Reconstruction of gas chromatograms from digitally filtered Fourier

Anal. Chem. 1989, 61, 1073-1079

1073

Reconstruction of Gas Chromatograms from Digitally Filtered Fourier Transform Infrared Interferograms Joanne M. Bjerga and Gary W. Small*

Department of Chemistry, The University of Iowa, Iowa City, Iowa 52242

Two methods have commonly been used to reconstruct gas chromatograms from interferograms collected during gas chromatography/Fourier transform infrared spectroscopy (GWFTIR) experiments. Gram-Schmidt orthogonalization has been applied to a section of the raw Interferograms, or the fast Fourler transform has been applied to the interferogram followed by absorbance calculations over specific frequency windows. The former method is preferred for Its ability to operate in real time, while the latter method is able to produce reconstructions selective for user-defined frequencles. I n the work presented here, the trade-offs between these methods are resolved through the use of digital filtering techniques. Band-pass dlgitai filters are mathematical transforms that can operate on Interferogram segments In a frequency-dependent manner. Effectively, a digital filter removes frequency information outside the range(s) of interest from the interferogram segment. The sum of squares of the filtered interferogram segment can then be plotted vs interferogram number to obtain the reconstructed chromatogram. Through this procedure, the selectivity of the fast Fourier transform method is obtained at a computational speed comparable to the Gram-Schmidt method. Gas chromatograms from a series of GWFTIR data sets are reconstructed by use of established techniques and compared to reconstructions produced with the digital filter technique. On the basis of the combination of seiectivtty, signal-to-noise ratio, and computational speed, the digital filtering approach Is shown to be the preferred reconstruction technique.

INTRODUCTION Modern computer-controlled instruments allow data to be collected at a faster rate than it can be analyzed. For example, Fourier transform infrared instruments are routinely capable of scanning the entire mid-infrared region 10 times or more per second. With increasing ability to collect greater amounts of data in less time, it becomes more and more important to be able to manage that data, i.e. to be able to discriminate between important and unimportant information. In the case of gas chromatography/Fourier transform infrared spectroscopy (GC/FTIR), the infrared data in the form of interferograms are collected during the entire chromatographic run, although mixture components will only be passing through the infrared sample cell (light pipe) a fraction of the total experimental time. Maximum efficiency in data collection can be achieved if data is stored only when a compound is actually in the light pipe. Information regarding the presence of mixture components at specific times is traditionally obtained from the trace produced by the gas chromatographic detector. The same information can be obtained from the infrared data during the experiment when the data is processed in such a way as to mimic the chromatographic trace. The resulting infrared-based chromatographic trace is termed a GC/FTIR reconstruction and is simply a plot of computed signal intensity vs interferogram number.

Reconstruction of GC/FTIR chromatograms is most commonly performed by use of the Gram-Schmidt orthogonalization procedure first described by DeHaseth and Isenhour ( I ) . This procedure operates directly on interferograms as they are collected and is performed in real time. The GramSchmidt orthogonalization procedure treats the same segment of each interferogram as a multidimensional vector. In their original work, DeHaseth and Isenhour used 100-point interferogram segments starting 60 points to the right of the center burst. As many as 60 and as few as 5 basis vectors were found to produce optimum signal-to-noise ratios for reconstruction peaks in two different data sets. In the Gram-Schmidt technique, interferogram segments taken from a known base-line region (Le. from interferograms collected when only carrier gas is present in the optical path of the spectrometer) are sequentially orthogonalized to form an orthonormal set of basis vectors. The offset, dimensionality, and number of basis vectors are variables that must be optimized. This basis set represents a base line of information to which all subsequent interferograms are compared. By use of the same segment from subsequent interferograms, the orthogonal distance, d , to the basis set is calculated as n

d = [I-I - C(I*BJ2]’/2 i=l

(1)

where I is the interferogram vector, Bi is the ith orthonormal basis vector, and n is the total number of basis vectors. Those interferograms containing information included in the basis set will not be very different and will thus have a reconstruction value around 0. Those interferograms containing analyte information will be different from the basis set and will give rise to the peaks in the reconstruction. The GramSchmidt orthogonalization has been adapted to produce compound-class selective reconstructions ( 2 , 3 )and has been used for quantitative analysis ( 4 ) . Attempts have been made to determine the optimum Gram-Schmidt parameters (5)and to determine the instrumental dependence of obtaining the optimum parameters (6, 7). Coffey, Mattson, and Wright (8)introduced a reconstruction method based on integrating absorbance values within frequency windows of the spectrum. In the original work, they coadded four interferograms, performed a low-resolution fast Fourier transform (FFT), phase-corrected the spectrum, ratioed the spectrlim to a background spectrum containing only carrier gas information, converted the resultant transmittance spectrum to absorbance, and integrated up to five spectral regions to produce a “chemigram”. Wide frequency windows were used to produce general reconstructions, while narrow windows, such as 1700-1600 cm-’ for C=O stretches, were used to produce selective reconstructions. The main disadvantages to the chemigram approach are the need to perform the FFT and the integration of large areas of base-line absorbance values when wide frequency windows are used. The latter operation has an overall negative effect on the signalto-noise ratio of the reconstruction. The maximum absorbance reconstruction algorithm described by Bowater et al. (9)is a modification of the integrated

0003-2700/89/0361-1073$01.50/00 1989 American Chemical Society

1074

ANALYTICAL CHEMISTRY, VOL. 61, NO. 10, MAY 15, 1989

absorbance method described above. A series of interferograms are coadded and apodized before Fourier transformation. The absorbances within frequency windows are monitored, but only the maximum absorbance values are plotted vs interferograms number to obtain the reconstruction. The maximum absorbance algorithm was evaluated under the following conditions: 10-12 interferograms coadded, 512-, 1024, 2048-point FFTs with one level of zero-filling, and Happ-Genzel apodization for 32-, 16-, and 8-cm-' spectral resolution, respectively. The best sensitivity for most compounds was obtained a t IFj-crn-' resolution. The frequency windows used to produce general reconstructions were 3 100-2850 ern..' and either 1800-1000 or 1800-700 cm-'. Selective reconstructions were obtained with 100-cm-' windows such as 3000-2900 and 1600--1500 cm-'. Bowater e t al. concluded that the maximum absorbance algorithm was superior t o the integrated absorbance algorithm for selective reconstructions and similar to the Gram-Schmidt procedure for general reconstructions. A reconstruction method based on factor analysis has also been reported (10). Many studies have also been reported comparing the different reconstruction techniques (11-13). Recently, Small e t al. (14) introduced the use of digital filters i n processing interferograms collected from a passive Fourier transform infrared spectrometer. The filters were generated in the spectral domain to be selective for a desired frequency (or range of frequencies) but were applied directly to a segment, of the raw interferogram. When applied to an interferogrm segment, the filters remove unwanted frequency information. In the passive infrared work, the sum of the squares of the points in the filtered interferogram segment were plotted vs interferogram number to detect the presence of a target analyte. This methodology is designed to be performed in real-time and can be extended to include applications such as reconstructing GCiFTIR chromatograms. The digital filter approach to reconstructions combines the speed of processing achieved by operating directly on the interferograms (as in Gram-Schmidt orthogonalization) with the selectivity of choosing which frequencies are monitored (as in the absorbance algorithms). The initial digital filtering work focused on narrow-bandpass filters designed to extract frequencies corresponding to individual absorption bands. This paper extends the filter methodology to include the combination of wider frequency ranges and multiple frequency ranges necessary for reconstructing GC/FTIR chromatograms. The filter methodology will be compared to the conventional reconstruction methods described above based on the signal-to-noise ratios of a series of reconstructed chromatograms. In addition, the number of computer operations required to produce a reconstruction from each of the methods will be compared.

EXPERIMENTAL SECTION Two sets of compounds were analyzed under varied gas chromatographic conditions, giving rise to a total of three data sets. The compounds in each of the samples are listed in Table I. The first data set was used in developing the digital filter reconstructions; the remaining data sets were used to test the optimized filter parameters. Each sample consisted of 0.500 gL of each analyte diluted to 10.0 mL with methylene chloride. Table I also lists the volume of sample injected, split ratios, and approximate amount of each analyte injected for each data set. The samples were analyzed with a GC/FTIR system consisting of a Hewlett-Packard 5890A gas chromatograph coupled to an IBM M98 FTIR spectrometer. The GC was equipped with a 15 m by 0.322 mm i.d. DB 5 capillary column. The injection port was kept a t 200 "C, the light pipe at 180 "C, and the flame ionization detector on the GC at 190 "C. The FTIR was equipped with a mediumrange Hg:Cd:Te detector. Data acquisition was controlled by the IBM M98 data system. Interferograms (2048 points) were collected at a rate of 3.5 per

Table I. GC/FTIR Data Sets

amt injected

weight

sample

split

( x I O ~ )I,,

compounds

(XlO+), g

1

1:lO

0.4

2

1:1

0.8

3

1:l

0.4

ethyl acetate penten-3-01 methylcyclohexanone ethyl acetate penten-3-01 rnethylcyclohexanone vinyl acetate 3-pentanone cyclohexanone acetyl acetone

1.80 1.78 1.84 36.0 35.6 36.8 18.6 18.2 19.6 19.2

second, with every third interferogram being saved to reduce the size of the data sets. After collection, the interferograms were transferred by serial communications link to a Prime 9955 interactive computer system operating a t the Gerard P. Weeg Computing Center a t the University of Iowa. The remaining computations reported here were performed on the Prime system. All computer software was written in Fortran 77. Fourier transform and multiple regression computations were performed by use of subroutines from the IMSL library (International Mathematical and Statistical Library, Houston, TX). Plots were generated by use of the TELLAGRAF interactive graphics system (Integrated Software Systems Corp., San Diego, CA) and with original software. A Hewlett-Packard 7475A digital plotter was used as the output device. Signal-to-noise ratios for the reconstructions were determined as S / N = ( I , - Zb)/RMS (2) where S I N is the computed signal-to-noiseratio, I , is the maximum reconstruction value for a given peak, Zb is the base-line value corresponding to Zp,and RMS is the root-mean-square noise of the base line. Sections of the reconstruction where there was no evidence of eluting compounds were used to compute a model for the reconstruction base line by simple linear regression. This model was used to compute Zb in eq 2. The root-mean-square noise (RMS) was computed as k

RMS =

[C(I,,L- I J 2 / ( k - l)]'/' I=1

(3)

where Zb,,is the computed base-line value described above and I, is the corresponding reconstruction value. The summation in eq 3 is carried out over the k reconstruction values used to define the base-line model. In other terms, eq 3 defines the standard deviation of the residuals in the regression computation.

RESULTS AND DISCUSSION Filter Development. The theory behind the digital filter approach to interferogram analysis was described by Small et al. (14). In general, a nonrecursive or finite impulse response digital filter has the form

(4) where Y*L,the filtered data point, is the sum of n + 1 terms based on Y,, the corresponding raw data point, and a series of raw data points preceding Yi.Each of the raw data points is weighted by a different coefficient, fo through f,,. The values of the coefficients determine which frequencies the filter will enhance and which frequencies it will suppress. In the present application, the data points in eq 4 are interferogram points that are filtered to remove information pertaining to unwanted frequencies. The purpose of developing the filter methodology is to be able to determine whether there is any analyte information within a specified spectral frequency range. We wish to make this determination directly from the interferogram, however. A useful way to illustrate how the filter works is to demonstrate how it is developed. Since the filter to be developed


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Flgure 1. Power spectrum (top) obtained by FFT of 128 co-added interferograms (collected when carrier gas was flowing through the GC/FTIR light pipe), followed by a plot of the Gaussian curves which include the fingerprint and C=O and C-H stretching regions of the spectrum and a plot of the power spectrum multiplied by the Gaussians. The spectrum is said to be “frequency filtered”, as the magnitude of frequency information outside the Gaussians is set to zero. The multipication also illustrates the varying sensitivity of the detector for the different regions of the spectrum.

has a frequency dependence, the filter generation must begin in the frequency domain. An interferogram is a signal that is a sum of frequencies. The intensity and frequency information is usually obtained by fast Fourier transforming the interferogram and inspecting the resulting spectrum. An effective method of filtering the spectrum would be to multiply the spectrum by a func m that would set the intensity of unwanted frequencies t o . 0. In principle, this function could be a square wave, a Gaus an curve, a Lorentzian curve, etc. This function is termed .he frequency response of the filter. Therefore, the first step in filter development is to generate the frequency response. The width of the frequency response determines the filter band-pass. In this study, Gaussian curves were chosen as the frequency response shape, as they have yielded the best filters in the work to date. The width of the Gaussian curve can be varied to produce a narrow band-pass or a wide band-pass filter. In this study, three Gaussian curves were generated centered in the fingerprint and carbonyl and C-H stretching regions of the IR spectrum. Figure 1 is a plot of a base-line power spectrum (top) and the filter function curves (middle). The coadded power spectrum, obtained by applying the FFT to a collected interferogram, is characteristic of the instrument being used to collect the data. If the power spectrum is multiplied by the Gaussian curves (the frequency filters), the bottom plot is obtained. The effect of the multiplication is to extract the spectral information in the frequency range covered by the Gaussian curves. The next goal is to develop digital filters that can mimic the action of the spectral multiplication when applied directly to interferograms.

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Figure 2. Inverse FFT of the three plots in Figure 1 plotted from 150 to 300 points to the right of the interferogram center burst. The points nearest the center bursr are excluded as their magnitude obscures trends in the region o f interest. Note that the inverse FFT of the Gaussians (middle) damps to zero while the inverse FFT of the frequency-filtered power spectrum (bottom) retains some information.

Given a desired frequency response, the most commonly used method for generating the filter coefficients in eq 4 is the Remez exchange algorithm (15). For the work described here, we have adopted an alternative approach to the filter generation that attempts to compute filter coefficients that are optimized for filtering interferograms. The rationale for this approach is that a filter to be used only for processing interferograms can have information built into it regarding the band-pass of the infrared detector used. For example, if the detector produces no response to frequencies higher than 6000 cm-’, the filter does not need to be able to remove these frequencies. Through this approach, we have found that filters can be generated with fewer coefficients than are required to produce equivalent filters by the Remez exchange algorithm. In our approach, the “frequency-filtered” power spectrum of Figure 1 is inverse Fourier transformed to the time domain. The resulting frequency-filtered interferogram is quite different from the original raw interferogram, as the frequency information outside the Gaussian curves has been eliminated. Figure 2 depicts a raw interferogram (top), the interferogram corresponding to the Gaussian curves (middle), and the frequency-filtered interferogram (bottom), plotted from 150 to 300 points to the right of the interferogram center burst. The frequency-filtered interferogram (bottom) clearly contains spectral information. However, the magnitude of the filtered segment relative to the raw interferogram segment (top) indicates that much overall information has been removed. As multiplication in the frequency domain is equivalent to convolution in the time domain, the effect of the multiplication of the Gaussian curve and the power spectrum can be achieved by evaluating the convolution integral of the corresponding time domain functions. However, directly evaluating the convolution integral of a 100-point segment of an interferogram would incur almost as great a computational cost as the

1076


FFT computer over an entire 1024-point interferogram. Instead, the digital filter approach approximates the convolution integral with a set of coefficients. Our approach to computing these coefficients is motivated by recognizing that eq 4 is a linear model composed of a dependent variable, Y*,and a set of independent variables, Y,. For such a model, the filter coefficients, fo-f,, can be obtained by use of multiple linear regression analysis. In this context, the filter coefficients are regression coefficients. For the regression analysis, the dependent variable is obtained by the FFT/Gaussian multiplication/inverse FFT procedure described above. The independent variables are defined by raw points in the same interferogram used in the generation of the dependent variable. A regression-based procedure is advantageous in that the individual terms in eq 4 can be evaluated for their statistical significance. For the work reported here, a stepwise regression procedure (16)was used to select those independent variables that define the most statistically significant regression model. The pool of potential independent variables was defined by setting the maximum value for n in eq 4. The stepwise regression algorithm begins with the selection of the single independent variable that has the highest correlation with the dependent variable. A one-term model is then formed based on this selected variable. Consecutive terms are added to the model in a stepwise manner. At each step, the variables remaining in the pool are evaluated for their correlation with the variance in the dependent variable that has not been explained by the terms previously selected. The variable chosen through this procedure is added to the model, and the process is repeated until no remaining variables in the pool meet a minimum standard of correlation. This correlation test is typically referenced to a statistical F distribution. The stepwise regression algorithm allows filters to be derived that provide maximum performance with a minimum number of filter terms. Once derived, the set of filter coefficients can be applied to a series of interferograms to remove unwanted frequency information without having to perform the FFT. Filter Optimization. Several factors must be optimized when developing a digital filter. The interferogram segment (i.e. the offset and dimensionality as in the Gram-Schmidt reconstruction procedure), the size of the pool of potential independent variables for the regression procedure, and the significance level for the entry of variables into the regression model must be selected. Selection of the best frequency filter, including shape and width, is analogous to selecting the best frequency window(s) in the absorbance reconstruction algorithms. In order to develop a filter for general reconstructions, a set of GC/FTIR interferograms were collected. The spectra of the solvent and the three compounds used are plotted in Figure 3. These compounds were chosen because of the different functionalities they possess. The spectra are plotted on different scales. The solvent spectrum (top) is clearly the strongest, while penten-3-01 (third from the top) is the weakest. For each of these spectra, there are absorptions in two of the three regions for which a Gaussian filter was constructed. The goal of this study'was to develop a filter which would be sensitive to three different areas of the IR spectrum. The Gaussian curves in Figure 1 (middle) were centered a t 1057, 1697, and 2954 cm-' in the fingerprint, carbonyl, and C-H stretching regions of the IR spectrum, respectively. The widths of the Gaussians were individually adjusted to account for the different ranges of frequencies a t which absorptions are typically observed in each of the three regions and not tailored to any one compound. The full-width a t half maximum (fwhm) of each of the Gaussians was 545.0,109.0, and 363.3 cm-', respectively. The fwhm is a useful measure of the

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Figure 3. Spectra of compounds used in developing the digital interferogram fitter (from top): methylene chbride (solvent), ethyl acetate, penten-3-01, methyl cyclohexanone.

filter band-pass. As can be seen in Figure 1, however, the Gaussian curves have appreciable area well past the fwhm range. However, absorptions at frequencies in the wings of the band-pass will not be detected as well as those occurring in the center of the band-pass. To optimize the filter development parameters, 50-, 75-, and 100-point interferogram segments were evaluated. The starting points for the segments were varied from points 21-251, at 10-point intervals. These starting points are relative to the center burst of the interferogram. The pool of independent variables for the stepwise regression analysis was varied between n = 50, 75, and 100 in eq 4. T o enter the regression models, variables had to be significant at the 90% probability level, based on the F distribution. The 90% significance level was kept fixed throughout the optimization study. A f d l factorial experimental design was conducted with the above parameters. All combinations of segment size, segment location, and independent variable pool size were tested. From this study, it was found that the best reconstruction peak signal-to-noise ratios were obtained with a 75-point interferogram segment starting 171 points to the right of the center burst. In this case, the pool of independent variables was defined by n = 75. The computed model contained 24 filter coefficients. The R2for the regression was 99.87% and the F value for the significance of the model was 1554. Figure 4 is a plot of reconstruction peak signal to noise (by compound) as a function of which 75-point segment was used in the model development. Figure 5 is a plot of four interferogram segments plotted from 150 to 300 points to the right of the center burst. The top interfergram segment was collected when ethyl acetate was present in the light pipe and the next when only carrier gas was present. The two bottom plots are the same ethyl acetate and carrier gas interferogram segments after filtering. The magnitude of the filtered carrier gas interferogram is clearly less than the filtered ethyl acetate interferogram. The


1077

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3 ETHYL ACCTAIE

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Figure 4. Reconstruction peak signal-to-noise ratio as a function of

interferogram segment used in the filter development and application.

INTERFEROGRAM NUMBER

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Figure 6. Reconstructions of data set 1 using (from top): (1) GramSchmidt orthogonalization (30 basis vectors, 100 interferogram points offset 10 points to the right of the burst), (2) maximum absorbance (1024 interferogram points, 1 level of zerofilling, Happ-Genzel apodization) monitoring three frequency windows (700-1500, 1520-1810, and 2700-3205 cm-'), and (3)digital filter reconstruction based on 24 coefficients, 75 interferogram points.

Table 11. Reconstruction Signal-to-Noise Values for Data Sets 1 and 2

___

S I N of sample 1 peaks 150

343,

200

250

300

INTERFEROGRAM POINT

reconstruction

1

2

3

4

Gram-Schmidt absorbance digital filter

285.9 247.9 190.6

254.4 195.8 372.7

29.8 28.6 14.9

77.8 84.2 133.1

reconstruction

1

2

3

4

Gram-Schmidt absorbance digital filter

220.9 296.7 263.6

285.2 334.4 654.3

38.1 49.3 17.1

83.9 154.3 297.1

S I N of sample 2 peaks __I

I50

200

250

300

INTERFEROGRAM POINT

Flgure 5. Interferogram segments from points 150 to 300 to the right of the interferogram center burst (from top): ethyl acetate, carrier gas,

ethyl acetate filtered, carrier gas filtered. Note the decrease in magnitude of the interferogram segment points after filtering. The sum of squares of the filtered ethyl acetate segment is clearly greater than the corresponding filtered carrier gas segment. sum of the squares of the points in the filtered interferogram segment can be plotted vs interferogram number to obtain the reconstruction. In addition, it was discovered that ratioing the sum-ofsquares of each filtered interferogram to the average sumof-squares of a series of filtered reference interferograms (usually lo), collected at the beginning of an experiment, resulted in an enhancement of single to noise in the reconstruction. Another advantage of ratioing is that a filter developed on a particular data set can be applied to another data set, as each newly filtered interferogram segment will be compared to a filtered reference interferogram segment from the same experiment. This eliminates the need to generate a new set of filter coefficients at the start of each experiment.

Signal-to-Noise Comparisons. Figures 6 and 7 are reconstructions of the first two data sets obtained by using Gram-Schmidt orthogonalization (top), the maximum absorbance alogrithm (frequency windows of 700-1500, 1520-1810, and 2700-3205 cm-', chosen to be similar to the Gaussians) (middle), and the filter methodology (bottom). The Gram-Schmidt parameters were optimized for 10-40 basis vectors (at intervals of lo), 50-150 interferogram points (at intervals of 25), and offsets from 10 to 100 (at intervals of 10) points to the right of the center burst. The optimum GramSchmidt parameters were 30 basis vectors of 100-point interferogram segments offset 10 points to the right of the center burst. The maximum absorbance reconstructions were based on a 1024-point F F T with one level of zero-filling and Happ-Genzel apodization. The signal-to-noise value for each peak is listed in Table I1 for each of the reconstruction methods. Across the two data

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Flguo 7. Reconstructions of data set 2 using the same conditions as in Figure 6. The digital filter used k based on the interferograms from data set 1.

Flgure 8. Spectra of compounds in data set 3 (from top): vinyl acetate, 3-pentanone, cyclohexanol, acetyl acetone. 2550-

Table 111. Reconstruction Signal-to-Noise Values for Data Set 3

S I N of sample 3 peaks 1

2

3

4

5

Gram-Schmidt absorbance

230.5 219.9 221.7

313.2 230.6 578.2

60.7 56.3 103.1

42.9 63.0 43.1

77.5 57.3 143.3

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sets, eight peaks are found. The first peak in each data set is the solvent peak. Of the eight peaks, the digital filtering procedure produced the highest signal-to-noise value in four cases. The Gram-Schmidt and maximum absorbance reconstructions each produced the highest signal to noise in two cases, one of which was the solvent peak. As can be seen in the spectra of the compounds in Figure 3, the GC/FTIR interface was not purged during the experiment, and COz was allowed to accumulate. This COz accumulation is manifested in the Gram-Schmidt reconstruction as a sloping base line. Neither the maximum absorbance nor the filter reconstructions are affected by the COz, as both exclude the COPfrequency range in their selection of frequency windows or filter band-pass. The same digital interferogram filter was used with each data set. The differences in the data sets are the concentrations of the compounds present. The filter was developed on the lowest concentration data and performs well for greater concentration data. Figure 8 is a plot of the spectra of the four compounds used to test the application of the filter to compounds not included in the optimization. Once again, each of the reconstruction methodologies was applied to the data. The three reconstructions are plotted in Figure 9. The same filter from the first data set was used to generate the filter reconstruction. Table I11 lists the signal-to-noise ratios for each of the peaks in this data set. The filter methodology outperforms both the

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Flgure 9. Reconstructions of data set 3 under the same cond%ions as in Figure 6. The digital filter used is the optimized filter developed for data set 1.

Gram-Schmidt and the maximum absorbance algorithms for four out of the five compounds. Computational Efficiency. Another useful means for comparing reconstruction methods is computational efficiency.


Table IV. Comparison of Computational Speed

reconstruction

conditions

Gram-Schmidt

30 basis vectors 100 point segment 10 points offset 1024 point FFT 1 level zero-filling

absorbance

Happ-Genzel apodization 1024 point FFT 24 coefficients 75 point segment

digital filter

171 points offset

computations

11901

51 200 22 528 3 675

The number of computations (multiplications, divisions, additions) required by the Gram-Schmidt orthogonalization procedure consists of two steps: the formation of the basis set and subsequent orthogonalization of interferograms to the basis as they are collected (1). The basis set formation requires 2N(2B + 1) + 1 operations where N is the number of interferogram points and B is the number of basis vectors. Each subsequent orthogonalization requires N(4B - 1)+ 1operations. Again, the optimized Gram-Schmidt parameters used in these experiments were 30 basis vectors, 100 interferogram points, and 10 points offset to the right of the interferogram center burst. Any reconstruction method relying on a fastFourier transform of the interferogram will require 2N logz 2N operations where N is the number of interferogram points used (and a power of two) ( I ) . The maximum absorbance reconstruction method will add the operations for finding the maximum absorbances within the designated frequency windows and summing these to obtain the reconstruction value. The “chemigram” approach will add the operations required to integrate frequency windows. For the sake of simplicity, only the operations required for the FFT will be used for comparisons. The digital filter reconstruction method also consists of two steps. The filter development requires (1) FFT of a raw background interferogram, 2N log, 2N, (2) multiplication of the frequency response by the background spectrum, 1/2N + 1,(3) the inverse FFT, 2IV log, 2N, and (4) stepwise multiple linear regression to obtain the regression coefficients, ( M 2),(N 3/2M), where M is the number of potential independent variables 1 and N is the size of the interferogram segment. Filtering subsequent interferograms requires (1) application of the filter as defined by eq 4, (2) computation of the sum of squares of the filtered points, and (3) ratio of the sum of squares to the average sum of squares computed over 10 filtered background interferograms for a total of N(2C + 2) 1 operations, where N is the number of interferogram points and C is the number of regression coefficients required to filter the interferogram segment. The filter used to generate the GC/FTIR reconstructions consisted of 24 coefficients applied to a 75-point interferogram segment. The number of computations required to form the GramSchmidt basis set from 30 interferogram segments of 100 points each is 12 201. The absorbance algorithms have no developmental computational requirement. Digital filter development requires 1234 499 computations, most of those a result of the stepwise regression (1188910 computations for 77 independent and dependent variables, and 75 interferogram points). However, the same filter can be used for multiple data sets as demonstrated earlier. The filter development takes place before any data is collected, as does forming the basis set for the Gram-Schmidt orthogonalization. To process individual interferograms, the Gram-Schmidt orthogonalization requires 11901 computations. Absorbance methods require 22 528 computations for a 1024-point FFT and up to 51200 to include apodization and one level of

+

+

+

+

1079

zero-filling. The application of filter coefficients to each interferogram requires 3675 computations, less than half as many as required for the next least computationally expensive method, the Gram-Schmidt orthogonalization. Therefore, the digital filter methodology is the fastest reconstruction algorithm to apply. These results are summarized in Table IV.

CONCLUSIONS The digital filter methodology gave recontructions superior to the other methods tested here, based on both signal-to-noise ratios and computational efficiency. Furthermore, the results presented here suggest that filters can be developed that are suitable to be used across data sets. If this conclusion holds, the computationally intensive development of the filters at the beginning of each experiment can be avoided. The integrated and maximum absorbance algorithms have been the best methods available for selective reconstructions but do require the computationally expensive FFT. Selective reconstructions can also be obtained with the filter methodology using the Gaussians from this study individually to generate three separate models. Small et al. (14) demonstrated clearly that a filter could be generated for a very narrow region of the spectrum. In this paper, it has been shown that a combination of narrow and wide spectral ranges can be combined to generate a filter capable of detecting absorbances within each of those ranges. Future research will explore the development of a series of filters specific for different areas of the spectrum for the generation of selective reconstructions. In addition, it will be determined whether it is preferable to generate general reconstructions with a combination of Gaussians as done for this study or to use a series of filters based on individual Gaussians. ACKNOWLEDGMENT D. M. Sevenich and H. B. Friedrich are acknowledged for their help in the GC/FTIR data collection. L1TERATUR.E CITED DeHasth, J. A.; Isenhour, T. L. Anal. Chem. 1977, 49, 1977-1981. Wieboldt, R. C.; Hohne, B. A.; Isenhour, T. L. Appl. Spectrosc. 1980, 34, 7-14. Hohne, B. A.; Hangac, G.; Small, G. W.; Isenhour, T. L. J . Chromatogr. S d . 1981. 19, 283-269. Sparks, D. T.; Lam, R. B.; Isenhour, T. L. Anal. Chem. 1982, 54, 1922- 1926. White, R. L.; Giss, G. N.; Brissey, G. M.; Wilkins, C. L. Anal. Chem. 1983, 55,998-1001. Brissey, G. M.; Henry, D. E.; Giss, G. N.; Yang, P. W.; Griffiths, P. R.; Wilkins, C. L. Anal. Chem. 1984, 56,2002-2006. Sparks, D. T.; Owens, P. M.; Williams, S. S.; Wang, C. P.; Isenhour, T. L. Appl. Spectrosc. 1985, 39, 288-296. Coffey, P.; Mattson, D. R.; Wright, J. C. A m . Lab. 1978, 10, 126-132. Bowater, I.C.;Brown, R. S.; Dooper, J. R.; Wilkins, C. L. Anal. Chern. 1986, 58,2195-2199. Owens, P. M.; Lam, R. 8.; Isenhour, T. L. Anal. Chem. 1982, 54, 2344-2347. Hanna, D. A.; Hangac, G.; Hohne, B. A.; Small, G. W.; Wieboldt, R. C.; Isenhour, T. L. J . Chrornatogr. Sci. 1979, 1 7 , 423-427. White, R. L.; Giss, G. N.; Brissey, G. M.; Wilkins, C. L. Anal. Chem. 1981, 53, 1778-1782. Wang, C. P.; Sparks, D. T.; Williams, S. S.; Isenhour, T. L. Anal. Chem. 1984, 56, 1268-1272. Small, G. W.; Kroutil, R. T.; Ditillo, J. T.; Loerop, W. R. Anal. Chem. 1988. 6 0 , 264-269. McClellan, J. H.; Parks, T. W.: Rabiner, L. R. If€€ Trans. Audio N e c troacoust. 1973, AU-21, 506-525. Draper, N. R.; Smith, H. Applied Regression Analysis, 2nd ed.; WileyInterscience: New York, 1981; pp 307-312.

RECEIVED for review November 15,1988. Accepted February 23,1989. This work was supported by the US. Army Chemical Research, Development, and Engineering Center, Edgewood, MD, under Contract DAAA15-86-C-0034, Portions of this work were presented at the 1988 Pittsburgh Conference and Expostion on Analytical Chemistry and Applied Spectroscopy, New Orleans, LA, February 22, 1988.

Reconstruction of gas chromatograms from digitally filtered Fourier

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