Anal. Chern. 1980, 52, 1090-1094
1090
Convolution-Based Phase Correction in Fourier Transform Mass Spectrometry E. B.
Ledford, Jr., R.
L. White,
Sahba Ghaderi, M. L. Gross,” and C. L. Wilkins”
Department of Chemistry, University of Nebraska-Lincoln, Lincoln, Nebraska 68588
I n this paper, we report the first experimental demonstration
of the technique of convolution-based phase correction in Fourier transform mass spectrometry. The technique eliminates phase distortions which have previously been minimized by computatlon of magnitude spectra, rather than absorption-mode spectra. It is shown that expected increases in mass resolution using the new phase correction procedure are achieved.
Fourier transformation is a powerful tool for the study of resonant systems. In such studies, one is concerned with the response of the system to an excitation of some form. Fourier transformation of the system‘s temporal response often provides a simple spectral characterization of it. I n Fourier transform mass spectrometry (FT/ MS) ( I ) , translational excitation of ions stored in a static magnetic trap is accomplished by a radio frequency “chirp” in which the frequency of an oscillatory excitation signal is scanned rapidly across the band containing resonance frequencies of interest. Because different ion species are excited a t different times, “phase distortion” appears in FT spectra unless measures are taken to eliminate it (2). Recently, Marshall has suggested a procedure which allows computation of phase-corrected absorption- (rather than magnitude-) mode spectra in F T / M S ( 3 ) . We have implemented this procedure and report here t h e first measurements verifying that the method yields phase-corrected spectra with improved mass resolution as predicted from the theory ( 3 ) . THEORY Mass analysis in F T / M S , as in pulsed ion cyclotron resonance (ICR) spectroscopy ( 4 ) ,is based upon the orbital motion of ions in a strong and uniform magnetic field ( I ) . Ions are formed by pulsing an electron beam through a trapped ion analyzer cell (4)(see Figure 1). Following a variable period during which ions and neutral molecules may react, the ions are translationally excited by applying a time-varying potential (rf “chirp”) to the transmitter plate. The coherently excited ions generate “image currents” in the circuit connecting the receiver plates (5). The consequent signal voltage, developed across a n RC load, has the form of a composite harmonic oscillation with one major frequency component for each excited ion species. The composite oscillation (transient) is damped by processes (e.g., collisions) which degrade the coherent motion of uniformly accelerated ions. After observation of t h e signal for a suitable time period, the cell is cleared by imposing a “quench pulse” which sends all ions to one of the t r a p plates. When the time-varying potential applied to the transmitter plate is a frequency-swept sine wave. ion species of different mass to charge ratios are excited sequentially, rather than simultaneously, as would be the case if the excitation were an ideal Dirac delta impulse. Consequently, the ion image oscillations for different mass to charge ratios have nonzero phases, causing asymmetry in the computed mass spectra (Figure 2b). Therefore, an important consideration in Fourier 0003-2700/80/0352-1090$01 O O / O
transform mass spectrometry (as in all forms of Fourier transform spectroscopy) is elimination of this asymmetry or “phase distortion”. A common approach is to compute a magnitude spectrum (2, 3 ) . This results in a clean and interpretable spectral representation. However, this method has its drawbacks. One may show that the full width a t half-height of a magnitude mode peak is twice that of an absorption peak. Furthermore, combining dispersion and absorption spectra in quadrature results in broad “skirting” a t the base of the magnitude peak, which significantly degrades resolution near the base line (Figure 2b). Another approach to phase correction employs the convolution theorem, as proposed by Marshall, for F T / M S ( 3 ) . In this paper we describe an experimental implementation of his proposal and, in addition, develop a more extended theory of convolution-based phase correction. The Convolution Theorem. A simple model of a trapped ion analyzer cell is that of a generalized two terminal “black box,” which, when subjected to electrical excitation, produces an electrical response related to the excitation by a linear differential equation ( 5 ) . Let the excitation “signal” be denoted &), a function of time, and likewise, the response signal g ( t ) . The convolution theorem relates the response to the excitation (2, 3) thus:
dt)= h ( t ) * e ( t )
(1)
where * denotes a convolution operation and h ( t ) is the impulse response, Le., the response that would be obtained if an ideal Dirac delta excitation were applied. T h e theorem further states that convolution of time domain functions corresponds to multiplication of their Fourier transforms, i.e.,
G ( f )= H ( f ) * E ( f ) (2) in which G ( f ) , H ( n , and E ( n are the Fourier transforms of the response, impulse, and excitation functions, respectively. By complex division, the result in Equation 3 is obtained.
m -- H ( f l G(f)
(3)
The Fourier transform of the impulse response is the complex ratio of the Fourier transforms of the device response and the applied excitation. H ( f l is a frequency spectrum which is inherently free of phase distortion. T h e Need for B a l a n c e d B r i d g e Detection. In general, the functions h ( t ) ,and e ( t ) ,start from t = 0. Practical application of convolution-based phase correction requires that A/D conversion of the ion image signal, g ( t ) , begin a t t = 0. Here an experimental difficulty arises. The excitation signal feeds through the capacitances between the transmitter and receiver plates of the FT/MS analyzer cell. T h e amplitude of this feedthrough signal is very much larger than that of the ion image signal, making it impossible to observe the latter during excitation. A practical solution to this problem is the use of a balanced bridge detector. By incorporating the receiver plates of the analyzer cell into a bridge circuit, rf feedthrough during excitation can be nulled, permitting observation of the ion image signal from t = 0. The bridge can be thrown out of balance to reestablish feedthrough for the .s’ 1980 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 52, NO. 7, JUNE 1980
,-RECEIVER
ai/
1091
-T
PLATE
U
J
RAP
FILAYEWT
Figure 1. Cubic analyzer cell used in the present study
purpose of digitizing the excitation waveform. Convolution-Based Phase C o r r e c t i o n i n t h e D i r e c t Mode. Two signal acquisition modes are employed in FT/MS ( 5 , 6). In the “direct mode“, the amplified ion image signal is directly digitized by a high speed A/D converter and stored in a fast buffer memory prior to discrete Fourier transformation. The convolution-based phase correction is particularly straightforward when the spectrometer is configured for direct mode signal acquisition, since Equation 3 may be applied directly to the Fourier transforms obtained. Convolution-Based P h a s e C o r r e c t i o n i n t h e Mixer Mode. In the second mode of operation, the “mixer mode”, data acquisition time is increased by frequency translating the ion image oscillation to lower frequencies. Frequency translation is accomplished by multiplying (mixing) the ion image signal with a reference sine wave and passing the resulting heterodyne signal via a band pass filter to the A / D converter. T h e digitized heterodyne signal is subjected to discrete Fourier transformation. Convolution-based phase correction is possible in the mixer mode, but special measures must be taken, as a simple analysis will show. Consider that the reference signal r ( t ) is mixed with the ion image signal g ( t ) , to form the product r(t).g(t),and fed t o a band-pass filter. A linear band-pass filter, like the F T / M S analyzer cell, has a characteristic impulse response, which we shall call h b ( t ) to distinguish it from the ion impulse response h ( t ) . In accordance with the convolution theorem, the action of the filter is to convolve its impulse response with the mixer output, to form g(t).r(t)*hb(t).Therefore, the ion signal presented to the A/D converter, call it s , ( t ) ,is given by
s , ( t ) = h(t)*c(t).r(t)*h,(t)
(4)
in which the order of operations is strictly left to right. Similarly, when the excitation signal is fed through the mixer/filter combination, there results a t the A/D input a function, designated s,(t), given by s , ( t ) = e(tbr(t)*h,(t)
(5)
where again the order of operations is strictly left to right. Let S,(f, and S,cf, be the Fourier transforms of the ion image and excitation signals, respectively. Then, by the convolution theorem,
S,CR = H ( f ) * E ( J ) * R ( f ) . H b ( f ) S,(f) = E(f)*R(f).Hb(f)
(6)
left to right, because this reflects the direction of signal flow through a train of electrical devices. Consequently, the phase-corrected spectrum is “buried” in a convolution integral. In general, simple division of S,(f) into Si(f),by analogy to direct mode processing, does not isolate H ( f )on the right hand side. There is, however, a special case which permits simplification of Equations 6 and 7 . Assume the reference signal is a true sine wave, Le., r ( t ) = A sin (2afrt) (8) where A is the base to peak amplitude and f , is the reference frequency. T h e Fourier transform of Equation 8 is
R ( f ) = l/jA[-6(f - f r ) + S(f + f r ) l (9) in which S ( f ) is the Dirac delta function and j = 6. Dirac delta functions convolve in a particularly simple fashion ( 7 ) ,thus: HO*S(f)= H(f) (10) (11) Equation 8 may be substituted into Equations 6 and 7 , and ff(f)*S(f
fr)
= H(f - f r )
Equation 11 applied to obtain:
(13) Se(f) = ‘ / j A E [ - E ( f- f r ) + E(f + f r ) I’Hb(f) T h e ratio of Equations 12 and 13 cannot be taken to cancel out Hb(f)because the signals s , ( t ) and s,(t) are band-pass filtered prior to digitization and discrete Fourier transformation. The effects of the low-pass filter are therefore manifest in S,(f) and S,(f). We must carry out the multiplication by the filter function H b ( f )before taking the ratio of S , ( f )and S , ( f ) . An ideal band pass filter may be described as a frequency window spanning a frequency band of interest (7): O ; - fbl fo ffb(f) = 1/22; If - fbl = f o If f b l < f o (14) where f b is the center frequency of the window and f o is its half width. Substitution of Equation 13 into Equations 11 and 1 2 yields, over the interval If - f b l < f,,,
If
’
fbbl
fr)I; < f o (15)
+ E(f + f r ) I ; If - fbl < f o
(16)
S,(f) = 1/2iA[--E(f- f r ) * H (-f f r ) +E(f + fr).H(f
If
(7)
Again, the order of operations in Equations 6 and ’i is strictly
-
S,(f) = l / j A [ - E ( f- f r )
--
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ANALYTICAL CHEMISTRY, VOL. 5 2 , NO. 7, JUNE 1980
TIME DOMAIN
a
Figure 3. Graphical frequency domain representation of the analog signal processing involved in mixer mode FT/MS. The mixer convolves the ion signal (a) with the reference sinewave (b) to form a mixer output (c), which is a sum of beat and heterodyne signals. An ideal bandpass filter (d) is used to select the heterodyne peak (e), which is to close approximation a frequency translated version of the ion signal, (a) '\ \
1
' .
CONVOLVED
120" PHASE ANGLE
li
Re
I 1!
DE-CONVOLVED
I
Figure 2. (a) Graphical representation of an exponentially damped sine wave and the effect of phase shift upon its appearance. (b) The effect of phase angle on the appearance of the real and imaginary parts of the Fourier Transforms of the sine waves in (a). Below is a magnitude mode spectrum showing degraded resolution, especially near the base line
T h e action of the band pass filter is to place a frequency restriction, If - jbl < f,, on the mixer output. Physically, f b a n d f, are knob settings on an electronic filter whose characteristics are designed to be as close as possible to the description in Equation 14. From Equations 15 and 16, it is apparent that no matter what the filter setting, both heterodyne (terms in f g - j,) and beat (terms in f g + j,) signal components are present in S,cf, and S,(fl. The reason for this is illustrated in Figure 3. As a result of mixing response and excitation signals with a sine wave, each spectral function, SlM and S,cf,, exhibits two peaks, one centered a t the heterodyne frequency, fg - f,, and the other centered at the beat frequency, f g + f,. These peaks approach the frequency axis asymptotically, i.e., they tail into each other (see Figure 3). However, provided the peaks are well separated. the contribution of the beat peak to t h e heterodyne peak is negligible. Therefore, when the center frequency of the low-pass filter is in the vicinity of the heterodyne frequency, i.e., j b = Ifg - f J ,con-
I
II
I
ll
I"+----------+-.. ,
117
119
-
I
' 1
L
~
I
c, 1--
131 1 3 3 ' 3 5
121
MASS l a m u )
Figure 4. Convolution-based phase correction in direct mode for the mukipeak Fourier transform mass spectrum of l , l ,1,24etrachloroethane Similar results are obtained in mixer mode
tributions from beat terms may be neglected. T o close approximation, then,
S,(f,
-1
se(J)
A E ( f- fr)*ff(f =
-J AE(f 2
-
- fr); fh
fr): f b
Ifg -
lfg - frl
frI
(17)
(18)
whence
From Equation 18, it is clear that division of the mixed and filtered ion image transform by the mixed and filtered excitation transform yields a frequency shifted version of the
ANALYTICAL CHEMISTRY, VOL. 52, NO. 7, JUNE 1980
-~
226
~~~
236
246
frequency ( K H ? )
+ lm2G(f)
JRe'GIf
~
236
226
246
frequency ( K H z )
Figure 5. (a) A direct-mode transient signal derived from benzene radical cation ( m l e 78) and its magnitude spectral representation. (b) The same signal derived in mixer-mode, and its magnitude representation, showing increased S/N and increased resolution
phase-corrected spectrum H ( i ) over the restricted frequency range determined by the low pass filter. T h e crucial step in arriving at Equation 18 is the use of a zero phased sine wave as a mixer reference function. This makes possible the explicit evaluation of the convolutions in Equation 6 (the same result is obtained if the mixer reference function is a cosine wave), owing t o t h e simple convolution properties of Dirac impulse functions. Reference functions whose Fourier transforms are not expressible in terms of Dirac impulse functions may not lead t o simple evaluation of the convolutions, with the result that t h e phase-corrected spectrum cannot be isolated by simple manipulation of Equation 6. It is apparent from this analysis that convolution-based phase-correction in the mixer mode requires two waveforms t o be simultaneously applied to different points in the signal train: (1)the excitation potential applied to the cell, and (2) t h e sine wave applied to the mixer. Thus, two separate rf voltage sources are needed for convolution-based phase correction in mixer mode FT/MS.
EXPERIMENTAL Instrumentation. Experimental tests of convolution-based phase correction were carried out on a modified Varian ICR-9
ion cyclotron resonance mass spectrometer. A trapped ion cell with cubic geometry (plate separation 0.0254 meter) was used in place of the original drift cell (Figure 1). Ion excitation was provided by a Rockland Systems 5110 programmable frequency synthesizer under computer control. The mixer reference sine wave was provided by a second Rockland synthesizer operated manually. Ion image signals were amplified by a Tektronix 5A22N high-speed differential oscilloscope amplifier. For balanced bridge detection, nulling capacitors were wired into two small metal boxes with BNC connectors a t either end. The boxes were inserted between the 5A22N amplifier and its input cables using the BNC connectors. The nulling capacitors were of the transistor radio type, 13 to 332 pf, as measured with a capacitance meter. Plastic knobs were attached to the shafts of the capacitors to prevent noise pickup when touched by the operator. Nulling of the cell was readily accomplished over a wide frequency range. An excitation voltage amplitude of 7 V peakto-peak produced a feedthrough signal of less than 20 p V peakto-peak (the noise level on the oscilloscope) when the capacitors were properly adjusted. Rapid balancing and unbalancing of the bridge was possible either by adjusting the null capacitors or by pushing the GROUND button on one ofthe differential amplifier inputs. The mixer employed was an OEI model 5050 high-speed four quadrant analog multiplier module. For band pass filtering, a Krohn-Hite model 3200 filter was employed. Data Processing. For data acquisition and reduction, a 40K word Nicolet 1180 minicomputer, equipped with a high-speed buffered digitizer capable of data acquisition of up to 5 MHz (9-bit resolution) or 2.5 MHz (12-bit resolution), was used. For data storage and display, the data system was equipped with a Diablo Systems Model 40 five million word disk drive, a Tektronix 4662 digital plotter, a Tektronix 4010 display terminal, and a Texas Instruments 810 impact printer. Typical F T / M S Conditions. Sample pressure (benzene) of 4 x lo-.' Torr as measured by an ion pump current gauge was used. The cell was operated with the quench pulse turned off. Thus, it functioned as a static ion accumulator. Transient, signals lasting about 200 ms were obtained. Two hundred ensemble-averaged transients were acquired prior to discrete Fourier transformation, which resulted in a signal-to-noise ratio of about 200:l in FT spectra. Typical conditions were: ionizing electron energy, 15 eV; trapping voltage difference, 0.86 V; electron beam duration, 5 ms; 20-nA instantaneous emission current; magnetic field, 1.2 Tesla. In direct mode operation, a radiofrequency excitation from f = 0 Hz to f = 300 KHz with peak-to-peak amplitude of 7 V was applied to one transmitter plate (the other was grounded). Sweep rate was about lo6 H z j s , with the synthesizer stepped 1 kHz every
DIRECT MODE
CONVOLVED
I
I
' I
_-
--
f DE-CONVOLVED
.~ --i ' s -
, 2 34
,
238
L 236
Frequency
238
(kHz1
1093
234
236
Frequency
Re
Figure 6. Convolution-based phase correction in direct mode for benzene molecular ion
Im
(kHz1
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ANALYTICAL CHEMISTRY, VOL. 52, NO. 7, JUNE 1980
microsecond. The AID converter was operated at a BOO-kHz conversion rate, and 16K data points were sampled, with obvious truncation of the ion image signal. In the mixer mode, a narrower rf excitation from f = 226 kHz to f = 246 kHz, and a sweep rate of 5 X lo4 Hz/s (steps of 50 Hz per microsecond) was employed. Amplitude of this excitation was 0.84 V to peak-to-peak. AID conversion rate was 40 kHz, and 8K data points were sampled with no apparent truncation of the ion image signal. The excitation signal was fed to the acquisition system by either detuning the balanced bridge detector or grounding one input of the differential amplifier. Ions were removed from the cell during acquisition of the excitation signal by turning off the heated filament used as a source of ionizing electrons. The appearance of FT mass spectra due t o a single ion resonance derived from benzene radical cation, m / e 78, were examined with and without convolution-based phase correction. Convolution-based phase correction was performed after completion of the signal acquisition and discrete Fourier transformation processes. Software written in BASIC was used for this purpose. Tests of convolution-based phase correction were also carried out on spectra having many peaks (see Figure 4).
RESULTS AND DISCUSSION T h e result of direct a n d mixer mode signal acquisition of a single resonance derived from benzene molecular ion at m / e 78 (resonance frequency nominally 236 kHz at 1.2 tesla) is shown in Figure 5. This illustrates the results obtained in conventional F T / M S operation. Broad “skirting” at the base of each magnitude peak is apparent. When the discrete Fourier transform of the direct mode ion image signal is divided by the transform of the ion excitation signal (Figure 6) (note the division is that of two complex functions, not a ratio of magnitude spectra), a phase-corrected spectral representation results. In the absorption spectrum (Figure 6) one may note the disappearance of phase distortion a n d improvement in mass resolution over magnitude mode (Figure 5a), especially near the base line. In practice, a small residual feedthrough signal is often observed (see Experimental). This gives rise to feedthrough contributions to the “nulled” spectrum as in Figure 4. T h e deconvolution procedure effectively eliminates this contribution. Similar results are obtained for mixer mode spectra when a zero-phased sine wave is applied to the mixer reference terminal. T h e resolution in the mixer mode absorption
spectrum is significantly improved (31 000 full width at half height definition) over that of the direct mode absorption spectrum (Figure 6) owing to the longer observation time (A/D sampling period) available in mixer mode. Spectra of 1,1,1,2-tetrachloroethanedemonstrate t h e phase correction procedure is applicable to wider bandwidths containing multiple peaks (Figure 4).
CONCLUSIONS A general phase-correction technique based on the convolution theorem has been experimentally demonstrated for both direct and mixer modes in Fourier transform mass spectrometry. By elimination of phase distortions which result from practical ion excitation methods, a factor of two in mass resolution is gained (full width a t half-height definition), and the broad skirting near the bases of peaks inherent in magnitude mode display is attenuated. Two modifications of conventional F T / M S instrumentation are required in order to implement convolution-based phase correction. First, a balanced bridge detector circuit is needed to null excitation feedthrough into t h e detection circuitry, since signal acquisition must begin at the start of excitation. Second, a separate radiofrequency voltage source must be used to provide a zero-phased, pulsed sine wave to t h e mixer reference terminal during mixer mode signal acquisition.
ACKNOWLEDGMENT Helpful discussions with Donald Rempel are appreciated.
LITERATURE CITED (1) Comisarow, M. B.; Marshall, A . G. Cbem. Pbys. Lett. 1974, 25, 2 8 2 . (2) Comisarow, M. 8.;Marshall, A. G. Can. J . Cbem. 1974, 52, 1997. (3) Marshall, A. G. Cbem. Pbys. Len. 1979, 6 3 , 575. (4) Mclver, R. T., Jr. Rev. Sci. Instrum. 1970, 4 1 , 555. (5) Comisarow, M. B. J . Cbem. Pbys. 1978, 69, 4097. (6) McIver, R . T., Jr.; Ledford, E. B., Jr.; Hunter, R. L. J . Chem. Phys. in
press. (7) Brigham, E. Oran. “The Fast Fourier Transform”; Prentice-Hall: Englewood Cliffs, N.J., 1974; pp 50-74.
RECEIVED for review October 29, 1979. Accepted March 17, 1980. Support of this research by the National Science Foundation (Grant No. CHE-77-03964) and a grant from Gulf Research Foundation is gratefully acknowledged.
