4Fourier transformation and data
processing In the previous chapter we have en 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 .
frequency Fig.4.1Fourier 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 pha and lineshapes,which are cloly associated with
the Fourier transform and then go on to explore some uful manipulations of
NMR data such as nsitivity and resolution enhancement.
4.1The FID In ction 3.6we saw that the x and y components of the free induction sig-
牧童图片nal could be computed by thinking about the evolution of the magnetization
during the acquisition time.In that discussion we assumed that the magneti-
zation started out along the −y axis as this is where it would be rotated to by
a 90◦pul.For the purpos of this chapter we are going to assume that magnetization starts out along x ;we will e later that this choice of starting
position is esntially arbitrary.
Fig.4.2Evolution of the magnetization over time;the offt 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
4–2Fourier transformation and data processing
x Fig.4.3The x and y components of the signal can be thought of as arising from the rotation of a vector S0at frequency .
If the magnetization does indeed start along x then Fig.3.16needs to be redrawn,as is shown in Fig.4.2.From this we can easily e that the x and y components of the magnetization are:
M x=M0cos
t
M y=M0sin
t.
The signal that we detect is
proportional to the magnetizations.The con-
stant 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,S x(t)and设计质量
S y(t)as
x=S0cos t and S y(t)=S0sin t
where S0gives is the overall size of the signal and we have reminded ourlves
that the signal is a function of time by writing it as S x(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 S x and S y,
as is illustrated in Fig.4.3.
S
x
(
t
)
o
r
r
e
a
l
p
a
r
t
S
(
t
)
o
r
i
m
a
g
.
p
a
r
t
Fig.4.4Illustration of a typical
FID,showing the real and
imaginary parts of the signal;
both decay over time.
As a conquence of the way the Fourier transform works,it is also con-
venient to regard S x(t)and S y(t)as the real and imaginary parts of a complex
signal S(t):
S(t)=S x(t)+i S y(t)
=S0cos t+i S0sin t
=S0exp(i t).
We need not concern ourlves too much with the
but just note that the time-domain signal is complex,with the real and imag-
inary parts corresponding to the x and y components of the signal.
We mentioned at the start of this ction that the transver magnetization
decays over time,and this is most simply reprented by an exponential decay
with a time constant T2.The signal then becomes
S(t)=S0exp(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/T2and so S(t)becomes
S(t)=S0exp(i t)exp(−R2t).(4.2)
The shorter the time T2(or the larger the rate constant R2)the more rapidly
the signal decays.
4.2Fourier transformation
Fourier transformation of a signal such as that given in Eq.4.1gives the fre-
quency 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
4.2Fourier transformation 4–3part of the spectrum shows what we call an absorption mode line,in fact in
the ca of the exponentially decaying signal of Eq.4.1the line has a shape
known as a Lorentzian ,or to be preci 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.
frequency
Fig.4.6Illustration of the fact that the more rapidly the FID decays the broader the line in the corresponding
spectrum.A ries 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 decreas.
Fig.4.5Illustration 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
T 2)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;
the points are illustrated in Fig.4.6.
If the size of the time domain signal increas,for example by increasing
S 0the height of the peak increas in direct proportion.The obrvations
lead to the very important conquence that by integrating the lines in the
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spectrum we can determine the relative number of protons (typically)which
contribute to each.
The dispersion line shape is not one that we would choo to u.Not
only is it broader than the absorption mode,but it also has positive and nega-
tive parts.In a complex spectrum the 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 line-
shape which is traditionally ud to plot ESR spectra.Although the two
lineshapes do look roughly the same,they are not in fact related to one an-
other.
Positive and negative frequencies
As we discusd in ction 3.5,the evolution we obrve is at frequency
positive or negative and,as we will e 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 the with zero in the middle.
4–4Fourier transformation and data processing
Several lines
What happens if we have more than one line in the spectrum?In this ca,
as we saw in ction 3.5,the FID
will be the sum of contributions from each
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line.For example,if there are three lines
S (t )will be:S (t )=S 0,
1exp (i 1t )exp −t T (1)2 +S 0,2exp (i 2t )exp −t T (2)2 +S 0,3exp (i 3t )exp −t T (3)2
.where we have allowed each line to have a parate intensity,S 0,i ,frequency, i ,and relaxation time constant,T (i )2.如何开公司
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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 tho 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 the lines –which is the complete
spectrum,just as we require it.
4.3Pha So far we have that at time zero (i.e.at the start of the FID)S x (t )is a maximum and S y (t )is zero.However,in general this need not be the ca –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 pha shifted or that
it has a pha error .The situation is portrayed in Fig.4.7.
In Fig.4.7(a)we e the situation we had before,with the signal starting
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out along x and precessing towards y .The real part of the FID (corresponding
to S x )is a damped cosine wave and the imaginary part (corresponding to S y )
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 e the effect of a pha shift,φ,of 45◦.S y 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 pha shift is 90◦.Now it is S y which takes the form of a
damped cosine wave,whereas S x is a sine wave.The Fourier transform gives
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a spectrum in which the absorption mode signal now appears in the imaginary
part.Finally in (d)the pha shift is 180◦and this gives a negative absorption
mode signal in the real part of the spectrum.
What we e is that in general the appearance of the spectrum depends on
the position of the signal at time zero,that is on the pha of the signal at time
zero.Mathematically,inclusion of this pha shift means that the (complex)
signal becomes:S (t )=S 0exp (i φ)exp (i t )exp −t .(4.3)
4.3Pha
4–5S x S y S x
S y
S x S y S x S y
x
x
x x
real imag imag
imag
imag real real real (a)(b)(d)(c)Fig.4.7Illustration of the effect of a pha 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 pha shift,φ,of 45
◦;the real and
imaginary parts of the spectrum are now mixtures of absorption and dispersion.In (c)the pha shift is 90◦;now the absorption mode appears in the imaginary part of the spectrum.Finally in (d)the pha shift is 180◦giving a negative time It pears cannot be predicted,so in any practical situation there is an unknown pha 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 sorption lineshape.This is undesirable as for the best resolution we an absorption mode lineshape.Luckily,restoring the spectrum to the absorption mode is easy.with take the FID,reprented by Eq.4.3,and multiply it by exp (i φcorr ):exp (i φcorr )S (t )=exp (i φcorr )× S 0exp (i φ)exp (i t )exp −t T 2
.This is easy to do as by now the FID is stored in computer memory,so the
