Anal. Chem. 1992, 64, 2770-2774
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Application of Linear Prediction to Fourier Transform Ion Cyclotron Resonance Signals for Accurate Relative Ion Abundance Measurements Thomas C. Farrar,' John W. Elling, and Mark D. Krahling Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706
Pulse Ion cyclotron resonance (ICR)tlme domain dgnals arklng from a mlxture of Ion specles typically do not exhlbit the smooth exponential decay one normally observes In FTIR and puke nuclear magnetlc resonance (NMR) tlme domaln dgnals. This nonexponentlal decay Is caused by a number of factors whlch leads to a tlme varlatlon in the relatlve Ion Intensities of the species present and to large errors In the accurate experlmental measurement of those relatlve abundances. If pulse ICR rlgnals of shod tkne duratlon are used (0.g. the flrst 1K data polnts out of a total of 64K data points), the accuracy of the Ion abundance measurements k greatly Improved. It is shown that the h e a r predlctlon (LP) method glves more accurate Intenslty values for each frequency component of the time domaln slgnal than those obtained from the more standard Fourler transform (FT) method, and for short acquldtlon tlmes the reooiutlon obtalned with LP methodrkmuchgreatwthanforFT. Inaddltlon,thelntewltles determlned by LP are much less sendtlve to Instrument operating condltlons. The advantages of LP methods are partlcularly relevant to laser desorptlon ICR spectroscopy.
prediction rather than a Fourier transform allows one to use time domain signals of very short duration and avoids the time-dependent distortions. This leads to much more accurate ion abundance information. FTICR spectroscopy is typically done with ions trapped with parallel electric and magnetic fields in an analyzer cell. Ions undergo characteristic cyclotron motion in the plane perpendicular to the magnetic field. Ideally, the frequency of the ion cyclotron motion is related only to the strength of the magnetic field and the mass to charge ratio of the ion. The cyclotron motion of the trapped ions is excited with a sinusoidalexcitationvoltage applied across the magnetic field. Each species of ions in the cell is excited as a coherent ion packet that cyclotronsin the cell at the characteristiccyclotron frequency. The excited cyclotroning packets of ions induce a sinusoidal signal on the cell detection plates, with a characteristic frequency which is dependent upon the mass to charge ratio, mlz. In FTICR, the digitized time domain signal is converted to the frequency domain by means of the Fourier transf ~ r m . ~ *The J ~ Fourier transformation produces the familiar mass (or frequency) spectrum in the frequency domain, F(w)
INTRODUCTION Fourier transform ion cyclotron resonance mass spectroscopy (FTICR MS or FTMS) is now a widely used analytical tool for making mass measurements with very high resolution and mass accuracy. This work and other applications have been reviewed in the The accurate measurement of ion abundance8 with FTICR has, however, proven to be particularly difficult. These difficulties have limited the utility of FTICR for quantitative measurements. Several articles addressing the problem of the accuracy and precision in ion abundance measurements have recently appeared in the l i t e r a t ~ r e . ~In - ~ this work, the difficulty in obtaining accurate relative abundance information is attributed to the averaging process of the Fourier transform methods conventionally used to convert the observed time domain signal into a frequency, or mass, spectrum. The use of linear
* To whom communications should be sent.
(1)Wilkins, C. L.; Chowdhury, A. K.;Nuwaysir, L. M.; Coates, M. L. Mass Spectrom. Reu. 1989,8,67-92. (2)Asamoto,B. Spectroscopy 1989,3,3&46. See also: Asamoto, B.; Dunbar, R. C. Analytical Applications of Fourier Transform Ion Cyclotron Resonance Mass Spectrometry; VCH: New York, 1991. (3)Laude, D. A.;Johlman, C. L.; Brown, R. S.; Weil, D. A.;Wilkins, C. L. Mass Spectrom. Rev. 1986,5, 107-166. (4)Huang, S.K.; Rempe1,D. L.; Gross, M. L. Am. SOC.Mass Spectrom. 1984,596. (5) Poretti, M.; Rapin, J.; Gaumann, T. Int. J.Mass Spectrom. Ion Processes 1986, 72, 187-194. Rapin, J.; Poretti, M.; Gaumann, T. Spectrosc. Int. J. 1984,3,124-128. (6)Liang, Z.;Marshall, A. G. Anal. Chem. 1990,62,70-75. (7)de Koning, L.J.; Kort,C. W. F.; Pinkse, F. A.;Nibbering, N. M. M. Int. J. Mass Spectrom. Ion Processes 1989,95,71-92. (8)Mitchel1,D. W.; DeLong, S. E. Int. J. Mass Spectrom. IonProcesses 1990,96,1-16.
where w is the cyclotron frequencyof the different ion epecies. In practice, the integral in eq 1 is replaced by a summation over a discrete number of data points and one obtains N-1
~(w,= ) p
( t k ) exp(-iwjtk/l\?
j = 0,1,...,N - 1 (2)
=O
where F(w;)is the intensity of the j t h element in the frequency domain, g(tk) is the amplitude of the kth element in the time domain, and N is the number of data points. The discrete Fourier transform (DFT) given by eq 2 is mathematically not very efficient (it requires N2 mathematical operations) and has been superceded by the much more efficient fast Fourier transform (FFT) developed by Cooley and Tukey.Qb Because the phase of the signal for each of the ion species is a complex function of ion mass and experiment timing, phase corrections for the Fourier transformation are neither linear nor simple. Consequently, the mass spectra are conventionally displayed as magnitude-mode frequency domain spectra. Sincemagnitude-mode peaks do not have finite integrals, the frequency domain spectra do not accurately represent the ion abundance. In most of the FTICR spectrometers currently in use the peaks are digitally integrated over a fixed frequency range. The resulting area depends on both the initial time domain amplitude, Aj, of the jth component of the time domain signal and the time domain (9)(a) Fourier, J. B. Theorie Anal. C h l e u r 1822. (b) Cooley, J. W.; Tukey, J. W. Math Comput. 1976,19,9. (10) Marshall, A. G.; Comisarow, M.B.; P a r i d , G. J. Chem. Phys. 1979,71,No. 11,4434-4444.See also: Marshall, A. G.; Grosshans, P. B. Anal. Chem. 1990,63,215A-229A.
0003-2700/92/0384-2770$03.00/0 0 1992 Amerlcan Chemical Society
ANALYTICAL CHEMISTRY, VOL. 64, NO. 22, NOVEMBER 15, 1992
decay rate, T j , of that same j t h component of the time domain signal (see eq 3 below). Using the magnitude-mode peak heights to represent A j requires that each peak have an identical line width. Since the magnitude mode peak shapes are determined by Tj, this implies that the ~j values for each signal component must be identical and A j must depend only on the number of ions of each species trapped in the cell. Since neither of these two basic requirements is met, inaccuracies in ion abundance measurements made with FTICR arise. The initial amplitude of each ion signal depends not only on the number of ions in the cell but also on the radius of its cyclotron orbit. For the mass spectrum to accurately reflect ion abundance, all the ions must be excited to the same cyclotron radius. This is difficult to achieve. The distribution of excitation power in “burst”,“chirp”,and tailored waveform excitation techniques is not uniform over the excited frequency range.11-14 The uneven frequency distribution of the excitation power and the mass dependence of the excited radius results in the excitation of ions of different masses to different radii and reduces the correlation between peak area and ion abundance. Further distortions in the relation between peak area or peak height to ion abundance arise from the unique T, value associated with each frequency component in the composite signal (see eq 3). The frequency domain peak shape and, consequently, the peak area, depends on the ~j of each frequency component.1l The ~j value of each component depends on ion mass, relative ion abundance, charge density, and the mass distribution of the neutral backgroundand varies in time during a transient.7J”l” For magnitude-mode spectra, the differences in ~j among components of the same signal reduce the correlation between peak height or peak areas and relative ion abundance. The nonexponential decay of the signal further reduces the correlation between peak shape and the time domain signal intensity. Various techniques have been developed to correct for the variations in T , between components of the same signal.7~8In these techniques, information about the time dependence of the power in the signal is retained by transforming parts of the time domain signal separately, Comparison of the transformed segments yields information about how the intensity of each signal changes as a function of time. For example, a time domain signal consisting of 64K (1K = 1024) data points may be divided into eight 8K segments.8 Each segment is Fourier-transformed, and the amplitudes of the (11) Comisarow, M. B.; Marshall, A. G.Chem. Phys. Lett. 1974,26, No. 4, 489-491. (12) Noest, A. J.; Kort, C. W. F. Comput. Chem. 1983,7, No. 2,81-86. (13) Chen, L.; Wang, T. C.; Ricca, T. L.; Marshall, A. G. Anal. Chem. 1987,59, 449-445. (14) (a) Tomlinson, B. L.; Hill, H. D. W. J. Chem. Phys. 1973,59,2775. (b) Marshall. A. G.: Wane. T. C.: Chen. L.: Ricca. T. L. In Fourier TransformMass Spectrometry; Bdchank, M. V., Ed.; ACS Symposium Series No. 359; American Chemical Society: Washington, DC, 1987; pp 21-33. (15) Dunbar, R. C. Znt. J. Mass Spectrom. Ion Processes 1984,56, 1. (16) Comisarow, M. B. Lect. Notes Chem. 1982, 31, 484. (17) Laukien, F. H. Int. J. Mass Spectrom. Zon Processes 1986,73,81. (18) Loo, J. F. Thesis, University of Wisconsin-Madison, Madison, Wisconsin, 1989. (19) Loo, J. F.; Krahling, M. D.; Farrar, T. C. Rapid Commun. Mass Spectrom. 1990,4, 297-299. (20) Kumaresan, R.; Tufts, D. W. ZEEE Trans. 1982, ASSP-30,671675. (21) Barkhuijsen, H.; de Beer, R.; Bovee, W. M. M. J.; van Ormondt, D. J. Mag. Reson. 1986,61,465-481. (22) Tirendi, C. F.; Martin, J. F. J. Mag. Reson. 1989, 85, 162-169. (23) Tang, J.; Norris, J. R. J. Mag. Reson. 1988, 79, 190-196. (24) McLafferty, F. W.; Stauffer, D. B.; Loh, S. W.; Williams, E. R. Anal. Chem. 1987,59,2213-2216. (25) Cody, R. B.; Bjarnason, A.; Weil, D. A. In Lasers in Mass Spectrometry; Lubman, D. M., Ed.; Oxford university: New York, 1990; pp 316-340.
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resulting low-resolution lines are obtained. The eight amplitudes of each of the signals in each 8K segment are plotted as a function of time and extrapolated to zero time. This zero-time amplitude has been found to be a much more accurate representation of the true amplitude than any of the amplitudes in the various 8K segments. We have analyzed time domain signals using linear prediction (LP) methods rather than the more conventional Fourier transformation methods.26 As seen below, the LP method avoids some of the problems associated with Fourier transformation of long, time domain signals.18Jg Linear prediction is an autoregressive technique and is based on the important assumption that the time domain signal is a composite of a number of different, damped cosine signals. If this is true then a faithful representation of the time domain signal is given by the expression n
j=1
where n ( t )is the amplitude of the time domain signal at time, t, n is the number of signal components present, Aj is the initial amplitude of the jth signal component, ~j is the decay time, w j is the frequency, and $j is the initial phase. As can be seen here, for each signal present, there are four degrees of freedom, an amplitude, a decay time (or line width), a frequency, and a phase. In other words, the time domain signal defined by eq 3 above is a superposition of Lorentzians. The linear prediction method is simply a way of obtaining the least squares best-fit values for the four parameters associated with each frequency component. One popular implementation of the linear prediction method, linear prediction with singular valued decomposition (LPSVD), was introdpced by Kumaresan and Tufts.20 This method and other recently developedmore efficient ones have been applied with great success to NMR spe~troscopy.2~-~3 All of these procedures involve the use of an autoregressive algorithm used to fit the observed transient to a known mathematical model of the signal. The ideal model of the pulse-mode ICR time domain signal is presented by eq 3 given above. In autoregressive methods one attempts to reconstruct the entire time response from a small fraction of the total number of data points, thus the amplitude, Xn of the nth data point may be obtained from a knowledge of the preceding k data points (4) Note that each data point, Xn, is represented as a combination of the M previous data points, weighted by a AR coefficient, aj, plus a residual 2,. Equation 3 is solved by determining the AR coefficients needed to model the known data with minimum residuals. The solution technique developed by Kumaresan and Tufts20 uses the singular value decomposition (SVD) linear least squares procedure to solve eq 4 in matrix form by minimizing the residuals of the fit of the data matrix to the predicted values. The LPSVD technique yields a vector of AR Coefficients. Without noise in the data, only two AR coefficientsare needed to describe each of the K sinusoidal components. To model real data sets that include noise, as many as 0.75N AR coefficients are used.20 In the case of transients with a high signal to noise ratio, the signal-related AR coefficients can be distinguished from the coefficients describing noise by their (26) Farrar, T. C.; Loo, J. F.; Krahling, M. D.; Elling, J. W. U.S. Pat. No. P89083US.
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magnitude. The AR coefficients describing sinusoids form a polynomial whose roots represent the frequencies and decay constants of the sinusoids making up the signal. The amplitude and phase of the sinusoids are calculated by substituting the frequency and decay constants into the model equation and doing another least squares fit to the data. Altogether then, a total of four parameters is required to define each sinusoid. The resolution is not limited by the totalsampling time; this is the strength of the linear prediction method. The price one pays for this advantage is that the LPSVD procedure is computationally intensive, requiring up to "2 calculations compared to only N log2 N calculations for an FFT. The advantage of using LPSVD is that the calculations may be carried out on very small data sets (64-1024 points). These data sets can be collected over a very short period of time, minimizing the effect of differences in 7 j for the various signal components and for the variation of the different 7,as a function of time. The signal amplitudes calculated with LPSVD are therefore less susceptible to the time-dependent processes that distort the signal-to-abundance relationship when the FFT is used to analyze long transients.
EXPERIMENTAL SECTION Experiments were performed with an Extrel FTMS 2001 Fourier transform ion cyclotron resonance mass spectrometer equipped with a 3.0 T superconducting magnet and used in the single-cellconfiguration.24 A pulsed TEA COz laser was used for laser desorpti~n.~~ Krypton gas was introduced to the vacuum system through the batch inlet at varying pressures and ionized with 70-eV electrons. Krypton of 99.999% purity was purchased from Spectra Gasses, Inc. A highly frequency-selective, tailored excitation waveform was used to apply excitation power such that only ions in the mass range from mlz 70 to mlz 90 were excited.13J4 The time domain signal resulting from the krypton ions was mixed with a 615-kHz reference signal and digitized at a rate of 175.7 kHz for a period of 1.49s. The signal to noise ratio (SIN) was increased by averaging (coadding) 500 time domain signals. Laser desorption experimenta were carried out using a lead sample. The lead standard reference materialwas obtained from the National Institute of Standards and Technology (SRM No. 981). The lead ions were excited with a rapid frequency sweep chirp excitation from 0 Hz to 2.6 MHz at 1.0kHz/ps. The excited signal was mixed with a 244-kHz reference signal and digitized at a rate of 66.6 kHz for 0.98 s. Ions generated by a single laser desorption experiment were excited and detected without signal averaging. The LPSVD software was received from Drs. Barkhuijsen and de Beer, from the Technische Hogeschool Delft, to whom requests for copies of the program should be sent.20
RESULTS AND DISCUSSION The magnitude of time-dependent variations in the signal decay rate was investigated. To this end, krypton gas at 0.7 X 10-8 Torr was ionized and approximately 6000 ions were trapped to minimize the effects of space charge on the ion signal. The time-dependent charge in intensity during the 1.49-ssignal from the krypton ions was assessed by performing an FFT on sequential 8K segments of the 256K points acquired. The magnitude-mode FFT of each 8K (47 ms) segment yielded the intensity of each frequency component in that segment of the total signal. Prior to transformation, each 8K segment was zero-fiied four times to minimize "picket fence" errors. The frequency resolution produced by an FFT of 8K points was sufficient to provide baseline resolution of the Kr isotope peaks. In Figure 1the natural log of the heights of the five major Kr isotope peaks is plotted as a function of the time between the beginning of the signal acquistion and the middle of the
I
84 miz
[L
! 01
0
100
200 300 400 Segment Offset Time (ms)
500
600
Flgurs 1. Naturallog of the peak areas In each sequential 8K eegment of the time domaln signal versus the time between the start of the signal and the center of the segment. The line oorrespondlng to the O*Kr+ Isotope signal overliesthe line for the "Kr+ isotopeand Isomitted for clarity.
8K segment. On the logarithmic scale in Figure 1,a purely exponential decay would result in a linear drop in the intensity in time. Figure 1 illustrates that the variations in signal intensity from this ideal linear relationship are severe and grow with time. Similar experimenta on signals generated a t higher background pressures were conducted. At higher pressures the signals decay faster but in a more exponential fashion. Techniques that perform segmented transformations of the transient798 assume a constant signal decay rate in the models that extrapolate signal intensity back to the time of signalexcitation. The accuracy of these techniques are limited by the nonideal behavior illustrated in Figure 1a t the low pressures needed to obtain high signal resolution. In order to evaluate the performance of LPSVD on ICR transients, three different procedures were used to determine the relative abundance of the five most abundant Kr isotopes from a series of time domain signals generated a t low Kr background pressures. (1) The LPSVD procedure was applied to the first 512 points in the 256K point signal. The calculated amplitudes of each frequency component were used to represent relative ion abundance. (2) The entire 1.49-s transient was Fourier-transformed without apodization. The peak areas were used to represent relative ion abundance. (3) The first 64K points of the time domain signal were transformed in 8K segments (see Figure 1). A line was fit to the natural log of the intensity of each peak versus the segment offset time from the beginning of the signal.8 The linear equation describing the signal decay in time was used to extrapolate each signal amplitude to the time the ICR frequency was excited. The extrapolated intensities were used to represent ion abundances. The results in Table I illustrate the accuracy of the relative ion abundances determined with each of these methods. The data in Table I were generated from analysis of seven Kr+ signals generated under identical conditions. The error in isotopic abundance is reported as the average error in the normalized signal intensities (5) where Ei is the natural abundance of the WKr, 82Kr, =Kr, &Kr,or sBKrions and Oi is the observed normalized intensity of each signal. The average error in all the signal intensities (27) CRC Handbook, 59th ed.; CRC Press: Boca Raton, FL.
ANALYTICAL CHEMISTRY, VOL. 64, NO. 22, NOVEMBER 15, 1992
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Table I. Normalized Signal Intensity of the Krypton Isotopes and the MSE from Various Treatments of the Same Seven Time Domain Signals normalized % intensity (std dev) method" mJz 80 mlz 82 mlz 83 mJz 84 mlz 86 av error LPSVD 2.24 (0.40) 11.98 (0.70) 11.36 (0.85) 57.88 (0.91) 16.54 (0.71) 0.044 128K FFT 2.88 (0.31) 9.61 (0.79) 9.63 (0.63) 60.84 (1.25) 14.64 (0.48) 0.166 segmented FFT
2.18 (0.21)
10.65 (0.37)
10.80(0.33)
60.15 (0.38)
16.21 (0.28)
NISTb
2.27
11.56
11.55
56.90
17.37
0.166 0.102 0.102
See Results and Discussion. See ref 27. I
-'--
I
Table 11. Three LD/ICR Experiments on Lead Standard Reference Material No. 981. methoda apodizeda 128K FFT 1KLPSVD
NISTb "
0
02
04
06
1 12 Pressure (torr) (Times 10E-7)
08
14
16
18
2
Figure 2. Effect of pressure and space charge on relative lon abundances in the magnitude-rode krypton mass spectra. The upper line Is the average error in the relative abundance of the five krypton laatopesdetermined from the peak areas in the mass spectra obtained with an FFT. The lower line is the average error in relative abundance determined from the intensitiesreported by the LPSVD caiculatlons on the same slgnais.
determined by LPSVD is 0.044. The average error of all the relative peak areas in the seven mass spectra is 0.166. The segmented FFT procedure produced an error of 0.102. The failure of the segmented FFT technique to produce accurate isotope abundances is expected, given the nonideal behavior shown in Figure 1. This experimentillustrates that the initial intensities of the signals are a better representation of ion abundance than the average intensity obtained by Fourier transformation of the entire signal. High background pressure of neutrals and high space charge during the FTICR experiment increases the error in relating signal power to ion abundance. The increased error results from the increased collisional dampening which increases the mass-dependent differences in 71.16 Rapid dampingalso limits the mass resolution, causing peak overlap in the transformed spectrum. With constant ionization conditions, increased neutral pressurealso leads to the generation of a large number of ions, the space charge from which can both further increase damping and cause peak distortions." The effect of pressure and space charge on the error in Kr isotopic abundance determination is illustrated in Figure 2. In all cases the ionization conditions remained constant while the background pressure of Kr neutrals was varied. Procedures 1and 2 described above were used to analyze each time domain signal colleded at each different background pressure. Figure 2 illustrates both the superior accuracy of the LPSVDdetermined signal intensities and the resistance of the LPSVD technique to changes in peak ratios with changing experimental conditions. At krypton background pressures higher than those shown in Figure 2, distortions in peak shapes became severe and the areas could not be integrated for procedure 2. Even when the peaks resulting from Fourier transformation were split, the LPSVD technique was able to
exP no.
1 2 3 1 2 3
208pb
normalized % intensity "Pb mPb mPb
57.17 64.83 39.14 54.05 53.3 54.79 52.3470
17.4 9.50 29.54 20.05 21.16 21.94 22.0833
23.03 20.14 26.27 24.82 23.63 22.34 24.1142
2.43 5.52 5.04 1.07 1.91 0.93 1.4255
av % error 24.7 96.2 80.5 10.0 10.5 11.9
"The isotopic abundances determined by apodized FFT and LPSVD on the time domain signal are compared. A three-term Blackman Harris apodization function. e See ref 27. produce accurate abundance information from the signal intensities. The accuracy of the LPSVD technique and resistance of the LPSVD calculations to changes in pressure and number of ions is especially valuable in laser desorption experiments. The large, irreproducible pressure bursts and variable ionization resulting from laser impact with a sample result in widely varying relative peak ratios for repeated LD/FTICR experiments. In contrast, the signal intensities that are calculated with the LPSVD method are not strongly dependent on the variable pressure and ionization that characterize the laser desorption experiment. Several laser desorption FTICR experiments were carried out on a lead standard reference material. In Table 11, the MSE in intensity ratio calculated for the a P b , MPb, MPb, and 2mPb peaks are shown for spectra resulting from a full 64K apodized transformation of the signal and a LPSVD performed on the first 1024 points. Of particular interest in Table I1 is the tremendous variation of relative peak area between LD-FTICR experiments. It is clear from the LPSVD results that these variations do not arise from variations in ion abundances and so must represent extremes in pressure and space charge distortions.
CONCLUSIONS The use of LPSVD instead of the standard Fourier transformto evaluate ICR transients can produce significantly more accurate relative ion abundance measurements. The superior performance is especially critical when LPSVD is applied to transients collected under nonideal operating conditions such as high background pressure and large numbers of ions. The LPSVD results are more accurate than the peak areas in spectra produced by Fourier transforms and the accuracy is available over a wider range of instrument operating parameters. The LPSVD results are also resistant to variation in the relative peak areas with changes in operating conditions. The later advantageis particularly valuable when the signal-to-signal variations are uncontrollable, as in laser desorption ionization.
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The advantages of LPSVD in ion abundance determination outweigh the computational disadvantage of autoregressive techniques. As refmementa are made to the LPSVD algorithm and the power of the data acquisition computers increase, it will be possible to add LPSVD as an option for routine evaluation of transients.
Zirbel was a great help with some of the mathematics. J.W.E. is grateful to Nicolet Instruments and Extrel FTMS for a graduate fellowship. This work was supported in part by a grant from the National Science Foundation, Grant No. 8802313, and by the Wisconsin Alumni Research Foundation.
ACKNOWLEDGMENT The authors would like to thank Dave Weil, Extrel FTMS, for carrying out the laser desorption experiments. Susan J.
RECEIVEDfor review May 18, 1992. Accepted August 24, 1992.
