Anal. Chem. 1992, 64, 1601-1605
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Spectral Analysis Based on Bipolar Time-Domain Sampling: A Multiplex Method for Time-of-Flight Mass Spectrometry Michael A. May and Alan G. Marshall’J Department of Chemistry, The Ohio State University, 120 West 18th Avenue, Columbus, Ohio 43210
Hermann Wollnik’
ZZ. Physikalisches Zwtitut, Uniuersitiit Giessen, Heinrich-Buff-Ring 16, 6300 Giessen, Germany
I n thk paper, we present a novel non-Fourler transform muttlpkx meam for sampllng and frequency analyzlnga slgnal condrtlng of a sum of exponentlally damped slnusolds of different frequency. The methodk Illustratedby a hypothetkal torddal or hear reflective tkneof-fllght mato spectrometer In whkh lonk mato k related to the repetltlon frequency of puM4n)ected analyte Ions. Tlme-domaln data k generated from the voltage recorded by each of two detector plates placed 180’ apart at opporlte ends of the detector system, where ll Is advantageow, to record succeorlve maximal voltages from the two detecton, alternately as posltlve and negathre data vs thne. Slgnal component frequencles, amplitudes, phases, and tlmedomaln exponential damping constants are determlned by pattern recognltlon of lndlvldual a n a l frequency tlme-domaln components. Unllke Fourler transform (FT) sampllng, In whlch the sum of all of the component dnwolds k sampled at equa/&spaced tlme Increments, the present b@o/ar satnpllng records only the m a x h m of each perlodlc slgnal to ykld a (non-equa/& spaced) tlme-domaln data ret In which each frequency component k In prlnclple detected independently. Blpolar sampllng achleves the multlplex advantage of FT analyrk, requlresmlnhal dlgltal storage capaclty, and yklds excellent lonlc frequency-ecak accuracy In “zero-nolse”slmulatlons.
INTRODUCTION Conventional Sampling and Frequency Analysis of a Time-DomainSignal. Nuclear magneticresonance (NMR), infrared (IR),and ion cyclotron resonance (ICR)mass spectra (amongothers)are now conventionallygenerated by (discrete) fast Fourier transformation (FFT) of time-domain data sampled at equally spaced time intervals.’ Since each timedomain data point represents a linear combination of all of the frequency-domaincomponents,such an FT spectrometer effectively detecta an entire N-point spectrum in 1/N the time required to scan the frequency-domain components one at a time, to yield the so-called “multiplex” or “Fellgett” advantage. The time-domain signal,f(t)(free-inductiondecay in NMR, interferogram in IR, free-ion decay in ICR), consistsof a sum of the signals from all of the component oscillators (nuclear spins, vibrating molecules, ions)
in which Ai, vi, &, and 7j are the amplitude, frequency,phase,
* To whom correspondence may be addressed.
Also a member of the Department of Biochemistry. (1) Marshall,A. G.; Verdun, F. R. Fourier Transform in NMR, Optical,
t
and Mass Spectrometry: A User’s Handbook; Elsevier: Amsterdam, 1990; 460 pp. 0003-2700/02/0364-1601$03.00/0
and exponential damping time constant for the jth linearlypolarized oscillator signal. Bipolar Sampling. A different (“bipolar”)way of sampling a time-domain signal is to sample oscillators of a given frequency individually. Here the passage of a packet of ions of a given mass-to-charge ratio, mlq, past a given point is detected as the maximum charge induced on a conductor placed at that point (see Figure 1). In order that the detected signal be periodic, it is necessary that the ion packet pass repeatedlythrough the detector at mass-dependent intervals. For example, the ion trajectory could be circular,2-5 as in a toroidal time-of-flight(TOF) mass spectrometer (see Figure 2a), or linear,6J as in a dual reflectronTOF mass spectrometer (see Figure 2b). Both types of time-of-flightinstruments are a~ailable.~-~08 The detected signal is rendered bipolar by placing detectors at two locations differing by about 180° in phase of oscillation (linear TOF) or rotation (toroidal TOF) and storing their respective signal maxima as positive and negative data for later analysis. Alternatively, one could record only time intervals between consecutive ion packets rather than the arrival times themselves. The advantages of time-of-flightmass spectrometry (TOF/ MS) have been recently reviewed799JOand include high upper mass limit, high sensitivity, rapid data acquisition, high ion throughput, and design simplicity. Detectors for TOF/MS are generally of the ion collector type (e.g., channel electron multiplier)mainly due to their high sensitivity-approaching single-iondetection per second (GalileoElectro-OpticsCorp., Sturbridge, MA). However, induced-charge detectors can approach single-ion detection sensitivity if repeated measurements are available, as in McLafferty et al.’s recent multiple remeasurement of the ICR signal from high-mass peptide i0ns.l’ Bipolar sampling has the advantage that timedomain data need be stored only twice per cycle of each component signal frequency,by samplingeach oscillator only at ita maximum (and minimum) signal value, and thus is particularly attractive for sparse broad-band spectra. (2) Wollnik, H. Nucl. Inatrum. Methods Phya. Res. 1987, AH&?, 289296. (3) Wollnik, H. Nucl. Inatrum. Methods Phys. Res. 1987, B26, 267272. (4) Sakurai,T.;Fujita,Y.;Matsuo,T.;Matsuda,H.;Katakuse,I.;Miaeki, K. Int. J. Mass Spectrom. Ion Proc. 1985,66,283-290. (5) Sakurai, T.;Matauo, T.; Matauda, H. Int. J. Mass Spectrom. Ion ROC. 1985,63, 273-287. (6) Wollnik, H.; Przewloka, M. Int. J.Mass Spectrom. IonProc. 1990, 96,267-274. (7) Price, D.; Milnes, G. J. Int. J.Mass Spectrom. Ion Roc. 1990,99, 1-39. (8) Kutscher, R.; Grix, L. R.; Wollnik, H. Int. J. Mass Spectrom. Zon ROC. 1991,103, 117-128. (9) Cottar, R. Biomed. Enuiron. Mass Spectrom. 1989, 18, 613-632. (10) BruneB, C. Int. J. Mass Spectrom. Zon Roc. 1987, 76, 12E-237. (11) Williams, E. R.; Henry, K. D.; McLafferty, F. W. J. Am. Chem. Soc. 1990,112, 6157-6162.
0 1002 American Chemical Soclely
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 14, JULY 15, 1992 Detection Ring Ion Packet
I
h
300000
++++
L
++
L
.-.-ti
Torus Radius = 0.5m
c
200000-
n
a L
Time Domain Signal
100000
.Stored data value
I f
Induced charge
I + time
Figure 1. Schematic monopolar sampling of an ion packet. Top: A packet of ions of a given mlqvaiue passes through a thin ring electrical conductor. Bottom: The charge inducedon the conductor is monitored as a function of time, and only its maximum (+) value is sampled and stored. Bipolar sampling involves the storage of one positive (+) and one negative (-) peak per cycle in the mass analyzer. (a) Toroidal Time of Flight Mass Analyzer
45" Torus Condenser
nM .
Ion MasdCharge (We) Figure 3. Plot of ion rotational frequency in a toroidal timeof-flight mass spectrometer as a function of ionic mass-to-charge ratio (log scale) for ions of 5 0 0 0 4 translational energy circulating at a radius of 0.5 m.
different (constant) velocities. After a short flight period, ions of different m/qbecome spatiallyseparated and therefore arrive at different times at a remote detector. Our proposed linear (Figure 2a) or toroidal (Figure 2b) TOF geometry constrains all analyte ions to isochronous orbits such that the ion flight times depend only upon the m/q ratio independent of their energy spreads. We shall assume here the following: (i) Analyte ions of all m/q ratios are formed simultaneously with zero dispersion in initial position and velocity and are accelerated before injection into the system. (ii) Once in the system, analyte ions of different m/q separate according to their respective velocities into spatially and temporally coherent ion packets which then pass periodically through a given detector at repetition frequency, v. This frequency, v, is independent of the spread in ion energies since the ion flight time in the linear system or the torus is assumed to be energy isochronous. For a length L (meters) of the flight path per period, this frequency v is given in hertz as
(b)
Reflector 1
Detector
L.
in which m is ion mass in kilograms, q is ion charge in Coulombs, V is the electrostatic potential (volts) through which ions are accelerated before injection into the system. (iii)The ion path length through the detector is much shorter than length L. (iv) Residual gas pressure within the system can be neglected since the ion scattering angles are small at kilovolt ion energies, so that ion flight times for most ions remain almost unaffected. In more familiar units
Reflector 2
Linear TOF Mass Analyzer Flgure 2. Schematic sampling of a packet of ions of a given m/9 value, by a toroidal (a) or linear reflective (b) timeof-flight mass spectrometer. As the ion packet transits each detector,the maximum induced charge is stored as a positive or negative discrete value (see text).
THEORY Repeated Linear or Toroidal Time-of-Flight Mass Spectrometry. A fundamental principle of time-of-flight mass spectrometry is the extraction of a closely packed ensemble of ions formed at time zero. If the ion ensemble is isoenergetic and spatially coherent, ions of different massto-charge ratio (mlq)will migrate through the drift region at
in which v is in kilohertz, q is in multiples of the electronic charge, V is in volts, m is in u, and L is in meters. Under these assumptionsthe frequency of singly charged ions with masses 100 OOO-100u range between 3 and 100kHz for a realistic accelerating potential of V = 5000 V and a repetition length of 1.0m (see Figure 3). The digital sampling frequency at each detector should be significantlyhigher than the highest ion repetition frequency in order to obtain an accurate measure of the maximum signal. For a repetition length of 1.0 m (i.e., for a linear TOF of 0.5-m length or a torus of 1.0-m circumference) and an ion packet spatial dispersion of -1 mm, a singly-charged ion packet passes through a plane perpendicular to the ion path in 10ns (for 100 u) to -300 ns (for 100 OOO u). These values can feasibly be achieved by a detector of 4 - m m length, the signal of I
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 14, JULY 15, 1992
which is fed to a commercially available analog-to-digital converter.12 Form of the Discrete Bipolar Signal. As spatially coherent ion packets of a particular m/q ratio travel through a thin ring-shaped conductor, they induce a potential difference in this conductor and thus a current. The detected signal from an ion packet of one m/q ratio should appear as a series of narrow peaks. It is worth noting that the analytical form of the time-domainsignal (e.g., sawtooth, Gaussian, etc.) is not known and need not be known because it does not affect the present method. For bipolar detection, two detectors are placed in the system, one yielding a positive signal (+)and the other a negative signal (-). Only the (time, amplitude) data pair corresponding to a given signal extremum is recorded and stored (see Figure 1,bottom, for which only the positive (+) half of the signal is shown). For a packet of ions of a single m/q, the bipolar time-domain signal for a singleionic mass thus consistsof a seriesof equullyspaced spikes of alternating polarity (+/-/+/-), whose amplitudes are proportional to the number of ions passing through the detector f ( t ) = A exp(-At/.r),
-A exp(-2At/.r), A exp(-3At/~), -A exp(-4At/~), (3) in which At = 1/2v, where v is given by eq 2. When ions of N distinct m/q are present, the time-domain data F a y will consist of a superposition of N series of data, eachdthe form of eq 3, with different values of A, 7, and At. Although the data from ions of any one m/q is a set of equally-spaced values, the superposition of data from ions of different m/q is in general not equally-spaced (see Figure 4a). For simulated data, we of course know that equally spaced extrema of the bipolar signal, f ( t )of eq 3, must occur at ( Z z v t + 4 = nr), or t = (nr4)/2zv,n = 0, 1,2, ... That is, simulated peak maxima were computer generated by use of a cosine function; the peak maxima are periodic regardless of the exact form of the time-domain signal. For actual data, it is necessary to locate the signal extrema by use of a suitable peak-finding algorithm,'3 applied to real-time sampling of the continuous signal (Figure 1, bottom) at a frequency much higher than the TOF repetition frequencies of interest. Recognition of Spectral Components. For actual data, we must identify the periodic components of the unequally spaced raw data (e.g., Figure 4a). We here consider two approaches: (i) direct pattern recognition and (ii) interpolation of the time-domain data to yield a continuous function which could then be sampled at equally-spaced intervals and subjected to discrete FT to yield a frequencydomain spectrum. Pattern Recognition. Pattern recognition for characterization of one-dimensional (i.e., oscillatory waveforms) or multidimensional (i.e., satellite reconnaissance) data is often appropriate for composite data sets whose components share a common coordinate (e.g., time, space) in which each component has repetitive internal features.14 Bipolarsampled signals offer just such an application, since each packet of ions of a given m/q generates a periodic time-domain data set. In this work, we have devised a corresponding pattern recognition algorithm (seeQuickBASIC15algorithm available on request). First, find the highest-amplitude time-domain data point, then locate the nearest (in time) adjacent extre-
Tlme-domelndate:
0-0.001sec
1603
Component frequency: 13941 Hz
Component frequency: 1977 Hz Amplitude: 100.00
Component frequency: 42948 Hz Amplitude: 99.955
Component frequency: 4420 Hz Amplitude: 99.995
Component frequency: 139828 HZ
...
(12) LeCroy Model 9400A Digital Oscilloscope Operator's Manual, October 1988. (13) Press, W.H.; Flannery, B. P.; Teukolaky, S. A.; Vetterling,W. T. Numerical Recipes; Cambridge University Press: New York, 1986. (14) Ahuja, N.; Schachter, B. J. Pattern Models; John Wiley & Sons: New York, 1983. (15) Microsoft QuickBASICUser's Guide (Macintosh Systems),1986.
figure 4. Simulated blpolar-sampled noiseless data for a signal with components of five different frequencies (1976,4420,13 941,42 948, and 139 828 Hz, Correspondingto singly-chargedionsof 25 0 13,5003, 503, 53, and 5 u, with 50004' kinetic energy, rotating In a 0.5-mradius toroidal time-of-flight mass spectrometer (Figure 2a) and equal amplitudes, phases, and time-domain exponentialdamping constants. (a) Composite bipolar-sampled signal from all five components. (b-9 Single-frequency signals extractedby pattern recognition (see text) for each of the five components.
mum, and then calculate the time interval (At) between the two. From the (known) total acquisition period, the number of theoretically possible extrema for the signal component at frequency, (2At)-l Hz, is calculated. The algorithm then searchesthe experimentaldata set at all predicted time values, and stores the signal values from those times. The search is considered valid if the number of located extrema exceeds at least 95% of the possible number expected for that frequency component. Once a spectral component has been identified (for a given At), those data are stored independently and then removed (i.e., amplitudes set to zero) from the original time-domain data. The remaining data is then searched for its highest amplitude component, and so on. If a spectral component is not found for the originally specified At, then a new At interval is calculated as the time between the highest amplitude extremum and the second nearest adjacent extremum in time. This identification/characterization/removal strategy is continued until the highest amplitude signals remaining in the time-domain data are at the noise level. Interpolation Followed by FFT Analysis. Direct discrete FT analysis of bipolar-sampled data is impossible because the data are unequally spaced in time. However, it is possible to interpolate the bipolar-sampled data with a cubic-spline procedure13J6to yield equally spaced data to which discrete FT processing1 could be applied.
RESULTS AND DISCUSSION
It is important to distinguish between the present bipolar sampling and simple "clipping" of a conventionaltime-domain signal to 1bit, as previously applied to FT/NMRl7and FT/ ~ ~ _ _ _ _
~____
(16) Meisel, D. D. Astron. J. 1978,83, 538-545. (17) Larsen, R. D.; Crawford, E. F. Anal. Chem. 1977,49, 508-510.
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 14, JULY 15, 1992
ICR.18 In the latter cases, the original unclipped signal is of the form given by eq 1,whereas in bipolar sampling, the signal from only one frequency component (eq 3) is detected at a time. Bipolar sampling also differs from binary delta modulation of analog signals in long-range communications theory.lg Pattern Recognition of Bipolar-Sampled Time-Domain Noiseless Signals. Our pattern recognition algorithm was applied to the identification of five signal components of different frequency (139 828-1977 Hz, corresponding to singlycharged ions of 5-25 013u) and equal magnitude,phase, and time-domain damping constant from their bipolarsampled noise-free time-domain data. In all simulations,the input masses were assigned integer values for simplicity. As visually evident from Figure 4, the algorithm successfully isolated the five frequency components (Figure 4b-f) from the simulated “raw” bipolar-sampled time-domain data (Figure4a). No spurious “extra” extrema are present in any of the extracted component data sets, and none is missing. This high-component isolation efficiency is especially significant in view of the wide mass range presented in Figure 4, although in practice the isolation efficiency will be limited by time-scale errors. The pattern-recognized frequency, relative magnitude, phase, and time-domain exponential damping constant for each component were highly accurate. Time-scale accuracy was limited only by computer rounding errors. The theoretical mass scale accuracy was of the order of lO-l5 relative error. The limit of mass resolution was tested by simulating (not shown) singly charged ions of mass 5.000 OOO 000 OOO and 5.000 OOO OOO 005 u; the pattern recognition algorithm successfully resolved (resolving power of 1012) the frequencies (and phases) of both input components. Ion relative abundance accuracy was about In any case, high mass resolution from direct pattern recognition of bipolar-sampled data appears feasible, in principle. The present algorithm is simple and rapidly executable (- 5 min for the data of Figure 4) on a standard personal computer by use of a compiled QuickBASIC program. Limitations of Pattern Recognition of Bipolar-Sampled Noiseless Time-DomainData: Time-of-FlightMass Spectrometry Aspects. The validity of the analysis illustrated for toroidaltime-of-fight mass spectrometry in Figure 4 is limited by several practical considerations. First, in any time-of-flight mass spectrometric experiment, a packet of ions of a given mlq ratio will have a kinetic energy spread and thus a velocity spread, due to initial ion thermal energy and spatial and temporal variation in ion formation. Also, ion detection efficiency will be mass-dependent, and ion-ion repulsions may interfere with focusing. Mass resolving power of >10 000O : >20 000,21 and recently >35 O0Oz2 have been reported by means of various combinations of narrow initial ion pulse width (1.5 X lo+ s by means of laser desorption), a supersonicion beam source,and gridleasreflectron detection. Those experimental results suggest that ion packets with sufficientlynarrow energy and spatial coherence for bipolar sampling should be feasible. We further assumed that the ion detector is sufficiently thin that the total (time-domain)width of the detected signal (Figure 1,bottom) is not significantlybroadened by passage ~
~~~
(18) Hsu, A. T.;Ricca, T. L.; Marshall,A. G. Anal. Chim. Acta 1986,
178,27-41. (19) Steele, R. Delta Modulation Systems; John Wiley & Sons: New York, 1975. (20) Walter, K.; Boesl, U.; Schlag, E. W. Znt. J. Mass Spectrom. Zon Proc. 1986,71,309-313. (21) Grix, R.; Gruener, U.; Li, G.; Kutacher, R.; Wollnik, H.Rapid Commun. Mass Spectrom. 1988,2, 83-85. (22) Bergmann, T.; Martin, T. P.; Schaber, H.Reo. Sci. Znstrum. 1989, 60,792-793.
*
E
.-
Repetltlon Frequency
3
10-4
139,828 HZ
4
3b 2 x 1 0 - 4 1
-a e
LL
0
li
m n
1;-8
0
II
2 x‘10-8
RMS N o h Level
Flgure 5. Relative error in pattern-recognized frequency of bipolarsampled data, as a function of the rms wldth of Qausslan-dMbuted Jitter in the sampling times.
through the detector. More generally, we assume that no two packets of ions of different m/q pass through a given detector at the same time. In addition to the temporal dispersion due to detector thickness, temporal broadening also depends on the velocity (and mlq value) of the ion packet. For example, under the simulated experimental conditions of 5000eV ion injectionenergy and 1 V - mdetector thickness, the signal broadening is estimated to range from 10-loto lo” s for singly charged ions of 1-10 OOO u. Thus, for a packet of singly charged ions of 100u, detector broadening should be comprable to temporal broadening arising from pulsed ion formation (104 s). Although the present pattern recognition algorithm was designed for speed and simplicity,its lack of complexity can lead to complications. For instance, suppose that an ion packet contains both doubly and singly charged ions of the same mass which happen to arrive at the detector at the same time and thus generate simultaneous detection signals. The present algorithm would treat such a data set as a single frequency component. If ion packet time-domain amplitudes aswell as arrivaltimes were included, then the algorithmcould resolve various harmonics from each other. The proposed experiments require that ions be able to continue with spatial coherence for many cycles of the lowest repetition frequency (corresponding to the lowest mlz) of interest. An advantage of this type of time-of-flight mass spectrometry is kilovolt ion injection energy, so that ions are not easily lost via “near-collisions*, as is the case for some low-energy mass analyzers. Effect of Time-Domain Noise on the Pattern Recognition Algorithm. Since the pattern recognition of bipolarsampled time-domain data depends on precise periodicityin the original signal frequency Components, random errors in the measurement time scale are potentially critical. Therefore, to simulatethe effectof such errors,bipolar time-domain sampling of a simulated 139 828-Hzsignal was subjected to a Gaussian-distributedjitter of specified rms width. Figure 5 shows the increase in relative error in pattern-recognized signal frequency (i.e., the difference between the actual and calculated frequenciesrelative to the actual frequency) as a function of time-domain jitter in the sampling times. The figure shows that sub-part-per-million frequency accuracy requires time-domain rms jitter below - 1 0 - 1 O s, or about 1% of the experimentally achievable sampling interval of 1O-a s. However, in order to “flag” at least 95% of the observable extrema for a given signal frequency component, the acceptable time-scalejitter should be -10-l1 s. Also, the pattern recognition algorithmcan be adjusted torecognizedata within a predetermined time interval of the At value predicted from
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 14, JULY 15, 1992
the difference between the initially sampled extrema (see Theory). Uncertainty in the amplitude of a bipolar-sampled timedomain signal will shift the ion packet maxima. The larger the noise amplitude, the larger will be the error in determining the instant of passage of that ion packet through the detector. If the analytical form of the time-domain signal were known, it would be possible to quantitate the effect of amplitude noise. However, as noted above, the present detectionscheme is independent of time-domain peak shape, and records only the time-domain peak maxima. Thus, the error introduced by such amplitude noise is not easily quantitated. C u b i c Spline I n t e r p o l a t i o n / F F T Analysis of B i p o l a r S a m p l e d Noiseless Data. As an alternative to the pattern recognition approach, a cubic spline function was used to interpolate nonuniformly spaced bipolar-sampled timedomain data into uniformly spaced data for subsequent discrete FT data reduction. The cubicspline was chosen due to its continuous nature and its roubu~tness.~3 Typically(see Figure 6a,b), cubic spline interpolation of a simulated singlefrequency time-domain signal followed by FFT produces a spectrum exhibiting a major peak at the fundamental frequency, with some additional minor peaks elsewhere in the spectrum. However, the presence of as few as two signal components of different frequency results in an unacceptably distorted frequency-domain spectrum (Figure 6c). In retrospect, it is clear that the FT approach should not be expected to work for bipolar-sampled polychromatic signals. FT data reduction is intended for a time-domain signal whose individualfrequency componentsare first added and then sampled. In bipolar sampling, on the other hand, the individual frequency components are first sampled individually,and then arrayed in a single (unequallyspaced) row for analysis. Stated another way, FT treatment applies to time-domain “interferograms”,in which all of the signal components add (“interfere with”) each other, whereas the (23)Frigieri, P.F.;Rossi, F.B. Anal. Chem. 1979,51,54-57.
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Magnitude-Mode Spectra: 0 250,000 Hz 512 data points per spectrum
Flgwe 6. Absorption-mode spectra obtained by discrete Fourier transformation of a cubic-spilne interpolated bipolargempled data set. Note that FT data reduction gives a good representatlon of singlefrequency tlmedomain signals: (a) 139 828 Hz; (b) 42 948 Hs; but introduces unacceptable distortion when the sarne two components are simultaneously present in the timedomain signal (c).
bipolar sampling removes such interference before sampling.
ACKNOWLEDGMENT We thank P. B. Grosshans for valuable discussions. This work was supported by grants (to A.G.M.) from the USA. National Science Foundation (CHE-9021058) and The Ohio State University.
RECEIVED for review January 30, 1992. Accepted April 13, 1992.
