2094
Anal. Chem. 1983,55,2094-2096
Zoom Transform for Mass Measurement Accuracy in Fourier Transform Mass Spectrometry Thomas J. Francl, Richard L. Hunter, and Robert T. McIver, Jr.*
Department of Chemistry, University of California, Irvine, California 9271 7
Fourler transform mass spectrometry (FT-MS) provldes ultrahigh mass resolution, exceeding that of the best double focusing sector mass spectrometers. One of the problems encountered, however, Is not havlng enough data points across a peak to calculate accurately the peak height and centroid. I n this paper a numerlcal analysls method called the zoom transform is described. I t Is Implemented by applying forward and Inverse Fourier transforms to FT-MS data which have been previously acqulred and stored In a computer. The zoom transform procedure greatly Increases the number of data points across a peak, improves the mass measurement accuracy, and Is faster and easler lo calculate than the zero-fllllng method used prevlously.
In Fourier transform mass spectrometry (FT-MS) ions are stored inside an analyzer cell that is situated in a strong magnetic field. The ions are constrained by the magnetic field to move in circular orbits having a frequency o = qB/m (1) where w is the cyclotron frequency, Is is the magnetic field strength, and m l q is the mass-to-charge ratio. One of the features of FT-MS is that all ions in the analyzer cell can be detected simultaneously by accelerating them with a radiofrequency pulse and measuring their cyclotron frequencies with a sensitive amplifier (1-4). Since frequency is the primary parameter measured in FT-MS, high mass measurement accuracy and unmatched stability can be achieved. A new calibration equation has recently been developed in our laboratory which provides a mass measurement accuracy of 0.8 parts-per-million (pprn) over a 2 mass unit range (5). Previously, Ledford and coworkers demonstrated a 3 ppm error over the range m / z 116-135 (6). White and co-workers have shown that the high field stability of a superconducting magnet allows accurate mass measurements to be made in the absence of internal calibration standards (7). Accurate mass measurement is best accomplished by using the narrow-band detection method (also called heterodyne or mixer mode). However, a major limitation of the narrow-band mode is that only a small range (typically a few atomic mass units) of the mass spectrum is detected a t a time. T o get accurate mass assignments for several peaks, a separate narrow-band experiment must be performed for each peak of interest. Unfortunately, this defeats one of the main advantages of FT-MS which is to detect all ions simultaneously. A different mode of operation, called broad-band detection, must be used to acquire a complete mass spectrum. When this is done, however, the discrete nature of the FT-MS spectrum severely limits the accuracy of determining peak heights and centroids. For example, in a typical broad-band acquisition the spacing between data points is 0.017 amu a t m / z 158, which could give rise to mass measurement errors of as large as 54 ppm. In this paper a numerical analysis procedure we call the zoom transform is described that increases the number of data
points across a peak and allows for greatly improved mass measurement accuracy. With the zoom transform many of the advantages of the narrow-band mode can now be realized with the broad-band mode of FT-MS. DISCRETE NATURE OF FT-MS SPECTRA To illustrate the limitations of the broad-band mode, a computer-generated FT-MS transient consisting of equal amplitude signals at m / z 78.04640, m / z 157.9549, and m l z 233.8688 is shown in Figure 1. The transient signal consists of a total of 64K (65 536) data points and lasts for 93.62 ms. Taking the Fourier transform of the data gives a spectrum which, with the use of eq 1,has been converted to the mass spectrum in Figure 2a. Despite the fact that the amplitudes of the three transient FT-MS signals are equal, the mass spectrum in Figure 2a shows peak heights that are not the same. The reason for this is made apparent by an expanded portion of the same spectrum, Figure 2b. The peak height and the centroid of the m / z 158 peak cannot be determined from this spectrum because there is only one data point across the top half of the peak. In this region of the mass spectrum data points are spaced by 0.017 amu, which corresponds to 110 ppm. High mass resolution is an important feature of FT-MS, but the example above illustrates the commonly encountered difficulty of not having enough data points across a peak to calculate accurately its height and centroid. Two methods are currently used to deal with this problem. Comisarow and co-workers have shown that three data points across a peak can be fit to a theoretical line shape, from which the centroid and peak height can be calculated (8,9). This method works well if the peaks in the FT-MS spectrum are widely spaced, but it breaks down if two peaks are close together. In addition, the accuracy of the method is decreased if there is noise on the signal or if the actual line shape is different from the assumed theoretical line shape. A second approach for improving mass measurement accuracy is to use a method called zero-filling to reduce the spacing between the data points ( 1 0 , l l ) . Zero-filling artifically extends the length of the transient signal by adding a string of zeros at the end of the data. For example, zero-filling four times on the data in Figure 1would extend the length of the transient by a factor or 24 from 93.62 ms to 1.498 s, and this would be sufficient for part-per-million mass measurement accuracy. The problem with this approach, however, is that the size of the time domain data would be 1048576 data points (1 megaword), and calculation of the fast Fourier transform (FFT) would take approximately 11 h on a typical minicomputer. In addition, such a large calculation would probably have to be done in double precision to reduce the effect of rounding errors. ZOOM TRANSFORM The zoom transform is a numerical analysis procedure that provides more data points across a peak, but is far faster and easier to calculate than the zero-filling method described above. To implement the zoom transform, broad-band FT-MS data, as in Figure 1, is collected and a 64K FFT is calculated to get the mass spectrum shown in Figure 2a. A small section
0003-2700/83/0355-2094$01.50/00 1983 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 55, NO. 13, NOVEMBER 1983 * 20!85
of the peak at a spacing of 0.0011 amu or 6.9 ppm. Better mass measurement accuracy can be obtained by fitting the data points across the top of the peak to a parabolla. A parabola is used because it is very rapid to calculate and because the ideal FT-MS line shape, a magnitude Lorentzian, is approximately parabolic at the top. A parabolic fitting function can be written as
NUMBER OF DATA POINTS
BK 16K ;?4K 32K 4 0 K U 1 W
0
48K
56K
64K
I = Am2 + Bm -k C
T W V !I'
-I 0-
0
IO
?r
'*
20
50
40 50 60 T I M E (rnsec)
70
SO
100
90
Figure 1. FT-MS transient signal generated by a computer for three signals corresponding to m l z 78.04640, m l z 157.9549, and m f z 233.8688 at a magnetic field strength of 1 T.
!
I
I
50
i
u x
100
I5O m/z
250
r - 1578 7 -1580m/z m 1582 157.6
.... 1584
+ Bm, + C
(4)
330
-
0 0-
and the peak height is Applying eq 2 to the data in Figure 3 gave a mass measurement error of only 9.8 X lo4 amu (0.062 ppm), which is 100 times smaller than the spacing between data points. The zoom transform can also be applied to other peaks in the FT-MS mass spectrum. Use of the same procedure on the first peak in Figure 2a gave an error of only 1.2 X lo4 m u (0.015 ppm), and the amplitudes of the two peaks agreed to within 0.047%, instead of the 30% error which is apparent in Figure 2a. It is clear, therefore, that the zoom transform followed by the parabolic fitting routine very accurately extracts the true resonant frequencies and peak heights, even from relatively low resolution broad-band FT-MS data.
o,6-i
".?
where I is the signal height a t position m on the mass scale. Only three data points are needed to solve for the parameters A, B , and C, but we prefer to use a linear least-squares fit to at least ten points across the peak to minimize the effects of random noise. In addition, calculational rounding errors are minimized by using relative signal heigha and relative mas8es in eq 2. Manipulation of eq 2 shows that the centroid of the peak is given by m, = -B/2A (3)
I,, = Am:
I
200
(2)
158.6
Figure 2. (a) FT-MS mass spectrum computed from a 64K complex FFT on the transient data generated in Flgure 1 followed by use of eq 1 to convert from the frequency domain to a mass spectrum; (b) A f 0.5 amu section of the mass spectrum in Figure 2a plotted as
discrete points. of the spectrum, for example a 1 amu window around m / z 158, is then selected by the computer for an extended numerical analysis. The first step is a complex inverse FFT on the selected section of data to give a 51%-pointtime domain transient which consists of all cyclotron frequencies in the window. Next, zero-filling four times yields an 8K data set that is subjected to a forward FIT. The #pacing between data points in the resulting mass spectrum is the same as that obtained by zero-filling four times on the data in Figure 1, but the advantage of the zoom transform is that it can be calculated very quickly. An example of the use of the zoom transform is shown in Fibwre 3. Instead off only one data point, as in Figure 2b, there are now 25 points acrms the upper half
DISCUSSION The zoom transform can be applied to selected sections of the mass spectrum whenever high mass measurement accuracy and decreased spacing between the data points are needed. One problem that results is the appearance of oscillations, or lobes, on the sides of the peak. This is not an artifact caused by the zoom transform but results instead from the rectangular windowing function superimposed on the signal in Figure 1. The FFT of a rectangular window is the (sin x ) / x function which is approximately the shape of the peak in Figure 3. The lobes are not observed in Figure 2 because the spacing between lobes is exactly the same as the spacing between the data points. One method for removing the lobes is to use a different windowing function on the time domain data. In our laboratory the transient signal is multiplied by a 50% cosine taper of the form G(t) = 1, for 0 € t € T/2 G(t)
[l -- cos (n - 2nt/T)]/2, for T/2 € t
