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ANALYTICAL CHEMISTRY, VOL. 51, NO. 13, NOVEMBER 1979
to the 22.5-24 ppm band as well as the 18-22.5 band. The 18-20.5 and 20.5-22.5 ppm bands, attributed respectively tQ methyl a tQ an aromatic ring adjacent t~ one (e.g., 1-methylnaphthalene) or no ring or group (e.g., 2-methylnaphthalene) are of similar intensity for all the extracts. This fact should be taken into account when average structures are constructed to represent the large number of individual compounds in these complex materials ( 5 ) . The band between 15 and 18 ppm has been assigned to P-CH, in ethyl groups, although methyl groups, for example, in disubstituted hydroaromatic and naphthenic rings may also contribute. The intensity of this band accounts for less than 50% of that of the 18-22.5 ppm band for all the extracts and therefore it is likely that there are at least twice as many methyl as ethyl groups. The band between 11 and 15 ppm has been assigned to methyl a to an aromatic ring shielded by 2 adjacent groups or rings (e.g., 9-methylanthracene) and to terminal methyl in alkyl side chains (2C3). The intensity of this band for extract d is far larger than that of the 15-18 ppm ethyl band, and it is thought that the major contributor is terminal methyl because this extract probably contains the most long alkyl chain aromatics. For the other extracts, the intensity of the 11-15 ppm band is similar to that of the 15-18 band and it is thought that in these the 11-15 ppm band contains similar amounts of shielded methyl and terminal methyl.
CONCLUSIONS A detailed consideration of the 13Cchemical shifts in modo1 compounds has been made which has enabled a I3C chemical shift assignment scheme to be developed for proton decoupled spectra of coal-derived materials. This scheme has yielded valuable structural information on the oxygen groups and on the distribution of aliphatic substituents in extracts from low-rank British coals. ACKNOWLEDGMENT The authors thank I. Stenhouse of the PCMU, Harwell, for
the 13Cspectra and T. G. Martin and W. F. Wyss for providing the extracts.
LITERATURE CITED (1) I. Schwager, P. A. Farmanion, and T. F. Yen, hepr., Div. Pet. Chem., Am. Chem. Soc., 22(2), 677 (1977). (2) K. D. Bartle, T. G. Martin, and D. F. Williams, Chem. Ind. (London),313, 119751. (3) H. L. detcofsky and R. A. Friedel "Spectrometry of Fuels", Plenum, New York, 1970, Chapter 8. (4) R. J. Pugmire, D. M. Grant, K. W. Ziim, L. L. Anderson, A. G. Obhd, and R. E. Wood, fuel, 58, 295 (1977). (5) K. D. Bartle, W. R. Ladner, C. E. Snape, T. G. Martin, and D. F. Williams, fuel, 58, 413 (1979). (6) R . A. Friedel and H. L. Retcofsky, Chem. Ind. (London), 455 (1966). (7) S. A. Knight, Chem. Ind. (London), 1923 (1967). (8) H. L. Retcofsky, F. K. Schweighardt, and M. Hough, Anal. Chem., 49, 585 (1977). (9) J. K. Brown and W. R. Ladner, Fuel, 39, 87 (1960). (IO) J. K. Stothers, "'% NMR Spectroscopy", Academic Press, New York, 1972, Chapter 3. (11) E. Breitmaier, and W. Voeker, "'% NMR Spectroscopy", V&g Chemie, Weinheim/Bergst, Germany, 1974. (12) K. D. Bartle, A. Caiimli, D. W. Jones, R. S. Matthews, A. Olcay, H. Pakdel, and T. Tugrul, fuel, 58, 423 (1979). (13) J. N. Shoolery and W. L. Budde, Anal. Chem., 48, 2146 (1976). (14) W. R. Ladner and C. E. Snape, fuel, 57, 658 (1978). (15) K. S. Seshadri, R. G. Ruberto, D. M. Jewell, and H. P. Malone, fuel, 57, 111 (1978). (16) D. K. Dailing, K. H. Ladner, D. M. Grant, and W. R. Woolfenden, J. Am. Chem. Soc., 99, 7142 (1977). (17) R. S. Ozubko and G. W. Buchanan, Can. J . Chem., 52, 2493 (1974). (18) J. B. Stothers, C. T. Tan, and N. K. Wilson, Org. h4agn. Reson., 9, 408 11977). (19) D. M.'Grant and E. G. Paul, J . Am. Chem. Soc., 86, 2984 (1964). (20) D. K. Dalling and D. M. Grant, J . Am. Chem. Soc., 8g, 6612 (1967). (21) D. K. Dalling and D. M. Grant, J . Am. Chem. Soc., 95, 3718 (1973). (22) M. Farcasiu, fuel, 56, 9 (1977). (23) K. S. Seshadri, R. G. Ruberto, D. M. Jeweli, and H. P. Malone, fuel, 57, 549 (1978). (24) B. M. Benjamin, V. F. Raaen, P. H. Maupin, L. L. Brown, and C. J. Collins, Fuel, 57, 269 (1978).
RECEIVED for review March 8, 1979. Accepted July 11, 1979. Permission to publish this work is given by the National Coal Board, United Kingdom, and the views expressed are those of the authors and not necessarily those of the Board.
Error Estimates for Finite Zero-Filling in Fourier Transform Spectrometry Melvin B. Comisarow" and Joe D. Melka Chemistry Department, University of British Columbia, Vancouver, British Columbia Canada V6T 1W5
A systematic procedure is developed for estimating the maximum peak height error and the maximum frequency error which results from Fourier transformation of a zero-filled, truncated, exponentially damped sinusoid. The errors are functions of the number of zero-fillings and the ratio of the acquisition period to the relaxation time of the sinusoid. The error estimates are given in both analytical form and graphical form for both absorption mode and magnitude mode Fourier transform spectra. It is concluded that four zero-fillings for the absorption mode and three zero-fillings for the magnitude mode will usually suffice to reduce the peak height error to less than 2 %. Applications to Fourier transform nuclear magnetic resonance spectrometry and Fourier transform ion cyclotron resonance spectrometry are briefly discussed.
Within the past 15 years, a new method called the Fourier transform (FT) method has been developed for obtaining
spectral data ( I ) . In the FT method as applied to nuclear magnetic resonance (NMR) spectrometry (2-5), microwave spectrometry ( 6 ) , and ion cyclotron resonance (ICR) spectrometry (7-I6), the motion of the entire sample is excited at once and a time domain signal which is a composite signal from all of the excited motion is sampled and stored. In this manner the FT method can produce a time domain transient signal, which is characteristic of the entire spectrum, in the amount of time which a conventional scanning spectrometer would require to observe just a single peak in the spectrum ( I , 1 7 ) . The experiment may be repeated several times to produce a time domain signal of increased signal to noise ratio ( I ) . The conventional frequency spectrum is derived from the transient time domain signal by the mathematical process called Fourier transformation ( I 7-21). Since for any linear system the Fourier transform of the time domain response is identical with the frequency spectrum (20), the above procedure provides a powerful technique for either producing spectra very quickly or for producing spectra of high sig-
0003-2700/79/0351-2198$01.00/00 1979 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 51, NO. 13, NOVEMBER 1979 Absorption
k L
b\
ld
Q.0 Q=1 f
t
l
t
l
l
t
t
t
l
l
t l
t
t
l
A f
t
t
t
t
t
t
t
t
f
t
t
t
t
t
t
t
t
Figure 1. Continuous and discrete absorption mode spectral peaks. Curves A and B are continuous spectral peaks of identical line shape and intensity which were calculated from Equation 7 with a value of T / T = 1.0. The maximum of curve A falls exactly on one of the frequencies of the discrete frequency spectrum labeled n = 0. The maximum of curve B falls exactly half way between two points of the discrete frequency spectrum labeled n = 0. Curves A‘ and B‘ are discrete frequency spectral line shapes formed by s t r a w line connof the points in the discrete spectrum labeled n = 0. The discrete frequency spectrum labeled n = 1 has twice the resolution of the n = 0 discrete spectrum. The open circles are the values of curve A in the n = 1 discrete spectrum
nal-to-noise ratio (1). In actual experimental practice, however, the continuous time domain response is not analytically transformed to produce a continuous frequency spectrum. Rather, the continuous time domain response is sampled at a finite number of particular instants of time to yield a discrete time domain response. The discrete time domain response is then numerically transformed to produce a discrete frequency spectrum. This discrete frequency spectrum is defined a t M specific frequencies, f , given by f = m/THz, m = O , l , 2 . . .M - 1 (1) where T i s the acquisition period of the time domain signal. Now unless the signal frequency to be determined happens to be exactly one of the particular discrete frequencies given by Equation 1, the intensities in the discrete frequency spectrum will not correspond to the peak maxima in the continuous frequency spectrum. This situation is illustrated in Figure 1. Figure 1 shows two continuous line shapes, A and B, where the maximum of A is exactly o n one of the discrete frequencies of a discrete frequency spectrum and the maximum of B is half way between two discrete frequencies. Straight line connnection of the amplitudes in the discrete frequency spectrum leads to the discrete line shapes A’ and B’. I t is obvious that A’ provides an exact estimate of the amplitude and the frequency of A. A’ also provides a reasonable approximation to the line width of A. However, the peak height, frequency, and line shape of B are poorly estimated by B’. With experience, the distorted line shapes and incorrect amplitudes occurring in FT spectrometry can be recognized. However, the occurrence of this distortion can be obscured by the manner in which the data system indicates peak maxima and locations. The peak heights and the peak locations in FT spectrometry are usually determined with a “peak peaking” algorithm run under computer control. If this algorithm examines only the values of the discrete spectrum, the output intensities and frequencies from the algorithm will usually not correspond to the (true) maxima and frequencies in the continuous spectrum. Furthermore, the discrete graphical spectra produced by straight line connection of the points in the discrete frequency spectrum will usually not give an accurate description of the continuous line shape (cf. Figure 1).
The problem illustrated by Figure 1 has been well recognized in the literature. One solution (4,18,21-28) is to extend the time domain data table by adding zeros to the end of the
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sampled transient signal prior to Fourier transformation. Usually zeros are added until the data table length has been extended by a factor of 2”, where n = 1, 2 , 3 . . ., but the procedure is valid for any positive value of n. Since the effective “zero-filled acquisition time”, T,, is now longer, T , = (T)P
-
the spacing in the discrete frequency spectrum is now reduced to l/Tzfrom 1/T. In the limit, n 03, the numerical Fourier transform becomes identical with the analytical Fourier transform and the discrete frequency spectrum becomes identical to the continuous frequency spectrum. A further advantage of this particular interpolation method is that the first set of added zeros increases the signal-to-noise ratio of the final absorption spectrum (24). Further zero-fillings, however, will only interpolate to the continuous line shape. Another solution to the problem illustrated by Figure 1is to use an interpolating procedure which fits three or more points from a discrete line shape to an assumed analytic line shape. For cases in which the continuous line shape is known in advance and is analytically simple (i.e., Gaussian, Lorentzian, etc.), this procedure has the advantage of shorter computation time when compared with extended zero-filling. These curve fitting procedures will give the peak height and the peak maximum and addition will provide the complete continuous line shape if desired. For cases where the line shape is more complicated, the discrete line shape can be fitted to a simpler analytical form. For example, the experimental line shapes in high resolution FT-NMR spectrometry, whose analytical form is given by Equation 7 of this work, may be fitted to a parabolic equation. Fitting the discrete points from a complex line shape to a simple analytical function yields interpolated peak maxima and peak frequencies, which, while closer to the true maxima, are still in error because of the mismatch between the (complex) continuous line shape and the (simple) fitting function. A significant advantage of the extended zero-filling method of interpolation is that the method will always produce the true continuous line shape whatever that line shape may be. While the effect of zero-filling has been known for some time, we are unaware of any quantiative criteria for determining the number of zero-fillings which are required to achieve a particular accuracy. These criteria are required because the computation time for the fast Fourier transform (FFT) rapidly increases with increasing length of the data table being transformed (21). I t is thus advantageous to zero-fill only until the desired accuracy is achieved. More extended zero-filling merely increases the computation time. The error resulting from finite zero-filling of Fourier transform faradaic admittance data and quantitative criteria for reducing this error have recently been studied by Smith (29). Closely related to the objective of the present work is the study of Horlick (27) which examined the error in peak maximum measurement as a function of the number of points above the half-maximum for a number of different line shapes. The residual error for finite zero-filling of a Lorentzian line shape (as is found in pressure-broadened Fourier transform infrared (FT-IR) spectrometry) can be readily derived from the tables in ref 27. In the following section, the maximum frequency error resulting from finite zero-filling is given. The continuous frequency spectrum of a truncated exponentially damped sinusoid is given in two forms: the absorption spectrum (the cosine transform of the time domain signal) and the magnitude spectrum (the square root of the sum of the squares of the cosine transform and the sine transform). A systematic procedure is developed for estimating the maximum error in peak height which results after a specific number of zerofillings. The procedure is applied to both absorption mode
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spectra and magnitude mode spectra for different values of T/s, where T i s the acquisition period and T is the relaxation time of the time domain signal. The error estimates are presented in both analytical and graphical form. Use of the error equations and the error graphs as well as applications to FT-NMR spectrometry and FT-ICR spectrometry are discussed.
ERROR ESTIMATES FOR ZERO-FILLED FT SPECTRA (a) Frequency Errors. It follows from Equations 1 and 2 that the only allowed values, A f , for the distance between the points in the discrete frequency spectrum are given by 1 A f = - HZ (3) T if no zero-filling is done prior to Fourier transformation, and 1 1 Af = - = - Hz; n G O , 1, 2 , . , (4) T, 2"T if the time domain signal is zero-fied n times prior to Fourier transformation. The maximum error in frequency determination will occur when the maximum for the continuous line shape falls exactly half way between two adjacent points in the discrete spectrum. Since the spacing in the discrete spectrum is halved for every power of 2 of zero-filling (Equation 4), the maximum error in frequency determination as a function of the number zero-fillings may be stated by Equation 5. max frequency error in Hz after n zero fillings =
f l T2""
n=0,1,2
" Z=O it t I Il l I n t f
The continuous absorption mode frequency spectrum (Le., the analytical Fourier transform) of Equation 6 is (29)
tI t t t 'f t
't
t
tf
t
fi 1 'I 1
t
Flgure 2. Continuous and discrete magnitude mode spectral peaks. Curves A and B are continuous spectral peaks of identical line shape and intensity which were calculated from Equation 8 with a value of T l s = 1.0. The maximum of curve A falls exactly on one of the frequencies of the discrete frequency spectrum labeled n = 0. The maximum of curve B falls exactly half way between two points of the discrete frequency spectrum labeled n = 0. Curves A' and B' are discrete frequency spectral line shapes fcfmed by straight line connection of the points in the discrete spectrum labeled n = 0. The discrete frequency spectrum labeled n = 1 has twice the resolution of the n = 0 discrete spectrum. The open circles are the values of curve A in the n = 1 discrete spectrum spectrum are given by Equation 3 if no zero-filling is done prior to Fourier transformation, and Equation 4 if the time domain signal is zero-filled n times prior to Fourier transformation. For a continuous spectral peak which happens to fall exactly on one of the discrete frequencies of the discrete frequency spectrum, as, for example, curve A in Figure 1,the intensity of the line shape will be defined only at the discrete frequency values given by
. . . (5)
Equation 5 gives the maximum error in determination of the frequency of a peak maximum as a function of acquisition period T, and n, the number of zero-fillings. The error may be positive or negative. (b) Intensity Errors. Consider a continuous time domain signal of the form F ( t ) = exp(-t/T) cos ut 0Ct C T (6)
A w = -P 2R 2"T
P = 0, f l , f 2 . .
.
where n is the number of zero-fillings. For a value of P = +1, the intensity of the peak at the first discrete frequency above the peak maximum will be obtained. Substituting Equation 9 with a value of P = +1 into Equation 7 and Equation 8, gives
and
T
(1 + 1 (AO)~T~ e x p [ - T / ~ ] ( A u ) s sin ( ( A w ) T )- cos ((Au)~')) (7)
A(Aw) =
AA Magnitude
+
where
The continuous magnitude mode frequency spectrum of Equation 6 is and
Y = 2R/2"
2 e x p [ - T / ~ ] cos ( ( A w ) T ) + e x p [ - 2 T / ~ ] ) ' / ~(8) In Equations 6-8, s is the relaxation time of the oscillation in seconds, Aw is the frequency distance from the peak maximum in radians/second, and T is the time period in seconds over which the damped oscillation (Equation 6) was observed. Equations 6-8 are very general equations characteristic of any damped oscillation. They are applicable to many forms of spectrometry and in particular NMR spectrometry and ICR spectrometry. The line shape described by Equation 7 is sometimes called a "convolved sinc function", but the phrase "Lorentzian convolved with a sinc" is probably more apt. Now when Equation 6 is sampled at a series of discrete times, the discrete frequency spectrum resulting from Fourier transformation will only exist a t the particular frequencies given by Equation 1. The only allowed values for the distance between the points in the discrete frequency
(13) Equation 10 gives the discrete intensity value for the first discrete frequency above the peak maximum (i.e., P = l),for an absorption mode line shape as a function of 7, the relaxation time of the time domain signal, X,the ratio of the acquisition period T to the relaxation time 7, and n, the number of zero-fillings. Equation 11 is the corresponding equation for a discrete magnitude mode line shape. Examination of Figure 1or 2 leads to a systematic procedure for determining the maximum amplitude error due to the finite frequency resolution of a discrete frequency spectrum. Curve A in Figure 1 (Figure 2) is a continuous absorption mode (magnitude mode) line shape calculated from Equation 7 (Equation 8) for T/s = 1.0. Curve B in Figure 1 or 2 is a continuous line shape which is identical with Curve A but is located a t a different frequency. Now if the spectral peak A were obtained by sampling a time domain signal (Equation 6) and Fourier transformation, and if the maximum of curve
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A was exactly on one of the points in the discrete frequency spectrum, the discrete magnitude mode line shape obtained by connecting the points in the discrete frequency spectrum would be curve A’. If the sampling conditions were appropriate for curve A, then the maximum of curve B will be incorrectly indicated by the discrete magnitude line shape, curve B’. Since the maximum for the continuous peak B falls exactly half way between two of the discrete frequencies of the nonzero-filled (n = 0) discrete frequency spectrum, measurement of spectral peak B’ will lead to the worst possible estimate for the frequency and amplitude of curve B. If peak B were less than half way between two frequencies of the n = 0 discrete spectrum, B’ would have an amplitude closer to that of peak B. If the sampled transient which leads to curve A‘ were zero-filled once ( n = 1)prior to Fourier transformation, the points indicated as open circles in Figure 1 or 2 would be obtained for spectral peak A. Comparison of curve B’ with the open circles of curve A leads to the following conclusion: The minimum amplitude for a discrete peak which falls between two points of a non-zero-filled spectrum will be equal to the amplitude of the first (i.e., P = 1)point away from the maximum of the “zero-filled once” (n = 1)spectrum of a peak whose maximum falls exactly on one of the frequencies of the non-zero-filled ( n = 0) spectrum. The fractional error after no zero-filling is given by 1.0 less the peak maximum in the discrete, n = 0 spectrum divided by the maximum in the continuous spectrum. Since the (true) continuous spectrum maximum is obtained after an infinite number of zero-fillings, the maximum fractional error after no zero-filling is given by the formulas max fractional error obtained after n o zero- filling Equation 10 ( n = 1) = (1.0 Equation 10 ( n = a) (14) max fractional error obtained after n o zero-filling Equation 11 ( n = 1) = (1.0 Equation 11 ( n = m ) (15) Equation 14 gives the maximum fractional error for a discrete absorption mode line shape which was obtained after no zero-filling as a function of the ratio X (Equation 12). Equation 15 is the corresponding equation for the magnitude mode line shape. Note that Equations 14 and 15 are dependent upon the ratio X but, unlike Equations 10 and l l , are independent of the absolute values of the acquisition period, T , and the relaxation time, T . The preceding ideas may be generalized to max fractional error obtained after n zero-fillings Equation 10 ( n = n + 1) = (1.0Equation 10 ( n = a) (16) max fractional error obtained after n zero-fillings Equation 11 ( n = n 1) = (1.0Equation 11 ( n = a) (17) Equation 16 gives the maximum fractional error for an absorption mode line shape as a function of n, the number of
(
r
)
)
)
+
)
Absorption mode
F1\
, F ( T ) = 14 FlOl
F ( T ) = F(O1
h
40
L
0
1
2
3
4
k
0 0
I
2
3
c
5 0
1
2
3
4
Number of zero fillings
1
2
3
4
Figure 3. Relative error due to noninfinite zero-filling for an absorption mode line shape. Each curve gives the maximum percentage error
for a particular ratio of T / T as a function of the number of zero-fillings. The value of the time domain function (Equation 6) at the end of the acquisition period is indicated on each graph. The errors at integral values of n are the most important but the cwves are valid for nonintegral values of n also. The curves were calculated from Equation 16. Note that the error scale is different than that of Figure 4 Moqnitude mode gbermr/il 2o
F ( T ) = 0 5 F(0)
p
T T T/Zl
F ( T ) = 0 2 F(0)
F(Tl=OIF(0’~
r-7 1
F(T)=SxIO”F(Oi
Number of zero fillings Figure 4. Relative error due to noninfinite zero-filling for a magnitude mode line shape. Each curve gives the maximum percentage error for a particular ratio of TIT as a function of the number of zerefillings. The value of the time domain signal (Equation 6) at the end of the acquisition period is indicated on each graph. The errors at integral values of n are the most important but the curves are valid for nonintegal values of n also. The curves were calculated from Equation 17. Note that the error scale is different than that of Figure 3
zero-fillings, and X, the ratio of the acquisition period to the relaxation time of the time domain signal, which when Fourier transformed, gives the absorption mode line shape. Equation 17 is the corresponding equation for the magnitude mode line shape. Equations 16 and 17 like Equations 14 and 15 depend upon the ratio X but, unlike Equations 10 and 11, are independent of the absolute magnitudes of T and T . The residual error calculated from Equations 16 and 17 for various values of n are graphically displayed in Figures 3 and 4. Figure 3 shows the percentage amplitude error resulting from finite zero-filling for the absorption mode line shape. As expected, extended zero-filling prior to Fourier transformation rapidly reduces the error. As the time domain signal is relaxed during the acquisition period (increasing T/ T ) , the error for fixed n also becomes less. Figure 4 shows the same information as Figure 3 but for the magnitude mode line shape. The general dependence of the error is the same as for the absorption mode but the error is less for any given values of n and T / T . This lower error for the magnitude mode arises because of the broader line shape of the magnitude mode (3, 12, 29); cf. Figures 1 and 2.
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The peak maximum in the discrete frequency spectrum is always less than the peak maximum in the continuous spectrum. The amplitude errors described by Equations 16 and 17 and Figures 3 and 4 are therefore always negative. On the other hand, the frequency error resulting from measurement of the peak frequency in a discrete spectrum may be positive or negative (Equation 5).
DISCUSSION The treatment of the previous section shows that the maximum error in the determination of both the location and amplitude of a spectral peak described by a discrete frequency spectrum occurs when the maximum of continuous peak falls exactly half way between two adjacent points in the discrete frequency spectrum. This simple statement for the Occurrence of worst case allows the derivation of the simple analytical expressions, Equations 16 and 17, for the maximum amplitude error. The derivation of the error expressions, Equations 16 and 17, shows that the amplitude error in a discrete frequency spectrum depends upon the T / Tbut not the absolute values of T or 7. As a consequence, the errors described by Equations 16 and 17 and displayed in Figures 3 and 4 can be presented as functions of only two independent variables: n, the number of zero-fillings and the ratio T / r . This is very convenient because the relatively few graphs in Figures 3 and 4 can be applied to a very wide range of experimental values of T and 7.
Usually, provision is made in the E T experiment for display of the (accumulated) time domain signal prior to Fourier transformation and the ratio T / T can be visually obtained from this signal by noting the initial and final values, F(0) and F(T),of the time domain signal. The final value, F(T) as a function of the initial value, F(O), is indicated on the appropriate graphs of Figure 3 and 4. Once the ratio T / r is known and the desired error limit is chosen, the required number of zero-fillings can be determined quickly from the appropriate graph in Figure 3 or 4. One conclusion which follows directly Figures 3 and 4 is that four zero-fillings for the absorption mode and three zero-fillings for the magnitude mode are sufficient to reduce the peak height error to less than 2%. In most experimental practice, the time domain signal will consist of a sum of components of the form, Equation 6. These spectral components could have differing relaxation times and differing errors associated with their spectral peaks. Determination of an overall T / Tratio for the composite signal will not necessarily give the correct T / T ratio for any particular spectral component. However, it follows from Figures 3 and 4 that, after three or four zero-fillings, the error is essentially independent of the ratio, TIT. Thus even for cases where the relaxation time differs from peak to peak, Figures 3 and 4 provide a useful guide to the required number of zero-fillings. Of c o m e the Occurrence of differing relaxation times is readily apparent in zero-filled spectra from the differing line widths of the various spectral components. For cases where one line is much narrower than others it is probably best to repeat the zero-filling procedure with more zero fillings to ensure that no significant errors remain. As noted above, the error resulting from finite zero-filling depends upon the ratio TIT. This ratio is often under control of the experimentalist and deserves some comment. In general it is desirable to make the acquisition time T as long as possible as this maximizes the inherent resolution of the F T experiment. In practice, however, this is not always possible because of limitations in the size of the available memory. According to the Nyquist criterion, the transient signal must be sampled as a rate which exceeds twice the highest frequency in the signal. For a sampling rate of S, the acquisition period will be given by
T=N/S
(18) where N is the number of words of available computer memory. For example, for a transient signal which contained frequency components from dc to 1 kHz, the sampling rate must exceed 2 kHz. With a computer data memory of 16 384 words, the memory will be filled up in 8 s (Equation 18). If the relaxation time T was, for example, 1s, the ratio T/swould be 8 and only two zero-fillings would be required to reduce the error to less than 2%. On the other hand, if the transient signal contained components up to 20 kHz, a computer memory of 16 384 words would be filled up in only 0.41 s. For the same value of 7 as the above example the ratio T/Tis 0.41 and four zero-fillings are required to achieve an error of less than 2%. In general then, as the band width increases and as the relaxation time increases, more zero-filling will be required to achieve any particular error minimization. The data table length to be transformed is of course limited by the size of the total memory (semiconductor plus magnetic disk) in the data system. The access time of disk memory is such that the maximum rate at which data can be written onto the disk is about 20 kHz and the band width for a “direct to disk data acquisition” is therefore limited by the Nyquist criterion to less than 10 kHz. The access time of semiconductor memory is much shorter and acquisition rates to semiconductor memory can be as high as 100 MHz. For wide spectral band widths then, the acquisition time T will be limited by the size of the available semiconductor memory as described in the previous paragraph, but the number of zero-fillings will be limited by the size of the total (semiconductor plus disk) memory. In NMR spectrometry it is most common to present frequency data in the absorption mode (3). This is true for both scanning and Fourier transform NMR spectrometry. The most common FT-NMR procedure is to zero-fill the sampled transient once (n = 1)to obtain the maximum possible absorption mode sensitivity as originally recommended by Berthodi and Ernst (24). However, it follows from Figure 3 that, for cases where the transient signal has not decayed significantly during the acquisition time, significant amplitude errors will be still present unless the data me further treated by, for example, curve fitting. For example, it is not until the transient has been sampled for greater than four relaxation times (Figure 3) that the finite resolution of the n = 1discrete frequency spectrum is great enough to reduce the amplitude error to less than 5%. In FT-ICR spectrometry, the most common spectral presentation is the magnitude mode (12, 13). The spectral band width is large and the Nyquist criterion requires high sampling rates (14).For example, a band width of 1 MHz (corresponding to a mass range of 30 to m at a magnetic field of 20 kG) requires a sampling rate of at least 2 MHz, and a computer memory of 16 384 words will be filled in only 8 ms. Since FT-ICR transient signals often last for several tens of milliseconds (12), wide mass range FT-ICR signals are characterized by values of T / Twhich are less than 1.0. For these cases several zero-fillings are necessary to reduce the frequency and amplitude errors accruing from the finite resolution of the FT-ICR spectrum. LITERATURE CITED (1) P. R. G r i f f i , “Transform Techniques in Chemistry", Plenum R e s , New York, 1978. (2) R. R. Ernst and W. A. Anderson, Rev. Sci. InStrUm., 37, 93 (1966). (3) D. Shaw, “Fourier Transform NM( Spectroscopy”, Elsevier, Amsterdam, 1976. (4) T. C. Farrar, Chapter 8 in ref 1. (5) J. W.Cooper, Chapter 9 in ref 1. (6) J. C. McGurk, T. G. Schmalz, and W. H. Flygare, Adv. Chem. Phys., 25, 1, 1974. (7) M. 8. Comisarow, Chapter 10 in ref 1. (8) M. E. Comisarow, Adv. Msss Spectfosc., 7 , 1042 (1978). (9) C. Wilkins, Anal. Chem., 50, 493A (1978).
ANALYTICAL CHEMISTRY, VOL. 51, NO. 13, NOVEMBER 1979 (10) (1 1) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)
M. B. Comisarow and A. G. Marshall, Chsm. Wy5. Lett., 25, 282 (1974). M. B. Comisarow and A. G. Marshall, Chem. Wy5. Lett., 26,489 (1974). M. B. Comisarow and A. G. Marshall, Can. J. Chem., 52, 1977 (1974). M. B. Comisarow and A. G. Marshall, J. Chem. Phys., 82, 293 (1975). M. 8. Comisarow and A. G. Marshall, J. Chem. fhys., 64, 110 (1976). M. Comisarow, G. Parisod, and V. Grassi, Chem. fhys. Lett., 57, 413 (1978). M. B. Comisarow, J. Chem. fhys., 89, 4097 (1978). A. G. Marshall and M. B. Comisarow, Chapter 3 in ref 1. J. W. Cooper, Chapter 4 in ref 1. Charles T. Foskett, Chapter 2 in ref 1. D. C. Champeney, "Fouier Transforms arid Thei Applications", Academic Press, New York, 1973. E. 0. Brigham, "The Fast Fourier Transform", Prentice-Hall, Englewood Cliffs, N.J., 1974. R. D. Larsen, Chapter 13 in ref 1.
P.
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R. Griffiths, Chapter 5 in ref 1.
E. Bertholdi and R . Ernst, J. Magn. Reson., 11, 9 (1973). P. R. Griffiths, Appl. Spectrosc., 29, 11 (1975). R. T. Pajer and I. M. Armitage, J. Magn. Reson., 21, 485 (1976). G. Horlick and W. K. Yuen, Anal. Chem., 48, 1643 (1976). P. C. Kelly and G. Horllck, Anal. Chem.. 45, 518 (1973). R. J. O'Halloran and D. E. Smith, Anal. Chem., 50, 1391 (1978). A. G. Marshall, M. 6.Comisarow, and G. Parisod, J. Chem. fhys., in Dress.
RECEIVEDfor review January 22,1979. Accepted July 23,1979. This research was supported by the Natural Sciences and Engineering Research Council of Canada and by the Research Corp.
Simulation of Nuclear Magnetic Resonance Spin Lattice Relaxation Time Measurements for Examination of Systematic and Random Error Effects T. Phil Pitner" and Jerry F. Whidby" Philip Morris USA Research Center, P.O.
Box 26583, Richmond, Virginia 2326 1
Systematic and random errors in NMR spin lattice relaxation time ( T , ) measurements are investigated by simulating relaxation data using various experimental parameters: pulse width, recovery delay, waiting times, and signal-to-noise. T,'s are calculated from the hypothetical data by linear least squares, two-parameter exponential, three-parameter exponential, and four-parameter exponential analyses to explore the suitability of these analyses. The computed T,'s and standard deviations are discussed in terms of random and systematk errors. Experimental data are presented to illustrate the applicability of the calculations.
When faced with measuring spin lattice relaxation times (T,),the NMR spectroscopist is confronted by a myriad of confusing opinions, almost amounting to myth and folklore, as to the proper choice of measurement method, experimental parameters, and data analysis method. Several studies have appeared in the literature, aimed at delineating and clarifying this problem (1-19). The need in this laboratory for accurate and reproducible T , measurements led us to investigate some of the various methods by simulating experimental relaxation data and subjecting these data to several analyses to determine how well the calculated Tl's compare with 5"''s used in generating the data. We hope the results presented in this paper will aid and give insight to experimentalists who desire to have a reasonable degree of confidence in the Tl's they measure.
EXPERIMENTAL 'H (12.29 MHz) NMR spectra were obtained with a Bruker WP-80 spectrometer at ambient probe temperature. T,'s were measured for the 2H resonance of a 5% solution of DzO(Merck & Co.) in HzO doped with a trace of copper sulfate. RESULTS AND DISCUSSION The most common method of measuring T,'s involves applying a perturbing rf pulse to the nuclear spins, waiting for a range of times 7 to allow the spins to relax, and sampling 0003-2700/79/035 1-2203$01 .OO/O
the extent of relaxation with an observing pulse. The peak intensity A , after waiting time 7 is given for steady-state conditions by
A, = A , sin /3
1 - [I - cos
cy
(1- e-D/T1)]e-r/T1
1 - [cos cy cos /3 e-D/T1]e-.IT1
(1)
(2, 14, 18) where A , is the equilibrium intensity; a , the perturbing pulse flip-angle; p, the observing pulse flip-angle; and D , the recovery delay between the observing pulse and the next perturbing pulse. For the standard inversion-recovery experiment a = 180" and /3 = 90" (1.2). Equation 1 assumes rf homogeneity, no offset effects (Le., (pulse width)
