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Signal processing of transient atomic absorption signals
SPECTROCHIMICA ACTA PART B
ELSEVIER
Spectrochimica Acta Part B 50 (1995) 1531-1541
Signal processing of transient atomic absorption signals Uma Kale, Edward Voigtman* Chemist~. Department, Lederle GRC Tower, University of Massachusetts, Box 34510, Amherst, MA 01003-4510, USA Received 3 February 1995; Accepted 2 June 1995
Abstract
This article is an electronic publication in Spectrochimica Acta Electronica (SAE), the electronic section of Spectrochimica Acta Part B (SAB). The hard copy text is accompanied by a disk with a demonstration version of the simulation program, libraries of electronic and optical component blocks, simulation models, manual, and other files. Absorbance signals for electrothermal atomization atomic absorption spectroscopy (ETA-AAS) were generated digitally and the effect of various types and sources of noise upon the precision of the absorbance measurement was evaluated by numerical calculation. Peak area measurement, peak height measurement, and matched filtering were used for processing these signals. The performance of these three techniques in the presence of various types of noises and the sensitivity of each to small variation in the atomization conditions was calculated. It is demonstrated that significant improvement in signalto-noise ratios can be realized by application of appropriate signal processing methods. The results also indicate that one of the principal causes for loss of precision could be the variation in the heating characteristics of the furnace. Keywords: Electrothermal atomization atomic absorption spectroscopy; Noise; Precision; Signal-tonoise ratio
1. I n t r o d u c t i o n
The extensive use of microcomputers in the development of analytical instrumentation provides the opportunity for application of signal processing techniques which may be unsuitable for implementation with conventional analog instrumentation. In many cases, the most commonly employed or the most convenient signal processing methods are not necessarily the optimum methods, and significant improvement in the precision of the measurement can be achieved by application of the appropriate signal processing scheme. For example, the relative advantages of peak height and peak area measurement techniques for transient signals have
* Corresponding author. * This article is an electronicpublicationin SpectrochimicaActa Electronica (SAE), the electronicsection of Spectrochimica Acta Part B. The accompanyingdisk is identifiedas "ETA-AAS Simulations",Spectrochimica Acta Electronica, 50B (1995) 1531. Readers of this journal are permitted to copy the contents of the disk for their personal use. Note "Copyright" and "Disclaimer" at the end of this article, and the "Instructions for Authors", published elsewhere in this issue. 0584-8547/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0584-8547(95)01380-6
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[ 1,2]. Although peak area measurements are known to provide superior signal-to-noise (S/N) performance when the peak is in additive white noise [3], peak height measurement is widely employed due to its ease of implementation. The appropriate signal processing scheme for a particular signal is dictated by the nature of the signal and the noise, and requires knowledge of the behavior of both. Unfortunately, the dominant source and type of noise in an experiment are seldom quantitatively characterized. The aim of the investigation reported here was to demonstrate that the statistical behavior of the noise strongly influences the performance of any signal processing scheme, and cannot be ignored as is most commonly the case. The focus of this investigation was the optimum measurement, with respect to S/N ratio, of electrothermal atomization atomic absorption spectroscopy (ETA-AAS) signals. The noise which limits the precision of the measurement of these signals may be separated into two types. The first type results in random fluctuation of the signal induced by the noise on the light source and the noise contributed by the electronic components. The origin of the second type of noise lies in the changes in the signal profile caused by changes in the characteristics of the atomizer, atomization conditions or the sample matrix. For example, the atomization temperature varies for different analytes, analyte concentrations and the matrix composition. It is also known that excessively high atomization temperatures lead to interference due to the vaporization of less volatile matrix compounds and may also degrade the surface of the atomizer. All these factors can cause variation, from one run to the next, in the shape of the signal, its area, peak location and peak height [4]. Three different signal process schemes. namely. peak area measurement by gated integration, peak height measurement and matched filtering were applied to the measurement of ETA-AAS signals. The sensitivity of each technique to modest changes in the shape of the signal was investigated. The effect of various kinds of noises on the S/N ratio of the measurement was determined in an attempt to identify the major factors contributing to loss in precision. All results were obtained by numerical calculation using commercially available software (Extendr”, Imagine That, Inc., San Jose, CA and Lightstone LabsTM). A comprehensive description of the simulation software may be found in previously published papers and references cited therein 15-71. long been debated
2. Experimental 2.1. Signal generation The time evolution of a plausible ETA-AAS signal is described by the following equation given by Dawson and co-workers [S]
differential
Here k,, is an experimentally determined factor (s-l) related to the rate constant for atom formation, A. is the absorbance if all analyte atoms were simultaneously present in the cell, B is the ratio of activation energy for producing a free atom and the gas constant K, C is the linear heating rate (K s-l), r, is the atomizer temperature corresponding to the first appearance of the absorption signal and k, is rate constant for atom removal (s-l) from the furnace. Transient ETA-AAS signals were generated by solving this equation numerically, using the same values for each of the variables as were used by Dawson et al. [8] in their paper which described the simulation of ETA-AAS signals for Ag. Here, kO = 20 s-l, A0 = 0.5 (corresponding to 50 pg of Ag), B = 21000 K, C = 33 000 K s-r, r, = 1500 K and k2 = 6.7 s-l. In addition, the code for solving the differential equation was written so as to allow k,, and k2 to be varied within specified limits, to simulate variation in the atomization conditions from one run to the next. For instance, formation of stable compounds prior to atomization can cause variation in the rate constant for atom formation. The rate constant for atom removal also depends on several factors and could vary due to changes in flow rate of the inert gas or
U. Kale, E. Voigtman/Spectrochimica Acta Part B 50 (1995) 1531-1541
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changes in diffusion of the atom cloud caused by degeneration of the furnace surface. Fig. 1 shows five signals obtained by using different values of ko and k2 for each signal. The variation of the ko and k2 values was 3.0% relative to the specified values of 20 and 6.7 s-t, respectively. 2.2. Noise generation Noise with white and band-limited llf power spectral densities was generated digitally and summed with the signal. This allowed generation of signals with additive white or 1/f noise. Signals with multiplicative noise, i.e., noise proportional to the magnitude of the signal, were generated by multiplying the noise and the signal, and then summing the product with a clean, noiseless signal. Thus, the noise magnitude was maximum at the signal peak and least in the temporal wings of the transient. A different situation arises when the noise on the light source intensity is the dominant noise which limits the S/N ratio. In this case, the noise is expected to be least at the signal peak, where the transmitted light intensity is least, and increases as the signal magnitude decreases. However, since an absorbance measurement involves a ratio of two intensities, the dominant noise would be the additive noise from the reference intensity measurement. Thus, as long as the signal is small, this situation is not distinct from the additive noise case. For completeness, this case was also simulated, using the Mueller optical calculus to model the light source, but the results are only briefly summarized in the next section. In all cases the probability distribution function of the noise was Gaussian, its mean was zero and the standard deviation was 0.004. For the non-white noise, the power spectrum was l/f between 1 mHz and 50 Hz and it had a white power spectrum from 50 Hz to 500 Hz, the Nyquist frequency. 2.3. Gated integration Peak area measurements were performed by integrating the signal between specified time limits. The start and stop times for gated integration were selected so as to correspond to integration thresholds of 99.8%, 97.2%, 79.3%, 61.6%, 33.3% and 8.8% of the peak maximum. These integration limits correspond to intervals of 0.011, 0.094, 0.276, 0.405, 0.614 and 0.930 s, respectively. Fig. 2 shows the placement of the different gates around the peak maximum. In addition, the optimum gate width for gated integration of an ETA-AAS signal in Gaussian additive white noise was evaluated by using 100 different gate widths and a clean, noiseless signal with no variation in its ko or k2 values. The narrowest gate width was the full width at 016 0.140120.100.08 .El
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99.0% of the peak height, the next smallest one was the full width at 98.0% of the peak height and so on, and the widest gate integrated the entire peak. Absolute S/N ratios were not calculated but the peak areas obtained by integration were divided by the square root of the integration time to give relative S/N ratios. 2.4. Peak height measurement
Peak height measurement was implemented simply by recording the signal magnitude at the peak maximum known to occur at 0.475 s. In situations where the rate constants ko and k 2 were allowed to vary, peak detection was performed by recording the largest positive value generated. The peak height was measured in this manner because, as can be seen from Fig. 1, the peak maximum was not located at 0.475 s, owing to variations in the k0 and k2 values. 2.5. Matched filtering
Dawson and co-workers have reported that for ETA-AAS signals cross-correlation does not perform much better than gated integration [8], and is also a more complex procedure. It is, however, an extremely efficient tool for recovery of signals from data that would otherwise be useless. Here cross-correlation was implemented as matched filtering by using a timereversed copy of a digitally generated noiseless signal as the impulse response of a finite impulse response filter. The output of the filter was obtained by convolving the ETA-AAS signal with the impulse response of the filter. Fig. 3 shows the impulse response and the output of the matched filter on each simulation step. The output of the filter on the last simulation step, which is the optimum output corresponding to maximum overlap between the two functions, was recorded. Matched filtering is known to be the optimum signal processing scheme for signals in additive white noise if the signal shape is known [9]. It can be easily implemented on a computer and it imposes few timing constraints on the measurement process. For signals with an unknown shape, adaptive matched filtering was performed by using time reversed and averaged previous signals as the evolving impulse response of the filter. It was desirable to know how adaptive matched filtering performed in comparison to matched filtering so as to predict the applicability of this technique to signals of unknown shape. Matched filtering and adaptive matched filtering were performed only for the additive white noise case.
U. Kale, E. Voigtman/Spectrochimica Acta Part B 50 (1995) 1531-1541
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Fig. 3. The impulse response and typical output of the matched filter.
2.6. Simulation parameters Each simulation was run in 1401 steps with a time interval of 0.001 s between successive steps. The precision of each measurement was evaluated by repeating the simulation 1000 times. The mean signal was divided by the rms noise, i.e., the standard deviation obtained from the thousand measurements, to obtain the S/N ratio. All simulations were also run with a 3.0% rsd on the specified values of ko and k2 and were repeated at a S/N ratio worse by a factor of approximately 10, i.e., by using noise with a standard deviation of 0.04. This was done mainly to ensure that the S/N ratios scaled appropriately. The simulations were performed on a Macintosh Ilfx computer (Apple Computer, Inc., Cupertino, CA) with 8 M RAM.
3. Results and discussion The S/N ratios obtained by gated integration of the signal in the presence of additive white and l/f noise are plotted as a function of the integration gate width in Fig. 4. In additive white 700
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U. Kale, E. Voigtman/Spectrochimica Acta Part B 50 (1995) 1531-1541
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Fig. 5. Relative S/N ratio obtained for 100 different gate widths, |or signals in additive white noise.
noise an optimum integration time that is less than the full width of the signal and above which the S/N ratio remains almost constant, clearly exists. For this case the relative S/N ratio is plotted as a function of integration time in Fig. 5. The optimum integration interval, within 1.0% of the true optimum, was evaluated to be the full peak width at 38% of the peak height. This result is in good agreement with the rule of thumb for the additive white noise case which gives the optimum integration time to be the full width at one third of the peak maximum [10]. The curve is also rather flat around the maximum, indicating that rigid control of gate width is not crucial. As expected, and confirmed by the data in Fig. 4, additive low frequency noise has a seriously detrimental effect on the S/N ratio. A slight preference for narrow gates is evident, but no dramatic improvement in S/N ratios can be expected by use of the optimum gated integration parameters, unlike the corresponding white noise case for which the S/N ratio was a factor of 17 greater than that obtained by peak height measurement. The dependence of the S/N ratios upon the integration time, obtained in multiplicative white and l/f noise is shown in Fig. 6. In the presence of multiplicative white noise the precision improves with increased integration times, and integration of the entire signal peak is most 7000-
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1537
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desirable, as can be expected. The same is true for the l/f noise case although the curve is much flatter. Therefore for multiplicative noise, the spectral density function determines the absolute S/N ratio, but not the optimum integration gate width. The effect on S/N ratio of different integration gate widths, in the presence of 3% random variation in the specified ko and k2 values, is plotted in Figs. 7 and 8 respectively. The dependence of the integrated absorbance i.e., signal, on kn was more clearly evident in the presence of white noise than 1/f noise. In white noise the S/N ratio decreased nearly 28% compared with the additive noise case when ko was constant. The optimum gate width also changed and Fig. 7 indicates that if the signal shape varies, careful control of the gate width and location is necessary. For a changing signal in l/f noise the S/N ratios, and their dependence on gate width, were comparable to the additive l/f noise case. The rate constant for atom removal k2 showed a much stronger influence on the temporal signal profile. It is seen that in white noise, gated integration is very sensitive to small changes in the shape of the signal, resulting in a precipitous decrease in S/N ratios. This is because any decrease in the noise, achieved by integration of the signal, if offset by variation in the measured peak area resulting from the changing signal shape. 45 43
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U. Kale, E. Voigtman/Spectrochimica Acta Part B 50 (1995) 1531-1541
In the presence of additive l/f noise the S/N ratio appears to be relatively immune to changes in the signal shape. The shape of the curve and the S/N ratios, which are similar to the results in Figs. 4 and 7, indicate that additive l/f noise is dominant among the noise sources which limit the precision of the absorbance measurement. The optimum parameters for peak area measurement varied somewhat, but this is inconsequential in consideration of the insignificant improvement in S/N ratio that is achieved by use of optimum gate widths in the presence of l / f noise. The performance of peak height measurement, matched filtering and adaptive matched filtering is described in Table 1. In all cases, regardless of the noise statistics and type, peak height measurement was inferior to gated integration. This was true even in cases where the signal peak was known to have least noise at the maximum, i.e., the source noise limited case. The simulations showed that the source noise limited case was, for practical purposes, indistinguishable from the additive noise case as it was expected to be. With 1.0% rms noise on a source intensity of 0.001 roW, the S/N ratios were slightly lower than those for the additive noise case, but showed similar dependence on integration gate width. However, the S/N ratio for peak height measurement in 1/f noise was, in the worst case, lower than the S/N ratio obtained by gated integration by a factor of 3.7, and under these circumstances, peak height measurement may still be a viable measurement technique owing to its simplicity. For the white noise case, gated integration is clearly the technique of choice between the two. The maximum S/N ratio for signals in additive white noise, obtained by matched filtering, was 6.8 x 102. Thus, optimum gated integration is expected to yield S/N ratios within 90% of the maximum value achieved by matched filtering. Adaptive matched filtering also performed remarkably well, giving a S/N ratio of 6.6 x 102. This S/N ratio was obtained from a thousand repeated simulations. Matched filtering was also severely affected by variation in signal shape, giving a S/N ratio of 1.9 x 102 for a 3.0% deviation from the specified values of ko. A 3.0% variation in the rate constant for atom removal resulted in a S/N ratio of 39 for matched filtering, which is substantially lower than the precision obtained by gated integration. Similar results were obtained for adaptive matched filtering. On the other hand, peak height measurement was rather insensitive to the change in signal shape, decreasing by not more than a factor of two in the worst case, that of additive 1/f noise. All simulations were run with the noise level increased by a factor of ten, and the results obtained were consistent with the conclusions stated above. The S/N ratios were an order of magnitude lower but there was no change in the manner in which they changed with gate width and the rate constants.
Table 1 Signal-to-noise ratios for peak height measurements and matched filtering Noise type
Additive white Constant ko, k2 Variable ko Variable k2 Additive l/f Constant ko, k2 Variable ko Variable k2 Multiplicative white Multiplicative l/f
S/N ratio Peak detection
Matched filter
Adaptive matched filter
Gated integration ~' S/N ratio (gate width)
36 25 22
6.8 x 102 1.9 x 102 39
6.6 × 102 1.9 x 102 39
6.2 × 102 (0.614s) 4.5 × lO 2 (0.930 s) 47 (0.094 s)
28 12 l3 2.6 × 102 2.0 x 102
These S/N ratios are for the optimum gate width of the six gates used.
49 (0.276 44 (0.276 32 (0.094 6.8 x 103 3.9 x 102
s) s, 0.405 s) s, 0.276 s) (0.930s) (0.930 s)
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4. Conclusions It is seen from the preceding discussion that the scheme used for measurement of a signal has a strong impact on the quality of that measurement, and the choice cannot be arbitrary. Many factors influence it, the most important also being the one most frequently overlooked, namely the noise behavior. In general, it is accurate to say that peak area measurements are preferable over peak height measurements. Unfortunately, they are also sensitive to changes in the peak shape. Although Piepmeier [3] has pointed out that integrating only part of the signal gives improved performance, a major concern has been that this technique could be unreliable if the signal peak is unpredictably influenced by changing analysis conditions, sample matrix, and so on. From the results described above, this would not be a complication for peak height measurements. However, it can be seen that by integrating the entire peak this problem can be circumvented to obtain much better precision, as compared to peak height measurement, although the measurement may not be optimum. In some cases peak height measurement may provide additional information not accessible by peak area measurement [2] but if the prime consideration is precision of the measurement, peak area measurements are always superior regardless of the type and source of noise or its statistics. Although ETA-AAS signals were used in this investigation, the results are applicable to similar peak shapes i.e., single monotonically increasing and decreasing peaks, such as chromatographic peaks and FIA peaks. For ETA-AAS spectrometry in particular, two noise sources may be identified which limit the precision. The first is l / f noise from the light source and the electronic components or other sources and the second is variable atomization conditions. Increased precision in this case may be achieved by careful control of the heating characteristics of the furnace to avoid variation in the shape and appearance time of the signal. Another possible source of noise is imprecision in sample dosing. Although this would affect the peak area and height, the peak location would remain unchanged. If a small error in sample dosing causes an equal error in the peak area, then gated integration would not be significantly better than peak height measurement. For example, if the true absorbance varies 3% because of variation in sample size and this induces a 3% variation in the peak area, then the S/N ratio would be only 33, irrespective of gate width. This was confirmed by simulation, for 3% variation in the absorbance being the only noise source, the S/N ratio was ca. 34 for the various gate widths and for peak height measurement. Thus it is evident that the factors which cause changes in the area, peak height and location of the transient maximum may collectively be a more serious noise source than random fluctuation of the signal, especially if the latter is non-white in nature.
Acknowledgement This work was supported by the National Science Foundation through grant CHE-9108707.
Appendix The simulation software and files present on the disk ETA-AAS Simulations are described in this Appendix. Additional information about running simulations may be found in a manual on the disk. Note that the software is for use on Macintosh computers only. Extend T M is a commercially available Macintosh based simulation program. Briefly described, it allows users to model complex processes by creating a block diagram of it, where each block is programmed to represent one part of the process. Additionally, users may program their own blocks. The demonstration version of the program is included on the accompanying disk with the permission of Imagine That, Inc. None of Imagine That's libraries of blocks that accompany the program are included on the disk ETA-AAS Simulations as those blocks are not required for the situations described here. For additional information about Extend, contact Imagine That Inc., 6830 Via Del Oro, Suite 230, San Jose, CA 95119-1353, USA.
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U. Kale, E. Voigtman/Spectrochimica Acta Part B 50 (1995) 1531-1541
Table A I The extracted folder ETA-AAS Simulations contains the following files File name
File Format
Description
I 2 3 4 5 6 7 8 9 10
Extend demo Folder Extend model Extend model Extend model Extend model Extend model MS Word 4.0 MS Word 4.0 MS Word 4.0
Demonstration version of ExtendT M 2.0.6 Set of four libraries (demo) Simulation model Simulation model Simulation model Simulation model Simulation model Manual for running simulations Manuscript of the paper Title and Abstract
ExtendTM Demo Lightstone Labs Lite TM ETA-AAS Signals Additive White Noise Gated Integration Matched Filtering Adaptive Matched Filtering Manual Manuscript Title and Abstract
Lightstone Labs TM is a set of six libraries, and a complete manual describing their use, fully compatible with Extend TM 2.0.6 or higher. Collectively, the libraries contain nearly 260 optical and electronic component blocks which can be used for realistic simulation of spectrometric techniques, as has been described in previous papers [5-7]. Lightstone Labs TM can be used for simulation of instrumental analysis techniques which use a polychromatic or monochromatic light source. A demonstration version of Lightstone Labs Lite TM, which is included with this paper, is a subset of Lightstone Labs TM, and contains blocks to be used in monochromatic simulations only. Information about Lightstone Labs TM or Lightstone Labs Lite TM, both commercially available, may be obtained from the corresponding author who may be contacted at the given address or at the Internet address [email protected]. The files contained on the disk E T A - A A S Simulations accompanying this electronic publication have been compacted into a self-extracting archive named E T A - A A S Simulations.sea. The simulation software (demonstration version of Extend TM 2.0.6, libraries and models) are for use on a Macintosh TM system only. A Macintosh 11 or higher system is preferable, but the minimum requirement is a Macintosh Plus with 4 M RAM, operating system 6.0.3 and at least 6 M free storage space on the hard disk. The models used to generate the results are present on the disk but the figures and tables in the manuscript are not included. It is strongly suggested that the manual on disk be read
completely, and perhaps be printed for reference, prior to running any simulations. The files may be extracted from the archive onto the hard disk in the following manner. (1) Ensure that the disk is locked and that at least 6 M of space is available on the hard disk for storage. (2) Turn off any virus protection program. (3) Copy the file E T A - A A S Simulations.sea on to the hard disk. (4) To recover the flies from the archive, double-click on this file and click the Extract button in the window which appears. (5) After the files have been de-compacted delete the file E T A - A A S Simulations.sea. Any virus protection programs may be re-activated now.
Copyright The demonstration files, most of the code in the library blocks, the manual, and the hard copy text, in their totality published as a paper in Spectrochimica Acta Electronica, are copyrighted by the corresponding author. The demonstration version of Extend 2.0.6, and portions of the code of six library blocks, as indicated in their individual help file text, is copyrighted by Imagine That, Inc. and is used by permission. Readers of Spectrochimica Acta Electronica are permitted by the Publisher, Elsevier Science B.V. to make a copy of the material on the disk for their own private, non-commercial use, and to run the program according to the instructions provided by the authors. No charge for any copies may be requested, neither may the program or any modified version of it be sold or used for commercial purposes.
U. Kale, E. V o i g t m a n / S p e c t r o c h i m i c a Acta Part B 50 (1995) 1531 1541
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Those who wish to use the program and demonstration files in a commercial environment should contact the authors at the address given in the hard copy paper. Programs of which the source code is made available by the authors may be freely modified by the readers. However, if a modified version is brought into the public domain, the original author and the journal reference should be clearly stated in all subsequent use and dissemination.
Disclaimer Neither the Authors nor the Publisher warrant that the program and demonstration files are free from defects, that it operates as designed, or that the documentation is accurate. Neither the Authors nor the Publisher are liable for any damage of whatever kind sustained through copying the disk and/or using the program and the demonstration files. By copying and/or using the program the reader of Spectrochimica Acta Electronica, acting as a user of an electronic publication published therein, agrees to the above terms and conditions.
References II1 [2] [31 [4] [51 16] 17] [81
B. Welz, Spectrochim. Acta Part B, 47 (1992) 1043. P.S, Doidge, Spectrochim. Acta Part B, 48 (1993) 473. E.H. Piepmeier, Anal.Chem, 48 (1976) 1296. J.D. Ingle and S.R. Crouch, Spectrochemical Analysis. Prentice-Hall, New Jersey, 1988. E. Voigtman, Anal. Chem. 65 (1993) 1029A. U. Kale and E. Voigtman, Analyst, 120 (1995) 325. E. Voigtman, A.I.Yezefovsky and R.G. Michel, Spectrochim. Acta Part B, 49 (1994) 1629. J.B. Dawson, R.J. Duffield, P.R. King, M. Hajizadeh-Saffar and G.W.Fisher, Spectrochim. Acta Part B, 43 (1988) 1133. 191 H.J. Blinchikoff and A.I. Zverev, Filtering in the Time and Frequency Domains, John Wiley and Sons, New York, 1987. [10] E. Voigtman, Appl. Spectrosc., 45 (1990) 237.
ELSEVIER
Spectrochimica Acta Part B 50 (1995) 1531-1541
Signal processing of transient atomic absorption signals Uma Kale, Edward Voigtman* Chemist~. Department, Lederle GRC Tower, University of Massachusetts, Box 34510, Amherst, MA 01003-4510, USA Received 3 February 1995; Accepted 2 June 1995
Abstract
This article is an electronic publication in Spectrochimica Acta Electronica (SAE), the electronic section of Spectrochimica Acta Part B (SAB). The hard copy text is accompanied by a disk with a demonstration version of the simulation program, libraries of electronic and optical component blocks, simulation models, manual, and other files. Absorbance signals for electrothermal atomization atomic absorption spectroscopy (ETA-AAS) were generated digitally and the effect of various types and sources of noise upon the precision of the absorbance measurement was evaluated by numerical calculation. Peak area measurement, peak height measurement, and matched filtering were used for processing these signals. The performance of these three techniques in the presence of various types of noises and the sensitivity of each to small variation in the atomization conditions was calculated. It is demonstrated that significant improvement in signalto-noise ratios can be realized by application of appropriate signal processing methods. The results also indicate that one of the principal causes for loss of precision could be the variation in the heating characteristics of the furnace. Keywords: Electrothermal atomization atomic absorption spectroscopy; Noise; Precision; Signal-tonoise ratio
1. I n t r o d u c t i o n
The extensive use of microcomputers in the development of analytical instrumentation provides the opportunity for application of signal processing techniques which may be unsuitable for implementation with conventional analog instrumentation. In many cases, the most commonly employed or the most convenient signal processing methods are not necessarily the optimum methods, and significant improvement in the precision of the measurement can be achieved by application of the appropriate signal processing scheme. For example, the relative advantages of peak height and peak area measurement techniques for transient signals have
* Corresponding author. * This article is an electronicpublicationin SpectrochimicaActa Electronica (SAE), the electronicsection of Spectrochimica Acta Part B. The accompanyingdisk is identifiedas "ETA-AAS Simulations",Spectrochimica Acta Electronica, 50B (1995) 1531. Readers of this journal are permitted to copy the contents of the disk for their personal use. Note "Copyright" and "Disclaimer" at the end of this article, and the "Instructions for Authors", published elsewhere in this issue. 0584-8547/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0584-8547(95)01380-6
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[ 1,2]. Although peak area measurements are known to provide superior signal-to-noise (S/N) performance when the peak is in additive white noise [3], peak height measurement is widely employed due to its ease of implementation. The appropriate signal processing scheme for a particular signal is dictated by the nature of the signal and the noise, and requires knowledge of the behavior of both. Unfortunately, the dominant source and type of noise in an experiment are seldom quantitatively characterized. The aim of the investigation reported here was to demonstrate that the statistical behavior of the noise strongly influences the performance of any signal processing scheme, and cannot be ignored as is most commonly the case. The focus of this investigation was the optimum measurement, with respect to S/N ratio, of electrothermal atomization atomic absorption spectroscopy (ETA-AAS) signals. The noise which limits the precision of the measurement of these signals may be separated into two types. The first type results in random fluctuation of the signal induced by the noise on the light source and the noise contributed by the electronic components. The origin of the second type of noise lies in the changes in the signal profile caused by changes in the characteristics of the atomizer, atomization conditions or the sample matrix. For example, the atomization temperature varies for different analytes, analyte concentrations and the matrix composition. It is also known that excessively high atomization temperatures lead to interference due to the vaporization of less volatile matrix compounds and may also degrade the surface of the atomizer. All these factors can cause variation, from one run to the next, in the shape of the signal, its area, peak location and peak height [4]. Three different signal process schemes. namely. peak area measurement by gated integration, peak height measurement and matched filtering were applied to the measurement of ETA-AAS signals. The sensitivity of each technique to modest changes in the shape of the signal was investigated. The effect of various kinds of noises on the S/N ratio of the measurement was determined in an attempt to identify the major factors contributing to loss in precision. All results were obtained by numerical calculation using commercially available software (Extendr”, Imagine That, Inc., San Jose, CA and Lightstone LabsTM). A comprehensive description of the simulation software may be found in previously published papers and references cited therein 15-71. long been debated
2. Experimental 2.1. Signal generation The time evolution of a plausible ETA-AAS signal is described by the following equation given by Dawson and co-workers [S]
differential
Here k,, is an experimentally determined factor (s-l) related to the rate constant for atom formation, A. is the absorbance if all analyte atoms were simultaneously present in the cell, B is the ratio of activation energy for producing a free atom and the gas constant K, C is the linear heating rate (K s-l), r, is the atomizer temperature corresponding to the first appearance of the absorption signal and k, is rate constant for atom removal (s-l) from the furnace. Transient ETA-AAS signals were generated by solving this equation numerically, using the same values for each of the variables as were used by Dawson et al. [8] in their paper which described the simulation of ETA-AAS signals for Ag. Here, kO = 20 s-l, A0 = 0.5 (corresponding to 50 pg of Ag), B = 21000 K, C = 33 000 K s-r, r, = 1500 K and k2 = 6.7 s-l. In addition, the code for solving the differential equation was written so as to allow k,, and k2 to be varied within specified limits, to simulate variation in the atomization conditions from one run to the next. For instance, formation of stable compounds prior to atomization can cause variation in the rate constant for atom formation. The rate constant for atom removal also depends on several factors and could vary due to changes in flow rate of the inert gas or
U. Kale, E. Voigtman/Spectrochimica Acta Part B 50 (1995) 1531-1541
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changes in diffusion of the atom cloud caused by degeneration of the furnace surface. Fig. 1 shows five signals obtained by using different values of ko and k2 for each signal. The variation of the ko and k2 values was 3.0% relative to the specified values of 20 and 6.7 s-t, respectively. 2.2. Noise generation Noise with white and band-limited llf power spectral densities was generated digitally and summed with the signal. This allowed generation of signals with additive white or 1/f noise. Signals with multiplicative noise, i.e., noise proportional to the magnitude of the signal, were generated by multiplying the noise and the signal, and then summing the product with a clean, noiseless signal. Thus, the noise magnitude was maximum at the signal peak and least in the temporal wings of the transient. A different situation arises when the noise on the light source intensity is the dominant noise which limits the S/N ratio. In this case, the noise is expected to be least at the signal peak, where the transmitted light intensity is least, and increases as the signal magnitude decreases. However, since an absorbance measurement involves a ratio of two intensities, the dominant noise would be the additive noise from the reference intensity measurement. Thus, as long as the signal is small, this situation is not distinct from the additive noise case. For completeness, this case was also simulated, using the Mueller optical calculus to model the light source, but the results are only briefly summarized in the next section. In all cases the probability distribution function of the noise was Gaussian, its mean was zero and the standard deviation was 0.004. For the non-white noise, the power spectrum was l/f between 1 mHz and 50 Hz and it had a white power spectrum from 50 Hz to 500 Hz, the Nyquist frequency. 2.3. Gated integration Peak area measurements were performed by integrating the signal between specified time limits. The start and stop times for gated integration were selected so as to correspond to integration thresholds of 99.8%, 97.2%, 79.3%, 61.6%, 33.3% and 8.8% of the peak maximum. These integration limits correspond to intervals of 0.011, 0.094, 0.276, 0.405, 0.614 and 0.930 s, respectively. Fig. 2 shows the placement of the different gates around the peak maximum. In addition, the optimum gate width for gated integration of an ETA-AAS signal in Gaussian additive white noise was evaluated by using 100 different gate widths and a clean, noiseless signal with no variation in its ko or k2 values. The narrowest gate width was the full width at 016 0.140120.100.08 .El
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Fig. I. Effect of typical changes in atomization conditions (3.0% variation in ko and k2) on the signal shape.
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99.0% of the peak height, the next smallest one was the full width at 98.0% of the peak height and so on, and the widest gate integrated the entire peak. Absolute S/N ratios were not calculated but the peak areas obtained by integration were divided by the square root of the integration time to give relative S/N ratios. 2.4. Peak height measurement
Peak height measurement was implemented simply by recording the signal magnitude at the peak maximum known to occur at 0.475 s. In situations where the rate constants ko and k 2 were allowed to vary, peak detection was performed by recording the largest positive value generated. The peak height was measured in this manner because, as can be seen from Fig. 1, the peak maximum was not located at 0.475 s, owing to variations in the k0 and k2 values. 2.5. Matched filtering
Dawson and co-workers have reported that for ETA-AAS signals cross-correlation does not perform much better than gated integration [8], and is also a more complex procedure. It is, however, an extremely efficient tool for recovery of signals from data that would otherwise be useless. Here cross-correlation was implemented as matched filtering by using a timereversed copy of a digitally generated noiseless signal as the impulse response of a finite impulse response filter. The output of the filter was obtained by convolving the ETA-AAS signal with the impulse response of the filter. Fig. 3 shows the impulse response and the output of the matched filter on each simulation step. The output of the filter on the last simulation step, which is the optimum output corresponding to maximum overlap between the two functions, was recorded. Matched filtering is known to be the optimum signal processing scheme for signals in additive white noise if the signal shape is known [9]. It can be easily implemented on a computer and it imposes few timing constraints on the measurement process. For signals with an unknown shape, adaptive matched filtering was performed by using time reversed and averaged previous signals as the evolving impulse response of the filter. It was desirable to know how adaptive matched filtering performed in comparison to matched filtering so as to predict the applicability of this technique to signals of unknown shape. Matched filtering and adaptive matched filtering were performed only for the additive white noise case.
U. Kale, E. Voigtman/Spectrochimica Acta Part B 50 (1995) 1531-1541
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Fig. 3. The impulse response and typical output of the matched filter.
2.6. Simulation parameters Each simulation was run in 1401 steps with a time interval of 0.001 s between successive steps. The precision of each measurement was evaluated by repeating the simulation 1000 times. The mean signal was divided by the rms noise, i.e., the standard deviation obtained from the thousand measurements, to obtain the S/N ratio. All simulations were also run with a 3.0% rsd on the specified values of ko and k2 and were repeated at a S/N ratio worse by a factor of approximately 10, i.e., by using noise with a standard deviation of 0.04. This was done mainly to ensure that the S/N ratios scaled appropriately. The simulations were performed on a Macintosh Ilfx computer (Apple Computer, Inc., Cupertino, CA) with 8 M RAM.
3. Results and discussion The S/N ratios obtained by gated integration of the signal in the presence of additive white and l/f noise are plotted as a function of the integration gate width in Fig. 4. In additive white 700
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Fig. 4. Dependence of additive noise limited S/N ratio on the integration time: (11) white noise; (0) l/f noise, plotted to y-axis on the right.
U. Kale, E. Voigtman/Spectrochimica Acta Part B 50 (1995) 1531-1541
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Fig. 5. Relative S/N ratio obtained for 100 different gate widths, |or signals in additive white noise.
noise an optimum integration time that is less than the full width of the signal and above which the S/N ratio remains almost constant, clearly exists. For this case the relative S/N ratio is plotted as a function of integration time in Fig. 5. The optimum integration interval, within 1.0% of the true optimum, was evaluated to be the full peak width at 38% of the peak height. This result is in good agreement with the rule of thumb for the additive white noise case which gives the optimum integration time to be the full width at one third of the peak maximum [10]. The curve is also rather flat around the maximum, indicating that rigid control of gate width is not crucial. As expected, and confirmed by the data in Fig. 4, additive low frequency noise has a seriously detrimental effect on the S/N ratio. A slight preference for narrow gates is evident, but no dramatic improvement in S/N ratios can be expected by use of the optimum gated integration parameters, unlike the corresponding white noise case for which the S/N ratio was a factor of 17 greater than that obtained by peak height measurement. The dependence of the S/N ratios upon the integration time, obtained in multiplicative white and l/f noise is shown in Fig. 6. In the presence of multiplicative white noise the precision improves with increased integration times, and integration of the entire signal peak is most 7000-
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1537
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desirable, as can be expected. The same is true for the l/f noise case although the curve is much flatter. Therefore for multiplicative noise, the spectral density function determines the absolute S/N ratio, but not the optimum integration gate width. The effect on S/N ratio of different integration gate widths, in the presence of 3% random variation in the specified ko and k2 values, is plotted in Figs. 7 and 8 respectively. The dependence of the integrated absorbance i.e., signal, on kn was more clearly evident in the presence of white noise than 1/f noise. In white noise the S/N ratio decreased nearly 28% compared with the additive noise case when ko was constant. The optimum gate width also changed and Fig. 7 indicates that if the signal shape varies, careful control of the gate width and location is necessary. For a changing signal in l/f noise the S/N ratios, and their dependence on gate width, were comparable to the additive l/f noise case. The rate constant for atom removal k2 showed a much stronger influence on the temporal signal profile. It is seen that in white noise, gated integration is very sensitive to small changes in the shape of the signal, resulting in a precipitous decrease in S/N ratios. This is because any decrease in the noise, achieved by integration of the signal, if offset by variation in the measured peak area resulting from the changing signal shape. 45 43
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Fig. 8. Effect of variation in k2 on the dependence of S/N ratio upon gate width: (11) white noise: (Q) 1(/" noise, plotted to y-axis on the right.
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In the presence of additive l/f noise the S/N ratio appears to be relatively immune to changes in the signal shape. The shape of the curve and the S/N ratios, which are similar to the results in Figs. 4 and 7, indicate that additive l/f noise is dominant among the noise sources which limit the precision of the absorbance measurement. The optimum parameters for peak area measurement varied somewhat, but this is inconsequential in consideration of the insignificant improvement in S/N ratio that is achieved by use of optimum gate widths in the presence of l / f noise. The performance of peak height measurement, matched filtering and adaptive matched filtering is described in Table 1. In all cases, regardless of the noise statistics and type, peak height measurement was inferior to gated integration. This was true even in cases where the signal peak was known to have least noise at the maximum, i.e., the source noise limited case. The simulations showed that the source noise limited case was, for practical purposes, indistinguishable from the additive noise case as it was expected to be. With 1.0% rms noise on a source intensity of 0.001 roW, the S/N ratios were slightly lower than those for the additive noise case, but showed similar dependence on integration gate width. However, the S/N ratio for peak height measurement in 1/f noise was, in the worst case, lower than the S/N ratio obtained by gated integration by a factor of 3.7, and under these circumstances, peak height measurement may still be a viable measurement technique owing to its simplicity. For the white noise case, gated integration is clearly the technique of choice between the two. The maximum S/N ratio for signals in additive white noise, obtained by matched filtering, was 6.8 x 102. Thus, optimum gated integration is expected to yield S/N ratios within 90% of the maximum value achieved by matched filtering. Adaptive matched filtering also performed remarkably well, giving a S/N ratio of 6.6 x 102. This S/N ratio was obtained from a thousand repeated simulations. Matched filtering was also severely affected by variation in signal shape, giving a S/N ratio of 1.9 x 102 for a 3.0% deviation from the specified values of ko. A 3.0% variation in the rate constant for atom removal resulted in a S/N ratio of 39 for matched filtering, which is substantially lower than the precision obtained by gated integration. Similar results were obtained for adaptive matched filtering. On the other hand, peak height measurement was rather insensitive to the change in signal shape, decreasing by not more than a factor of two in the worst case, that of additive 1/f noise. All simulations were run with the noise level increased by a factor of ten, and the results obtained were consistent with the conclusions stated above. The S/N ratios were an order of magnitude lower but there was no change in the manner in which they changed with gate width and the rate constants.
Table 1 Signal-to-noise ratios for peak height measurements and matched filtering Noise type
Additive white Constant ko, k2 Variable ko Variable k2 Additive l/f Constant ko, k2 Variable ko Variable k2 Multiplicative white Multiplicative l/f
S/N ratio Peak detection
Matched filter
Adaptive matched filter
Gated integration ~' S/N ratio (gate width)
36 25 22
6.8 x 102 1.9 x 102 39
6.6 × 102 1.9 x 102 39
6.2 × 102 (0.614s) 4.5 × lO 2 (0.930 s) 47 (0.094 s)
28 12 l3 2.6 × 102 2.0 x 102
These S/N ratios are for the optimum gate width of the six gates used.
49 (0.276 44 (0.276 32 (0.094 6.8 x 103 3.9 x 102
s) s, 0.405 s) s, 0.276 s) (0.930s) (0.930 s)
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4. Conclusions It is seen from the preceding discussion that the scheme used for measurement of a signal has a strong impact on the quality of that measurement, and the choice cannot be arbitrary. Many factors influence it, the most important also being the one most frequently overlooked, namely the noise behavior. In general, it is accurate to say that peak area measurements are preferable over peak height measurements. Unfortunately, they are also sensitive to changes in the peak shape. Although Piepmeier [3] has pointed out that integrating only part of the signal gives improved performance, a major concern has been that this technique could be unreliable if the signal peak is unpredictably influenced by changing analysis conditions, sample matrix, and so on. From the results described above, this would not be a complication for peak height measurements. However, it can be seen that by integrating the entire peak this problem can be circumvented to obtain much better precision, as compared to peak height measurement, although the measurement may not be optimum. In some cases peak height measurement may provide additional information not accessible by peak area measurement [2] but if the prime consideration is precision of the measurement, peak area measurements are always superior regardless of the type and source of noise or its statistics. Although ETA-AAS signals were used in this investigation, the results are applicable to similar peak shapes i.e., single monotonically increasing and decreasing peaks, such as chromatographic peaks and FIA peaks. For ETA-AAS spectrometry in particular, two noise sources may be identified which limit the precision. The first is l / f noise from the light source and the electronic components or other sources and the second is variable atomization conditions. Increased precision in this case may be achieved by careful control of the heating characteristics of the furnace to avoid variation in the shape and appearance time of the signal. Another possible source of noise is imprecision in sample dosing. Although this would affect the peak area and height, the peak location would remain unchanged. If a small error in sample dosing causes an equal error in the peak area, then gated integration would not be significantly better than peak height measurement. For example, if the true absorbance varies 3% because of variation in sample size and this induces a 3% variation in the peak area, then the S/N ratio would be only 33, irrespective of gate width. This was confirmed by simulation, for 3% variation in the absorbance being the only noise source, the S/N ratio was ca. 34 for the various gate widths and for peak height measurement. Thus it is evident that the factors which cause changes in the area, peak height and location of the transient maximum may collectively be a more serious noise source than random fluctuation of the signal, especially if the latter is non-white in nature.
Acknowledgement This work was supported by the National Science Foundation through grant CHE-9108707.
Appendix The simulation software and files present on the disk ETA-AAS Simulations are described in this Appendix. Additional information about running simulations may be found in a manual on the disk. Note that the software is for use on Macintosh computers only. Extend T M is a commercially available Macintosh based simulation program. Briefly described, it allows users to model complex processes by creating a block diagram of it, where each block is programmed to represent one part of the process. Additionally, users may program their own blocks. The demonstration version of the program is included on the accompanying disk with the permission of Imagine That, Inc. None of Imagine That's libraries of blocks that accompany the program are included on the disk ETA-AAS Simulations as those blocks are not required for the situations described here. For additional information about Extend, contact Imagine That Inc., 6830 Via Del Oro, Suite 230, San Jose, CA 95119-1353, USA.
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Table A I The extracted folder ETA-AAS Simulations contains the following files File name
File Format
Description
I 2 3 4 5 6 7 8 9 10
Extend demo Folder Extend model Extend model Extend model Extend model Extend model MS Word 4.0 MS Word 4.0 MS Word 4.0
Demonstration version of ExtendT M 2.0.6 Set of four libraries (demo) Simulation model Simulation model Simulation model Simulation model Simulation model Manual for running simulations Manuscript of the paper Title and Abstract
ExtendTM Demo Lightstone Labs Lite TM ETA-AAS Signals Additive White Noise Gated Integration Matched Filtering Adaptive Matched Filtering Manual Manuscript Title and Abstract
Lightstone Labs TM is a set of six libraries, and a complete manual describing their use, fully compatible with Extend TM 2.0.6 or higher. Collectively, the libraries contain nearly 260 optical and electronic component blocks which can be used for realistic simulation of spectrometric techniques, as has been described in previous papers [5-7]. Lightstone Labs TM can be used for simulation of instrumental analysis techniques which use a polychromatic or monochromatic light source. A demonstration version of Lightstone Labs Lite TM, which is included with this paper, is a subset of Lightstone Labs TM, and contains blocks to be used in monochromatic simulations only. Information about Lightstone Labs TM or Lightstone Labs Lite TM, both commercially available, may be obtained from the corresponding author who may be contacted at the given address or at the Internet address [email protected]. The files contained on the disk E T A - A A S Simulations accompanying this electronic publication have been compacted into a self-extracting archive named E T A - A A S Simulations.sea. The simulation software (demonstration version of Extend TM 2.0.6, libraries and models) are for use on a Macintosh TM system only. A Macintosh 11 or higher system is preferable, but the minimum requirement is a Macintosh Plus with 4 M RAM, operating system 6.0.3 and at least 6 M free storage space on the hard disk. The models used to generate the results are present on the disk but the figures and tables in the manuscript are not included. It is strongly suggested that the manual on disk be read
completely, and perhaps be printed for reference, prior to running any simulations. The files may be extracted from the archive onto the hard disk in the following manner. (1) Ensure that the disk is locked and that at least 6 M of space is available on the hard disk for storage. (2) Turn off any virus protection program. (3) Copy the file E T A - A A S Simulations.sea on to the hard disk. (4) To recover the flies from the archive, double-click on this file and click the Extract button in the window which appears. (5) After the files have been de-compacted delete the file E T A - A A S Simulations.sea. Any virus protection programs may be re-activated now.
Copyright The demonstration files, most of the code in the library blocks, the manual, and the hard copy text, in their totality published as a paper in Spectrochimica Acta Electronica, are copyrighted by the corresponding author. The demonstration version of Extend 2.0.6, and portions of the code of six library blocks, as indicated in their individual help file text, is copyrighted by Imagine That, Inc. and is used by permission. Readers of Spectrochimica Acta Electronica are permitted by the Publisher, Elsevier Science B.V. to make a copy of the material on the disk for their own private, non-commercial use, and to run the program according to the instructions provided by the authors. No charge for any copies may be requested, neither may the program or any modified version of it be sold or used for commercial purposes.
U. Kale, E. V o i g t m a n / S p e c t r o c h i m i c a Acta Part B 50 (1995) 1531 1541
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Those who wish to use the program and demonstration files in a commercial environment should contact the authors at the address given in the hard copy paper. Programs of which the source code is made available by the authors may be freely modified by the readers. However, if a modified version is brought into the public domain, the original author and the journal reference should be clearly stated in all subsequent use and dissemination.
Disclaimer Neither the Authors nor the Publisher warrant that the program and demonstration files are free from defects, that it operates as designed, or that the documentation is accurate. Neither the Authors nor the Publisher are liable for any damage of whatever kind sustained through copying the disk and/or using the program and the demonstration files. By copying and/or using the program the reader of Spectrochimica Acta Electronica, acting as a user of an electronic publication published therein, agrees to the above terms and conditions.
References II1 [2] [31 [4] [51 16] 17] [81
B. Welz, Spectrochim. Acta Part B, 47 (1992) 1043. P.S, Doidge, Spectrochim. Acta Part B, 48 (1993) 473. E.H. Piepmeier, Anal.Chem, 48 (1976) 1296. J.D. Ingle and S.R. Crouch, Spectrochemical Analysis. Prentice-Hall, New Jersey, 1988. E. Voigtman, Anal. Chem. 65 (1993) 1029A. U. Kale and E. Voigtman, Analyst, 120 (1995) 325. E. Voigtman, A.I.Yezefovsky and R.G. Michel, Spectrochim. Acta Part B, 49 (1994) 1629. J.B. Dawson, R.J. Duffield, P.R. King, M. Hajizadeh-Saffar and G.W.Fisher, Spectrochim. Acta Part B, 43 (1988) 1133. 191 H.J. Blinchikoff and A.I. Zverev, Filtering in the Time and Frequency Domains, John Wiley and Sons, New York, 1987. [10] E. Voigtman, Appl. Spectrosc., 45 (1990) 237.