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Application of Hartley and Hilbert transforms in Fourier transform ion cyclotron resonance mass spectrometry
Williams, Christopher Paul, Ph.D.
The Ohio State University, 1992
UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106
Application of Hartley and Hilbert Transforms in Fourier Transform Ion Cyclotron Resonance Mass Spectrometry
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Christopher Paul Williams, B.S., M.S.
* * * * *
The Ohio State University 1992
Dissertation Committee: Approved by Professor A. G. M arshall Professor E. J. Behrman Q h t . f k J M Professor R. T. Ross Adviser Professor G. S. Serif Department of Biochemistry DEDICATION
I dedicate this work to my wonderful wife Holly, whose encouragement and support has kept me going throughout my graduate work. Not many women would have shown the same patience and good humor as she did when "a couple of years to finish my degree before we have kids" turned into five years and three children.
ii ACKNOWLEDGMENTS
I must first express my thanks to Professor Alan Marshall for his guidance, insight, and assistance. I also wish to thank Professor E. J. Behrman for his help earlier in my graduate studies, and for his encouraging me to temporarily leave the business world and come back to finish my doctoral work. I thank all the members of Dr. Marshall's research group for their friendship and assistance over the years. In particular, Dr. Francis Verdun helped me considerably in understanding many of the finer points of the Fourier transform. I also thank Kaye Craggs, Dr. Tom Ricca, and Dr. Charles Cottrell for their help when I needed it. I also w ant to th an k my kids, Bethany (4) and M atthew (2), who have made my stint as a graduate student so much more enjoyable. Without their constant attention and affection my graduate studies could not have lasted nearly so long. I also thank my Mom and Dad for their support throughout my educational career. Finally, I must remember the one who truly deserves the credit for anything that I may may accomplish in life. "Great is the Lord and most worthy of praise; his greatness no one can fathom." "Not to us, O Lord, but to your name be the glory". Psalms 145:3; 115:1.
iii VITA
October 5, 1958 Bom - Akron, Ohio
1980 B.S., Summa cum laude. Biochemistry, The Ohio State University
1980-1981 National Science Foundation Fellow, The Ohio State University
1982 M.S., Biochemistry, The Ohio State University
1982-1984 Assistant Editor, Chemical Abstracts Service, Columbus, Ohio
1984-1986 Computer Programmer-Analyst, Chemical Abstracts Service, Columbus, Ohio
1986-1987 Ohio State University Fellow, The Ohio State University
1987-1990 Graduate Research Associate, The Ohio State University
PUBLICATIONS
Williams, ( '. P.; Marshall, A. G. "Hartley Transform Ion Cyclotron Resonance Mass Spectrometry", Analytical Chemistry, 1989, 61, 428- 431.
FIELDS OF STUDY
Major Field: Biochemistry TABLE OF CONTENTS
DEDICATION...... 11
ACKNOWLEDGMENTS...... Hi
VITA ...... iv
LIST OF TABLES...... 7 ...... viii
LIST OF FIGURES ...... x
CHAPTER PAGE
I. INTRODUCTION...... 1
II. PRINCIPLES AND FEATURES OF FOURIER TRANSFORM ION CYCLOTRON RESONANCE MASS SPECTROMETRY ...... 3 Magnetic Sector Mass Spectrometry ...... 3 Ion Cyclotron Resonance Spectroscopy ...... 4 Basics of Fourier Transform Ion Cyclotron Resonance Mass Spectrometry ...... 9 Ionization Methods ...... 17 Excitation Techniques ...... 25 Cell Design and Ion Trapping ...... 28 Data Manipulation ...... 34 Current Usage...... 36 R eferences ...... 39
III. BIOCHEMICAL APPLICATIONS OF FOURIER TRANSFORM ION CYCLOTRON RESONANCE MASS SPECTROMETRY...... 48 Special Concerns ...... 48 Carbohydrates ...... 49 P roteins ...... 53 Matrix-Assisted Laser Desorption of Proteins ...... 55 Electrospray of Proteins ...... 57 v Other Biochemical Applications ...... 60 References ...... 63
IV. APPLICATION OF THE HARTLEY TRANSFORM TO FT/ICR DATA...... 69 Introduction ...... 69 The Fourier Transform ...... 69 Phase Shift and Phase Correction ...... 76 Discrete Nature of Experimental D ata ...... 79 The Hartley Transform ...... 92 Experimental ...... 99 Results and Discussion ...... 100 Computer Implementation ...... 100 Spectral Display Modes ...... 104 Computational Speed Comparisons ...... 104 Precision of the FHT ...... 108 C onclusion ...... 114 R eferences ...... 115 Appendix ...... 117
V. A FASTER ALGORITHM FOR THE HILBERT TRANSFORM... 145 Mathematical basis of the Hilbert Transform ...... 145 A New Hilbert Algorithm ...... 146 Experimental ...... 148 Results and Discussion ...... 150 Computation of the Hilbert Transform ...... 150 Precision of the Hilbert Transform Algorithm ...... 150 C onclusion ...... 153 R eferences ...... 157 Appendix ...... 159
VI. THE HILBERT-TRANSFORMED DISPERSION SPECTRUM AND THE "ENHANCED" ABSORPTION SPECTRUM ...... 161 Introduction ...... 161 Theory ...... 163 Experimental ...... 165 Results and Discussion ...... 167 Standard deviation and root-mean-square noise in absorption and magnitude spectra ...... 167 Figures of Merit...... 168 Zero-Filling...... 171 vi Double-Forward FT ...... 173 Improved Identification of Weak Signals ...... 176 Improved Discrimination against Spurious Noise S p ik e s...... 179 Improved Resolving Power ...... 179 Correlated Noise ...... 182 Discrete Data Limitations ...... 184 Truncation of Peaks near Either Edge of the S p e c tru m ...... 185 C o n clu sio n ...... 189 R eferences ...... 190 Appendix ...... 191
BIBLIOGRAPHY...... 193
vii LIST OF TABLES TABLE PAGE
4-1 Comparison of the various frequency-domain line shapes obtained by Fourier or Hartley transformation of an exponentially damped infmite-duration time-domain sinusoidal signal (Eq. 4-1) ...... 105
4-2 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 16K...... 107
4-3 Average Absolute-Value Fractional Differences Between Corresponding Data Points after Two or Four Successive FHT's...... 113
5-1 Algorithms for the (forward) Hilbert and Inverse Hilbert transforms ...... 149
5-2 Average Absolute-Value Fractional Differences Between Corresponding Data Points after Application of the Hilbert and Inverse Hilbert Transforms ...... 155
6-1 Comparison of theoretical relative noise, signal-to-noise ratio (SNR), and resolving power for three FT spectral modes, in the Lorentz limit that the time-domain signal decays nearly to zero during the time-domain acquisition period. Noise is reported both as standard deviation from zero (S.D.) and as root-mean-square deviation (R.M.S.), and signal-to-noise ratio is reported with respect to each of the two measures of noise. See text for interpretation ...... 169
6-2 Ratio of relative signal-to-noise ratio (SNR) and resolving power for "enhanced" absorption mode relative to either (unenhanced) absorption mode or magnitude mode spectra, for noise evaluated either as standard deviation from the mean (S.D.) or root-mean- square deviation from zero (R.M.S.). The standard deviation of each computed ratio is listed, along with the theoretically viii predicted values (in parentheses). The data were obtained from the average of 11,870 independently simulated noisy 8K time- domain exponentially damped sinusoids, acquired for 10 time- domain exponential damping periods. Gaussian-distributed random noise with zero mean and absolute mean value of 5% of the maximum time-domain amplitude was added to the time- domain signal ...... 170
6-3 Comparison of noise, signal-to-noise ratio (SNR), and resolving power for experimental FT/ICR mass spectral data from N 2 + ions (~7 points per peak full width at half-maximum peak height). Noise is reported both as standard deviation from the mean (S.D.) and as root-mean-square deviation from zero (R.M.S.), and signal- to-noise ratio is reported with respect to each of the two measures of noise. See text for interpretation ...... 172
ix LIST OF FIGURES FIGURE PAGE
2-1 Illustration of the relationship between the velocity vector v of a moving ion in a magnetic field and the resultant Lorentz force vector qvB. (a) Positive ion. (b) Negative ion. Note th a t the magnetic force vector B is directed perpendicularly into the plane of the paper...... 6
2-2 Relationship between the mass-to-charge ratio m/z and the cyclotron frequency v of an ion at three commonly used magnetic field strengths in the absence of electric fields ...... 8
2-3 Mechanism of FT/ICR signal detection, (a) Incoherent ion packet which generates no detectable signal on the detector plates, (b) Excitation (and absorption of resonant rf energy) causes the ion packet spirals outward to a larger orbital radius, (c) At this larger radius, the ion's coherent motion induces a detectable sinusoidal image current on the detector plates, (d) Excessive excitation power excites the ion packet to a radius larger than the cell radius, causing ions to be lost upon colliding with the plates 11
2-4 Conversion of a frequency spectrum at 3 Tesla to a mass spectrum, (a) A frequency spectrum containing five equally shaped peaks, (b) The same data as in the previous figure, but re-scaled according to mass using the approximate relationship (for 3 Tesla) MASS(AMU) = 46,000,000/FREQUENCY(KHz). Note that the highest mass peaks (A and B) cover a much wider mass range (are less well resolved) than the lowest mass peaks (C, D and E) ...... 13
2-5 Generalized FT/ICR experimental event sequence (left), with the equivalent conventional chemical manipulations (right) ...... 15
2-6 Two methods of directing a laser beam onto a sample probe adjacent to the FT/ICR trapping cell, (a) Aluminum mirrors (M) direct the beam into the chamber, and a parabolic mirror (P) focuses the incident beam onto the probe, (b) A fiber optic cable directs the laser beam directly onto the probe. In both methods the desorbed neutrals and ions simply drift into the adjacent trapping cell. The trapping potential is temporarily dropped to zero for a short time after the desorption event to allow the ions to enter. The probes can be rotated to permit multiple laser shots at fresh sample areas...... 20
2-7 Simplified sketch of the electrospray process. Droplets containing the sample acquire an electric charge as they emerge from a high- voltage needle. They then pass through a drying gas and through one or more pumping stages while being focussed by electrostatic lenses into a quadrupole device ...... 23
2-8 Three excitation techniques that have been used in FT/ICR. (a) Single-frequency, (b) Frequency sweep or "chirp", (c) Stored waveform inverse Fourier transform or "SWIFT" used to excite a single frequency windows, (d) "SWIFT" used to excite two narrow frequency windows. (e) "SWIFT" used to excite two narrow frequency windows and to eject or over-excite ions in a third (wider) frequency window ...... 27
2-9 Upper mass limits for singly charged thermal ions in a 1" trapping cell at various typical strength homogeneous magnetic fields as a function of the trapping voltage VT. Calculations assume a quadrupolar electrostatic potential from the trapping plates. Upper arrows at VT = 1 V indicate the highest m /z ion which has a stable 1" orbit; lower arrows indicate the highest m/z for which 99% of the thermal ions (25 °C) have radii of 1" or less ...... 30
2-10 Various geometries of FT/ICR ion traps (cells). The cells all basically consist of three pairs of opposing plates: Trapping plates (T), Excitation plates (E), and Detection plates (D). Holes in the center of the trapping plates allow passage of an electric beam for electron ionization within the cell, as well as entrance of externally generated ions (such as from laser desorption), (a) Cubic, (b) Cylindrical, (c) Hyperbolic, (d) Multi-section, (e) Screened [grounded screens are denoted by (S)] ...... 32
2-11 Number of FT/ICR systems in use worldwide ...... 38
3-1 Some of the more common fragmentation pathways observed for underivatized oligosaccharides in positive and/or negative ion detection mode. Fragments A, C, and E retain the charge on the xi non-reducing end. Fragments B and D retain the charge on the reducing end. The stereochemistry of hydroxyl groups of specific sugar monomers can lead to additional structurally informative fragmentation ...... 50
3-2 FT/ICR mass spectrum of the tetrasaccharide stachyose using Nd/YAG laser desorption and negative ion mode detection. The molecular ion (665) and major fragment peaks are indicated. Traces of contaminating iodine caused the main peak at 127 ...... 52
3-3 High resolution FT/ICR mass spectrum of electrospray-ionized Cytochrome c. Figure provided by F. W. McLafferty ...... 59
4-1 The Fourier transform converts a time domain waveform (top) consisting of the sum of three cosines (middle) into a frequency domain spectrum (bottom) containing the correct frequency and magnitude information for each of the three components ...... 71
4-2 Origin of the Lorentzian "real" component and the dispersion "imaginary" component of Fourier-transformed exponentially damped causal (where f(t)=0 when t<0) cosine signals. Top row shows that the FT of the symmetrical exponentially damped time domain signal leads to just a Lorentzian real component. Middle row shows that the FT of an antisymmetrical exponential signal gives just a dispersion imaginary component. Because FT integrals are additive, addition of half of the top row plus half the middle row results in the bottom row. The FT of this causal signal has both real Lorentzian and imaginary dispersion com ponents ...... 73
4-3 Lineshapes associated with Fourier-transformed damped cosines. A cosine Fourier transform of a damped cosine gives an absorption lineshape. A sine Fourier transform of the same damped cosine gives a dispersion lineshape. The magnitude calculation combines the absorption and dispersion spectra into a phase-independent magnitude lineshape ...... 75
4-4 Effect of a time domain phase shift on the corresponding frequency domain real, imaginary, and magnitude spectra. The top row is for an unshifted cosine (phase-0o), which has a pure absorption shape for the real component and pure dispersion for the imaginary. As the phase is gradually shifted by 30° (second row) and 60° (third row), the real and imaginary components
xii exhibit mixed character. Finally, a sine (90° phase shift, bottom) results in a pure dispersion shape in the real component and a pure (negative) absorption shape in the imaginary component. Regardless of the phase shift, the magnitude spectrum is unchanged ...... 77
4-5 Effect of the discrete digital representation of extremely narrow absorption and dispersion lineshapes in the case where the true centroid of the absorption peak is located at various distances between data points. Spectra were generated from 64-point time domain data sets consisting of a single cosine at various frequencies from 15.5 to 16. The data was not exponentially damped; that is, T/x = 0. A real FT was then done on each of these time domain data sets to generate the 64-point spectra in Figures A through F. The 32 data points on the left side of each spectrum are the real part (absorption) of the FT, and the 32 points on the right are the imaginary part (dispersion) of the FT. The same horizontal and vertical scaling was used for each spectrum. Especially note that with a frequency placing the absorption maximum and the center of the dispersion exactly halfway between data points, the absorption is not observed and the dispersion is maximized (Figure A). At the other extreme, where the frequency is centered on a data point, the absorption is maximized and dispersion is not observed (Figure F). Frequency values between these extremes give intermediate absorption and dispersion lineshapes (Figures B-E). Although these intermediate spectra give the appearance of being phase shifted, that is not the case...... 83
4-6 Effect of time domain signal damping on the discrete digital representation of absorption and dispersion lineshapes in the case where the centroid of the absorption peak is exactly at a data point. Spectra were generated from a 64-point initial time domain data set consisting of a single cosine of frequency 16, exactly equal to Nyquist/2. The data was then exponentially damped with damping constants {T/x) ranging from 0 to 15. A real FT was then done on each of these differentially damped time domain data sets to generate the 64-point spectra in Figures A through I. The 32 data points on the left side of each spectrum are the real part (absorption) of the FT, and the 32 points on the right are the imaginary part (dispersion) of the FT. The vertical scaling was adjusted so that the maximum of each spectrum is the same relative height. Especially note that with no damping (Figure A)
xiii the digital resolution is such that the dispersion signal is not observed at all (because it falls completely between two points). Increased damping (Figures B through I) broadens both absorption and dispersion peaks so that they are better represented digitally ...... 85
4-7 Effect of time domain signal damping on the discrete digital representation of absorption and dispersion lineshapes in the case where the centroid of the absorption peak is exactly halfway between two data points. Spectra were generated from a 64-point initial time domain data set consisting of a single cosine of frequency 15.5 (half a point from Nyquist/2). The data was then exponentially damped with damping constants (T/x) ranging from 0 to 15. A real FT was then done on each of these differentially damped time domain data sets to generate the 64-point spectra in Figures A through I. The 32 data points on the left side of each spectrum are the real part (absorption) of the FT, and the 32 points on the right are the imaginary part (dispersion) of the FT. The vertical scaling was adjusted so that the maximum of each spectrum is the same relative height. Especially note that with no damping (Figure A) the digital resolution is such that the absorption signal is not observed at all (because it falls completely between two points). Increased damping (Figures B through I) broadens both absorption and dispersion peaks so that they are better represented digitally) ...... 88
4-8 The CAS function is simply the sum of the COS function and the SIN function ...... 94
4-9 Fourier and Hartley transformations of an exponentially damped sinusoidal signal of the type acquired in FT/ICR. Note that except for magnitude mode calculations (which lose phase information), the transformations are fully reversible. Thus the time domain signal, the Fourier real and imaginary spectra, and the Hartley spectrum contain equivalent information ...... 97
4-10 Graphs of a 512 point data set after each of the main stages (8 stages in this case, since 28 = 512) of the FHT transform. Starting with a simulated noiseless time domain data set consisting of a single frequency (A), the intermediate stages (B-I) can be seen to progress toward the final Hartley spectrum (J) ...... 101
xiv 4-11 An experimental FTMS time domain signal of N2+ ions (from air) at 3 tesla (8192 points, heterodyne mode, phase corrected by adjustment of the carrier frequency) was transformed with both the "real" fast Fourier transform and the fast Hartley transform (with subsequent conversion to the FT absorption spectral mode). Although the initial FTMS data was in integer format, all calculations were done using real numbers, with the final results rounded back to integers. The same 100 points surrounding the N2+ absorption peak are plotted for both the "real" FFT spectrum (TOP) and the FFT spectrum derived from the FHT (BOTTOM). At the absorption peak the relative difference between the two spectral values was only 4 parts in 31000 (0.013%). Almost 99% of the other corresponding (integer) data points were identical ...... 106
4-12 The same 256 point simulated time domain data set (one single damped cosine plus added noise) (top) was transformed using both the "real" fast Fourier transform (middle) (based on the complex FFT program FOUR1 in Reference 23, pages 394-395) and the fast Hartley transform with subsequent conversion to the FT spectral mode (bottom). For both the "real" FFT and the Hartley-derived FFT spectra, the 128 points on the left are the real data and the 128 on the right are the imaginary data. A point by point comparison of the two spectra showed the average (absolute value) relative difference in values to be only 0.265%. At the main absorption peak the relative difference between the two values was even less (0.000838%) ...... 110
4-13 Since the FHT is its own inverse, two (or four) successive applications of the FHT should recover the original waveform exactly. (A) Initial 1024 point time domain consisting of the sum of three damped cosines. (B) After one FHT. (C) After two successive FHT's. (D) After three successive FHT's. (E) After four successive FHT's...... 112
5-1 Stages of the (forward) Hilbert transform ...... 151
5-2 Stages of the inverse Hilbert transform ...... 152
5-3 Test of the reversibility of the Hilbert transform. Application of the Hilbert transform and followed by its inverse should recover the original data exactly. (A) Initial 512-point dispersion data. (B) After Hilbert transform is in absorption mode. (C) After a subsequent inverse Hilbert transform, is back to dispersion mode. xv (D) Another Hilbert transform converts back to absorption m ode ...... 154
6-1 Stages in the generation of an "enhanced" absorption mode spectrum (bottom) from a discrete time-domain signal (top). Following discrete Fourier transformation of the time-domain signal and phase correction to yield the conventional absorption and dispersion spectra, the dispersion data is subjected to a discrete Hilbert transform to yield a "pseudo-absorption" mode spectrum. The absorption and "pseudo-absorption" spectra are then averaged to yield the "enhanced" absorption mode sp ectru m ...... 164
6-2 Correlation of signal and noise in the absence of zero filling. A 512 point exponentially damped time domain signal containing three cosine frequencies was processed with a real FT. The dispersion part was then Hilbert-transformed to give a pseudo absorption spectrum. Top figure shows the original absorption and the pseudo absorption spectra plotted directly over top of each other. Bottom figure is a blow up of the region from points 25 to 100, with the two spectra artificially displaced in the vertical dimension to better distinguish them. The "Enhanced" absorption spectrum, which is the average of the absorption and pseudo absorption, is also shown. Note that with no zero filling the signal peaks are the same for both absorption and pseudo-absorption spectra, but that the baseline noise is uncorrelated ...... 174
6-3 Correlation of signal and noise with a single zero filling. A 512 point exponentially damped time domain signal containing three cosine frequencies and added noise was first zero-filled before it was processed with a real FT. The dispersion part was then Hilbert-transformed to give a pseudo-absorption spectrum. Top figure shows the original absorption and the pseudo absorption spectra plotted directly over top of each other. Bottom figure is a blow up of the region from points 50 to 200, with the two spectra artificially displaced in the vertical dimension to better distinguish them. Note that with one zero filling both the signal peaks and the noise are virtually the same for both absorption and pseudo absorption spectra. This demonstrates that a single zero-filling actually correlates the noise in the absorption and dispersion components of the spectra. As a result, the "enhanced" absorption spectrum would be virtually identical to these sp ectra...... 175 xv i 6-4 Flow chart for the Double Forward FT procedure employing the "enhanced" absorption instead of the ordinary absorption spectrum. Storage of either the "enhanced" absorption or the final N/2-point FID preserves all the useful signal information 177
6-5 Three different spectral representations of the same simulated time-domain sum of three exponentially damped sinusoids of relative amplitude 10:1:3, with added Gaussian-distributed random noise. Note that the "enhanced" absorption spectrum offers the best visual identification of the smallest-amplitude signal (denoted by an arrow in each spectrum) ...... 178
6-6 Three different spectral representations of the same simulated time-domain sum of three exponentially damped sinusoids of relative amplitude 10:1:3, with added Gaussian-distributed random noise. Note that the spurious noise spike (denoted by an arrow in each spectrum is least prominent in the "enhanced" absorption spectrum ...... 180
6-7 Two different overlaid spectral representations (with the same scale and baseline) of the data shown in Figure 6-6. Relative to the magnitude mode spectrum (upper trace), the "enhanced" absorption mode spectrum (lower trace) has -\/3 higher resolving power, the same noise standard deviation from the mean noise value, and just half the root-mean-square noise (see Tables 6-1 and 6-2) ...... 181
6-8 Three different spectral representations of the same experimental FT/ICR time-domain signal from N 2 + ions at 3.0 tesla. As for the theoretical (Table 6-1) and simulated (Table 6-2, Figures 6-5 - 6-7) data, the "enhanced" absorption spectrum offers higher resolution than magnitude mode and higher signal-to-noise ratio than (unenhanced) absorption mode ...... 183
6-9 Relative performance of the "enhanced" absorption mode spectrum (as measured by the relative ratio of signal to standard deviation noise) as a function of peak displacement away from the high- frequency (Nyquist frequency) end of the spectrum (measured in multiples of the peak full width at half-maximum peak height)...... 186
xvii 6-10 Relative performance of the "enhanced" absorption mode spectrum (as measured by the relative ratio of signal to standard deviation noise) as a function of peak displacement away from the zero- frequency end of the spectrum (measured in multiples of the peak full width at half-maximum peak height)...... 187
xviii C hapter I
INTRODUCTION
The relatively new analytical technique of Fourier transform ion cyclotron resonance mass spectrometry has undergone tremendous growth in its 17 years of existence. Its unique abilities have only recently been coupled with powerful ionization techniques that enable it to handle many of the largest and most complex biomolecules. Chapter II of this dissertation briefly reviews FT/ICR basics that are particularly relevant to biochemical research, while Chapter III goes on to touch upon a few of its many specific biochemical applications. Because the study of extremely large, complex molecules stretches current FT/ICR technology to its limits, every improvement to FT/ICR techniques and data interpretation is important. Chapter IV introduces the discrete Hartley transform as an alternative to the Fourier transform in processing the time domain data typically acquired in an FT/ICR experiment. Chapter V Introduces a new algorithm (using the Hartley transform) for performing the discrete Hilbert transform, which relates the absorption and dispersion mode spectra obtained by transformation of the FT/ICR time domain signal. Finally, Chapter VI presents a new "Enhanced Absorption Mode" spectra, produced by application of the Hilbert transform. This enhanced absorption spectrum, with the same
1 number of data points and the same signal peak widths, offers a true -\J2 improvement in signal-to-noise ratio and precision over the ordinary absorption spectrum. These spectral improvements come without the loss of resolution accompanying magnitude mode transformation and without doubling the number of data points of storage as is true with zero-filling. The new Hilbert algorithm and the enhanced absorption mode technique presented in this dissertation can conceivably be utilized in other biochemically important spectroscopic fields such as FT/NMR and FT/IR. C hapter II
PRINCIPLES AND FEATURES OF FOURIER TRANSFORM ION CYCLOTRON RESONANCE MASS SPECTROMETRY
Magnetic Sector Mass Spectrometry Magnetic sector mass spectrometry rapidly came to prominence at the beginning of this century with the work of J. J. Thompson [1]. His demonstration that the non-integral atomic weight of neon (20.2) likely resulted from the 10:1 relative abundances of two isotopes (weights 20 and 22) was an important milestone in modern atomic theory. F. W. Aston, initially a student of Thompson, considerably improved on the instrumentation and firmly established the existence of stable isotopes [2]. At the same time, A. J. Dempster utilized heated filaments to generate ions from salts, and developed instrumentation that has been the basis for some modern sector instruments [3]. This form of mass spectrometry, based on the differential deflection of particles with varying mass-to-charge ratios when passing through a magnetic field, was primarily used in the identification of isotopes. Ironically, after Dempster identified the isotopes of the last remaining stable atoms (Pt and Ir) in 1935 [4], Aston remarked during a talk that mass spectrometry had fulfilled its purpose, and would soon fade away [5]. This view of mass spectrometiy, which limited its utility mainly to the determination of atomic weights, was not uncommon at the time. Fortunately this prophecy proved false. In the 1940's reliable commercial instruments became available, designed primarily for the analysis of hydrocarbon components in petroleum mixtures [6], By the mid 1960's, magnetic sector mass spectrometry became a standard tool in the identification and structural analysis of more complex molecular species [7-9]. Modifications to the original types of instruments, such as adding an electrostatic field before the magnetic field and using various geometries, has produced double focussing instruments capable of spectra with moderately high mass resolution (about 40,000) [ 10]. With less fanfare, alternatives to magnetic sector mass spectrometry were developed. Time-of-flight [11] and quadrupole [12] instruments are more rugged and less expensive than sector instruments, but suffer from much poorer mass resolving power - m generally <; 1000. For most purposes these instruments offer at best unit mass resolution (Am s lu).
Ion Cyclotron Resonance Spectrometry No known physical property can be measured more accurately than frequency [ 13]. A frequency-based determination of mass, with its potentially ultra-high mass resolution, became possible with the discovery of the ion cyclotron resonance (ICR) principle in 1932 by Lawrence and Livingston [14]. They demonstrated that a charged particle moving in a circular path in a static magnetic field can absorb energy from an oscillating electric field if the frequency of the electric field and the ion's angular frequency are equal ("in resonance"). Interestingly enough, the absorption of energy only increases the ion's velocity and orbital radius; its angular frequency is unchanged. To a first approximation, the calculation of the frequency of an ion in a static magnetic field (and in the absence of an electric field) is rather simple. The Lorentz force FL exerted by a magnetic field B on an ion with electric charge q and with a velocity v is described by Equation 2-1.
Fl = qvB, in which v ± B (2-1)
This Lorentz force is actually a vector whose direction is perpendicular to both the velocity vector of the ion and the direction of the magnetic field. Therefore this force is directed inward along the radius of the ion's orbit (see Figure 2-1). Oppositely charged ions simply have their direction of motion reversed. Acting against this Lorentz force is a centrifugal force. The force, Fc, whose magnitude and direction is determined by the mass and velocity vector of the ion, is given by
m v2 Fc = — (2-2)
Combining Equations 2-1 and 2-2, we obtain Equation 2-3
m v2 —— = qvB in which v l B (2-3)
Rearrangement of this equation yields b)
Figure 2-1 Illustration of the relationship between the velocity vector v of a moving ion in a magnetic field and the resultant Lorentz force vector qvB. (a) Positive ion. (b) Negative ion. Note that the magnetic force vector B is directed perpendicularly into the plane of the paper. where go is angular frequency (radians per second) of the ion. Three points should be noted from this equation of cyclotron motion. First, the ion's natural cyclotron frequency is directly proportional to the magnetic field strength. For any given experimental observation period, a higher frequency can be measured more precisely than a lower frequency. Therefore it is desirable to use as large a magnetic field as possible. Second, increasing the number of charges on an ion also increases its frequency. Finally, as mentioned before, the angular frequency to is not dependent on the ion's velocity (energy) or the radius of its orbit, but is a constant determined the magnetic field strength and (inversely) by its mass-to-charge (m/z) ratio. Figure 2-2 shows the relationship between m /z for an ion and its cyclotron frequency at three commonly used magnetic field strengths. Because the ion frequency is inversely related to its mass-to- charge ratio (if the magnetic field is constant), this ICR principle could be used to build a new type of mass spectrometer. It was not until 1949, however, that such an instrument, the omegatron, was actually built by Hippie, Sommer, and Thomas [15-16]. In their application, a resonant radiofrequency (rf) excitation forced the ions into orbits with larger and larger radii until they ultimately collided into the detector. Sensitivity was limited, and the instrument did not prove generally useful as a mass spectrometer. An improved method of detection was employed by Wobschall, Graham, and Malone in 1963 [17-18]. A marginal oscillator [19] was used to measure the amount of rf power absorbed by on-resonance ions. 8
v c (H z) 100 ,000 ,000-1
10,000,000 ■
1,000,000- 7 .0 T
100, 00 0 -
10,000- 3 .0 T
1000 -
100 - 1.2 T
10 - m/z 100 1000 10,000 100,000
Figure 2-2 Relationship between the mass-to-charge ratio m/z and the cyclotron frequency v of an ion at three commonly used magnetic field strengths in the absence of electric fields. 9
A mass spectrum was obtained by either scanning the rf frequency at constant magnetic field or scanning the magnetic field at a constant rf frequency. In 1970 Mclver improved on earlier instruments with his invention of the ion "trap" [20]. This cell had two trapping plates perpendicular to the magnetic field. A small potential of up to a few volts could be applied to these plates, which kept ions from drifting out of the cell. This greatly increased the length of time that ions could be detected, thus improving the obtainable mass resolution. It also increased the ICR experimental period available for ion-molecule reactions at low pressures to a factor of nearly 1000 more than that for sector instruments. The study of such ion-molecule reactions was the most common application of the ICR at that time [21-23]. Because of the large length of time (about 30 minutes) needed to scan even the small mass range of 15 - 200 with a sufficient signal-to-noise ratio, ICR did not lend itself for general mass spectrometry work.
Basics of Fourier Transform Ion Cyclotron Resonance Mass Spectrometry The next significant ICR development came in 1974, when Comisarow and Marshall first applied Fourier transform (FT) techniques to ICR signal detection [24-25]. The Fourier transform had been known since the early 1900's, but it was not until 1965 that Cooley and Tukey's fast Fourier transform (FFT) algorithm made it possible for existing minicomputers to process large data sets of up to 10,000 points in less than a minute [26-27]. In a short time FFT techniques were applied to both infrared spectroscopy (FT/IR) [28] and to nuclear magnetic resonance spectroscopy (FT/NMR) [29]. Several years then passed before electronic analog-to-digital'converters (ADC) became fast enough to handle the megahertz digitizing rates necessary for collecting broadband ICR signals. Also, in 1973 Comisarow developed a technique to detect the ion-induced image currents on detector plates by passing the induced oscillating current through an impedance, producing a voltage that could be amplified and digitized. It took the combination of all these developments - the FFT algorithm, the fast ADC converter, and image current detection - to make FT/ICR possible. The basic FT/ICR technique is as follows. Ions are either formed in the ICR trapping cell or are transferred into the cell from some other source. The generalized cell consists of three pairs of opposing plates. A small potential (usually from +1 to +3 volts) is applied to the trapping plates to prevent the positive ions from leaking axially out either end of the cell. Reversing the trapping voltage to a negative voltage is all that is needed to convert to the negative ion mode. The excitation plates deliver the resonant rf excitation power needed to move the ions coherently outward as a "packet" to a larger orbital radii (see Figure 2-3). In effect, this packet of ions is a rotating electric monopole [30]. At sufficiently large radii, the ion packets come close enough to induce an ICR image current on the detection plates. This image current is converted to a voltage, which has an amplitude that is proportional to both the radius of orbit (larger orbits bring the charges closer to the detection plates) and to the number of charges, but which is independent of the mass of 11
a) b )
c) d )
Figure 2-3 Mechanism of FT/ICR signal detection, (a) Incoherent ion packet which generates no detectable signal on the detector plates, (b) Excitation (and absorption of resonant rf energy) causes the ion packet spirals outward to a larger orbital radius, (c) At this larger radius, the ion's coherent motion induces a detectable sinusoidal image current on the detector plates, (d) Excessive excitation power excites the ion packet to a radius larger than the cell radius, causing ions to be lost upon colliding with the plates. 12 the ions [31]. To prevent ion-molecule collisions from prematurely damping out this signal (which degrades the achievable resolution), extremely low pressures of 10'8 torr to 10'10 torr or better must be maintained in the cell. As a general rule of thumb, the signal duration for smaller ions is inversely related to the pressure:
2 x lO-8 Signal Duration (s) = Pressure(Torr) (2-5)
Thus a one second signal detection period requires a pressure of 2 x lO 8 Torr. Larger ions, with their greater collisional cross-sections, require even lower pressure. Assuming ions of all masses in the cell are excited simultaneously, which can be accomplished with a variety of excitation techniques, a broad-band image current detector can capture the induced signals of all frequencies (masses) simultaneously [32]. A sufficiently fast ADC converter turns the detected analog voltage signal into discrete digital time domain data that can be stored and manipulated on a computer. This digitized time domain signal consists of the superposition of decaying sinusoids of the frequencies of every mass present in the cell. The FFT (see Chapter IV) can convert this data into a frequency spectrum. Since mass is inversely related to cyclotron frequency, the frequency spectrum can easily be rescaled into a mass spectrum (see Figure 2-4; note the broadening of high mass peaks). As a result, the entire mass spectrum for all ions in the cell can be obtained in a single 13
a)
o 500 1000 15002000 2500 3000 FREQUENCY in KHz
b)
15 65 115 165 215 MASS IN A.M.U.
Figure 2-4. Conversion of a frequency spectrum at 3 Tesla to a mass spectrum, (a) A frequency spectrum containing five equally shaped peaks, (b) The same data as in the previous figure, but re-scaled according to mass using the approximate relationship (for 3 Tesla) MASS(AMU) = 46,000,000/FREQUENCY(KHZ). Note th a t the highest mass peaks (A and B) cover a much wider mass range (and are thus less well resolved) than the lowest mass peaks (D and E). 14 experimental sequence lasting from just milliseconds to at most a few seconds. Compared to the prior single-channel method of scanning and detecting each frequency (or mass) separately, the Fourier transform multi-channel technique delivers a multiplex (or Fellgett) advantage. That is, an FT spectrum of N frequencies can be obtained in 1/N the time of a scanned spectrum with a comparable signal-to-noise ratio. Alternatively, FT techniques permit N independent spectra to be signal averaged in the same amount of time needed for a single conventional scan of each N frequencies separately. Because signal averaging increases the signal by a factor of N but the noise by only a factor of >/n , the overall signal-to-noise ratio is improved by a/ n . Since an FT/ICR experiment is basicalfy a variable sequence of events under computer control, the technique is extremely flexible. A typical experiment consists of the ionization step, ejection of unwanted ions, a delay time for ion-molecule collisions and reactions, the excitation of ions to coherent orbits, ion detection, and a quench step to clear ions from the trap. An entire experiment may take as little as a few milliseconds. This permits signal averaging of hundreds of experiments in just a few seconds. Figure 2-5 shows the generalized FT/ICR experimental sequence. Fourier transform ion cyclotron resonance mass spectrometry (FT/ICR/MS; henceforth simply FT/ICR; also referred to as Fourier transform mass spectrometry, or FTMS), thus overcame the serious detection limitations of its ICR parent. At the same time, its frequency based mass determination offers tremendous improvement in resolution 15
Ion Synthesis Form ation
Ion Purification Selection
Ion-Molecule Chemical Collisions Reactions & Reactions
Excitation Product & Detection Analysis
Ion Removal Cleanup (Quench)
Figure 2-5 Generalized FT/ICR experimental event sequence (left), with the equivalent conventional chemical manipulations (right). 16
over the best sector Instruments. To get a perspective on why FT/ICR can provide superior resolution to even the best (biggest) double- focussing magnetic sector instrument, one need only look at the relative lengths of their flight paths. The largest magnetic sector flight paths are about 7 meters [33], while a singly charged ion of mass 50 in a 3 Tesla FT/ICR instrument with a 1" cell travels approximately 70 kilometers in a typical one second detection period. To date, FT/ICR instruments have reached ultrahigh mass resolving power of nearly 200,000,000 for 40Ar+ and 300,000,000 for ieO+ [34], and 400,000,000 for 18H20+ [35]. Since these measurements appear to be limited by pressure (which determines how long the signal can be detected), even a small improvement in vacuum systems could make FT/ICR mass resolution of more than a billion possible. One additional unique aspect of FT/ICR is that its detection of ions is nondestructive. A clever experiment by McLafferty's research group took advantage of this by replacing the "quench" event with a long delay (up to 2 minutes for m/z 1100 at 5 x 10 9 Torr), enabling ion- molecule collisions to gradually slow down and "relax" these large ions back to the center of the trapping cell [46]. Subsequently the ions were re-excited and detected. This whole process was repeated up to 25 times for ions from a single laser shot (increasing signal-to-noise fourfold), and up to 1000 times when done with additional replenishment of ions from an external 252Cf plasma desorption source (increasing the signal a hundredfold). Remeasurement efficiencies of 17
98% have been attained for ions of m/z 2000. Refinement of this technique may possibly enable the detection of a singly charged ion. The tremendous potential of FT/ICR was quickly grasped by numerous researchers throughout the world. Rapid advances in FT/ICR soon followed in such critical areas as ionization methods, cell design, excitation techniques, and data transformation.
Ionization Methods The ionization process is often the most crucial phase of a mass spectrometric analysis, especially for biomolecules, since they are often larger and less stable than simpler organic or inorganic analytes. A variety of different ionization techniques have been developed, each of which offers particular advantages [37-39]. A brief description of the main techniques will be given here. Specific biochemical applications based on some of these techniques will be discussed in Chapter 3. Electron ionization (El) is the simplest and most commonly used form of ionization for substances that can be sufficiently volatilized [6]. Since the pressures used in FT/ICR are 10'8 torr or lower, even compounds with fairly low vapor pressures often are amenable to simple El. A beam of electrons is generated from a heated filament with energies in the range of a few eV to 70 eV (a 50eV electron has a velocity of 4 x 108 cm s 1 and traverses a typical 1" cell in about 6 nanoseconds).
Typical currents used are 0.1-100 jaA with an electron beam duration of a few milliseconds. The electrons (being charged particles themselves) undergo a high frequency cyclotron motion along the axis of the magnet and pass through the holes in the center of the trapping plates through 18 the cell. El is often referred to as a "hard ionization" process, since the large energy imparted by 70 eV electrons to the target neutrals usually results in extensive fragmentation. Although such fragmentation is generally useful for structure determination, it is usually desirable to also generate a sufficient number of molecular ions, if only for molecular weight determinations. Another limitation of El is the requirement of sufficiently volatile neutrals. A heated probe can be used for some materials. Most small biomolecules (sugars, amino acids, etc.), because of their polarity or thermal instability, require a prior chemical derivatization step to create a sufficiently volatile (and stable) sample. Larger biomolecules generally cannot be made sufficiently volatile even with derivatization. Furthermore, the loss of sample and the addition of impurities that accompany the derivatization process limit the usefulness of El when only small amounts of sample are available. A "softer" form of ionization, which induces less fragmentation and increases the yield of molecular ions, is chemical ionization (Cl) [40]. An excess of reagent gas such as methane or ammonia is added along with the volatilized sample. Because of the predominance of reagent molecules over sample, the electron beam generates primarily ions from the reagent gas. Collisions between the reagent ions and sample neutrals can result in proton or hydride transfers. Much less energy is imparted to the sample molecule than by electron impact, so there is much less fragmentation and significantly more molecular ions. Added reagent gas is not always needed. If there is a high enough pressure of the sample molecule (M) or sufficient reaction time is allowed after 19 electron bombardment, some of the sample molecules will fragment into daughter ions. If the daughter ions have a lower proton affinity than their parent (which is often the case), these ions can act as reagent molecules. This process is referred to as self-CI, and results in an increased abundance of (M+H)+ ions [41-42]. Self-CI can also be used along with many of the other ionization techniques. Perhaps the most generally useful ionization technique in FT/ICR for biomolecules is laser desorption. The laser beam is directed onto a sample probe by mirrors or by using fiber optics [43] (see Figure 2-6). Various laser frequencies ranging from infrared (IR) to ultraviolet (UV) can be used. The laser desorption process itself is not fully understood, although it appears to utilize the same mechanisms as other energy- intensive desorption process such as fast atom bombardment (FAB) [44] and plasma desorption (PD) [45]. Somehow the impact of a high energy pulse of photons or particles is transmitted throughout the target matrix, knocking intact particles from the matrix directly into the gas phase [46-47]. It has been compared to throwing a bowling ball onto one end of a trampoline to bounce off a feather at the other end. Some arguments have been presented favoring the desorption of "preformed" ions directly from the matrix [48], More evidence, however, seems to support the importance of gas-phase ionization and cationization reactions in the high-pressure "selvedge" produced by the energetic desorption event [49-51]. In particular, FAB desorption studies on equimolar mixtures of pairs of substances such as aniline (higher gas- phase basicity) and ammonia (higher aqueous basicity) showed aniline 2 0
A [ LASER
a ) r l PROBE VACUUM CHAMBER
FIBER
— ~ ) CELL b ) w k r
PROBE VACUUM CHAMBER
Figure 2-6 Two methods of directing a laser beam onto a sample probe adjacent to the FT/ICR trapping cell, (a) Aluminum mirrors (M) direct the beam into the chamber, and a parabolic mirror (P) focuses the incident beam onto the probe, (b) A fiber optic cable directs the laser beam directly onto the probe. In both methods the desorbed neutrals and ions simply drift into the adjacent trapping cell. The trapping potential is temporarily dropped to zero for a short time after the desorption event to allow the ions to enter. The probes can be rotated to permit multiple laser shots at fresh sample areas. 21
to strongly suppress the ammonia peaks in the mass spectrum [52]. Another study of a number of similar pairs likewise showed the suppression of ions of the substance with lower gas-phase basicities (with the one exceptional case being possibly explained by surface enrichment) [53]. Further support of the gas-phase ionization model is that increasing the concentration of particular ions in the matrix does not increase the observed ion current for that species [54]. Since laser desorption usually generates a 1000 to 10,000-fold excess of neutrals over ions, the process can be enhanced by subsequent electron bombardment [55] or chemical ionization [56] to generate more ions from these neutrals. Traces of alkali metal salts are often present as contaminants in the sample, and gas phase cationization results in the formation of (M+Na)+ or (M+K)+ pseudomolecular ions. Such alkali salts as KBr can be intentionally added to the sample to enhance the formation of these pseudomolecular ions when normal molecular ions are not observed. Matrix-assisted laser desorption is an extremely important recent development which shows tremendous potential for biochemical research [57-58]. Typically the analyte is mixed with an excess of a matrix compound having a strong absorption in the UV region. A laser frequency in resonance with the matrix absorption band is then used. The matrix molecules therefore absorb nearly all the incident laser power, and gently transfer this energy to the analyte molecules. In the process at least some of the matrix is sacrificed. Intact sample molecules with molecular weights of up to several hundred thousand 22 have been successfully volatilized, ionized and observed. In general, this technique produces predominantly singly-charged molecular ions with little fragmentation and is usually not affected by the matrix. Various host matrices, such as nicotinic, cinnamic, sinapinic, ferulic, and caffeic acid, have been found to work at different laser wavelengths in the UV, visible, and even IR regions [59-61]. There are possibly different mechanisms operating for the UV (electronic excitation) as opposed to the IR (thermal excitation) methods [62]. Another recent advance of particular relevance to biochemistry is the coupling of electrospray ionization to FT/ICR [63-64]. Electrospray is an extremely soft ionization process [65]. Figure 2-7 gives a simplified sketch of the electrospray design. A solution containing the analyte is forced through a small needle at a relative potential of several kilovolts to generate small charged droplets. Electrodes force, these charged droplets through a hot (300K), high pressure (up to 800 torr) drying gas where most of the solvent is evaporated and charge repulsion separates the remaining analyte molecules. These multiply charged molecules pass through capillaries and skimmer nozzles through two or more stages of pumping. They are subsequently directed by electrostatic or rf quadrupole lenses into the mass spectrometer. An electrospray source has been successfully linked to an FT/ICR instrument via an intermediate rf-only quadrupole and five stages of pumping. As with matrix-assisted laser desorption, extremely large biomolecules have been volatilized and ionized intact. One particular benefit of electrospray ionization is that these large molecules can each contain up 23
Electrostatic Lenses
r Quadrupole L J 0 ) Liquid I d Sample ■
nFirst Second Pumping Pumping Stage Stage
Drying Gas
Figure 2-7 Simplified sketch of the electrospray process. Droplets containing the sample acquire an electric charge as they emerge from a high-voltage needle. They then pass through a drying gas and through one or more pumping stages while being focussed by electrostatic lenses into a quadrupole device. 2 4 to 100 or more charges - far more charges than can be obtained by any other current form of ionization, including the closely related thermospray (TS) [66-67] and aerospray (AS) [68] techniques. This extensive charging greatly reduces the ion's mass-to-charge ratio, moving it to significantly higher ICR frequencies that can be more easily and accurately measured. Although electrospray generally is used to produce positive ions, reversing the electrode potentials can produce negative ions with multiple charges. Because electrospray requires such little sample (protein concentrations of 0.01 to 1.0 pg/[AL with flow rates of a few pL/min.), it is possible to feed in the sample from the effluent of a liquid chromatograph. FT/ICR has borrowed other ionization techniques from its sister areas of mass spectrometry, such as continuous flow fast atom bombardment (FAB) [69], cesium-ion secondary ion mass spectrometry (SIMS) [70], SF6 fast neutral beam (FNB) [71-72], laser (UV, visible, and
IR) photodissociation [73-75], and 252Cf plasma desorption (PD) [76-77]. Even some chromatographic techniques, such as gas chromatography and supercritical fluid chromatography, have been hybridized with FT/ICR (GC/FT/ICR [78], SFC/FT/ICR [79]). The trick in many of these cases involves introducing externally generated atoms or ions at relatively high pressure into the very low pressure environment of the ICR trapping cell in the center of an extremely powerful magnetic field. As many as five stages of differential pumping are required to bridge the several orders of magnitude pressure differential between the external source and the ICR analyzer cell. If ions rather than neutrals are to be 25 introduced from external sources, their paths must be carefully aligned along the z-axis of the magnet, usually by some type of electrostatic lens [80-81] or rf quadrupole lens [82-84]. Otherwise the "magnetic mirror" effect caused by inhomogeneities in the magnetic field will deflect the ions and prevent their entrance into the center of the magnet [85]. As these techniques for interfacing high pressure external ion sources with FT/ICR continue to improve, one can expect the unique analytic capabilities of FT/ICR to be applied to an increasing number of interesting biomolecules.
Excitation Techniques In the original FT/ICR experiments, a very short single-frequency rf pulse was used to excite ions of a narrow frequency bandwidth into coherent cyclotron orbits [24]. Although the Fourier transform of a theoretical time domain impulse (an single frequency pulse of infinitely brief duration) excitation is a uniform broadband excitation in the frequency domain, it is not possible to approximate this in the real world because of the tremendous instantaneous power required. Because a finite duration is required to deliver adequate excitation power, the best that can be obtained from a single frequency pulse is a broad "sine" excitation in the frequency domain. A major improvement soon followed when Comisarow and Marshall introduced frequency sweep (also referred to as "chirp") excitation, which excites a broad range of frequencies [25, 86]. Most FT/ICR experiments to date have used this excitation technique. 26
Unfortunately, this type of excitation is not very uniform in power and cannot easily be tailored to excite at specific frequency windows. Currently, the most flexible, selective, and uniform excitation method available for FT/ICR is the stored waveform inverse Fourier transform (SWIFT) developed by Marshall and co-workers [87-89]. First, the experimenter selects the exact frequency windows where excitation (or ejection) is desired. Then a discrete inverse Fourier transform is done on this desired frequency spectrum to generate the necessary time domain excitation waveform required. Phase modulation of the frequency spectrum before the inverse FT reduces the dynamic range of the resultant time domain waveform so that it can be handled by existing electronics. Apodization may also be performed to reduce distortions. SWIFT excitation devices, available commercially, can potentially provide any desired excitation waveform. Figure 2-8 compares the three types of excitation discussed. It should be noted that ejection of unwanted ions can be accomplished simply by over-exciting at the appropriate ICR frequencies so that those ions will spiral out and collide with the cell walls. Using SWIFT, a single time domain excitation waveform can easily handle multiple excitation and ejection windows simultaneously. This makes multiple MS/MS/MS... (or MSn) event sequences relatively straightforward in FT/ICR [90-91], a tremendous advantage over conventional tandem sector or quadrupole MS/MS experiments requiring multiple instruments. Such FT/ICR M S11 event sequences consist of a series of selective excitation/ejection events, interspersed 2 7
M(co)
CD
CO
Ejection Ml co
Figure 2-8 Three excitation techniques that have been used in FT/ICR. (a) Single-frequency, (b) Frequency sweep or "chirp", (c) Stored waveform inverse Fourier transform or "SWIFT" used to' excite a single frequency windows, (d) "SWIFT" used to excite two narrow frequency windows, (e) "SWIFT" used to excite two narrow frequency windows and to eject or over-excite ions in a third (wider) frequency window. 28 with appropriate delays for Cl or self-CI to induce the desired fragmentation. These experiments are extremely useful in the complete structural elucidation of larger molecules. In cases where the fragmentation of many daughter ions is to be studied, Hadamard encoding allows a series of experiments to achieve much higher signal- to-noise ratios (or the same S/N ratio in less time) than would be obtained from studying each daughter fragmentation path separately [92].
Cell Design and Ion Trapping The design of the FT/ICR trapping cell is critically important in maximizing resolution and the observable upper mass limit [93]. The frequency of an ion is affected both by the magnetic field as well as any by any electric fields present in the cell. The magnetic field of a typical superconducting solenoid is effectively homogeneous to within 1 part in 105 in the confines of a small 1" to 2" cell. Ideally one would also prefer having no electric fields at all in the cell. Unfortunately, it is necessary to apply a trapping potential along the z-axis to keep ions from escaping the cell. This potential introduces an electric field within the cell, which has two serious consequences. First, any electric field inhomogeneities cause ions of the same m/z ratio but in different regions of the cell to be shifted in their cyclotron frequency. This frequency broadening degrades resolution. Second, the radial component of this electric field (even if perfectly homogeneous) reduces the effective magnetic field, which therefore increases the cyclotron orbital radii for ions of a given m/z ratio. In the absence of radial electric fields, the cyclotron radius of 29 a thermal ion in equilibrium with its surroundings is given by Equation
2 - 6 :
1 /mkT = qB V mkT 2 (2-6) where r is the cyclotron orbital radius, q is the charge, B is the magnetic field, m is the mass, T is the absolute temperature, and k is the Boltzmann constant. Since the maximum cyclotron orbital radius is limited to half the radius of the trapping cell (rmax = 2 ~rceil), rearrangement of Equation 2-6 yields the upper mass limit for singly charged thermal ions:
q2B2r2 u m upper= 2kT (2-7)
Solving this equation one finds that in the absence of radial electric fields the upper mass limit for a singly charged thermal ion in a 1" diameter cell in a 7 Tesla magnetic field is over 60 million amu! Even allowing for a smaller radius to so the ion packets can develop spatial coherence, the upper mass limit at 7 Tesla is still above 5 million. However, the presence of radial electric fields severely reduces this limit (see Figure 2-9). A 1 V trapping potential in a quadrupolar cell reduces the upper mass in the above case down to just 200,000 amu. In a 3 Tesla magnetic field, the upper mass limit drops to 38,000 amu. The maximum upper mass achievable in practice will be even less, since the 30
m/z 10,000,000 Upper Mass Limit
1,000,000 : ,276 ku (100% ejection) 2 0 5 ku (1% ejection)
100,000 :
10,000 - 6 ku 3.0 T
1000
100
Figure 2-9 Upper mass limits for singly charged thermal ions in a 1" trapping cell at various typical strength homogeneous magnetic fields as a function of the trapping voltage VT. Calculations assume a quadrupolar electrostatic potential from the trapping plates. Upper arrows at VT = IV indicate the highest m/z ion which has a stable 1" orbit; lower arrows indicate the highest m/z for which 99% of the thermal ions (25 °C) have radii of 1" or less. 31
Incoherent Ion packet has to have an initial radius smaller than the cell radius if it is to be excited to generate a detectable signal. A number of design modifications have been made to the traditional cubic trapping cell to minimize the inhomogeneities and magnitude of electric fields produced by the trapping plates. Figure 2- 10 shows illustrates some of the more important FT/ICR cells being studied. A hyperbolic cell maintains a uniform quadrupolar potential throughout the cell, unlike the cubic cell which is approximately quadrupolar only in the center of the cell [94]. Cylindrical cells have also been used [95]. Guard wires have been used to shim the electric field for even better homogeneity [96]. For extending the upper mass limit, though, it is necessary to reduce or eliminate the radial component of the electric field. The placement of grounded screens just inside the trapping plates can reduce the electric field within the cell by a factor of up to 100 [97]. Elongated multi-section (tetragonal) cells also reduce the effective electric field in the center of the cell [95]. In addition to reducing the radial electric field, several other factors from Equation 2-7 are currently being played with to increase the upper mass limit. This includes increasing q by using multiply charged ions, increasing B with a stronger magnet or addition of an appropriately oriented electric field, increasing the trap radius r, and cooling the ions to decrease T. Appropriately designed trapping cells can also be used to maintain the ultra-low pressure needed for high resolution work while permitting ion formation or accumulation to occur at much higher pressure. The 32
Bo
(a) (b) t T
E or D D or E
HI
Figure 2-10 Various geometries of FT/ICR ion traps (cells). The cells all basically consist of three pairs of opposing plates: Trapping plates (T). Excitation plates (E). and Detection plates (D). Holes in the center of the trapping plates allow passage of an electric beam for electron ionization within the cell, as well as entrance of externally generated ions (such as from laser desorption), (a) Cubic, (b) Cylindrical, (c) Hyperbolic, (d) Elongated m ulti-section, (e) Screened [grounded screens are denoted by (S)]. 33 dual cell, available on commercial instruments from Extrel, has a high pressure source side separated by a gas conductance limit plate from the low pressure analyzer side [98]. Differential pumping on each side permits the source side pressure to be from 100 to 500 times higher than on the analyzer side. Dropping the voltage to 0 on the conductance limit plate can allow ions from the source to equilibrate with the analyzer side by passing through a tiny hole in the center of that plate along the z-axis. Longer detection periods, leading to higher mass resolving power, can then take place on the low pressure analyzer side. This dual cell is particularly useful when interfacing FT/ICR with high pressure external ion sources such as gas chromatography. In a discussion of ion trapping, it is necessaiy to keep in mind the number of ions involved. When more than 100,000 ions are in a 1" trap, space charge repulsion between ions distorts signal measurements. On the other hand, it is difficult to detect fewer than 100 charges with existing electronics. This means the effective dynamic range in sensitivity is only about 1000:1 in a single spectrum, much less than magnetic sector mass spectrometry. If a compound (or compounds) of interest is present in very small amounts in a mixture, it is necessary to employ some selectivity in the ionization or excitation process so as to increase its relative ion abundance. Alternatively, ejection of unwanted ions or their removal by other techniques (such as temporarily dropping the trapping potential to preferentially reduce the concentration of certain ions) can be employed. Recent work in the precise measurement of the cyclotron radii has enabled ions to be carefully excited to their 34 optimal radii, allowing the detection of as few as 40-50 singly charged ions with a signal-to-noise ratio of ~3:1 [99]. This is particularly exciting since it indicates that with optimal ionization and excitation it should be possible to detect a single highly charged ion, especially if combined with multiple remeasurements. It should be mentioned that quantitative measurements are somewhat difficult to do with current FT/ICR technology. The electric fields and space charges within the cell introduce mass-dependent effects on observed signal amplitudes (since signal amplitude is determined by both the number of charges and the orbital raduis). Even if it is someday possible to accurately measure the abundances of all the ions present in the trapping cell, it does not guarantee that those reflect the actual abundances in the original sample. Both the ionization process and the ion trapping process can introduce discrimination among ions on the basis of chemical properties and/or molecular weight. The best hope for FT/ICR quantitative studies is in isotope abundance measurements, where both chemical and mass differences are minimized.
Data Manipulation The FT/ICR discrete time domain data contains amplitude, phase, and frequency information for eveiy detected ion packet (as well as a variable amounts of noise). For the most part, though, the time domain data format is not directly useful. Some type of transformation must be employed to convert these data into an understandable frequency spectrum (which can then be easily rescaled to a mass spectrum). 35
The first and still almost universally used transformation process in FT/ICR is the fast Fourier transform (FFT) [100-110]. Although best suited for handling both real and imaginary data as input, a "real" FFT for handling FT/ICR time domain data can be obtained by adding several data shuffling operations. The frequency domain output from the FFT has two parts, one real and one imaginary. When phase corrected, the real data are in the absorption mode, and the imaginary data are in the dispersion mode. If phase correction is difficult, the real and imaginary data can be combined into a magnitude mode spectrum. The fast Hartley transform (FHT) is similar in operation to the FFT, but is better suited to the real-valued time domain data of FT/ICR. The FHT can be used to produce spectra (real, imaginary, and magnitude) identical to that obtained from the FFT. If some information is known about the spectrum, such as the number of peaks present and the average noise levels, other non-FT methods of spectral estimation can give superior results. Bayesian analysis, based on maximum entropy methods, has been used for FT/ICR data [111]. However, because it is an iterative process requiring numerous FFT operations, it can be up to 10,000 times slower than the FFT. Mention should be made of apodization, a commonly employed process that can alter the frequency domain lineshapes and baseline by applying one of many weighting functions to the time domain data before transformation [112-113]. Although apodization can "smooth" the spectrum and reduce wiggles and noise, it accomplishes this at the expense of spectral resolving power. 36
Further details of FT /ICR data manipulations and lineshape analysis involving the Fourier, Hartley, and Hilbert transforms are discussed in Chapters IV though VI.
Current FT/ICR Usage Nicolet Analytical Instruments introduced the first commercial FT/ICR instrument, the FTMS-1000, in 1981. This instrument used a 3 Tesla superconducting magnet. This was followed by the CMS47 from Spectrospin AG with a 4.7 Tesla magnet, and the Nicolet (now Extrel) FTMS-2000 with either a 3 or 7 Tesla magnet. Ionspec (Irvine, CA) also sells a complete FT/ICR instrument. A research consortium in Florida is developing funding for a proposed FT/ICR instrument to be equipped with a 14 Tesla magnet. The increased usefulness of FT/ICR as an analytical tool over the past 17 years is reflected in the growing number of instruments in use worldwide (see Figure 2-11). This growth is particularly impressive considering the relatively large cost ($300,000 to $700,000) for a typical FT/ICR instrument. Even though the number of FT/ICR systems is but a small fraction (less than 1%) of the total number of sector, quadrupole, time-of-flight, and all other mass spectrometers in use, it is interesting that FT/ICR research contributed more than 12% of the total papers at the 1989 annual conference of the American Society for Mass Spectrometry [114]. A number of reviews give further details of FT/ICR principles, techniques, and applications [41,115-129]. FT/ICR's combination of features, in particular its potentially ultra-high resolution (for precise determination of chemical formulas), its ability to trap ions (for ion-molecule reactions, controlled fragmentation, and multi-stage MS/MS/MS...), its flexible experimental sequences, and its adaptability to numerous ionization methods, promises a bright future and continued growth. 38
125 r
100
1982 1986 1990
Figure 2-11. Number of FT/ICR systems in use worldwide. 39
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BIOCHEMICAL APPLICATIONS OF MASS SPECTROMETRY
Special Concerns As mentioned briefly in the preceding chapter, biomolecules in general often present special challenges to mass spectrometry. All but the simplest biocompounds are usually involatile in their underivatized form, and most become thermally unstable at elevated temperatures before they volatilize. Many of the compounds of interest are quite complex structurally. Certain classes of biomolecules, in particular proteins and DNA, have among the highest molecular weights of any substances. Finally, the samples to be studied are frequently available only in trace amounts. Fortunately, mass spectrometric techniques have been developed, or are being developed, to address these concerns. In general, the problem in most of the cases involves the ionization process. Although some of the applications to be mentioned have been devised in connection with magnetic sector or time-of-flight instruments, these applications can all potentially be ported to FT/ICR. The extremely high mass resolving power and the ability to do structural studies using MSn are just two incentives to move to FT/ICR for mass analysis. Although the potential utility of FT/ICR for biochemical and biomedical work has been recognized for many years [1], many recent developments have led to explosive growth in this field. Following is a
48 49 cursory glance at some of the interesting mass spectral applications for biomolecules which have relevance to FT/ICR.
Carbohydrates Mass spectrometry offers tremendous assistance in determining the structure of complex carbohydrates (oligo- and polysaccharides) [2- 9]. Unlike common proteins and nucleic acids, typical oligosaccharides cannot be sequenced using generalized enzymatic or chemical procedures. Oligosaccharide structure is also potentially far more complex, involving numerous possible linkage positions and stereochemical orientations, branching, and numerous types of modified saccharide residues. Often, too, these oligosaccharides are part of glycoproteins or glycolipids, adding further difficulty to any structural elucidation. Although mass spectrometry is not generally useful in determining stereochemical linkages (that is usually best done by NMR), it can potentially offer the most structural information of any modern technique for large carbohydrates present in trace amounts.
Traditionally, even small oligosaccharides and monosaccharides, because of their polarity, have had to be derivatized for mass spectral analysis, usually by permethylation or peracetylation [10-12]. The fairly recent techniques of FAB and LD allow native sugars to be studied directly. Some typically observed fragmentation pathways (both positive and negative ion mode) for underivatized oligohexoses are shown in Figure 3-1. Unique stereochemical arrangements of particular hexoses can add further fragmentation possibilities. FT/ICR laser desorption followed by electron ionization and negative ion mode detection 50 Carbohydrate Fragmentation Pathways
ch2oh ch2oh
+ Hor
+ Hor OH OH
CHoOH CH2OH + Hor
O ^ CH2OH
OH OH
OH OH ch2oh CH2OH
+ H or
+ H or OH OH
FIGURE 3-1. Some of the more common fragmentation pathways observed for underivatized oligosaccharides in positive and/or negative ion detection mode. Fragments A, C, and E retain the charge on the non-reducing end. Fragments B and D retain the charge on the reducing end. The stereochemistry of hydroxyl groups of specific sugar monomers can lead to additional structurally informative fragmentation. demonstrates some of the structural information that can be obtained on oligosaccharides (Figure 3-2). In many cases the negative ion mode was found to provide more informative fragmentation than positive ion mode. Individual underivatized monomeric aldohexose and deoxyaldohexose isomers have been differentiated by their relative abundances of particular fragment peaks in both negative and positive ion mode [13]. Another study on underivatized di- and trisaccharides (using FAB and positive ion mode) suggests that the identity of certain individual hexose residues (fructose, glucose, etc.) can be determined [14]. Other negative ion FAB studies using MS/MS have been used to differentiate l->2, l->3, l->4, and l->6 linkage positions [15-16], and to locate the position of sulfate substituents [17]. An LD-FT/ICR study has even indicated that fragmentation of oligosaccharides can be indicative of a particular anomeric (a or (3) configuration [18]. Also, chemical ionization studies using an optically active collision gas to resolve enantiomers is being attempted [19]. Admittedly most of these structural and stereochemical determinations on such small monosaccharides and oligosaccharides could be accomplished more readily and with more certainty using NMR. However, larger oligosaccharides as well as those available only in picomole amounts [20] are likely to require the high sensitivity and MSn capabilities offered by FT/ICR. Techniques developed with these small molecules can eventually be extended to far more complex oligo- and polysaccharides. LD-FT/ICR has been used to sequence bacterial capsular polysaccharides with a partial determination of linkages [21]. Matrix- assisted laser desorption ionization using a dihydroxybenzoic acid or 52
O’
STACHYOSE
HO
OH OH OH HO
HO OH OH OH OH
a-O-QaHl ->6}-a-0-GaH1 *>2}-^-0-Fru
(Hex-H)" 179
(2HaiCAO>H)' 383
100 200 400 500 600 700 MASS IN A.M.U.
FIGURE 3-2. FT/ICR mass spectrum of the tetrasaccharide stachyose using Nd/YAG laser desorption and negative ion mode detection. The molecular ion (665) and major fragment peaks are indicated. Traces of contaminating iodine caused the main peak at 127. 53 caffeic acid matrix has been shown to successfully volatilize intact underivatized dextrans containing more than 40 glucose units (MW >6000) [22], Although FAB and LD ionization methods permit underivatized sugars to be volatilized and ionized directly, there is likely to be continued need for derivatization as an aid in structure determination [23-25]. Often permethylation or acetylation adds stability to carbohydrate rings and results in simpler and more informative fragmentation between rings. This type of derivatization also assists in determining the reducing terminal residue and branch points. The attachment of positively or negatively charged "tags" to the reducing end of sugars increases ionization efficiency (and thus sensitivity), and helps in deciphering the fragmentation of oligomers built of uniform weight m onom ers [26].
Proteins Not too many years ago it would have seemed unlikely that FT/'ICR (or any form of mass spectrometry) would ever be capable of analyzing large intact protein molecules with masses over 100,000. At the time there were no ionization techniques capable of gently volatilizing such huge molecules. "High-mass" meant over 2500, and "very-high mass" was anything above 8000 [2]. In 1983 252Cf and 127I plasma desorption (with time-of-flight detection) was first used to generate single and doubly charged peptides and small proteins with weights up to 14,000 [27]. Over the next four years use of nitrocellulose, glutathione, and other matrices "softened" the plasma 5 4 desorption process and enabled proteins as large as pepsin (34,630) to be observed intact [28-30]. Although intact protein ions had now been created and observed by time-of-flight instruments, it wasn't clear at first whether these large organic ions (with masses above a few thousand) would be stable enough to be seen with FT/ICR. Magnetic sector and time-of-flight instruments require ions to remain stable for at most a microsecond or so before they are detected, whereas FT/ICR detection requires intervals ranging from milliseconds to a few seconds. To make matters worse, high mass ions, with their low cyclotron frequencies, require longer detection periods for high resolution. Nearly all the early time-of-flight mass spectral studies of species such as chlorophyll a and small proteins suggested that these large biomolecular ions were metastable (having lifetimes on the order of a few microseconds), thus making them unobservable on FT/ICR's millisecond time scale [31-34]. Soon, however, it was shown that chlorophyll ions were indeed stable during observation by LD-FT/ICR [35], and that with proper experimental conditions ("soft" desorption that imparted less energy to the ions) other high-mass organic ions such as synthetic polymers were stable for at least milliseconds [36]. Final evidence that protein ions were stable on at least a microsecond time scale came in 1987 when FT/ICR detected intact bovine and porcine insulin (m/z -5700) and horse cytochrome c (m/z 12,384) [37]. Despite the limited success of PD, the ionization of large intact proteins still proved elusive. In 1986, mass spectrometrist Frank Field of Rockefeller University wrote "... the mass region of real interest for 55 proteins is m/z 40,000-100,000, and one can only speculate as to whether such monster gaseous ions can be produced. My personal feeling is that to do so may well require the discovery of some new technique..." [38].
Matrix-Assisted Laser Desorption of Proteins Not one, but two revolutionary techniques appeared on the scene in the next two years. The first major breakthrough came in 1987, when Karas and Hillenkamp introduced matrix-assisted laser desorption [39-42]. Instantly the upper mass limit for intact proteins shot up to 100,000. Before long, a host of different proteins (over 300) with masses going up to 350,000 could be detected in femtomole quantities with mass accuracies of lO3 to 1CH [43-48]. Several other particularly advantageous features of matrix-assisted laser desorption add further to its utility [49]. Fragmentation peaks are usually not more than a few percent of the molecular ion abundance, increasing sensitivity in molecular weight determinations. Cluster ions (dimers, trimers, etc) and multiply charged species are also generally present in lower abundance than the singly charged molecular ion. The proteins are observed "clean" of water and most other contaminants. Extremes of protein hydrophobicity or hydrophilicity seem to have very little effect in suppressing desorption. In some cases quaternary structure is maintained - that is, non-covalently bound assemblages are desorbed and protonated intact. Finally, glycoproteins are observed with their carbohydrate moieties intact. 56 By comparison, the most commonly used current methods for protein molecular weight determinations, such as sodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS-PAGE), require at least picomolar sample quantities and are seldom accurate to within more than a few percent. Because electrophoretic measurements are only indirectly related to the mass, masses of proteins with unusual composition or modifications (such as glycosylation) can often be in error by several percent. These conventional techniques are virtually useless in distinguishing single amino acid substitutions and deletions. The tremendously better accuracy in molecular weight determination offered by matrix-assisted laser desorption, as well as its speed, assures it a major role in the future of protein chemistry. Already it has proven quite useful in confirming the structures of synthetic and recombinant proteins and in identifying impurities [50]. Although intact molecular ions are the predominant species produced by matrix-assisted laser desorption, it is possible to induce fragmentation of peptides and proteins by multiphoton ionization (MUPI) at a frequency that is absorbed by the molecule [51-53]. Tuning the frequency and intensity of the laser can produce different degrees of fragmentation, which offers a method of generating a variety of daughter fragments for MSn structure analysis and sequencing by already established techniques [54]. Peptide sequences have been determined by partial digestion with carboxypeptidase followed by matrix-assisted laser desorption measurement of the progressively shortened peptides [55-56]. A particular advantage of this technique is that it is not affected by 57 contamination with individual amino acids. At the same time, it is just as sensitive as Edman degradation with HPLC detection [57]. A further advantage of mass spectrometric techniques is its superior handling of unusual peptide modifications, glycosylated proteins, or those with blocked N-termini [58-59]. Derivatization of the N-terminus has even enabled discrimination between isomeric leucine and isoleucine residues [60]. FT/ICR using matrix-assisted laser desorption is currently being applied to peptide studies [61 ].
Electrospray of Proteins Although FT/ICR has been used with matrix-assisted laser desorption, current FT/ICR technology limits its usefulness at m /z values much above 30,000 - 50,000 (as discussed in the previous chapter). Research suggests that these limitations can be overcome in the near future, but for the present it would be advantageous to shift high mass (low frequency) ions to higher cyclotron frequencies where they can be more easily and accurately detected with existing FT/ICR techniques. Electrospray ionization is ideally suited to this task. Although the beginnings of the technique go back more than 20 years [62], it caught biochemists by surprise when first applied to proteins in 1988 [63]. Proteins ranging in weight from 5,000 to 76,000 were volatilized intact with up to 60 charges, pushing them down to a m /z window of 500 to 1500. It appears that typical electrospray of most proteins, no matter what their size, will deal out to them all a similar stable "charge density" of about one charge for every four to twelve amino acid residues. It is not clear if this charge capacity is directly 58 related to the number of basic Arg, Lys, and His residues, as it is in the case of field desorption ionization [64]. In any case, if there are too few charges per unit length (less than about one charge per every twelve residues) the protein is not be able to overcome solvation forces and "lift off' into the gas phase. Likewise, there appears to be a maximum charge density (approximately one charge per four residues) where the protein breaks away from the droplet before it can grab any additional charges. The few exceptional proteins which exhibit lower charge densities (only one charge per 20-25 residues, resulting in a m /z window extending up to 3000) are those prevented by disulfide linkages from "stretching out" and separating the charges [65]. Even these "worst case" proteins fall well within the mass range of FT/ICR detection. Thus electrospray ionization coupled to FT/ICR can potentially provide high resolution mass measurements for virtually any protein of any size. Electrospray (with quadrupole detection) has already produced intact multiply charged proteins with masses over 100,000 [66-67]. Figure 3-3 illustrates the incredible capabilities of electrospray FT/ICR in protein mass determination, as done by the research group of F. W. McLafferty [68]. The mass of equine cytochrome c (12,358.34) was determined to within | of an a.m.u. Only FT/ICR detection can provide this ultra-high resolving power (greater than 63,000) at such high mass (12,000, with m /z = 800) and with such sensitivity (3 femtomoles).
Other larger proteins such as albumin (66,000) have also been successfully detected with high resolution by FT/ICR [69-70]. Although electrospray of proteins is a relatively new technique, it is sure to 59
Cytochrome c, 3 fmols, 1000 s ion cooling, 2.8 T Single scan, m/z 400—► 16000, 256K data MW, calculated: 12,358.34 MW, measured: 12,358.22 (MW 1141 external standard) Resolving Power (average) = 63,000
(M + 14H)
1000 1200
(M + 17H (M + 16H) (M + 15H)
825.0
FIGURE 3-3. High resolution FT/ICR mass spectrum of electrospray- ionized Cytochrome c. Figure provided by F. W. McLafferty. 6 0
become a powerful tool for biochemists. It has already been applied to the detection of pyruvate as an N-terminal blocking group on human hemoglobin variants [71]. The ability to induce fragmentation of multiply-charged proteins by collision-induced dissociation holds promise for MSn structure studies [72]. Other features and biochemical applications of electrospray mass spectrometry have been reviewed [73].
Other Biochemical Applications Numerous additional applications have been found for various types of mass spectrometry in biochemical research. Although DNA and RNA sequencing by chemical means is firmly established and quite unlikely to be supplanted, such techniques are not well suited for detection of chemically modified residues. Oligonucleotides have been successfully studied with a variety of ionization methods in both positive and negative ion modes [74-77]. Matrix-assisted laser desorption of frozen aqueous solutions has successfully volatilized intact double stranded DNA with masses over 100,000, enabling mass spectrometric techniques to complement conventional sequencing in those cases involving modified purine or pyrimidine bases [78]. Stable isotope enrichment measurements play an important role in a wide variety of studies in nutrition and metabolism [79-82]. GC-MS of various bodily fluids has numerous biomedical purposes, including the chemical diagnosis of inherited metabolic diseases [83]. GC-FT/ICR has also identified components of other complex natural product mixtures such as Eucalyptus oil [84]. One final interesting application is the identification of bacterial species by the characteristic mass 61 spectra of lipids laser-desorbed directly from the intact membranes [85- 87]. Ratios of phosphatidylethanolamine, phosphatidylglycerol, lysylphosphatidylglycerol, phosphatidylinositol, diphosphatidylglycerol, and diglycoslydiglyceride were found to be distinctive for different species of both Gram-positive and Gram-negative bacteria. Different techniques (LD, FAB, PD, SIMS) have been shown to preferentially desorb particular lipids from membranes [88]. Some of the countless other biochemical applications of mass spectrometry have been reviewed [4, 89-92]. It is clear that mass spectrometry in general, and FT/ICR in particular, will play a major role in future biochemical research. There is still a need for improvement in the ionization processes involving biomolecules, such as laser desorption and electrospray (including the introduction of externally produced ions). Just as important are improvements to FT/ICR in its ability to handle high mass ions such proteins. Because studies of proteins and other macromolecules will always test the limits of FT/ICR sensitivity and resolution, any means of improving the signal-to-noise ratio and resolution of the FT/ICR mass spectrum (especially at higher masses) will be welcomed. The huge data sets (256K to 1M data points or more) generated for a single high- resolution protein spectrum, as well as the tremendous amount of data generated by GC-FT/ICR, call out for a more compact method for storing spectral information. With these goals in mind, the remainder of this thesis will introduce alternative data reduction procedures to FT/ICR. Using the Hartley and Hilbert transforms, these new procedures lead to an "enhanced" absorption spectrum offering several important 62 advantages over conventional FT/ICR absorption and magnitude mode spectra. 63
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APPLICATION OF THE HARTLEY TRANSFORM TO FT/ICR DATA
Introduction
The Fourier Transform July 27, 1794 was to be the last full day on earth for Jean Baptiste Joseph Fourier, a scientist whose background and politics had earned him a death sentence from the French revolutionaries. Fortunately for him, Robespierre fell from power that day. Fourier's life was spared. Thirteen years later, in 1807, the study of heat conduction led Fourier to develop the series expansion of sines and cosines that bears his name. Fourier's announcement of his discovery at the French Academy of Sciences was greeted with skepticism by many celebrated mathematicians (including Lagrange, Laplace, and Poisson) who had difficulty believing that an infinite summation of a sinusoid and its higher-frequency harmonics could converge to any arbitrary waveform, especially a discontinuous one. Questioning of his work delayed its publication eight years, and Fourier's approach was not fully described until the 1822 publication of The Analytical Theory of Heat [1]. It was in 1899 that a proof by Josiah Willard Gibbs finally established among mathematicians the validity of Fourier's methods. Engineers and physicists had accepted Fourier's methods more readily than the
69 70 mathematicians, and put them to work in a variety of applications, including the prediction of tides. Only quite recently, though, has usage of the Fourier transform mushroomed due to the developments of powerful digital computers and the fast Fourier transform algorithm [2]. The Fourier transform (FT) converts an arbitrary waveform into a frequency spectrum specifying the frequency, amplitude, and phase of each of the component sinusoids needed to add up to that waveform. Figure 4-1 illustrates how the FT absorption spectrum describes a time domain waveform consisting of three damped cosines of different frequencies and amplitudes. The examples used here and in the remainder of this paper will likewise deal strictly with the transformations of exponentially damped cosines, since they best approximate the ion-induced image current signals detected in FT/ICR experiments
j[t) = cos(2ttv0t)e-(t/'d = cos(co0t)e-(t /T) , 0 s t< oo (4-1)
However, the general principles and procedures to be discussed should apply just as well to other forms of signals. The frequency domain produced by the FT contains two values for each frequency. These values specify both the amplitude and the phase for the sinusoid component of that particular frequency, although the equivalent information can be given by other pairs of numbers. Complex notation, with real and imaginary components, has historically been used for this purpose. The complex Fourier transform (Equation 71
Initial Time Domain / 1 \
Frequency=32 Frequency=20 Frequency=10 Amplitude=20 Amplitude= 100 Amplitude=50 \ 1 /
100
50
20
20 Frequency Spectrum
FIGURE 4-1 The Fourier transform converts a time domain waveform (top) consisting of the sum of three cosines (middle) into a frequency domain spectrum (bottom) containing the correct frequency and magnitude information for each of the three components. 72 4-2a) and its inverse (Equation 4-2b) differ only by a sign in the complex exponent:
+00 +00 F(v) = / J[t) e '2ltfv£ dt = f J[t) [cos(2jivt) - * sin(2nvt)] dt (4-2a)
-00
+ 00 +00 Jit) = f F(v) e2ltfv£ dv = J F(v) [cos(2rcvt) + i sin(2nvt)] dv (4-2b)
-00 .00
For the forward transform, the cosine component of the FT gives the real part of the frequency spectrum, and the negative of the sine component of the FT gives the imaginary part. For simplification of these and subsequent equations involving frequency, one can substitute angular frequency, co, for 2 jw.
In experimental work one deals with causal signals, that is, signals which start at time zero and which are usually defined as having a value of zero for all times before the start of detection at t0. It may not be immediately obvious, but the Fourier transform of such causal real signals has both real and imaginary components. Figure 4-2 gives one graphic explanation why this is so. It is important to remember that exactly half of the information describing a causal signal is present in the real part of the FT spectrum, and half in the imaginary part, even though these lineshapes appear quite different. The real and imaginary parts of the FT can be combined into magnitude (Equation 4-3) and phase (Equation 4-4) spectra:
M(co) = yj Real(F(oj)]2 + Imaginary[F(u>)]2 (4-3) 73
Time Domain FT Frequency Domain fit) Re[F(v)J + Im[F(v)] JL J\y
2A*a f(t)= A exp(-a| t | ) F(v)= = A(v) [a2 + (2jcv )2]
f(t)= A exp(-a 111); t>0 -i2A(2jtv) flt)= -A exp(-a 111); t<0 F(v)= = D(v) [a2 + ( 2 j i v ) 2 ]
f(t)= A exp(-a 111); t>0 A(a-i2jiv) A(v) + D(v) flt)= 0; t<0 F(v)= [a2 + ( 2 j i v ) 2 ]
FIGURE 4-2 Origin of the Lorentzian "real" component and the dispersion "imaginary" component of Fourier-transformed exponentially damped causal (where f(t)=0 when t<0) cosine signals. Top row shows that the FT of the symmetrical exponentially damped time domain signal leads to just a Lorentzian real component. Middle row shows that the FT of an antisymmetrical exponential signal gives just a dispersion imaginary component. Because FT integrals are additive, addition of half of the top row plus half the middle row results in the bottom row. The FT of this causal signal has both real Lorentzian and imaginary dispersion components. 74 Imaginary(F(co)l] ^
The magnitude spectrum is frequently used in spectroscopic work, especially in FT/ICR, because in most cases, with the exception of overlapping peaks, it is independent of the initial phase of the sinusoid components of the time domain. The phase spectrum, on the other hand, is practically useless in the presence of even small amounts of noise. This is because most of the calculations for the phase spectrum involve very small numbers in the denominator, so that even tiny perturbations due to noise dramatically affect the final phase values. Figure 4-3 shows the typical lineshapes involved in the Fourier transform of causal, phase-corrected damped cosines. The absorption and dispersion lineshapes are both Lorentzian lineshapes, whose normalized forms are described by Equation 4-5:
Absorption! w) = — ------— (4-5a) l+(to0-to)2X^
(to0-to)x2 DispersionM = (4-5b)
in which w0 is the cyclotron angular frequency and x is the relaxation or decay time for the signal. Although in appearance these lineshapes are quite different, they both contain the same information about the signal. The frequency w0 is at the maximum for the absorption and at the zero crossing for the dispersion lineshape. Also, the full width at half- 75
Time Domain Cosine FT / \ Sine FT
Absorption [A(to)] Dispersion [D(oo)] M agnitude \ Calculation /
M agnitude [-\/A(co)2+ D( co)2]
FIGURE 4-3 Lineshapes associated with Fourier-transformed damped cosines. A cosine Fourier transform of a damped cosine gives an absorption lineshape. A sine Fourier transform of the same damped cosine gives a dispersion lineshape. The magnitude calculation combines the absorption and dispersion spectra into a phase- independent magnitude lineshape. 76 maximum height for the absorption peak and the peak-to*peak separation for the dispersion lineshape are both equal to 2 / t rad/s. In fact, the absorption and dispersion spectra can be interconverted using the Hilbert transform (see Chapter V).
Phase Shift and Phase Correction When discussing the phase of a time domain signal, a cosine wave (with maximum amplitude at £0) is considered to have a phase of zero (or no phase shift). A sine wave has a phase of 90° or n /2 radians. As the phase of a damped sinusoid goes from zero to n/2, the FT real component gradually changes from the absorption to the dispersion lineshape. At the same time, the FT imaginary component gradually changes from a dispersion to an absorption lineshape. Figure 4-4 shows this effect. Because of the distortion of these intermediate lineshapes it is necessary to phase correct the spectrum or else use the phase- independent magnitude mode spectrum. Individual frequency domain peaks can be phase corrected visually by adjusting the phase shift until the real component has the purely symmetrical absorption lineshape and the imaginary component has the purely antisymmetrical dispersion lineshape. Equation 4-6 specifies how the phase correction is applied
Absorption(oo) = Real[F(w)] cos
Dispersion(a>) = Real[FMl sin <|> + ImaginaiytFM] cos p — v- p - ------A \t— ^ = ^ 3 0 ° \k r - r _____ A \K_ Y FIGURE 4-4 Effect of a time domain phase shift on the corresponding frequency domain real, imaginary, and magnitude spectra. The top row is for an unshifted cosine (phase=0°), which has a pure absorption shape for the real component and pure dispersion for the imaginary. As the phase is gradually shifted by 30° (second row) and 60° (third row), the real and imaginary components exhibit mixed character. Finally, a sine (90° phase shift, bottom) results in a pure dispersion shape in the real component and a pure (negative) absorption shape in the imaginary component. Regardless of the phase shift, the magnitude spectrum is unchanged. 78 where <]> is the phase shift needed to effect the correction for a peak at a particular frequency. What makes FT/ICR phase correction more difficult than in FT/NMR is that its phase shift is not linear across the frequency spectrum. Because of the higher FT/ICR frequency range involved, even the very short delay between excitation and detection (required in order that excitation power can fully go to zero so it won't spill over into the detected signal) permits many cycles of phase to accumulate across the spectrum. Additional quadratic phase variation is added if frequency-sweep excitation is used. Although manual phase correction of individual peaks is easily implemented by computer and is useful when only a few FT/ICR peaks need phase correcting, it is preferable to have a more objective and automatic means of phase correction. DISPA (dispersion-vs.-absorption) plots provide an objective means for automatic phase correction of peaks [3-8]. When the detected signal is mixed with a sine or cosine wave in order to lower the frequency bandwidth (heterodyne mode detection), it is possible to phase correct using convolution-based methods [9-10]. Non-linear phase shifts introduced by the excitation waveform can be eliminated by making the excitation waveform phase coherent using SWIFT excitation [11-12], Additional methods of automatic phase correction developed for other areas of spectroscopy, such as FREIGHT-CARS (Fourier analysis of real and imaginary data going through Hilbert transforms of real data for coherent anti-Stokes Raman spectroscopy) can potentially be adapted to FT/ICR spectra [13]. There are several important advantages to using the absorption instead of the magnitude spectrum, including the \f3 improvement (or better) in mass resolving power because of the 79 narrower absorption peaks, the minimization of distortion when adjacent peaks overlap, and the ability to spot folded-back (aliased) absorption peaks by their anomalous phase. Although generally considered phase-independent, closely spaced (partially overlapping) magnitude mode peaks are in fact distorted to various degrees depending on the phase shifts of the two frequencies [14]. The magnitude mode spectrum, on the other hand, offers the advantages of yj2 improvement in signal-to-(standard deviation) noise (and thus a \J2 improvement in precision). The magnitude spectrum is also simpler to calculate due to its (relative) phase independence. Because the normal absorption spectrum does not offer overwhelming superiority in all areas over the magnitude spectrum, simplicity of calculation has led to the vast majority of FT/ICR spectra being converted to magnitude mode. The "enhanced" absorption spectrum introduced in Chapter VI offers clear advantages over the magnitude (and the ordinary absorption) spectrum in every regard, and justifies the extra effort needed to first phase correct the FT spectrum. Discrete Nature of Experimental Data An important aspect to remember in dealing with experimental data is its discrete nature. We are only able to sample the time domain signal at certain maximum rate. This sampling interval At, or dwell time, is limited by the speed of the electronic analog-to-digital converter and the amount of digital storage available, and is defined as A t= (4-7) 80 where T is the total sampling time and N is the total number of data points sampled. A uniform sampling rate is generally assumed, because the Fourier transform of non-uniformly spaced data involves very complicated additional processing in order to obtain the transform of the true signal [15]. The Fourier integrals (Equations 4-1) apply only to continuous mathematical functions. However, one may use a Fourier series in place of the Fourier integral for both the forward (Equation 4-8) and inverse (Equation 4-9) transform. IV-1 N N (4-8a) n=0 where Xnm = exp Hwm£n) = exp(-f2jtnm/A0 (4-8b) N-l N N (4-9a) m=0 where X^-i = (l/N)exp((2jtnm/JV) (4-9b) For N data points, the discrete Fourier transform based on the above equations requires approximately N2 multiplications. Even with the best current computers this algorithm becomes prohibitively slow at large values of N. Fortunately, there is considerable redundancy in this algorithm. In 1965, Cooley and Tukey took advantage of this 81 redundancy in developing their fast Fourier transform (FFT) algorithm [2]. The FFT slashed the number of multiplications required from N 2 down to N\og2N. This reduced the time for a thousand-point transform by a factor of 100, and for a million-point transform by factor of nearly 50,000. Generalized computer algorithms for the fast Fourier transform usually require the number of data points to be some power of 2 (radix-2 implementations), although it is possible to come up with specialized versions for other values of N as well. Another factor to consider when using discrete data is the sampling theorem. This states that the highest frequency that can be correctly determined must be sampled at least twice per cycle. If the maximum frequency to be observed is 1 KHz, the sampling rate should be at least 2 KHz. The highest frequency that can be correctly observed at a given sampling rate is known as the Nyquist frequency. Frequencies higher than the Nyquist frequency are aliased, or folded back, and appear at lower frequencies in the spectrum. A low-pass filter can be used before the Fourier transform to remove unwanted frequencies above the Nyquist limit. The precision obtainable in the measurement of spectral parameters is dependent on both analog and digital resolution in the frequency spectrum. The analog spectrum is the theoretical continuous lineshape that would be obtained if the frequency domain data points were infinitely close together. Analog resolution can be defined as COn (4-10) AO) AV 82 where to0 is the angular frequency in rad/s, v0 is the frequency in Hz, and ao ) and a v are peak widths at some specified percentage of the peak height, usually 50% (full width at half height or FWHH), but sometimes 10% or even 1%. For exponentially damped cosines of the form cos a>0texp(-t/x), the analog spectral resolution increases as the experiment time T increases. The maximum analog resolution (within a few percent) is obtained when T is at least 4x. When dealing with discrete data, the digital resolution, or the closeness of data points in the frequency domain, is „ , 2*Bandwidth(Hz) ...... Frequency-domain spacing (Hz/point) = ------(4-11) where the spectral bandwidth is the highest minus the lowest frequency. Digital resolution is also increased by longer signal acquisition time. The importance of adequate digital resolution can not be underemphasized. If the actual analog frequency of a very narrow absorption peak falls between data points, the lineshape can be distorted or may even disappear! [16] Figure 4-5 shows how the appearance of both absorption and dispersion lineshapes can be severely distorted if digital resolution is poor. In such cases it is highly desirable to increase the acquisition time to at least 2 or 3 times x, even if it means the bandwidth must be narrowed in order to reduce the number of data points needed. Figures 4-6 and 4-7 illustrate the effect that increasing the time domain damping constant, x, at a constant experiment time T, Figure 4-5 Effect of the discrete digital representation of extremely narrow absorption and dispersion lineshapes in the case where the true centroid of the absorption peak is located at various distances between data points. Spectra were generated from 64-point time domain data sets consisting of a single cosine at various frequencies from 15.5 to 16. The data was not exponentially damped; that is, T /x = 0. A real FT was then done on each of these time domain data sets to generate the 64- point spectra in Figures A through F. The 32 data points on the left side of each spectrum are the real part (absorption) of the FT, and the 32 points on the right are the imaginary part (dispersion) of the FT. The same horizontal and vertical scaling was used for each spectrum. Especially note that with a frequency placing the absorption maximum and the center of the dispersion exactly halfway between data points, the absorption is not observed and the dispersion is maximized (Figure A). At the other extreme, where the frequency is centered on a data point, the absorption is maximized and dispersion is not observed (Figure F). Frequency values between these extremes give intermediate absorption and dispersion lineshapes (Figures B-E). Although these intermediate spectra give the appearance of being phase shifted, that is not the case. 83 84 A) Frequency = 15.5 B) Frequency = 15.6 C) Frequency = 15.7 D) Frequency = 15.8 ______E) Frequency = 15.9 -OOOOOOOOOOOOO^X F) Frequency = 16.0 Figure 4-5 Figure 4-6 Effect of time domain signal damping on the discrete digital representation of absorption and dispersion lineshapes in the case where the centroid of the absorption peak is exactly at a data point. Spectra were generated from a 64-point initial time domain data set consisting of a single cosine of frequency 16, exactly equal to Nyqui»t/2. The data was then exponentially damped with damping constants (T/x) ranging from 0 to 15. A real FT was then done on each of these differentially damped time domain data sets to generate the 64-point spectra in Figures A through I. The 32 data points on the left side of each spectrum are the real part (absorption) of the FT, and the 32 points on the right are the imaginary part (dispersion) of the FT. The vertical scaling was adjusted so that the maximum of each spectrum is the same relative height. Especially note that with no damping (Figure A) the digital resolution is such that the dispersion signal is not observed at all (because it falls completely between two points). Increased damping (Figures B through I) broadens both absorption and dispersion peaks so that they are better represented digitally. 85 86 A) T/x = 0 30000 -80000 B) T/x = 1 50000 -50000 C) T/x = 2 35000 -35000 D) T/x = 3 25000 -25000 E) T/x = 4 20000 -20000 Figure 4-6 87 Figure 4-6 (continued) F) T/x = 6 15000 -15000 G) T/x= 9 10000 -10000 H) T/x= 12 8000 -8000 I) T/x - 15 6500 -6500 Figure 4-7 Effect of time domain signal damping on the discrete digital representation of absorption and dispersion lineshapes in the case where the centroid of the absorption peak is exactly halfway between two data points. Spectra were generated from a 64-point initial time domain data set consisting of a single cosine of frequency 15.5 (half a point from Nyquist/2). The data was then exponentially damped with damping constants (T/x) ranging from 0 to 15. A real FT was then done on each of these differentially damped time domain data sets to generate the 64-point spectra in Figures A through I. The 32 data points on the left side of each spectrum are the real part (absorption) of the FT, and the 32 points on the right are the imaginary part (dispersion) of the FT. The vertical scaling was adjusted so that the maximum of each spectrum is the same relative height. Especially note that with no damping (Figure A) the digital resolution is such that the absorption signal is not observed at all (because it falls completely between two points). Increased damping (Figures B through I) broadens both absorption and dispersion peaks so that they are better represented digitally. 88 89 A) T/x= 0 50000 -50000 B) T/x = 1 30000 -30000 C) T/x= 2 20000 -20000 D) T/x = 3 15000 -15000 E) T/x= 4 15000 -15000 Figure 4-7 90 Figure 4-7 (continued) F) T/x= 6 12000 -12000 G) T/x= 9 10000 -10000 H) T/x= 12 7500 -7500 I) T/x = 15 6500 -6500 91 has on the frequency domain lineshapes. Since the ratio T/x determines the lineshape, an equivalent result would be obtained by increasing the acquisition time T at a constant x (although the increased digital resolution from a longer T means that resolution would remain the same, in spite of the apparent digital "broadening" of the lineshape). It can be seen that acquisition times of T s: 4x (T/x 2: 4) provide the best defined lineshapes. if it is not possible or desirable to increase the acquisition time, then apodization of the signal or use of the magnitude mode spectrum can compensate for poor digital resolution by broadening the signal sufficiently so that the lineshape is better defined. Apodization functions also reduce the sine function side lobes that are present when the acquisition time T is less than 1 or 2x [17-18], However, these last methods can also reduce the obtainable mass resolution. A better method of handling poor digital resolution, although more involved, involves interpolating the (known) true continuous analog peak shape using at least three discrete data points [19]. Zero-filling, or the addition of (2n -1 )N zeros to the end of an Ap point time domain data set, increases the apparent acquisition time and thus the digital resolution [20]. The first zero filling, in addition to doubling the digital resolution, provides a yj2 improvement in absorption mode signal-to-noise ratio by effectively combining the information of the absorption and dispersion spectra. Subsequent zero fillings only increase digital resolution by interpolating, and actually decrease the observable resolution by distorting the lineshape [21-22]. Other drawbacks to zero-filling are the increased computation time for FFT and the larger data storage requirements. 92 The Hartley Transform It is taken for granted by most Fourier transform spectroscopists that the Fourier transform is the preferred method for obtaining a frequency-domain spectrum from a discrete time-domain data set. However, the standard (mathematically) complex FFT algorithm is not well suited for treating the linearly-polarized time-domain signals acquired in ICR mass spectrometry that are represented by mathematically real data. Although the mathematically imaginaiy 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. The iV-point complex FFT requires both N real and N complex data points of input. It is possible to calculate the transforms of two independent (real) iV-point data sets simultaneously by treating one of them as the imaginary component. The complex FFT of such data is accompanied by considerable post-transform shuffling of data to recover the two independent spectra [23]. By exploiting various recursion relations between elements of the discrete complex Fourier code matrix, an IV-point real data set can be broken up into two iV/2-point data sets. This is accomplished by first "unshuffling" the original data into two halves consisting of the even- and odd-numbered original data points. These halves can then be simultaneously processed by a N/2-point complex FFT. FT symmetry and recursion relations can then be used to reconstitute the desired spectrum (3). While this "real FFT" method allows mathematically real data to be processed approximately twice as 93 fast by the complex FFT, the fundamental algebra and algorithm still employ mathematically complex variables. However, if the initial time-domain data set is inherently real, it should not be necessary to resort to a transform employing complex notation. In 1942, R. V. L. Hartley, a radio engineer, first published a time-to-frequency transform employing only mathematically real notation [24]. Unfortunately, the Hartley transform was little used and the technique lay virtually dormant until the 1980's when Bracewell worked out a fast algorithm (the FHT) for its discrete representation [25]. Because the Hartley transform is designed for mathematically real functions, it is inherently twice as fast as 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, it remains nevertheless conceptually simpler because it does not require the use of complex variables. One nice advantage the Hartley transform (and FHT program) does offer is that, unlike the Fourier transform, it is its own inverse. Thus, when the FHT is used in convolution programs or similar work where data is moved back and forth between time and frequency domains, there is no need to keep track of which version (forward or inverse) is needed to jump back to the other domain. The Hartley transform is based on the cas function cas(0) = cos(0)+sin(0) = >/2cos(0-|) = ^/2sin(0+^j (4-12) Figure 4-8 compares the cos, sin, and cas functions. In the Fourier transform, the real and complex axes serve to separate the (real) cosine 94 COS(0) SIN(8) CAS(0) = COS(0) + SIN(0) = V2COS(0-J) = ^ S IN (0 + t) FIGURE 4-8 The CAS function is simply the sum of the COS function and the SIN function. 95 and (imaginary) sine components of the transform. However, this separation is unnecessary, since the cosine and sine functions are inherently orthogonal (that is. the sum of a cosine and sine can always be unambiguously restored to the separate components). Therefore, the same real spectrum can be obtained without resorting to complex notation using the Hartley transform. The Hartley transform and its inverse are given by + 00 +00 H(v) = Jf(t) [cos(2jzvt) + sin(2jtvt)] dt = J/( t) cas(2jtv£) d t (4-13a) -00 -00 + 00 +00 fit) = J H[v) [ c o s ( 2 j i v £) + sin(2jrv£)l dv = J H(v) cas(2;wt) dv (4- 13b) It should be noted that, unlike the Fourier transform, the Hartley transform and its inverse are identical. The discrete Hartley transform is given by 1 N N H(vn) = cas(2nvt/iV) , m = , . . . , ^ (4-14) n=0 The inverse transform is identical (just H(v) and/(£) are reversed). The arbitrary normalization (scaling) factor \/-\[N is chosen simply to make both the forward and inverse Hartley transform identical, as opposed to the somewhat conventional use of 1 and 1/N as normalization factors for the respective forward and inverse Fourier transforms. 96 Further insight into the nature of the Hartley transform can be gained by comparison to the Fourier transform. If the FT is considered to give the projection of a frequency-domain vector (whose length is the magnitude and whose angle is the phase) onto two orthogonal (cosine and sine) axes, then the Hartley transform is the projection of that same vector onto a single axis (the cas axis) bisecting the sine and cosine axes. Then, if the cas function of the Hartley transform is expressed in terms of a phase-shifted cosine (Equation 4-12), then the Hartley transform (Equation 4-13) is recognized as a time-shifted Fourier Transform [26]: + 00 +00 H(v) = J*/(t) cas(2jtvt) dt = J/U) cos(2irvt-(jr/4)) dt (4-15) -00 -00 For mathematically real time domain data, the Hartley frequency spectrum can be thought of as the sum of the Fourier absorption and dispersion spectra, and contains the same information as the complex FT spectrum (Figure 4-9). Both the absorption and the dispersion components of the complex FT can easily be recovered as the even and odd components, E(v) and O(v), of the Hartley spectrum + 00 J/(t) cos(2jtvt) dt (4 -16a) -00 97 Time Domain Signal (FID) l01i time — ► FHT Inverse Fourier Spectrum FHT (Real) Hartley Spectrum Even FFT-to-FHT Part Fourier Spectrum (Imaginary) FHT-to-FFT Odd Part f FHT FFT ' Magnitude Magnitude Mode Mode Magnitude Mode Spectrum Figure 4-9 Fourier and Hartley transformations of an exponentially damped sinusoidal signal of the type acquired in FT/ICR. Note that except for magnitude mode calculations (which lose phase information), the transformations are fully reversible. Thus the time domain signal, the Fourier real and imaginary spectra, and the Hartley spectrum contain equivalent information. 98 + 00 H(v) - H(-v) . 0(v) = ------2 ------= / sin(2jrvt) dt (4-16b) It is also possible to calculate the magnitude and phase spectra, M(v) and H(v)2 + H(-v)2 M(v) = (4-17) -O(v) H(-v) - H(v) When processing a mathematically real iV-point time domain data set, the complex FFT will take about twice as long as a (real) FHT, assuming both programs are similarly optimized (e.g., precalculation of trigonometric identities). However, a "real" FFT (discussed above) which is based upon a N/ 2-point complex FFT will run in about half the time, and is thus roughly equivalent in speed to the FHT. Many additional aspects of the Hartley transform are presented in detail by Bracewell [27]. Since the vast majority of FT/ICR time domain signals consist purely of mathematically real data (with the less commonly used quadrature detection being the main exception), an investigation was made as to the suitability of using the Hartley transform as a replacement for the "real" Fourier transform. The speed of the FHT (relative to the "real" FFT) and its precision in handling both simulated 99 and experimental data were examined. Some of these results have previously been reported [28]. Experimental Experimental time-domain ICR signals (1K-8K integer data points) of electron-ionized (70eV) air (pressure s 10'8 Torr) were acquired in both heterodyne and direct mode by use of a Nicolet FTMS-2000 spectrometer. The data was transferred for analysis to an Atari Mega-4 (8MHz 68000 cpu, 4 megabytes directly-addressable RAM; equivalent in speed and memory to an Apple Macintosh SE). Programs for handling both the experimental and simulated data were written and compiled using GFA Basic v2.0, a highly optimized language which allows arrays to be as large as is permitted by available memory, and which uses an extended 6-byte real data with 10-11 digits precision (as opposed to the standard 4-byte single-precision numbers with only 7 digits precision). Simulated data (exponentially damped cosines) ranged from IK to 16K. Because the Basic algorithm for the FHT listed in reference 27 contains several minor (but very important!) transcriptional errors, the corrected source code version used on the Atari ST is listed in the Appendix to this chapter. The real and complex FFT programs used were written based on the programs listed in reference 23. 100 Results and Discussion Computer Implementation Fully tested source code of the Hartley transform (and related routines to convert to the Hartley spectrum to the complex Fourier and magnitude spectra) for several commonly used computer systems are listed in the appendix. The programs are based on listings in reference 27, with corrections made based on conversation with the author. Appendix 4-1 contains integer Fortran routines for the Nicolet 1280 processor. Appendix 4-2 contains single-precision Microsoft Quick Basic routines for the Apple Macintosh. Appendix 4-3 contains single precision PowerBasic 2.0 (formerly TurboBasic) routines for IBM PC compatible computers. Appendix 4-4 contains single-precision GFA Basic 2.0 routines for Atari ST computer. To avoid the transcription errors typically found in published computer listings (such as were found after much frustration and wasted time in the two previously published FHT listings examined), the tested source code from each machine was transferred in ascii format to a PC by either modem or disk file conversion, and was incorporated directly into this document. The actual stages in the operation of the fast Hartley transform can be seen in Figure 4-10. A 512 point simulated noiseless time domain data set consisting of a single damped frequency was graphed after each of the main stages (8 stages in this case, since 28 = 512) of the FHT transform, illustrating the progression through various intermediate stages to the final Hartley spectrum. FIGURE 4-10 Graphs of a 512 point data set after each of the main stages (8 stages in this case, since 28 = 512) of the FHT transform. Starting with a simulated noiseless time domain data set consisting of a single frequency (A), the intermediate stages (B-I) can be seen to progress toward the final Hartley spectrum (J). 101 102 A) Initial time domain data. B) After the Permuting step. C) After Stage 1. D) After Stage 2. E) After Stage 3. Figure 4-10 103 Figure 4-10 (continued) A A A h ^ ’\j -\JI \ f Y \ T Y \ n 1 F) After Stage 4. G) After Stage 5. H) After Stage 6. I) After Stage 7. J) After Stage 8. FHT completed. 104 Spectral Display Modes The analytic forms of the various lineshapes resulting from Hartley or Fourier transformation of a time domain sinusoid of infinite duration (Equation 4-1) and their relative peak widths are listed in Table 4-1. The Hartley frequency domain peak, as shown in Figure 4-9, is about 2.4 times wider than a phase-corrected absorption mode Lorentzian line shape. As a result, one would not normally display the raw Hartley spectrum, even if phase-corrected. Fortunately, the FT real and imaginary spectra can be simply computed from the Hartley spectrum using Equations 4-15a and b (also see the appropriate programs in the appendix). Likewise, the magnitude mode spectrum generally used in FT/ICR spectrometry [29] can also readily be calculated directly from the Hartley spectrum using Equation 4-16 (also see the programs in the appendix). FT/ICR absorption mode mass spectra from a heterodyne mode 8K integer time-domain signal of electron-ionized air (which was phase corrected by adjustment of the heterodyne mixing frequency) were obtained by both Fourier and Hartley methods employing calculations with real numbers (see Figure 4-11). Visually the two spectra are indistinguishable. Similarly indistinguishable spectra were obtained when using the magnitude mode. Computational Speed Comparisons Comparisons of the execution speed of optimized radix-2 FHT and FFT programs for various size data sets are shown in Table 4-2. The veiy slight speed advantage of the FHT program over the real FFT 105 Table 4-1 Comparison of the various frequency-domain line shapes obtained by Fourier or Hartley transformation of an exponentially damped infinite-duration time-domain sinusoidal signal (Equation 4-1). Transform Frequency-domain Spectrum Peak Width3 FFT Absorption-mode 2 (cosine transform) A(0j) = [l+( W0 - ( 0 ) 2 x 2 1 X FFT Dispersion-mode ( co0 - CO h2 2 (sine transform) D M — r 1 / ^9 9 1 [ 1 +( 0)0 - CO Vx2 ] X Hartley-mode X + ( (On - CO )x 2 2(1 +yl2) 4.828 ("cas" transform) [ 1 +( coC) - CO )2X2 ] X X Magnitude Mode 2\/3 3.464 (Hartley or Fourier) M(co) = ,------1 + ( COQ - CO )1T2 X ~ X a Peak width denotes full width at half-maximum peak height, except for the dispersion mode, where it is the peak-to-peak separation between maximum and minimum amplitudes. 106 Absorption Spectrum (from the "real" FFT) Absorption Spectrum (derived from the FHT) FIGURE 4-11 An experimental FTMS time domain signal of N2+ ions (from air) at 3 tesla (8192 points, heterodyne mode, phase corrected by adjustment of the carrier frequency) was transformed with both the "real" fast Fourier transform and the fast Hartley transform (with subsequent conversion to the FT absorption spectral mode). Although the initial FTMS data was in integer format, all calculations were done using real numbers, with the final results rounded back to integers. The same 100 points surrounding the N2+ absorption peak are plotted for both the "real" FFT spectrum (TOP) and the FFT spectrum derived from the FHT (BOTTOM). At the absorption peak the relative difference between the two spectral values was only 4 parts in 31000 (0.013%). Almost 99% of the other corresponding (integer) data points were identical. 107 Table 4-2 Computational Times (in seconds) for Corresponding Transforms of Three Arbitrarily Different FID Time-Domain signals (A, B, and C) of D ata Sizes of up to 16Ka Computational Time (seconds) "Real" FFT FHT Number of Time- Domain Points A B C AB C 128 0.37 0.38 0.37 0.35 0.35 0.35 256 0.79 0.78 0.78 0.75 0.74 0.74 5 12 1.70 1.66 1.66 1.60 1.60 1.61 1024 3.61 3.55 3.54 3.46 3.47 3.46 20 4 8 7.70 7.59 7.58 7.45 7.49 7.47 4096 16.38 16.16 16.13 15.98 16.11 16.08 8192 34.75 34.32 34.27 34.23 34.54 34.49 16384 73.51 72.66 72.59 72.89 73.62 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 6-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 essentially the same. 108 program Is most likely due to differences in the calculation of trigonometric identities, which were handled differently by the two programs. In any case, both programs ran at essentially the same speed for the largest size (16K) data set. Although radix-2 implementations of the Hartley transform were employed in this work due to their programming simplicity and general applicability to any 2N data set, it is also possible to construct Hartley transforms of other radices (4, 8, etc.) that would be more efficient in transforming data sets of less general size (4N, 8N, etc.), as is also the case for the Fourier transform. It is also useful to note that it is not difficult to interconvert between FFT and FHT algorithms [30]. Thus any optimization "tricks" that may be found in the future for either method can be readily applied to the other [31]. Precision of the FHT Mere visual similarity between the experimental FHT spectrum and FFT spectrum (Figure 4-11) is not sufficient proof of the precision of the FHT computer algorithm. A examination of the integer data of the two spectra showed that at the absorption maximum, the relative difference between the two FHT and FFT spectral values was only 4 parts in 31,000 (0.013%). Almost 99% of the other corresponding (integer) 8K data points were identical, with the few non-identical data differing on average by only one part in several thousand. This indicates that transformation of integer data by both the FHT and FFT programs (using floating-point rather than integer arithmetic calculations) produces spectra, when rounded back to integer format, are indeed 109 virtually identical. Even slight variations in the calculation of trigonometric identities (which was handled differently by the routines) could account for the tiny discrepancies. A better check on the precision of the FHT is to use floating point data in comparison with an accepted FFT routine. The same 256-point simulated time domain data set (consisting of a single damped cosine plus added noise) was transformed using both the "real" fast Fourier transform [23] and the fast Hartley transform (with subsequent conversion to the FT real/imaginary spectral mode). Figure 4-12 shows the two spectra (real data on left half, imaginary on right). A point by point comparison of the two spectra showed the average (absolute value) relative difference in values to be just one part in 400 (0.265%). At the main absorption peak the relative difference between the two values was even less (0.000838%). As with the integer data, the differences for floating point data are minute enough to be ignored for even the most precise FT /ICR experimental work being done today. Because the Hartley transform is its own inverse, one final check on its precision can be done by doing successive FHT's to see if the original data are recovered exactly. Figure 4-13 illustrates the test employed on an initial 1024-point simulated time domain consisting of the sum of three damped cosines. The data and calculations employed floating-point numbers. Table 4-3 shows the differences (the average of the absolute values of the fractional differences) for corresponding data sets after either two or four successive FHT's. For the time domain data after two successive FHT's, nearly all the corresponding data points agreed to between eight or nine significant digits. Since the calculations FIGURE 4-12 The same 256 point simulated time domain data set (one single damped cosine plus added noise) (top) was transformed using both the "rear fast Fourier transform (middle) (based on the complex FFT program FOUR1 in Reference 23, pages 394-395) and the fast Hartley transform with subsequent conversion to the FT spectral mode (bottom). For both the "real" FFT and the Hartley- derived FFT spectra, the 128 points on the left are the real data and the 128 on the right are the imaginary data. A point by point comparison of the two spectra showed the average (absolute value) relative difference in values to be only 0.265%. At the main absorption peak the relative difference between the two values was even less (0.000838%). 110 I l l Initial Time Domain Data Real FFT Spectrum FT (from FHT) Spectrum Figure 4-12 112 A) 1 FHT B) j FHT C) -11111 lly 11^ I FHT D) | FHT E) Figure 4-13 Since the FHT is its own inverse, two (or four) successive applications of the FHT should recover the original waveform exactly. (A) Initial 1024 point time domain consisting of the sum of three damped cosines. (B) After one FHT. (C) After two successive FHT's. (D) After three successive FHT's. (E) After four successive FHT's. 113 Table 4-3 Average Absolute-Value Fractional Differences Between Corresponding Data Points after Two or Four Successive FHT's3 Initial Final Number of Transformation Average Absolute Data Set3 Data Set3 FHT's Typeb Fractional Difference0-*1 A C 2 T=>F=>T 1.16135E-08 CE 2 T=>F=>T 1.16135E-08 A E 4 t =>f =>t =*f =>t 2.33712E-08 B D 2 f =>t =*f 6.26801E-09 3 Data sets used are shown in Figure 4-13. All data sets were IK floating point numbers. The initial time domain data set contained three exponentially-damped cosines of different amplitudes plus Gaussian-distributed random noise with an average magnitude of 5% of the maximum time domain value. b Transformation type refers to the format of the initial, intermediate, and final data set. "T" indicates time domain, and "F" indicates frequency domain. "T=>F=»T" refers to an initial time domain going to the frequency domain and ending back in the time domain. ° The absolute average value of the fractional differences was calculated as follows. The difference between two corresponding data points was divided by the value of the first of those points to get the fractional difference. The absolute values of these fractional differences were then averaged. d After two successive FHT's nearly all the corresponding data points agreed to between 8-9 significant digits. Since the FHT calculations were carried out using 6-byte floating point numbers with 10 digits of precision, it appears that approximately one significant digit is lost for each pair of successive FHT's. This effect is cumulative. 114 were carried out using 6-byte floating point numbers with about 10 digits of precision, it appears that approximately one significant digit is lost for each pair of FHT's. This effect is cumulative, as can be seen in the loss of about two significant digits if four successive FHT's are done. When going from the frequency domain through two FHT's back to the frequency domain, the average differences appear a little smaller, possibly because the majority of the data is baseline with relatively small magnitude. Again, the precision shown in these tests of the FHT is more than sufficient for typical FT/ICR work. 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Academic Press: New York. 1973, p2. 27. Bracewell, R.N. The Hartley Transform. Oxford University Press: New York. 1986. 28. Williams, C. P.; M arshall, A. G. Anal. Chem. 1989, 61, 428. 29. Craig, E. C.; Santos, I.; Marshall, A. G. Rapid Commun. Mass Spectrom. 1987, 1, 33. 30. Bunem an, O. SIAM J. Sci. Stat. Comput. 1986, 7, 624-638. 31. Nussbaumer, H.J. Fast Fourier Transform and Convolution Algorithms. Springer-Verlag: New York. 1982. 117 APPENDICES APPENDIX 4-1 NICOLET (EXTREL) 1280 COMPUTER - FORTRAN CODE Linker Instructions for the FHT Program (FHT.LOD): :E ROOT.EXT[FTMS]:J FTMS. EXT[ FTMS] : J :B=1000-1400000 FHT.REL:L FHTSUB.REL:L TABLES.REL:L GETREGS. REL[ GRT] : L DATESTAMP. REL[ FTMS] : L FORRUN. L IB [ HLIB] : R FTMUTL. L IB [ HLIB] :R FCOM.LIB[HLIB]:R TIOS1 6 . L IB [ HLIB ] :R NICSYS. L IB [ HLIB] :R /-TT:U FHT/FMSFOR.FHT:S /FHT.MAP:M : Q FHT.ASC c c c FHT Fast Hartley Transform c c C Chris Williams C 2 / 3 / 8 8 C This routine does a discrete Fast Hartley C Transform (FHT) on the NDP data points in the C active SDATA array of the FTMS program. C This FORTRAN program can be compiled on the C Nicolet 1280 computer with the NICOS command: C FORTRAN FHT.ASC:C 118 C After compiling this routine and all its C subroutines, they can be linked into an C executable FTMS routine with the NICOS command: RELOAD FHT. LOD: X C After compilation and linking, the executable C p rogram FMSFOR.FHT ca n be in v o k e d from t h e FTMS C program to process the active spectrum by typing C t h e 3 - l e t t e r command "FHT". • C Subroutines called: FHTSUB.ASC, TABLES.ASC C For linker instructions, see FHT.LOD. PROGRAM FHT C COMMON PARAMETERS INTEGER SDATA(65 5 3 5 ) COMMON / SDATA/ SDATA INTEGER NDP, NTP, KSPECT, FFTNRM COMMON /D E F SIZ / NDP COMMON /NTP/NTP COMMON /KSPECT/ KSPECT COMMON /FFTNRM/FFTNRM COMMON /BWIDTH/BWIDTH REAL LOFREQ, HIFREQ,REFFRQ COMMON /LOFREQ/ LOFREQ COMMON /HIFREQ/ HIFREQ COMMON /REFFRQ/ REFFRQ COMMON /HZPPT/ HZPPT REAL LFFREQ, RTFREQ COMMON /LFFREQ/ LFFREQ COMMON /RTFREQ/ RTFREQ C Allow for NDP/4 sines and tangents REAL SIN E ( 8 1 9 2 ) , TANGENT(8 1 9 2 ) COMMON /S IN E / SINE COMMON /TANGENT/ TANGENT C Powers of 2 up to NDP INTEGER POWER( 2 0 ) , EXPONENT COMMON /POWER/ POWER COMMON /EXPONENT/ EXPONENT C ======*: C Program Mainline C ======: C Do the FHT on the first NDP points of SDATA CALL FHTSUB(NDP) 119 C Redraw the Window CALL FCOMNO ( ’NM X ',*900) GOTO 999 C ------C Error Statements C ------ 900 CONTINUE PRINT * , 'ERROR IN FCOM. ’ CALL FCOMNO ( 'E R R ',*999) 999 CONTINUE CALL EXIT END FHTSUB.ASC c c c c c c Chris Williams c 2 / 3 / 8 8 C This routine is invoked by the mainline routine C FHT.ASC C Subroutines called: TABLES.ASC C This routine carries out the Fast Hartley C Transform (FHT). SUBROUTINE FHTSUB(NDP) C COMMON PARAMETERS C ------INTEGER SDATA(6553 5) COMMON /SDATA/ SDATA C Allow for NDP/4 sines and tangents REAL SIN E( 8 1 9 2 ) , TANGENT(8 1 9 2 ) COMMON /S IN E / SINE Set u structure table l b a t e r u t c u r t s up t e S C oo on oo o non oo NDP o t up 2 f o ers Pow C PROGRAM PARAMETERS PARAMETERSCLOCK TR O PROGRAM OF START A( 1) =1) A( 1 = 0) A( 0 Q=Q+MOD( ) EXPONENT,2 Q=EXPONENT/2 N9=2**(EXPONENT-2) C2=POWER(Q) C6=EXPONENT-1 C5=N-1 N=4*N9 .) PI=4*ATAN(1 ers pow CLOCK) and TIME1=CTIME( , s t n e g n a t , s e n i s et G D to transform r o f s n a r t o t NDP r te u erm P t s a F (NDP) TABLES CALL CSTART(CLOCK) CALL CRESET(CLOCK) CALL EL ,PI H, REAL s e l b a i r a v ry ra o p tem , e t a n i d r o l l e C EXPONENT /EXPONENT/ POWER /POWER/ COMMON COMMON O M N TNET TANGENT /TANGENT/ COMMON E L IE TIME2 , TIME1 REAL NEE D, I, K, N, Q, U, ,Y ,X ,U ,S 9 ,Q ,C 9 8 ,N ,C 7 V9 ,N ,C , 6 8 ,L V ,C , 5 ,K 7 V ,C ,J , 4 6 ,I V ,C 3 , ,E 5 ,C D 2 ,V 4 ,C 1 ,V 3 ,C 0 ,V C 2 ,V 1 ,V 0 V INTEGER ) 6 5 (2 A INTEGER INTEGER INTEGER NDP INTEGER NEE* CLOCK INTEGER*2 EXPONENT , ) 0 POWER( 2 INTEGER 120 121 DO 100 1=2,Q DO 110 J=0,POWER(1-1)-1 A(J)=2*A(J) A(J+POWER(1 -1 ))=A(J)+l 110 CONTINUE 100 CONTINUE C------______C P erm u te DO 200 1=1, C2-1 V 4= C 2*A (I) V5=I V6=V4 V7=SDATA(V5) SDATA(V5)=SDATA(V6) SDATA(V6)=V7 DO 210 J=1,A (I)-1 V5=V5+C2 V6=V4+A(J) V7=SDATA(V5) SDATA(V5)=SDATA(V6) SDATA(V6)=V7 210 CONTINUE 200 CONTINUE C - C First Stages C Two E lem en t DHT DO 300 1=0,N-2,2 V6=SDATA(I)+SDATA(1+1) V7=SDATA(I)-SDATA(1+1) SDATA( I )=V6 SDATA(1+1)=V7 300 CONTINUE C Four Element DHT DO 310 1=0, N-4,4 V6=SDATA(I )+SDATA(1+2) V7=SDATA(1+1)+SDATA(1+3) V8=SDATA(I)-SDATA(1+2) V9=SDATA(1+1)-SDATA(1+3) SDATA( I )=V6 SDATA(1+1)=V7 SDATA( 1 + 2 )=V8 SDATA(1 + 3 )=V9 310 CONTINUE C------C Last Stages (3,4,5...) 122 U = C 6 S=4 DO 400 L=2,C6 V2=2*S U = U -1 V3=POWER(U-l) DO 410 Q=0,C5,V2 I=Q D=I+S V6=SDATA(I )+SDATA(D) V7=SDATA(I)-SDATA(D) SDATA(I) =V6 SDATA(D)=V7 K=D-1 DO 411 J=V3,N9,V3 1= 1+1 D=I+S E=K+S V9=SDATA(D)+ (SDATA(E)*TANGENT(J ) ) X=SDATA(E)-(V9*SINE(J ) ) Y =( X * TANGENT( J ) ) +V9 V 6 = S DATA( I )+Y V7=SDATA(I )-Y V8=SDATA(K)-X V9=SDATA(K)+X SDATA( I ) =V6 SDATA(D)=V7 SDATA(K) =V8 SDATA(E)=V9 K=K-1 411 CONTINUE E=K+S 410 CONTINUE S=V2 400 CONTINUE TIME2=CTIME(CLOCK) PRINT * , ' ’ PRINT *,'Fast Hartley Transform' * 1 PRINT 9 NDP : ' r NDP PRINT *, ' Lookup Table ( s e c ): ' , TIMEl TIME1=:TIME2-TIME1 PRINT *, ' FHT Time ( s e c ): ' , TIMEl PRINT * , ' T o t a l Time ( s e c ): ' , TIME2 PRINT *,'' RETURN END TABLES. TABLES. ASC nn n o o ooo nnnoooonnnooo COMMON PARAMETERS TR O PROGRAM OF START n Po r Tabl ed in FHTSUB.ASC n i d se u e s in le t b u a o T r b u s ers ow P e h t and by d e k o v in s i e n i t u o r is h T i none : d e l l a c s e tin u o r b u S I4AA( .) PI=4*ATAN(1 EL PI REAL VO H, REAL l D/ sines ad s t n e g n a t and s e n i s NDP/4 r o f w llo A e ar s e l b a i r a v ry ra o p tem , e t a n i d r o l l e C m r fo s n a tr o t NDP o r of 2 p NDP o t up 2 f o ers Pow TANGENTCOMMON /TANGENT/ O M N EPNN/ EXPONENTCOMMON /EXPONENT/ POWERCOMMON /POWER/ EL NE( TNET8192) 1 SINE ,TANGENT(8 ) 2 / 9 E COMMON 1 IN (8 /S E IN S REAL s routine generates the Sine, t, n e g n a T , e n i S e h t s e t a r e n e g e n i t u o r is h T NEE C0, C2, C4, C6, C8, 9 ,C 8 ,C 7 ,C 6 ,C 5 K , ,C 4 J ,C , 3 ,C I 2 INTEGER ,C 1 ,C V4 0 , C 3 INTEGER ,V 2 V , I V INTEGER s Wilams m illia W is r h C NEE PWR EXPONENT , ) 0 POWER( 2 INTEGER NEE NDP INTEGER 8 8 / 3 / 2 URUIE ALS (NDP) TABLES SUBROUTINE HSB ASC FHTSUB. TABLES 123 124 Get Exponents EXPONENT=0 I=NDP WHILE ( I .GT. 1 ) DO EXPONENT=EXPONENT+l 1=1/2 ENDWHILE C Get Powers of 2 DO 200 1 = 0 ,EXPONENT POWER(I )=2 * *I 200 CONTINUE C Get S in es N9=2**(EXPONENT-2) SINE(N9)=1. DO 300 1=1,3 SINE((I*N9)/4 )=SIN(FLOAT(I )*Pl/8.) 300 CONTINUE H=0.5/COS(PI/16.) C Fill Sine table C4=EXPONENT-4 DO 310 1 = 1 ,EXPONENT-4 C4=C4-1 V0 = 0. DO 320 J=POWER(C4) , ( N9-POWER(C4) ) , POWER(C4+1) Vl=J+POWER(C4) SINE(J)=H*(SINE(VI)+V0) V0=SINE(VI) 320 CONTINUE C Half secant Recursion H=1. / SQRT( 2 . + 1 . /H) 310 CONTINUE C Get Tangents C0=N9-1 DO 400 1=1,N9-1 TANGENT(I )=(1.-SINE(CO))/SINE(I ) C0=C0-1 400 CONTINUE TANGENT(N9)=1 RETURN END 125 Linker Instructions for the HMG Program (HMG.LOD): :E ROOT.EXT[FTMS]:J FTMS.EXT[FTMS]:J :B=1000-1400000 HMG.REL:L GETREGS. REL[GRT] : L DATESTAMP. REL[FTMS] : L FORRUN.LIB[HLIB]:R FTMUTL.LIB[HLIB] : R FCOM.LIB[HLIB]:R TIOS16.LIB[HLIB] :R NICSYS.LIB[HLIB] :R /-TT:U HMG/FMSFOR. HMG:S /HMG.MAP:M :Q HMG.ASC c c c c c c Chris Williams c 8 /9 /8 8 C This routine converts the Hartley spectrum C generated by the FHT program into a C Magnitude mode spectrum. C Linker instructions are in HMG.LOD. PROGRAM HMG C COMMON PARAMETERS INTEGER SDATA(65535) COMMON /SDATA/ SDATA INTEGER NDP COMMON /DEFSIZE/ NDP REAL T l , T2 INTEGER I,N2 126 C Program N2=NDP/2 DO 100 I=1,N2 Tl=FLOAT(SDATA(N2-I))*FLOAT(SDATA(N2-I)) T2=Tl+(FLOAT(SDATA(N2+I))*FLOAT(SDATA(N2+I))) Tl=T2/2 T2=SQRT(T1) SDATA(N2+I)=T2 100 CONTINUE DO 200 1=1,N2 SDATA(I)=SDATA(N2+I) 200 CONTINUE CALL EXIT END APPENDIX 4-2 MACINTOSH - MICROSOFT QUICK BASIC CODE FHT Program ______ 'FHT - Fast Hartley Transform 'Macintosh Microsoft QuickBasic Version for Single ' .P r e c is io n Real Number Data 'This FHT routine was coded based on the routine ' FHTSUB on page 118 of "The Hartley Transform” by ' Ronald N. Bracewell. Corrections to typographical '..errors in this reference have been made based on a ' call made to the author. Pi=4 *ATN(1 .0 ) OPTION BASE 0 ' DATA INPUT: Ndp%, S.Data(Ndp%) ' Replace the following sample data generation routine ' with a routine to generate or load the desired data. 'Generate Sample Data Ndp%=1024 'Number o f Data P o in ts DIM S.Data(Ndp%) FOR 1=0 TO Ndp%-1 127 S.Data(I)=1000*COS(Pi/3+100*I*Pi/CSNG(Ndp%)) NEXT I Power%=0 'Power of 2 for Ndp I%=Ndp% WHILE I%>1 Power%=Power%+l I%=I%/2 WEND N9=2A(Power%-2) N=4*N9 C5=N-1 C6=Power%-l DIM M(Power%) 'Get Powers of 2 FOR I%=0 TO Power% M(I%)=2AI% NEXT 1% DIM S(Ndp%/4) 'Get S in es S(N9)=1 FOR 1=1 TO 3 S(I*N9/4)=SIN(I*Pi/8) NEXT I H=. 5 / COS(P i / 1 6) ' Initial half secant C4=Power%-4 ' Fill sine table FOR I%=1 TO Power%-4 C4=C4-1 V0=0 FOR J%=M(C4) TO (N9-M(C4 ) STEP M(C4+1) V1=J%+M(C4) S(J%)=H*(S(V1)+V0) V0=S(V1) NEXT J% H=1/SQR(2+1/H) ' Half secant recursion NEXT 1% DIM T (Ndp%/4 ) 'Get Tangents C0=N9-1 FOR I%=1 TO N9-1 T(I%)=(1-S(C0))/S(I%) C0=C0-1 NEXT 1% T(N9)=1 Q%=Power%/2 'F a st Permute C2=M(Q%) Q%=Q%+(Power% MOD 2) DIM A (1024) 'Cell ordinate A(l)=l FOR I%=2 TO Q% 128 FOR J%=0 TO M(I%-1)-1 A(J%)=2*A(J%) A(J%+M(I%-1) )=A(J%)+l NEXT J% NEXT 1% FOR I%=1 TO C2-1 'Permute V4=C2 *A(1%) V5=I% V6=V4 V7=S.Data(V5) S.Data(V5)=S.Data(V6) S.Data(V6)=V7 FOR J%=1 TO A (1%)-1 V5=V5+C2 V6=V4+A(J%) V7=S.Data(V5) S.Data(V5)=S.Data(V6) S.Data(V6)=V7 NEXT J% NEXT 1% FOR I%=0 TO n-2 STEP 2 'First Stages (1 and 2) V6=S.Data(1%)+S.Data(I%+1) ' 2-Element DHT V7=S.Data(1%)-S .Data(I%+1) S . D a ta (1%)=V6 S.Data(I%+1)=V7 NEXT 1% FOR I%=0 TO n-4 STEP 4 ' 4-E lem ent DHT V6=S.Data(I%)+S.Data(I%+2) V7=S.Data(I%+1)+S.Data(I%+3) V8=S.Data(1%)-S .Data(I%+2) V9=S. Data(I% + 1)-S . Data(I%+3) S.Data(1%)=V6 S.Data(I%+1)=V7 S.Data(I%+2)=V8 S.Data(I%+3)=V9 NEXT 1% U%=C6 'Last Stages (3 and up) S%=4 FOR L%=2 TO C6 V2=2*S% U%=u%-1 V3=M(U%-1) FOR Q%=0 TO C5 STEP V2 I%=Q% D%=I%+S% V6=S.Data(1%)+S.Data(D%) V7=S.Data(1%)-S .Data(D%) S.Data(I%)=V6 S .Data(D%)=V7 129 K%=D%-1 FOR J%=V3 TO N9 STEP V3 I%=I%+1 D%=I%+S% E%=K%+S% V9=S.Data(D%)+(S .Data(E%)*T(J%)) X=S.Data(E%)-V9*S(J%) Y=X*T(J%)+V9 V6=S.Data(1%)+Y V7=S. D a ta (1%)-Y V8=S. D a ta (K%)-X V9=S.Data(K%)+X S .Data(1%)=V6 S . D a ta (D%)=V7 S . D a ta (K%)=V8 S . D a ta (E%)=V9 K%=K%-1 NEXT J% E%=K%+S% NEXT Q% S%=V2 NEXT L% Norm.factor=l/SQR(Ndp%) 'Normalize FOR I%=0 TO Ndp%-1 S.Data(1%)=Norm.factor*S.Data(1%) NEXT 1% END APPENDIX 4-3 IBM PC - POWER BASIC V2.0 FHT.BAS Program FHT - Fast Hartley Transform (for real Time Domain Data) IBM PC Ver. 1.04 Chris Williams 10/8/90 Written for PowerBasic Ver 2.0 (TurboBasic upgrade) This FHT routine was coded based on the routine FHTSUB on page 118 of "The Hartley Transform" by Ronald N. Bracewell, 1986. Numerous errors in that 130 ' published routine have been corrected. ' Initializations l DEFEXT A,Z ' INPUT DATA: NDP, Sdata(NDP) ' Load or generate data to be transformed in the ' array Sdata. Set NDP to the correct number of ' points to be transformed. Dim Sdata(NDP) Pi=3.14159265359 Twopi=2*Pi NDPold=-l 'First time flag t F h t : ' Fast Hartley Transform ' Input: SDATA(NDP) (real data) Print "*** Fast Hartley Transform in Progress ***" Stime=Timer 'Optional timer If NDPoNDPold then Print "Status: Initializing" ' Calculate Powers of 2 Power%=0 I%=NDP W hile I%>1 Power%=Power%+l I%=I%/2 Wend ' Define Arrays If NDPold=-l then Dim A (1024) ' Cell ordinate Dim S (1 ),T (1 ),M (1 ) ' Will be erased and End i f ' redimensioned later N9=2A(Power%-2) N=4*N9 C5=N-1 C6=Power%-l Gosub Getpowers Gosub Getsines Gosub Gettangents 131 Lookuptime=Int(Timer-Stime) NDPold=NDP End i f Stime=Timer Gosub Fastpermute Gosub Permute Gosub Firststages Gosub Laststages Gosub Normalize Fhttime=Int(Timer-Stime) 'Time used in seconds Optional Routines for processing the Hartley Spectrum now in Sdata (uncomment your choice) I) To go to m agnitude mode d i r e c t l y : Gosub Hartley2Magnitude II) Or go to magnitude mode via the FT spectrum: Gosub Hartley2FT 'This gives the FT Real & Imaginary Spectrum Gosub FT2Magnitude 'Converts FT->Magnitude End I G etpow ers: Erase M Dim M(Power%) For II%=0 To Power% M(11%)=2 A11% Next 11% Return G e t s in e s : Erase S Dim S(NDP/4) S(N9)=1 For II%=1 To 3 S(II%*N9/4)=Sin(II%*Pi/8) Next 11% H=0.5/Cos(Pi/16) 'Initial half secant ' Fill Sine table C4=Power%-4 For II%=1 To Power%-4 C4=C4-1 132 vo=o For JJ%=M(C4) To (N9-M(C4)) Step M(C4+1) V1=JJ%+M(C4) S(JJ%)=H*(S(V1)+V0) V0=S(V1) Next JJ% H=l/Sqr(2+1/H) ' Half secant recursion N ext 11% Return Gettangents: Erase T Dim T(NDP/4) C0=N9-1 For II%=1 To N9-1 T(II%)=(1-S(C0))/S(II%) C0=C0-1 N ext 11% T(N9)=1 Return Fastpermute: Q%=Int(Power%/2) C2=M(Q%) Q%=Q%+(Power% Mod 2) ' Set up structure table A (0)=0 A ( l ) = l For II%=2 To Q% For JJ%=0 To M (II% -1)—1 A(JJ%)=2*A(JJ%) A(JJ%+M( II% -1) ) =A(JJ%)+1 Next JJ% N ext 11% Return Perm ute: For II%=1 To C2-1 V4=C2*A(11%) V5=II% V6=V4 V7=Sdata(V5) Sdata(V5)=Sdata(V6) Sdata(V6)=V7 For JJ%=1 To A (11%)-1 V5=V5+C2 V6=V4+A(JJ%) V7=Sdata(V5) Sdata(V5)=Sdata(V6) 133 Sdata(V6)=V7 N ext JJ% Next 11% Return Firststages: ' Stages 1 and 2 ' 2-Element DHT For II%=0 To N-2 Step 2 V 6=Sdata(11%)+Sdata(II% +1) V7=Sdata(11%)-Sdata(II%+1) S d a ta (11%)=V6 Sdata(II%+1)=V7 Next 11% 1 4-Element DHT For II%=0 To N-4 Step 4 V6=Sdata(11%)+Sdata(II%+2) V7=Sdata(II%+1)+Sdata(II%+3) V8=Sdata(11%)-Sdata(II%+2) V9=Sdata(II%+1)-Sdata(II%+3) Sdata(11%)=V6 Sdata(II%+1)=V7 Sdata(II%+2)=V8 Sdata(II%+3)=V9 Next 11% Return Laststages: U%=C6 S%=4 For LL%=2 To C6 1 Stages 3/4/5/6/7... V2=2 *S% U%=U%-1 V3=M(U%-1) For Q%=0 To C5 Step V2 II%=Q% D%=II%+S% V 6=Sdata(11%)+Sdata(D%) V7=Sdata(11%)-Sdata(D%) Sdata(11%)=V6 Sdata(D%)=V7 KK%=D%-1 For JJ%=V3 To N9 Step V3 II%=II%+1 D% = II%+S % E%=KK%+S % V9=Sdata(D%)+(Sdata(E%)*T(JJ%)) 134 X =Sdata(E% )-V9*S( JJ%) Y=X*T(JJ%)+V9 V 6= S data(11%)+Y V 7= S data(11%)-Y V 8=Sdata(KK%)-X V9=Sdata(KK%)+X Sdata(11%)=V6 S d a ta (D%) =V7 S d a t a (KK%)=V8 Sdata(E% )=V9 KK%=KK%-1 Next JJ% E%=KK%+S% Next Q% S%=V2 Next LL% Return N o r m a liz e : ' Using this normalization factor will make the 1 forward and reverse FHT identical. Normfactor=l/Sqr(NDP) ' Normalization factor For II%=0 To NDP-1 Sdata(11%)=Normfactor*Sdata(11%) Next 11% Return Hartley2Magnitude: ' Convert the Hartley Spectrum to Magnitude Mode N2=NDP/2 for I%=(N2-1) to 0 step -1 Sdata (N2-I% -l)=sqr( Sdata (N2-I% ) /'2+Sdata(N2+I% ) A2 ) n ext 1% NDP=N2 Return Hartley2FT: ' Convert the Hartley Spectrum to the Fourier Real & ' Imaginary Spectrum Dim Temp(NDP) N2=NDP/2 for I%=0 to N2-1 'Use Temporary work array 'R eal part Temp(I%)=(Sdata(N2 + I%)+Sdata(N2-I%))/2 'Imaginary part Temp(N2+I%)=-(Sdata(N2+I%)-Sdata(N2-I%))/2 next 1% for I%=0 to NDP-1 Sdata(1%)=Temp(1%) 'Restore to original array next 1% Return FT2Magnitude: ' Convert the Fourier Real & Imaginary Spectrum to ' Magnitude Mode N2=NDP/2 for I%=0 to N2-1 S d a ta (1% )=sqr ( Sdata (1% ) /v2+Sdata(N2 + I% ) "2 ) n ext 1% NDP=N2 Return APPENDIX 4-4 A TA R I S T - GFA BASIC V2.0 FHT. BAS Program ______ FHT - Fast Hartley Transform (Real) Ver. 1.04 Chris Williams 2/12/88 The FHT subroutine was coded based on the routine FHTSUB on page 118 of "The Hartley Transform" by Ronald N. Bracewell, 1986. Note: numerous errors in that routine have been corrected. This routine automatically loads in the single p r e c is io n data f i l e c a l l e d "DATA.SNG". I f this file doesn't exist, a file selector box appears showing all ".SNG" files (6-byte single precision) available to choose from). Several optional subroutions (Graph, 136 Hartley2Magnitude, Hartley2Fourier, and Fourier2Magnitude) are listed toward the end of the code. They can be called with the GOSUB command. No explicit parameters are required. They all operate (as does the FHT routine) on an NDP-point single precision array S_data (with elements ranging from 0 to (Ndp-1)). Initializations Rez=Xbios(4) ICheck screen resolution If Rez=0 ! LOW (16 colors) Xres%=320 !X resolution Yres%=200 !Y resolution E ndif I f Rez=l ! MED (4 c o lo r s ) Xres%=640 !X resolution Yres%=200 !Y resolution E ndif If Rez=2 ! HIGH (Black/White) Xres%=640 !X resolution Yres%=400 !Y resolution E ndif Hand=3 ! Mouse p o in te r form Defmouse Hand Twopi=2*Pi Let Datasize=6 ! Using single precision Option Base 0 ! Start arrays at element 0 Dim S _ d a ta (l) Ndp_old=-l ! First time flag for FHT subroutine ' This routine loads a single precision data file; ' it can be commented out and replaced by other code ' to generate the Ndp data points in array S_data(Ndp) ' (starting with element S_data(0)). Gosub Load_data I ' Graph the input data before transforming Gosub Graph I ' S_data now has the Ndp input data points Gosub Fht ' Do the FHT ' Save the Hartley transformed data Gosub Save data ' Graph the results Gosub Graph t 137 PROGRAMMING OPTIONS (J u st un-comment th e Goto or Gosub command l i n e ) You can optionally go back and do another FHT without having to re-initialize with: Gosub Fht To Graph any S_data array of Ndp points (0->Ndp-l): Gosub Graph To convert the Hartley spectrum to Magnitude Mode: Gosub Hartley2Magnitude To convert the Hartley spectrum to the positive Fourier Real and Imaginary spectrum: Gosub Hartley2Fourier To convert the positive Fourier spectrum to M agnitude Mode: Gosub Fourier2Magnitude End Procedure Fht I 1 Fast Hartley Transform - Main Subroutine ' Input: S_DATA(NDP) (r e a l data) I Print At(21,8);"Fast Hartley Transform in Progress" f S_time=Timer i If Ndp<>Ndp_old 'Only recalculate if needed Print At(30,13);"Status: Initializing I ’ Calculate Power of 2 Power%=0 I%=Ndp W hile I%>1 Power%=Power%+l I%=I%/2 Wend f ' Define Arrays If Ndp_old=-l ! Only do this the first time Dim A(1024) ! Cell ordinate ! Will erase and redimension these later: 138 Dim S (1 ),T (1 ),M (1 ) E ndif Ndp_old=Ndp t N9=2/v (Power%-2) N=4*N9 C5=N-1 C6=Power%-l f Gosub Get_powers Gosub Get_sines Gosub Get_tangents I Lookup_time=Int((Timer-S_time)/2 )/100 E ndif t Print At(3 0,13);"Status: Permuting Gosub Fast_permute Gosub Permute Print At(30,13);"Status: First Stages" Gosub First_stages Print At(30,13);"Status: Last Stages " Gosub Last_stages Print At(30,13);"Status: Normalizing " I 3_max=-10000 S_min=10000 S_sum=0 t Norm_factor=l/Sqr(Ndp) ! Normalization factor I For I%=0 To Ndp-1 S_data(1%)=Norm_factor*S_data(1%) 1 Keep track of max & min values If S_data(1%)>S_max Then S_max=S _data(1%) E ndif If S_data(1%) Procedure Get_powers 139 L ocal 1% Erase M () Dim M(Power%) I For I%=0 To Power% M(1%)=2 A1% N ext 1% Return Procedure Get_sines Local 1,1%,J% Erase S () Dim S(Ndp/4) S(N9)=1 For 1=1 To 3 S(I*N9/4)=Sin(I*Pi/8) Next I H=0.5/Cos(Pi/16) ! Initial half secant I ' Fill sine table C4=Power%-4 For I%=1 To Power%-4 C4=C4-1 V0 = 0 For J%=M(C4) To (N9-M(C4)) Step M(C4+1) V1=J%+M(C4) S(J%)=H*(S(V1)+V0) V0=S(V1) N ext J% H=l/Sqr(2+1/H) ! Half secant recursion Next 1% I Return I I Procedure Get_tangents L ocal 1% Erase T () Dim T(Ndp/4) C0=N9-1 For I%=1 To N9-1 T(I%)=(1-S(CO) )/S(I%) Dec CO Next 1% T(N9)=1 Return Procedure Fast_permute L ocal I%,J% Q%=Power%/2 140 C2=M(Q%) Q%=Q%+(Power% Mod 2) l ' Set up structure table A( 0 )=0 A ( l ) = l For I%=2 To Q% For J%=0 To M( I% -1)-1 A(J%)=2*A(J%) A(J%+M(I%-1) )=A(J%)+1 N ext J% N ext 1% I Return Procedure Permute L ocal I%,J% For I%=1 To C2-1 V4=C2*A(1%) V5 = I% V6=V4 V7=S_data(V5) S_data(V5)=S_data(V6) S_data(V6)=V7 For J%=1 To A (1%)-1 V5=V5+C2 V6=V4+A(J%) V7=S_data(V5) S_data(V5)=S_data(V6) S_data(V6)=V7 N ext J% Next 1% Return Procedure First_stages L ocal 1% ' 2-Element DHT Stage 1 For I%=0 To N-2 Step 2 V6=S_data(1%)+S_data(I%+1) V7=S_data(1%)-S_data(I%+1) S _ d a ta (1%)=V6 S_data(I%+1)=V7 Next 1% I ' 4-Element DHT Stage 2 For I%=0 To N-4 Step 4 V6=S_data(1%)+S_data(I%+2) V7=S_data(I%+1)+S_data(I%+3) V8=S_data(1%)-S_data(l%+2) V9=S_data(I% + 1)-S_data(I%+3) 141 S _ d a ta (1%)=V6 S _ d a ta ( I%+1)=V7 S _ d a ta ( I%+2)=V8 S _ d a ta ( I%+3)=V9 N ext 1% l Return Procedure Last_stages ' S ta g e s 3,4,5,... I L ocal 1%,J%, K%,L% U%=C6 S%=4 For L%=2 To C6 ! Stage = L%+1 V2=2*S% Dec U% V3=M(U%-1) For Q%=0 To C5 Step V2 I%=Q% D%=I%+S% V6=S_data(I%)+S_data(D%) V7=S_data(1%)-S_data(D%) S_data(I%)=V6 S _ d a ta (D%)=V7 K%=D%-1 For J%=V3 To N9 Step V3 Inc 1% D%=I%+S% E%=K%+S% V9=S_data(D%)+(S_data(E%)*T(J%)) X=S_data(E%)-V9 *S(J%) Y=X*T(J%)+V9 V6=S_data(1%)+Y V 7= S _data(1%)-Y V8=S_data(K%)-X V9=S_data(K%)+X S_data(1%)=V6 S _ d a ta (D%)=V7 S _ d a ta (K%)=V8 S_data(E%)=V9 Dec K% Next J% E%=K%+S% N ext Q% S%=V2 Next L% t Return l 142 Procedure Load_data C ls I f E x i s t ( "DATA.SNG") Filename$="DATA. SNG" E lse Print "Select data file for FHT:" Fileselect "*.sng","",Filename$ If Exist(Filename$)=False Cls End ! Q uit i f no f i l e g iv en E ndif E ndif C ls Hidem ! Hide mouse cursor Print At(30,13);"Status: Loading data Open "I " ,# 1 ,Filenam e$ Fsize=Lof(#1) C lo se #1 Ndp=Fsize/Datasize Erase S_data() ' Must create an array of appropriate type & size Dim S_data(Ndp) ' Load data directly into the array S_data Bload Filename$,Varptr(S_data(0)) Return Procedure Save_data ' This routine directly (and quickly) saves the ' array data to a file. The data file can be ' reloaded to an existing array using BLOAD (see ' the Load_data routine). P r in t Print " FHT data saved in file < FHT_OUT.SNG >. " Bsave "FHT_OUT.SNG",Varptr(S_data(0 )),Ndp*Datasize Return Procedure Graph ' Scale & graph the Ndp (0->Ndp-l) S_data array C ls I f Ndp<=0 Print "No data to graph." E lse S_max=-100 000 00 S_min=1000 000 0 For I%=0 To Ndp-1 ' Keep track of max & min values If S_data(1%)>S_max Then S_max=S_data(I %) I_max%=I% E ndif 143 If S_data(1%) Procedure Hartley2Magnitude For I%=0 To N2-1 S_data(N2+I%)=Sqr((S_data(N2-I%)"2+ (continuation line) S_data (N2 + I% ) A2 ) /2 ) . Next 1% For I%=0 To N2-1 S_data(1%)=S_data(N2+I%) Next 1% Ndp=N2 Return Procedure Hartley2Fourier ' Convert Hartley to Positive FT Real/Imaginary Dim T_data(Ndp) ' Temporary work array N2=Ndp/2 For I%=0 To N2-1 1 Real/Even Part T_data(1%)=(S_data(N2+I%)+S_data(N2-I%) )/2 ’ Imaginary/Odd Part T_data(N2+I%)=-((S_data(N2+I%)-S_data(N2-I%)) 12) Next 1% For I%=0 To Ndp-1 S_data(1%)=T_data(1%) 144 Next 1% Erase T_data() Return Procedure Fourier2Magnitude ' Convert Positive Real/Imaginary FT to Magnitude N2=Ndp/2 For I%=0 To N2-1 S_data(1%)=Sqr(S_data(1%)A2+S_data(N2+I%)A 2) Next 1% Ndp=N2 Return C hapter V A FASTER ALGORITHM FOR THE HILBERT TRANSFORM Mathematical Basis of the Hilbert Transform There are often occasions when it is desirable to convert spectral frequency data from dispersion mode to absorption mode, or vice versa. Such conversions have been required in many diverse areas of study, including DISPA line shape analysis in various spectroscopic fields [1-2], optical refraction in semiconductors [3] and optical pulse studies [4], microwave line shape analysis [5], resonance Raman excitation studies [6], low-energy kaon-nucleon scattering [7-8], and analysis of low- frequency vibrational signals from nuclear reactors [9]. Fortunately, there is a direct mathematical relation between these two spectral forms known as the Hilbert transform: (5-la) -00 + 00 (5-lb) -00 145 146 Equation 5-la Is the forward Hilbert transform; Equation 5-lb Is the inverse Hilbert transform. The Hilbert transform can be thought of as phase shifting all frequencies by jt/2. In spectroscopy, this transform is also known as the Kramers-Kronig relation [ 10-11], and in the context of servomechanisms as the Bode relation [12]. Even earlier this relationship had been demonstrated for complex refractive index by Sommerfeld [13] and Brillouin [14]. The relationship between causality and the absorption and dispersion forms related by the Hilbert transform has been discussed in depth by Toll [15] (also see Figure 4-2). A New Hilbert Algorithm Although direct numerical evaluation of the Hilbert transform can be tedious [16-18], the matter is much simplified if the integral of Equation 5 - la is viewed instead as a convolution (*) of two functions: + 00 -1 D(co) * (5-2) One can then take advantage of the convolution theorem of Fourier transforms, which states that the FT of a convolution of two functions is equal to the product of the FT's of the two functions [19]: + 00 If fit) = h(t) * e(t) = J h(t') e(t-t') dt' -00 = the convolution of h(t) with e(t) 147 and F(w) = the Fourier transform of fft) H(w) = the Fourier transform of h(t) E(a>) = the Fourier transform of e(t) then F((o) = H(co) • E M (5-3) The result is that a fairly complex integration in one domain is converted to a much simpler multiplication in the other Fourier domain. In this case, the FT of the function 1 / juo is simply the signum, or sign function: sgn(t) = +t , for t 3: 0 = -t , for t < 0 (5-4) Taken together, the above relations reveal that the Hilbert transformation of DM to AM can be computationally accomplished use of two FT's and an application of the signum function to half of the intermediate data. An analogous procedure has been previously used in an electron paramagnetic resonance (EPR) study to convert an absorption spectrum into a dispersion spectrum, enabling construction of a dispersion-vs.-absorption (DISPA) plot [20-21]. For real time-domain data (i.e., the usual situation in FT/ICR mass spectrometry and optical interferometiy), one drawback of using the forward and inverse Fourier transforms to move back and forth between domains is that an initial zero-fill of the original real data is required to provide the necessary imaginary input to the FT (since even the "real" FFT is unable to transform a lone dispersion or absorption 148 component). The memory size and computational time for both the forward and inverse FT's is thereby doubled. Once the final FT is completed, the resulting imaginary data are thrown away. Use of the Hartley transform [22-23] in place of the FT avoids this redundancy. Since the Hartley transform operates directly on real data, there is no need to zero-fill the real input data to create imaginary input data. The Hilbert transform is thus twice as fast when performed by the Hartley rather than the Fourier transform, Also, since the Hartley transform is its own inverse, the overall computation of the Hilbert transform simplifies to four steps, two of which are identical (the fast Hartley transform) and two of which are trivial (an inversion in data element order and an intermediate sign change for half of the data). Table 5-1 outlines the complete algorithm for this simpler and faster implementation of the Hilbert transform. Note that the forward and inverse Hilbert transform differ only in the order of the four previously mentioned steps. The usefulness of this new algorithm for the Hilbert transform was examined using computer analysis of simulated data. Experimental Appendix 5-1 lists the Atari ST Basic computer source code for this implementation of the Hilbert transform (Table 5-1), which can be easily converted to other programming languages for the Nicolet 1280 Computer, the Apple Macintosh, and the IBM PC. Simulated time- domain exponentially damped sinusoid data sets, 512 <; IV <; 16,384, were generated and processed on an Atari Mega-4 computer (Motorola 68000 processor, 8 MHz CPU speed, 4 megabytes RAM) with programs 149 Table 5-1 Algorithms for the (forward) Hilbert and Inverse Hilbert transforms. (Forward) Hilbert Dispersion Absorption l. Invert Data Element Order 2. FHT 3. Signum function 4. FHT Inverse Hilbert Absorption -» Dispersion l. FHT 2. Signum function 3. FHT 4. Invert Data Element Order 150 compiled using the GFA Basic Compiler Ver. 2.0. Gaussian distributed random noise of varying amplitude was generated from appropriate computer routines [24]. The Hartley transform algorithm used here is from Appendix 4-4. Output data were written to disk files and transferred to an IBM 386 PC for further analysis by use of the Microsoft Excel for Windows ver. 3.0 and Borland Quattro Pro ver. 3.0 spreadsheet programs. Results and Discussion Computation of the Hilbert Transform The actual stages in the operation of the Hilbert transform algorithm of Table 5-1 can be seen in Figure 5-1. A 512 point simulated time domain data set consisting of a three damped cosines and added Gaussian-distributed random noise was Fourier transformed to produce the normal absorption and dispersion spectra. The dispersion data was graphed after each of the four stages of the Hilbert transform. The final output is in absorption mode. Figure 5-2 illustrates the four stages of the inverse Hilbert transform, which converts the absorption mode data to the dispersion mode. Precision of the Hilbert Transform Algorithm Although the visual appearance of the data in Figures 5-1 and 5-2 indicate that the forward and inverse transforms have been implemented correctly, more objective confirmation is needed. Because successive application of the Hilbert transform and its inverse (in either order) should recover the starting data exactly, this provides a means of 151 , __4 . . J f ...... ' Initial Dispersion Mode j Invert data element order N----- L. 'w ...... 1 | First FHT | Reverse sign of second half of data | Second FHT . . JL Final Absorption Mode Figure 5-1 Stages of the (forward) Hilbert transform. 152 ...... 1 ...... Initial Absorption Mode I First FHT j Reverse sign of second half of data | Second FHT .. 1 | Invert data element order ..J v v »"w v v » ,4Ai f ...... Final Dispersion Mode Figure 5-2 Stages of the inverse Hilbert transform. 153 checking their precision. An initial IK time domain data set containing three exponentially-damped cosines of different amplitudes plus Gaussian-distributed random noise was Fourier-transformed to generate the initial dispersion data. Figure 5-3 illustrates the data after two sets of successive forward and inverse Hilbert transforms. All calculations employed floating-point numbers. Table 5-2 shows the differences (the average of the absolute values of the fractional differences) for corresponding data sets after either forward/inverse or inverse/forward Hilbert transforms. After successive Hilbert and inverse Hilbert transforms in either order, nearly all the corresponding data points agreed to 8 significant digits, or to within one part in 40 million. Since the calculations were carried out using 6-byte floating point numbers with 10 digits of precision, it appears that approximately two significant digits are lost for each forward and inverse Hilbert pair. Since each Hilbert transform (or inverse transform) involves two FHT's, this corresponds well with the finding that each successive FHT pair loses one significant digit, and that this effect is cumulative (see Figure 4-13 and Table 4-3). The precision shown in these tests of the Hilbert transform should prove sufficient for most spectroscopic work. Should higher precision be needed, one could convert the Hilbert program (including the FHT subroutine) to use double-precision floating point calculations. Conclusion A simplified, faster version of the Hilbert transform based on the fast Hartley transform has been demonstrated for the conversion of 154 A) .... - y r " r 1 c...... | Hilbert B) •sr+rr+ *\i \J*i j Inverse Hilbert C) I Hilbert D) Figure 5-3 Test of the reversibility of the Hilbert transform. Application of the Hilbert transform and followed by its inverse should recover the original data exactly. (A) Initial 512-point dispersion data. (B) After Hilbert transform is in absorption mode. (C) After a subsequent inverse Hilbert transform, is back to dispersion mode. (D) Another Hilbert transform converts back to absorption mode. 155 Table 5-2 Average Absolute-Value Fractional Differences Between Corresponding Data Points after Application of the Hilbert and Inverse Hilbert Transforms3 Initial Final Transformation Average Absolute D ata Set3 Data Set3 Typeb Fractional Differencecd AC D=^A=>D 2.34600E-08 BD A=>D=>A 2.02182E-08 3 Data sets used are shown in Figure 5-3. All data sets had 512 floating point numbers. An initial IK time domain data set containing three exponentially-damped cosines of different amplitudes plus Gaussian- distributed random noise with an average magnitude of 5% of the maximum time domain value was Fourier-transformed to generate the initial dispersion data. b Transformation type refers to the format of the initial and final data set. "A" indicates absorption mode, and "D" indicates dispersion mode. "D=>A=>D" refers to an initial dispersion mode Hilbert-transformed to absorption mode, then inverse Hilbert-transformed back to dispersion mode. c The absolute average value of the fractional differences was calculated as follows. The difference between two corresponding data points was divided by the value of the first of those points to get the fractional difference. The absolute values of these fractional differences were then averaged. d After successive Hilbert and inverse Hilbert transforms (in either order), nearly all the corresponding data points agreed to 8 significant digits. Since the calculations were carried out using 6-byte floating point numbers with 10 digits of precision, it appears that approximately two significant digits are lost for each forward and inverse Hilbert pair. Since each Hilbert transform (or inverse transform) involves two FHT's, this corresponds well with the finding that each successive FHT pair loses one significant digit, and that this effect is cumulative (see Figure 4-13 and Table 4-3). 156 dispersion mode spectra to absorption mode, and vice versa. Because the algorithm presented here is a general one, and is not concerned with the form of the data to be transformed, it is not limited to spectroscopic applications. 157 REFERENCES 1. Marshall, A. G. in Marshall, A. G., Ed. Fourier .Hadamard, and Hilbert Transforms in Chemistry. Plenum: New York. 1982. 2. Herring, F. G.; M arshall, A. G.; Phillips, P. S.: Roe, C. D. J. Mag. Res. 1980, 37. 293. 3. Forouhi, A. R.; Bloomer. 1. Phys. Rev. B. 1988, 38, 1865. 4. Sonajalg, H.; Gorokhovskii, A.: Kaarli, R.; Palm, V.; Ratsep, M.; Saari, P. Opt. Commun. 1989, 71, 377. 5. Mehrotra, S. C. J. Quant. Spectrosc. Radiat. Transfer. 1984, 32, 169. 6. Cable, J. R.; Albrecht. A. C. J. Chem. Phys. 1986, 84, 4745. 7. Conti, A. Nucl. Phys. B., 1972, 42, 607. 8. Conti, A. Nucl. Phys. B., 1974, 70 116. 9. Saxe, R. F. Ann. Nucl. Energy 1986, 13, 399. 10. Kramers, H. A. Estratto dagli Atti del Congresso Intemazionale de Fisici Como; Nicolo Zonichelli: Bologna. 1927. 11. Kronig, R. Ned. Tijdschr. Natuurk 1942, 9, 402. 12. Bode, H. W. Network Analysis and Feedback Amplifier Design; D. Van Nostrand Company: NY. 1940. 13. Sommerfeld, A.: Ann. Physik 1914, 44. 177. 14. Brillouin, L. Ann. Physik 1914, 44, 203. 15. Toll, J. S. Phys. Rev. 1956, 104, 1760. 158 16. E rnst, R. R. J. Magn. Reson. 1969, I, 7. 17. Ohta, K.; Ishida, H. Appl. Spectrosc. 1988, 42, 952. 18. Wang, T.-C. L.; M arshall, A. G. Anal. Chem. 1983, 55, 2348. 19. Marshal], A. G.; Verdun, F. R. Fourier Transforms in NMR, Optical, and Mass Spectrometry: A User's Handbook. Elsevier: Amsterdam. 1990. 20. M arshall, A. G. Chemometrics & Intelligent Lab. Systems. 1988, 3, 261. 21. Herring, F. G.; M arshall. A. G.; Phillips, P. S.; Roe, D. C. J. Mag. Res. 1980, 37. 293. 22. Bracewell, R. N. The Hartley Transform. Oxford University Press: New York. 1986. 23. Williams, C. P.; M arshall. A. G. Anal. Chem. 1989, 61, 428. 24. Press, W. H.; Flannery, B. P.: Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes. Cambridge University Press: New York. 1986. 159 APPENDIX 5-1 A TA R I S T - GFA BASIC V2.0 HILBERT.LST Subroutine HILBERT.LST H ilbert Transform This routine calls the FHT subroutine listed in Chapter IV. This routine can be merged into that program code listing. This routine performs a Hilbert transform on the NDP single-precision data points in S_DATA(NDP) (NDP and S_data(NDP) are assumed to be valid) It is invoked by: GOSUB HILBERT(DIRECTION) The p a r a m e te r DIRECTION i s an i n t e g e r t e l l i n g whether to do a forward or inverse Hilbert tr a n s f o r m . I f DIRECTION >= 0, th e n t h e fo rw a r d H i l b e r t i s d one If DIRECTION <0, then the inverse Hilbert is done Procedure Hilbert(Direction) ! Hilbert transform N2=Ndp/2 If Direction>=0 ! Forward Hilbert Only F or I%=0 To N2-1 ' Invert data element order for Convolution Temp=S_data(1%) S_data(1%)=S_data(Ndp-I%) S_data(Ndp-I%)=Temp N e x t 1% E n d if i Gosub Fht ! Fast Hartley Transform I 1 Signum function (Reverse sign for half the data) For I%=Ndp/2 To Ndp-1 S_data(1%)=-S_data(1%) N ex t 1% Gosub Fht ! Fast Hartley Transform l If DirectioncO ! Inverse Hilbert Only F or I%=0 To N2-1 ' Invert data element order for Convolution Temp=S_data(1%) S_data(1%)=S_data(Ndp-I%) S_data(Ndp-I%)=Temp N e x t 1% E n d if l R etu rn (It should prove straightforward to convert this routine to any other language). C hapter VI THE "ENHANCED" ABSORPTION SPECTRUM Introduction In Fourier transform (FT) spectroscopy, such as ion cyclotron resonance (ICR), nuclear magnetic resonance (NMR), or infrared (IR) interferometry, an initial time-domain or interferogram discrete signal of JV data points is converted into a frequency-domain spectrum by a discrete (fast) Fourier transform (FFT), as discussed in Chapter IV. The FFT produces, after phase correction (see Chapter IV), an N /2 point absorption mode, (A(to)], and an N /2 point dispersion mode, [D(oj)], spectrum. Each of these spectra contains half of the total spectral information [1], assuming that the noise in the initial time domain signal is bandwidth limited: i.e., the signal is not filtered or otherwise correlated, as by zero-filling. The information in the dispersion format, however, is not as well suited for direct analysis as that in the absorption format, because the dispersion spectrum is broader than the absorption spectrum and is odd rather than even (thereby complicating the appearance of overlapped peaks). As mentioned earlier, two common ways of recovering the dispersion-mode information are frequency-domain magnitude calculation and time-domain zero-filling before FFT. 161 162 The magnitude mode (also known as absolute-value) spectrum, M(co), is often used to combine the absorption and dispersion information (especially when it is difficult to phase the data). The precision in determination of magnitude mode spectral peak position (or width, or area) is higher than that of the component absorption mode spectrum [2-3], but magnitude mode spectral resolving power (namely, o j/A c o , in which to is the signal frequency and A to is the peak full width at half-maximum peak height) is reduced by a factor of between v/3 and 2 (dependent on the degree of damping) for unapodized data compared to that for the absorption mode spectrum [4], Alternatively, zero-filling the initial N-point time domain data set (i.e., adding N zeroes to increase its size to 2N data points) before FFT effectively puts all of the spectral information into each of the absorption and dispersion data sets [1,3] : i.e., both the absorption and dispersion spectra are exactly equivalent in information content (apart from roundoff errors [5] introduced by the FFT itself). The N-point absorption spectrum obtained from a once zero- filled data set retains the original absorption mode resolving power along with improved precision, but takes up twice as much data storage. In this chapter, a new method is offered for extracting the dispersion information, based on the Hilbert transform (discussed in Chapter V) of the dispersion spectrum into an absorption spectrum having an equivalent information content. This "pseudo-absorption" spectrum can then be averaged with the original absorption spectrum. The resultant N /2 point "enhanced" absorption spectrum combines the best features of both zero-filling and magnitude computation, namely, higher resolving power, higher precision, and higher signal-to-noise 163 ratio, while retaining the compact data size of the original (unenhanced) absorption mode spectrum. Figure 6-1 outlines the procedure. The theoretical basis and computational methods for obtaining the "enhanced" absorption mode will be presented, along with a demonstration of the technique with both simulated and experimental ICR time-domain data. Limitations of the method will also be examined and discussed. Within these limitations, the "enhanced" absorption technique can also be applied to advantage in other fields using Fourier transform spectroscopy (e.g., nuclear magnetic resonance, interferometry, etc.) [4]. Theory Random uncorrelated noise present in a causal time-domain noisy signal has both a random amplitude and phase distribution in the absence of filtering, zero-filling or other similar correlative processes. In other words, after Fourier transformation, the noise is independently distributed between the mathematically real and imaginary spectra; that is, there is no correlation between the noise present in the real spectrum and that in the imaginary spectrum. Phasing the frequency-domain complex spectrum does not change the situation: noise (and signal) are present equally and independently in the absorption mode and dispersion mode FT spectra. The discrete Hilbert transform of D(co) yields an lV/2-point "pseudo-absorption" spectrum which is then averaged with the original A(w) to yield an N/2-point "enhanced" absorption spectrum with the same peak width and same number of data points, but with peak- 164 Time Domain Signal (FID) FFT (Phase-Corrected) Absorption Spectrum Dispersion Spectrum Hilbert Transform Add, then Pseudo-Absorption" Divide by 2 Spectrum "Enhanced" Absorption Spectrum Figure 6-1 Stages in the generation of an "enhanced" absorption mode spectrum (bottom) from a discrete time-domain signal (top). Following discrete Fourier transformation of the time-domain signal and phase correction to yield the conventional absorption and dispersion spectra, the dispersion data is subjected to a discrete Hilbert transform to yield a "pseudo-absorption" mode spectrum. The absorption and "pseudo absorption" spectra are then averaged to yield the "enhanced" absorption mode spectrum. 165 height-to-noise ratio improved by a factor of \J2 over the original JV/2- point absorption spectrum. In effect, this is the same as signal averaging two independent spectra. The peak signal(s), which are the same in both the absorption and pseudo-absorption spectra, are unchanged both in magnitude and in shape. The random noise from the two different spectra, however, partially cancels out and is effectively reduced by a factor of \]2. As a result, the new procedure yields a spectrum with enhanced signal-to-noise ratio and precision, without any attendant loss in resolving power (as for magnitude-mode display) and without any increase in the number of stored data points (as for zero- filling). The three essential prerequisites for this technique are that the absorption and dispersion data are phase-corrected (which is required of the regular absorption and zero-filled absorption spectra as well), that there is sufficient digital resolution to properly define the absorption and dispersion line shapes (which is also necessary for the regular absorption and zero-filled absorption spectra), and that there has been no filtering or zero-filling done to correlate the absorption and dispersion noise (which is usually easy to arrange). A possible limitation to be examined is that of peaks located very near either the zero-frequency or Nyquist-frequency ends of the spectrum, since the broad "tails" of the dispersion data are likely to be truncated. The effect of zero-filling will also be examined. Experimental Simulated time-domain exponentially damped sinusoid data sets, 512 *s N s 16,384, were generated and processed on an Atari Mega-4 166 computer (Motorola 68000 processor, 8 MHz CPU speed, 4 megabytes RAM) with programs compiled using the GFA Basic Compiler Ver. 2.0. Gaussian distributed random noise of varying amplitude was generated from appropriate computer routines [6]. The Hartley transform program used here is that of Appendix 4-4, and the Hilbert program is the same as given in Appendix 5-1. Output data were written to disk files and transferred to an IBM 386 PC for further analysis by use of the Microsoft Excel for Windows ver. 3.0 and Borland Quattro Pro ver. 3.0 spreadsheet programs. A phased ICR spectrum (i.e., pure absorption mode and pure dispersion mode) was obtained for the single resonance of electron- ionized (70 eV electron beam) N 2 on an Extrel FTMS-2000 instrument operating at 3.0 tesla by use of single-frequency on-resonance excitation of a single time- domain transient, for ions trapped in a cubic ICR cell at ~1 V/inch trapping potential). The pressure was kept deliberately high in order to damp the signal sufficiently to yield a large number of data points per peak width [2-31. The time-domain signal was analog-filtered to contain bandwidth limited noise, for a Nyquist bandwidth of 129 kHz. Phasing was achieved simply by vaiying the frequency of the carrier heterodyne frequency until the real and imaginary FT spectra became pure absorption mode and pure dispersion mode. The 8K time-domain data were subjected to FFT without apodization. The resulting absorption and dispersion data sets (4K each) were transferred to an Atari Mega-4. With the methods and algorithm presented earlier, a Hilbert transform was performed on the dispersion data. Appendix 6-1 lists the Atari ST Basic computer source code for generating the 167 enhanced absorption spectrum by means of the Hilbert transform. This program is relatively simple and can be easily converted to other programming languages for the Nicolet 1280 Computer, the Apple M acintosh, and the IBM PC. Results and Discussion Standard deviation and root-mean-square noise in absorption and magnitude spectra. The relative resolving powers of absorption mode and magnitude mode spectra are well-known [4]: for unapodized time-domain (or interferogram) data, absorption mode resolving power is higher by a factor ranging from 2 to -\/3 according to the ratio of the acquisition period, T, to the time-domain exponential damping constant, x. However, in order to understand the predicted advantages of "enhanced" absorption mode spectra, we must first review the nature and measurement of noise in conventional absorption mode and magnitude mode spectra. First, since absorption (or "enhanced" absorption) noise is randomly Gaussian-distributed about a zero mean value, the noise standard deviation from the mean (S.D.) and root-mean-square deviation from zero (R.M.S.) are the same. However, because magnitude mode data are necessarily positive-valued (since it is based on the sum of squares), magnitude mode noise follows a Rayleigh rather than Gaussian distribution [3,4,7], and has a non-zero mean value. In fact, magnitude-mode noise can be shown experimentally to have a standard deviation noise which is approximately half its R.M.S. noise magnitude. Thus, although magnitude mode R.M.S. noise is yj2 higher than 168 absorption mode R.M.S. noise, magnitude mode S.D. (from non-zero mean) noise is only *\/2/2 that of absorption mode S.D. noise. From the above facts, we quickly infer that the "enhanced" absorption mode spectrum must have S.D. noise approximately equal to that of magnitude mode (since two independent data sets have been combined to produce either spectrum), whereas R.M.S. noise for "enhanced" absorption is only half that of magnitude mode (because the "enhanced" absorption noise is distributed about zero mean whereas the magnitude mode noise has non-zero mean value). The spectral peak height (neglecting any noise component, and assuming the spectrum is baseline corrected) should be the same for absorption, magnitude, and "enhanced" absorption modes. Spectral precision in determination of peak height, width, frequency, or area is directly proportional to the ratio of signal to S.D. noise [2-3]. On the other hand, in trying to identify a low-amplitude peak in the presence of baseline noise, R.M.S. noise is the more relevant measure. In summary, compared to magnitude mode, the "enhanced" absorption mode offers roughly equal precision but superior visualization of weak peaks. Table 6-1 lists the relative values of noise (both S.D. and R.M.S.), signal-to-noise ratio, and resolving power for the absorption, magnitude, and "enhanced" absorption spectra, with and without zero-filling. Figures of merit Table 6-2 compares the predicted relative resolving power, and signal-to- noise ratio (both S.D. and R.M.S.) for "enhanced" absorption relative to absorption or magnitude spectra, computed from almost 169 Table 6-1 Comparison of theoretical relative noise, signal-to-noise ratio (SNR), and resolving power for three FT spectral modes, in the Lorentz limit that the time-domain signal decays nearly to zero during the time- domain acquisition period. Noise is reported both as standard deviation from zero (S.D.) and as root-mean-square deviation (R.M.S.), and signal- to-noise ratio is reported with respect to each of the two measures of noise. See text for interpretation. No Zero F illin g Mode Noise SNR Noise SNR R esolving (S.D.) (S.D.) (R.M.S.) (R.M.S.) Power Absorption v/2 V2 V2 V3 M agnitude 1 2 2 1 1 E nhanced 1 2 1 2 N/3 Absorption One Zero-Fill Mode Noise SNR Noise SNR R esolving (S.D.) (S.D.) (R.M.S.) (R.M.S.) Power Absorption 1 >/2 1 a/3 Magnitude 1A/2 2 i 1 Enhanced 1 V 2 i a/2 A/3 Absorption 170 Table 6-2 Ratio of relative signal-to-noise ratio (SNR) and resolving power for "enhanced" absorption mode relative to either (unenhanced) absorption mode or magnitude mode spectra, for noise evaluated either as standard deviation from the mean (S.D.) or root-mean-square deviation from zero (R.M.S.). The standard deviation of each computed ratio is listed, along with the theoretically predicted values (in parentheses). The data were obtained from the average of 11,870 independently simulated noisy 8K time-domain exponentially damped sinusoids, acquired for 10 time-domain exponential damping periods. Gaussian-distributed random noise with zero mean and absolute mean value of 5% of the maximum time-domain amplitude was added to the time-domain signal. Resolving SNR SNR Power (S.D.) (R.M.S.) Enhanced Absorption 0.995±0.007 1.404±0.032 1.420±0.033 Unenhanced Absorption 11 (1.414) (1 .4 1 4 ) Enhanced Absorption 1.677±0.020 0.927±0.022 2.001 ±0.034 M ag n itu d e (1.732) ( 1.0 0 0 ) (2 .0 0 0 ) 171 12,000 independently simulated noisy data sets. It is clear that the results from simulated data agree very closely with those predicted from Table 6-1. Table 6-3 shows a similar comparison, this time based on experimental FT/ICR mass spectra of N 2 +. Again, the agreement with the predictions of Table 6-1 is quite good: signal-to-S.D.-noise enhancements of 1.33 (vs. theoretical 1.41) relative to absorption and 0.88 (vs. theoretical 1.00) relative to magnitude mode; signal-to-R.M.S. noise enhancements of 1.35 (vs. theoretical 1.41) relative to absorption and 1.88 (vs. theoretical ~2) relative to magnitude mode; resolving power enhancements of 0.88 (vs. theoretical 1.00) relative to absorption and 1.58 (vs. theoretical 1.73) relative to magnitude mode; and relative spectral peak heights of absorption:magnitude:"enhanced" absorption of 1.00:1.03:0.93 (vs. theoretical 1:1:1). The relatively minor discrepancies in each case may arise from imperfect phasing of the original spectra, limited digital resolution, and incomplete damping of the time-domain signal. Zero-Filling The effect of zero-filling on spectral signal-to-noise ratio and precision for both absorption and magnitude mode spectra is discussed at length in reference 3 and the literature cited therein. Briefly, although zero-filling does not change the ratio of signal to standard deviation noise (see Table 6-1), one zero-fill (i.e., addition of N zeroes to an N-point time-domain data set before FT) does improve absorption mode (or magnitude mode) precision in determination of spectral peak 172 Table 6-3 Com parison of noise, signal-to-noise ratio (SNR), and resolving power for experimental FT/ICR mass spectral data from N 2 + ions (~7 points per peak full width at half-maximum peak height). Noise is reported both as standard deviation from the mean (S.D.) and as root- mean-square deviation from zero (R.M.S.), and signal-to-noise ratio is reported with respect to each of the two measures of noise. See text for interpretation. Mode Noise SNR Noise SNR R esolving Peak (S.D.) (S.D.) (R.M.S.) (R.M.S.) Power H eight Absorption 6 9 5 46 6 9 7 4 6 5 6 3 1 .8 K M agnitude 4 7 4 6 9 9 7 8 33 31 3 2 .6 K E nhanced 4 81 61 481 6 2 4 9 2 9 .6 K Absorption 173 height, width, or frequency by a factor of \J2 because precision is proportional to the square root of the number of spectral data points per line width, and there are twice as many data points per line width in an FT spectrum of once zero-filled time-domain data. Additional zero-fills do not improve spectral precision. Note that after one zero-fill, the "enhanced" absorption spectrum is identical to the conventional absorption spectrum, as expected, since both the (unenhanced) absorption and dispersion spectra now contain the full spectral information. Visual proof that a single zero-filling correlates the noise in the absorption and dispersion data is found in the comparison of Figure 6-2 (no zero filling) and Figure 6-3 (one zero-filling), which both show graphs of the absorption spectrum alongside the "pseudo-absorption" (Hilbert-transformed dispersion) spectrum. Double-Forward FT At this stage, it may be worth digressing to discuss the "double forward FT" procedure (8], in which an N-point time-domain data set is subjected to FFT, and the A//2-point absorption data is then zero-filled and subjected to an inverse FFT to yield two lV/2-point "pseudo-FID's", either of which contains all of the original N/2-point absorption mode (but none of the N/2-point original dispersion mode) information. This procedure permits regeneration of a time domain signal of half the original length (and with half the original information content, not all as is mistakenly stated in that paper). This new time domain can subsequently be reprocessed by various techniques such as apodization or even non-FT methods. In contrast, an inverse FFT applied to an 174 Absorption Hilbert- transformed D ispersion 50 100 150 200 250 Absorption Hilbert- transformed D ispersion E nhanced Figure 6-2 Correlation of signal and noise in the absence of zero filling. A 512 point exponentially damped time domain signal containing three cosine frequencies was processed with a real FT. The dispersion part was then Hilbert-transformed to give a pseudo-absorption spectrum. Top figure shows the original absorption and the pseudo absorption spectra plotted directly over top of each other. Bottom figure is a blow up of the region from points 25 to 100, with the two spectra artificially displaced in the vertical dimension to better distinguish them. The "Enhanced" absorption spectrum, which is the average of the absorption and pseudo absorption, is also shown. Note that with no zero filling the signal peaks are the same for both absorption and pseudo-absorption spectra, but that the baseline noise is uncorrelated. 175 Absorption Hilbert- transformed Dispersion 100 200 300 400 500 Absorption Hilbert- transformed Dispersion Figure 6-3 Correlation of signal and noise with a single zero filling. A 512 point exponentially damped time domain signal containing three cosine frequencies and added noise was first zero-filled before it was processed with a real FT. The dispersion part was then Hilbert- transformed to give a pseudo-absorption spectrum. Top figure shows the original absorption and the pseudo absorption spectra plotted directly over top of each other. Bottom figure is a blow up of the region from points 50 to 200, with the two spectra artificially displaced in the vertical dimension to better distinguish them. Note that with one zero filling both the signal peaks and the noise are virtually the same for both absorption and pseudo-absorption spectra. This demonstrates that a single zero-filling actually correlates the noise in the absorption and dispersion components of the spectra. As a result, the "enhanced" absorption spectrum would be virtually identical to these spectra. 176 "enhanced" absorption mode spectrum which has first been padded with another N /2 zeroes will generate two identical N/2-point "pseudo-FID's", either of which contains all of the "enhanced" absorption mode spectral information. This procedure (Figure 6-4) appears to let us store all the information of the original N-point time domain in a final time domain of just half the size. In reality, by throwing away half of the time domain data points we are effectively losing all the phase information of the original data, and are only keeping the frequency amplitude data. However, for normal FT/ICR and FT/NMR spectra this phase information is simply an artifact of the excitation/detection process and is not at all useful. Thus, the data storage advantage of the present procedure may be realized by storing either the "enhanced" absorption spectrum or the lV/2-point "pseudo-FID" obtained by FFT of a once zero- filled "enhanced" absorption spectrum. (In fact, zero-filling is not even necessary if a fast Hartley transform rather than FFT is used). This reduction of data without the loss of useful information is particularly advantageous for storing the huge amounts of data generated by techniques such as GC-FT/ICR. The enhanced absorption data (or its pseudo-FID) can be further compressed by as much as 50% using any of several general purpose commercial or public domain data compression program s (such as ARC, PKZIP, or LHARC for the IBM PC). Improved Identification of Weak Signals The enhanced signal-to-noise ratio predicted in Table 6-1 and confirmed in Tables 6-2 and 6-3 is perhaps most dramatically evident from visual inspection of the spectra themselves (see Figure 6-5). In 177 => FT Phase C orrect " E n h a n c e d " < = j J L I FT Z ero F ill |j, Truncate T o S to ra g e To S to ra g e Figure 6-4 Flow chart for the Double Forward FT procedure employing the "enhanced" absorption instead of the ordinary absorption spectrum. Storage of either the "enhanced" absorption or the final N/2-point FID preserves all the useful signal information. 178 Absorption Magnitude Enhanced Absorption Figure 6-5 Three different spectral representations of the same simulated time-domain sum of three exponentially damped sinusoids of relative amplitude 10:1:3, with added Gaussian-distributed random noise. Note that the "enhanced" absorption spectrum offers the best visual identification of the smallest-amplitude signal (denoted by an arrow in each spectrum) 179 that example, the weakest of the three signals is much more clearly distinguished from noise in the "enhanced" absorption spectrum than in either the conventional absorption or magnitude mode spectra. Again, it is worth noting that even though the "enhanced" absorption S.D. noise is the same as that of the magnitude spectrum, the R.M.S. noise of the "enhanced" absorption spectrum is only half that of the magnitude spectrum. Thus, R.M.S. is a better noise measure than S.D. in visualizing weak signals in a noisy spectrum. Automated peak-finding programs will also do better picking small peaks out from the baseline noise when using the "enhanced" spectrum. Improved Discrimination against Spurious Noise Spikes Another way of visualizing the signal-to-noise advantage of the "enhanced" absorption display is shown in Figure 6-6. In the conventional absorption spectrum, there is a large noise spike which is nearly as large as two of the three signal peaks. The relative height of the spike is somewhat reduced in the magnitude spectrum, but drops down nearly to the baseline noise level in the "enhanced" absorption spectrum . Improved Resolving Power Figure 6-7 shows an exact overlay of magnitude and "enhanced" absorption spectra for the simulated data of Figure 6-6. Apart from the predicted (and observed) reduction in line width, the two-fold reduction in R.M.S. noise of the "enhanced" absorption relative to magnitude mode 180 Absorption Magnitude Enhanced Absorption Figure 6-6 Three different spectral representations of the same simulated time-domain sum of three exponentially damped sinusoids of relative amplitude 10:1:3, with added Gaussian-distributed random noise. Note that the spurious noise spike (denoted by an arrow in each spectrum is least prominent in the "enhanced" absorption spectrum. 181 Magnitude Enhanced Absorption Figure 6-7 Two different overlaid spectral representations (with the same scale and baseline) of the data shown in Figure 6-6. Relative to the magnitude mode spectrum (upper trace), the "enhanced" absorption mode spectrum (lower trace) has -\[3 higher resolving power, the same noise standard deviation from the mean noise value, and just half the root-mean-square noise (see Tables 6-1 and 6-2). 182 is clearly apparent. This spectrum illustrates the importance of including the baseline in magnitude-mode spectra. Finally, Figure 6-8 shows absorption, magnitude, and "enhanced" FT/ICR mass spectra of N2 +. The theoretical advantages claimed in Table 6-1 and observed for the simulated data of Table 6-2 and Figures 6-5 - 6-7 are clearly evident in the experimental spectra of Figure 6-8. Although the observed enhancements in signal-to-noise ratio and resolving power are somewhat less (see Table 6-3) than the theoretical limits of Table 6-1, the enhancements are clearly evident. Having examined the various advantages of "enhanced" absorption mode display, we shall next consider the various limitations of the method. Correlated Noise Any filtering of noise before collection of the time-domain signal may introduce a partial to complete correlation between the noise in the absorption and dispersion spectra. Zero-filling, as previously discussed, also results in complete correlation of absorption and dispersion noise. Any such correlation leads to a corresponding reduction in the cancellation of noise when the Hilbert-transformed dispersion (i.e., "pseudo-absorption) spectrum is added to the absorption spectrum. In the worst case of completely correlated noise, there is no reduction in the final standard deviation noise, and therefore no improvement in signal-to-noise ratio or precision in the "enhanced" absorption spectrum compared to the original (unenhanced) absorption spectrum). 183 N2+ FT/ICR/MS Absorption Magnitude Enhanced Absorption Figure 6-8 Three different spectral representations of the same experimental FT/ICR time-domain signal from N 2 + ions at 3.0 tesla. As for the theoretical (Table 6-1) and simulated (Table 6-2, Figures 6-5 - 6- 7) data, the "enhanced" absorption spectrum offers higher resolution than magnitude mode and higher signal-to-noise ratio than (unenhanced) absorption mode. 184 Discrete Data Limitations Any manipulation of discrete data (including this one) is ultimately limited by the discrete nature of the data. If the time-domain acquisition period is truncated before the exponentially damped time- domain signal has decayed by less than a few (about 3) damping periods, then there will be very few points per absorption or dispersion spectral peak. The fewer points per peak, the less precise is the determination of the center of the peak and the overall lineshape [2-3]. Depending on the precise center of a very narrow frequency peak, either the absorption or dispersion component can be severely distorted (see Figures 4-6 and 4-7). Since the present method depends on accurate location and representation of peaks in the dispersion spectrum, the Hilbert transform of a poorly defined dispersion peak will produce a "pseudo-absorption" peak which may be shifted and/or distorted relative to its corresponding peak in the original absorption spectrum. Although noise will still be reduced by -^2 (Table 6-1), in the worst case the "enhanced" absorption signal may be degraded to such an extent as to nullify that gain. It should be noted, however, that this same limitation also affects ordinary absorption spectra. In fact, depending on the exact frequency, a narrow peak in an "enhanced" absorption spectrum can be many times larger than the corresponding absorption peak. Digital resolution can be improved by increasing the acquisition time, and, if necessary, extremely narrow, poorly defined peaks can first be broadened by apodization. 185 Truncation of Peaks near Either Edge of the Spectrum Perhaps the most important limitation of the present method concerns peaks located within a few linewidths of either end of the spectrum. Because the dispersion lineshape is much broader than the absorption lineshape, and because the domain of integration of a Hilbert transform should extend to -20 line widths (in units of full-width at half height) on either side of a given frequency-domain peak [91, the accuracy of the Hilbert transform is degraded severely for peaks near the edge of the spectrum. Thus, Hilbert-transformed peaks near either edge of the spectrum will exhibit reduced absorption mode amplitude and/or degraded lineshape. (However, the resolving power of the "enhanced" absorption is undiminished even to within one line width of either edge of the spectrum.) In order to test for the effect of truncation of the dispersion signal on the quality of the "enhanced" absorption spectrum, the signal-to- noise ratio for "enhanced" absorption spectra produced from highly- dam ped (T/t =8) simulated data was examined in cases where the peak was separated by a variable number of line widths from either edge of the spectrum. As expected, the improvement in signal-to-noise ratio of the enhanced versus the absorption spectrum is degraded when the signal frequency approaches either the zero-frequency or Nyquist- frequency end of the spectrum. Figure 6-9 shows the results for the high-frequency end of the spectrum. The results for the low-frequency end of the spectrum were similar, but slightly worse (Figure 6-10). In the extreme case that a highly peak is located within about five linewidths from either end of the spectrum, the "enhanced" absorption 186 2 ■ 4—>2 2 * 1 V to o n 0.5 c < CO o JZ 0.0 0 20 40 60 80 Linewidths from High-Frequency End of Bandwidth Figure 6-9 Relative performance of the "enhanced" absorption mode spectrum (as measured by the relative ratio of signal to standard deviation noise) as a function of peak displacement away from the high- frequency (Nyquist frequency) end of the spectrum (measured in multiples of the peak full width at half-maximum peak height). 187 O a 0.5 < 0.0 0 10 20 3040 50 6070 80 Linewidths from Zero-Frequency End of Bandwidth Figure 6-10 Relative performance of the "enhanced" absorption mode spectrum {as measured by the relative ratio of signal to standard deviation noise) as a function of peak displacement away from the zero- frequency end of the spectrum (measured in multiples of the peak full width at half-maximum peak height). 188 signal-to-noise ratio actually becomes worse than that of the conventional absorption mode. When the data of Figures 6-9 and 6-10 are replotted as a function of the logarithm of the number of line widths from the end of the spectrum, the degradation in signal-to-noise ratio is nearly linear from 0-20 linewidths away from the high-frequency end of the spectrum, and from 0-30 linewidths away from the low frequency end. In general, as long as any signal peaks are at least 20 linewidths away from the high-frequency end or 30 linewidths from the low- frequency end of the spectrum, the "enhanced" absorption spectral improvement in signal-to-noise ratio is undiminished. For the 4K data set examined, 30 linewidths of the highly damped signal corresponded to just 80 frequency domain data points, or less than 2% of the total frequency domain data. Only peaks within five linewidths (about 12 frequency domain points, or 0.3% of the total spectrum) were actually degraded compared to normal absorption peaks. This is likely to be the "worst case" scenario. Typical ICR experimental data are seldom observed for much longer than T/x=4 (since by this time the signal has decayed to less than 2% of its initial amplitude, and is usually overwhelmed by noise), which means their peak line widths (and the spectral region where truncation occurs) would be only half that of the values given for the simulated data above. Therefore, truncation effects are not likely to be a problem for most experimental data, and can be avoided in some cases by simple adjustment of the frequency bandw idth. 189 Conclusion Whenever the FT absorption and dispersion data meet the necessary conditions - phase-corrected spectrum, uncorrelated noise, digitally well defined peak shape (e.g., data acquired for several damping periods, or apodized), and peaks well-removed from either end of the spectrum - the "enhanced" absorption spectrum offers various advantages over conventional absorption mode or magnitude mode or zero-filled absorption mode spectra. The combination of absorption mode resolving power with magnitude mode precision, along with the reduced R.M.S. baseline noise and compact data length, makes the "enhanced" absorption mode advantageous for most forms of FT spectroscopy. 190 REFERENCES 1. Bartholdi, E.; Ernst, R. R. J. Magn. Reson. 1973, 11, 9. 2. Chen, L.; Cottrell, C. E.; M arshall, A. G. Chemometrics and Intelligent Laboratory Systems 1986, I, 51. 3. Liang, Z.; M arshall, A. G. Appl. Spectrosc. 1990, 44, 766. 4. Marshall, A. G.; Verdun, F. R. Fourier Transforms in NMR, Optical, and Mass Spectrometry: A User's Handbook. Elsevier: Amsterdam. 1990. 5. Verdun, F. R.; Giancaspro, C.; M arshall, A. G. Appl. Spectrosc. 1988, 42, 715. 6. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes. Cambridge University Press: New York. 1986. 7. D unbar, R. C. Appl. Spectrosc. 1990, 44, 1547. 8. Roth, K.; Kimber, B. J. Magn. Reson. Chem. 1985, 23, 832. 9. Herring, F. G.; M arshall, A. G.; Phillips, P. S.; Roe, D. C. J. Mag. Res. 1980, 37, 293. 191 APPENDIX 5-1 ATARI ST - GFA BASIC V2.0 HILBERT.LST Subroutine ENHANCED.LST Generate Enhanced Absorption from Phase- Corrected (and Baseline-Corrected) Absorption-Dispersion data This routine calls the FHT subroutine listed in Chapter IV. This routine can be merged into that program code listing. This routine converts a phase- and baseline- corrected NDP-point spectrum in S_DATA(NDP) to an NDP/2-point Enhanced Absorption spectrum. The original absorption data is in the first NDP/2 points of S_DATA(NDP), and the original dispersion data is in the last NDP/2-points of S_DATA(NDP). NDP and S_data(NDP) are assumed to be v a l i d . It is invoked by: GOSUB ENHANCED The output value of NDP is cut in half. The output Enhanced Absorption spectrum is in S_DATA(NDP). Procedure Enhanced I N2=Ndp/2 ' Switch the original absorption and dispersion data F or I%=0 To N 2-1 T em p = S _d ata(1%) S_data(1%)=S_data(N2+I%) S_data(N2+I%)=Temp N e x t 1% I ' Invert the dispersion data element order 192 For I%=0 To N 2-1 T em p = S _d ata(1%) S_data(1%)=S_data(Ndp-I%) S_data(Ndp-I%)=Temp N e x t 1% f Ndp=N2 ! Do FHT only on the half of S_DATA() with the dispersion data Gosub F ht l ' Signum function (invert sign on half the data) For I%=Ndp/2 To Ndp-1 S _ d a t a (1%)= - S _ d a t a (1%) N e x t 1% I Gosub F ht f ' Dispersion data is now in Absorption mode ' Now add back the original Absorption data and ' take the average For I%=0 To Ndp-1 ! Remember, Ndp is now N2 S_data(I% )=(S_data(1%)+S_data(Ndp+I%))/2 N e x t 1% I ' Done! Note: NDP is half its original value. I R etu rn (It should prove straightforward to convert this routine to any other language). 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