ANALYTICAL CHEMISTRY, VOL. 51, NO. 9, AUGUST 1979
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Selection of Optimum Parameters for Pulse Fourier Transform Nuclear Magnetic Resonance Edwin D. Becker,” James A. Ferretti, and Prem N. Gambhir’ National Institutes of Health, Bethesda, Maryland 200 74
Factors involved in the acquisition and processing of data in pulse Fourier transform NMR are re-examined and results presented graphically. Included in the discussion are the effects on signal/noise of data acquisition time and exponential filtering, the characteristics of analog filters, and the use of quadrature phase detection. Particular attention is given to the choice of optimum pulse repetitiontime and magnetization flip angle. The effects of zero-filling on signalhoise, resolution, and lineshape are analyzed.
Pulse and Fourier transform methods have been widely adopted in NMR, particularly as a means for enhancing signal-to-noise (S/N) through coherent time averaging; and equipment for carrying out pulse FT studies has become standard in many laboratories. The chemist approaching pulse F T studies for the first time, however, often encounters a rather bewildering array of parameters that must be selected for each experiment. On the basis of intuition, advice from colleagues, and ultimately experience, he normally arrives at reasonable sets of conditions for carrying out different types of experiments. Of course, the parameters needed for such studies are subject to mathematical analysis; indeed, a number of papers have analyzed in detail the effect of various parameters on S / N , resolution, lineshape, and other aspects of pulse FT studies. In many cases, however, the results of these calculations have been presented in a manner not readily conducive to the extraction of pertinent information by a typical NMR spectrometer operator. We felt that it would be desirable to bring together in one place a summary of some of the earlier, more rigorous treatments that exist in the literature, to add a few new points not previously discussed, and to present all of this in a relatively nonmathematical manner that we hope will be of practical value in determining optimum conditions. We have not considered any aspects of instrument design, and we assume in this treatment that a commercial instrument of more or less optimum configuration is normally employed. Likewise, we have not entered into a discussion of the choice of computer, either in terms of optimal size of memory for a given cost or in terms of the word length that might be employed. Again we assume that the usual commercial computers of 16- to 24-bit word length and of 8 to 32K data memory would normally be used.
DATA ACQUISTION TIME Figure l a shows a typical free induction decay (FID) following a pulse. The signal decays with a time constant T2*, which is determined by both the spin-spin relaxation time of the nuclei and the magnetic field inhomogeneity. The instrumental noise, on the other hand, remains constant with time. Thus it can be seen that the S / N in the FID decreases with increasing time. Our interest is generally in obtaining On International Atomic Energy Agency Fellowship from Nuclear Research Laboratory, I.A.R.I., New Delhi 110012, India.
the best possible S,”, not in the FID, but in the Fourier transformed spectrum, as illustrated in Figure Ib. Clearly it is unwise to continue data acquisition of the FID for a very long time, since one is just continuing to accumulate noise that will be Fourier transformed along with the signal. On the other hand, premature truncation of the FID is known to cause a deterioration in resolution, with truncation after an acquisition time T leading in the first approximation to a line width in the transformed spectrum of about 1 / T Hz ( I ) . Without truncation, an FID of time constant Tz* g’ives on Fourier transformation a line width of 1 / r T 2 * ,or approximately 1/3T2*. Hence, data acquisition beyond about 3Tz* provides little gain in resolution, but causes considerable deterioration in S,”. At time 3T2*after the pulse, the signal has decreased to about 5% ( l / e 3 )of its initial value, and in practical cases is usually less than the noise level, while the line width (as we show later) is within 10% of its true value. Figure 2a illustrates the effects of acquisition time on the Fourier transformed spectrum. For short acquisition times the spectrum is relatively noise-free but broadened by truncation, while for long acquisition times it becomes noisy.
EXPONENTIAL FILTERING The foregoing description and examples apply to data as actually obtained without any mathematical manipulations. In addition to the advantage of rapid acquisition of data, an additional virtue of acquiring data in the time domain is that filtering is accomplished by simple multiplication (as opposed to more complicated operations which would be required in the frequency domain). Although any multiplicative function can in principle be used, an exponential is of theoretical significance since NMR line shapes are usually Lorentzian in character, and the Fourier transform of a Lorentzian is an exponential decay. (Other, so-called “apodization functions” are sometimes employed to remove unwanted features arising from instrumental imperfections. While useful, they do introduce distortions in line shape.) When an exponential is used, it can be chosen either to enhance resolution or to improve S/N. For S / N improvement, which is normally what is desired, it is known that the optimum exponential filter function is the so-called “matched filter” (2). A matched filter is simply an exponential with the same time constant as the FID itself. Since the product of two exponentials is an exponential with a sum of the reciprocal time constants, the resultant function is one which on Fourier transformation will still give a Lorentzian line shape but now with twice the width it would have had in the absence of the filtering. We now explore quantitatively S / N obtained in the transformed spectrum with the use of various exponential multipliers, including the matched filter. The peak signal after a 90” pulse can easily be calculated from the Fourier transform definition:
To simplify the calculation. we take a single line at o = 0, which makes f ( t ) = Moe-’/T2’, where M, is the macroscopic magnetization. Since f ( t ) = 0 before application of the pulse
This article not subject to U.S. Copyright. Published 1979 by the American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 9, AUGUST 1979 I
-?-
a
f
b
i
Flgure 1. Typical free induction decay (FID) (a) and the resultant Fourier transformed spectrum (b). Signal height and peak-to-peak noise are indicated. Root mean square (rms) noise is usually smaller than peak-to-peak noise by a factor of about 2.5-5 la1 No Exponential Filter
l b I Matched Filter
T T2* 1 /2
-_ -
S / N at the peak. The limit of integration T represents the data acquisition time. Clearly if T i s very short, S / N in the free induction decay will be high, but because of truncation the resultant Fourier transformed spectral line will be broad and hence will have a relatively low peak S/N. On the other hand, a long acquisition time as we have seen, leads to the accumulation of an excessive amount of noise and thus again decreases the signal to noise in the transformed spectrum. These effects are illustrated in Figure 2a. Exponential filtering can substantially alter these results, since the filter decreases both signal and noise as time increases. We use a filter time constant T f = T2*/n, so that the more general form of Equations 2-7 becomes
1
-
(10) 2
S/N=
4
-
8
Figure 2. (a) Truncation of the FID of Figure 1 by acquisition of data for the times shown, followed by Fourier transformation. (b) Same as (a) except that a matched exponential filter has been applied before Fourier transformation
(i.e., t < O ) , we can integrate from 0 to T , the end of the data acquisition period. Then
S = F(0) =
s 0
T
M&iT2*dt
(2)
= M0T2*(1- e-TiTz*) (3) Because of the random character of noise, we must determine its root mean square (rms) value, which is (4) Here u ( t ) is the instantaneous noise voltage. Under the usual assumptions of random white noise and fluctuations in u ( t ) whose periods are small compared with the time of integration, T , we can replace u ( t ) with an rms average ii ( 2 ) to give
[
N = &Ta2dt]''2
(5)
This expression gives S / N in terms of the ratio T / T 2 * .As expected, the peak S / N as expressed here depends on Tz*, since an FID with large T2*yields a sharp line with higher
M0(2nTz*)'/2(1 - e-(n+l)TiTz* ~ ( +n1) (1 - e-2nTiT2*)1/2
(12)
(As n 0, Equation 1 2 approaches Equation 7 as a limit.) With the use of a matched exponential filter the appearance of the spectrum after truncation changes, as illustrated in Figure 2b. When T >> T2*,filtering reduces noise considerably, while the peak height is reduced by less than a factor of 2. For T I T2*,on the other hand, the exponential filter has little effect on the already badly broadened line. We can obtain a quantitative measure of line broadening due to exponential filtering and truncation by considering the Fourier transform of the real part of the FID for a single peak at zero frequency. Thus from Equation 1 we have
F ( w ) = ~TMoe-t/T~*e-nt/T2*cos wt d t
(13)
Here w is the angular frequency given by w = 2 m . Integration of Equation 13 gives
+
F(w) = {(n 1)T2*+ T.2 *e-("+1)TIT2*[aTZ* sin wT (IZ+ 1) COS W ~ ] / [ ( W T+~ (IZ * )+' 1123 (14) from which the linewidth in Hz at half-maximum, u l p , can be computed for selected values of T and n. The effects on S/Nand on linewidth of truncation and exponential multiplication are summarized in Figure 3. The unfiltered response (n = 0) provides optimum S/Nwith an acquisition time of about 1.2 Tz*. A matched filter corresponds to n = 1 and gives the curves so labeled in Figure 3. Here S / N increases monotonically with increasing acquisition time but is essentially constant after about 2 T2*. The linewidth also reaches its minimum value (twice that of the unfiltered line) near 2 Tz*,so that it is unneccessary from the standpoint of resolution to acquire data beyond this point. In fact, even at 1.5 T2*the line is broadened only 10% above the minimum, and SIN is 97% of its possible maximum value. (We discuss later other effects on the line shape that may be produced by truncation.) Where the spectrum consists of a number of lines with differing line widths, it is not possible to choose an optimum filter for all lines simultaneously. Thus an attempt to use an optimum filter may in fact result in the selection of an exponential filter that is off by a factor of two or even three.
ANALYTICAL CHEMISTRY, VOL. 51, NO. 9, AUGUST 1979
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Spectral W i d t h 1 KHz Filter 5 KHz I
I
I
LL
2
06
k ‘ -
9
f-
Spectral W i d t h 1 KHz Filter KHz
04
LL 01
1
2
3
4
5
6
20
3
‘s,
Figure 4. Effect of the foldover of high frequency noise as a result of poor choice of digitization rate and/or analog filter. (a)Spectral width = 1 kHz (2000 data points/s), filter = 5 kHz. (b) Spectral width = 1 kHz, filter = 1 kHz. Noise level in (a) is theoretically 5’” times as great as that in (b)
T Ti
2
),
7
4
5
6
7
T Ti
Figure 3. Top: S/N in the Fourier transformed spectrum after data acquisition of the FID for time T . Curves are given for several exponential filter functions e-’’Tf, where the filter time constant T, = T,’/n. ( n = 0 corresponds to no filter; n = 1 to a matched filter). Curves are plotted from Equations 7 and 12, with M,‘ T 2 * /P2 arbitrarily set equal to 2 in order to normalize maximum relative S/N to 1.0. Bottom: Linewidth for various exponential filters. Ordinate is the linewidth (full width at half-maximum), v , ~ in~ Hz, , normalized by multiplication by T z * . Curves from Equation 14
We have calculated the results of application of exponential filters with n = l/*, 2, and 3, and these results also are given in Figure 3. Note that with a truncation in the range of 2 T2*the loss in sensitivity by choosing n either too large or too small by a factor of 2 is only 5% or less. Thus the choice of the optimum matched filter can usually be done reasonably well with the knowledge of expected line widths or a rough approximation of the time constant of the FID. Resolution Enhancement. If a rising exponential function ( n < 0) is chosen as the “filter”, the free induction decay that is finally Fourier transformed will persist longer than the true FID and thus will provide an artificial enhancement of resolution. Such applications of exponential filtering have been employed with considerable success (3). I t should be recognized, of course, that resolution enhancement has an unavoidable concomitant of a reduction in S/N. This feature is illustrated also in Figure 3 for values of n = -1 and - I / * . Application of an exponential filter with n = -1 would precisely compensate the actual decay of the experimental response and would lead to a function to be transformed that did not decay a t all. Clearly in this case S / N suffers quite badly. In addition, the abrupt truncation of the resulting function would introduce severe distortions of the sort that we take up later. I t is seldom wise to use functions with n more negative than and even in this case rather high S / N in the original data is required for useful results. In order to lessen problems associated with abrupt truncation of the FID, functions other than a simple exponential are normally used for resolution enhancement ( 3 ) . Provision for exponential filtering is made on all commercial NMR spectrometers, but the terminology used and parameters needed vary from one manufacturer to another. “Line broadening“, “sensitivity enhancement”, and “resolution enhancement” all refer to the exponential filtering process.
RELATION OF S/N IN THE TIME AND FREQUENCY DOMAINS It is apparent from a comparison of an FID with its Fourier transformed spectrum (e.g., Figure 1)that S / N is usually far better in the latter than the former. Qualitatively this difference arises from the fact that in the FID the observed noise represents the composite of noise at all frequencies within the bandwidth of the spectrometer, whereas the FT process sorts out the noise according to frequency. Thus a t each discrete frequency in the transformed spectrum only the noise characteristic of that frequency appears. For simplicity we continue to treat a spectrum of only one line. As indicated in Figure 1, the (S/N)FIDis the ratio of the initial height of the FID to the rms noise; it is measured before application of any exponential filter. I t is easy to show that (S/N)FREQ/(S/N)FID
= [(FT2*/2)(1 - e-TiT,*)]li? (15)
where F is the bandwidth of the spectrometer, and the calculated ( S / N ) F R E Q includes the effect of a matched filter. Except for the factor F1I2,the right hand side of Equation 15 is essentially the same as that of Equation 12 (with n = 1) and accounts for the effect of truncation a t time T. For T 2 3 T2* the exponential factor approaches zero. The dependence on bandwidth results from the sorting effect on the noise, as indicated above. The interpretation of the bandwidth F requires some further consideration in the following paragraphs. We note now that difference in S / N between the two domains can be substantial. For example, for T,* = 1 s and F = 1 kHz, the ratio is about 22. Equation 15 is useful for estimating ( S / N ) F R E Q while data are being accumulated as FIDs when the FID is dominated by a single line, but obviously for a multiline spectrum the relation is not directly applicable. Analog Filters. In order to perform the computations necessary for Fourier transformation of the FID, the time response data must be digitized. I t is well known that the rate of sampling of the FID determines the maximum spectral width that can be displayed. Any spectral lines outside this range appear to be folded back into the spectrum. Whether or not there are folded lines, high frequency noise also folds back in the same mannelca fact that is sometimes overlooked. If not dealt with, the final S I N in the spectrum deteriorates, as indicated in Figure 4. The solution to this problem is to limit the bandwidth of the spectrometer by applying a low-pass analog filter before digitization of the data. Electronic filters have characteristics of the sort displayed in Figure 5 . A simple RC filter attenuates signals by an amount that varies with frequency a t a rate of only about 6 dB/octave (where an octave refers to a doubling of the frequency). Most FT NMR spectrometers utilize the more efficient 4-pole Butterworth filter, which has the characteristics shown in Figure 5 , with a falloff above the cutoff frequency of 24 dB/octave,
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 9, AUGUST 1979 a
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4
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Figure 5. Analog filter characteristics. (a) RC filter. (b) 4-Pole Butterworth filter. (c) 6-Pole Butterworth Filter. Filter equations from reference 4
four times that of the “one-pole’’ RC filter. Still higher order filters (e.g., a 6-pole Butterworth) can also be used, but have the disadvantage of introducing substantial nonlinear phase distortions, which cannot be corrected with the usual software supplied with commercial spectrometers. (Redfield and Gupta ( 5 ) have pointed out that both amplitude and phase corrections-linear and nonlinear-can easily be made by first storing in the computer a Fourier transformed response to an ideal delta function impulse. Surprisingly, this approach has never been adopted for commercial instruments.) As a specific illustration of the effect of the filter, consider the 4-pole Butterworth filter response in Figure 5. This filter has a “cutoff’ frequency of 1 kHz, which means that at 1 kHz it has already dropped by 3 dB (i.e., by about 30%) from its maximum transmission. At 2 kHz it still transmits 6%, and a t 3 kHz it is down to 1% transmission. The “cutoff’ frequency of a filter is commonly designated in electronics literature as the 3-dB point, as we have done here. Some NMR manufacturers (e.g., Nicolet) use this notation, while others (e.g., Varian) specify a filter by stating a frequency a t which it still has 93% transmission. Varian, for example, would call the “1 kHz” filter of Figure 5 a “800 Hz” filter. Phase Detection. The use of a low-pass filter does not completely ensure against foldover of noise. In the detection process with a single phase-sensitive detector, the radiofrequency signals (and noise) emanating from the probe are compared in frequency and phase with a reference signal a t the transmitter frequency. As shown in Figure 6a, in order to cover a spectrum that lies between u f i (the pulse frequency) and vrf + F , a spectral width of F Hz (or a digitization rate of 2F data points per second) must be chosen. After detection the signals are in the audio range, so that a low-pass filter excludes high frequency noise, as indicated. However, it is apparent from Figure 6a that the actual range of frequencies passed by an “F Hz filter” is 2F ( h F about zero). In the digitization process noise a t these “negative” frequencies cannot be distinguished from that a t positive frequencies, so it is folded into the region 0-F, thus increasing the rms noise in this region by 21’2. This additional noise can be eliminated by use of a sharp cutoff radio frequency filter before detection. The result, as shown in Figure 6b, is that the desired spectral region alone is isolated. Thus the use of an rf crystal filter (or “single sideband detection”) improves ( S / N ) F R E Q by approximately 21/2,but there is some loss in flexibility since the pulse frequency must be kept near the rf filter cutoff frequency. An alternative, and generally superior, way of avoiding the foldover of noise a t “negative” frequencies is to use two phase-sensitive detectors set at 90” phase (in quadrature) with respect to each other. Thus positive and negative frequencies can be distinguished from each other, and the total range -F to F is properly displayed without foldover, as illustrated in Figure 6c, with improvement in ( S / N ) F R E Q just equivalent to that obtained with the rf crystal filter. Clearly there is no point in displaying the spectral range 0 to -F, which consists
d
i
1
I C
i
,F F
I
.-.-
urn- F q u s n c y Before 9 e l e m o n 0
Frequency Alter Detecticn
Y
,
F
F
Figure 6. Effect of analog filters and quadrature phase detection on signal/noise. (a) Spectrum obtained with single phase detection and a low-pass audio fiker of FHz cutoff. The spectral WM is FHz, relative to pulse frequency at zero. (b) Same as (a), except that an RF crystal filter is also applied. Note decrease in noise level. (c) Use of QPD to distinguish “positive” from “negative” frequencies, with resultant doubling of spectral width and use of double the number of data points. Noise level is the same as in (b). (d) Same as (c), except that pulse frequency is placed near the center of the spectrum, spectral width is restricted to fF/2, and F / 2 filter is used. Noise level is the same as in (b) and (c). Filter transmissions are approximate
only of noise. As shown in Figure 6d, the pulse frequency can be set near the middle of the desired range, with the spectral width and audio filter set to fF/2. Thus QPD also permits more effective use of rf power. Furthermore, effective procedures can be employed for reducing the intensity of unwanted strong solvent signals ( 5 ) .
CHOICE OF PULSE REPETITION TIME AND FLIP ANGLE In most FT experiments, S/Nis enhanced by coherently averaging the FID signals after a large number of repetitive pulses. A single 90” pulse rotates the entire magnetization into the xy plane and produces maximum signal in the FID. However, when the pulse is repeated, the signal can be much smaller if insufficient time has intervened to permit complete spin-lattice relaxation (characterized by the time TI). A time interval of 5 TI allows 99.3% recovery, which is usually regarded as “complete” for practical purposes. In cases where Tl >> T2*,an interval as long as 5 T1implies that most of the experimental time is spent waiting rather than acquiring data, and the overall efficiency of the experiment suffers. In such cases the optimum efficiency can be obtained by using smaller flip angles than 90” and by waiting a shorter time between pulses than 5 T1. We discuss briefly the two approaches that have been made toward the determination of optimum conditions, with the usual assumption that the xy component of the magnetization has disappeared before application of following pulses and that echo effects are suppressed. (1) Optimization of Repetition Rate at 90” Flip Angle. If the repetition time is less than 5 T,, signals after the first FID will be diminished in magnitude, but this reduction may be more than compensated by the increased number of signals that can be obtained in a given time. Waugh (6) first analyzed this situation and showed that with about four pulses a steady state is set up, after which the signal/noise obtained during the repetition time T is given by
S/N is easily shown to be a maximum when T = 1.27 T I . Under these conditions the amplitude of each FID is 0.72 M,, rather than M o itself, but in a given total experimental time
ANALYTICAL CHEMISTRY, VOL. 51, NO. 9, AUGUST 1979
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Z
G
MO 0
I
0
8-
25 T T)
50
Figure 8. Gain in overall S/N by optimization of flip angle and repetition time. G is the factor by which S/N is improved through the optimization relative to what would be obtained in the same total time by waiting 5 T , between successive pulses. (a) Optimization of flip angle by Equation 18. (b) Optimization of waiting time for 90' pulse ( T = 1.27
I --.)
Mxy = Mo s i n 8
TI)
Nutation of magnetization M through a "flip" angle 0. Projection in xy plane is M, sin 0, and projection on z axis is M, cos Figure 7.
8 the coherently added FIDs give a S/Nthat is 1.43 times as large as that obtained with 7 = 5 T1. Thus a 43% gain in sensitivity (or equivalently a twofold saving in experiment time) is produced. (2) Optimization of the Flip Angle. If a flip angle less than 90" is used, the resulting FID will be reduced in proportion to sin 8, where 0 is the flip angle, as illustrated in Figure 7. However, the z component of the magnetization is now reduced to only Mo cos 8, rather than to zero; hence less time is required for recovery to nearly its initial value. This situation has been analyzed by Ernst and Anderson (71, and later by Jones and Sternlicht (8). Again a steady state is reached after about four pulses, and the signal/noise is then given by
s -- M o ( l - e-r/T1)sin0 _ N
(1 - e - r / T 10)~ ~ ~
(17)
is regarded as fixed and set equal to the acquisition time T (usually 3 T2*or 1.5 - 2 T2*if a matched filter is to be used). By differentiating with respect to 6, it is easily shown that T
cos Oopt = e-T/T1
(18)
Relative to a 90" flip angle with repetition time of 5 T1,the optimum flip angle experiment gives an overall gain in S/N that can be as large as 1.58, as indicated in Figure 8. This maximal gain occurs when the data acquisition time is much less than T I ,but even for T T 1the gain is over 50%. Figure 8 demonstrates that the optimum flip angle approach is always better than the 90" flip angle with longer waiting time. In practice the optimum flip angle method has been almost universally adopted, and we consider only this method in detail in the following discussion. Clearly the major deficiencies in these procedures for optimizing signal/noise is that they depend on a knowledge of T I . The question then arises as to just how sensitive the optimization criterion (Equation 18) is; that is, if one misestimates T I ,how badly will S/N be reduced? Fortunately the optima are found to be rather flat as indicated in Figure 9. Here we have plotted relative S/N vs. flip angle for various ratios of T 1to acquistion time. For each value of T 1 / T ,the dark dot represents the optimum value of 0, as found from Equation 18. For example, if one mistakenly estimated T 1 / T = 1, when actually T I I T = 3, one would carry out the ex-
-
0
50
90
F p A v o e Deg e 2 s
Figure 9. Plot of S/N vs. flip angle for selected values of T J T , from Equation 17. Dots show optimum values of 0, from Equation 18. See text for discussion of the use of these curves
periment at a flip angle of 68O, instead of the true optimum angle of 44". However, the reduction of S / N resulting from this mistaken estimate of T I (as read from the T I /T = 3 curve) would be only 12% (reduced from 41 at the optimum to 36 at the chosen angle of 68'). Figure 9 illustrates also the more serious reduction in S / N resulting from the presence of differing T,'s in the sample. For example, two different nuclei with T 1 / T = 1 and T J T = 3 would have relative intensities differing by a factor of 1.46 if measured under optimum conditions for the longer T1,(44" flip angle). At a flip angle of 68", which is optimum for the shorter T 1 , the disparity in relative intensities is even greater-a factor of 1.89. For quantitative intensity relations among nuclei of different Tl's a flip angle of 90" and a repetition time of at least 5 times the longest TI is normally used. However, Figure 9 shows that for qualitative purposes an experiment optimized on the longest anticipated T1usually results in intensity distortions that are acceptable. For example, even with a ratio of 10 between the longest and shortest Tl's the relative intensities are in the ratio of 1.8 if optimized for T 1 / T = 10; but if optimized for T 1 / T = 1, the intensity ratio is 5.1. In instances where T I is known accurately Equation 17 can be used to correct for the effect of rapid pulsing. This situation is seldom, if ever, encountered in qualitative analysis or structure elucidation studies. However, in quantitative analysis of well defined samples that are to be run frequently (e.g., quality control applications) it may pay to determine Tl's and thus be able to use the time-saving optimization of flip angle, with corrections applied via Equation 17.
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 9, AUGUST 1979 a
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Figure 11. Effects of zero-filling and of data point placement. (a, b ) Fourier transform of same (simulated)FID except for slight frequency displacement. 1K data points acquired in both cases. (c, d) Same Figure 10. Effects of zero-filling. (a) Spectrum without zero-filling, 1K data points. (b) 1K of zeroes added before Fourier transformation. (c) Same as (b) except that the additional 500 data points in the spectrum have been displaced by 1 / 2 T Hz and added to existing data points. Improved SIN is evident in (c)
as (a)and (b), respectively, except that 1K zeroes were added in each case before Fourier transformation. (e, f ) Same as (a) and (b), respectively, except that 3K zeroes were added before transformation. Note that peak obscured in (b) shows clearly in (d) and (f)
ZERO-FILLING The number of data points acquired during a free induction decay is the product of the acquistion time and the rate at which data points are obtained. The data rate (points/s) required to avoid foldover is well known to be just twice the spectral width (Hz);it is thus normally set by the requirements of the experiment. The desired acquisition time for the FID may be as long as 4 T2*when maximum resolution is required (or even longer if resolution enhancement is employed), but, as we have seen, for optimum S / N it may often be chosen to be smaller-often about 2 T2*. In some instances, for a given data rate it is the limitation of computer memory that sets the maximum possible acquisition time; but in most cases the memory is more than adequate for the number of data points acquired. The data should then be supplemented by adding a string of zeroes to the end of the FID (zero-filling) prior to Fourier transformation. The reason that zero-filling improves the quality of the spectrum stems from the nature of the discrete Fourier transform process. After Fourier transformation of N data points, N/2 constitute the “real” spectrum, while the other N / 2 form an “imaginary“ spectrum which is 90’ out of phase with the real part. The usual phase correction procedure results in a final spectrum which is a linear combination of real and imaginary parts, but it too has only N / 2 points. Bartholdi and Ernst (9) have shown that the N/2 points that are “lost” in this process are indeed independent points, which contain additional information. The easiest way to retrieve these points is to add a string of N zeroes to the FID that already contains N points. The resulting transformed spectrum will then have N points, the extra N/2 being interleaved with the original N / 2 , thereby improving definition of the line shape. Since the additional points are independent of the original set, the integrated S/Nover a line turns out to be 2lI2 times as large as with the original set alone, but the peak S / N , which is determined by only one point, remains unchanged. I t is possible to shift the new points in frequency by 1/2T Hz, so that they coincide with the original points, and coherently add them to improve the peak S / N by at most 2’’’. Figure 10 shows an example of such zero-filling and co-addition. However, the use of a matched filter on the FID largely anticipates this gain in S/N. Furthermore, the lineshape advantage associated with the additional N / 2 points is lost on co-addition, as illustrated in Figure 11.
The line in Figure l l a results from Fourier transformation of 1024 (1K) data points acquired over a period of approximately 5 T2* for this simulated, noise-free example. The spectrum is plotted on a greatly expanded abscissa scale to make clear the individual data points used by the recorder in plotting the spectrum. In this case we have carefully adjusted the frequency so that a point appears almost at the peak of the spectral line. Figure l l b differs from l l a only in that the frequency has been altered slightly, so that no point appears at the peak, and the spectral line is flattened. Figures l l c and l l d show the corresponding spectra when 1K of zeroes are added to the FID before Fourier transformation. In Figure l l d the peak position is now defined. The remaining parts of Figure 11 show the effect of adding still more zeroes. These additional points are not independent of the rest and theoretically contain no new information; they are merely interpolated between the previous points. Because of the nature of the F T process, the interpolation function is not a straight line, as might be imagined, but rather a (sin x ) / x function, as we discuss in the next section. In some instances these additional points permit better definition of the line and indicate the peak position more precisely. (Most commercial NMR software permits a listing of peak positions to be obtained, in which a parabolic or exponential function is fit to the data points in order to estimate peak positions as precisely as possible, but the plotting routines merely connect the points with straight line segments.) The spectra in Figure 11 were obtained without the application of any exponential filter. If the FID is acquired for a period of about 3 T2*to 5 T2*and a matched exponential filter then applied to the N data points that have been acquired, the last N / 2 points become almost noise-free zeroes. Adding another N zeroes does in principle add new, independent data points, as we have seen, but these additional points cannot retrieve information lost as a result of the line broadening caused by application of the exponential filter. For example, Figure 12 shows the effect of zero-filling on a simulated spectrum that consists of two closely spaced lines. (Again the frequency has been carefully selected to place the data points in pre-selected positions.) Figure 12a shows the Fourier transformed spectrum with no zeroes added, while Figures 12b and 12c show the Fourier transformed spectrum with different amounts of zero-filling. Figure 12d shows the spectrum after application of a matched filter and Figures 12, e-f show the corresponding spectra with zero filling. Without
ANALYTICAL CHEMISTRY, VOL. 51, NO. 9, AUGUST 1979
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Flgure 12. Effects of zero-filling and exponential filtering. (a) Fourier
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transformed spectrum of a simulated FID formed from two closely spaced lines. 1K data points acquired. (b) Same as (a) except that 1K zeroes were added before transformation. (c) Same as (a) except that 3K zeroes were added before transformation. (d) Same as (a) except that a matched filter was applied before transformation. (e) Same as (d) except that 1K zeroes were added before transformation. (f) Same as (d) except that 3K zeroes were added before transformation
the exponential filter, the zero-filling clearly shows the presence of two lines, thus again illustrating the independence of the additional data points. In the filtered FID the information at long acquisition times, which is essential for the resolution of the two lines, is lost and cannot be retrieved by zero-filling.
TRUNCATION AND THE APPEARANCE OF SIDE LOBES Since we cannot acquire data on the free induction decay out to an infinite time, every pulse FT experiment results in some truncation of the FID. Mathematically, Fourier transformation of a truncated FID is equivalent to convolving the Lorentzian line resulting from the transformation of the infinite FID with a function of the form (sin oT)/wT. This function, which is illustrated in Figure 13a, is the Fourier transform of the rectangular “window” that represents the time during which data are acquired. One effect of this convolution is to introduce side lobes, of the sort shown in Figure 13a. However, such side lobes will be observed only when the following conditions are met: (1)zero-filling provides enough data points; and ( 2 ) the S/N at the point of truncation of the FID is sufficiently large. If zero-filling is not employed, the Fourier transformed spectrum consists of a data point each 1/T Hz, since for a spectral width W (Hz), 2 W data points/s must be acquired to avoid foldover. With an acquisition time T , 2 W T data points are thus acquired and transformed, but only half of these are “real” points, W T points spread over W Hz. In Figure 13, b-d, we show the Fourier transformation of a rectangular window. With no zero filling, or even with one added set of zeroes, information on the presence of side lobes is lost. Only when additional zero-filling is used, then, do we need to consider the possibility of the appearance of side lobes. Even when zero-filling provides the points to define side lobes, these features may not be very pronounced. In the (sin wT)/oT function itself the peak-to-peak amplitude of the first side lobe is nearly 25% of the height of the central peak, but after convolution with a Lorentzian line of moderate width the side lobes are less apparent. In fact, whether such side lobes can be seen a t all depends on the extent to which the free induction signal has decayed at the point of truncation (Le., the ratio T / T 2 * )and the noise level of the final transformed spectrum. For example, Figure 14 shows greatly amplified Fourier transforms of a simulated, noise-free FID that has been truncated at T = 3 T2*and 5 Tz*. The side lobes are unobservable in the latter, in which the FID has decayed to only 0.7% of its initial height, while in the former
Figure 13. Effects of truncation of the FID. (a) (sin wT)/wTfunction. (b) Fourier transform of a rectangular function of width Twith 1K data points acquired. Note that data points occur only every 1ITHz. (c) Same as (b) except that 1K zeroes were added to the rectangular time function before transformation. Points now occur at every 1/2THz. (d) Same as (b) except that 3K zeroes were added before transformation. Points now occur at every 114THz and show presence of side lobes
Flgure 14. (a) Fourier transform of a simulated, noise-free FID that has been truncated after T = 3T2*. (b) Same as (a) except that truncation occurred at T = 5 T,’
(truncated at 5% height) the side lobes are evident. In real cases, where noise is present in the transformed spectrum, even these side lobes would be largely masked. Although side lobes of intense, sharp lines are occasionally troublesome in partially obscuring weak lines, they can always be eliminated by using a sufficiently long acquisition time or by applying an appropriate exponential filter.
CONCLUSIONS In situations where long pulse-FT experiments are needed because of poor signal (e.g., nuclei of low sensititivity and/or low natural abundance; samples of low concentration), attention to the factors discussed here can provide significant improvements in S / N , or even more dramatic savings in instrument time for a given S/N. We have noticed that there is often a tendency on the part of inexperienced operators to use an acquisition time that is too long, to omit the use of a matched filter, and to pay inadequate attention to the optimization of flip angles. Consider two examples. First, take T I = 3 s, and Tz* = s (a linewidth of 1 Hz).Under op-
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 9, AUGUST 1979
timized conditions ( T = 2 T2*= 0.67 s, use of matched filter, and 8 = 37') the relative S / N is 40 (from the product of numbers given in Figures 3 and 9 divided by T':'). On the other hand, if data were acquired for 2 s without a matched filter and 0 were set equal to 90°, the relative S / N would be only 20. Thus 4 times as long would be required to obtain the same S / N under these non-optimum conditions. As a second example, consider a quadrupolar nucleus in which T I = Tz = Tz* = 50 ms. Optimum conditions (T = 1.5 T2*)would give a relative S / N of 282. An operator inexperienced with nuclei of such short relaxation times often tends to acquire data for far too long a period, and if a matched filter is not applied, serious reduction in S / N can occur. If T i s set equal to 0.5 s (10 T2*)and 8 is taken a t 90°, the relative S / N is only 63, thus requiring 20 times as much experimental time to obtain equivalent signal/noise! Failure to use an appropriate analog filter can provide equally significant losses in S/N. Fortunately most modern NMR spectrometers provide automatic selection of filter matched to spectral width unless manually overridden by the operator. Other factors that we have discussed have less effect
on S/h' but are significant in reproducing line shapes. ACKNOWLEDGMENT We than D. H. Hoult and H. Shindo for their help in obtaining Figures 6 and 11-14. LITERATURE CITED (1) Farrar T. C.; Becker, E. D. "Pulse and Fourier Transform NMR"; Academic Press: New York, 1971: p 71. (2) Ernst, R. R. Adv. Magn. Reson. 1966, 2 , 1. (3) Ferigge A. G.; Lindon, J. C . J. Magn. Reson. 1978, 31, 337; DeMarco A,; Wuthrich, K. b i d . 1976, 24, 201. (4) "Reference Data for Radio Engineers", 6th ed.; Howard W. Sams and Co.: Indianapolis, Ind., 1975. (5) Redfield, A. G.;Gupta, R. K . Adv. Magn. Reson. 1971, 5, 81. (6) Waugh, J. S. J . Mol. Spectrosc. 1970, 3 5 , 298. (7) Ernst R. R.: Anderson, W. A. Rev. Sci. Instrum. 1966, 3 7 , 93. ( 8 ) Jones D. E.;Sternlicht, H. J . Magn. Reson. 1972, 6. 167. (9) Bartholdi E.; Ernst, R. R. J . Magn. Reson. 1973, 7 7 , 9.
RECEIVED for review January 26, 1979. Accepted April 23, 1979. This paper was presented in part a t the 17th Experimental NMR Conference, Pittsburgh, Pa., April 1976. P.N.G. thanks the International Atomic Energy Agency for fellowship support.
Quality of Mass and Intensity Measurements from a High Performance Mass Spectrometer K. D. Kilburn, P. H. Lewis," and J. G. Underwood' B.A.T. Company Ltd., Regents Park Road, Southampton, England
S. Evans and J. Holmes' Kratos Lid., Barton Dock Road, Urmston, Manchester, England
M. Dean Ferranti Ltd., Wythenshawe, Manchester, England
Using a double focusing mass spectrometer, a series of scans was obtained at resolutions of 10 000 and 40 000 (10 % valley definition). The data were analyzed to assess the accuracy and reproducibility of the mass and ion intenstty measurements and the results were significantly better than those reported previously. A comparison was made with theoretical values obtained from the estimated numbers of ions in each peak and differences were adequately described by ion statistics.
The assignment of elemental compositions to the peaks in a mass spectrum is assisted both by accurate mass measurement, which allows the elimination of those compositions with similar masses, and by accurate intensity measurement, which facilitates the use of isotopic distribution information. Present address, Insurance Technical Bureau, Petty France, London, England. 2Present address, VG Organic Ltd., Tudor Road, Altrincham, England.
For organic analysis, it is also important to maintain high accuracy when scanning over a wide range of mass values and, since statistical errors may be reduced by averaging across scans, the reproducibility of good data must be maintained. For these reasons, a continuous effort is made to improve the performance of high resolution mass spectrometers and several reports of the quality of mass and intensity data have been published (1-4). In this paper we present an evaluation of the performance of a high resolution mass spectrometer when scanning over a wide range in mass.
EXPERIMENTAL The instrumentation used in this study consisted of a Kratos MS5074 mass spectrometer coupled directly to a data system operating on a Ferranti Argus 500 computer ( 5 ) .
