Selective Data Acquisition in NMR. The Quantification of Anti-phase Scalar Couplings

JOURNAL OF MAGNETIC RESONANCE, ARTICLE NO.

Series A 120, 18–30 (1996)

0096

Selective Data Acquisition in NMR. The Quantification of Anti-phase Scalar Couplings P. HODGKINSON,* K. J. HOLMES,

AND

P. J. HORE†

Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford OX1 3QZ, United Kingdom Received October 5, 1995; revised January 22, 1996

Almost all time-domain NMR experiments employ ‘‘linear sampling,’’ in which the NMR response is digitized at equally spaced times, with uniform signal averaging. Here, the possibilities of nonlinear sampling are explored using anti-phase doublets in the indirectly detected dimensions of multidimensional COSY-type experiments as an example. The Crame´r–Rao lower bounds are used to evaluate and optimize experiments in which the sampling points, or the extent of signal averaging at each point, or both, are varied. The optimal nonlinear sampling for the estimation of the coupling constant J, by model fitting, turns out to involve just a few key time points, for example, at the first node ( t Å 1/J) of the sin( p Jt) modulation. Such sparse sampling patterns can be used to derive more practical strategies, in which the sampling or the signal averaging is distributed around the most significant time points. The improvements in the quantification of NMR parameters can be quite substantial especially when, as is often the case for indirectly detected dimensions, the total number of samples is limited by the time available. q 1996 Academic Press, Inc.

An alternative to this ‘‘nonuniform signal averaging’’ is to allow irregular t1 increments using, for example, a high sampling density at short t1 where the signal is strong, with a smaller number of samples at longer times ( 3, 4 ) . Given the huge variety of possible sampling schemes, we require some method of judging the effectiveness of a sampling pattern. Previous work in this area has generally employed the maximum entropy method (MEM) to obtain a spectrum, which is then assessed by eye as a measure of the success of the sampling method, which is often chosen in an ad hoc fashion using intuitive arguments (3–6). While MEM can be invaluable in providing a general purpose spectrum consistent with the data, the NMR intensities are usually biased (7, 8), making it difficult to draw general conclusions. What is clearly required is an objective, unbiased measure of the quality of a spectrum or the data from which it is derived, which permits the design and optimization of the sampling scheme. ‘‘Monte Carlo’’ methods ( 9 ) involve repeating a quantification procedure many times on the same data set perturbed by different sets of random noise. The deviation of the observed parameters from their true values can then be used to characterize the experimental protocol. Although such methods have the advantages of directness and generality, they tend to be time consuming and provide little insight into why a particular experimental procedure works ( or does not work ) well. Fortunately, there is a more direct approach which is particularly appropriate when model-fitting is being used to extract parameter values from a data set. It can be shown that the ‘‘Crame´ r – Rao lower bounds’’ ( also known as ‘‘minimum-variance bounds’’ ) are fundamental limits on the accuracy of parameters derived from experiments ( 10, 11 ) ; for any unbiased estimation method, the error in the derived parameter ( as measured by its standard deviation ) must be greater than or equal to the lower bound. In particular, for maximum-likelihood estimators, such as model fitting, the standard deviation equals the Crame´ r – Rao lower bound ( 12 ) ; hence, we can calculate directly, for a given sampling strategy, the standard deviations of the parameters that would be measured by model fitting a large number of

INTRODUCTION

The vast majority of NMR experiments involve digitizing a continuous time-domain signal at equally spaced intervals, with uniform signal averaging at each time point, a process that may be called linear sampling. To do otherwise ( nonlinear sampling ) in one-dimensional NMR, or for the directly detected dimension of a multidimensional NMR experiment, offers no advantage at the expense of considerable inconvenience. There is, however, much greater scope for nonlinear sampling of the indirectly detected dimensions. For example, one could easily use irregularly spaced values of the time variable t1 in a two-dimensional pulse sequence, or vary the number of free-induction decays accumulated for each t1 . Regular sampling in the t1 dimension could be retained, but the number of transients recorded at each t1 could be concentrated at those t1 values felt to be particularly significant, cf. the use of ‘‘tailored acquisition’’ in the context of in vivo spectroscopy ( 1, 2 ) . * Present address: Chemistry Department, University of California, Berkeley, California 94720. † To whom correspondence should be addressed. 18

1064-1858/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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Monte Carlo data sets. 1 These bounds provide a quantitative measure of the information present about a particular parameter in a given data set, irrespective of the inversion method, and so trends in the Crame´ r – Rao lower bounds are generally matched by trends in the results from biased estimates of spectral parameters such as those from maximum entropy and linear prediction ( 13 ) . In this article, we extend the preliminary work of van Ormondt et al. ( 14 ) who used the Crame´ r – Rao lower bounds to assess the efficacies of various sampling patterns [ note also the use of Crame´ r – Rao lower bounds in the context of nonlinear sampling in Ref. ( 15 ) ] . Timedomain model fitting is assumed to be the inversion method of choice, which allows the Crame´ r – Rao bounds to be used directly as a criterion of efficiency, as well as permitting an unrestricted choice of time-domain sampling pattern ( provided, of course, that there are sufficient data points to permit fitting to the chosen model function ) . Although many of the conclusions are of general application, we focus for simplicity on the quantification of antiphase multiplets which are ubiquitous in coherence-transfer experiments, with the aim of identifying the factors that determine the optimal choice of sampling patterns for the quantification of active coupling constants from COSY-type experiments.

a

∑ xn S 0a

0a

Fj k Å

∑ nxn S 2n

a2

2 n

ap

.

[3]

v

∑ nxn SnCn

0

n

∑ n 2 xn S 2n

0 a 2p

∑ n 2 xn SnCn

0

n

∑ nxn SnCn 0 a 2p ∑ n 2 xn SnCn

a 2p 2

n

0

D

Ordering the four parameters as ( u1 , u2 , u3 , u4 ) Å ( a, l, J, v ), it is easily shown that F is the real symmetric matrix

n

n

S

1 ÌyP * n ÌyP n Wn Re 2 ∑ Ìuj Ìuk s n

J

∑ nxn S

[2]

i.e., the square root of the kth diagonal element of the inverse of F . If the noise is Gaussian, the elements of F are given by (10)

n

n

ap

q

s( uk ) Å ( F01 )kk ;

l 2 n

n

1 s2

The sampling times, tn , are expressed as multiples of a basic dwell time Dt (i.e., tn Å nDt), and v, J, and l are dimensionless variables expressed as fractions of the spectral width, 1/ Dt. We suppose that each point in the experimental NMR response has been divided by the number of samples acquired at that point, Wn . Defined in this way, signal averaging does not affect the signal, but reduces the noise level; if s is the noise standard deviation in the absence of signal averaging (Wqn Å 1), then the standard deviation sn at point n will be s / Wn after signal averaging. The Crame´r–Rao lower bound for a parameter uk is its standard deviation, s( uk ), calculated from the Fisher information matrix, F (10):

∑ n 2 xnC 2n

,

[4]

0

n

0

0

a2

∑ n 2 xn S 2n n

with THE CRAME´R–RAO LOWER BOUNDS

We start with a suitable (time-domain) model function for an isolated weakly coupled anti-phase doublet, parameterized by amplitude a, frequency v, coupling J, and damping rate l ( Å1/T * 2 ): yP n Å ae ivn sin( p Jn)e 0 ln .

[1]

1 Note that we assume that any systematic error involved in model fitting is small in comparison to the random errors due to noise, and that the noise does not so dominate the signal that the fitting becomes unreliable. This is equivalent to the requirement that the value of parameter is significantly larger than its standard deviation, which is only reasonable for a meaningful experiment.

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xn Å Wn e 02l n

[5]

Sn Å sin( p Jn)

[6]

Cn Å cos( p Jn).

[7]

Elements of F01 may be obtained from the corresponding cofactor and the determinant of F (16): ( F01 )kk Å cof Fkk /det F .

[8]

It follows that the Crame´r–Rao lower bounds have the form

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HODGKINSON, HOLMES, AND HORE

s( u ) Å

2 s(T * 2 ) Å (T * 2 ) s( l ) Å

s

gu(JT * 2 )

u Å J, v [9]

sT * 2 gl (JT * 2 ) a

[10]

aT * 2

s( a ) Å sga(JT * 2 ),

[11]

where the four functions, gu ( u Å a, l, J, v ), are dependent only on JT * 2 and the sampling pattern, i.e., the set of Wn and tn . Note that l is used as a time-scaling factor which allows the sums of Eq. [3] to be expressed as functions of JT * 2 and Wn = only, where Wn = is a function of the reduced time variable n * Å n/T * 2 . Several important conclusions can be drawn from these expressions. (i) The quantification of all four parameters is independent of v, the position of the doublet within the spectrum. (ii) The amplitude of the signal a and the noise parameter s are not important other than as scaling factors. (iii) The standard deviations of l, J, and v are inversely proportional to the signal-to-noise ratio a / s (although model fitting will, however, break down if a / s becomes too small). (iv) The accuracy with which frequency parameters, such as J and v, can be determined improves with increasing T 2* (i.e., decreasing linewidth). (v) The only nonlinear terms in the expressions for the lower bounds are the functions gu , which are dependent only on the degree of overlap of the components of the doublet, parametrized by JT * 2 ( Å J/ l ), and by the choice of sampling pattern, Wn and tn . It could be argued that analysis in terms of model fitting overlooks experimental deficiencies. For instance, roll-off of the noise filters at the edges of spectrum will qualify the claim that quantification is independent of spectral position, but in so far as such problems do not compromise the model used, they are not significant. It is clearly unwise, however, to rely on a model which does not adequately fit the data, e.g., if the system deviates markedly from the monoexponential relaxation assumed above. The more the experiment is tailored toward a defective model, the less meaningful the results. LINEAR SAMPLING

To illustrate the use of the Crame´r–Rao lower bounds, we consider initially a free-induction decay which has been regularly sampled (i.e., tn Å nDt, n Å 0, 1, 2, . . . , N 0 1) with uniform signal averaging, 2 that is, Wn Å W. It is easily seen from Eqs. [2] and [3] that the Crame´r–Rao lower bounds for all model parameters are proportional to q 1/ W —the familiar signal-to-noise improvement expected from signal averaging (17). 2 A glossary of terms used in this article can be found following Conclusions.

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FIG. 1. Crame´r–Rao lower bounds as a function of the acquisition time N expressed as a multiple of T *2 , for (a) directly detected and (b) indirectly detected dimensions of a multidimensional NMR experiment. T *2 Å 25.6, J Å 0.1. The vertical scales are arbitrary; in general, s(J) ! s( a ) õ s(T *2 ).

Figure 1a shows the lower bounds for the amplitude, relaxation time, and coupling constant as a function of the number of samples N (the duration of the free-induction decay), for the case JT * 2 Å 2.56 (i.e., J/linewidth approximately equals 0.8). As expected, the three standard deviations decrease monotonically as the length of the free-induction decay increases, and level off once the acquisition time NDt is larger than about 2T * 2 . Clearly, there is little to be gained from continuing to acquire signal beyond this point. On the other hand, there is little to lose by prolonging the acquisition, given the need for a relaxation delay before acquiring the next transient. The acquisition should not, however, be extended too far down into the noise level; it is

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unwise to rely too heavily on the assumption of Gaussian noise. The situation is more interesting for the indirectly detected dimensions of a multidimensional NMR experiment, such as t1 in 2D NMR. The form of the signals is the same as in the directly detected dimension, t2 —it would be necessary to modify the model function if the data were not acquired in hypercomplex mode (18) —but it is now necessary to include the time taken to record each transient in the assessment of experimental efficiency. If the level of ‘‘t1 noise’’ (or rather its deviation from Gaussian behavior) is small compared to true random noise, the noise distribution can also be assumed to be identical. The complete definition of sensitivity is signal-to-noise per square root unit time (17). It is necessary, therefore, to ‘‘normalize’’ the Crame´r–Rao q lower bounds by multiplying them by Ttotal , where Ttotal is the total time taken by the experiment. Alternatively, we can define the efficiency of the quantification of parameter u as the ‘‘reliability,’’ u /s( u ) per square root unit time. If, for instance, the experiment is repeated and the results averaged, the size of the error bars (asq measured by the standard deviation) drops by a factor of 2. The total time taken has doubled, however, and so the efficiency of the experiment is the same. In directly detected dimensions, Ttotal is fixed by the time taken for the system to return sufficiently close to equilibrium such that saturation effects are avoided; i.e., Ttotal is independent of the nature of the sampling. In indirectly detected dimensions, however, Ttotal is dependent on the sampling pattern, and in particular, the number of data points N . We can assume that the time taken to acquire each transient is constant and is independent of the value of t1 , and so Ttotal is simply proportional to N , the number of t1 values sampled. Hence, it is appropriate to multiply the lowerqbounds for indirectly detected dimensions by a factor of N . As illustrated in Fig. 1b, this leads to a minimum in the resulting ‘‘normalized’’ Crame´r–Rao lower bounds as a function of N. This optimal acquisition length, Nopt , corresponds to the best linear sampling for a given parameter and represents the point at which the information gained from acquiring further t1 increments is exactly offset by the data that would be gained from signal averaging instead. Nopt is conveniently expressed in terms of T * 2 ; in this case, for instance, the optimum acquisition time for determining J is NDt Å 1.68T * 2 Å 4.3/J. The position of this minimum will vary with the degree of overlap of the two anti-phase components. Figure 2 plots the optimal FID length, Nopt , as a function of overlap, JT * 2 . Also shown are the results for the corresponding in-phase doublet. For very poorly resolved lines (JT * 2 ! 1), the optimal . This is unrealistic in acquisition time tends toward 6T * 2 practice, as the signal will have all but vanished by this point, but reflects the necessity of acquiring points very far

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FIG. 2. Optimal linear samplings for measurement of J. The solid and dotted lines give the optimal acquisition time Nopt , as a multiple of T *2 (scale on left) for in-phase and anti-phase doublets, respectively. Also shown, on a log scale (arbitrary units) are the associated normalized Crame´r–Rao lower bounds (i.e., the standard deviations) for the in-phase (circles) and anti-phase (diamonds) cases.

into the decay in order to resolve such tiny splittings. In this limit, the different interference characteristics of the in- vs anti-phase summations are especially apparent. For the inphase doublet, the optimal FID length tends toward 4.1T 2* , with the gradient of the standard deviation log–log plot showing that s(J) } J 01 . For the anti-phase doublet, the destructive interference further reduces the accuracy of quantification, and now s(J) } J 02 , with Nopt r 6.0T * 2 . This corresponds to the fractional error in J varying as J 03 , i.e., the quantification of anti-phase doublets rapidly becomes impractical as J drops below a critical point of JT * 2 É 0.6, Fig. 3b. Once the standard deviation of J becomes comparable with the splitting itself, the determination of J is effectively impossible. It is noteworthy that the accuracy of quantification cannot be deduced simply from the visual appearance of the spectrum. For instance, the in-phase doublet of Fig. 3c is so strongly overlapped that no splitting is visible. Nevertheless, J can still be determined more accurately from this data set than from the corresponding anti-phase doublet, even though the doublet character is clearly visible in this case. Indeed, the quantification of anti-phase doublets is consistently less accurate than the quantification of the corresponding inphase doublet. Several important NMR experiments, however, give rise to anti-phase doublets, most notably the COSY family of experiments, and it not always possible, or desirable, to restrict quantification to data from experiments such as TOCSY (19) which generate only in-phase cross peaks. As JT 2* becomes larger than about 0.1, the optimum sampling length drops and the accuracy with which J can be

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HODGKINSON, HOLMES, AND HORE

* * FIG. 3. In-phase (dotted lines) and anti-phase (solid lines) doublets as a function of overlap: (a) JT * 2 Å 3.0, (b) JT 2 Å 0.6, and (c) JT 2 Å 0.1.

determined improves (Fig. 2). In the range 0.3 ( JT * 2 ( 1.0, the optimum length is roughly 1.3/J, i.e., between the first and second nodes of the sin( p Jt) modulation caused by the coupling. When JT * 2 is larger than about 1.0, the components of the anti-phase doublet are well resolved, Fig. 3(a), and the lower bound becomes independent of the degree of overlap, although the fractional error in J will decrease as J becomes larger. The ‘‘sawtooth’’ pattern in the plot of Nopt is a direct result of the undulations in s(J) visible in Fig. 1b: as J increases, the global minimum in s(J) jumps from one local minimum to the next giving the discontinuities in Fig. 2. As J increases, the effect of these oscillations becomes less noticeable and Nopt tends toward 1.7T * 2 . Reassuringly, this is identical to the optimal linear sampling for the determination of the frequency of a single isolated resonance; it is shown in a later section that well-resolved features can be quantified independently. OPTIMAL NONLINEAR SAMPLING

While the optimal linear samplings are useful guides for improving conventional experiments and provide suitable benchmarks against which to compare other sampling schemes, our primary interest is in exploring the possibilities offered by nonlinear sampling. Choosing sampling patterns on an intuitive, ad hoc basis rarely reveals why one particular sampling performs more efficiently than another. Instead, we would like to evaluate ideal sampling patterns and, armed with some insight as to why they work well, use them to design practical experiments. We can define an optimal nonlinear sampling pattern as the set of sampling times tn , and the associated number of acquisitions Wn , that minimizes the Crame´r–Rao lower bound of a given parameter. Since it is only possible to derive optimal sampling patterns when there is a penalty for acquiring data points with little information content, we concentrate exclusively on data from indirectly detected dimensions of multidimensional experiments. Following the arguments in the previous section, the bounds are therefore q normalized by multiplying by (nWn .

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The results of minimizing s(J) as a function of the sampling pattern are shown in Fig. 4. For a well-resolved doublet (JT * 2 @ 1), the optimum reduces to sampling exclusively at the node of sin( p Jt) that lies closest to t Å T * 2 (which is the optimum nonlinear sampling for determining the frequency of an isolated resonance). As JT * 2 is reduced, the optimal sampling ‘‘jumps’’ from one nodal point to the next around t Å T * 2 : for JT * 2 ( 1.4, the first node (at t Å 1/J) is the closest to t Å T * 2 . As JT * 2 is further reduced, however, the position of this node moves away from t Å T 2* and the sampling quickly becomes inefficient, until at JT * 2 É 0.3, the optimum becomes a three-point sampling. In the limit of very small JT 2* , the three sampling times are * , with relative weights (Wn ) of 0.31T * 2 , 1.65T * 2 , and 4.71T 2 23, 23, and 54% respectively. Although about 30% more efficient than the best linear sampling, this is of marginal interest as the bounds are still too large for practical quantification. Note that the optimal nonlinear samplings for the in-phase doublet are similar, although the limiting threepoint sampling for strongly overlapped lines is different; for practical cases, JT * 2 ™ 0.2, the sampling again follows nodal points—at the zero crossings of cos( p Jn) rather than sin( p Jn). The interesting region of Fig. 4 is 0.3 ( JT * 2 ( 1.0 in which the doublet is neither so well resolved that measurement of J is trivial, nor so poorly resolved that determination of the splitting is almost impossible. In this region, the best solution, sampling exclusively at t Å 1/J, gives lower bounds for J that are Ç60% better than for the best linear sampling. Since the signal-to-noise ratio rises as the square root of the time taken, this means that the linearly sampled experiment would take a factor of 2.5 longer than the nonlinearly sampled experiment to achieve the same quality of results. Nodal sampling, at points where there should be no signal, may seem perverse. The positions of the nodes are, however, extremely sensitive to the value of J and so it is to be expected that the sampling should be concentrated at these points; the definition of F (Eq. [3]) indicates that it is the derivative of the model function with respect to the parame-

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known in advance in order to determine the sampling scheme! Even if we were to overcome the problem of insufficient data points by adding sampling points in the immediate vicinity of the node, the experiment would still be rather sensitive to the actual value of J; J values close to the estimated value would be determined very accurately, but if the true value were very different from its estimate, the error bars could be enormous. One solution, which is appropriate when the total number of acquisitions, Ntotal , is large, is to use both the optimal linear and nonlinear samplings. Acquiring a linearly sampled data set (which can, if necessary, be Fourier transformed) and acquiring the remaining transients at the optimal nonlinear sampling times (preferably deduced on the basis of the linearly sampled data set) allows the efficiency of the nonlinear sampling to be combined with the ‘‘stability’’ of the linear sampling. This does, however, require the acquisition of a relatively large number of transients; in the following sections, we derive sampling patterns that are suitable when the total time of the experiment is restricted. NONUNIFORM SIGNAL AVERAGING

For linear sampling, the signal averaging pattern, Wn , is a step function Wn Å

H

1 nõN 0 n § N,

[12]

whereas for the optimal (‘‘sparse’’) nonlinear sampling just described, it takes the form of a limited set of nonzero values, e.g., Wn Å

FIG. 4. Optimal nonlinear samplings for the quantification of J for an anti-phase doublet. (a) Sampling points, tn , in multiples of T *2 and (b) the corresponding weightings, Wn . Note that the best sampling strategy involves three time points for JT *2 ( 0.3 and a single point for JT *2 ) 0.3. (c) The efficiency of the optimal nonlinear samplings relative to the best linear sampling.

ter in question that is significant, rather than the value of the function itself. The sampling patterns which perform best are predicted to be those which concentrate intensity at the nodal points (the maxima of the derivative with respect to J of the sin( p Jt) modulation) rather than at the maxima of the signal itself. There are two objections to this minimalist sampling scheme. First, one data point is insufficient to permit model fitting. Second, and more fundamentally, it requires J to be

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H

1 tn Å 1/J 0 otherwise.

[13]

An intermediate between these two limits is to allow Wn to follow some functional form: for example, ‘‘exponential sampling’’ (3, 4) can be represented Wn } e 0 l=n ,

[14]

where the rate constant l* may (or may not) be chosen to match the decay of the signal. In the case of linear sampling, the only variable parameter is the length of the sampling, N. Most nonlinear sampling functions contain parameters other than the length of acquisition, such as the decay rate constant l* above. Given that such samplings can never be optimal from a simple efficiency viewpoint, insofar as they differ from the optimal samplings derived above, it is only realistic to consider a limited range of sampling patterns chosen on an intuitive rather than systematic basis. Such samplings may offer sig-

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HODGKINSON, HOLMES, AND HORE

FIG. 5. Nonuniform signal averaging patterns (b–i) (see Table 1) to be tested against the trial time-domain signal (a) with J Å 0.004, T * 2 Å 100. The vertical lines in (b–i) mark the sampling points (tn /T *2 ), and their heights the extent of signal averaging (Wn ).

nificant pragmatic advantages over the efficient, but somewhat delicate, sparse nonlinear sampling patterns. A variety of acquisition patterns are shown in Figs. 5b– 5i. The vertical scale is the sampling density (Wn ) which, by definition, must be positive. These eight sampling patterns were tested against the NMR signal (JT * 2 Å 0.4) of Fig. 5a by finding the number of sampling points, Nopt , that minimized the Crame´r–Rao lower bound of J. The relative efficiencies of the resulting optimal samplings are shown in Table 1, where the efficiency is measured as the ratio of the normalized Crame´r–Rao lower bound from linear sampling to that from the nonlinear sampling; a value ú1.0 corre-

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sponds to an experiment that is more efficient than the best linear sampling. In all cases, the optimum number of sampling points is sufficiently large ( ú300 ) that the effect of digitizing the time axis can be neglected; i.e., the sampling is sufficiently dense to capture accurately all the variations in the weighting functions. Hence, the results can be taken as a good characterization of the sampling patterns for this value of JT * 2 ( 0.4 ) . It is significant that the optimal acquisition times all cluster around the optimal linear sampling length of 336 points. This is unsurprising given that the optimal acquisition time should mark the point where fur-

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TABLE 1 Relative Efficiencies of the Nonuniform Signal-Averaging Patterns of Fig. 5 Sampling

Samples

Efficiency

(b) Linear (c) Ésin(pJn)É (d) Écos(pJn)É (e) Ésin(pJn/2)É (f) exp(0n/T*2 ) (g) exp(n/T*2 ) (h) Ésin(pJn)Éexp(0n/T*2 ) (i) n3exp(03n/T*2 )

336 349 329 330 365 307 374 372

1.00 0.83 1.12 1.15 0.59 1.32 0.52 0.66

ther signal acquisition contributes ever diminishing information returns. Previous work on nonlinear sampling of NMR data has concentrated almost exclusively on sampling patterns that are matched to the envelope of the signal, i.e., concentrating sampling at points in the time domain with the highest ‘‘signal-to-noise ratio’’ (3–6, 20). While such samplings may, in the manner of a matched filter, improve the signal-tonoise ratio and appearance of spectra, it is clear from Table 1 that such samplings are unsuitable from the viewpoint of parameter quantification. The Ésin( p Jn)É modulated sampling (c), as proposed in Ref. (5), the exponentially decaying exp( 0n/T * 2 ) sampling (f ) (3, 4), and the combined sampling Ésin( p Jn)exp( 0n/T * 2 )É (h) —which matches the envelope of the signal exactly—all perform less well than simple linear sampling. Better samplings are those such as (d) and (e) which are concentrated at the extrema of the derivative of the signal envelope modulation. The best sampling of Table 1 is (g) which contains a rising (rather than decaying) exponential and is the most effective at concentrating sampling at the first nodal point (t Å 1/J), which is the optimal nonlinear sampling for this value of JT 2* . Indeed, if the time constant of the exponential exp(n/T * 2 ) and Nopt are allowed to vary freely, they tend to zero and 1/J respectively, resulting in an acquisition pattern that rises sharply and terminates at the first nodal point; such an acquisition pattern is, however, unrealistic. Sampling (i) concentrates the sampling at T * 2 , which is expected to be most effective when the doublet is well resolved; in this case, with JT 2* Å 0.4, it performs poorly since the nodal point (1/J Å 2.5T * 2 ) is far from T * 2 . Figure 6 shows the performance of some nonuniform signal-averaging patterns as the degree of overlap changes. The sampling patterns fall into two categories. The ‘‘trigonometric’’ samplings [cos( p Jn) etc.] are primarily dependent on the estimate of J. Since these sampling should be chosen to concentrate on the nodal points, the worst case will occur when the estimate of J is such that the sampling pattern misses the nodes. This sampling cannot, however, be worse than the sin ( p Jn ) which only concentrates sampling at

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the anti-nodes. Hence, the performance of a trigonometric sampling in practice will lie somewhere between the curves for the cos ( p Jn ) and sin ( p Jn ) samplings. In contrast, the n 3 exp ( 03n / T * 2 ) sampling, a special case of n k exp( 0kn / T * 2 ) which concentrates sampling at T * 2 , is independent of J and relies solely on the T * 2 estimate. As k is increased, the sampling tends (albeit slowly) toward a delta function at T 2* . This improves the efficiency of the sampling (provided that the optimal nodal point is not too far from T* 2 ) at the cost of decreased stability. Where J is unknown, or a single sampling pattern suitable for a range of J values is required, these ‘‘modified exponential’’ samplings are much more robust than the trigonometric forms. They soon cease to be effective, however, when the optimal nodal point is far from T * 2 , as is the case when JT * 2 ( 0.6. Clearly, nonlinear sampling patterns designed to improve parameter quantification should not be chosen on the basis of time-domain signal-to-noise arguments. Although a better rule of thumb would be to follow the derivative of the signal envelope, it is necessary to find the optimal (sparse) nonlinear sampling in order to understand why any particular sampling performs well. While nonuniform signal-averaging patterns are a useful intermediary between sparse and more robust nonlinear samplings, they are not, in and of themselves, very practical since they require the acquisition of a large number of transients; because Wn must, by definition, be integral, the total number of transients, Ntotal must be significantly larger than Nopt for the errors in ‘‘quantizing’’ Wn to become negligible. IRREGULAR SAMPLING

Rather than encoding the density of sampling in terms of the number of acquisitions at each time point, the spacing

FIG. 6. Performance of various nonuniform signal-averaging patterns as a function of the degree of overlap, JT *2 . The vertical axis gives, in arbitrary units, the reliability of the coupling constant defined as J/s(J) where s(J) is the normalized Crame´r–Rao lower bound (the standard deviation) of J.

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FIG. 7. Schematic illustration of irregular sampling, where the black bars mark the positions of sampling points along the time axis: (c) linear sampling; (b) irregular sampling ( dt small); and (a) the same sampling density function Y (t) as (b) but with sampling points constrained to multiples of the dwell time ( dt Å Dt).

between time points can be varied to achieve the same effect. Starting from a target density function Y (t), for example, an exponential decay, M acquisitions are distributed over the time interval 0 to T. The integral of Y, multiplied by a scaling constant a, over the acquisition time T must equal M a

*

T

Y (t)dt Å M,

[15]

0

and the integral of the scaled Y between two sampling points tk and tk /1 must be unity: a

*

tk /1

Y (t)dt Å 1.

[16]

tk

Thus,

*

tk /1

Y (t)dt Å

tk

1 M

*

T

Y (t)dt.

[17]

0

The right-hand side of this expression is fixed and so tk /1 can be found starting from the previous sampling point, tk . Since t 0 Å 0, the sampling can be found by numerically integrating the scaled Y (t) using a time step of dt, and sampling whenever the integral of aY (t) crosses an integer value. A special case is when dt equals the dwell time of the conventionally sampled signal. Of course, if dt is large, then the sampling points will only be correlated loosely with their true positions as determined from Eq. [17]. Despite the extra complication this introduces into the analysis of such samplings, they do have the practical advantages of being more easily inverted, cf. Fig. 9 and discussion. This is illustrated schematically in Fig. 7; Fig. 7b represents the ‘‘ideal’’ irregular sampling as dt r 0 while Fig. 7a shows the result of constraining the samples to lie on a regular grid defined by the linear sampling, Fig. 7c.

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As the number of sampling points, M, increases, the results from this dense irregular sampling are expected to tend toward those of nonuniform signal averaging, with the determining factor being the quality of digitization of Y in both cases. For the former sampling method, the density of the (equally weighted) sampling points is tailored to Y, while the latter involves varying the number of acquisitions at regularly sampled values of Y. Constraining dense irregular sampling to a regular grid introduces further quantization problems, but, provided M ! T/ dt, these will not be too pronounced. Hence, the results from related sampling patterns are expected to be comparable (although the interacting quantization errors require numerical evaluation of the bounds in each case); if a sampling function performs poorly as a nonuniform signal averaging pattern, it will still perform poorly in its irregular sampling incarnation, and vice versa. This assumes, however, that the total number of acquisitions is unrestricted. In particular, nonuniform signal-averaging is unsuitable when the number of acquisitions is relatively small. Irregular sampling, however, requires far fewer scans. In particular, if the sampling points are constrained to the time steps of the conventional time-domain signal, the nonlinear samplings will, by definition, extend further into the decay than the conventional linear sampling. In the quantification of J, where such ‘‘late’’ data points are required to resolve the doublet, the balance may shift in favor of irregular samplings. Figure 8 illustrates some irregular samplings, derived from a subset of the patterns used earlier. The samplings are constructed by distributing 128 acquisitions over the first 512 points of the time-domain signal of Fig. 8a, with the sampling points constrained to the grid of the linear sampling. The vertical bars denote the sampling points, with the density of sampling given by the functions Y. Table 2 lists the efficiencies of the samplings relative to the sin( p Jt/2) sampling which is the best in this case. The linear sampling, of just the first 128 points of the signal, is very unsatisfactory as it

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27

FIG. 8. Irregular sampling patterns (b–g) (see Table 2) to be tested against the trial time-domain signal (a) with J Å 0.004, T * 2 Å 100. Each sampling contains 128 sampling points distributed over the first 512 points of the signal. The sampling times ( tn /T *2 ) are indicated by the vertical lines superimposed on the corresponding sampling density function, Y.

fails to reach the nodal point at 1/J Å 250, which results in a very inefficient experiment. The nonlinear samplings, because they sample more of the signal, perform much bet-

TABLE 2 Relative Efficiencies of the Irregular Sampling Patterns of Fig. 8

(b) (c) (d) (e) (f) (g)

Sampling

Efficiency

Uniform Ésin(pJt/2)É Ésin(pJt)É Écos(pJt)É exp(0t/T *2) Ésin(pJt)Éexp(0t/T *2)

0.13 1.00 0.69 0.92 0.56 0.65

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ter. The trends followed by these irregular samplings mirror those of the nonuniform acquisition patterns of Table 1, and the best samplings are those, such as (c) and (e), that concentrate sampling on the nodal points rather than following the envelope of the decay, such as (d) and (g). INVERSION OF IRREGULARLY SAMPLED DATA

Dense irregular samplings have the advantage over sparse nonlinear samplings of containing enough data points to enable an inversion of the time-domain data to obtain a spectrum. Because the data record is incomplete, however, it is necessary to use inversion methods such as MEM or linear prediction rather than the trusty Fourier transform. It is straightforward to apply MEM to such data sets especially

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HODGKINSON, HOLMES, AND HORE

FIG. 9. Inversion of irregularly sampled data sets. (a, d, f) represent the same time-domain signal sampled in different ways: (a) the first 128 points (i.e., linear sampling); (d) 128 points sampled out of the first 512 using Y Å sin( p Jt/2); (f ) 48 points out of the first 512 using the same Y. (b, c) are respectively the Fourier-transform and maximum-entropy spectra of (a). (e, g) are the maximum-entropy spectra corresponding to (d, f). The figures beside the spectra are the peak-to-peak separations of the doublet components in points; each spectrum contains 1024 points.

when the sampling points are constrained to a regular grid: if this is not the case, the FFT algorithm cannot be used and the Fourier transforms must be performed in full. Note that conventional MEM requires the spectrum to be purely positive, as the entropy is defined on the logarithm of the spectrum. Hence, spectra containing negative features, such as anti-phase doublets, must be handled by a modified entropy function, more suitable for NMR data, such as that introduced by Daniell and Hore (21). Figure 9 shows the result of applying MEM to the noisefree signal of Fig. 8 truncated to the first 128 points (Fig. 9a). It is possible to apply the FT to this data set since the artifacts introduced (‘‘sinc wiggles’’) create only minor distortions in the spectrum (Fig. 9b), which are localized around the signals. Although MEM efficiently suppresses the sinc wiggles (Fig. 9c), the data are too heavily truncated for J to be estimated with any confidence. If, however, the 128 sample points are distributed using a nonlinear sampling

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function, such as sin( p Jt/2), then the data record extends further and, crucially, as far as the first nodal point. Fourier transforming such data sets, after replacing unmeasured data points with zeroes, gives grossly distorted spectra (not shown). MEM, however, can still provide a largely artifactfree spectrum (Fig. 9e), although these artifacts, negligible in comparison to those in the FT spectrum, may distort weak spectral features, and nonlinear sampling should be used with caution on signals of large dynamic range ( 5). As the sampling becomes more sparse, for instance, when 48 samples are distributed over 512 points (Fig. 9f), the artifacts in the spectrum become more pronounced (Fig. 9g). Unlike sinc wiggles, these artifacts are not solely in the neighborhood of their source signals which makes the effect of the nonlinear sampling difficult to predict. Nevertheless, the original doublet is reproduced with much greater accuracy than from the truncated data set, and, provided not too much is expected of these spectra, they are useful in providing a fair representation of the desired spectrum. If it is possible to obtain a reasonable spectrum, J can be estimated from the peak-to-peak separation of the components of the doublet. While this is subject to systematic error (22), it may provide useful crude estimates for subsequent model fitting. The peak-to-peak separations beside the spectra in Fig. 9 show that heavy truncation of the time-domain data leads to overestimation of J, even when MEM is used to reduce truncation artifacts; the ideal value of 4 (0.004 1 1024) is reproduced by the MEM spectra of the nonlinearly sampled data sets. This is not, however, an accurate test of the quality of the sampling method as this result is reproduced by all sampling patterns, including random ones (6), that sample far enough into the decay to characterize the first nodal point. Note that the linear sampling could be ‘‘stretched’’ by increasing the time between samples so that the first nodal point was reached after 128 samples. However, this would violate the Nyquist criterion and result in extensive folding of the spectrum. COMPLEX SPECTRA

In practice, even in the context of multidimensional NMR, it is unlikely that the interferograms being quantified will consist of a single resonance. It is important to understand how the presence of other spectral features will affect the quantification. If we consider a model function of two independent signals at different frequencies ( v1 and v2 ): yP n Å a1 exp(iv1n)R1 (n) / a2 exp(iv2n)R2 (n), [18] where R1 and R2 are functions describing the signal modulation, e.g., exp( 0 ln)sin( p Jn) in the case of an anti-phase doublet. It is easily seen from Eq. [3] that off-diagonal elements of the information matrix F , which link parameters for the two lines, are functions of an oscillation at the difference

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frequency v2 0 v1 . For example, the element of F connecting the amplitude of line 1 and the amplitude of line 2, a1 and a2 , is proportional to

∑ Wn Re n

Å

S

ÌyP * n ÌyP n Ìa 1 Ìa 2

D

∑ Wn R1 R2 Re[e i ( v20 v1) n ],

[19]

n

while elements connecting parameters of the same line do not depend on v2 0 v1 in this way, e.g., the diagonal term in F for a1 : 1 ∑ Wn R 21 . s2 n

[20]

If the sums in expressions such as Eq. [19] run over several cycles of the v2 0 v1 oscillation, the elements of F linking the two lines will be small in comparison to those that do not. Hence, if T * 2 ( v2 0 v1 ) @ 1, F becomes block diagonal and the problem reduces to the single-component case considered above. Note how the doublet itself can be resolved into separate signals; when JT * 2 @ 1, the mean of the terms in Eq. [4] become » C 2n … Å » S 2n … Å 12 and » SnCn … Å 0, and so s(J) tends toward s( v ), implying that the optimal sampling patterns for J quantification tend toward those appropriate for simple frequency determination. Hence, for linear samplings, the quantification of a spectral feature is independent of the complexity of spectrum, providing the overlap with other features is not significant. The cancellation of cross terms will be imperfect, however, for most nonlinear samplings; it will usually be necessary to find optimal nonlinear samplings on a case-by-case basis rather than simply summing sampling patterns appropriate to individual spectral features. CONCLUSIONS

Linear sampling (regularly spaced time points and uniform signal averaging) is used to the almost complete exclusion of other (nonlinear) sampling schemes in time-domain NMR. Such sampling ensures a uniform response across the spectral bandwidth and the applicability of the discrete Fourier transform and, as such, is the method of choice when the frequency spectrum is of primary importance. This is no longer the case when the goal of the experiment is parameter quantification. Provided the acquisition length is set correctly, linear sampling can approach the limit of optimal efficiency. In general, however, linear sampling is suboptimal as some time-domain data points prove to be more ‘‘informative’’ than others, most notably when parameters are estimated using model fitting. By concentrating the sampling for indirectly detected dimensions at these key data points,

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29

it is possible to improve the efficiency of experiments significantly. There are two limiting cases where nonlinear sampling may prove to be useful. When there is sufficient time for (or poor sensitivity demands) extensive signal averaging, it is desirable to concentrate the sampling at the significant points of the ‘‘optimal nonlinear sampling,’’ e.g., at the first node of the sin( p Jt) for an anti-phase doublet with 0.3 ( JT * 2 ( 1.4. Such sparse nonlinear samplings are not ‘‘stable’’ in isolation and cannot be inverted, and so require stabilization either by combination with a linear sampling or by spreading the signal averaging, Wn , and/or the sampling times, tn , around the key points. The improvements in precision are relatively modest, corresponding in the most favorable cases to a twofold reduction in experimental time for the same standard deviation on the coupling constant. Evaluation of various strategies using the (normalized) Crame´r– Rao lower bounds confirms that the intuitively sensible policy of matching the density of sampling to the envelope of the signal is a poor choice for a quantification experiment. The other limiting situation, which is more often encountered in multidimensional NMR, is when the number of sampling points is restricted by time constraints, such that the linear sampling would result in a strongly truncated signal. Irregular sampling spreads the sampling points along the NMR response, which improves effective resolution and the accuracy with which small frequency differences, such as coupling constants, can be measured. While it is no longer possible to apply the Fourier transform to such data sets, spectra can still be obtained using nonlinear processing methods such as the MEM. In these cases, the improvements over linear sampling can be dramatic, e.g., the factor of eight for the example considered in Table 2 and Fig. 8. To summarize: although improvements in experimental sensitivity, e.g., the use of optimal flip-angles (17), lead to linear improvements in the quality of both spectra and parameter quantification, we have shown that a sampling strategy, such as linear sampling, which is optimal for spectral sensitivity is not necessarily also optimal for the quantification of NMR parameters. Indeed, the optimal samplings, such as the highly sparse samplings described above, differ substantially from conventional linear samplings. By tailoring the acquisition of the data, it is possible to improve the efficiency of quantification experiments, allowing parameters such as coupling constants to be measured with significantly greater accuracy. GLOSSARY

Efficiency. The ‘‘sensitivity’’ of a particular experimental protocol toward a given parameter, as measured by the ‘‘reliability’’ of that parameter per square root unit time. Exponential sampling. A sampling pattern (usually ‘‘irregular’’) which concentrates the sampling at the beginning

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HODGKINSON, HOLMES, AND HORE

of the decay and whose sampling density falls exponentially, cf. Fig. 8f and Refs. (3, 4). Irregular sampling. A sampling pattern in which the intervals between sampling points are no longer regular, cf. Fig. 8. Dense irregular samplings have sufficient data points to be inverted to a spectrum using techniques such as linear prediction or maximum entropy; the Fourier transform cannot be used on irregularly sampled data. Linear sampling. Conventional sampling using equally spaced time intervals (i.e., regular sampling) and uniform signal averaging. Nonlinear sampling. A sampling pattern in which the sampling points and/or the number of signals averaged at each point is allowed to vary. Nonuniform signal-averaging. Data acquisition using a regular sampling pattern where the number of signals averaged at each data point is allowed to vary, cf. Fig. 5. Normalized Crame´r–Rao lower bound. The Crame´r–Rao lower bound associated with a given parameter and experimental protocol multiplied by the square root of the time taken to acquire the data. This allows the efficiencies of differing experiments to be compared meaningfully. Random samplings. An irregular sampling where the sampling points are randomly distributed along a time interval, cf. Ref. (6). Reliability. A dimensionless measure of the precision of a parameter u, defined by u /s( u ) where s( u ) is the error on u given by its standard deviation. Note that if u is defined on a periodic domain (e.g., frequencies and phases) it is more appropriate to define the reliability simply as the reciprocal of the standard deviation. Sparse nonlinear sampling. A sampling pattern in which only a few data points are acquired, usually just sufficient to allow model fitting, cf. Fig. 4. Such data sets are often highly efficient but cannot be inverted to a spectrum.

REFERENCES 1. J. Pekar, J. S. Leigh, Jr., and B. Chance, J. Magn. Reson. 64, 115 (1985). 2. P. Hodgkinson and P. J. Hore, J. Magn. Reson. B 106, 261 (1995). 3. J. C. J. Barna, E. D. Laue, M. R. Mayger, J. Skilling, and S. J. P. Worrall, J. Magn. Reson. 73, 69 (1987). 4. J. C. J. Barna and E. D. Laue, J. Magn. Reson. 75, 384 (1987). 5. P. Schmieder, A. S. Stern, G. Wagner, and J. C. Hoch, J. Biol. NMR 3, 569 (1993). 6. P. Schmieder, A. S. Stern, G. Wagner, and J. C. Hoch, J. Biol. NMR 4, 483 (1994). 7. J. A. Jones and P. J. Hore, J. Magn. Reson. 92, 276 (1991). 8. J. A. Jones and P. J. Hore, J. Magn. Reson. 92, 363 (1991). 9. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, ‘‘Numerical Recipes in C,’’ 2nd ed., Cambridge Univ. Press, Cambridge, 1992. 10. J. P. Norton, ‘‘An Introduction to Identification,’’ Academic Press, San Diego, 1986. 11. A. van den Bos, in ‘‘Handbook of Measurement Science’’ (P. H. Sydenham, Ed.), Wiley, New York, 1982. 12. A. van den Bos, IEEE Trans. Instrum. Meas. 38, 1005 (1989). 13. H. Barkhuijsen, R. de Beer, and D. van Ormondt, J. Magn. Reson. 67, 371 (1986). 14. R. de Beer, D. van Ormondt, W. W. F. Pijnappel, and J. W. C. van der Ween, in ‘‘Maximum Entropy and Bayesian Methods ’’ (J. Skilling, Ed.), Kluwer Academic, Dordrecht, 1989. 15. Y. Manassen and G. Navon, J. Magn. Reson. 79, 291 (1988). 16. B. Noble and J. W. Daniel, ‘‘ Applied Linear Algebra,’’ 3rd ed., Prentice-Hall, Englewood Cliffs, New Jersey, 1988. 17. R. R. Ernst, G. Bodenhausen, and A. Wokaun, ‘‘Principles of Nuclear Magnetic Resonance in One and Two Dimensions,’’ Oxford Univ. Press, Oxford, 1987. 18. K. Schmidt-Rohr and H. W. Spiess, ‘‘ Multidimensional Solid-State NMR and Polymers,’’ Academic Press, San Diego, 1994. 19. L. Braunschweiler and R. R. Ernst, J. Magn. Reson. 53, 521 (1983).

ACKNOWLEDGMENTS

20. M. Robin, M.-A. Delsuc, E. Guittet, and J.-Y. Lallemand, J. Magn. Reson. 92, 645 (1991).

P.H. thanks the Queen’s College, Oxford for a Senior Scholarship and the Engineering and Physical Sciences Research Council for a Research Studentship.

22. D. Neuhaus, G. Wagner, M. Vas˘a´k, J. H. R. Ka¨gi, and K. Wu¨thrich, Eur. J. Biochem. 151, 257 (1985).

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21. G. J. Daniell and P. J. Hore, J. Magn. Reson. 84, 515 (1989).

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