- + e— )/2 

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

2 sin(ax) = (e*» - e"^)/(2i) 

v^ V 4a 4/ 

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

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

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

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

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

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

■g$&™ -^if-«) -^■^-m -7^9^: 

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

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

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

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

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

Fourier transform 

£ S(x-nT) 

E'- f 



309. This time the 

3 ourier 




function k 

x) is the Heaviside 

nit step f 


follows fro 

m rules 101, 301, an 

d 312. 


function i 

known as the Dirac 

comb lu 


result can 

be derived from 302 

and 102 

£ e™ = 2n £ 6(x + 27vk)., 



V 7 ! - 4tt 2 C 2 

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


The function ./ (\ 

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

Two-dimensional functions 

ordinary frequency 

;.,{,) = J[f(x,y)e-" <s "+ e «<d 

„,^ = lJJf<,.y)e-"^>* 

angular frequency 

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


1 -("1/ 

2v\ab\ e 

|afe| e 

Fourier transform 

Formulas for general n -dimensional functions 


Fourier transform 
unitary, ordinary frequency 

Fourier transform 
unitary, angular frequency 

Fourier transform 

non-unitary, angular 




fc) = j^me-^rx 

^=^L f{x)e ^ dnx 

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

Fourier transform 


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

^r(5 + i)|-| 


on of 

the ii 







function of 
the first 
kind with 

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

102. (Stein 
& Weiss 
1971, Thm. 


See Riesz 


also holds 

but then the 




as suitably 







Fourier transform 

See also 

Fourier series 
Fast Fourier transform 
Laplace transform 
Discrete Fourier transform 

• DFT matrix 

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

• Hartley transform 

• Hankel transform 


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Fourier transform 

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

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

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

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

External links 

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

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

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

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

• Fourier Transform Module by John H. Mathews 

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

• An Interactive Flash Tutorial for the Fourier Transform 






[5] liarv/aux intlrans.htm 

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

[7] s/c2003/K>uriei TransformMod.html 

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

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

Discrete Fourier transform 

Discrete Fourier transform 

Fourier transforms 

Continuous Fourier transform 

Fourier series 

Discrete Fourier transform 

Discrete-time Fourier 

I ran-- form 

Related transforms 

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

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

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


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: 


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

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

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

of x in an orthonormal basis.) 

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

The inverse discrete Fourier transform (IDFT) is given by 

Discrete Fourier transform 

-V£ x * Vhn 

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

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

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

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

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


The discrete Fourier transform is an invertible, linear transformation 

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


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

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

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

Discrete Fourier transform 

The Plancherel theorem and Parseval's theorem 

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

n =0 VV fc=0 

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


JV-1 i JV-1 

These theorems are also equivalent to the unitary condition below. 


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

JV-1 JV-1 JV-1 

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

n=0 n=0 1 n=0 

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

The shift theorem 

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

then p({ Xn . e ¥— }) fc = Xk _ m 

and H{Xn-m})k = X k ■ e~'^ km 

Circular convolution theorem and cross-correlation theorem 

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

N ' 


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

JV-1 JV-1 / 1 JV-1 


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

Discrete Fourier transform 

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

iV-1 oo / oo 

E X M E »»-«-pJV = ( X *yN)n 

1=0 \p=-oo 

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


Similarly, it can be shown that: 


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

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

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

Convolution theorem duality 

It can also be shown that: 


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 

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

Discrete Fourier transform 

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

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

The unitary DFT 

Another way of looking at the DFT i 
Vandermonde matrix: 


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





iV N 



, ,1-1 

. i-(JV-i) 

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


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): 


If Xi s defined as the unitary DFT of the vector xthen 


and the Plancherel theorem is expressed as: 

JV-1 JV-1 

E *»»; = E x ^: 

71=0 fc=0 

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

Discrete Fourier transform 

well — this is just Parseval's theorem: 

JV-1 JV-1 

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

n,=0 fc=0 

Expressing the inverse DFT in terms of the DFT 

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

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

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

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

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

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

Eigenvalues and eigenvectors 

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

unique, and are the subject of ongoing research. 

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

m,n Vn n Vn 

This matrix satisfies the equation: 

U 4 = I. 

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

A 4 = l. 

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

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

Discrete Fourier transform 

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

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


l = +l 

l = -l 

/. = -I 

l = +i 





m- 1 

4m +1 





Am + 2 





Am + 3 



m + l 


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

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

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

r, the follow 

ing !• 

of DFT: 

F(m) — sin I 




II l cos ( ~ m I — cos ( 

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

The real-input DFT 

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

x k = x* N _ k . 

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

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

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

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

+ Nyquist element of the complex output X. 

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


Discrete Fourier transform 

Generalized/shifted DFT 

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

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

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

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

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

Multidimensional DFT 

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

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

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

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

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

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

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

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

Discrete Fourier transform 42 

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

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

The real-input multidimensional DFT 

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

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

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

£= 1,2,. ..,d). 

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

Spectral analysis 

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

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

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

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

Discrete Fourier transform 

Data compression 

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

Partial differential equations 

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

Polynomial multiplication 

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

c = a* b 

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

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

But convolution becomes multiplication under the DFT: 

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

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

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

Discrete Fourier transform 

Multiplication of large integers 

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

Some discrete Fourier transform pairs 

Some DFT pairs 

JV fc=0 

Xk = y- Xne -i2wk n /N 



x n e' 2 ^ N 

X k - e 

Shift theorem 

x n -i 

X k e- aM ' N 


x k = X* N _ k 

Real DFT 


J N if a = ^ k l N 

from the geometric progression formula 

ll-ae— 2 **/" ° CrVV1 " C 


(l + e-™'*)"- 1 

from the binomial theorem 

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

( 1 if k = 

1 — . /t t \ otherwise 

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

Derivation as Fourier series 

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

Representation theory 

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

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

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

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

by the «-element set Z/nZ. 

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

representation theory of finite groups. 

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

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

Discrete Fourier transform 

Other fields 

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

Other finite groups 

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

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

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


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

See also 

• DFT matrix 

• Fast Fourier transform 

• List of Fourier-related transforms 
. fftw 


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

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

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

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

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

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

Discrete Fourier transform 

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

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

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

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

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

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

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

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

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

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


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 


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

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

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

| ! i hllp://\\ \\ u Courier 




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 
complex-number arithmetic to group theory and number theory; this article gives an overview of the available 
techniques and some of their general properties, while the specific algorithms are described in subsidiary articles 
linked below. 

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

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

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

Definition and speed 

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


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


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

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

Fast Fourier transform 

Computational issues 

Bounds on complexity and operation counts 

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

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

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

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


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

2007; Lundy and Van Buskirk, 2007). 

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

Accuracy and approximations 

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

Fast Fourier transform 

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

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

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

Cooley-Tukey algorithm 

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

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

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


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

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

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


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

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

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

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

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

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

such as those described below. 

Other FFT algorithms 

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

Fast Fourier transform 5( 

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

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

FFT algorithms specialized for real and/or symmetric data 

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

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

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

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

Fast Fourier transform 

Multidimensional FFTs 

As defined in the multidimensional DFT article, the multidimensional DFT 

x k = J2 € ~ 2wik ' HN)x " 

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

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

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

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

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

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

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

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

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

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

FFTs is: 

N N 

—0(N 1 \ 0g N 1 ) + - 

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

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

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

Fast Fourier transform 

Other generalizations 

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

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

See also 

Split-radix FFT algorithm 

Prime-factor FFT algorithm 

Bruun's FFT algorithm 

Rader's FFT algorithm 

Bluestein's FFT algorithm 

Butterfly diagram - a diagram used to describe FFTs. 

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

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

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

Spectrum analyzers - Devices that perform an FFT 

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

more dimensions. 

Time Series 

Math Kernel Library 



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

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

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

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

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

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

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

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

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

Fast Fourier transform 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Fast Fourier transform 

External links 

.[16] , 

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

• Links to FFT code and information online. 

• National Taiwan University - FFT 

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

• Online documentation, links, book, and code. 

• Using FFT to construct aggregate probability distributions 

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

• FFT Basics and Case Study Using Multi-Instrument 

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

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


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

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

Conceptual introduction 
Measuring an emission spectrum 

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






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 


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 


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

Continuous wave Michelson or Fourier transform spectrograph 

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

light source 


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

general have been supplanted by imaging Fabry-Perot 

Fourier transform spectroscopy 

Extracting the spectrum 

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

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

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

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


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

Pulsed Fourier transform spectrometer 

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

Examples of Pulsed Fourier transform spectrometry 

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

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

The Free Induction Decay 

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

Stationary Forms of Fourier Transform Spectrometers 

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

Fourier transform spectroscopy 

Fellgett Advantage 

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

Converting spectra from time domain to frequency domain 

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

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

See also 

• Applied spectroscopy 

• 2D-FT NMRI and Spectroscopy 

• Forensic chemistry 

• Forensic polymer engineering 

• nuclear magnetic resonance 

• Infrared spectroscopy 

External links 


• Description of how a Fourier transform spectrometer works 

• The Michelson or Fourier transform spectrograph 

• Internet Journal of Vibrational Spectroscopy - How FTIR works 

• Fourier Transform Spectroscopy Topical Meeting and Tabletop Exhibit 


[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. 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] 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 thin-walled glass 
tube - an NMR tube. 

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

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

Chemical shift 

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

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

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

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

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

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

NMR Spectroscopy 

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



Intensity Ratio 

Singlet (s) 


Doublet (d) 


Triplet (t) 


Quartet (q) 








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

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

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

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

NMR Spectroscopy 

Second-order (or strong) coupling 

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

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

Magnetic inequivalence 

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

Correlation spectroscopy 

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

Solid-state nuclear magnetic resonance 

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

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

NMR Spectroscopy 

currently being used in NMR spectroscopy. 

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

NMR spectroscopy applied to proteins 

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

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

See also 

• distance geometry 

• In vivo magnetic resonance spectroscopy 

• Low field NMR 

• Magnetic Resonance Imaging 

• Nuclear Magnetic Resonance 

• NMR spectra database 

• NMR tube - includes a section on sample preparation 

• Protein nuclear magnetic resonance spectroscopy 

• NMR spectroscopy of stereoisomers 

External links 

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

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

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

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

• NMR spectroscopy for organic chemistry 

• The Spectral Game NMR spectroscopy game. 

NMR Spectroscopy 

Spectra libray NMR spectroscopy library 

y [9] N 

• Obtaining dihedral angles from J coupling constants 

• Another Javascript-like NMR coupling constant to dihedral 

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

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

• CARA - resonance assignment software developed at the Wuthrich group 

• NMRShiftDB - open database and NMR prediction website 

• Spinworks [16] 

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

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


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

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

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

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

| ! ] htlp://\\ \\ \\\ inc/ 


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

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





[12]\\ ;ki/NMR_spectroscopy 




[16] > /nmr/spinw orks/ 

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

[18] http ://www. php 


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


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

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

1 , 1. 

u . 



Ill [,„ 


■ f h 

3roton-COSY experiment o 

i Pn\ 

tM s ; 

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


COSY-90 is the most common COSY 

experiment. In COSY-90, the sample is 

irradiated with a radio frequency pulse, pi, 

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

the sample is allowed to freely precess 

during an evolution period (tl). A second 

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

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

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

cross-peaks appear in the spectrum. 

Cross-peaks result from a phenomenon 

called magnetization transfer. In 

COSY, magnetization transfer occurs 

through the chemical bonds rather than 

through space. 

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

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

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

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


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

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


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

See also 

• Exclusive correlation spectroscopy 

• Two dimensional correlation analysis 

User:Bci2/2D-FT NMRI and Spectroscopy 

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

User:Bci2/2D-FT NMRI and Spectroscopy 

NMRI Principle 

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

linical dunjnoxtio am! biomedical re 

Chemical Shifts 

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

NMR Imaging Principles 

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

User:Bci2/2D-FT NMRI and Spectroscopy 

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

Two-dimensional Fourier transform imaging and spectroscopy 

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

2D-FT Definition 

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

User:Bci2/2D-FT NMRI and Spectroscopy 

Example 1 

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

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

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

Example 2 

Atomic Structure from 2D-FT STEM Images of electron distributions in a high-temperature cuprate 

superconductor "paracrystal' reveal both the domains (or "location') and the local symmetry of the 'pseudo-gap' in the 
electron-pair correlation band responsible for the high— temperature superconductivity effect (obtained at Cornell 
University). So far there have been three Nobel prizes awarded for 2D-FT NMR/MRI during 1992-2003, and an 
additional, earlier Nobel prize for 2D-FT of X-ray data ("CAT scans'); recently the advanced possibilities of 2D-FT 
techniques in Chemistry, Physiology and Medicine received very significant recognition. 

Brief explanation of NMRI diagnostic uses in Pathology 

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

See also 

User:Bci2/2D-FT NMRI and Spectroscopy 

Nuclear magnetic resonance (NMR) 
Medical imaging 

Protein nuclear magnetic resonance spe 
Kurt Wuthrich 

Chemical shifl 

Computed tomography (CT) 

Fourier transform spectroscopy FTS) 

il I IglK li 

Solid-state NMR 
Herbert S. Gutowsky 
John S. Waugh 
Charles P. Slichter 
Magnetic re 

Earth's field NMR (EFNMR) 
Robinson oscillator 

Reference list 

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

[2] [ Howstuffworks "How MRI Works" 

[3] Peter Mansfield. 2003. Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI ( 

[4] Antoine Ait igam / u ' ' / pp.. Cambridge Uni\ei i i n I 1 

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

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

phosphates" ( Proc. Natl. Acad. Sci. U.S.A. 96 (4): 1574-8. 

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

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

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

[II] Charles P. Slichter. 1996. Principles of Magnetic Resonance. Springer: Berlin and New York, Third Edition., 65 1pp. ISBN 0-387-50157-6. 

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 

[15]' < 20Infarcl ' i 2()Short%20Axis%20Cine%204 


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

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

• Kurt Wuthrich. 1986, NMR of Proteins and Nucleic Acids., J. Wiley and Sons: New York, Chichester, Brisbane, 
Toronto, Singapore. ( Nobel Laureate in 2002 for 2D-FT NMR Studies of Structure and Function of Biological 
Macromolecules ( 

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

• 2D-FT NMRI Instrument image: A JPG color image of a 2D-FT NMRI "monster' Instrument (http://upload. 

User:Bci2/2D-FT NMRI and Spectroscopy 

• Richard R. Ernst. 1992. Nuclear Magnetic Resonance Fourier Transform (2D-FT) Spectroscopy. Nobel Lecture 
(, on December 9, 1992. 

• Peter Mansfield. 2003. Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI (http://www. 

• D. Benett. 2007. PhD Thesis. Worcester Polytechnic Institute. PDF of 2D-FT Imaging Applications to NMRI in 
Medical Research. ( 

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

• Paul Lauterbur. 2003. Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI. ( 

• Jean Jeener. 1971. Two-dimensional Fourier Transform NMR, presented at an Ampere International Summer 
School, Basko Polje, unpublished. A verbatim quote follows from Richard R. Ernst's Nobel Laureate Lecture 
delivered on December 2, 1992, "A new approach to measure two-dimensional (2D) spectra." has been proposed 
by Jean Jeener at an Ampere Summer School in Basko Polje, Yugoslavia, 1971 (Jean Jeneer,1971 }). He 
suggested a 2D Fourier transform experiment consisting of two $\pi/2$ pulses with a variable time $ t_l$ between 
the pulses and the time variable $t_2$ measuring the time elapsed after the second pulse as shown in Fig. 6 that 
expands the principles of Fig. 1. Measuring the response $s(t_l,t_2)$ of the two-pulse sequence and 
Fourier-transformation with respect to both time variables produces a two-dimensional spectrum $S(0_1,0_2)$ 
of the desired form. This two-pulse experiment by Jean Jeener is the forefather of a whole class of $2D$ 
experiments that can also easily be expanded to multidimensional spectroscopy. 

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

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

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

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

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

External links 

• Cardiac Infarct or "heart attack" Imaged in Real Time by 2D-FT NMRI ( 

• 3D Animation Movie about MRI Exam ( 

• Interactive Flash Animation on MRI ( - Online Magnetic Resonance Imaging physics and 
technique course 

• International Society for Magnetic Resonance in Medicine ( 

• Danger of objects flying into the scanner ( 

User:Bci2/2D-FT NMRI and Spectroscopy 

Related Wikipedia websites 

Medical imaging 

Computed tomography 

Magnetic resonance microscopy 

Fourier transform spectroscopy 


Magnetic resonance elastography 

Nuclear magnetic resonance (NMR) 

Chemical shift 


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 (, which is 
licensed under the GFDL. 

Solid-state nuclear magnetic resonance 

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


Basic concepts A spin interacts with a 

magnetic or an electric field. Spatial 

proximity and/or a chemical bond between 

two atoms can give rise to interactions 

between nuclei. In general, these 

interactions are orientation dependent. In 

media with no or little mobility (e.g. 

crystals, powders, large membrane vesicles, 

molecular aggregates), anisotropic 

interactions have a substantial influence on 

the behaviour of a system of nuclear spins. 

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

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

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

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

Solid-state nuclear magnetic resonance 

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

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

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

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

High-resolution conditions in solids (in a 

wider sense) can be established using magic 

angle spinning (MAS), macroscopic sample 

orientation, combinations of both of these 

techniques, enhancement of mobility by 

highly viscous sample conditions, and a 

variety of radio frequency (RF) irradiation 

patterns. While the latter allows decoupling 

of interactions in spin space, the others 

facilitate averaging of interactions in real 

space. In addition, line-broadening effects 

from microscopic inhomogeneities can be 

reduced by appropriate methods of sample 


mia MAS re 
8 kHz, 3.2 n 

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

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

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

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

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

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

Solid-state nuclear magnetic resonance 

Nuclear spin interactions in the solid phase 
Chemical shielding 

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

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

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

molecular environments usually experience unique local fields from this effect. 

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

chemical shielding is removed, leaving the isotropic chemical shift. 


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

Dipolar coupling 

Main article: Dipolar coupling (NMR) 

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


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

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

Quadrupolar interaction 

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

Solid-state nuclear magnetic resonance 

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

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

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

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

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

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

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

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

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

Other interactions 

Paramagnetic substances are subjec 


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

spectroscopy in general. 

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

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


Modern solid-state NMR spectroscopy 

Methods and techniques 


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

Basic example 

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

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

NMR spectroscopy. Given its importance, a pulse sequence 

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

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

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





^ l^ 



(Frequency channel 2 
e.g. ,3 Q 

CP pulse seque 

J \J time 

Solid-state nuclear magnetic resonance 


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

isolate spin systems. 

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

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

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

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


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

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

interatomic distances in isotopically labeled molecules. 

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

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

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

andSchaefer, 1989]. 


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


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


[1] http://nmr900.c; 

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

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

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

Suggested readings for beginners 

• High Resolution Solid-State NMR of Quadrupolar Nuclei ( 
GrandinettiRMCBruker2007.pdf) Grandinetti ENC Tutorial 

• David D. Laws, Hans-Marcus L. Bitter, and Alexej Jerschow, "Solid-State NMR Spectroscopic Methods in 
Chemistry", Angewandte Chemie International Edition (engl.), Vol. 41, pp. 3096 (2002) 

• Levitt, Malcolm H, Spin Dynamics: Basics of Nuclear Magnetic Resonance, Wiley, Chichester, United Kingdom, 
2001. (NMR basics, including solids) 

• Duer, Melinda J., Introduction to Solid-State NMR Spectroscopy, Blackwell, Oxford, 2004. (Some detailed 
examples of SSNMR spectroscopy) 

Solid-state nuclear magnetic resonance 

Advanced readings 

Books and major review articles 

• McDermott, A, Structure and Dynamics of Membrane Proteins by Magic Angle Spinning Solid-State NMR (http:/ 
/arjourn 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. 

• Schmidt-Rohr, 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. 


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., "Proton-enhanced NMR of dilute spins in solids" J. Chem. Phys. 59, 569-90, 

• Purcell, Torrey and Pound (1945). 

• Schaefer, J. and Stejskal, E. O., "Carbon-13 Nuclear Magnetic Resonance of Polymers Spinning at the Magic 
Angle," Journal of the American Chemical Society 98, 1031 (1976). 

• Gullion, T. and Schaefer, J., "Rotational-Echo, Double-Resonance NMR," J. Magn. Reson., 81, 196 (1989). 

External links 

• NMRWiki.ORG ( NMR resource you can edit 

• SSNMRBLOG ( Solid-State NMR Literature Blog by Prof. Rob Schurko's 
Solid-State NMR group at the University of Windsor 

• ( Rocky Mountain Conference on Solid-State NMR 

• Varian Inc ( 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 


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 

• 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 

• Sensitivity: Because the voxels for MRM can be 1/100,000 of those in MRI, the signal will be proportionately 

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 micrometre-scale 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. 


[1] P. Glover and P. Mansfield, Limits to magnetic resonance microscopy, Rep. Prog. Phys. 65 1489-15 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 

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. 


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 sub-discipline 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 sub-discipline 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 X-ray 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 X-Ray 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 


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 real-time images of internal structures of the body in a similar fashion to radiography, but 
employs a constant input of x-rays, 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 image-guided 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 x-rays, 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 radio-opaque 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 time-varying (on the order of 
1 kHz) field(s) for spatial encoding, called the gradient field(s); and a 
weak radio-frequency (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 single-slice, 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 long-term 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 X-ray 

and CT. However, there are well-identified 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, X-rays 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 soft-tissue 
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 function-based 
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 two-dimensional images from the radiation 
emitted by the radiopharmaceuticals. For example, technetium-labeled 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 X-Ray, CT, or 

• Positron emission tomography (PET) uses coincidence detection to image functional processes. Short-lived 
positron emitting isotope, such as F, is incorporated with an organic substance such as glucose, creating 
F18-fluorodeoxyglucose, 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 non-invasive diagnosis and patient 

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 quasi-diffusive 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 pre-cancerous tissue and the area surrounding a developing breast cancer is almost always higher than in normal 
breast tissue. Cancerous tumors require an ever-increasing 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 high-resolution 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 pre-cancerous state of the 
breast [5] . 


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 X-ray 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 X-rays. 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 X-Ray projections 
but in this case, the patient is enclosed in a surrounding ring of detectors assigned with 500-1000 scintillation 

. This being the fourth-generation X-Ray CT scanner geometry. Previously in older generation scanners, the X-Ray 
beam was paired by a translating source and detector. 


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 real-time, 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 three-dimensional 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 

• A-scan 

• B-scan 

• 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 state-of-the-art 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. 

Non-diagnostic 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 imaging-based 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 


Digital Mammography and PACS 

EMMI European Master in Molecular Imaging 


Full-body scan 

Magnetic field imaging • Pneumoencephalogram 

Medical examination • Radiology information s\ stem 

Medical radiography • Segmentation (image processing) 

Medical test • Signal-to-noise ratio 

Neuroimaging • Society for Imaging Science and Technology 

Non invasive (medical) • Tomogram 

PACS • Virtopsy 

JPEG 2000 


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. 1007/978-1-84628-968-2. ISBN 978-1-84628-379-6. 

• Baert, Albert L., ed (2008). Encyclop i >fDu nt I maging.I thn: Springer- Verlag. 
doi: 10. 1007/978-3-540-35280-8. ISBN 978-3-540-35278-5. 

• 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 0897-1889. 

• Using JPIP for Standard-Compliant 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 

• What is JPIP? 


[I] Society of Nuclear Medicine ( 

|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; McGraw-Hill Concise Dictionary of 

Modern Medicine, 2002 by The McGraw-Hill Companies 
[3] Singh, Namrala Singh. Clinical K\ a] nation of Radiolabeled Drugs for Tubercular Imaging. LAP Lambert Academic Publishing (2010). 

ISBN-13: 978-3838378381 
[4] Dhawan P, A. (2003). Medical Imaging Analysis. Hoboken, NJ: Wiley-Interscience Publication 

[7] Dhawan P, A. (2003). Medical Imaging Analysis. Hoboken, NJ: Wiley-Interscience Publication 
[8] Dhawan P, A. (2003). Medical Imaging Analysis. Hoboken, NJ: Wiley-Interscience 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://\\ \\ \\>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] arc. cum/iniaging/w hitepapcrs.hlni 

[16] http://www.dmoz.Org/Health/Medicine/Imaging// 

I i l\ 


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) 

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 cross-sectional 

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 X-ray 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 non-medical devices operating on 
the same principles. The term magnetic resonance tomography (MRT) 
is also sometimes used. 

Sagittal MR image of the knee 

Para-sagittal 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 spin-down 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 field-strength 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. 


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 
non-ionizing radiation in the radio frequency range, unlike CT scans and traditional X-rays, which both use ionizing 

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 
fluid-containing tissues are bright (most modern T sequences are actually fast T sequences) and fat-containing 
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 5-20 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 MP-RAGE 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 diffusion-weighted 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 5-10 minutes of the onset of stroke symptoms (as compared with 

computed tomography, which often does not detect changes of acute infarct for up to 4-6 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 inversion-recovery 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 

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 "run-off"). 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 
"flow-related enhancement" (e.g. 2D and 3D time-of-flight 
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 (MR-GILD) 

Magnetic resonance gated intracranial cerebrospinal fluid (CSF) or liquor dynamics (MR-GILD) 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 2-3 seconds). Increases in 
neural activity cause changes in the MR signal via T changes; this 
mechanism is referred to as the BOLD (blood-oxygen-level 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 well-suited 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 

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 three-dimensional view of the target tissue, allowing for precise focusing of ultrasound energy. 
The MR imaging provides quantitative, real-time, 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 
helium-3, carbon-13, fluorine-19, oxygen-17, sodium-23, phosphorus-31 and xenon-129. 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. O-water, C-glucose 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, high-resolution, 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 spin-echo (T 
TSE MRI), double inversion recovery MRI (DIR-MRI) or phase-sensitive inversion recovery MRI (PSIR-MRI), all 
of them able to improve imaging of the brain lesions . Another example is MP-RAGE 

(magnetization-prepared 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 X-rays, 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 non-ionizing radio frequency (RF) signals to acquire its images and is best 
suited for non-calcified 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 

Both CT and MRI scanners are able to generate multiple two-dimensional cross-sections (slices) of tissue and 
three-dimensional reconstructions. Unlike CT, which uses only X-ray 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 cross-sectional 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 multi-detector CT scanners with near-isotropic 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 

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 


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 



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. 


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 cardioverter-defibrillators, 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 nano-coating 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 radio-frequency 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 x-ray to be performed on 
anyone suspected of having metal fragments in their eyes, something not uncommon in metalworking. 
Because of its non-ferromagnetic nature and poor electrical conductivity, titanium and its alloys are useful for long 
term implants and surgical instruments intended for use in image-guided 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 'black-hole 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 10-20 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: 

MR-Safe — The device or implant is completely non-magnetic, 
non-electric ally conductive, and non-RF reactive, eliminating all of the 
primary potential threats during an MRI procedure. 


MR-Conditional — A device or implant that may contain magnetic, 
electrically conductive or RF-reactive 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'). 

MR-Unsafe — Nearly self-explanatory, 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 

regulatory-approved medical devices, it is being applied to all manner 

of items, appliances and equipment intended for use in the MR 


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. 

Missile-effect 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 non-ferromagnetic 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 missile-effect 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 off-center 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 high-field machines and 
rapid-imaging techniques in which sound intensity can reach 120 dB(A) (equivalent to a jet engine at take-off) 
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 10-fold 
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. 


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 non-toxic, 

their physical properties present specific hazards. 

An unintentional shut-down 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 

time-consuming. 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 X-ray radiography or CT. Anaphylactoid reactions are 
rare, occurring in approx. 0.03-0.1%. Of particular interest is the lower incidence of nephrotoxicity, compared 
with iodinated agents, when given at usual doses — this has made contrast-enhanced MRI scanning an option for 
patients with renal impairment, who would otherwise not be able to undergo contrast-enhanced 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 gadolinium-containing 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 
gadolinium-containing 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. 


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 pre-surgical, in-utero 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 

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 Head-mounted 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. 


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 

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 Non-Ionizing 
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. 

Three-dimensional (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 

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 3-dimensional 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 semi-transparent, 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 Urbana-Champaign 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 full-page 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 one-dimensional 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) 
2D-FT NMRI and Spectroscopy 
Robinson oscillator 
Rabi cycle 


Further reading 

• Simon, Merrill; Mattson, James S (1996). The pioneers of NMR and magnetic resonance in medicine: The story of 
MRI. Ramat Gan, Israel: Bar-Ilan University Press. ISBN 0-9619243-1-4. 

• 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 0-471-35128-8. 

• Lee, S. C. et al., (2001). One Micrometer Resolution NMR Microscopy. J. Magn. Res., 150: 207-213. 

External links 

Listing of MRI Types [68] - All Inclusive Listing of MRI Types [ 9] - MRI-TUTORIAL.COM I A free learning repository about neuroimaging L J MRI step-by-step, interactive course on magnetic resonance 


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 single-core general MRI Simulator 

Professor Laurance Hall - Daily Telegraph obituary - Online MRI physics textbook. 

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[36] Randal C. Archibokl. " Hospital Details Failures Leading: to M.R.I. Fatalih (http://quer> .n\ 

html?res=9C04E4D91731F931A1575BC0A9679C8B63)", The New York Times. August 22, 2001 
[37] Donald G. McNeil Jr, " M.R.I.'s Strong Magnets Cited in Accidents (", 

The New York Times, August 19, 2005. 
|3S| hit] i pi drh I'd I'MAUDl irch.CFM 


[40] http://ww\\ n | i i i ii n i i i i I i ' _IO_niri_| ucnt_questions.html 

[41] ?Sec=sup&Sub=mri&Pag=dis&ItemId=74489 

[44] http://www.Medno\us m It nload v \_MRI li n i nude 08 pdf 'MRI 
[45] Price DL, de Wilde JP, Papadaki AM, Curran JS, Kitney RI (January 2001). "Investigation of acoustic noise on 15 MRI scanners from 0.2 T 

to 3 T.". Journal of Magnetic Resonance Imaging 13 (2): 288-293. doi:10.1002/1522-2586(200102)13:2<288::AID-JMRI1041>3.0.CO;2-P. 

PMID 11169836. 
[46] The Open University 2007: Understanding Cardiovascular Diseases, course book for the lesson SKI 21 Understanding cardiovascular 

diseases (, printed by University Press, Cambridge, ISBN 9780749226770 (can 

be found at OUW (, pages 220 and 224. 
[47] Kanal E, Barkovich A], Bell C, et al. (2007). "ACR Guidance Document for Safe MR Practices: 2007" ( 

I I 1 I I 1 H ill III I 1 II I I 1 1 

doi:I0.2214/AJR.06.1616. PMID 17515363. . page 22. 
[48] International Hleclrotechnical Commission 200X: hh dieal tie: trical Equipment Pan 2 33: Particular requirements for basic safety and 

essential perform i I i i I Ii i i ndai up // 

Webstore/webstore.nsf/0/EC11496F487C406DC125742C000B2805), published by International Electrotechnical Commission, ISBN 

2-8318-9626-6 (can be found for purchase at ). 
[49] Murphy KJ, Brunberg JA, Cohan RH (1 October 1996). "Adverse reactions to gadolinium contrast media: A review of 36 cases" (http:// AJR Am J Roentgenol 167 (4): 847-9. PMID 8819369. . 
[50] "ACRguidelin hit] uid lin t iiinman ummai in it 1- i 01 

[51] H.S. Thomsen, S.K. Morcos and P. Dawson (N ( I H ill I i I idolinium-based 

contrasl media and tl lopment of nephrogei i I ' < I 05-( 

doi:10.1016/j.crad.2006.09.003. PMID 17018301. 
[52] " FDA Pnl I Health Kisoi doliniuin ntainii ( lira t nts for M i I ' nan lin lg (hit] lei 

dnig/ad\i son, /gadoli niuin_agents.htm)' 1 
[54] MRI Questions and Answers ( 


[56] Ibrahim A. Alorainy, Fahad B. Albadr, Abdullah H. Abujamea (2006). "Attitude towards MRI safety during pregnancy" (http://www. Ann Saudi Med 26 (4): 306-9. PMID 16885635. . 
[57] Siemens Introduces First 1.5 Tesla Open Bore MRI (http://www. medical. Siemens. coin/w ebapp/wcs/stores/servlet/ 


[59] http://ww\\ | I did i i il i < i i \l in 

[60] http//ww\\ \ i i i in i i hi i mi i _im i p 
[61] Bassen, H; Schaefer, D J. ; Zaremba, L; Bushberg, J; Ziskin, M [S]; Foster, K R. (2005). "IEEE Committee on Man and Radiation 

(COMAR) Icchnical information -.lalemcnl "Kxposurc of medical personnel to electromagnetic fields from open magnetic resonance imagine 

systems"". Health Physics 89 (6): 684-9. doi:10.1097/01.HP.0000172545.71238.15. PMID 16282801. 
[62] HSE 2007, RR570:Assessmcnl of electromagnetic fields around magnetic resonance (MRI) equipment ihflp:// 

rrpdf/rr570.pdf). MCL-T Ltd, London 
[63] Filler, AG (2009) "Thi h ti i\, de\elopment, and impact ol ni] l n n n ui logicaldiagi a m ii i\ CT, MRI, DTI". 

Nature Precedings. doi: 10.1038/npre.2009.3267.2. 
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[66] "Best Regards to Alfred Nobel" ( . Retrieved 2009-10-16. 
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I -I Imp unpl |'h i ■■in 1 1 in/ "ii i l I 

[79] ?v=PF_mDsdxSsg 


[81] http://ww\\ t l i i I il I i i i i i mi i 111 Liunnce Ftall.html 



ESR Spectroscopy and Microspectroscopy 


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. 


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 
so-called g-factor (see also the Lande g-factor) 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 field-frequency 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 double-arrow 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 Maxwell-Boltzmann 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. 


3348 3350 3352 3354 

Magnetic Field Strength (G) 

Because of electron-nuclear 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.) 

Maxwell-Boltzmann 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 
Maxwell-Boltzmann 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, X-band 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 ~ 1-5- 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 

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 g-factor, causing it 
to differ from g . This is especially significant for chemical systems with transition-metal ions. 

2. If an atom with which an unpaired electron is associated has a non-zero nuclear spin, then its magnetic moment 
will affect the electron. This leads to the phenomenon of hyperfine coupling, analogous to J-coupling 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 g-factor 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 g-factor 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 g-factor, 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 
spin-orbit coupling. Because the mechanisms of spin-orbit 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, multi-lined 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 





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 
multi-line EPR spectrum and assigning the 
various spacings to specific nuclei, is more 

In the oft-encountered 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 12-line 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 


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 high-energy 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 spin-label molecules of interest. 
These reagents are particularly useful in biological systems. Specially-designed nonreactive radical molecules can 
attach to specific sites in a biological cell, and EPR spectra can then give information on the environment of these 
so-called spin-label or spin-probes. 

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 a-alanine(the alanine deamination radical, the hydrogen 
abstraction radical, and the (CO"(OH))=C(CH )NH radical) . This method is suitable for measuring gamma and 
x-rays, electrons, protons, and high-linear 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 free-radicals 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 x-rays, work that sometimes leads 
to activation energies and rate constants for radical reactions. 

The study of radiation-induced 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 

, 141 |5| 

Radiation-sterilized 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. 

High-field high-frequency measurements 

High-field-high-frequency 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 W-band EPR spectrometer was produced as a small commercial line by the German Bruker 

Company, initiating the expansion of W-band EPR techniques into medium-sized academic laboratories. Today there 
still are only a few scientific centers in the world capable of high-field-high-frequency 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 


le radical as a function of frequency. Note the improvement in 
resolution from left to right. [7] 





































































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 g-factor 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 second-order 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 zero-field splitting due to the larger microwave quantum energy h v . 

5. The higher spectral resolution over g-factor, 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 
spin-modified 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 cross-relaxation of paramagnetic centers decreases dramatically at high magnetic fields, making it easier to 
obtain more-precise and more-complete information about the system under study. 

See also 

• Ferromagnetic resonance 

• Spin labels 

• Site-directed 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 2-mm Range for Chemical Research". Instrum. 
Experim. Techn. 20 (4): 1229. 

• Krinichnyi, V. I. (1995). 2-mm Wave Band EPR Spectroscopy of Condensed Systems. Boca Raton, Florida: CRC 

• Lebedev, Y. S. (1994). "2". High-Field 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 Electron-Spin 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: Wiley-Interscience. 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: McGraw-Hill. ISBN 0070694540. 

• Protein structure elucidation by EPR: Steinhoff, H.-J. (2002). "Methods for study of protein dynamics and 
protein-protein interaction in protein-ubiquitination by electron paramagnetic resonance spectroscopy". Frontiers 
in Bioscience 7: 97-110. doi:10.2741/stein. 

External links 

• NMRWiki.ORG [6] project 

• Electron Magnetic Resonance Program National High Magnetic Field Laboratory 


[1] Odom, B.; Hanneke, D.; D'Urso, B.; and Gabrielse, G. (2006). "New Measurement of the Electron Magnetic Moment Using a One-Electron 
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/S0969-8043(00)00351-l. 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" ( Radiation Protection Dosimetry 84: 515-520. . 
[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): 199-212. doi:10.1163/1568552053750242. 
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