4 Fourier transformation and data
processing
In the previous chapter we have seen how the precessing magnetization can
be detected to give a signal which oscillates at the Larmor frequency – the
free induction signal. We also commented that this signal will eventually
decay away due to the action of relaxation; the signal is therefore often called
the free induction decay or FID. The question is how do we turn this signal,
which depends on time, into the a spectrum, in which the horizontal axis is
frequency.
time
frequency
Fourier
transformation
Fig. 4.1 Fourier transformation
is the mathematical process
which takes us from a function
of time (the time domain) – such
as a FID – to a function of
frequency – the spectrum.
This conversion is made using a mathematical process known as Fourier
transformation. This process takes the time domain function (the FID) and
converts it into a frequency domain function (the spectrum); this is shown in
Fig. 4.1. In this chapter we will start out by exploring some features of the
spectrum, such as phase and lineshapes, which are closely associated with
the Fourier transform and then go on to explore some useful manipulations of
NMR data such as sensitivity and resolution enhancement.
4.1 The FID
In section 3.6 we saw that the x and y components of the free induction signal
could be computed by thinking about the evolution of the magnetization
during the acquisition time. In that discussion we assumed that the magnetization
started out along the −y axis as this is where it would be rotated to by
a 90◦ pulse. For the purposes of this chapter we are going to assume that the
magnetization starts out along x; we will see later that this choice of starting
position is essentially arbitrary.
x
y
time
My
Mx
Fig. 4.2 Evolution of the magnetization over time; the offset is assumed to be positive and the magnetization
starts out along the x axis.
Chapter 4 “Fourier transformation and data processing” c James Keeler, 2002
4–2 Fourier transformation and data processing
Sx
Sy
S0
Ωt
Fig. 4.3 The x and y
components of the signal can
be thought of as arising from
the rotation of a vector S0 at
frequency .
If the magnetization does indeed start along x then Fig. 3.16 needs to be
redrawn, as is shown in Fig. 4.2. From this we can easily see that the x and y
components of the magnetization are:
Mx = M0 cos t
My = M0 sin t.
The signal that we detect is proportional to these magnetizations. The constant
of proportion depends on all sorts of instrumental factors which need not
concern us here; we will simply write the detected x and y signals, Sx(t) and
Sy(t) as
Sx(t) = S0 cos t and Sy(t) = S0 sin t
where S0 gives is the overall size of the signal and we have reminded ourselves
that the signal is a function of time by writing it as Sx(t) etc.
It is convenient to think of this signal as arising from a vector of length S0
rotating at frequency ; the x and y components of the vector give Sx and Sy,
as is illustrated in Fig. 4.3.
time
Sx(t)orrealpartSy(t)orimag.part
Fig. 4.4 Illustration of a typical
FID, showing the real and
imaginary parts of the signal;
both decay over time.
As a consequence of the way the Fourier transform works, it is also convenient
to regard Sx(t) and Sy(t) as the real and imaginary parts of a complex
signal S(t):
S(t) = Sx (t) + i Sy(t)
= S0 cos t + i S0 sin t
= S0 exp(i t).
We need not concern ourselves too much with the mathematical details here,
but just note that the time-domain signal is complex, with the real and imaginary
parts corresponding to the x and y components of the signal.
We mentioned at the start of this section that the transverse magnetization
decays over time, and this is most simply represented by an exponential decay
with a time constant T2. The signal then becomes
S(t) = S0 exp(i t) exp
−t
T2
. (4.1)
A typical example is illustrated in Fig. 4.4. Another way of writing this is to
define a (first order) rate constant R2 = 1/T2 and so S(t) becomes
S(t) = S0 exp(i t) exp(−R2t). (4.2)
The shorter the time T2 (or the larger the rate constant R2) the more rapidly
the signal decays.
4.2 Fourier transformation
Fourier transformation of a signal such as that given in Eq. 4.1 gives the frequency
domain signal which we know as the spectrum. Like the time domain
signal the frequency domain signal has a real and an imaginary part. The real
4.2 Fourier transformation 4–3
part of the spectrum shows what we call an absorption mode line, in fact in
the case of the exponentially decaying signal of Eq. 4.1 the line has a shape
known as a Lorentzian, or to be precise the absorption mode Lorentzian. The
imaginary part of the spectrum gives a lineshape known as the dispersion
mode Lorentzian. Both lineshapes are illustrated in Fig. 4.5.
time
frequency
Fig. 4.6 Illustration of the fact that the more rapidly the FID decays the broader the line in the corresponding
spectrum. A series of FIDs are shown at the top of the figure and below are the corresponding spectra,
all plotted on the same vertical scale. The integral of the peaks remains constant, so as they get broader
the peak height decreases.
frequency
Ω
absorption
dispersion
Fig. 4.5 Illustration of the
absorption and dispersion mode
Lorentzian lineshapes.
Whereas the absorption
lineshape is always positive, the
dispersion lineshape has
positive and negative parts; it
also extends further.
This absorption lineshape has a width at half of its maximum height of
1/(πT2) Hz or (R/π) Hz. This means that the faster the decay of the FID
the broader the line becomes. However, the area under the line – that is the
integral – remains constant so as it gets broader so the peak height reduces;
these points are illustrated in Fig. 4.6.
If the size of the time domain signal increases, for example by increasing
S0 the height of the peak increases in direct proportion. These observations
lead to the very important consequence that by integrating the lines in the
spectrum we can determine the relative number of protons (typically) which
contribute to each.
The dispersion line shape is not one that we would choose to use. Not
only is it broader than the absorption mode, but it also has positive and negative
parts. In a complex spectrum these might cancel one another out, leading
to a great deal of confusion. If you are familiar with ESR spectra you might
recognize the dispersion mode lineshape as looking like the derivative lineshape
which is traditionally used to plot ESR spectra. Although these two
lineshapes do look roughly the same, they are not in fact related to one an-
other.
Positive and negative frequencies
As we discussed in section 3.5, the evolution we observe is at frequency
i.e. the apparent Larmor frequency in the rotating frame. This offset can be
positive or negative and, as we will see later, it turns out to be possible to
determine the sign of the frequency. So, in our spectrum we have positive and
negative frequencies, and it is usual to plot these with zero in the middle.
4–4 Fourier transformation and data processing
Several lines
What happens if we have more than one line in the spectrum? In this case,
as we saw in section 3.5, the FID will be the sum of contributions from each
line. For example, if there are three lines S(t) will be:
S(t) =S0,1 exp(i 1t) exp
−t
T
(1)
2
+ S0,2 exp(i 2t) exp
−t
T(2)
2
+ S0,3 exp(i 3t) exp
−t
T (3)
2
.
where we have allowed each line to have a separate intensity, S0,i , frequency,
i , and relaxation time constant, T
(i)
2 .
The Fourier transform is a linear process which means that if the time
domain is a sum of functions the frequency domain will be a sum of Fourier
transforms of those functions. So, as Fourier transformation of each of the
terms in S(t) gives a line of appropriate width and frequency, the Fourier
transformation of S(t) will be the sum of these lines – which is the complete
spectrum, just as we require it.
4.3 Phase
So far we have assumed that at time zero (i.e. at the start of the FID) Sx(t) is
a maximum and Sy(t) is zero. However, in general this need not be the case
– it might just as well be the other way round or anywhere in between. We
describe this general situation be saying that the signal is phase shifted or that
it has a phase error. The situation is portrayed in Fig. 4.7.
In Fig. 4.7 (a) we see the situation we had before, with the signal starting
out along x and precessing towards y. The real part of the FID (corresponding
to Sx) is a damped cosine wave and the imaginary part (corresponding to Sy)
is a damped sine wave. Fourier transformation gives a spectrum in which the
real part contains the absorption mode lineshape and the imaginary part the
dispersion mode.
In (b) we see the effect of a phase shift, φ, of 45◦. Sy now starts out at a
finite value, rather than at zero. As a result neither the real nor the imaginary
part of the spectrum has the absorption mode lineshape; both are a mixture of
absorption and dispersion.
In (c) the phase shift is 90◦. Now it is Sy which takes the form of a
damped cosine wave, whereas Sx is a sine wave. The Fourier transform gives
a spectrum in which the absorption mode signal now appears in the imaginary
part. Finally in (d) the phase shift is 180◦ and this gives a negative absorption
mode signal in the real part of the spectrum.
What we see is that in general the appearance of the spectrum depends on
the position of the signal at time zero, that is on the phase of the signal at time
zero. Mathematically, inclusion of this phase shift means that the (complex)
signal becomes:
S(t) = S0 exp(iφ) exp(i t) exp
−t
T2
. (4.3)
4.3 Phase 4–5
Sx Sy
Sx SySx Sy
Sx Sy
Sx
Sy
Sx
Sy
φ
Sx
Sy
φ
Sx
Sy
φ
real imag imag
imagimag
real
realreal
(a) (b)
(d)(c)
Fig. 4.7 Illustration of the effect of a phase shift of the time domain signal on the spectrum. In (a) the
signal starts out along x and so the spectrum is the absorption mode in the real part and the dispersion
mode in the imaginary part. In (b) there is a phase shift, φ, of 45◦; the real and imaginary parts of the
spectrum are now mixtures of absorption and dispersion. In (c) the phase shift is 90◦; now the absorption
mode appears in the imaginary part of the spectrum. Finally in (d) the phase shift is 180◦ giving a negative
absorption line in the real part of the spectrum. The vector diagrams illustrate the position of the signal at
time zero.
Phase correction
It turns out that for instrumental reasons the axis along which the signal appears
cannot be predicted, so in any practical situation there is an unknown
phase shift. In general, this leads to a situation in which the real part of the
spectrum (which is normally the part we display) does not show a pure absorption
lineshape. This is undesirable as for the best resolution we require
an absorption mode lineshape.
Luckily, restoring the spectrum to the absorption mode is easy. Suppose
with take the FID, represented by Eq. 4.3, and multiply it by exp(iφcorr):
exp(iφcorr)S(t) = exp(iφcorr) × S0 exp(iφ) exp(i t) exp
−t
T2
.
This is easy to do as by now the FID is stored in computer memory, so the
4–6 Fourier transformation and data processing
multiplication is just a mathematical operation on some numbers. Exponentials
have the property that exp(A) exp(B) = exp(A + B) so we can re-write
the time domain signal as
exp(iφcorr)S(t) = exp(i(φcorr + φ)) S0 exp(i t) exp
−t
T2
.
Now suppose that we set φcorr = −φ; as exp(0) = 1 the time domain signal
becomes:
exp(iφcorr)S(t) = S0 exp(i t) exp
−t
T2
.
The signal now has no phase shift and so will give us a spectrum in which
the real part shows the absorption mode lineshape – which is exactly what we
want. All we need to do is find the correct φcorr.
It turns out that the phase correction can just as easily be applied to the
spectrum as it can to the FID. So, if the spectrum is represented by S(ω) (a
function of frequency, ω) the phase correction is applied by computing
exp(iφcorr)S(ω).
Such a correction is called a frequency independent or zero order phase correction
as it is the same for all peaks in the spectrum, regardless of their offset.Attempts have been made over
the years to automate this
phasing process; on well
resolved spectra the results are
usually good, but these
automatic algorithms tend to
have more trouble with
poorly-resolved spectra. In any
case, what constitutes a
correctly phased spectrum is
rather subjective.
In practice what happens is that we Fourier transform the FID and display
the real part of the spectrum. We then adjust the phase correction (i.e.
the value of φcorr) until the spectrum appears to be in the absorption mode –
usually this adjustment is made by turning a knob or by a “click and drag”
operation with the mouse. The whole process is called phasing the spectrum
and is something we have to do each time we record a spectrum.
In addition to the phase shifts introduced by the spectrometer we can of
course deliberately introduce a shift of phase by, for example, altering the
phase of a pulse. In a sense it does not matter what the phase of the signal
is – we can always obtain an absorption spectrum by phase correcting the
spectrum later on.
Frequency dependent phase errors
We saw in section 3.11 that if the offset becomes comparable with the RF
field strength a 90◦ pulse about x results in the generation of magnetization
along both the x and y axes. This is in contrast to the case of a hard pulse,
where the magnetization appears only along −y. We can now describe this
mixture of x and y magnetization as resulting in a phase shift or phase error
of the spectrum.
Figure 3.25 illustrates very clearly how the x component increases as the
offset increases, resulting in a phase error which also increases with offset.
Therefore lines at different offsets in the spectrum will have different phase
errors, the error increasing as the offset increases. This is illustrated schematically
in the upper spectrum shown in Fig. 4.8.
If there were only one line in the spectrum it would be possible to ensure
that the line appeared in the absorption mode simply by adjusting the phase
4.3 Phase 4–7
phase frequency0
frequency0
(a)
(b)
(c)
Fig. 4.8 Illustration of the appearance of a frequency dependent phase error in the spectrum. In (a) the
line which is on resonance (at zero frequency) is in pure absorption, but as the offset increases the phase
error increases. Such an frequency dependent phase error would result from the use of a pulse whose
RF field strength was not much larger than the range of offsets. The spectrum can be returned to the
absorption mode, (c), by applying a phase correction which varies with the offset in a linear manner, as
shown in (b). Of course, to obtain a correctly phased spectrum we have to choose the correct slope of the
graph of phase against offset.
in the way described above. However, if there is more than one line present
in the spectrum the phase correction for each will be different, and so it will
be impossible to phase all of the lines at once.
Luckily, it is often the case that the phase correction needed is directly
proportional to the offset – called a linear or first order phase correction. Such
a variation in phase with offset is shown in Fig. 4.8 (b). All we have to do is
to vary the rate of change of phase with frequency (the slope of the line) until
the spectrum appears to be phased; as with the zero-order phase correction the
computer software usually makes it easy for us to do this by turning a knob
or pushing the mouse. In practice, to phase the spectrum correctly usually
requires some iteration of the zero- and first-order phase corrections.
The usual convention is to express the frequency dependent phase correction
as the value that the phase takes at the extreme edges of the spectrum.
So, for example, such a correction by 100◦ means that the phase correction is
zero in the middle (at zero offset) and rises linearly to +100◦ at on edge and
falls linearly to −100◦ at the opposite edge.
For a pulse the phase error due to these off-resonance effects for a peak
with offset is of the order of ( tp), where tp is the length of the pulse. For
a carbon-13 spectrum recorded at a Larmor frequency of 125 MHz, the maximum
offset is about 100 ppm which translates to 12500 Hz. Let us suppose
that the 90◦ pulse width is 15 µs, then the phase error is
2π × 12500 × 15 × 10−6
≈ 1.2 radians
which is about 68◦; note that in the calculation we had to convert the offset
from Hz to rad s−1 by multiplying by 2π. So, we expect the frequency dependent
phase error to vary from zero in the middle of the spectrum (where
the offset is zero) to 68◦ at the edges; this is a significant effect.
For reasons which we cannot go into here it turns out that the linear phase
4–8 Fourier transformation and data processing
correction is sometimes only a first approximation to the actual correction
needed. Provided that the lines in the spectrum are sharp a linear correction
works very well, but for broad lines it is not so good. Attempting to use a
first-order phase correction on such spectra often results in distortions of the
baseline.
4.4 Sensitivity enhancement
Inevitably when we record a FID we also record noise at the same time. Some
of the noise is contributed by the amplifiers and other electronics in the spectrometer,
but the major contributor is the thermal noise from the coil used
to detect the signal. Reducing the noise contributed by these two sources
is largely a technical matter which will not concern use here. NMR is not
a sensitive technique, so we need to take any steps we can to improve the
signal-to-noise ratio in the spectrum. We will see that there are some manipulations
we can perform on the FID which will give us some improvement in
the signal-to-noise ratio (SNR).
By its very nature, the FID decays over time but in contrast the noise just
goes on and on. Therefore, if we carry on recording data for long after the FID
has decayed we will just measure noise and no signal. The resulting spectrum
will therefore have a poor signal-to-noise ratio.
(a) (b) (c)
Fig. 4.9 Illustration of the effect of the time spent acquiring the FID on the signal-to-noise ratio (SNR) in the
spectrum. In (a) the FID has decayed to next to nothing within the first quarter of the time, but the noise
carries on unabated for the whole time. Shown in(b) is the effect of halving the time spent acquiring the
data; the SNR improves significantly. In (c) we see that taking the first quarter of the data gives a further
improvement in the SNR.
This point is illustrated in Fig. 4.9 where we see that by recording the
FID for long after it has decayed all we end up doing is recording more noise
and no signal. Just shortening the time spent recording the signal (called
the acquisition time) will improve the SNR since more or less all the signal
is contained in the early part of the FID. Of course, we must not shorten the
acquisition time too much or we will start to miss the FID, which would result
in a reduction in SNR.
Sensitivity enhancement
Looking at the FID we can see that at the start the signal is strongest. As
time progresses, the signal decays and so gets weaker but the noise remains at
the same level. The idea arises, therefore, that the early parts of the FID are
4.4 Sensitivity enhancement 4–9
“more important” as it is here where the signal is the strongest.
This effect can be exploited by deliberately multiplying the FID by a function
which starts at 1 and then steadily tails away to zero. The idea is that this
function will cut off the later parts of the FID where the signal is weakest, but
leave the early parts unaffected.
weighting functionweighting functionoriginal FID
weighted FID weighted FID
(f) (g) (h)
(i) (j) (k)
(b)(a)
(d) (e)
(c)
Fig. 4.10 Illustration of how multiplying a FID by a decaying function (a weighting function) can improve
the SNR. The original FID is shown in (a) and the corresponding spectrum is (f). Multiplying the FID by a
weighting function (b) gives (c); Fourier transformation of (c) gives the spectrum (g). Note the improvement
in SNR of (g) compared to (f). Multiplying (a) by the more rapidly decaying weighting function (d) gives
(e); the corresponding spectrum is (h). The improvement in SNR is less marked. Spectra (f) – (h) are all
plotted on the same vertical scale so that the decrease in peak height can be seen. The same spectra
are plotted in (i) – (k) but this time normalized so that the peak height is the same; this shows most clearly
the improvement in the SNR.
A typical choice for this function – called a weighting function – is an
exponential:
W(t) = exp(−RLBt) (4.4)
where RLB is a rate constant which we are free to choose. Figure 4.10 illustrates
the effect on the SNR of different choices of this decay constant.
Spectrum (g) shows a large improvement in the SNR when compared to
spectrum (f) simply because the long tail of noise is suppressed. Using a more
rapidly decaying weighting function, (d), gives a further small improvement
in the SNR (h). It should not be forgotten that the weighting function also
acts on the signal, causing it to decay more quickly. As was explained in
section 4.2 a more rapidly decaying signal leads to a broader line. So, the use
4–10 Fourier transformation and data processing
of the weighting function will not only attenuate the noise it will also broaden
the lines; this is clear from Fig. 4.10.
Broadening the lines also reduces the peak height (remember that the integral
remains constant) – something that is again evident from Fig. 4.10.
Clearly, this reduction in peak height will reduce the SNR. Hence there is a
trade-off we have to make: the more rapidly decaying the weighting function
the more the noise in the tail of the FID is attenuated thus reducing the noise
level in the spectrum. However, at the same time a more rapidly decaying
function will cause greater line broadening, and this will reduce the SNR.
It turns out that there is an optimum weighting function, called the matched
filter.
Matched filter
Suppose that the FID can be represented by the exponentially decaying function
introduced in Eq. 4.2:
S(t) = S0 exp(i t) exp(−R2t).
The line in the corresponding spectrum is of width (R2/π) Hz. Let us apply a
weighting function of the kind described by Eq. 4.4 i.e. a decaying exponen-
tial:
W(t) × S(t) = exp(−RLBt) S0 exp(i t) exp(−R2t) .
The decay due to the weighting function on its own would give a linewidth
of (RLB/π) Hz. Combining the two terms describing the exponential decays
gives
W(t) × S(t) = S0 exp(i t) exp (−(RLB + R2)t) .
From this we see that the weighted FID will give a linewidth of
(RLB + R2)/π.
In words, the linewidth in the spectrum is the sum of the linewidths in the
original spectrum and the additional linebroadening imposed by the weighting
function.
It is usual to specify the weighting function in terms of the extra line
broadening it will cause. So, a “linebroadening of 1 Hz” is a function which
will increase the linewidth in the spectrum by 1 Hz. For example if 5 Hz of
linebroadening is required then (RLB/π) = 5 giving RLB = 15.7 s−1.
It can be shown that the best SNR is obtained by applying a weighting
function which matches the linewidth in the original spectrum – such a
weighting function is called a matched filter. So, for example, if the linewidth
is 2 Hz in the original spectrum, applying an additional line broadening of 2
Hz will give the optimum SNR.
We can see easily from this argument that if there is a range of linewidths
in the spectrum we cannot find a value of the linebroadening which is the
optimum for all the peaks. Also, the extra line broadening caused by the
matched filter may not be acceptable on the grounds of the decrease in resolution
it causes. Under these circumstances we may choose to use sufficient
line broadening to cut off the excess noise in the tail of the FID, but still less
than the matched filter.
4.5 Resolution enhancement 4–11
4.5 Resolution enhancement
We noted in section 4.2 that the more rapidly the time domain signal decays
the broader the lines become. A weighting function designed to improve the
SNR inevitably leads to a broadening of the lines as such a function hastens
the decay of the signal. In this section we will consider the opposite case,
where the weighting function is designed to narrow the lines in the spectrum
and so increase the resolution.
The basic idea is simple. All we need to do is to multiply the FID by a
weighting function which increases with time, for example a rising exponen-
tial:
W(t) = exp(+RREt) RRE > 0.
This function starts at 1 when t = 0 and then rises indefinitely.
×
× ×
=
=
(a)
(b) (e)
(f)
(g)
(h)
(i)
(j)
(a)
(c) (d)
Fig. 4.11 Illustration of the use of weighting functions to enhance the resolution in the spectrum. Note that
the scales of the plots have been altered to make the relevant features clear. See text for details.
The problem with multiplying the FID with such a function is that the
noise in the tail of the FID is amplified, thus making the SNR in the spectrum
very poor indeed. To get round this it is, after applying the positive exponential
we multiply by a second decaying function to “clip” the noise at the
4–12 Fourier transformation and data processing
tail of the FID. Usually, the second function is chosen to be one that decays
relatively slowly over most of the FID and then quite rapidly at the end – a
common choice is the Gaussian function:
W(t) = exp(−αt2
),
where α is a parameter which sets the decay rate. The larger α, the faster the
decay rate.
Lorentzian
Gaussian
Fig. 4.12 Comparison of the
Lorentzian and Gaussian
lineshapes; the two peaks have
been adjusted so that their peak
heights and widths at half height
are equal. The Gaussian is a
more “compact” lineshape.
The whole process is illustrated in Fig. 4.11. The original FID, (a), contains
a significant amount of noise and has been recorded well beyond the
point where the signal decays into the noise. Fourier transformation of (a)
gives the spectrum (b). If (a) is multiplied by the rising exponential function
plotted in (c), the result is the FID (d); note how the decay of the signal has
been slowed, but the noise in the tail of the FID has been greatly magnified.
Fourier transformation of (d) gives the spectrum (e); the resolution has clearly
been improved, but at the expense of a large reduction in the SNR.
Referring now to the bottom part of Fig. 4.11 we can see the effect of
introducing a Gaussian weighting function as well. The original FID (a) is
multiplied by the rising exponential (f) and the decaying Gaussian (g); this
gives the time-domain signal (h). Note that once again the signal decay has
been slowed, but the noise in the tail of the FID is not as large as it is in (d).
Fourier transformation of (h) gives the spectrum (i); the resolution has clearly
been improved when compared to (b), but without too great a loss of SNR.
Finally, plot (j) shows the product of the two weighting functions (f) and
(g). We can see clearly from this plot how the two functions combine together
to first increase the time-domain function and then to attenuate it at longer
times. Careful choice of the parameters RRE and α are needed to obtain the
optimum result. Usually, a process of trial and error is adopted.
If we can set RRE to cancel exactly the original decay of the FID then the
result of this process is to generate a time-domain function which only has
a Gaussian decay. The resulting peak in the spectrum will have a Gaussian
lineshape, which is often considered to be superior to the Lorentzian as it
is narrower at the base; the two lineshapes are compared in Fig. 4.12. This
transformation to a Gaussian lineshape is often called the Lorentz-to-Gauss
transformation.
Parameters for the Lorentz-to-Gauss transformation
The combined weighting function for this transformation is
W(t) = exp(RREt) exp(−αt2
).
The usual approach is to specify RRE in terms of the linewidth that it would
create on its own if it were used to specify a decaying exponential. Recall that
in such a situation the linewidth, L, is given by RRE/π. Thus RRE = π L.
The weighting function can therefore be rewritten
W(t) = exp(−π Lt) exp(−αt2
).
For compatibility with the linebroadening role of a decaying exponential it
is usual to define the exponential weighting function as exp(−π Lt) so that
4.6 Other weighting functions 4–13
a positive value for L leads to line broadening and a negative value leads to
resolution enhancement.
From Fig. 4.11 (j) we can see that the effect of the rising exponential and
the Gaussian is to give an overall weighting function which has a maximum
in it. It is usual to define the Gaussian parameter α from the position of this
maximum. A little mathematics shows that this maximum occurs at time tmax
given by:
tmax = −
Lπ
2α
(recall that L is negative) so that
α = −
Lπ
2tmax
.
We simply select a value of tmax and use this to define the value of α. On some
spectrometers, tmax is expressed as a fraction, f , of the acquisition time, tacq:
tmax = f tacq. In this case
α = −
Lπ
2 f tacq
.
4.6 Other weighting functions
Many other weighting functions have been used for sensitivity enhancement
and resolution enhancement. Perhaps the most popular are the sine bell are
variants on it, which are illustrated in Fig. 4.13.
tacq
tacq
/20
0 π/4π/8 π/2phase
Fig. 4.13 The top row shows sine bell and the bottom row shows sine bell squared weighting functions for
different choices of the phase parameter; see text for details.
The basic sine bell is just the first part of a sin θ for θ = 0 to θ = π;
this is illustrated in the top left-hand plot of Fig. 4.13. In this form the function
will give resolution enhancement rather like the combination of a rising
exponential and a Gaussian function (compare Fig. 4.11 (j)). The weighting
function is chosen so that the sine bell fits exactly across the acquisition time;
mathematically the required function is:
W(t) = sin
πt
tacq
.
4–14 Fourier transformation and data processing
The sine bell can be modified by shifting it the left, as is shown in
Fig. 4.13. The further the shift to the left the smaller the resolution enhancement
effect will be, and in the limit that the shift is by π/2 or 90◦ the function
is simply a decaying one and so will broaden the lines. The shift is usually
expressed in terms of a phase φ (in radians); the resulting weighting function
is:
W(t) = sin
(π − φ)t
tacq
+ φ .
Note that this definition of the function ensures that it goes to zero at tacq.
The shape of all of these weighting functions are altered subtly by squaring
them to give the sine bell squared functions; these are also shown in
Fig. 4.13. The weighting function is then
W(t) = sin2 (π − φ)t
tacq
+ φ .
Much of the popularity of these functions probably rests of the fact that
there is only one parameter to adjust, rather than two in the case of the
Lorentz-to-Gauss transformation.
4.7 Zero filling
Before being processed the FID must be converted into a digital form so that
it can be stored in computer memory. We will have more to say about this
in Chapter 5 but for now we will just note that in this process the signal is
sampled at regular intervals. The FID is therefore represented by a series
of data points. When the FID is Fourier transformed the spectrum is also
represented by a series of data points. So, although we plot the spectrum as a
smooth line, it is in fact a series of closely spaced points.
tacq
tacq
tacq
(a) (b) (c)
Fig. 4.14 Illustration of the results of zero filling. The FIDs along the top row have been supplemented
with increasing numbers of zeroes and so contain more and more data points. Fourier transformation
preserves the number of data points so the line in the spectrum is represented by more points as zeroes
are added to the end of the FID. Note that the FID remains the same for all three cases; no extra data has
been acquired.
This is illustrated in Fig. 4.14 (a) which shows the FID and the corresponding
spectrum; rather than joining up the points which make up the spectrum
we have just plotted the points. We can see that there are only a few data
points which define the line in the spectrum.
4.8 Truncation 4–15
If we take the original FID and add an equal number of zeroes to it, the
corresponding spectrum has double the number of points and so the line is
represented by more data points. This is illustrated in Fig. 4.14 (b). Adding a
set of zeroes equal to the number of data points is called “one zero filling”.
We can carry on with this zero filling process. For example, having added
one set of zeroes, we can add another to double the total number of data
points (“two zero fillings”). This results in an even larger number of data
points defining the line, as is shown in Fig. 4.14 (c).
Zero filling costs nothing in the sense that no extra data is required; it
is just a manipulation in the computer. Of course, it does not improve the
resolution as the measured signal remains the same, but the lines will be better
defined in the spectrum. This is desirable, at least for aesthetic reasons if
nothing else!
It turns out that the Fourier transform algorithm used by computer programs
is most suited to a number of data points which is a power of 2. So,
for example, 214 = 16384 is a suitable number of data points to transform,
but 15000 is not. In practice, therefore, it is usual to zero fill the time domain
data so that the total number of points is a power of 2; it is always an option,
of course, to zero fill beyond this point.
4.8 Truncation
In conventional NMR it is virtually always possible to record the FID until it
has decayed almost to zero (or into the noise). However, in multi-dimensional
NMR this may not be the case, simply because of the restrictions on the
amount of data which can be recorded, particularly in the “indirect dimension”
(see Chapter X for further details). If we stop recording the signal before
it has fully decayed the FID is said to be “truncated”; this is illustrated in
Fig. 4.15.
Fig. 4.15 Illustration of how truncation leads to artefacts (called sinc wiggles) in the spectrum. The FID
on the left has been recorded for sufficient time that it has decayed almost the zero; the corresponding
spectrum shows the expected lineshape. However, if data recording is stopped before the signal has fully
decayed the corresponding spectra show oscillations around the base of the peak.
4–16 Fourier transformation and data processing
As is shown clearly in the figure, a truncated FID leads to oscillations
around the base of the peak; these are usually called sinc wiggles or truncation
artefacts – the name arises as the peak shape is related to a sinc function. The
more severe the truncation, the larger the sinc wiggles. It is easy to show that
the separation of successive maxima in these wiggles is 1/tacq Hz.
Clearly these oscillations are undesirable as they may obscure nearby
weaker peaks. Assuming that it is not an option to increase the acquisition
time, the way forward is to apply a decaying weighting function to the FID so
as to force the signal to go to zero at the end. Unfortunately, this will have the
side effects of broadening the lines and reducing the SNR.
Highly truncated time domain signals are a feature of three- and higherdimensional
NMR experiments. Much effort has therefore been put into finding
alternatives to the Fourier transform which will generate spectra without
these truncation artefacts. The popular methods are maximum entropy, linear
prediction and FDM. Each has its merits and drawbacks; they all need to be
applied with great care.
4.9 Exercises 4–17
4.9 Exercises
E 4–1
In a spectrum with just one line, the dispersion mode lineshape might be acceptable
– in fact we can think of reasons why it might even be desirable
(what might these be?). However, in a spectrum with many lines the dispersion
mode lineshape is very undesirable – why?
E 4–2
Suppose that we record a spectrum with the simple pulse-acquire sequence
using a 90◦ pulse applied along the x axis. The resulting FID is Fourier transformed
and the spectrum is phased to give an absorption mode lineshape.
We then change the phase of the pulse from x to y, acquire an FID in the
same way and phase the spectrum using the same phase correction as above.
What lineshape would you expect to see in the spectrum; give the reasons for
your answer.
How would the spectrum be affected by: (a) applying the pulse about −x;
(b) changing the pulse flip angle to 270◦ about x?
E 4–3
The gyromagnetic ratio of phosphorus-31 is 1.08 × 108 rad s−1 T−1. This
nucleus shows a wide range of shifts, covering some 700 ppm.
Suppose that the transmitter is placed in the middle of the shift range and
that a 90◦ pulse of width 20 µs is used to excite the spectrum. Estimate the
size of the phase correction which will be needed at the edges of the spectrum.
(Assume that the spectrometer has a B0 field strength of 9.4 T).
E 4–4
Why is it undesirable to continue to acquire the FID after the signal has decayed
away?
How can weighting functions be used to improve the SNR of a spectrum?
In your answer described how the parameters of a suitable weighting function
can be chosen to optimize the SNR. Are there any disadvantages to the use of
such weighting functions?
E 4–5
Describe how weighting functions can be used to improve the resolution in
a spectrum. What sets the limit on the improvement that can be obtained in
practice? Is zero filling likely to improve the situation?
E 4–6
Explain why use of a sine bell weighting function shifted by 45◦ may enhance
the resolution but use of a sine bell shifted by 90◦ does not.
E 4–7
In a proton NMR spectrum the peak from TMS was found to show “wig-
4–18 Fourier transformation and data processing
gles” characteristic of truncation of the FID. However, the other peaks in the
spectrum showed no such artefacts. Explain.
How can truncation artefacts be suppressed? Mention any difficulties with
your solution to the problem.