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Systematic errors in the discrete integration of FT NMR spectra
JOURNAL
OF MAGNETIC
RESONANCE
84,490-500
( 1989)
Systematic Errors in the Discrete Integration of FT NMR Spectra KARENMCLEODAND
MELVINB. COMISAROW
Department of Chemistry, University ofBritish Columbia, Vancouver, British Columbia, Canada V6T 1 Y6 Received July 20, 1988; revised February 7, 1989 The systematic error for digital integration of FT NMR spectra is theoretically derived. The error is evaluated as a function of the finite limits of integration, the number of zero fillings, and T/T;“, the ratio of the acquisition time to the relaxation time of the free induction decay. Finite integration limits are inherent in digital integration and give rise to an integration error which is most often negative. Under properly chosen conditions, the relative error, the difference in the error from peak to peak in the spectrum, can be made lessthan the absolute error. For typical conditions, integration limits which include all contributions in excess of 1% of the largest discrete intensity will leave a residual systematic error in the determination of the relative number of nuclei of less than 10% if the FID is zero filled once. Two levels of zero filling will reduce this relative error to about 2%.
0 1989 Academic Press, Inc.
THEORY
ContinuousNMR spectra.Nuclear magnetic resonance differs from many other forms of spectroscopy in that the extinction coefficient for NMR absorption from a resonant electromagnetic field is constant for all nuclei of a given type. It follows then that in an NMR spectrum the relative number of nuclei at each chemical shift can be determined directly from the NMR peak, i.e., without a prior determination of the individual “extinction coefficients.” In high resolution, conventional, scanning NMR (I ) the lineshape is usually taken (slow passage, low power conditions) to be the Lorentzian lineshape, L( w) ( I ), L(w) =
KTT 1 + (o - w~)~T: ’
where o. is the resonant frequency, Tz is the experimental transverse relaxation time, and K is a constant proportional to the number of nuclei with resonant frequency oo. In the limit w + cd0
[21
L(wo) = KT,*
[31
Eq. [ 1 ] reduces to
which gives the peak height of the Lorentzian peak. The peak height depends upon both Kand 7’: and since TT varies from peak to peak, the peak height in a multipeak 0022-2364/89$3.00 Copyright 0 1989 by Academic Press, Inc. All rights ofreproduction in any form reserved.
490
ERRORS
IN
INTEGRATION
OF
FI
491
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spectrum is unsuitable for meas8uring the relative number of nuclei at different chemical shifts. On the other hand, the integral of L( w ) , i-CC L( w)dw = nK, ]41 s -cc is proportional to K but is independent of Tz. Equations [ 31 and [ 41 provide the rationale for measuring relative NMR nuclear populations by integration rather than by peak height. FT NMR spectra. In Fourier transform nuclear magnetic resonance spectroscopy (2) the entire NMR sample is excited and the received signal, the free induction decay, is a sum of signals, each of the form f(t)
= 2aKcos(qt)exp(-t/T:),
O
]51
where T is the acquisition time of the signal. The frequency spectrum of Eq. [ 5 ] is given by its Fourier transform, F(w), F(w) = ~+cf(t)e-‘u’dt, -m
O
The real part of F( w ), the absorption spectrum, A ( w ) , is, neglecting negative frequency components (3,4),
/Lj(w)=
KT: 1 + (w - u,,)*T:~
[l + (ew(-T/T,*)) X ((0 - wO)T:sin((w
- wO)T) - cos((w - wO)T))].
[7]
In the limit [ 21, Eq. [ 71 reduces to (4, 5) A(w,)
= KT:[l
- exp(-T/T?)]
181
which gives the peak height of the absorption lineshape, Eq. [ 71. In the limit T-CO
[91
Eq. [ 7 ] reduces to L( w ) the Lorentzian lineshape, Eq. [ 11. For any time function, f( t), the continuous integral, CI, of its Fourier transform, F( w ), i.e., the area under the spectral lineshape, is CI = sfm F(w)dw
-‘m = “f(O).
= stm J+mf(t)eP”‘dtdw,
-00 -co
O
[lOI t111
Note that the integral depends on the initial value, f( 0)) of the time function but is otherwise independent off(t) . When applied to FT NMR spectroscopy, Eqs. [ 6 ] and [ 7 ] show that the integration of spectra will give the relative number of nuclei at each chemical shift even when the absorption lineshape is non-Lorentzian; i.e., limit [ 91 need not apply. Discrete FT NMR spectra. In FT NMR spectroscopy, the signal, Eq. [ 51, is sampled N times at a rate S for a time period T, called the acquisition time, given by
492
MCLEOD
AND COMISAROW
T= N/S
[I21
to give a discrete time signal. The discrete time signal may be zero filled (6, 7) with each zero filling doubling the length of the time-domain data set. Discrete Fourier transformation of this data set gives, after phase corrections, a discrete absorption spectrum and a discrete dispersion spectrum, each of 2 ‘-‘N data points, where n is the number of zero fillings. Each of the discrete absorption spectrum and the discrete dispersion spectrum is defined only at the A4 ( =2*-‘N) frequencies&, given by
fm=$T5
1,2 ,...,
m=O,
M-
1.
[I31
The channel spacing Af, in hertz, in the discrete spectrum is given by Af =+T.
[I41
Figure 1 illustrates a continuous absorption peak A, calculated from Eq. [ 71. The crosses in Fig. 1 are values in the non-zero-filled (y1= 0) discrete absorption spectrum at the frequenciesf,. The discrete angular frequencies o, are given by
The location, wo, of a continuous spectral peak which lies within *$Af (Eq. [ 91) of the mth point in the discrete spectrum can be specified in terms of the parameters of the discrete spectrum by wo =
2am + (b 2”T ’
n=0,1,2
)...)
where the frequency shift angle 4 varies from -r (when w. lies halfway between m and m - 1) to K (when o. lies halfway between m and m + 1) . For discrete spectra, integration is given by the product of the channel spacing and the sum of spectral intensities, M-l
DI=
2~Af
2 A,, m=O
[I71
where DI is the discrete integral. Note that each A, (Eq. [7]) is a function of both m (Eq. [15]and$(Eq. [16]). Because of the discrete nature of the spectrum, for any finite number of zero fillings DI is only an approximation to the true continuous integral CI (Eq. [ 11 I). In addition, the difference between CI and DI will also depend upon 4 (Eq. [ 16 ] ), i.e., exactly where the continuous peak falls between two frequencies in the discrete spectrum. This is illustrated in Figs. 1 and 2. Figure 1 shows a continuous absorption peak, calculated from Eq. [ 71, with T/ Tf = 0.7, which falls exactly on (q5 = 0.0) one of the frequencies of the discrete spectrum. The crosses in the figure are the values in the non-zero-filled discrete spectrum. Note that all of the discrete intensities are positive. Figure 2 shows a spectrum which is identical, but for the value of 4. Note that the
ERRORS IN INTEGRATION
T T=
493
OF FT SPECTRA
0.7
rad/sec
w urn-1 I frnL
m-4’
T
m-1
I m-2
Wm=Wo I
T
I m-2
urn+1
I m T
I
I
I
m
I m*2 T
I
I
I
In+2
I m+4 T
I
’ Hz
I
m+4
FIG. 1. Continuous absorption lineshape and non-zero-filled discrete absorption lineshape. Curve A is the continuous absorption lineshape calculated from Eq. [ 71 of the text with the ratio of acquisition time to relaxation time, T/ T: = 0.7. The continuous peak height is given by Eq. [ 8 1. The peak location, w0 ( Eq. [ 161)) falls exactly OFIthe channel m = m of the discrete spectrum. The crosses are the values in the non-zero-filled discrete absorption spectrum. Curve DA is the discrete absorption lineshape formed by straight-line connection ofthe values in the discrete spectrum. Note that all values in the discrete spectrum are positive.
discrete intensities in Fig. 1 not only differ from those in Fig. 2 but also include many negative values. Since C#J cannot be known in advance any study of digital integration errors must allow for this variance in 4. In experimental practice the discrete spectral intensities are summed and the analog equivalent of the running sum is displayed as a function of frequency. For any particular peak in the spectrum the running sum has a jump whose amplitude is the integral associated with the peak. Quantitative assessment of a particular jump requires the arbitrary setting of limits to the integration range associated with the jump. This is especially true when the running sum is partitioned into partial sums each of which generates a numerical output to be associated with a particular peak. Limits
494
MCLEOD
AND COMISAROW
,,!,
radlsec
W
f nl
ml
m-4
m-2
T
T
I
m-4
I
I
mr2 T
m
T
I
m-2
I m
I
I m+2
mr4 T
I
HZ
I m+4
FIG. 2. Continuous absorption lineshape and non-zero-filled discrete absorption lineshape. Curve A is the continuous absorption lineshape calculated from Eq. [ 71 of the text with the ratio of acquisition time to relaxation time, T/T,* = 0.7, and a value of the frequency shift factor b/T (Eq. [ 161) = -1 S. The continuous peak height is given by Eq. [ 81. The peak location w0 (Eq. [ 161) falls between two channels of the non-zero-filled discrete spectrum. The crosses are the values in the non-zero-filled discrete absorption spectrum. Curve DA is the discrete absorption lineshape formed by straight-line connection of the values in the discrete spectrum. Note that curve A is identical, except for location, to curve A in Fig. 1. Note also that the discrete lineshape, DA, has negative intensities and differs from curve DA in Fig. 1.
which are too narrow will miss some of the integral; limits which are too broad will include contributions from other peaks. The choice of limits for integration is thus a parameter which should be varied in a study of digital integration. In this study the limits of integration were chosen to include all spectral intensity in excess of 1,5, 10, or 20% ofA (m), the maximum discrete intensity. Thus, in practice, the discrete integral is given by Eq. [ 18 ] rather than Eq. [ 17 1, upper limit
DI=
27rAf
C
A,.
[If41
lower limit
The integration error of this work is given by the difference between the discrete integral, DI, and the continuous integral, Kr,
ERRORS IN INTEGRATION
OF FT SPECTRA
%errOr= DI W. t181- K71. (Eq.t41) K??
1oo
495 [I91
In usual experimental NMR practice, only the positive frequency spectrum is plotted and integrated. As a consequence the appropriate continuous integral to be used for comparison with the digital integral (Eq. [IS ] ) and used for error calculation (Eq. [ 19 ] ) is Kr, which is half the value of the FID (Eq. [ 5 ] ) at t = 0. RESULTS
The error (Eq. [ 19 1) was calculated for a particular
value of the frequency shift
angle 4 (Eq.t161), for a particular level y1of zero filling, a particular value of the integration limit, and a particular value of T/ Tz . The error calculation was then repeated for different values of 4 in the range 0 < 4 < r and the largest positive and negative errors were noted. Because of the symmetry of the lineshape about oo, it is redundant to examine the range - T < 4 < 0. These calculations were then repeated for other values of T/ TT in the range 0.0 < T/ T,* < 4.0. T/ T: = 0.0 corresponds to an undamped FID; T/ Tf = 4.0 corresponds to an FID which has decayed to 1.8% of its initial value and is essentially completely damped. Our results then cover essentially all cases from undamped to completely damped FIDs. Figure 3 graphically displays typical results derived from Eq. [ 19 1. Figure 3 displays the most positive and the most negative errors, for whatever value of 4 gave those errors, for a non-zero-filled discrete absorption spectrum as a function of T/ Tz . The negative errors result from values of 4 which gave lineshapes with negative discrete intensities; cf. Fig. 2. Note that the error decreases as T/ T,* increases. This is characteristic of all cases and is a consequence of the greater linewidth as T: decreases. The acquisition time T( Eqs. [ 61, [ 121) is held constant in this work and Tf must decrease for T/ Tz to increase. As the relaxation time T: decreases, both the linewidth and the number of discrete intensities across a particular peak will increase. This makes the digital summation DI (Eq. [ 17 or 181 a better approximation to the continuous integral, CI (Eq. [lo]). Another characteristic of all figures is that for larger values T/ Tf , where the discrete integral approaches the continuous integral, the remaining error is always negative. This is a consequence of the finite limits of the digital integration; i.e., not all of the peak is included in the smmmation. The four graphs in Fig. 3 display the errors which result from summation limits which include spectral intensity in excess of 1, 5, 10, and 20%, respectively, of the largest discrete intensity. Note that this negative error at T/ TT = 4.0 becomes more negative as the spectral intensity included in the summation decreases (Fig. 3a to Fig. 3d.) Figures 4 and 5 display results like those of Fig. 3, but for zero-filled spectra. Note that for three levels of zero filling (Fig. 5 ), the digital error is essentially independent of both T/ Tz and 4. This is because three levels of zero filling provide a discrete spectrum in which the intensities differ from the continuous intensities by less than 3% for T/ T: = 0.0 and less than 1% for T/ Tc = 4.0 (4, 7). Three levels of zero filling thus give a discrete spectrum which is a good approximation to the continuous spectrum. As a consequence, the difference between the continuous integral CI (EQ. [ lo] ) and the discrete integral DI (Eq. [ 17 ] ) becomes negligible.
496
MCLEOD
l&‘,rirEit
AND COMISAROW
(Cl
% Error 100.0
iK,iEAit(d)
T/T ;
FIG. 3. Discrete integration error as a function of T/ T,* for non-zero-filled discrete absorption spectra. The errors, which vary with $J/T ( Eq. [ 161)) were calculated from Eq. [ 191 of the text. The largest positive error and the largest negative error for each value of T/ T:, found by numerical searching, are displayed. The four graphs, (a)-(d) , display the errors for integration limits which include all spectral intensity in excessof 1, 5, 10, and 20%, respectively, of the largest discrete intensity (A (m = m)) in the non-zero-filled discrete spectrum. Note that for all cases the errors are greater than the integration errors for zero-filled spectra (Figs. 4-6).
DISCUSSION
The errors which are discussed in this work are systematic errors which result from the discrete nature of experimental FT NMR spectra. Unlike random errors which are characterized by their root-mean-square values, systematic errors should be characterized by the largest possible error which can occur. Since the location o. of a spectral peak with respect to a discrete frequency f, (Eq. [ 131) is unknowable in advance, 4 must be varied to find the largest error which can occur. These largest possible errors, which derive from the worst cases which could occur in experimental practice, are shown in Figs. 3-6. It is evident from Figs. 3-6 that some spectra will give rise to a digital integral which is greater than the true integral (positive integration error) ; others give rise to a digital
ERRORS IN INTEGRATION
-100.0
-100.0 Er%or 100.0
OF FT SPECTRA
l&=A
(”
2X
Er%r 100.0
lilnit
(d)
II,
50.0
-
1 .o
2.0
3.0
4.0 1
T/T;
‘;-
100.0
1
FIG. 4. Discrete integration error as a function of T/ TT, for once-zero-filled discrete absorption spectra. Theerrors,whichvarywith+/T(Eq.[16]), were calculated from Eq. [ 191 of the text. The largest positive error and the largest negative error for each value of T/T:, found by numerical searching, are displayed. The four graphs, (a)-(d), display the errors for integration limits which include all spectral intensity in excess of 1,5, 10, and 20%, respectively, ofthe largest discrete intensity (A( m = m)) in the once-zero-filled discrete spectrum. Note that for all cases the errors are less than the integration errors for non-zero-filled spectra (Fig. 3) and exceed the errors for twice (Fig. 5 ) or three-time-zero-filled (Fig. 6) spectra.
integral which is less than the true integral (negative integration error). Figure 1 shows a spectrum which would have a positive integration error. Figure 2 shows a spectrum which would have a negative integration error. Note that these spectra differ only in their positions relative to discrete frequencies of the discrete spectrum. Although T, the acquisition time (Eqs. [ 61 and [ 12]), is chosen by the operator and is known in advance, the value of T/ Tf , the ratio of the acquisition time to the relaxation time of the FID, is unknown prior to running an FT NMR experiment. Typically, T/ TT is of the order of 3.0 for ‘H NMR and is somewhat less for resonances of other magnetic nuclei. T/ Tz can also vary from peak to peak in a given spectrum. Because the systematic integration error is a strong function of T/ TT the integration errors are presented as a function of T/ TT in Figs. 3-6, so that the results will cover all cases which could occur in practice.
498
MCLEOD
AND COMISAROW
%
Error 100.0
&Zit
(b)
4
50.0
-
0
50.0 3.0
4.0I
-
T/T; 0
-5o.o-
-50.0
-
-lOO.O-
-100.0
-
ErTor 100.0
50.0 i
el._l\:.-.--
ll&Ti,“it
(‘I
EZor 100.0
2X
IAt
(d)
50.0 I
T/T;;_ j;;,;i:,,:_._O
T/T;
FIG. 5. Discrete integration error as a function of T/ Tf for twice-zero-filled discrete absorption spectra. The errors, which vary with 4 / T ( Eq. [ 16 ] ) , were calculated from Eq. [ 19 ] of the text. The largest positive error and the largest negative error for each value of T/T,*, found by numerical searching, are displayed. The four graphs, (a)-(d) , display the errors for integration limits which include all spectral intensity in excess of 1, 5, 10, and 20%, respectively, of the largest discrete intensity (A (m = m)) in the twice-zerofilled discrete spectrum. Note that for all cases the errors are less than those of Figs. 3 and 4 but greater than those of Fig. 5.
Ever since the publication of Barthodi and Ernst (6), it has been accepted that one level of zero filling will enhance the information content of absorption-mode NMR spectra over that contained in the corresponding non-zero-filled absorption spectra. Further zero fillings, however, merely interpolate toward the continuous lineshape. The present work shows that integration accuracy is also enhanced by zero filling prior to integration. Integration is used to measure the relative number of nuclei at each chemical shift. It follows then that the most significant error indicated in Figs. 3-6 is not the absolute error but rather the error range, the difference between the largest positive error and the largest negative error which can occur in a given spectrum. For example, consider Fig. 6a for T/ Tc > 3.0, where the absolute error (Eq. [ 18 ] ) is - lo%, but the error range is negligible. For an FID whose acquisition time T was long enough such T/ T: > 3.0 for every peak in the spectrum, the digital integrals of each peak will be
ERRORS IN INTEGRATION
499
OF FT SPECTRA
% Error
&kfit
100.0
lb)
II
%
,&=,i,“it
Error 100.0
(cl
% Error 100.0
50.0
50.0 i o]\c
-50.0
- 100.0
i 3/.0
4i0
T/T;
Oj~\~O
T/T;
-50.0
I
-100.0
i
FIG. 6. Discrete integration error as a function of T/T: for three-time-zero-filled discrete absorption spectra. The errors, which vary with I$/ T (Eq. [ 16 ] ) , were calculated from Eq. [ 191 of the text. The largest positive error and the largest negative error for each value of T/T:, found by numerical searching, are displayed. The four graphs, (a)-(d), display the errors for integration limits which include all spectral intensity in excess of 1, 5, 10, and 2070, respectively, of the largest discrete intensity (A( WI = m)) in the three-time-zero-filled discrete spectrum. Note that for all casesthe errors are less those of Figs. 3-5.
less than the true value by 10% but the relative error will be negligible. Digital integration of this type of spectrum will give a measure of the relative number of nuclei which has virtually no error arising from the discrete nature of the spectrum. In contrast, Fig. 3a describes the case where the relative error at T/ Tf = 3.0 will be about 10%. The differences among Figs. 3-6 show the effect of increased zero filling upon integration accuracy. Two trends are evident. First, the error range, the difference between the largest positive error and the largest negative error, is dramatically reduced by zero filling. Second, the error range is little reduced by zero filling beyond the 12 = 2 level. For NMR spectra with T/ T,* > 1.O, which would include the majority of high-resolution NMR spectra, one level of zero filling (Fig. 4) will reduce the relative integration error to less than 10% if a 1% integration limit is used and less than 20%
500
MCLEOD
AND
COMISAROW
if a 20% integration limit is used. Two levels of zero filling (Fig. 5 ) will leave a residual systematic error in determining the relative number of nuclei of typically 2% for a 1% integration limit and 6% for a 20% integration limit. In most cases one level of zero filling will be sufficient to reduce the systematic integration error to a low level, providing a 1% integration limit is used. However, in cases where it is important to exclude contributions from nearby peaks, a higher integration limit, say 20%, may be desirable. In this situation more than one level of zero filling will permit accurate digital summation. This follows from noting the reduction in error in the figure sequence 3d, 4d, 5d, 6d. ACKNOWLEDGMENT
This research was supported by the Natural Sciences and Engineering Research Council of Canada. REFERENCES 1. J. A. POPLE, W. G. SCHNEIDER,AND H. J. BERNSTEIN, “High-Resolution Nuclear Magnetic Resonance,” McGraw-Hill, New York, 1959. 2. R. R. ERNST AND W. A. ANDERSON, Rev. Sci. Instrum. 37,93 ( 1966). 3. A. G. MARSHALL, M. B. COMISAROW, AND G. PARISOD, J. Chem. Phys. 71,4434 ( 1979). 4. S. 0. CHAN AND M. B. COMISAROW, J. Magn. Reson. 51,252 ( 1983). 5. C.GIANCASPROANDM.B.COMISAROW, Appl.Spectrosc.37,153(1983). 6. E. BARTHOLDI AND R. R. ERNST, J. Magn. Reson. 11,9 ( 1973). 7. M. B. COMISAROW AND J. D. MELKA, Anal. Chem. 51,2198 ( 1979).
OF MAGNETIC
RESONANCE
84,490-500
( 1989)
Systematic Errors in the Discrete Integration of FT NMR Spectra KARENMCLEODAND
MELVINB. COMISAROW
Department of Chemistry, University ofBritish Columbia, Vancouver, British Columbia, Canada V6T 1 Y6 Received July 20, 1988; revised February 7, 1989 The systematic error for digital integration of FT NMR spectra is theoretically derived. The error is evaluated as a function of the finite limits of integration, the number of zero fillings, and T/T;“, the ratio of the acquisition time to the relaxation time of the free induction decay. Finite integration limits are inherent in digital integration and give rise to an integration error which is most often negative. Under properly chosen conditions, the relative error, the difference in the error from peak to peak in the spectrum, can be made lessthan the absolute error. For typical conditions, integration limits which include all contributions in excess of 1% of the largest discrete intensity will leave a residual systematic error in the determination of the relative number of nuclei of less than 10% if the FID is zero filled once. Two levels of zero filling will reduce this relative error to about 2%.
0 1989 Academic Press, Inc.
THEORY
ContinuousNMR spectra.Nuclear magnetic resonance differs from many other forms of spectroscopy in that the extinction coefficient for NMR absorption from a resonant electromagnetic field is constant for all nuclei of a given type. It follows then that in an NMR spectrum the relative number of nuclei at each chemical shift can be determined directly from the NMR peak, i.e., without a prior determination of the individual “extinction coefficients.” In high resolution, conventional, scanning NMR (I ) the lineshape is usually taken (slow passage, low power conditions) to be the Lorentzian lineshape, L( w) ( I ), L(w) =
KTT 1 + (o - w~)~T: ’
where o. is the resonant frequency, Tz is the experimental transverse relaxation time, and K is a constant proportional to the number of nuclei with resonant frequency oo. In the limit w + cd0
[21
L(wo) = KT,*
[31
Eq. [ 1 ] reduces to
which gives the peak height of the Lorentzian peak. The peak height depends upon both Kand 7’: and since TT varies from peak to peak, the peak height in a multipeak 0022-2364/89$3.00 Copyright 0 1989 by Academic Press, Inc. All rights ofreproduction in any form reserved.
490
ERRORS
IN
INTEGRATION
OF
FI
491
SPECTRA
spectrum is unsuitable for meas8uring the relative number of nuclei at different chemical shifts. On the other hand, the integral of L( w ) , i-CC L( w)dw = nK, ]41 s -cc is proportional to K but is independent of Tz. Equations [ 31 and [ 41 provide the rationale for measuring relative NMR nuclear populations by integration rather than by peak height. FT NMR spectra. In Fourier transform nuclear magnetic resonance spectroscopy (2) the entire NMR sample is excited and the received signal, the free induction decay, is a sum of signals, each of the form f(t)
= 2aKcos(qt)exp(-t/T:),
O
]51
where T is the acquisition time of the signal. The frequency spectrum of Eq. [ 5 ] is given by its Fourier transform, F(w), F(w) = ~+cf(t)e-‘u’dt, -m
O
The real part of F( w ), the absorption spectrum, A ( w ) , is, neglecting negative frequency components (3,4),
/Lj(w)=
KT: 1 + (w - u,,)*T:~
[l + (ew(-T/T,*)) X ((0 - wO)T:sin((w
- wO)T) - cos((w - wO)T))].
[7]
In the limit [ 21, Eq. [ 71 reduces to (4, 5) A(w,)
= KT:[l
- exp(-T/T?)]
181
which gives the peak height of the absorption lineshape, Eq. [ 71. In the limit T-CO
[91
Eq. [ 7 ] reduces to L( w ) the Lorentzian lineshape, Eq. [ 11. For any time function, f( t), the continuous integral, CI, of its Fourier transform, F( w ), i.e., the area under the spectral lineshape, is CI = sfm F(w)dw
-‘m = “f(O).
= stm J+mf(t)eP”‘dtdw,
-00 -co
O
[lOI t111
Note that the integral depends on the initial value, f( 0)) of the time function but is otherwise independent off(t) . When applied to FT NMR spectroscopy, Eqs. [ 6 ] and [ 7 ] show that the integration of spectra will give the relative number of nuclei at each chemical shift even when the absorption lineshape is non-Lorentzian; i.e., limit [ 91 need not apply. Discrete FT NMR spectra. In FT NMR spectroscopy, the signal, Eq. [ 51, is sampled N times at a rate S for a time period T, called the acquisition time, given by
492
MCLEOD
AND COMISAROW
T= N/S
[I21
to give a discrete time signal. The discrete time signal may be zero filled (6, 7) with each zero filling doubling the length of the time-domain data set. Discrete Fourier transformation of this data set gives, after phase corrections, a discrete absorption spectrum and a discrete dispersion spectrum, each of 2 ‘-‘N data points, where n is the number of zero fillings. Each of the discrete absorption spectrum and the discrete dispersion spectrum is defined only at the A4 ( =2*-‘N) frequencies&, given by
fm=$T5
1,2 ,...,
m=O,
M-
1.
[I31
The channel spacing Af, in hertz, in the discrete spectrum is given by Af =+T.
[I41
Figure 1 illustrates a continuous absorption peak A, calculated from Eq. [ 71. The crosses in Fig. 1 are values in the non-zero-filled (y1= 0) discrete absorption spectrum at the frequenciesf,. The discrete angular frequencies o, are given by
The location, wo, of a continuous spectral peak which lies within *$Af (Eq. [ 91) of the mth point in the discrete spectrum can be specified in terms of the parameters of the discrete spectrum by wo =
2am + (b 2”T ’
n=0,1,2
)...)
where the frequency shift angle 4 varies from -r (when w. lies halfway between m and m - 1) to K (when o. lies halfway between m and m + 1) . For discrete spectra, integration is given by the product of the channel spacing and the sum of spectral intensities, M-l
DI=
2~Af
2 A,, m=O
[I71
where DI is the discrete integral. Note that each A, (Eq. [7]) is a function of both m (Eq. [15]and$(Eq. [16]). Because of the discrete nature of the spectrum, for any finite number of zero fillings DI is only an approximation to the true continuous integral CI (Eq. [ 11 I). In addition, the difference between CI and DI will also depend upon 4 (Eq. [ 16 ] ), i.e., exactly where the continuous peak falls between two frequencies in the discrete spectrum. This is illustrated in Figs. 1 and 2. Figure 1 shows a continuous absorption peak, calculated from Eq. [ 71, with T/ Tf = 0.7, which falls exactly on (q5 = 0.0) one of the frequencies of the discrete spectrum. The crosses in the figure are the values in the non-zero-filled discrete spectrum. Note that all of the discrete intensities are positive. Figure 2 shows a spectrum which is identical, but for the value of 4. Note that the
ERRORS IN INTEGRATION
T T=
493
OF FT SPECTRA
0.7
rad/sec
w urn-1 I frnL
m-4’
T
m-1
I m-2
Wm=Wo I
T
I m-2
urn+1
I m T
I
I
I
m
I m*2 T
I
I
I
In+2
I m+4 T
I
’ Hz
I
m+4
FIG. 1. Continuous absorption lineshape and non-zero-filled discrete absorption lineshape. Curve A is the continuous absorption lineshape calculated from Eq. [ 71 of the text with the ratio of acquisition time to relaxation time, T/ T: = 0.7. The continuous peak height is given by Eq. [ 8 1. The peak location, w0 ( Eq. [ 161)) falls exactly OFIthe channel m = m of the discrete spectrum. The crosses are the values in the non-zero-filled discrete absorption spectrum. Curve DA is the discrete absorption lineshape formed by straight-line connection ofthe values in the discrete spectrum. Note that all values in the discrete spectrum are positive.
discrete intensities in Fig. 1 not only differ from those in Fig. 2 but also include many negative values. Since C#J cannot be known in advance any study of digital integration errors must allow for this variance in 4. In experimental practice the discrete spectral intensities are summed and the analog equivalent of the running sum is displayed as a function of frequency. For any particular peak in the spectrum the running sum has a jump whose amplitude is the integral associated with the peak. Quantitative assessment of a particular jump requires the arbitrary setting of limits to the integration range associated with the jump. This is especially true when the running sum is partitioned into partial sums each of which generates a numerical output to be associated with a particular peak. Limits
494
MCLEOD
AND COMISAROW
,,!,
radlsec
W
f nl
ml
m-4
m-2
T
T
I
m-4
I
I
mr2 T
m
T
I
m-2
I m
I
I m+2
mr4 T
I
HZ
I m+4
FIG. 2. Continuous absorption lineshape and non-zero-filled discrete absorption lineshape. Curve A is the continuous absorption lineshape calculated from Eq. [ 71 of the text with the ratio of acquisition time to relaxation time, T/T,* = 0.7, and a value of the frequency shift factor b/T (Eq. [ 161) = -1 S. The continuous peak height is given by Eq. [ 81. The peak location w0 (Eq. [ 161) falls between two channels of the non-zero-filled discrete spectrum. The crosses are the values in the non-zero-filled discrete absorption spectrum. Curve DA is the discrete absorption lineshape formed by straight-line connection of the values in the discrete spectrum. Note that curve A is identical, except for location, to curve A in Fig. 1. Note also that the discrete lineshape, DA, has negative intensities and differs from curve DA in Fig. 1.
which are too narrow will miss some of the integral; limits which are too broad will include contributions from other peaks. The choice of limits for integration is thus a parameter which should be varied in a study of digital integration. In this study the limits of integration were chosen to include all spectral intensity in excess of 1,5, 10, or 20% ofA (m), the maximum discrete intensity. Thus, in practice, the discrete integral is given by Eq. [ 18 ] rather than Eq. [ 17 1, upper limit
DI=
27rAf
C
A,.
[If41
lower limit
The integration error of this work is given by the difference between the discrete integral, DI, and the continuous integral, Kr,
ERRORS IN INTEGRATION
OF FT SPECTRA
%errOr= DI W. t181- K71. (Eq.t41) K??
1oo
495 [I91
In usual experimental NMR practice, only the positive frequency spectrum is plotted and integrated. As a consequence the appropriate continuous integral to be used for comparison with the digital integral (Eq. [IS ] ) and used for error calculation (Eq. [ 19 ] ) is Kr, which is half the value of the FID (Eq. [ 5 ] ) at t = 0. RESULTS
The error (Eq. [ 19 1) was calculated for a particular
value of the frequency shift
angle 4 (Eq.t161), for a particular level y1of zero filling, a particular value of the integration limit, and a particular value of T/ Tz . The error calculation was then repeated for different values of 4 in the range 0 < 4 < r and the largest positive and negative errors were noted. Because of the symmetry of the lineshape about oo, it is redundant to examine the range - T < 4 < 0. These calculations were then repeated for other values of T/ TT in the range 0.0 < T/ T,* < 4.0. T/ T: = 0.0 corresponds to an undamped FID; T/ Tf = 4.0 corresponds to an FID which has decayed to 1.8% of its initial value and is essentially completely damped. Our results then cover essentially all cases from undamped to completely damped FIDs. Figure 3 graphically displays typical results derived from Eq. [ 19 1. Figure 3 displays the most positive and the most negative errors, for whatever value of 4 gave those errors, for a non-zero-filled discrete absorption spectrum as a function of T/ Tz . The negative errors result from values of 4 which gave lineshapes with negative discrete intensities; cf. Fig. 2. Note that the error decreases as T/ T,* increases. This is characteristic of all cases and is a consequence of the greater linewidth as T: decreases. The acquisition time T( Eqs. [ 61, [ 121) is held constant in this work and Tf must decrease for T/ Tz to increase. As the relaxation time T: decreases, both the linewidth and the number of discrete intensities across a particular peak will increase. This makes the digital summation DI (Eq. [ 17 or 181 a better approximation to the continuous integral, CI (Eq. [lo]). Another characteristic of all figures is that for larger values T/ Tf , where the discrete integral approaches the continuous integral, the remaining error is always negative. This is a consequence of the finite limits of the digital integration; i.e., not all of the peak is included in the smmmation. The four graphs in Fig. 3 display the errors which result from summation limits which include spectral intensity in excess of 1, 5, 10, and 20%, respectively, of the largest discrete intensity. Note that this negative error at T/ TT = 4.0 becomes more negative as the spectral intensity included in the summation decreases (Fig. 3a to Fig. 3d.) Figures 4 and 5 display results like those of Fig. 3, but for zero-filled spectra. Note that for three levels of zero filling (Fig. 5 ), the digital error is essentially independent of both T/ Tz and 4. This is because three levels of zero filling provide a discrete spectrum in which the intensities differ from the continuous intensities by less than 3% for T/ T: = 0.0 and less than 1% for T/ Tc = 4.0 (4, 7). Three levels of zero filling thus give a discrete spectrum which is a good approximation to the continuous spectrum. As a consequence, the difference between the continuous integral CI (EQ. [ lo] ) and the discrete integral DI (Eq. [ 17 ] ) becomes negligible.
496
MCLEOD
l&‘,rirEit
AND COMISAROW
(Cl
% Error 100.0
iK,iEAit(d)
T/T ;
FIG. 3. Discrete integration error as a function of T/ T,* for non-zero-filled discrete absorption spectra. The errors, which vary with $J/T ( Eq. [ 161)) were calculated from Eq. [ 191 of the text. The largest positive error and the largest negative error for each value of T/ T:, found by numerical searching, are displayed. The four graphs, (a)-(d) , display the errors for integration limits which include all spectral intensity in excessof 1, 5, 10, and 20%, respectively, of the largest discrete intensity (A (m = m)) in the non-zero-filled discrete spectrum. Note that for all cases the errors are greater than the integration errors for zero-filled spectra (Figs. 4-6).
DISCUSSION
The errors which are discussed in this work are systematic errors which result from the discrete nature of experimental FT NMR spectra. Unlike random errors which are characterized by their root-mean-square values, systematic errors should be characterized by the largest possible error which can occur. Since the location o. of a spectral peak with respect to a discrete frequency f, (Eq. [ 131) is unknowable in advance, 4 must be varied to find the largest error which can occur. These largest possible errors, which derive from the worst cases which could occur in experimental practice, are shown in Figs. 3-6. It is evident from Figs. 3-6 that some spectra will give rise to a digital integral which is greater than the true integral (positive integration error) ; others give rise to a digital
ERRORS IN INTEGRATION
-100.0
-100.0 Er%or 100.0
OF FT SPECTRA
l&=A
(”
2X
Er%r 100.0
lilnit
(d)
II,
50.0
-
1 .o
2.0
3.0
4.0 1
T/T;
‘;-
100.0
1
FIG. 4. Discrete integration error as a function of T/ TT, for once-zero-filled discrete absorption spectra. Theerrors,whichvarywith+/T(Eq.[16]), were calculated from Eq. [ 191 of the text. The largest positive error and the largest negative error for each value of T/T:, found by numerical searching, are displayed. The four graphs, (a)-(d), display the errors for integration limits which include all spectral intensity in excess of 1,5, 10, and 20%, respectively, ofthe largest discrete intensity (A( m = m)) in the once-zero-filled discrete spectrum. Note that for all cases the errors are less than the integration errors for non-zero-filled spectra (Fig. 3) and exceed the errors for twice (Fig. 5 ) or three-time-zero-filled (Fig. 6) spectra.
integral which is less than the true integral (negative integration error). Figure 1 shows a spectrum which would have a positive integration error. Figure 2 shows a spectrum which would have a negative integration error. Note that these spectra differ only in their positions relative to discrete frequencies of the discrete spectrum. Although T, the acquisition time (Eqs. [ 61 and [ 12]), is chosen by the operator and is known in advance, the value of T/ Tf , the ratio of the acquisition time to the relaxation time of the FID, is unknown prior to running an FT NMR experiment. Typically, T/ TT is of the order of 3.0 for ‘H NMR and is somewhat less for resonances of other magnetic nuclei. T/ Tz can also vary from peak to peak in a given spectrum. Because the systematic integration error is a strong function of T/ TT the integration errors are presented as a function of T/ TT in Figs. 3-6, so that the results will cover all cases which could occur in practice.
498
MCLEOD
AND COMISAROW
%
Error 100.0
&Zit
(b)
4
50.0
-
0
50.0 3.0
4.0I
-
T/T; 0
-5o.o-
-50.0
-
-lOO.O-
-100.0
-
ErTor 100.0
50.0 i
el._l\:.-.--
ll&Ti,“it
(‘I
EZor 100.0
2X
IAt
(d)
50.0 I
T/T;;_ j;;,;i:,,:_._O
T/T;
FIG. 5. Discrete integration error as a function of T/ Tf for twice-zero-filled discrete absorption spectra. The errors, which vary with 4 / T ( Eq. [ 16 ] ) , were calculated from Eq. [ 19 ] of the text. The largest positive error and the largest negative error for each value of T/T,*, found by numerical searching, are displayed. The four graphs, (a)-(d) , display the errors for integration limits which include all spectral intensity in excess of 1, 5, 10, and 20%, respectively, of the largest discrete intensity (A (m = m)) in the twice-zerofilled discrete spectrum. Note that for all cases the errors are less than those of Figs. 3 and 4 but greater than those of Fig. 5.
Ever since the publication of Barthodi and Ernst (6), it has been accepted that one level of zero filling will enhance the information content of absorption-mode NMR spectra over that contained in the corresponding non-zero-filled absorption spectra. Further zero fillings, however, merely interpolate toward the continuous lineshape. The present work shows that integration accuracy is also enhanced by zero filling prior to integration. Integration is used to measure the relative number of nuclei at each chemical shift. It follows then that the most significant error indicated in Figs. 3-6 is not the absolute error but rather the error range, the difference between the largest positive error and the largest negative error which can occur in a given spectrum. For example, consider Fig. 6a for T/ Tc > 3.0, where the absolute error (Eq. [ 18 ] ) is - lo%, but the error range is negligible. For an FID whose acquisition time T was long enough such T/ T: > 3.0 for every peak in the spectrum, the digital integrals of each peak will be
ERRORS IN INTEGRATION
499
OF FT SPECTRA
% Error
&kfit
100.0
lb)
II
%
,&=,i,“it
Error 100.0
(cl
% Error 100.0
50.0
50.0 i o]\c
-50.0
- 100.0
i 3/.0
4i0
T/T;
Oj~\~O
T/T;
-50.0
I
-100.0
i
FIG. 6. Discrete integration error as a function of T/T: for three-time-zero-filled discrete absorption spectra. The errors, which vary with I$/ T (Eq. [ 16 ] ) , were calculated from Eq. [ 191 of the text. The largest positive error and the largest negative error for each value of T/T:, found by numerical searching, are displayed. The four graphs, (a)-(d), display the errors for integration limits which include all spectral intensity in excess of 1, 5, 10, and 2070, respectively, of the largest discrete intensity (A( WI = m)) in the three-time-zero-filled discrete spectrum. Note that for all casesthe errors are less those of Figs. 3-5.
less than the true value by 10% but the relative error will be negligible. Digital integration of this type of spectrum will give a measure of the relative number of nuclei which has virtually no error arising from the discrete nature of the spectrum. In contrast, Fig. 3a describes the case where the relative error at T/ Tf = 3.0 will be about 10%. The differences among Figs. 3-6 show the effect of increased zero filling upon integration accuracy. Two trends are evident. First, the error range, the difference between the largest positive error and the largest negative error, is dramatically reduced by zero filling. Second, the error range is little reduced by zero filling beyond the 12 = 2 level. For NMR spectra with T/ T,* > 1.O, which would include the majority of high-resolution NMR spectra, one level of zero filling (Fig. 4) will reduce the relative integration error to less than 10% if a 1% integration limit is used and less than 20%
500
MCLEOD
AND
COMISAROW
if a 20% integration limit is used. Two levels of zero filling (Fig. 5 ) will leave a residual systematic error in determining the relative number of nuclei of typically 2% for a 1% integration limit and 6% for a 20% integration limit. In most cases one level of zero filling will be sufficient to reduce the systematic integration error to a low level, providing a 1% integration limit is used. However, in cases where it is important to exclude contributions from nearby peaks, a higher integration limit, say 20%, may be desirable. In this situation more than one level of zero filling will permit accurate digital summation. This follows from noting the reduction in error in the figure sequence 3d, 4d, 5d, 6d. ACKNOWLEDGMENT
This research was supported by the Natural Sciences and Engineering Research Council of Canada. REFERENCES 1. J. A. POPLE, W. G. SCHNEIDER,AND H. J. BERNSTEIN, “High-Resolution Nuclear Magnetic Resonance,” McGraw-Hill, New York, 1959. 2. R. R. ERNST AND W. A. ANDERSON, Rev. Sci. Instrum. 37,93 ( 1966). 3. A. G. MARSHALL, M. B. COMISAROW, AND G. PARISOD, J. Chem. Phys. 71,4434 ( 1979). 4. S. 0. CHAN AND M. B. COMISAROW, J. Magn. Reson. 51,252 ( 1983). 5. C.GIANCASPROANDM.B.COMISAROW, Appl.Spectrosc.37,153(1983). 6. E. BARTHOLDI AND R. R. ERNST, J. Magn. Reson. 11,9 ( 1973). 7. M. B. COMISAROW AND J. D. MELKA, Anal. Chem. 51,2198 ( 1979).