Signal averaging of photoacoustic FTIR data

Injrared Phys. Vol. 27, No. 5, pp. 287-296, Printed in Great Britain

SIGNAL

1987

0020-0891/87 $3.00 + 0.00 Pergamon Journals Ltd

AVERAGING FTIR

OF PHOTOACOUSTIC DATA

I. COMPUTATION OF SPECTRA FROM DOUBLE-SIDED LOW RESOLUTION INTERFEROGRAMS K.

Energy,

Mines and Resources

Canada,

H.

MICHAELIAN

CANMET, Coal Research Alberta TOC lE0, Canada

Laboratory,

P.O. Bag 1280, Devon,

(Received 19 February 1987) Abstract-Five different strategies for calculating average photoacoustic infrared spectra were applied to a series of 10 double-sided low resolution interferograms recorded for fusinite (a separated coal maceral). Both multiplicative phase correction and computation of modulus (power) spectra were considered. It was found that the sequence chosen for averaging and Fourier transformation of the interferograms makes only slight differences in the average spectra. When calculating average modulus spectra, the position of the interferogram centreburst must be constant in order to produce accurate results, a criterion that always applies to averaging interferograms. In multiplicative phase correction, it was found that the number of interferogram points used to calculate the phase function can vary over a significant range without producing any observable change in the computed spectrum.

INTRODUCTION

The majority of photoacoustic (PA) spectra acquired with rapid-scan Fourier Transform IR (FTIR) spectrometers are calculated from single-sided interferograms that have a small number (512 or less) of data points at negative path difference. Phase correction of these interferograms is usually accomplished with the Mertz multiplicative method, (I) although it has been statedc2) that this procedure is incapable of correcting for the significant variation in phase across PA absorption bands. The accuracy of the PA spectrum calculated using multiplicative phase correction can be expected to increase with the number of interferogram points used: as this number increases, more of the total intensity appears in the real spectrum, and less is lost to the imaginary spectrum.“) Computation of modulus (power) PA spectra is an alternative which might be considered; the principal drawback in this approach is, of course, the requirement of double-sided interferograms, with a concomitant increase in data acquisition time. In order to investigate the feasibility of using modulus PA spectra while avoiding this increase in measurement time, we have obtained PA interferograms in which the number of data points on the negative side of zero path difference is increased from the usual value of 5 12 to 1024, at the expense of a corresponding number of points at positive path difference. Truncation of these 4 K interferograms after 2 K points yields double-sided interferograms that allow calculation of low resolution modulus spectra, and also permit an increase in the number of points used for multiplicative phase correction. Because PA spectra of solids recorded with FTIR spectrometers tend to be somewhat weak and noisy, the question of signal averaging of PA data takes on added importance. When a number of double-sided interferograms are recorded under the same conditions, several approaches to averaging exist. These include the following: (a) averaging of the interferograms, followed by calculation of the modulus spectrum; (b) averaging of the interferograms, followed by Fourier transformation and multiplicative phase correction; (c) computation of individual modulus spectra, which are subsequently averaged; (d) calculation of individual spectra using multiplicative phase correction, followed by averaging; and (e) calculation of the “average modulus spectrum” (described further below) according to the method of Birth.(4) 287

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In this paper, we present results of a study in which all five of these methods have been applied to a series of ten double-sided PA interferograms recorded for fusinite, a separated coal maceral (optically homogeneous organic unit with distinctive physical and chemical properties). Criteria affecting the accuracy of method (e) are examined, as is the effect of variation of the number of interferogram points used for multiplicative phase correction. EXPERIMENTAL PA interferograms were recorded using a Bruker 113 V FTIR spectrometer, and a Princeton Applied Research Corporation model 6003 cell and model 6005 amplifier. The operation of this apparatus has been described previously. (‘) Standard Bruker software allowed collection of 4 K interferograms with the centreburst at the 1024th point (sampling at the fundamental He-Ne laser wavelength of 0.6329 pm), and a maximum retardation of approximately 0.2 cm. A Fortran program was used to reduce these interferogram files to a length of 2048 points, corresponding to a retardation of about 0.06 cm and a resolution of - 16 cm- ‘. Other programs developed in our laboratory were used to average interferograms, to average spectra and to calculate modulus spectra. Multiplicative phase correction is implemented in the standard Bruker procedure for computing a spectrum from an interferogram. RESULTS

AND

DISCUSSION

I. Characteristics of the interferograms A typical PA interferogram of fusinite, from which the last 2 K points have already been discarded, is shown in Fig. 1. The asymmetry of this interferogram is manifested in the corresponding phase function, which is plotted in Fig. 2. This phase dispersion is typical for MIR PA interferograms recorded in our laboratory and its general characteristics are observed for carbon black (an opaque substance whose PA spectrum is frequently used to correct for spectrometer response), kaolinite@’ and other clays, coals and polymers that we have investigated. While the overall shape of the phase curve is independent of the sample, it is important to note that the phase varies significantly across certain absorption bands, as is illustrated in Fig. 3. Distinct minima are observed at approximately 750, 800 and 870 cm- ‘, frequencies of characteristic absorption bands known to arise from out-of-plane vibrations of hydrogen atoms on substituted aromatic rings in coal. (‘I Another minimum is observed at 1600 cm -‘, the frequency of the ubiquitous “coal band” whose assignment has traditionally been the subject of debate.‘“’ Moreover, in addition to PA absorption bands, other factors can also produce similar phase behaviour: an important example occurs at - 520 cm-’ (Fig. 2), corresponding to a modulation frequency of 120 Hz (second harmonic of the line frequency). Another important characteristic of the PA interferogram is the exact location of the centreburst. For weakly absorbing samples, this peak location can vary slightly from one interferogram file to the next-an effect which has proven to be unimportant in routine work. In the current investigation, the central interferogram peak occurred at either the 1023rd or the 1024th point. As long as the actual peak location is correctly recorded, this slight variation is insignificant in multiplicative phase correction; in contrast, calculation of the average modulus spectrum is affected by changes in the position of the centreburst, as is shown below. Similarly, the averaging of interferograms must take into account any shift of the centreburst if meaningful results are to be obtained. 2. Averaging of interferograms vs averaging of spectra 2.1. Multiplicative phase correction. The spectra to be compared here are obtained by procedures (b) and (d) mentioned in the Introduction: phase correction and Fourier transformation are carried out using Bruker software, while averaging is accomplished in separate programs. 512 phase interferogram points are used for phase correction; the influence of this number on the results is discussed in Section 4. The CH stretching region (2700-3200cm-‘) of the two spectra is shown in Fig. 4; the trends illustrated in this region occur in the complete MIR spectra. The upper curve illustrates the result

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Fig. 1. (a) Double-sided PA interferogram for fusinite, a maceral (homogeneous organic unit) of coal; (b) central part of interferogram.

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Fig. 2. Phase angle, in units of TLradians, of the interferogram in Fig. 1

of averaging the individual spectra, whereas the lower curve is the spectrum obtained from the average interferogram. Subtraction readily shows that these spectra are virtually the same, their difference being of the order of 1% or less at any frequency. This result suggests that the sequence chosen for averaging and Fourier transformation is not significant when multiplicative phase correction is used.* There is a general tendency for the bands in the upper spectrum in Fig. 4 to be smoother (i.e. to contain less noise) than those in the lower one, which contains more inflections; consequently, the bands in the lower curve appear to have slightly greater definition. The slightly different band shapes that result contribute a small degree of uncertainty to some band positions. However, the differences between these average spectra are not very great, and one can conclude that methods (b) and (d) are essentially equivalent. 2.2. Modulus spectra. These spectra are obtained by procedures (a) and (c); no explicit calculation of the phase is required in the computation of modulus spectra. In contrast with the results in Section 2.1, the sequence chosen for averaging and Fourier transformation has a noticeable-though small-effect on the final modulus spectrum: the two results differ by a multiplicative factor of 1.02, the modulus spectrum calculated from the average interferogram being slightly more intense than the average of the individual modulus spectra. (A similar result has been observed for resinite (a different coal maceral), where the multiplicative factor is 1.03.) As is the case with multiplicative phase correction, smoother band shapes are obtained by averaging individually calculated modulus spectra, although at the expense of slightly greater overlap of neighbouring bands. Thus, the general trend is that averaging spectra produces a slightly smoother result than that obtained by first averaging interferograms, the difference between the two approaches being rather small. Therefore, the choice of averaging strategies might depend on relative computation times. 3. The average ~od~~~~ spectrum l3irch(4)suggested that, instead of averaging individually computed modulus spectra, the cosine and sine components of the complex spectrum be separately averaged as a first step in the *Recent investigation has shown that this conclusion does not always apply to PA interferogram centreburst is located at the 512th point. These results will be discussed more fully elsewhere.“)

files in which the

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Fig. 3. Phase variation across PA absorption bands. (a) Out-of-plane carbon-hydrogen “coal band”.

calculation of the average modulus spectrum. Adopting Birch’s notation, S(v) is: S(V) = c(v) f Q(V) + i(S(V) + C,(V)>

vibrations; (b)

the complex spectrum (I)

where c(v) and s(v) are the cosine and sine components of the spectrum, and E,(V) and 4(v) are the cosine and sine components of the noise spectrum respectively. The average of the individually calculated modulus spectra [method (c)J is: jj INF

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1 E(c(v)+ E,(v))2+ {s(v)+ c,(v))2]l’?

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Fig. 4. Comparison of CH stretching regions of average spectra calculated using multiplicative phase correction. Upper curve: average of individual spectra; lower curve: spectrum computed from average interferogram. Spectra are displaced vertically for clarity, and have not been ratioed against a reference carbon black spectrum.

Birch proposed calculation of an alternative average modulus spectrum [method (e)] according to:

which was expected to lead to a reduction of noise in the averaged spectrum, It is easily shown that equation (3) is mathematically equivalent to averaging interferograms prior to transfo~ation; this point has already been mentioned by Lowe et al. (‘O)Application of Birch’s method requires a stable interferometer;‘4~‘0’ the phase should be constant and the point about which the transform is taken must correspond to the same path difference in each interferogram.“‘) The relative importance of these two criteria in the calculation of average modulus PA spectra is shown below. As mentioned previously, ten fusinite interferogram files were recorded; the centreburst was located at the 1023rd point for 6 files and at the 1024th point for the others. This variation was taken into account in the calculation of the average interferogram in methods (a) and (b) by aligning interferogram centrebursts prior to averaging. (If this is not done, an inaccurate average interferogram is produced and the corresponding spectrum is weak and distorted.) In view of the comments of Lowe et aI., it is important to determine the effect of this variation in interferogram peak location on the calculation of the average modulus spectrum. A comparison of average modulus spectra is given in Fig. 5. The lower curve is the spectrum calculated according to method (e) using all 10 interferogram files, while the upper curve shows the modulus spectrum calculated from the average interferogram [procedure (a)]. Although the two spectra superimpose near 2000 cm-‘, they differ significantly in the region shown-frequencydependent behaviour reminiscent of that reported by Birch, (4)who attributed the discrepancy at low signal levels to a reduction in the noise component of the average modulus spectrum calculated according to equation (3). When procedure (e) is applied either to the four interferograms for which the centreburst occurs at the 1024th point or to the six interferograms in which the peak is located at the 1023rd point, a spectrum that virtually superimposes on the upper curve in Fig. 5 is obtained. This result illustrates the fact that the location of the interferogram centreburst must be constant when calculating the average modulus spectrum if accurate results are to be obtained. This is consistent with the discussion of BirchC4’and Lowe et aE.,“*)as well as our observations regarding averaging

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Fig. 5. Upper curve: modulus spectrum calculated from average interferogram; lower curve: average modulus spectrum calculated using all ten interferograms, i.e. both those with the centreburst at the 1023rd point and those having the centreburst at the 1024th point (see text). Spectra have been calculated without zero-filling or apodization.

interferograms; the general conclusion is that interferograms must be properly aligned (the centrebursts must correspond to the same optical path difference) before any averaging procedure is implemented. The phase spectra of the ten interferogram files studied in this work are illustrated in Fig. 6. These phase curves are quite similar, but certainly do not agree exactly with each other. Thus, with respect to calculation of average modulus spectra, constancy of the phase of the interferograms is not as important a criterion as the requirement that the interferogram centrebursts occur at the same path difference. 4. Further examination

of multiplicative

phase correction

As mentioned in the Introduction, an increase in the number of data points at negative path difference makes possible a corresponding increase in the number of interferogram points used for multiplicative phase correction. In this final section, the effect of variation of this number is examined. The greatest number of phase interferogram points (Bruker software parameter “PIP”) that can be used in multiplicative phase correction is the largest multiple of 2 less than or equal to twice the number of points up to and including the centreburst. Lower multiples of 2 are also permissible, and it is of interest to determine the minimum value of PIP that yields an acceptable result. Thus the six files in which the interferogram peak is at the 1023rd point can be transformed using PIP = 1024, 512, 256 etc. On the other hand, the four files having a peak location at the 1024th point-or the average of all 10 files, in which this peak was arbitrarily located at the same place-permit use of a PIP value of 2048 or less. This gives a wider possible range (2048-2) of points for multiplicative phase correction. (The use of only two points leads to the incorrect conclusion that the phase angle is 0 at all frequencies. Although the interferogram is still multiplied by a ramp function”) prior to transformation, errors in band intensities and/or positions can be expected in such a case.) In order to ascertain the lowest useful value of PIP, the average interferogram was transformed a number of times using the Bruker software [procedure (b)], varying the number of points from 2048 to 2. Careful examination of the 600-1000 cm-’ and 270&3200 cm-’ regions of the resulting

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spectra revealed that the spectra produced when PIP is between 2048 and 64 are the same. Use of 32 or less points for phase correction caused noticeable errors in the spectra. In contradistinction with the first result is the observation that the phase curves differed as PIP decreased from 2048 to lower values. Calculation of the phase can safely be assumed to be more accurate when more points are used; therefore, over a substantial range, small errors in the phase angle do not make any observable difference in the calculation of the PA spectrum. This conclusion, together with the fact that the spectra from method (b) are very similar to the modulus spectra [methods (a) and (e)], appears to contradict the assertion c2)that multiplicative phase correction is inadequate for PA spectra. Further investigation of experimental or synthetic spectra is required before this question can be answered more definitely. Figure 7 displays the dependence of the CH stretching region on the number of points used for multiplicative phase correction. The three curves correspond to the use of 2048, 16 and 2 points in the calculation. It can be concluded that the primary error introduced by the use of an insufficient number of points is in the intensity of the spectrum; there is little difference in the shapes or the positions of the bands. This somewhat surprising result can be attributed to the fact that the phase angle is slowly varying and not very different from zero in this region (Figs 2 and 6). Therefore, inaccurate phase correction does not produce major errors. In the region below 1000 cm-‘, the use of small PIP values does lead to greater inaccuracy, because phase correction is much more important when the phase angle is far from zero and changes rapidly with frequency.

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Fig. 7. Effect of variation of number of points used in multiplicative phase correction on calculated spectra of fusinite in the CH stretching region. Upper curve: 2048 points; middle curve: 16 points; lower curve: 2 points (effectively no phase correction). Spectra calculated using 64 or more points superimpose with the upper curve. Analogous results are obtained in the 600- lOOOcm- region.

CONCLUSIONS

Signal averaging and phase correction of a series of 10 double-sided low resolution PA interferograms of fusinite have been investigated in this work. Spectra were calculated using standard Bruker software, which incorporates multiplicative phase correction; alternatively, modulus (power) spectra were computed in a separate program. Five different methods of averaging spectra or interferograms obtained under similar conditions have been examined. In addition, the effect of variation of the number of points used in multiplicative phase correction has been examined with regard to the PA spectra. The principal results of this investigation are as follows: 1. Averaging ~nter~rogr~ms or spectra: The sequence chosen for averaging and Fourier transformation is not important when multiplicative phase correction is used. On the other hand, the modulus spectrum calculated from the average interferogram is slightly more intense than the average of the individual modulus spectra. For both types of spectra, the average spectrum is slightly smoother than the spectrum calculated from the average interferogram. 2. Average modulus spectra: Separate averaging of the cosine and sine components of the complex spectrum’4’ yields an average modulus spectrum that agrees with the modulus spectrum calculated from the average interferogram. The location (apical path difference) of the centreburst must be constant in order to achieve meaningful averaging of interferograms, or for the (mathematically equivalent) averaging of the cosine and sine components. 3. Multiplicative phase correction: The number of phase interferogram points used in multiplicative phase correction of PA spectra can vary over a wide range without causing an observable effect on the calculated spectrum. Thus, discrepancies in the calculated phase do not necessarily give rise to differences in the corresponding spectra. Finally, it should be pointed out that the convolution (symmetrization) method of phase correction”‘) has not been considered in the present work. Preliminary investigations in our laboratory have shown that PA spectra calculated with this latter method differ slightly from those obtained using the more common multiplicative phase correction and it remains to be demon-

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strated whether the convolution method is a more accurate means of calculating PA spectra. This subject will be discussed in a future publication. Acknowledgement-The

author is grateful to Dr W. I. Friesen for several helpful discussions and suggestions.

REFERENCES I. L. Mertz, Transformations in Optics. Wiley, New York (1965); L. Mertz, Infrared Phys. 7, 17 (1967). 2. D. W. Vidrine, in: Fourier Transform Infrared Spectroscopy, Vol. 3, (Edited by J. R. Ferraro and L. J. Basile). Academic Press, New York (1982). 3. C. R. Anderson and D. R. Mattson, Proc. SPlE 191, 101 (1979). 4. J. R. Birch, Infrared Phys. 20, 349 (I 980). 5. J. C. Donini and K. H. Michaelian, Infrared Phys. 24, 157 (1984). 6. W. I. Friesen and K. H. Michaelian, Infrared Phys. 26, 235 (1986). 7. B. Riessex, M. Starsinic, E. Squires, A. Davis and P, C. Painter, Fuel 63, 1253 (1984). 8. P. C. Painter, R. W. Snyder, M. Starsinic, M. M. Coleman, D. W. Kuehn and A. Davis, Appl. Spectrosc. 35,475 (1981). 9. K. H. Michaelian, Appl. Specirosc. In preparation. IO. R. P. Lowe, R, J. Niciejewski and D. N. Turnbull, Infrared P&s. 21, 189 (1981). 11. M. L. Forman, W. H. Steel and G. A. Vanasse, J. opt. Sot. Am. 56, 59 (1966).