420
Anal. Chem. 1909, 61 428-431
(25) Burkert, W. G.; Owensby, C. N.; Hinze, W. L. J. Liq. Chrometogr. 1981, 4 , 1065. (26) Debowski, J.: Grasslni-Strazza, G.; Dybilska, D. J. Chrometogr. 1985, 349, 131. (27) Pagington, J. S. Chem. Brit. 1987, 2 3 , 457. (28) Pharr, D. Y.; Fu, 2 . S.; Smith, T. K.; Hinze, W. L. Anal. Chem., in press. (29) Armstrong, D. W.; Bui, K. H. J . Liq. Chrometogr. 1982, 5 , 1043 (30) Faupei, M.; Von Arx, E. J. Chmmetogr. 1981, 211, 262. (31) Armstrong, D. W.; Stlne, G. Y. J. Am. Chem. Soc. 1983. 105, 2962. (32) Armstrong, D. W.; Nome, F.; Splno, L. A.; Goiden, T. 0. J. Am. Chem. SOC.1986, 108, 1418. (33) Skoog, D. A. Principles of Instrumental Analysis, 3rd ed.; Saunders College Publishing: New York, 1985; pp 839. 840. (34) Barone. G.; Castronuovo, G.; Eiia, V.; Muscetta, M. J. Solutlon Chem. 1986, 15, 129. (35) Abrams, W. R.; Kallen, R. 0.J . Am. Chem. SOC. 1978, 98, 7789. (36) Woolley, E. M.; Hepler. L. G. Anal. Chem. 1972, 44, 1520. (37) Lee, F. M.; Lahtl, L. E. J . Chem. €4. Data 1972, 17, 304. (38) Vogel, L.; Figurskl, G.; Vohland, P. Wiss. Z.-Martin-Luther-Uni. Halle-Wmenberg, Meth.-Naturwiss. Reihe 1985, 3 4 , 147; Chem. Abstr. 1965, 102, 2101432. (39) Shlmizu, H.; Kalto, A,; Hatano, M. Bull. Chem. SOC.Jpn. 1979, 5 2 , 2678. (40) Lewis, E. A.; Hansen, L. D. J. Chem. SOC., Perkin Trans. 2 1973, 2061. (41) Buvari, A.; Barcza, L. J. Chem. Soc., Perkin Trans. 2 lS88, 543. (42) Harata, K. Bull. Chem. SOC.Jpn. 1979, 5 2 , 1807.
(43) Hamilton, J. A.; Sabesan, M. N.; Steinrauf, L. K. Carbohydr. Res. 1981, 8 9 , 33. (44) Takemoto, K.; Sonoda, N. I n Inclusion Compounds; Atwood, J. L., Davies, J. E. D., MacNlcol, D. D., Eds.; Academic Press: New York, 1984; Voi. 2, Chapter 2, p 47. (45) Spiridonova, M. E.; Gol'tsev, V. 0. Zh. Obshch. Khim. 1976, 4 6 , 2186. (46) Chebib, H.; Jambon, C.; Merlin, J. C. J. Chim. Phys. Phys.-Chim. Biol. 1983, 8 0 , 281. (47) Rymden, R.; Carifors, J.; Stllbs, P. J. Incl. Phenom. 1983, 1, 159. (48) Patonay, G.; Fowler, K.; Shapka, A,; Nelson, G.; Warner, I. M. J. Inci. Phenom. 1907, 5 , 717. (49) Armstrong. D. W.; He, F. Y.; Han, S. M., submined for publication in J. Chromatogr .
RECEIVED for review July 19, 1988. Accepted November 16, 1988. Support of this work by a Susan Greenwall Foundation, Inc., Grant of the Research Corporation is gratefully acknowledged. Z.S.F. acknowledges support in the form of a fellowship from the American Chinese Education Foundation of Indiana. This work was presented in part at the 78th Annual Meeting of the North Carolina Academy of Science, Charlotte, NC, April 4, 1981 (Abstr. No. 30 in J. Elisha Mitchell Sci. SOC.1981, 97, 255).
Hartley Transform Ion Cyclotron Resonance Mass Spectrometry Christopher P. Williams and Alan G. Marshall*
Department of Biochemistry and Department of Chemistry, T h e Ohio State University, Columbus, Ohio 43210
The HafiJey transform offers a useful akematlve to the Fourier transform for the conversion of a time-domain Ion cyclotron resonance (ICR)slgnal into Its corresponding frequencydomaln m a s spectrum. The Hartley transform has the advantage that It efimlnates the need for complex varlables, when (as for linearly polarized signals) the tbne-domaln slgnal can be represented by a mathematlcally real function. Moreover, the Hartley transform produces the same spectra (absorption mode, disperskn mode, magnitude mode) as does the Fowler transform. I n particular, the dlscrete fast Hartley transform (FHT) produces the same spectrum at twice the speed of a complex fast Fourier transform (FFT), making the FHT equivalent In speed l o a "real" FFT. Hartley and Fourier transform methods in ICR mass spectrometry are compared and demonstrated experlmentally. Essentlally the same advantages and compulatlonal methods shoutd apply to the use of the Hartley transform In place of the Fourier transform In other forms of spectrometry (e.g., nuclear magnetlc resonance, Infrared, etc.).
INTRODUCTION It is taken for granted by most Fourier transform spectroscopists that the Fourier transform [in particular, its fast discrete fast Fourier transform (FFT) algorithm ( I ) ] is the preferred method for obtaining a frequency-domainspectrum (absorption mode, dispersion mode, magnitude mode, or power) from a discrete time-domain data set (2). Unfortunately, the standard (mathematically)complex FFT algorithm is inefficient for treating linearly polarized time-domain signals
[as in ion cyclotron resonance (ICR) mass spectrometry or infrared interferometry] that are represented by mathematically real data. Although the mathematically imaginary half of the data is zero, the complex FFT algorithm nevertheless requires the same number of computational steps as if the time-domain data were mathematically complex. It is possible to exploit various recursion relations between elements of the discrete complex Fourier code matrix to speed the complex FFT computation by about a fador of 2. Briefly, the method consists of "unshuffling" the original data into two halves, consisting of the even- and odd-numbered original data points, and then treating those two halves as the real and imaginary parts of a complex data set for conventional complex FFT processing. F T symmetry and recursion relations can then be used to reconstitute the desired spectrum (3). However, the fundamental algebra and algorithm still employ mathematically complex variables. However, if the (time-domain) data set is inherently real, it is conceptually simpler to choose a time-to-frequency transform which avoids mathematically complex notation altogether. The Hartley transform, first published in 1942 ( 4 ) ,is just such a method. Unfortunately, the Hartley technique lay virtually dormant until Bracewell worked out a fast algorithm [fast Hartley transform (FHT)] for its discrete representation (5). Because the Hartley transform is inherently designed for mathematically real functions, it holds an inherent speed advantage of a factor of 2 over a complex FFT of the same number of (real) time-domain data points. Although the Hartley transform holds no speed advantage over a "real" FFT algorithm, the Hartley transform is nevertheless conceptually simpler because it does not require the use of complex variables.
0003-2700/89/0361-0428$01.50/00 1989 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 61, NO. 5. MARCH 1, 1989
In this paper, we introduce the Hartley transform to ICR mass spectrometry and use its fast (FHT) algorithm to generate the usual absorption-mode, dispersion-mode, and magnitude-mode spectra familiar to Fourier transform spectroscopists. F H T spectra are generated from both simulated and experimental ICR time-domain discrete transients, and the predicted computational speed is confirmed.
THEORY In FT-ICR mass spectrometry, a linearly polarized (i.e., mathematically “real”) time-domain signal, f (t ) ,is converted to its corresponding frequency-domain spectrum, F(v) (which is rescaled to give a mass spectrum) by use of a complex Fourier transform. The complex Fourier transform and its inverse differ by a sign in the complex exponent
I + - f ( t ) [ c o s (2nvt)
-i sin (2avt)l dt (1)
whose angle = phase) onto two orthogonal (cosine and sine) axes, then the Hartley transform is the projection of that same vector onto a single axis bisecting the sine and cosine axes cas (0) = cos
(e) + sin (e) = 2’/2 cos [e - (7r/4)] = 2’12 sin [e + (7r/4)]
+ i sin (27rut)l dv
(2)
In the usual complex Fourier transform of eq 1, the mathematically complex notation serves to separate the cosine and sine components of the transform. However, cosine and sine functions are inherently orthogonal. Therefore, there is no need to use complex notation in order to separate them. Thus, the same spectrum can be obtained without resorting to mathematically complex notation, by use of the “Hartley” transform, to give the Hartley frequency-domain spectrum,
H(v)
H(v) = J I m f ( t ) [ c o s (27rvt) + sin (2nut)l dt =
JS(t)
cas(27rvt) dt (3)
in which the “cas” function denotes a sum of cosine and sine functions of the same argument. For a real time-domain function, f ( t ) , the Hartley spectrum (see Figure 1)can thus be thought of as the sum of the usual absorption and dispersion spectra, and contains the same information as the conventional complex FT spectrum. For example, the real and (negative of the) imaginary components of the FT spectrum may be readily recovered from the even and odd components, E(v) and O(u), of the Hartley spectrum
E(u) =
O(v) =
H(v) + H(-v)
H(v) =
r +-
J - m
f ( t ) cas (27rvt) dt =
2 1 / 2 1 + m f ( t ) ( ~ o(27rvt s - (7r/4))] dt (7)
It is also interesting to note that, unlike the Fourier transform, the Hartley transform and its inverse are identical in form
H(v) - H(-v) 2
=
(8)
For mathematically real data, a complex FFT takes twice as long as a (real) FHT, when both are similarly optimized (e.g., precalculation of trigonometric identities). However, a “real” FFT algorithm (see above) of an N-point time-domain real data set effectively relies on a complex FFT of N/2-point subsets of the original data set, and thereby also reduces computational time by a factor of about 2 compared to a complex FFT of the N time-domain real data points.
EXPERIMENTAL SECTION Experimental time-domain ICR signals (lK-8K real data points) of electron-ionized (70 eV) air (-lo* Torr) were acquired by use of a Nicolet FTMS-2000 spectrometer and subsequently transformed on its Nicolet 1280 computer by Fortran integer implementationsof the fast Fourier transform (FFT) and the fast Hartley transform (FHT). Speed comparisons between an optimized radix-2 fast Hartley transform (7)and an optimized radix-2 “Real” fast Fourier transform (8)were performed on an Atmi 1040ST (8-MHz M68000 cpu, l-megabyte RAM; equivalent in speed and memory to a Macintosh SE) using compiled GFA Basic. Both transforms processed computer-generated 6-byte (11digits precision) real data representing exponentially decaying sinusoids of the type acquired in ICR mass spectrometry. The number of data points (NDP)ranged from 1K to 16K.Because the FHT Basic algorithm listed in ref 7 contains several minor (but important) transcriptional errors, a corrected version is listed in Chart I.
f ( t ) sin (27rut) dt (4b)
f ( t ) = cos ( 2 ~ v ~ t ) e - (=~ cos / ~ ) ( ~ ~ t ) e - ( 0~ -< / ~t) < ,
[ -Ob) x] [ = arctan
-m
f ( t ) cos (27rvt) dt (4a)
Jm +Jm
Similarly, the magnitude and phase spectra, M(v) and +(v), may be computed as easily from the H T spectrum as from the FT spectrum
@(v) = arctan
S + - H ( v ) cas (27rvt) dv
RESULTS AND DISCUSSION Spectral Display Modes. The various spectral line shapes resulting from Hartley and Fourier transformation of a time-domain sinusoid of infinite duration
+m
=
2
(6)
Thus, if the “cas” function of the Hartley transform is expressed in terms of a phase-shifted cosine (eq 6), then the Hartley transform of eq 3 is recognized as a time-shifted Fourier transform (6)
f(t) =
~ + m F ( v ) [ c o(27rvt) s
429
H(-v) - H ( v ) H(v) + H(-v)
]
(5b)
Further insight into the nature of the Hartley transform follows from its connection to the Fourier transform. If the Fourier transform is considered to give the projection of a frequency-domain vector (whose length = magnitude and
(9)
are listed in Table I. Fourier transformation of a time-domain signal, followed by appropriate phase correction, yields the familiar absorption-mode and dispersion-mode Lorentzian spectra shown in Figure 1. The frequency-domain spectrum, H(v), resulting directly from the Hartley transform of eq 3 consists of the sum of the absorption and dispersion spectra. Because of its dispersion component, the Hartley raw spectrum is wider by a factor of about 2.4 than the corresponding absorption-mode spectrum, for a Lorentzian line shape (see Table I). Thus, one would not normally display the raw Hartley spectrum, whether phase-corrected or not. Fortunately, the desired absorption-mode spectrum can be recovered from the even and odd components of the Hartley spectrum, as shown in eq 4. For FT-ICR spectrometry (9), in which spectra are normally reported in magnitude mode
ANALYTICAL CHEMISTRY, VOL. 61, NO. 5, MARCH 1, 1989
431
Table I. Comparison of the Various Frequency-DomainLine Shapes Obtained by Fourier or Hartley Transformation of an Exponentially Damped Infinite-Duration Time-Domain Sinusoidal Signal (Eq 9)o
frequencydomain spectrum
transform
FFT absorption mode absorption(w) = T / [l + (wg - o ) ~ T ~ ] (cosine transform) FFT dispersion mode dispersion(w) = (wo - a)?/ [ l + (a0- w ) ~ T ~ ] (sine transform) Hartley transform H ( w ) = [ T + (wo - w ) 7 2 ] / (“cas”transform) [l + (wo - w)2?] magnitude(w) = T / magnitude mode [ l + (ao- w)2?]1/2 (Fourier or Hartley) OPeak width denotes the full width at half-maximum peak height, except for dispersion-mode,for which the reported width is the peakto-peak separation between the maximum and minimum amplitudes. Table 11. Computational Times (in Seconds) for Corresponding Transforms of Three Arbitrarily Different FID Time-Domain Signals (A, B, and C) of Data Sizes of up to 16K0
no. of time-domain points 128 256 512 1024 2048 4096 8192 16384
computational time, s “real” FFT FHT A B C A B 0.37 0.79 1.70 3.61 7.70 16.38 34.75 73.51
0.38 0.78 1.66 3.55 7.59 16.16 34.32 72.66
0.37 0.78 1.66 3.54 7.58 16.13 34.27 72.59
0.35 0.75 1.60 3.46 7.45 15.98 34.23 72.89
0.35 0.74 1.60 3.47 7.49 16.11 34.54 73.62
r II
I&+
Mass in A.M.U.
Flgure 2. Magnitude-mode FT-ICR mass spectrum of an air sample, obtained by Fourier transformation of an 8K timedomain real data set.
C 0.35 0.74 1.61 3.46 7.47 16.08 34.49 73.58
a Both the “real” FFT and the FHT algorithms were radix-2 implementations that were optimized with respect to trigonometric calculations. The programs were written to handle &byte real data using compiled GFA Basic and were run on an 8-MHz M68000 Atari ST with 1 megabyte RAM. Note that the computational times for “real” FFT and FHT are essentiallv the same.
Massfn A.M.U.
Figwe 3. Magnitudemode FT-ICR mass spectrum obtained by Hartley transformation of the same timedomain data used to generate the Fourier transform spectrum in Figure 2.
Time Domain Sipnal (FID)
the use of mathematically complex quantities to describe physically “real” quantities.
ACKNOWLEDGMENT The authors thank R. N. Bracewell for helpful discussions. LITERATURE CITED Hartley Spectrum
(1) Cooley, J. W.; Tukey, J. W. Math. Comput. 1965, 10, 297-301.
Fwre 1. Transformations of an exponentially damped slnusddal signal of the type acquired In FT-ICR mass spectrometry.
(2) Fourlet‘, Hadamard, andrmaert Transforms In Chemistry; Marshall, A. G., Ed.; Plenum: New ‘fork, 1982: 562 pp. (3) Cooley, J. W.; Lewis, P. A. W.; Welch, P. D. J . Sound Vib. 1970, 12, 3 15-3 17. (4) Hartley. R. V. L. R o c . Inst. Radio Eng. 1942, 30, 144-150. (5) Bracewell, R. N. J . Opt. Soc. Am. 1989, 7 3 , 1832-1835. (6) Champeney, D. C. Fowier Transfonnatbns and Lhek Physicel Applice Uons; Academic Press: New ‘fork. 1973; p 2. (7) Bracewell, R. N. The Hartley Transform; Oxford University Press: New ‘fork, 1986. ( 8 ) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterllng, W. T. Numerlcal Reclpes : Cambridge University Press: Cambridge, 1986. (9) Marshall, A. 0. Acc. Chem. Res. 1985, 18, 316-322. (IO) Craig, E. C.; Santos, I.; Marshall, A. G. RapM Commun. Mass Spectrom. 1987, 1 , 33-37. ( 1 1 ) Buneman, 0.S I A M J . Scl. Stet. Comput. 1988, 7. 624-638.
data set (corresponding to a linearly polarized time-domain signal). In particular, the Hartley approach takes advantage of the natural orthogonality of cosine and sine functions and thereby avoids the confusion arising in Fourier analysis from
RECEIVED for review August 16, 1988. Accepted November 14,1988. This work was supported by grants (to A.G.M.) from the U.S.A. Public Health Service (N.I.H. GM-31683) and The Ohio State University.
Fourler Spectrum (Imaginary)
4
