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The maximum entropy method and fourier transformation compared
JOURNAL
OF MAGNETIC
RESONANCE
92,276-292
(
199 1)
The Maximum Entropy Method and Fourier Transformation Compared J.A. JONESANDP.J.HORE* Physical
Chemistv
Laboratory,
Oxford
University,
Oxford
OX1
3QZ, united
Kingdom
Received May 30, 1990; revised September 24, 1990 The relationship between NMR spectra obtained by the maximum entropy method and by conventional processing (Fourier transformation) is explored. In certain circumstances, the maximum entropy reconstruction is simply a nonlinearly amptied form of the Fourier transform spectrum and is therefore essentially worthless. More complex and interesting behavior is found under conditions more likely to be-met in practice. Using simple examples, it is argued that a maximum entropy reconstruction can reveal information that could not be obtained from a single Fourier transform spectrum. 0 1991 Academic Press, Inc. INTRODUCTION
In almost all NMR experiments, time-domain data are converted into frequencydomain spectra by discrete Fourier transformation. An alternative approach, which has attracted considerable interest recently, is the maximum entropy method ( l-35). Put simply, maximum entropy involves finding spectra that are consistent with the experimental data and rejecting all but the one with the minimum information content or equivalently the maximum entropy (2). By choosing the spectrum with the least structure one tries to ensure that it contains only features for which there is sufficient evidence in the data. The beauty of the method is that, unlike direct Fourier transformation, it can take into account known instrumental distortions and the fact that parts of the signal may be missing. Some of the original publications describing applications of maximum entropy to NMR contained optimistic statements about improvements in sensitivity or even simultaneous enhancement of sensitivity and resolution (3-6). Quite reasonably, such unsubstantiated claims were met with skepticism. Subsequent authors have been more circumspect, concentrating on matters other than sensitivity: truncated data ( 7-Z2), lineshape deconvolution (10, 13, 14), off-resonance effects (IS), exponential sampling ( 16, 17)) J deconvolution ( 18, 19), etc. Although the question of the quantitation of maximum entropy NMR spectra has yet to be seriously addressed, there now seems to be general agreement that the method has positive benefits over conventional processing in cases where data are incomplete or distorted in some well-defined way. A recent publication has reopened the debate on sensitivity enhancement by taking a radically different approach to the problem. Using a simple analytical approach, Hoch et al. (32) predict that the maximum entropy reconstruction of a free induction * To whom correspondence should be addressed. 0022-236419 1 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
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decay is, under certain circumstances, nothing more than a nonlinear amplification of the conventional Fourier transform spectrum. In practice, the amplification serves to suppress noise in regions of the spectrum where there are no significant signals without much affecting the noise where it really matters, on the peaks themselves. This result, if true more generally, implies that maximum entropy is a purely cosmetic operation and therefore largely worthless. In the following, we examine this prediction and demonstrate that while it is indeed correct under certain conditions, very different and much more interesting behavior is found in general. Furthermore, we argue that the conditions under which the two spectra are so trivially related are rather restrictive and unlikely to be important in practice. BACKGROUND
The principle (but not the practice) of the maximum entropy method as applied to NMR may be summarized as follows (see Ref. ( 20) for a more detailed description ) . Suppose we have an experimental free induction decay consisting of 7Vtnoisy, complex, time-domain data points dk, k = 1, 2, . . . , N,, from which we wish to obtain a spectrum. For convenience of notation, we arrange these points as a column vector d. The maximum entropy method seeks frequency-domain spectra x (which may be real or complex) that are consistent with the data d. This constraint is usually expressed in terms of a X2 parameter,
k=l
where y (yk, k = 1, 2, . . . , Nt) are the “mock data” predicted by the trial spectrum x and Ukis the statistical uncertainty in dk. A spectrum is judged to be acceptable when x2 equals the number of experimental data points, 2N, ( Nt real + Nt imaginary). In most NMR problems, the mock data are calculated from the trial spectrum (digitized at Nr regular intervals) by a simple linear operation, y = Rx, or Yk
=
2
Rk,X,,
k = 1, 2, . . . , N,,
J=l
where matrix R is independent of x. R is usually a complex Fourier transform but may also include convolution with, for example, an instrumental line-broadening function (10, 13). The idea is that R should transform a spectrum of the sort that might have been obtained from an ideal spectrometer into the corresponding free induction decay that would have been recorded on the imperfect spectrometer used to acquire the data. The final step is to choose, from among the acceptable spectra, the one with the minimum information content, that is, the maximum entropy (2). Entropy (S) is defined (36) in terms of a probability distribution p: SC -C
Pjlnpj.
[31
This expression may only be applied directly to NMR in the special case where all resonances have phase 0”. The real part of the complex NMR spectrum will then
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consist of positive, absorption-mode lines and, if suitably normalized, may be regarded as a probability distribution. Thus S =
-
2
141
(Xj/b)lIl(Xj/b),
j=l
where b, the background parameter, is a constant that takes care of the normalization (discussed in Ref. (13)). This entropy was used in most of the early applications of maximum entropy to NMR. When resonances of different phases are expected, a different approach is required: various entropies have been proposed on a variety of grounds (IO, 22, 22). The most satisfactory, in both theoretical and practical terms, is (30) s/2
= -,$,
{(Zj/b)lll(Zj/b
+
[l +
Zf/b2]“2)
- [l +
Zj/b2]1’2},
[51
where z is the modulus of a complex spectrum x (i.e., zj = I Xj () . The subscript indicates that S, ,2 is appropriate for spin-i nuclei. In fact, this expression is nothing more than the conventional entropy expression, Eq. [ 31, where p is the distribution of nuclear spin orientations (in a classical approach) or the spin density matrix (in a quantum mechanical approach) (30). Maximization of such an entropy subject to the usual constraints imposed by the data is exactly equivalent to maximization of Eq. [ 5 ] with respect to the real and imaginary parts of x, which represent the x and y components of the magnetization of the sample at the start of data acquisition. This new approach to the maximum entropy problem in NMR has been shown to cope satisfactorily with resonances of arbitrary phase (30, 33). To summarize, one seeks the (real or complex) spectrum x that maximizes the entropy (S or Sr ,2) subject to X2 = 24. This constrained maximum in the multidimensional space of the spectral intensities is located by an iterative procedure using search directions related to the gradients of the entropy and X2 with respect to the elements of x. An efficient numerical algorithm has been described by Skilling and Bryan (37). In a recent publication (32), Hoch considers a variety of entropies suitable for complex spectra. He shows that, under a restricted set of conditions, the x 2 constraint (Eq. [ 11) may be expressed in terms of the trial spectrum and the Fourier transform of the data. With this simplification, the maximum entropy problem can be solved analytically, entirely in the frequency domain, thus avoiding the necessity for a sophisticated and time-consuming iterative search algorithm. As described above, Hoch finds a particularly simple relation between the maximum entropy reconstruction of a free induction decay and its Fourier transform, for entropies defined in terms of complex spectra. Here we check these predictions using a Skilling/Bryan-type algorithm (kindly supplied by Dr. G. J. Daniell) and explore the relationship between the two types of spectra in the general case. As “experimental” data we use synthetic free induction decays with the general form
MAXIMUM
ENTROPY
dk = s a,exp[z$,]exp[2~i(k
AND FOURIER
- l)v,At]exp[-(k
279
TRANSFORMATION
- l)At/T]
+ ek, k=l,2
,...,
I?,,
[6]
that is, M complex sinusoids, exponentially damped with relaxation time T. The signal is digitized at Nt regular intervals, At; a,, &,, and Y, are respectively the amplitude, phase, and frequency of the nth oscillator; and the ek are random numbers (the noise) with real and imaginary parts drawn separately from a Gaussian distribution with standard deviation Q. The following dimensionless quantities are used for the frequency and relaxation time: v,At ( - 1 =Sv,At < f ) and T/ Trm, where TnD is the duration of the free induction decay, defined as N,At. Values of these parameters and others, for the various free induction decays used below, are summarized in Table 1. Fourier transform spectra are calculated using zero-filling so as to have the same digital resolution as the maximum entropy spectra with which they are compared. In the first three of the four examples discussed below, the simple -x In x entropy (Eq. [ 41) is used, the phases 4, are all zero, and the maximum entropy reconstruction is real. The final example involves resonances with different phases requiring S1,z and a complex reconstruction. First, we look at a simple example in which the data are not truncated and no resolution enhancement is required. A SIMPLE
EXAMPLE
Figure 1 compares spectra obtained from a synthesized free induction decay (No. 1, see Table 1) consisting of a single resonance digitized at 128 regular intervals ( Nt = 128). The free induction decay has no significant truncation ( TnD = 5 T) and its signal-to-noise ratio, defined as the maximum signal amplitude divided by the noise standard deviation, is 25. Conventional and maximum entropy spectra, both with Nr = 256 and with neither sensitivity nor resolution enhancement, are shown in Figs. la and 1b, respectively. TABLE 1 Free Induction Decays Number
M
a,
u,At
4.
TlTm
N
Nf
CT
1 2
1 3
256 1024
0.04 0.02
1.0
11
256
0.02
4
2
0 0 0 0 0 0 0 0 -*/2
128 128
3
0 -33/128 0 l/128 -l/l6 -l/16 0 -l/4 l/4
0.2 0.2
3
1 0.1 1.0 I.0 0.1 1.0 1.0 0.25 1.0
0.2
128
256
0.04
Note. M: number of resonances; an,v,, &: amplitude, frequency, phase of the nth resonance; T: relaxation time; TmD: duration of free induction decay; Nt: number of complex points in time domain; Nr: number of (real or complex) points in frequency domain; 0: standard deviation of the noise.
JONES AND HORE
280
Q
b
C
d
x2
-
FIG. 1. Spectra calculated from free induction decay 1 (see Table 1) . (a) Real part of the Fourier transform of the data after zero-filling from 128 to 256 complex points. (b) Maximum entropy reconstruction ( ak = 0.04, b = 10V8, T’, = co). (c) The difference between twice the conventional spectrum (a) and the maximum entropy reconstruction (b). (d) “Scissors” spectrum obtained from (a) by removing all intensity below a threshold level. Note the vertical expansions of (c) and (d) .
The two spectra differ in two obvious respects: the maximum entropy spectrum has less noise on the baseline and is approximately twice as intense as the conventionally processed spectrum. Signal-to-noise ratios, defined in the usual way (38), are 250 and 32 for maximum entropy and Fourier transformation, respectively. Apparent improvements of this kind have led some authors (3-6) to believe that maximum entropy delivers a sensitivity enhancement. In fact, the doubling is a red herring: it arises because Fourier transformation distributes signal intensity equally between the real and imaginary parts of the spectrum, while the maximum entropy algorithm forces everything into a real spectrum. The intensity doubling is essentially analogous to adding the real part of a conventional spectrum to the Hilbert transform of the imaginary part: neither process improves sensitivity. Similarly, the increase in signal-tonoise ratio is illusory; indeed the term has no real meaning for spectra obtained by nonlinear methods (38). Maximum entropy is very good at suppressing noise on the baseline, where there are no genuine signals, but very bad when it comes to the peaks themselves. This point is demonstrated by Fig. lc, which is the difference between twice the Fourier transform spectrum and the maximum entropy reconstruction. Around the position of the peak, the difference spectrum is almost noise-free (and only differs from zero because the maximum entropy spectrum is not quite twice the intensity of the conventional spectrum). This property of maximum entropy spectra can be revealed more clearly by plotting the intensity of each point in the maximum entropy reconstruction against the intensity of the corresponding point in the conventional spectrum (32). This is shown in Fig. 2 for the spectra discussed above (Figs. la and lb). All the points lie on a single smooth monotonically rising curve, indicating that the relationship between the two spectra is simply a nonlinear amplification. Thus, for any free induction decay, there exists a function 6 such that (32)
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0
Fourier
transform
intensity
xl00
FIG. 2. The intensity of points in the maximum entropy reconstruction of free induction decay 1 (Fig. lb) plotted against the intensity of the corresponding points in the conventional spectrum (Fig. la). The sloping line has gradient 2.
Xj =
6(J)~
[71
where f is the real part of the conventional Fourier transform spectrum. For small values of&, 6(A) is almost constant and close to zero, but beyond a certain threshold, 6(J) rises rapidly, becoming approximately linear with gradient close to 2. Ignoring the doubling, the operation of 6 on the conventional spectrum is very similar to what might be called “scissors processing,” in which a horizontal line is drawn across the spectrum skimming the more intense noise spikes, and everything below it is discarded. The result of this operation on Fig. la is shown in Fig. Id. The similarity to the maximum entropy reconstruction, Fig. lb, is striking. This simple relationship between the two types of spectra is always found, for synthetic and genuine data alike, irrespective of the number of peaks, provided three conditions are satisfied: (i) Nr = 2ZV,(i.e., as many points in the real spectrum as there are real + imaginary in the data); (ii) the noise level Uk is the same for all data points k; and (iii) the trial spectra x and the mock data y are related by Fourier transformation. These are identical to the conditions under which Hoch (32) was able to solve the maximum entropy problem analytically except that, for complex spectra, the first condition becomes: (i) Nr = Nt (i.e., as many complex points in the time as there are in the frequency domain). In fact, a fourth condition must be satisfied for real spectra: (iv) the free induction decay must not be significantly truncated (see the Appendix). To summarize, under the above conditions the maximum entropy method is purely cosmetic, producing a spectrum that is trivially related to the Fourier transform spec-
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trum, delivering neither resolution nor sensitivity enhancement, nor any other significant benefit. It is difficult to imagine reasons, apart perhaps from pedagogic, for preferring the maximum entropy reconstruction. This somewhat depressing result is in complete agreement with Hoch’s predictions (32). DISCUSSION
However, there is more to maximum entropy than the above would suggest. Conditions (i)- (iv) are rather restrictive and, as we now argue, unlikely to be acceptable in practice. We deal with the four conditions in turn. Condition(i). It is rare that one would be content to have no more points in the frequencydomain spectrum than in the time-domain data. In conventional processing, zero-filling is routinely used to increase digital resolution: this is especially important for truncated signals which would otherwise be very poorly digitized. Moreover, it is crucial (39, 40) to zero-fill at least once to take into account the absence of data prior to excitation (this point is discussed later). If this is not done, then one cannot focus attention on the real part of the conventional spectrum, as is customary, without losing the information present in the imaginary part. Finally, and more mundanely, Nr must normally be a power of 2 if a fast Fourier transform algorithm is to be used: zero-filling is routinely employed to achieve this. All these points apply equally to maximum entropy processing and require that Nr be greater than or equal to 2N,. Condition(ii). The standard deviation of the noise is usually uniform but is not necessarily so. For example, pulse breakthrough or acoustic ringing ( 11) may increase the uncertainty associated with the initial points in a free induction decay. Condition(iii). The desired spectrum is seldom related to the data by Fourier transformation alone. More commonly some degree of sensitivity or resolution enhancement is needed. Resolution enhancement in maximum entropy reconstructions is achieved by modifying the matrix (R) that relates the trial spectrum to the mock data. Specifically, R would first Fourier transform the trial spectrum and then multiply the resulting time-domain signal, point by point, by a weighting function, which might be a decaying exponential or, more satisfactorily, a damping function derived from a reference line in the spectrum (10, 13). Other applications of maximum entropy involve transformations in which R is not a simple Fourier transform, for example, J deconvolution, in which multiplet splittings are removed ( Z9), and a treatment of off-resonance effects, where R is based on the Bloch equations ( 1.5). Condition(iv). The most common case in which conventional processing is inadequate occurs when the time-domain data are truncated. This, almost certainly, is the single most common and important application of maximum entropy in NMR spectroscopy. It is certainly the aspect of maximum entropy processing that has generated the least controversy amongst NMR spectroscopists. In summary, experimental situations in which restrictions ( i ) - ( iv) are appropriate are likely to be few and far between. When these conditions are acceptable, it is probable that the Fourier transform spectrum would be more than adequate and more sophisticated data processing techniques redundant. We now examine, by means of two simple examples, the performance of maximum entropy under conditions more likely to be important in routine NMR.
MAXIMUM
ENTROPY RESOLUTION
AND
FOURIER
TRANSFORMATION
283
ENHANCEMENT
As mentioned above, maximum entropy spectra with enhanced resolution may be obtained by introducing a weighting function into transformation F?( 10). If F is the matrix that Fourier transforms a real spectrum into a complex time-domain signal, then, for resolution enhancement, R = WF, where W is a diagonal matrix whose elements are the weights for the Nt points in the mock data. For a free induction decay of the form of Eq. [ 61, a sensible choice of W would be a decaying exponential, Wjk= &jkexp[-(k-
l)At/T,],
[81
with the time constant, T,, matched to that of the data (i.e., T, = T). The idea is that trial spectra containing sharp lines are transformed by F into slowly decaying mock data, which in turn are weighted so that their decay parallels that of the actual data. To judge the ability of maximum entropy to enhance resolution, we must consider a spectrum consisting of at least two lines. This is because linewidth and resolution, terms which are often used interchangeably, are not necessarily related for spectra obtained by nonlinear methods (in the same way that signal-to-noise ratio and sensitivity are no longer synonymous (38)). Linewidth is a familiar and easily measured quantity; resolution, however, is more subtle and refers to the ability to separate overlapping resonances. This distinction may be illustrated by considering the nonlinear amplification of a doublet resonance in which the splitting is not resolved. We could imagine applying an amplification so nonlinear that the line is narrowed down to almost nothing. This operation will not cause the splitting to become apparent and so does not affect resolution. As an example of resolution enhancement by maximum entropy, we take a synthesized free induction decay (No. 2, Table 1) comprising a strong doublet and a weak singlet, plus noise. The splitting is chosen such that the two components of the doublet accumulate a phase difference of 27r by the end of the free induction decay. As before, the signal lasts five relaxation times ( TRD = 5 T) and is sampled at 128 regular intervals, but now Nr is chosen to be larger than 2N,. Figure 3 shows four spectra calculated from this free induction decay. Figure 3a is the conventional spectrum: the doublet is unresolved and the singlet just visible above the noise. Figure 3b is the maximum entropy reconstruction, with resolution enhancement, using as weighting function an exponential matched to the decay of the data, that is, Eq. [ 81 with T, = T = 0.2THD. The doublet splitting is now clear and the noise on the baseline essentially absent. Figure 3c is the Fourier transform spectrum, resolution enhanced by Lorentzian-to-Gaussian weighting: the exponential time constant was equal to T, and the Gaussian was chosen so as to make the linewidth of the doublet resonance similar to that in the maximum entropy spectrum. In this spectrum the singlet has all but disappeared into the noise. For comparison, Fig. 3d shows the Fourier transform spectrum with matched filtering for maximum sensitivity. Clearly, the maximum entropy reconstruction is not a nonlinear amplification of any of the Fourier transform spectra. In Figs. 3a and 3d, the doublet splitting is not resolved, and in Fig. 3c, the singlet is weaker than several of the noise spikes. The relationship between the spectra may be seen more clearly in Fig. 4. A plot of intensity
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FIG. 3. Spectra calculated from free induction decay 2 (see Table 1). (a) Real part of the Fourier transform of the data. (b) Maximum entropy reconstruction ( ak = 0.02, b = lo-‘, T, = T). (c) Real part of the Fourier transform of the data with Lorentzian-to-Gaussian resolution enhancement. (d) Real part of the Fourier transform of the data with matched filtering. Note the vertical expansions of(a), ( c) , and (d) Only half of each spectrum is displayed.
in the maximum entropy reconstruction against the corresponding intensity in the Fourier transform spectrum (without resolution enhancement) is shown in Fig. 4a. The points composing the small spur to the lower left correspond to the singlet while the major feature to the right-hand side is the doublet. The latter rises and then falls
a
+f *f+ ++ l
t
l
-1
4 Fourier
8 transform
12 intensity
16 xl000
-I
I
!
I
0 Fourier
I
,
I transform
,
,
2 intensity
,
,
3 xl00
FIG. 4. (a) The intensity of points in the maximum entropy reconstruction of free induction decay 2 (Fig. 3b) plotted against the intensity of the corresponding points in the conventional spectrum (Fig. 3a). (b) The same plotted against the resolution-enhanced conventional spectrum (Fig. 3~). The sloping line has gradient 2.
MAXIMUM
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285
in the diagram because the doublet is resolved by maximum entropy but not by Fourier transformation. More interesting is the plot of the maximum entropy reconstruction against the resolution-enhanced Fourier transform spectrum (Fig. 4b). The points in this diagram lie on or near two lines, one horizontal at zero intensity in the maximum entropy reconstruction and the other of gradient close to 2 running approximately through the origin. The former comprises baseline points which are noisy in the Fourier transform spectrum but near zero in the maximum entropy reconstruction. The latter is composed of points on the peaks, including some that are weaker than the more intense noise spikes in the Fourier transform spectrum. The scatter arises partly from the noise in the Fourier transform spectrum but also from the slightly different lineshapes in the two spectra. Thus, maximum entropy appears to be distinguishing between signal and noise. If true, this must be due to the inclusion of information on the shape and width of the resonances in the calculation. There is an obvious parallel here with matched filtering, in which knowledge of the lineshape is used to enhance sensitivity. Whether this apparent improvement is genuine is a matter currently under investigation. What can be said of Fig. 3b with reasonable confidence is that it reveals clearly, in a single spectrum, the existence of a weak singlet and a strong doublet with no evidence for further resonances. To obtain this information so convincingly by conventional methods, at least two Fourier transform spectra would be required-one with resolution enhancement to resolve the doublet splitting and another, preferably with sensitivity enhancement, to locate the singlet. There may be circumstances in which it will be advantageous to view a single spectrum containing all the information in the data, rather than several displaying different aspects of that information. TRUNCATION
The use of maximum entropy to circumvent truncation artifacts (sine wiggles) has been extensively described elsewhere ( 7-12). Briefly, one chooses Nf to be larger than 2N, (or Nt in the case of complex spectra) and the algorithm does the rest. Sine wiggles should not appear in the maximum entropy reconstruction because at no stage does one assume that all missing data points are equal to zero. To illustrate the difference between maximum entropy and Fourier transformation, a synthetic free induction decay was again used (No. 3, Table 1). As in the previous example, it comprises a strong doublet and a weak singlet, but now with severe truncation ( TFID = T). The splitting was chosen such that the two components of the doublet accumulate a phase difference of 11a/8 by the end of the data. Figure 5a shows the conventional spectrum with no weighting: the doublet is not resolved and the singlet cannot confidently be distinguished from the sine wiggles and the noise. Figure 5b is the same spectrum with strong exponential apodization: the singlet can just be discerned. The maximum entropy reconstruction (Fig. 5c), however, shows clear evidence for both features. Although the doublet components have different heights and widths (because of the noise), their integrated intensities are very similar. Once again there is no simple relationship between the maximum entropy and Fourier transform spectra, certainly nothing as simple as nonlinear amplification.
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FIG. 5. Spectra calculated from free induction decay 3 (see Table 1) . (a) Real part of the Fourier transform of the data. (b) Real part of the Fourier transform of the data with strong exponential apodization. (c) Maximum entropy reconstruction ( ak = 0.02, b = lo- 12, r, = T). Note the vertical expansions of (a) and(b).
DISPERSION-MODE
LINES
Broadly speaking the behavior described above for the -x In x entropy, Eq. [ 41, is also found for Si ,Z, Eq. [ 5 1. The maximum entropy reconstruction is indeed trivially related to the conventional spectrum (32) but only when conditions (i) to (iii) are satisfied. A new point concerning the handling of resonances of different phases arises when the spectra are complex. Ho&s treatment (32) shows that the nonlinear transformation 6, Eq. [ 71, should apply equally to all lines in the spectrum, whatever their phase. A consequence of this is that “maximum entropy reconstruction will not reduce a dispersive peak relative to an absorptive peak having smaller height” (32). This is certainly correct if conditions (i)-( iii) are met. However, when Nr 3 2N, very different behavior is found. At this point it is useful to consider briefly why dispersion-mode lines appear at all in conventional NMR spectra. At first sight they would seem to be inescapable, but they are simply artifacts of the way in which signals are normally recorded and processed. In most NMR experiments, the data are inherently one-sided, that is, signal is recorded for positive times (after excitation) but not for negative times (before excitation). It is the assumption that these missing data points are equal to zero (i.e., zero-filling) that produces dispersive contributions in the conventional spectrum. There is a clear parallel here with the conventional treatment of truncated data which leads to sine wiggles. To avoid dispersion-mode lineshapes, one needs a double-sided time-domain signal with an envelope symmetrical about time t = 0. Although not generally available experimentally, such signals can be generated by maximum entropy reconstructions provided Nf 2 2N,. Thus, a complex trial spectrum containing only absorption-mode lines would be transformed (by R ) into double-sided mock data, half of which would correspond to t < 0 and half to t > 0. The first Nt complex points of the latter are compared, point by point, with the actual data to determine X2. Thus, if Fourier transformation of a free induction decay such as Eq. [ 6 ] gives a spectrum in which
MAXIMUM
ENTROPY
AND FOURIER
real and imaginary parts are linear combinations dispersion-mode, D( v,) , lineshapes,
TRANSFORMATION
of absorption-mode,
C [A(h) + WbJlexp(k%A n
maximum
287
A( v,), and 191
entropy should produce a spectrum of the form 2
C A(hJexp(&),
1101
n
in which all the intensity of each resonance is concentrated into the absorption-mode line. In fact a small amount of dispersive contribution cannot be avoided, as discussed in Ref. (30). Figure 6 shows spectra obtained from a synthetic free induction decay (No. 4, Table 1) consisting of two resonances, one strong with phase -90” and the other weak with phase 0”. In the real part of the conventional spectrum (Fig. 6a) the absorption-mode line associated with the weaker resonance is smaller than the dispersion-mode peak of the stronger resonance. As shown in Fig. 6b, maximum entropy reconstruction using S1,2 can, and does, suppress the large dispersion-mode line in the real part of the spectrum relative to the smaller absorption-mode peak. Once again, plotting one spectrum against the other reveals what is happening (Fig, 7). The points lie on two lines, one for the absorption-mode peaks (gradient 1.6) and the other, with smaller gradient (0.27)) for the dispersion-mode lines, showing that maximum entropy processing does make a distinction between absorption-mode and dispersion-mode resonances. CONCLUSIONS
In the preceding pages we have compared the maximum entropy method with Fourier transformation, the conventional method of NMR data processing. In agreement with Hoch (32) we find that the maximum entropy reconstruction of a free
FIG. 6. Spectra calculated from free induction decay 4 (see Table 1). (a) Real (R) and imaginary (I) parts of the Fourier transform of the data. (b) Real and imaginary parts of the maximum entropy reconstruction (Ok = 0.04, b = lo-‘*, r, = co).
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al E
-.062 g--.osE -.lOs E .,-.12t -.14-.16-
/ I/
6
Fourier transform intensity FIG. 7. Intensity of points in the maximum entropy reconstruction of free induction decay 4 (Fig. 6b) plotted against the intensity of the corresponding points in the Fourier transform spectrum (Fig. 6a). The sloping lines have gradients 1.6 (absorption mode) and 0.27 (dispersion mode).
induction decay is nothing more than a simple nonlinear transformation of the conventional spectrum when a number ofconditions are met. These are: (i) that there is the same number of points in the spectrum as in the free induction decay; (ii) that the statistical uncertainty is the same for all data points; (iii) that the operation that relates the trial spectra to the corresponding mock data (R, Eq. [ 21) is simply a Fourier transform; and (iv) that the data are not significantly truncated (for real maximum entropy reconstructions only). The nature of the relationship between the two sorts of spectra is such that the maximum entropy reconstruction is essentially worthless. Under conditions more likely to be encountered in practice, we find no such correspondence between the maximum entropy and the Fourier transform spectra. Hoch’s suspicion (32) that “many of the qualitative features exhibited by the reconstructions in the special case” (i.e., under conditions (i)-( iii)) “are also present in the more general case” seems to be unduly pessimistic. A maximum entropy reconstruction can reveal spectral features that cannot be seen in a single conventional spectrum as illustrated by Figs. 3 and 5. Weak resonances can be detected above the baseline noise in the same spectrum in which small splittings in other peaks are resolved. To obtain the same information by conventional processing, two spectra would be needed, one with matched filtering to detect the weak signal and a second with resolution enhancement (e.g., Lorentzian-to-Gaussian conversion) to discern the splitting.
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289
TRANSFORMATION
It should be said, at this point, that the examples used here are in many ways ideal casesfor showing off the advantages of maximum entropy. They involve small numbers of relatively sharp resonances; the lineshapes are ideal Lorentzians, all with the same (known) width; the signal-to-noise ratios are quite high and the dynamic range fairly low; and the noise has an ideal Gaussian distribution, with no spurious correlations. In practice, there are ways to achieve some of these advantages with genuine data. For instance, the dynamic range and the number of resonances can be reduced by processing only a portion of the total spectrum (“zooming”). Linewidths can be measured, and it is possible to deconvolve the overall instrument response function, determined from some well-isolated peak ( 13, 34). Nevertheless, the application of maximum entropy to genuine data will seldom work out quite as nicely as in the examples discussed here. Whether Figs. 3b and 5c constitute an improvement in sensitivity or resolution is doubtful and is not a problem that this paper addresses. In our experience, it has proved impossible to reveal by maximum entropy a weak resonance that could not be located in the matched-filtered conventional spectrum. Similarly, we have not yet been able to resolve using maximum entropy a splitting that could not also be seen in a Fourier transform spectrum with Lorentzian-to-Gaussian weighting. If one desires a single, general-purpose spectrum displaying all the qualitative information present in the data, then one could do far worse than the maximum entropy reconstruction. Usually there will be better ways of extracting quantitative information, for instance, curve fitting. APPENDIX
The purpose of this Appendix is to demonstrate that condition (iv) must be satisfied (i.e., the free induction decay must not be significantly truncated) if the conventional spectrum and the reaZ maximum entropy reconstruction are to be related as in Eq. [71.
We start with the identity, which is valid for all sets of complex numbers dk (k = 0, l)andf,(n=0,...,2NI),
. . . ) 2N2N-I
C
2N-1
I dk -
k=O
2 fn exp(2hk/2N)
I2
n=O 2N-I
1
= 2N Rzo 2~
2N-1
2
dkexp(-2?rink/2N)
-h2,
L-411
k=O
which follows immediately from Parseval’s theorem for the discrete Fourier transform. Now assume that the fn (n = 0, . . . , 2N- l)arerealandthatdZN-k= dz (k= 1, . . . ) N - 1) . The left-hand side (L) of Eq. [Al] can now be written as 2N-1
L=
Ido-
c f,j2+2 n=O
N-l
2 k=l
ZN-I
ldk-
2 f,exp(2aink/2N)12 n=O 2N-I +
I&-
c
n=O
(-l)“fn12.
[A21
290
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dk (k = 0, . . . ) N - 1) represent the data in an NMR experiment, and let fn (n = 0 3 1 * * 2 2N - 1) be a real trial spectrum. Now X2 is given by
Let
N-l
ZN-1
x2 = C ldk-
2 f,exp(2aink/2N)12/&.
k=O
[A31
n=O
To simplify matters we let ffk = u for all k (0, . . . , N - 1) and set the imaginary part of the first data point, do, equal to zero. The latter is permissible because Im( do) simply determines the integral of the imaginary part of the spectrum, and we are only interested in obtaining real spectra. In practice, for data in which all resonances have phase O”, this point will be close to zero anyway. Combining [A21 and [A31 gives 2N-I
2a2X2 = L + [Re(do) -
2N-I
2 fn]* - IdNn=O
c
(-l)“fn]‘.
n=O
[A41
Turning to the right-hand side (R) of Eq. [A 11, we have, from the conjugate symmetry Of& 2N-1
dkexp( -2?rink/2N)
c k=O
N-l
= Re(do) + 2 Re c
+ (-l)“dN
dk exp(-2rink/2N)
[A51
k=l
and hence ZN-1
R=2N
whereD,(n=0,...,2N-
c
n=O
2
[A61
$ do + Ni’ dkexp(-2?rink/2N) I
D,, is the conventional Fourier transformation. = -$ ‘5’
(-1)”
1)isdefinedby D, = ;Re
X2
D,-s,+FdN,
.
spectrum produced from do, . . . , dN-, by zero-filling and Combining Eqs. [A 11, [ A4], [ A6], and [ A7 ] gives
D, -fn + $$
dN* + $
n=O
[Re(do) - 2E1h]2 n=o -$
,d,-‘;l
(-l)“fn,’ n=O
Maximum
respect
[A81
entropy processing involves maximizing Q=S-Ax’
with
[A71
1
k=l
[A91
tofk (i.e., solving aQ/afk = 0), where s = -c
(fk/b)ln(fk/b) k
[A101
MAXIMUM
ENTROPY
and X is a Lagrange multiplier 1 /b + ( 1/b)ln(f,/b)
AND FOURIER
such that Dk -f
- 5:
k
x2
TRANSFORMATION
291
= 2N. Thus
+ (-l)kWdN) 2N
-$[Re(&-2~1~]+-$[Re(&)-2~1(-l)~~](-l)k=0 n=O
[All] n=O
and hence
&=h+&j 11+ Whlb)l
- &VW&)
- 251Ll n=O C-1) k 2N-I -2~ c (-l)"fn. n=O
[A121
Thus the conventional spectrum Dk is a nonlinear mapping of the maximum entropy spectrum fk, but, because of the final term in Eq. [A 121, the mapping is different for the even and odd points. When this term is zero, the simple form shown in Eq. [ 71 is recovered. Now C,, (- 1)” fn is just the Nth point of the mock data predicted by the maximum entropy spectrum fn. The corresponding point in the actual data, dN, is not actually measured ( dN-, is the last recorded data point). Now since the mock data calculated from the maximum entropy reconstruction will resemble the actual data, we can expect the final term in Eq. [A 121 to be small if dN is itself small, which will only be true in general if the free induction decay is not signijkantly truncated. This conclusion should not be surprising. For a truncated free induction decay d(t), the conventional spectrum D, (n = 0, . . . , N - 1) is not a good approximation to the Fourier integral +CC
--oo d( t)e-jw’dt, s
[A131
which cannot be evaluated because d( t ) is measured for only a limited range oft . The conventional spectrum is consequently a distorted version of the desired spectrum, containing truncation artifacts. The maximum entropy reconstruction, however, is essentially free from such spurious signals and so much closer to the “true” spectrum. We are indebted to Dr. G. J. Daniel1 for this proof. ACKNOWLEDGMENTS It is a pleasure to thank Dr. G. J. Daniel1 for invaluable advice and guidance on many occasions and for generously supplying maximum entropy software. We are grateful to Dr. J. C. Hoch for sending a copy of Ref. (32) prior to publication and to a referee for unusually detailed and helpful suggestions. J.A.J. thanks St. John’s College, Oxford, for a North Senior Scholarship and the Science and Engineering Research Council for a Research Studentship. P.J.H. is a member of the Oxford Centre for Molecular Sciences. REFERENCES 1. D. S. STEPHENSON, Prog. NMR 2. S. F. GULL AND G. J. DANIELL,
Spectrosc. 20, 5 15 ( 1988). Nature (London) 272,686 ( 1978).
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3. S. SIBISI, Nature (London) 301, 134 ( 1983). 4. S. SIBISI, J. SKILLING, R. G. BRERETON, E. D. LAUE, AND J. STAUNTON, Nature (London) 311, 446 (1984). 5. F. NI, G. C. LEVY, AND H. SCHERAGA, J. Magn. Resort. 66,385 (1986). 6. A. R. MAZZEO AND G. C. LEVY, Comput. Enhanced Spectrosc. 3, 165 ( 1986). 7. P. J. HORE, J. Magn. Reson. 62, 561 ( 1985). 8. J. C. HOCH, J. Magn. Reson. 64,436 (1985). 9. J. F. MARTIN, J. Magn. Reson. 65,291 (1985). 10. E. D. LAUE, M. R. MAYGER, J. SKILLING, AND J. STAUNTON, .I. Magn. Resort. 68, 14 ( 1986). Il. E. D. LAUE, K. 0. B. POLLARD, J. SKILLING, J. STAUNTON, AND A. C. SUTKOWSKI, J. Magn. Reson. 72,493 (1987). 12. A. HEUER AND U. HAEBERLEN, J. Magn. Reson. 85,79 ( 1989). 13. S. J. DAVIES, C. BAUER, P. J. HORE, AND R. FREEMAN, J. Magn. Reson. 76,476 (1988). 14. A. I. GRANT AND K. J. PACKER, in “Maximum Entropy and Bayesian Methods” (J. Skilling, Ed.), pp. 251-259, Kluwer Academic, Dordrecht, 1989. 15. P. J. HORE AND G. J. DANIELL, J. Magn. Reson. 69,386 ( 1986). 16. J. C. J. BARNA, E. D. LAUE, M. R. MAYGER, J. SKILLING, AND S. J. P. WORRALL, J. Magn. Reson. 73,69(1987). 17. J. C. J. BARNA AND E. D. LAUE, J. Magn. Reson. 75,384 ( 1987). 18. R. A. JACKSON, J. Magn. Reson. 75, 174 (1987). 19. M. A. DEL.XJC AND G. C. LEVY, J. Magn. Reson. 76,306 ( 1988). 20. E. D. LAUE, J. SKILLING, J. STAUNTON, S. SIBISI, AND R. G. BRERETON, J. Magn. Reson. 62, 437 (1985). 21. E. D. LAUE, J. SKILLING, AND J. STAUNTON, J. Magn. Reson. 63,418 ( 1985). 22. K.M. WRIGHT AND P. S.BELTON,MOL Phys.58,485 (1986). 23. J. C. J. BARNA, E. D. LAUE, M. R. MAYGER, J. SKILLING, AND S. J. P. WORRALL, Biochem. Sot. Trans. 14, 1262 (1986). 24. M. A. DELSUC, F. NI, AND G. C. LEVY, J. Magn. Reson. 73, 548 (1987). 25. J. C. J. BARNA AND E. D. LAUE, Lab. Practice 36, 102 ( 1987). 26. M. L. WALLER AND P. S. TORTS, Magn. Reson. Med. 4,385 ( 1987). 27. R. H. NEWMAN, J. Magn. Reson. 79,448 ( 1988). 28. A. R. MAZZEO, M. A. DELSUC, A. KUMAR, AND G. C. LEW, J. Magn. Reson. 81, 512 ( 1989). 29. T. N. HUCKERBY, I. A. NIEDUSZYNSKI, G. H. COCKIN, J. M. DICKENSON, H. MORRIS, P. N. SANDERSON, AND D. J. THORNTON, Eur. Polym. J. 25,861 ( 1989). 30. G. J. DANIELL AND P. J. HORE, J. Magn. Reson. 84,5 15 ( 1989). 31. G. J. DANIELL AND P. J. HORE, in “Maximum Entropy and Bay&an Methods” (J. Skilling, Ed.), pp. 297-302, Kluwer Academic, Dordrecht, 1989. 32. J. C. HOCH, A. S. STERN, D. L. DONOHO, AND I. M. JOHNSTONE, J. Magn. Reson. 86,236 ( 1990). 33. P. J. HORE, D. S. GRAINGER, S. WIMPERIS, AND G. J. DANIELL, J. Magn. Reson. 89,415 (1990). 34. P. J. HORE, in “Maximum Entropy in Action” (B. Buck and V. A. Macaulay, &Is.), Oxford Univ. Press, Oxford, pp. 4 l-72. 35. S. DAVIES, K. J. PACKER, A. BARUYA, AND A. J. GRANT, in “Maximum Entropy in Action” (B. Buck and V. A. Macaulay, Eds.), Oxford Univ. Press, Oxford, in press. 36. C. E. SHANNON, Bell Syst. Tech. J. 27, 379 ( 1948). 37. J. SKILLING AND R. K. BRYAN, Mon. Not. R. Astron. Sot. 211, 111 ( 1984). 38. R. FREEMAN, “A Handbook of Nuclear Magnetic Resonance,” pp. 2 16-229, Longman, Harlcw, 1988. 39. E. BARTHOLDI AND R. R. ERNST, J. Magn. Reson. 11,9 (1973). 40. R. FREEMAN, “A Handbook of Nuclear Magnetic Resonance,” pp. 302-306, Longmap, Harlow, 1988.
OF MAGNETIC
RESONANCE
92,276-292
(
199 1)
The Maximum Entropy Method and Fourier Transformation Compared J.A. JONESANDP.J.HORE* Physical
Chemistv
Laboratory,
Oxford
University,
Oxford
OX1
3QZ, united
Kingdom
Received May 30, 1990; revised September 24, 1990 The relationship between NMR spectra obtained by the maximum entropy method and by conventional processing (Fourier transformation) is explored. In certain circumstances, the maximum entropy reconstruction is simply a nonlinearly amptied form of the Fourier transform spectrum and is therefore essentially worthless. More complex and interesting behavior is found under conditions more likely to be-met in practice. Using simple examples, it is argued that a maximum entropy reconstruction can reveal information that could not be obtained from a single Fourier transform spectrum. 0 1991 Academic Press, Inc. INTRODUCTION
In almost all NMR experiments, time-domain data are converted into frequencydomain spectra by discrete Fourier transformation. An alternative approach, which has attracted considerable interest recently, is the maximum entropy method ( l-35). Put simply, maximum entropy involves finding spectra that are consistent with the experimental data and rejecting all but the one with the minimum information content or equivalently the maximum entropy (2). By choosing the spectrum with the least structure one tries to ensure that it contains only features for which there is sufficient evidence in the data. The beauty of the method is that, unlike direct Fourier transformation, it can take into account known instrumental distortions and the fact that parts of the signal may be missing. Some of the original publications describing applications of maximum entropy to NMR contained optimistic statements about improvements in sensitivity or even simultaneous enhancement of sensitivity and resolution (3-6). Quite reasonably, such unsubstantiated claims were met with skepticism. Subsequent authors have been more circumspect, concentrating on matters other than sensitivity: truncated data ( 7-Z2), lineshape deconvolution (10, 13, 14), off-resonance effects (IS), exponential sampling ( 16, 17)) J deconvolution ( 18, 19), etc. Although the question of the quantitation of maximum entropy NMR spectra has yet to be seriously addressed, there now seems to be general agreement that the method has positive benefits over conventional processing in cases where data are incomplete or distorted in some well-defined way. A recent publication has reopened the debate on sensitivity enhancement by taking a radically different approach to the problem. Using a simple analytical approach, Hoch et al. (32) predict that the maximum entropy reconstruction of a free induction * To whom correspondence should be addressed. 0022-236419 1 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
276
MAXIMUM
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AND
FOURIER
TRANSFORMATION
277
decay is, under certain circumstances, nothing more than a nonlinear amplification of the conventional Fourier transform spectrum. In practice, the amplification serves to suppress noise in regions of the spectrum where there are no significant signals without much affecting the noise where it really matters, on the peaks themselves. This result, if true more generally, implies that maximum entropy is a purely cosmetic operation and therefore largely worthless. In the following, we examine this prediction and demonstrate that while it is indeed correct under certain conditions, very different and much more interesting behavior is found in general. Furthermore, we argue that the conditions under which the two spectra are so trivially related are rather restrictive and unlikely to be important in practice. BACKGROUND
The principle (but not the practice) of the maximum entropy method as applied to NMR may be summarized as follows (see Ref. ( 20) for a more detailed description ) . Suppose we have an experimental free induction decay consisting of 7Vtnoisy, complex, time-domain data points dk, k = 1, 2, . . . , N,, from which we wish to obtain a spectrum. For convenience of notation, we arrange these points as a column vector d. The maximum entropy method seeks frequency-domain spectra x (which may be real or complex) that are consistent with the data d. This constraint is usually expressed in terms of a X2 parameter,
k=l
where y (yk, k = 1, 2, . . . , Nt) are the “mock data” predicted by the trial spectrum x and Ukis the statistical uncertainty in dk. A spectrum is judged to be acceptable when x2 equals the number of experimental data points, 2N, ( Nt real + Nt imaginary). In most NMR problems, the mock data are calculated from the trial spectrum (digitized at Nr regular intervals) by a simple linear operation, y = Rx, or Yk
=
2
Rk,X,,
k = 1, 2, . . . , N,,
J=l
where matrix R is independent of x. R is usually a complex Fourier transform but may also include convolution with, for example, an instrumental line-broadening function (10, 13). The idea is that R should transform a spectrum of the sort that might have been obtained from an ideal spectrometer into the corresponding free induction decay that would have been recorded on the imperfect spectrometer used to acquire the data. The final step is to choose, from among the acceptable spectra, the one with the minimum information content, that is, the maximum entropy (2). Entropy (S) is defined (36) in terms of a probability distribution p: SC -C
Pjlnpj.
[31
This expression may only be applied directly to NMR in the special case where all resonances have phase 0”. The real part of the complex NMR spectrum will then
278
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consist of positive, absorption-mode lines and, if suitably normalized, may be regarded as a probability distribution. Thus S =
-
2
141
(Xj/b)lIl(Xj/b),
j=l
where b, the background parameter, is a constant that takes care of the normalization (discussed in Ref. (13)). This entropy was used in most of the early applications of maximum entropy to NMR. When resonances of different phases are expected, a different approach is required: various entropies have been proposed on a variety of grounds (IO, 22, 22). The most satisfactory, in both theoretical and practical terms, is (30) s/2
= -,$,
{(Zj/b)lll(Zj/b
+
[l +
Zf/b2]“2)
- [l +
Zj/b2]1’2},
[51
where z is the modulus of a complex spectrum x (i.e., zj = I Xj () . The subscript indicates that S, ,2 is appropriate for spin-i nuclei. In fact, this expression is nothing more than the conventional entropy expression, Eq. [ 31, where p is the distribution of nuclear spin orientations (in a classical approach) or the spin density matrix (in a quantum mechanical approach) (30). Maximization of such an entropy subject to the usual constraints imposed by the data is exactly equivalent to maximization of Eq. [ 5 ] with respect to the real and imaginary parts of x, which represent the x and y components of the magnetization of the sample at the start of data acquisition. This new approach to the maximum entropy problem in NMR has been shown to cope satisfactorily with resonances of arbitrary phase (30, 33). To summarize, one seeks the (real or complex) spectrum x that maximizes the entropy (S or Sr ,2) subject to X2 = 24. This constrained maximum in the multidimensional space of the spectral intensities is located by an iterative procedure using search directions related to the gradients of the entropy and X2 with respect to the elements of x. An efficient numerical algorithm has been described by Skilling and Bryan (37). In a recent publication (32), Hoch considers a variety of entropies suitable for complex spectra. He shows that, under a restricted set of conditions, the x 2 constraint (Eq. [ 11) may be expressed in terms of the trial spectrum and the Fourier transform of the data. With this simplification, the maximum entropy problem can be solved analytically, entirely in the frequency domain, thus avoiding the necessity for a sophisticated and time-consuming iterative search algorithm. As described above, Hoch finds a particularly simple relation between the maximum entropy reconstruction of a free induction decay and its Fourier transform, for entropies defined in terms of complex spectra. Here we check these predictions using a Skilling/Bryan-type algorithm (kindly supplied by Dr. G. J. Daniell) and explore the relationship between the two types of spectra in the general case. As “experimental” data we use synthetic free induction decays with the general form
MAXIMUM
ENTROPY
dk = s a,exp[z$,]exp[2~i(k
AND FOURIER
- l)v,At]exp[-(k
279
TRANSFORMATION
- l)At/T]
+ ek, k=l,2
,...,
I?,,
[6]
that is, M complex sinusoids, exponentially damped with relaxation time T. The signal is digitized at Nt regular intervals, At; a,, &,, and Y, are respectively the amplitude, phase, and frequency of the nth oscillator; and the ek are random numbers (the noise) with real and imaginary parts drawn separately from a Gaussian distribution with standard deviation Q. The following dimensionless quantities are used for the frequency and relaxation time: v,At ( - 1 =Sv,At < f ) and T/ Trm, where TnD is the duration of the free induction decay, defined as N,At. Values of these parameters and others, for the various free induction decays used below, are summarized in Table 1. Fourier transform spectra are calculated using zero-filling so as to have the same digital resolution as the maximum entropy spectra with which they are compared. In the first three of the four examples discussed below, the simple -x In x entropy (Eq. [ 41) is used, the phases 4, are all zero, and the maximum entropy reconstruction is real. The final example involves resonances with different phases requiring S1,z and a complex reconstruction. First, we look at a simple example in which the data are not truncated and no resolution enhancement is required. A SIMPLE
EXAMPLE
Figure 1 compares spectra obtained from a synthesized free induction decay (No. 1, see Table 1) consisting of a single resonance digitized at 128 regular intervals ( Nt = 128). The free induction decay has no significant truncation ( TnD = 5 T) and its signal-to-noise ratio, defined as the maximum signal amplitude divided by the noise standard deviation, is 25. Conventional and maximum entropy spectra, both with Nr = 256 and with neither sensitivity nor resolution enhancement, are shown in Figs. la and 1b, respectively. TABLE 1 Free Induction Decays Number
M
a,
u,At
4.
TlTm
N
Nf
CT
1 2
1 3
256 1024
0.04 0.02
1.0
11
256
0.02
4
2
0 0 0 0 0 0 0 0 -*/2
128 128
3
0 -33/128 0 l/128 -l/l6 -l/16 0 -l/4 l/4
0.2 0.2
3
1 0.1 1.0 I.0 0.1 1.0 1.0 0.25 1.0
0.2
128
256
0.04
Note. M: number of resonances; an,v,, &: amplitude, frequency, phase of the nth resonance; T: relaxation time; TmD: duration of free induction decay; Nt: number of complex points in time domain; Nr: number of (real or complex) points in frequency domain; 0: standard deviation of the noise.
JONES AND HORE
280
Q
b
C
d
x2
-
FIG. 1. Spectra calculated from free induction decay 1 (see Table 1) . (a) Real part of the Fourier transform of the data after zero-filling from 128 to 256 complex points. (b) Maximum entropy reconstruction ( ak = 0.04, b = 10V8, T’, = co). (c) The difference between twice the conventional spectrum (a) and the maximum entropy reconstruction (b). (d) “Scissors” spectrum obtained from (a) by removing all intensity below a threshold level. Note the vertical expansions of (c) and (d) .
The two spectra differ in two obvious respects: the maximum entropy spectrum has less noise on the baseline and is approximately twice as intense as the conventionally processed spectrum. Signal-to-noise ratios, defined in the usual way (38), are 250 and 32 for maximum entropy and Fourier transformation, respectively. Apparent improvements of this kind have led some authors (3-6) to believe that maximum entropy delivers a sensitivity enhancement. In fact, the doubling is a red herring: it arises because Fourier transformation distributes signal intensity equally between the real and imaginary parts of the spectrum, while the maximum entropy algorithm forces everything into a real spectrum. The intensity doubling is essentially analogous to adding the real part of a conventional spectrum to the Hilbert transform of the imaginary part: neither process improves sensitivity. Similarly, the increase in signal-tonoise ratio is illusory; indeed the term has no real meaning for spectra obtained by nonlinear methods (38). Maximum entropy is very good at suppressing noise on the baseline, where there are no genuine signals, but very bad when it comes to the peaks themselves. This point is demonstrated by Fig. lc, which is the difference between twice the Fourier transform spectrum and the maximum entropy reconstruction. Around the position of the peak, the difference spectrum is almost noise-free (and only differs from zero because the maximum entropy spectrum is not quite twice the intensity of the conventional spectrum). This property of maximum entropy spectra can be revealed more clearly by plotting the intensity of each point in the maximum entropy reconstruction against the intensity of the corresponding point in the conventional spectrum (32). This is shown in Fig. 2 for the spectra discussed above (Figs. la and lb). All the points lie on a single smooth monotonically rising curve, indicating that the relationship between the two spectra is simply a nonlinear amplification. Thus, for any free induction decay, there exists a function 6 such that (32)
MAXIMUM
ENTROPY
AND FOURIER
TRANSFORMATION
281
0
Fourier
transform
intensity
xl00
FIG. 2. The intensity of points in the maximum entropy reconstruction of free induction decay 1 (Fig. lb) plotted against the intensity of the corresponding points in the conventional spectrum (Fig. la). The sloping line has gradient 2.
Xj =
6(J)~
[71
where f is the real part of the conventional Fourier transform spectrum. For small values of&, 6(A) is almost constant and close to zero, but beyond a certain threshold, 6(J) rises rapidly, becoming approximately linear with gradient close to 2. Ignoring the doubling, the operation of 6 on the conventional spectrum is very similar to what might be called “scissors processing,” in which a horizontal line is drawn across the spectrum skimming the more intense noise spikes, and everything below it is discarded. The result of this operation on Fig. la is shown in Fig. Id. The similarity to the maximum entropy reconstruction, Fig. lb, is striking. This simple relationship between the two types of spectra is always found, for synthetic and genuine data alike, irrespective of the number of peaks, provided three conditions are satisfied: (i) Nr = 2ZV,(i.e., as many points in the real spectrum as there are real + imaginary in the data); (ii) the noise level Uk is the same for all data points k; and (iii) the trial spectra x and the mock data y are related by Fourier transformation. These are identical to the conditions under which Hoch (32) was able to solve the maximum entropy problem analytically except that, for complex spectra, the first condition becomes: (i) Nr = Nt (i.e., as many complex points in the time as there are in the frequency domain). In fact, a fourth condition must be satisfied for real spectra: (iv) the free induction decay must not be significantly truncated (see the Appendix). To summarize, under the above conditions the maximum entropy method is purely cosmetic, producing a spectrum that is trivially related to the Fourier transform spec-
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trum, delivering neither resolution nor sensitivity enhancement, nor any other significant benefit. It is difficult to imagine reasons, apart perhaps from pedagogic, for preferring the maximum entropy reconstruction. This somewhat depressing result is in complete agreement with Hoch’s predictions (32). DISCUSSION
However, there is more to maximum entropy than the above would suggest. Conditions (i)- (iv) are rather restrictive and, as we now argue, unlikely to be acceptable in practice. We deal with the four conditions in turn. Condition(i). It is rare that one would be content to have no more points in the frequencydomain spectrum than in the time-domain data. In conventional processing, zero-filling is routinely used to increase digital resolution: this is especially important for truncated signals which would otherwise be very poorly digitized. Moreover, it is crucial (39, 40) to zero-fill at least once to take into account the absence of data prior to excitation (this point is discussed later). If this is not done, then one cannot focus attention on the real part of the conventional spectrum, as is customary, without losing the information present in the imaginary part. Finally, and more mundanely, Nr must normally be a power of 2 if a fast Fourier transform algorithm is to be used: zero-filling is routinely employed to achieve this. All these points apply equally to maximum entropy processing and require that Nr be greater than or equal to 2N,. Condition(ii). The standard deviation of the noise is usually uniform but is not necessarily so. For example, pulse breakthrough or acoustic ringing ( 11) may increase the uncertainty associated with the initial points in a free induction decay. Condition(iii). The desired spectrum is seldom related to the data by Fourier transformation alone. More commonly some degree of sensitivity or resolution enhancement is needed. Resolution enhancement in maximum entropy reconstructions is achieved by modifying the matrix (R) that relates the trial spectrum to the mock data. Specifically, R would first Fourier transform the trial spectrum and then multiply the resulting time-domain signal, point by point, by a weighting function, which might be a decaying exponential or, more satisfactorily, a damping function derived from a reference line in the spectrum (10, 13). Other applications of maximum entropy involve transformations in which R is not a simple Fourier transform, for example, J deconvolution, in which multiplet splittings are removed ( Z9), and a treatment of off-resonance effects, where R is based on the Bloch equations ( 1.5). Condition(iv). The most common case in which conventional processing is inadequate occurs when the time-domain data are truncated. This, almost certainly, is the single most common and important application of maximum entropy in NMR spectroscopy. It is certainly the aspect of maximum entropy processing that has generated the least controversy amongst NMR spectroscopists. In summary, experimental situations in which restrictions ( i ) - ( iv) are appropriate are likely to be few and far between. When these conditions are acceptable, it is probable that the Fourier transform spectrum would be more than adequate and more sophisticated data processing techniques redundant. We now examine, by means of two simple examples, the performance of maximum entropy under conditions more likely to be important in routine NMR.
MAXIMUM
ENTROPY RESOLUTION
AND
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TRANSFORMATION
283
ENHANCEMENT
As mentioned above, maximum entropy spectra with enhanced resolution may be obtained by introducing a weighting function into transformation F?( 10). If F is the matrix that Fourier transforms a real spectrum into a complex time-domain signal, then, for resolution enhancement, R = WF, where W is a diagonal matrix whose elements are the weights for the Nt points in the mock data. For a free induction decay of the form of Eq. [ 61, a sensible choice of W would be a decaying exponential, Wjk= &jkexp[-(k-
l)At/T,],
[81
with the time constant, T,, matched to that of the data (i.e., T, = T). The idea is that trial spectra containing sharp lines are transformed by F into slowly decaying mock data, which in turn are weighted so that their decay parallels that of the actual data. To judge the ability of maximum entropy to enhance resolution, we must consider a spectrum consisting of at least two lines. This is because linewidth and resolution, terms which are often used interchangeably, are not necessarily related for spectra obtained by nonlinear methods (in the same way that signal-to-noise ratio and sensitivity are no longer synonymous (38)). Linewidth is a familiar and easily measured quantity; resolution, however, is more subtle and refers to the ability to separate overlapping resonances. This distinction may be illustrated by considering the nonlinear amplification of a doublet resonance in which the splitting is not resolved. We could imagine applying an amplification so nonlinear that the line is narrowed down to almost nothing. This operation will not cause the splitting to become apparent and so does not affect resolution. As an example of resolution enhancement by maximum entropy, we take a synthesized free induction decay (No. 2, Table 1) comprising a strong doublet and a weak singlet, plus noise. The splitting is chosen such that the two components of the doublet accumulate a phase difference of 27r by the end of the free induction decay. As before, the signal lasts five relaxation times ( TRD = 5 T) and is sampled at 128 regular intervals, but now Nr is chosen to be larger than 2N,. Figure 3 shows four spectra calculated from this free induction decay. Figure 3a is the conventional spectrum: the doublet is unresolved and the singlet just visible above the noise. Figure 3b is the maximum entropy reconstruction, with resolution enhancement, using as weighting function an exponential matched to the decay of the data, that is, Eq. [ 81 with T, = T = 0.2THD. The doublet splitting is now clear and the noise on the baseline essentially absent. Figure 3c is the Fourier transform spectrum, resolution enhanced by Lorentzian-to-Gaussian weighting: the exponential time constant was equal to T, and the Gaussian was chosen so as to make the linewidth of the doublet resonance similar to that in the maximum entropy spectrum. In this spectrum the singlet has all but disappeared into the noise. For comparison, Fig. 3d shows the Fourier transform spectrum with matched filtering for maximum sensitivity. Clearly, the maximum entropy reconstruction is not a nonlinear amplification of any of the Fourier transform spectra. In Figs. 3a and 3d, the doublet splitting is not resolved, and in Fig. 3c, the singlet is weaker than several of the noise spikes. The relationship between the spectra may be seen more clearly in Fig. 4. A plot of intensity
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FIG. 3. Spectra calculated from free induction decay 2 (see Table 1). (a) Real part of the Fourier transform of the data. (b) Maximum entropy reconstruction ( ak = 0.02, b = lo-‘, T, = T). (c) Real part of the Fourier transform of the data with Lorentzian-to-Gaussian resolution enhancement. (d) Real part of the Fourier transform of the data with matched filtering. Note the vertical expansions of(a), ( c) , and (d) Only half of each spectrum is displayed.
in the maximum entropy reconstruction against the corresponding intensity in the Fourier transform spectrum (without resolution enhancement) is shown in Fig. 4a. The points composing the small spur to the lower left correspond to the singlet while the major feature to the right-hand side is the doublet. The latter rises and then falls
a
+f *f+ ++ l
t
l
-1
4 Fourier
8 transform
12 intensity
16 xl000
-I
I
!
I
0 Fourier
I
,
I transform
,
,
2 intensity
,
,
3 xl00
FIG. 4. (a) The intensity of points in the maximum entropy reconstruction of free induction decay 2 (Fig. 3b) plotted against the intensity of the corresponding points in the conventional spectrum (Fig. 3a). (b) The same plotted against the resolution-enhanced conventional spectrum (Fig. 3~). The sloping line has gradient 2.
MAXIMUM
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TRANSFORMATION
285
in the diagram because the doublet is resolved by maximum entropy but not by Fourier transformation. More interesting is the plot of the maximum entropy reconstruction against the resolution-enhanced Fourier transform spectrum (Fig. 4b). The points in this diagram lie on or near two lines, one horizontal at zero intensity in the maximum entropy reconstruction and the other of gradient close to 2 running approximately through the origin. The former comprises baseline points which are noisy in the Fourier transform spectrum but near zero in the maximum entropy reconstruction. The latter is composed of points on the peaks, including some that are weaker than the more intense noise spikes in the Fourier transform spectrum. The scatter arises partly from the noise in the Fourier transform spectrum but also from the slightly different lineshapes in the two spectra. Thus, maximum entropy appears to be distinguishing between signal and noise. If true, this must be due to the inclusion of information on the shape and width of the resonances in the calculation. There is an obvious parallel here with matched filtering, in which knowledge of the lineshape is used to enhance sensitivity. Whether this apparent improvement is genuine is a matter currently under investigation. What can be said of Fig. 3b with reasonable confidence is that it reveals clearly, in a single spectrum, the existence of a weak singlet and a strong doublet with no evidence for further resonances. To obtain this information so convincingly by conventional methods, at least two Fourier transform spectra would be required-one with resolution enhancement to resolve the doublet splitting and another, preferably with sensitivity enhancement, to locate the singlet. There may be circumstances in which it will be advantageous to view a single spectrum containing all the information in the data, rather than several displaying different aspects of that information. TRUNCATION
The use of maximum entropy to circumvent truncation artifacts (sine wiggles) has been extensively described elsewhere ( 7-12). Briefly, one chooses Nf to be larger than 2N, (or Nt in the case of complex spectra) and the algorithm does the rest. Sine wiggles should not appear in the maximum entropy reconstruction because at no stage does one assume that all missing data points are equal to zero. To illustrate the difference between maximum entropy and Fourier transformation, a synthetic free induction decay was again used (No. 3, Table 1). As in the previous example, it comprises a strong doublet and a weak singlet, but now with severe truncation ( TFID = T). The splitting was chosen such that the two components of the doublet accumulate a phase difference of 11a/8 by the end of the data. Figure 5a shows the conventional spectrum with no weighting: the doublet is not resolved and the singlet cannot confidently be distinguished from the sine wiggles and the noise. Figure 5b is the same spectrum with strong exponential apodization: the singlet can just be discerned. The maximum entropy reconstruction (Fig. 5c), however, shows clear evidence for both features. Although the doublet components have different heights and widths (because of the noise), their integrated intensities are very similar. Once again there is no simple relationship between the maximum entropy and Fourier transform spectra, certainly nothing as simple as nonlinear amplification.
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FIG. 5. Spectra calculated from free induction decay 3 (see Table 1) . (a) Real part of the Fourier transform of the data. (b) Real part of the Fourier transform of the data with strong exponential apodization. (c) Maximum entropy reconstruction ( ak = 0.02, b = lo- 12, r, = T). Note the vertical expansions of (a) and(b).
DISPERSION-MODE
LINES
Broadly speaking the behavior described above for the -x In x entropy, Eq. [ 41, is also found for Si ,Z, Eq. [ 5 1. The maximum entropy reconstruction is indeed trivially related to the conventional spectrum (32) but only when conditions (i) to (iii) are satisfied. A new point concerning the handling of resonances of different phases arises when the spectra are complex. Ho&s treatment (32) shows that the nonlinear transformation 6, Eq. [ 71, should apply equally to all lines in the spectrum, whatever their phase. A consequence of this is that “maximum entropy reconstruction will not reduce a dispersive peak relative to an absorptive peak having smaller height” (32). This is certainly correct if conditions (i)-( iii) are met. However, when Nr 3 2N, very different behavior is found. At this point it is useful to consider briefly why dispersion-mode lines appear at all in conventional NMR spectra. At first sight they would seem to be inescapable, but they are simply artifacts of the way in which signals are normally recorded and processed. In most NMR experiments, the data are inherently one-sided, that is, signal is recorded for positive times (after excitation) but not for negative times (before excitation). It is the assumption that these missing data points are equal to zero (i.e., zero-filling) that produces dispersive contributions in the conventional spectrum. There is a clear parallel here with the conventional treatment of truncated data which leads to sine wiggles. To avoid dispersion-mode lineshapes, one needs a double-sided time-domain signal with an envelope symmetrical about time t = 0. Although not generally available experimentally, such signals can be generated by maximum entropy reconstructions provided Nf 2 2N,. Thus, a complex trial spectrum containing only absorption-mode lines would be transformed (by R ) into double-sided mock data, half of which would correspond to t < 0 and half to t > 0. The first Nt complex points of the latter are compared, point by point, with the actual data to determine X2. Thus, if Fourier transformation of a free induction decay such as Eq. [ 6 ] gives a spectrum in which
MAXIMUM
ENTROPY
AND FOURIER
real and imaginary parts are linear combinations dispersion-mode, D( v,) , lineshapes,
TRANSFORMATION
of absorption-mode,
C [A(h) + WbJlexp(k%A n
maximum
287
A( v,), and 191
entropy should produce a spectrum of the form 2
C A(hJexp(&),
1101
n
in which all the intensity of each resonance is concentrated into the absorption-mode line. In fact a small amount of dispersive contribution cannot be avoided, as discussed in Ref. (30). Figure 6 shows spectra obtained from a synthetic free induction decay (No. 4, Table 1) consisting of two resonances, one strong with phase -90” and the other weak with phase 0”. In the real part of the conventional spectrum (Fig. 6a) the absorption-mode line associated with the weaker resonance is smaller than the dispersion-mode peak of the stronger resonance. As shown in Fig. 6b, maximum entropy reconstruction using S1,2 can, and does, suppress the large dispersion-mode line in the real part of the spectrum relative to the smaller absorption-mode peak. Once again, plotting one spectrum against the other reveals what is happening (Fig, 7). The points lie on two lines, one for the absorption-mode peaks (gradient 1.6) and the other, with smaller gradient (0.27)) for the dispersion-mode lines, showing that maximum entropy processing does make a distinction between absorption-mode and dispersion-mode resonances. CONCLUSIONS
In the preceding pages we have compared the maximum entropy method with Fourier transformation, the conventional method of NMR data processing. In agreement with Hoch (32) we find that the maximum entropy reconstruction of a free
FIG. 6. Spectra calculated from free induction decay 4 (see Table 1). (a) Real (R) and imaginary (I) parts of the Fourier transform of the data. (b) Real and imaginary parts of the maximum entropy reconstruction (Ok = 0.04, b = lo-‘*, r, = co).
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al E
-.062 g--.osE -.lOs E .,-.12t -.14-.16-
/ I/
6
Fourier transform intensity FIG. 7. Intensity of points in the maximum entropy reconstruction of free induction decay 4 (Fig. 6b) plotted against the intensity of the corresponding points in the Fourier transform spectrum (Fig. 6a). The sloping lines have gradients 1.6 (absorption mode) and 0.27 (dispersion mode).
induction decay is nothing more than a simple nonlinear transformation of the conventional spectrum when a number ofconditions are met. These are: (i) that there is the same number of points in the spectrum as in the free induction decay; (ii) that the statistical uncertainty is the same for all data points; (iii) that the operation that relates the trial spectra to the corresponding mock data (R, Eq. [ 21) is simply a Fourier transform; and (iv) that the data are not significantly truncated (for real maximum entropy reconstructions only). The nature of the relationship between the two sorts of spectra is such that the maximum entropy reconstruction is essentially worthless. Under conditions more likely to be encountered in practice, we find no such correspondence between the maximum entropy and the Fourier transform spectra. Hoch’s suspicion (32) that “many of the qualitative features exhibited by the reconstructions in the special case” (i.e., under conditions (i)-( iii)) “are also present in the more general case” seems to be unduly pessimistic. A maximum entropy reconstruction can reveal spectral features that cannot be seen in a single conventional spectrum as illustrated by Figs. 3 and 5. Weak resonances can be detected above the baseline noise in the same spectrum in which small splittings in other peaks are resolved. To obtain the same information by conventional processing, two spectra would be needed, one with matched filtering to detect the weak signal and a second with resolution enhancement (e.g., Lorentzian-to-Gaussian conversion) to discern the splitting.
MAXIMUM
ENTROPY
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289
TRANSFORMATION
It should be said, at this point, that the examples used here are in many ways ideal casesfor showing off the advantages of maximum entropy. They involve small numbers of relatively sharp resonances; the lineshapes are ideal Lorentzians, all with the same (known) width; the signal-to-noise ratios are quite high and the dynamic range fairly low; and the noise has an ideal Gaussian distribution, with no spurious correlations. In practice, there are ways to achieve some of these advantages with genuine data. For instance, the dynamic range and the number of resonances can be reduced by processing only a portion of the total spectrum (“zooming”). Linewidths can be measured, and it is possible to deconvolve the overall instrument response function, determined from some well-isolated peak ( 13, 34). Nevertheless, the application of maximum entropy to genuine data will seldom work out quite as nicely as in the examples discussed here. Whether Figs. 3b and 5c constitute an improvement in sensitivity or resolution is doubtful and is not a problem that this paper addresses. In our experience, it has proved impossible to reveal by maximum entropy a weak resonance that could not be located in the matched-filtered conventional spectrum. Similarly, we have not yet been able to resolve using maximum entropy a splitting that could not also be seen in a Fourier transform spectrum with Lorentzian-to-Gaussian weighting. If one desires a single, general-purpose spectrum displaying all the qualitative information present in the data, then one could do far worse than the maximum entropy reconstruction. Usually there will be better ways of extracting quantitative information, for instance, curve fitting. APPENDIX
The purpose of this Appendix is to demonstrate that condition (iv) must be satisfied (i.e., the free induction decay must not be significantly truncated) if the conventional spectrum and the reaZ maximum entropy reconstruction are to be related as in Eq. [71.
We start with the identity, which is valid for all sets of complex numbers dk (k = 0, l)andf,(n=0,...,2NI),
. . . ) 2N2N-I
C
2N-1
I dk -
k=O
2 fn exp(2hk/2N)
I2
n=O 2N-I
1
= 2N Rzo 2~
2N-1
2
dkexp(-2?rink/2N)
-h2,
L-411
k=O
which follows immediately from Parseval’s theorem for the discrete Fourier transform. Now assume that the fn (n = 0, . . . , 2N- l)arerealandthatdZN-k= dz (k= 1, . . . ) N - 1) . The left-hand side (L) of Eq. [Al] can now be written as 2N-1
L=
Ido-
c f,j2+2 n=O
N-l
2 k=l
ZN-I
ldk-
2 f,exp(2aink/2N)12 n=O 2N-I +
I&-
c
n=O
(-l)“fn12.
[A21
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dk (k = 0, . . . ) N - 1) represent the data in an NMR experiment, and let fn (n = 0 3 1 * * 2 2N - 1) be a real trial spectrum. Now X2 is given by
Let
N-l
ZN-1
x2 = C ldk-
2 f,exp(2aink/2N)12/&.
k=O
[A31
n=O
To simplify matters we let ffk = u for all k (0, . . . , N - 1) and set the imaginary part of the first data point, do, equal to zero. The latter is permissible because Im( do) simply determines the integral of the imaginary part of the spectrum, and we are only interested in obtaining real spectra. In practice, for data in which all resonances have phase O”, this point will be close to zero anyway. Combining [A21 and [A31 gives 2N-I
2a2X2 = L + [Re(do) -
2N-I
2 fn]* - IdNn=O
c
(-l)“fn]‘.
n=O
[A41
Turning to the right-hand side (R) of Eq. [A 11, we have, from the conjugate symmetry Of& 2N-1
dkexp( -2?rink/2N)
c k=O
N-l
= Re(do) + 2 Re c
+ (-l)“dN
dk exp(-2rink/2N)
[A51
k=l
and hence ZN-1
R=2N
whereD,(n=0,...,2N-
c
n=O
2
[A61
$ do + Ni’ dkexp(-2?rink/2N) I
D,, is the conventional Fourier transformation. = -$ ‘5’
(-1)”
1)isdefinedby D, = ;Re
X2
D,-s,+FdN,
.
spectrum produced from do, . . . , dN-, by zero-filling and Combining Eqs. [A 11, [ A4], [ A6], and [ A7 ] gives
D, -fn + $$
dN* + $
n=O
[Re(do) - 2E1h]2 n=o -$
,d,-‘;l
(-l)“fn,’ n=O
Maximum
respect
[A81
entropy processing involves maximizing Q=S-Ax’
with
[A71
1
k=l
[A91
tofk (i.e., solving aQ/afk = 0), where s = -c
(fk/b)ln(fk/b) k
[A101
MAXIMUM
ENTROPY
and X is a Lagrange multiplier 1 /b + ( 1/b)ln(f,/b)
AND FOURIER
such that Dk -f
- 5:
k
x2
TRANSFORMATION
291
= 2N. Thus
+ (-l)kWdN) 2N
-$[Re(&-2~1~]+-$[Re(&)-2~1(-l)~~](-l)k=0 n=O
[All] n=O
and hence
&=h+&j 11+ Whlb)l
- &VW&)
- 251Ll n=O C-1) k 2N-I -2~ c (-l)"fn. n=O
[A121
Thus the conventional spectrum Dk is a nonlinear mapping of the maximum entropy spectrum fk, but, because of the final term in Eq. [A 121, the mapping is different for the even and odd points. When this term is zero, the simple form shown in Eq. [ 71 is recovered. Now C,, (- 1)” fn is just the Nth point of the mock data predicted by the maximum entropy spectrum fn. The corresponding point in the actual data, dN, is not actually measured ( dN-, is the last recorded data point). Now since the mock data calculated from the maximum entropy reconstruction will resemble the actual data, we can expect the final term in Eq. [A 121 to be small if dN is itself small, which will only be true in general if the free induction decay is not signijkantly truncated. This conclusion should not be surprising. For a truncated free induction decay d(t), the conventional spectrum D, (n = 0, . . . , N - 1) is not a good approximation to the Fourier integral +CC
--oo d( t)e-jw’dt, s
[A131
which cannot be evaluated because d( t ) is measured for only a limited range oft . The conventional spectrum is consequently a distorted version of the desired spectrum, containing truncation artifacts. The maximum entropy reconstruction, however, is essentially free from such spurious signals and so much closer to the “true” spectrum. We are indebted to Dr. G. J. Daniel1 for this proof. ACKNOWLEDGMENTS It is a pleasure to thank Dr. G. J. Daniel1 for invaluable advice and guidance on many occasions and for generously supplying maximum entropy software. We are grateful to Dr. J. C. Hoch for sending a copy of Ref. (32) prior to publication and to a referee for unusually detailed and helpful suggestions. J.A.J. thanks St. John’s College, Oxford, for a North Senior Scholarship and the Science and Engineering Research Council for a Research Studentship. P.J.H. is a member of the Oxford Centre for Molecular Sciences. REFERENCES 1. D. S. STEPHENSON, Prog. NMR 2. S. F. GULL AND G. J. DANIELL,
Spectrosc. 20, 5 15 ( 1988). Nature (London) 272,686 ( 1978).
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3. S. SIBISI, Nature (London) 301, 134 ( 1983). 4. S. SIBISI, J. SKILLING, R. G. BRERETON, E. D. LAUE, AND J. STAUNTON, Nature (London) 311, 446 (1984). 5. F. NI, G. C. LEVY, AND H. SCHERAGA, J. Magn. Resort. 66,385 (1986). 6. A. R. MAZZEO AND G. C. LEVY, Comput. Enhanced Spectrosc. 3, 165 ( 1986). 7. P. J. HORE, J. Magn. Reson. 62, 561 ( 1985). 8. J. C. HOCH, J. Magn. Reson. 64,436 (1985). 9. J. F. MARTIN, J. Magn. Reson. 65,291 (1985). 10. E. D. LAUE, M. R. MAYGER, J. SKILLING, AND J. STAUNTON, .I. Magn. Resort. 68, 14 ( 1986). Il. E. D. LAUE, K. 0. B. POLLARD, J. SKILLING, J. STAUNTON, AND A. C. SUTKOWSKI, J. Magn. Reson. 72,493 (1987). 12. A. HEUER AND U. HAEBERLEN, J. Magn. Reson. 85,79 ( 1989). 13. S. J. DAVIES, C. BAUER, P. J. HORE, AND R. FREEMAN, J. Magn. Reson. 76,476 (1988). 14. A. I. GRANT AND K. J. PACKER, in “Maximum Entropy and Bayesian Methods” (J. Skilling, Ed.), pp. 251-259, Kluwer Academic, Dordrecht, 1989. 15. P. J. HORE AND G. J. DANIELL, J. Magn. Reson. 69,386 ( 1986). 16. J. C. J. BARNA, E. D. LAUE, M. R. MAYGER, J. SKILLING, AND S. J. P. WORRALL, J. Magn. Reson. 73,69(1987). 17. J. C. J. BARNA AND E. D. LAUE, J. Magn. Reson. 75,384 ( 1987). 18. R. A. JACKSON, J. Magn. Reson. 75, 174 (1987). 19. M. A. DEL.XJC AND G. C. LEVY, J. Magn. Reson. 76,306 ( 1988). 20. E. D. LAUE, J. SKILLING, J. STAUNTON, S. SIBISI, AND R. G. BRERETON, J. Magn. Reson. 62, 437 (1985). 21. E. D. LAUE, J. SKILLING, AND J. STAUNTON, J. Magn. Reson. 63,418 ( 1985). 22. K.M. WRIGHT AND P. S.BELTON,MOL Phys.58,485 (1986). 23. J. C. J. BARNA, E. D. LAUE, M. R. MAYGER, J. SKILLING, AND S. J. P. WORRALL, Biochem. Sot. Trans. 14, 1262 (1986). 24. M. A. DELSUC, F. NI, AND G. C. LEVY, J. Magn. Reson. 73, 548 (1987). 25. J. C. J. BARNA AND E. D. LAUE, Lab. Practice 36, 102 ( 1987). 26. M. L. WALLER AND P. S. TORTS, Magn. Reson. Med. 4,385 ( 1987). 27. R. H. NEWMAN, J. Magn. Reson. 79,448 ( 1988). 28. A. R. MAZZEO, M. A. DELSUC, A. KUMAR, AND G. C. LEW, J. Magn. Reson. 81, 512 ( 1989). 29. T. N. HUCKERBY, I. A. NIEDUSZYNSKI, G. H. COCKIN, J. M. DICKENSON, H. MORRIS, P. N. SANDERSON, AND D. J. THORNTON, Eur. Polym. J. 25,861 ( 1989). 30. G. J. DANIELL AND P. J. HORE, J. Magn. Reson. 84,5 15 ( 1989). 31. G. J. DANIELL AND P. J. HORE, in “Maximum Entropy and Bay&an Methods” (J. Skilling, Ed.), pp. 297-302, Kluwer Academic, Dordrecht, 1989. 32. J. C. HOCH, A. S. STERN, D. L. DONOHO, AND I. M. JOHNSTONE, J. Magn. Reson. 86,236 ( 1990). 33. P. J. HORE, D. S. GRAINGER, S. WIMPERIS, AND G. J. DANIELL, J. Magn. Reson. 89,415 (1990). 34. P. J. HORE, in “Maximum Entropy in Action” (B. Buck and V. A. Macaulay, &Is.), Oxford Univ. Press, Oxford, pp. 4 l-72. 35. S. DAVIES, K. J. PACKER, A. BARUYA, AND A. J. GRANT, in “Maximum Entropy in Action” (B. Buck and V. A. Macaulay, Eds.), Oxford Univ. Press, Oxford, in press. 36. C. E. SHANNON, Bell Syst. Tech. J. 27, 379 ( 1948). 37. J. SKILLING AND R. K. BRYAN, Mon. Not. R. Astron. Sot. 211, 111 ( 1984). 38. R. FREEMAN, “A Handbook of Nuclear Magnetic Resonance,” pp. 2 16-229, Longman, Harlcw, 1988. 39. E. BARTHOLDI AND R. R. ERNST, J. Magn. Reson. 11,9 (1973). 40. R. FREEMAN, “A Handbook of Nuclear Magnetic Resonance,” pp. 302-306, Longmap, Harlow, 1988.