26
3 Multidimensional NMR Spectroscopy © Gerd Gemmecker, 1999
Models used for the description of NMR experiments
1. energy level diagram: only for polarisations, not for time-dependent phenomena
2. classical treatment (BLOCH EQUATIONS): only for isolated spins (no J coupling!)
3. vektor diagram: pictorial representation of the classical approach (same limitations)
4. quantum mechanical treatment (density matrix): rather complicated; however, using
appropriate simplifications and definitions  the product operators  a fairly easy and correct
description of most experiments is possible
3.1. BLOCH Equations
The behaviour of isolated spins can be described by classical differential equations:
x
dM/dt = Å‚M(t) B(t) - R{M(t) -M0} [3-1]
with M0 being the BOLTZMANN equilibrium magnetization and R the relaxation matrix:
x y z
1/Å„2 0 0
îÅ‚ Å‚Å‚
0 1/Å„2 0
R =
ïÅ‚ śł
0 0 1/Å„1
ðÅ‚ ûÅ‚
The external magnetic field consists of the static field B0 and the oscillating r.f. field Brf :
B(t) = B0 + Brf [3-2]
Brf = B1 cos(Ét + Ć)ex [3-3]
The time-dependent behaviour of the magnetization vector corresponds to rotations in space (plus
relaxation), with the Bx and By components derived from r.f. pulses and Bz from the static field:
dMz/dt = Å‚BxMy - Å‚ByMx -(Mz-M0)/T1 [3-4]
dMx/dt = Å‚ByMz - Å‚BzMy - Mx/T2 [3-5]
dMy/dt = Å‚BzMx - Å‚BxMz - My/T2 [3-6]
Product operators
To include coupling a special quantum mechanical treatment has to be chosen for description. An
operator, called the spin density matrix Á(t), completely describes the state of a large ensemble of
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spins. All observable (and non-observable) physical values can be extracted by multiplying the
density matrix with their appropriate operator and then calculating the trace of the resulting matrix.
The time-dependent evolution of the system is calculated by unitary transformations (corresponding
to "rotations") of the density matrix operator with the proper Hamiltonian H (including r.f. pulses,
chemical shift evolution, J coupling etc.):
Á(t') = exp{iHt} Á(t) exp{-iHt}
(for calculations these exponential operators have to be expanded into a Taylor series).
The density operator can be written als linear combination of a set of basis operators. Two specific
bases turn out to be useful for NMR experiments:
- the real Cartesian product operators Ix, Iy and Iz (useful for description of observable
magnetization and effects of r.f. pulses, J coupling and chemical shift) and
- the complex single-element basis set I+, I-, IÄ… and I² (raising / lowering operators, useful for
coherence order selection / phase cycling / gradient selection).
Cartesian Product operators
Lit. O.W. SÅ‚rensen et al. (1983), Prog. NMR. Spectr. 16, 163-192
Single spin operators
correspond to magnetization of single spins, analogous to the classical macroscopic magnetization
Mx, My, Mz.
Ix, Iy (in-phase coherence, observable)
Iz (z polarisation, not observable)
Two-spin operators
2I1xI2z , 2I1yI2z , 2I1zI2x , 2I1zI2y (antiphase coherence, not observable)
2I1z I2z (longitudinal two-spin order, not observable)
2I1xI2x , 2I1yI2x , 2I1xI2y , 2I1yI2y (multiquantum coherence, not observable)
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Sums and differences of product operators
2 I1xI2x + 2 I1yI2y = I1+ I2- + I1- I2+ zero-quantum coherence
2 I1yI2x - 2 I1xI2y = I1+ I2- - I1- I2+ (not observable)
2 I1xI2x - 2 I1yI2y = I1+ I2+ + I1- I2- double-quantum coherence
2 I1xI2y + 2 I1yI2x = I1+ I2+ - I1- I2- (not observable)
The single-element operators I+ and I- correspond to a transition from the mz = -1/2 to the mz = + 1/2
state and back, resp., hence "raising" and "lowering operator". Products of three and more operators
are also possible.
Only the operators Ix and Iy represent observable magnetization. However, other terms like antiphase
magnetization 2 I1x I2z can evolve into observable terms during the acquisition period.
Pictorial representations of product operators
(cf. the paper in Progr. NMR Spectrosc. by SÅ‚rensen et al.)
coherences
z
z
²²
²²
²Ä… ²Ä…
I I I
x 1 x 2 z
Ä…² Ä…²
x y
x y
Ä…Ä… Ä…Ä…
z z
²² ²²
²Ä… ²Ä…
I I I
y 1 y 2 z
Ä…² Ä…²
x y x y
Ä…Ä… Ä…Ä…
polarisations
²² ²² ²² ²²
I I
2I I
1 z 2 z I +I
1 z 2 z
²Ä… Ä…² ²Ä… Ä…² 1 z 2 z ²Ä… Ä…²
²Ä… Ä…²
Ä…Ä… Ä…Ä… Ä…Ä… Ä…Ä…
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In the energy level diagrams for coherences, the single quantum coherences Ix and Iy are
symbolically depicted as black and gray arrows. Both arrows in each two-spin scheme (for the
coupling partner being Ä… or ²) belong to the same operator; in the vector diagrams these two species
either align (for in-phase coherence) or a 180° out of phase (antiphase coherence). In the NMR
spectrum, these two arrows / transitions correspond to the two lines of the dublet caused by the J
coupling between the two spins. The term 2I1xI2z is called antiphase coherence of spin 1 with
respect to spin 2.
For the populations, filled circles represent a population surplus, empty circles a population deficit
(with respect to an even distribution). I1z and I2z are polarisations of one sort of spins only, I1z+I2z is
the normal BOLTZMANN equilibrium state, and 2 I1z I2z is called longitudinal two-spin order (with the
two spins in each molecule preferentially in the same spin state).
Evolution of product operators
Chemical shift
&!1tIz
I1x çÅ‚çÅ‚çÅ‚ I1xcos(&!1t) + I1ysin(&!1t) [3-7]
&!1tI1z
I1y çÅ‚çÅ‚çÅ‚ I1ycos(&!1t) - I1xsin(&!1t) [3-8]
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Effect of r.f. pulses
²Iy
I1z çÅ‚çÅ‚çÅ‚ I1zcos² + I1xsin² [3-9]
²Iy
I1x çÅ‚çÅ‚çÅ‚ I1xcos² - I1zsin² [3-10]
²Iy
I1y çÅ‚çÅ‚çÅ‚ I1y
The effects of x and z pulses can be determined by cyclic permutation of x, y, and z. All rotations
obey the "right-hand rule", i.e., with the thumb of the right (!) hand pointing in the direction of the
r.f. pulse, the curvature of the four other fingers indicate the direction of the rotation.
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Scalar coupling
Ä„JtI1zI2z
I1x çÅ‚çÅ‚çÅ‚çÅ‚çÅ‚ I1xcos(Ä„Jt) + 2I1yI2zsin(Ä„Jt) [3-11]
Ä„JtI1zI2z
I1y çÅ‚çÅ‚çÅ‚çÅ‚çÅ‚ I1ycos(Ä„Jt) - 2I1xI2zsin(Ä„Jt) [3-12]
Ä„JtI1zI2z
I1z çÅ‚çÅ‚çÅ‚çÅ‚çÅ‚ I1z
(i.e., I1z does not evolve J coupling!)
Ä„JtI1zI2z
2I1xI2z çÅ‚çÅ‚çÅ‚çÅ‚çÅ‚ I1ysin(Ä„Jt) + 2I1xI2zcos(Ä„Jt) [3-13]
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The antiphase term 2I1yI2z can be re-written using the single-element operators IÄ… und I²:
2I1yI2z = I1yI2Ä… - I1yI2² [3-14]
(2Iz = IÄ… - I², IÄ… + I² = 1; IÄ… und I² are called polarization operators )
1
The antiphase state 2I1yI2z consists of two separate populations: for one half of the molecules in the
ensemble spin 1 is in +y coherence (when spin 2 is in the Ä… state), for the other half spin 1 is in -y
coherence (with spin 2 in the ² state); "spin 1 is in antiphase with respect to spin 2".
Such an antiphase state can develop from I1x when spin 1 is J-coupled to spin 2. This leads to a
J
dublet for spin 1, i.e., it splits into two lines with an up- and downfield shift by /2, depending on the
1
spin state of the coipling partner, spin 2. If we wait long enough ( /2J ), then the frequency
difference of J between the dublet lines (I1x I2Ä… and I1x I2² ) has brought them 180° out of phase
("antiphase"), as shown in the vector diagram.
This is an oscillation between I1x in-phase coherence and 2I1y I2z antiphase coherence. The antiphase
component evolves with sin( Jt) and then refocusses back to -I1x in-phase coherence (after t= 1/J ).
Single-element operators
In some cases (phase cycling, gradient coherence selection) it is necessary to use operators with a
defined coherence order (Eigenstates of coherence order). Coherence order describes the changes in
quantum numbers mz caused by the coherence. A spin-1/2 system (no coupling) can assume two
coherent states: a transition from Ä… (mz=+1/2) to ² (mz=-1/2), i.e., a change (coherence order) of -1.
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This can be described by the lowering operator I- = |²><Ä…|, the coherent transition from the ² to Ä…
state by the raising operator I+ = |Ä…><²| (coherence order +1).
The real Cartesian operators Ix and Iy correspond to mixtures of both coherence orders, Ä…1, although
they are more useful for directly corresponding to the observable x and y components of the
magnetization. Their relationship with the complex IÄ… operators is simple:
I+ = Ix + iIy raising operator
Ix = 1/2 (I+ + I-)
I- = Ix - iIy lowering operator
Iy = -i/2 (I+ - I-)
IÄ… = 1/2 1 + Iz polarisation operator (Ä…)
Iz = 1/2 (IÄ… - I²)
I² = 1/2 1 - Iz polarisation operator (²)
1 = IÄ… + I²
The effect of r.f. pulses (here: an x pulse with flip angle Õ) on single-element operators is as follows:
Õx
I+/- çÅ‚çÅ‚çÅ‚I+/-cos2{Õ/2} + I-/+sin2{Õ/2} (+/- iIzsin{Õ}) [3-15]
Õx
I² çÅ‚çÅ‚çÅ‚I²cos2{Õ/2} + IÄ…sin2{Õ/2} + (1/2)sin{Õ}[I+ - iI-] [3-16]
Õx
IÄ… çÅ‚çÅ‚çÅ‚IÄ…cos2{Õ/2} + I²sin2{Õ/2} - (1/2)sin{Õ}[I+ - iI-] [3-17]
Generally it is easier to calculate the effects of r.f. pulses on Cartesian operators and then use the
conversion rules to get the single-element version.
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Signal phase, In-phase and antiphase signals
For a single spin I1 one gets the following signal during acquisition with receiver reference phase x:
Ix Ix cos (&!
&!t ) + Iy sin (&!t)
Iy Iy cos (&!t ) - Ix sin (&!
&!t)
These two signals are 90° out of phase (also after FT), which is indicated by one Ix component
having a sine, the other one a cosine modulation.
For a spin I1 coupled to another spin I2 one gets the following signal during acquisition (neglecting
chemical shift evolution):
Ix Ix cos (&!t) + Iy sin (&!t) Ix cos (&! Ä„Jt) + 2 I1y I2z cos (&!t) sin(Ä„Jt)
&!t) cos (Ä„
+ Iy sin (&!t) cos (Ä„Jt) - 2 I1x I2z sin (&!t)
sin(Ä„Jt)
the detected x component corresponds to an in-phase dublet with splitting J, i.e., lines with intensity
J J
1
/2 at positions (&! + /2) and (&! - /2) ( Ä„Jt =
J
2Ä„ /2 t ).
cos Ä… cos ² = 1/2 [cos (Ä…+²) + cos (Ä…-²)]
2 I1x I2z 2 I1x I2z cos (&!t) + 2 I1y I2z sin (&!t)
2 I1x I2z cos (&!t) cos(Ä„Jt) + Iy cos (&!t) sin (Ä„Jt)
+ 2 I1y I2z sin (&!t) sin(Ä„Jt) - Ix sin (&! Ä„Jt)
&!t) sin (Ä„
this tiem the x component corresponds to an anti-phase dublet with splitting J, i.e., lines with
J J
1
intensities of /2 and -1/2 at positions (&! + /2) and (&! - /2) :
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sin Ä… sin ² = 1/2 [cos (Ä…+²) - cos (Ä…-²)]
In the same way, Iy leads to an in-phase dublet 90° out of phase (=dispersive) and 2 I1y I2z to a
dispersive anti-phase dublet.
Some applications
1. For solvent signal suppression in 1D spectra, the Jump-Return sequence can be used:
90°(x) - Ä - 90°(x) - acquisition
Calculate the excitation profile with product operator formalism!
2. What happens to chemical shift evolution during this sequence, and what about J coupling?
Calculate!
90°(x) - Ä - 180°(x) - Ä -
90°(x) - Ä - 180°(y) - Ä -
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A simple 2D experiment (COSY)
Let's calculate the result of this two-pulse COSY sequence for a single spin I:
y y
The first 90°y pulse creates transverse magnetization, which then evolves under the influence of
chemical shift and J coupling during the delay t1, after the second 90°y pulse we get:
90°y &!t1 90°y
Iz çÅ‚çÅ‚ Ix çÅ‚çÅ‚ Ix cos(&!t1) çÅ‚çÅ‚ - Iz cos(&!t1)
+ Iy sin(&!t1) + Iy sin(&!
&!t1)
During the acquisition time t2 the first component is not detectable (polarization Iz), the other (bold)
term is a coherence which will evolve during t2 as follows:
&!t2
Iy sin(&!t1) çÅ‚çÅ‚ Iy sin(&!t1) cos(&!t2)
- Ix sin(&!t1) sin(&!t2)
If we compare this to a normal 1D spectrum (let's call the acquisition time again t2)
90°y &!t1
Iz çÅ‚çÅ‚ Ix çÅ‚çÅ‚ Ix cos(&!t2)
+ Iy sin(&!t2)
we see that both signals correspond the real and imaginary part of a precession with frequency &!
during the acquisition time t2. The only difference between the 1D and the COSY spectrum  besides
a 90° phase shift (for the 1D, the absorptive and dispersive components are Ix and Iy, for the COSY
they are Iy and -Ix, resp.)  is the factor sin( t1) in the COSY terms. So far, this is just a constant
with a value somewhere between -1 and +1, depending on the chosen t1 value (and on &!, of course).
However, the result of the COSY sequence contains very similar factors for t1 and t2, i.e., a sine or
cosine modulation with the argument &!ti. Instead of a t2 FT, we could also perform a FT in t1.
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Of course, we have only data for a single t1 value (the one we chose in the COSY sequence), but for
a FT we need to know a whole series of values of the oscillatory function, as we do for t2 (all the
TD2 time domain data points sampled during the acquisition time, for t2=0, t2=DW, t2=2DW, &
t2=AQ2).
We can re-run the COSY sequence with a different setting for t1, and another one, etc., starting from
t1=0, then t1=DW, t1=2DW, & t1=AQ1 (TD1 different t1 values). Our complete data set now consists
of TD1 FIDs (with TD2 data points each):
We can now perform a "normal" FT along t2, converting the series of FIDs into a series of spectra,
which are all identical (with a signal at &!), except for a modulation with sin( t1) along the t1
dimension:
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If we now read out single columns from our 2D data matrix, then we will generally get a pseudo-FID
A sin( t1) for each column, with A=0 where there is no signal in the F2 dimension (the frequency
dimension generated by the t2-FT) and with A`"0 where we have our signal (at &! in F2).
From these pseudo-FIDs we can again generate a frequency spectrum by FT (now along the t1
dimension), and will get a signal at the frequency &! in this F1 dimension  in the column at &! in F2:
This is a 2D COSY spectrum!  although not very interesting, since it contains just one diagonal
peak (=with identical chemical shift &! in both dimensions), no more information than a simple 1D
spectrum.
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In order to achieve a distinction between positive and negative &! values (&!=0 is in the center of
each dimension), a complex FT also in the indirect F1 dimension is required, i.e., the sine and the
cosine component. From our pulse sequence we get only
Iy sin(&!t1) cos(&!t2) - Ix sin(&!t1) sin(&!t2)
These are the two (real and imaginary) components for a t2 value, but only the sine component in t1.
We have to re-run our complete set of TD1 t1 increments with a slightly modified pulse sequence,
with the first pulse phase shifted by 90°:
90°-x &!t1 90°y &!t2
Iz çÅ‚ Iy çÅ‚ Iy cos(&!t1) çÅ‚ Iy cos(&!t1) çÅ‚ Iy cos(&! &!t2) - Ix cos(&! &!t2)
&!t1) cos(&! &!t1) sin(&!
-Ix sin(&!t1) + Iz sin(&!t1) + Iz sin(&!t1)
We now get exactly the same terms as for the first COSY pulse sequence, only with a cosine
modulation. Combining the two data sets, we get four data points for each t1/t2 combination:
Iy sin(&!t1) cos(&!t2) - Ix sin(&!t1) sin(&!t2)
Iy cos(&!t1) cos(&!t2) - Ix cos(&!t1) sin(&!t2)
This is called a hypercomplex data point, and a 2D matrix of such data points contains all sine/cosine
combinations needed for a hypercomplex 2D-FT, yielding the frequency &! including the correct sign
in both dimensions. There are different ways to acquire the equivalent of a hypercomplex data set:
- the STATES (-RUBEN-HABERKORN) method outlined here (first pulse phase x/y for each t1 point)
- the TPPI (Time Proportional Phase Incrementation), which is analogous to the REDFIELD trick
for single-channel acquisition (only real data points in t1, but twice as many, with half the time
increment
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- echo-antiecho quadrature detection, which does not sample the real and imaginary (Ix and Iy)
components separately, but rather I+ and I- (by coherence selction through phase cycling or
gradients)  which can then be easily converted into Ix and Iy by the computer
Magnitude mode spectra
Alternatively, one can only select either I+ or I- during t1 (i.e., only one data point per t1 value), again
by phase cycling or gradients. This corresponds to either the spectrum or its mirror image in the
indirect dimension, so no quad images will occur. However, according to I+ = Ix + iIy / I- = Ix - iIy
these are complex components with no pure phase. The resulting spectrum cannot be phased to pure
absorptive lineschapes in F1 and hence has to be displayed in absolute value/magnitude or power
mode.
This was quite popular many years ago, when data storage and processing capacity were limited.
However, the S/N is only 1/ (ca. 71 %) of a STATES or TPPI spectrum of equal measuring time
2
1
and digital resolution, because the r.f. pulses create Ix = /2 (I+ + I-) or Iy = -i/2 (I+ - I-) at the
beginning of t1, so only 50 % of the signal is selected by the phase cycle or gradients. Worse even,
the very unfavourable magnitude or power mode lineshapes greatly reduce the spectral resolution,
even with optimized apodization functions.
Some types of spectra, however, cannot be phase corrected because of J coupling evolution during
the pulse sequence, resulting in a mixture of absorptive/dispersive in-phase/antiphase signals
(Relayed-COSY, long-range COSY). One can get the S/N advantage of the phase sensitive version
(i.e., acquired in STATES or TPPI mode), but still has to convert the spectrum to absolute value mode
after the complex FT.
A COSY with crosspeaks
Let's calculate the result of the COSY sequence for two coupled spins I1 and I2:
90°y &!t1 Ä„Jt1
I1z çÅ‚çÅ‚ I1x çÅ‚çÅ‚ I1x cos(&!1t1) çÅ‚çÅ‚ I1x cos(&!1t1) cos(Ä„Jt1) + 2I1yI2z cos(&!1t1) sin(Ä„Jt1)
+ I1y sin(&!1t1) + I1y sin(&!1t1)
cos(Ä„Jt1) - 2I1xI2z sin(&!1t1) sin(Ä„Jt1)
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after the second 90°y pulse these four terms are converted into
90°y
çÅ‚çÅ‚ -I1z cos(&!1t1) cos(Ä„Jt1) + 2I1yI2x cos(&!1t1) sin(Ä„Jt1)
+ I1y sin(&!1t1) cos(Ä„ &! Ä„Jt1)
&! Ä„Jt1) + 2I1zI2x sin(&!1t1) sin(Ä„
What signal will be detected now during the acquisition time t2?
- the first two components are not detectable (polarization I1z and MQC 2I1yI2x)
- the other two (bold) terms are spin 1 in-phase coherence and spin 2 antiphase coherence, which
will evolve during t2 as follows (shown without the sine and cosine terms from t1):
&!t2 Ä„Jt2
I1y (& ) çÅ‚çÅ‚ I1y cos(&!1t2) çÅ‚çÅ‚ I1y cos(&!1t2) cos(Ä„
&! Ä„Jt2) - 2I1xI2z cos(&!1t2) sin(Ä„Jt2)
-I1x sin(&!1t2) - I1x sin(&!1t2) cos(Ä„
&! Ä„Jt2) - 2I1yI2z
sin(&!1t2) sin(Ä„Jt2)
The two detectable in-phase components are, in full length,
I1y sin(&!1t1) cos(Ä„ &! Ä„Jt2) - I1x sin(&!1t1) cos(Ä„ &! Ä„Jt2)
&! Ä„Jt1) cos(&!1t2) cos(Ä„ &! Ä„Jt1) sin(&!1t2) cos(Ä„
From the second SQC term 2I1zI2x sin( t1) sin( Jt1) present at the beginning of t2 we get:
1
&!t1 Ä„Jt1
2I1zI2x (& ) çÅ‚ 2I1zI2x cos(&!2t2) çÅ‚ 2I1zI2x cos(&!2t2) cos(Ä„Jt2) + I2y cos(&!2t2) sin(Ä„
&! Ä„Jt2)
+ 2I1zI2y sin(&!2t2) + 2I1zI2y sin(&!2t2) cos(Ä„Jt2) - I2x
sin(&!2t2) sin(Ä„
&! Ä„Jt2)
(since this is a spin-2 coherence, it will evolve chemical shift of spin 2, &!2 !).
The two detectable in-phase components are in full length:
I2y sin(&!1t1) sin(Ä„ &! Ä„Jt2) - I2x sin(&!1t1) sin(Ä„ &! Ä„Jt2)
&! Ä„Jt1) cos(&!2t2) sin(Ä„ &! Ä„Jt1) sin(&!2t2) sin(Ä„
According to the rules for in-phase and antiphase terms, we can now easily figure out what the 2D
spectrum will look like, remembering that the harmonics with t1 in the argument describe the signal
in the F1 domain (after t1-FT), and the ones with t2 correspond to the signals look in F2.
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1. I1y sin(&!1t1) cos(Ä„Jt1) cos(&!1t2) cos(Ä„ &! Ä„Jt2)
&! Ä„Jt2) - I1x sin(&!1t1) cos(Ä„Jt1) sin(&!1t2) cos(Ä„
This is an in-phase dublet signal at &!1 in F2 and at &!1 in F1, i.e., a diagonal peak again (as entioned
above, so far we have only recorded one component in t1, and we will have to repeat the whole 2D
experiment with a 90° shifted first r.f. pulse to get hypercomplex data points.
diagonal peak / F1 multiplett
1
sin(&!1t1)cos(Ä„Jt1) = /2 {sin(&!1 + Ä„J)t1 + sin(&!1-Ä„J)t1}
(dispersive)
diagonal peak / F2 multiplett
1
cos(&!1t1)cos(Ä„Jt1) = /2 {cos(&!1 - Ä„J)t1 + cos(&!1 + Ä„J)t1} (absorptive)
2. I2y sin(&!1t1) sin(Ä„Jt1) cos(&!2t2) sin(Ä„ &! Ä„Jt2)
&! Ä„Jt2) - I2x sin(&!1t1) sin(Ä„Jt1) sin(&!2t2) sin(Ä„
This describes again the absorptive (I2x) and dispersive parts (I2y) of a signal at frequency &!1 in F1
and at frequency &!2 in F2, i.e., a cross-peak with different resonance frequencies in the two
dimensions. Furthermore, it is an antiphase signal with respect to the J12 coupling in both
dimensions.
cross-peak / F1 /
sin(&!1t1)sin(Ä„Jt1) = 1/2 {-cos(&!1 + Ä„J)t1 + cos(&!1-Ä„J)t1}
(absorptive)
cross-peak / F2
1
cos(&!2t2)sin(Ä„Jt2) = /2 {sin(&!2 + Ä„J)t2- sin(&!2 - Ä„J)t2}
(dispersive)
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If we compare the expressions for the diagonal peak and the cross-peak, we can see that they are 90°
out of phase relativ to each other in both dimensions. E.g., for the diagonal peak we have an Ix
component which is dispersive in F1 and F2, while from the cross-peak we get an Ix component that
is absorptive (and vice versa for the Iy components). So, no matter what phase correction we choose
in each of the two (independently phase corrected) dimensions, always one of the two signals will be
dispersive.
Two ways of phase correcting a 2D COSY spectrum: diagonal peaks in-phase absorptive, cross-
peaks antiphase dispersive (left); or diagonal peaks in-phase dispersive, cross-peaks antiphase
absorptive (right).
No matter what phase corrrection is chosen, the dispersive tails always tend to obscure near-by
cross-peaks. Absolute value processing is no real solution either, since now the dispersive
components of both the diagonal and the cross-peaks contribute to the spectrum. Only the
employment of apodization functions with rigorous resolution enhancement and line narrowing
characteristics can yield a reasonable spectrum (although at the cost of losing S/N).