1927
Anal. Chem. 1982, 5 4 , 1927-1931
Cross-Correlation SignaVNoise Enhancement with Applications to Quantitative Gas Chromatography/Fourier Transform Infrared Spectrometry R. B. Lam,' D.
T. Siparks, and T. L.
Isenhour*
Department of Chemistry, Universiv of North Carolina, Chapel Hill, North Carolina 275 14
Cross-correlatlon slgmal/nolse ( S / N ) enhancement Is developed for qualitatlve and quantitativie applications in GC/ FTIR. Synthetlc Gausslan functlons are employed to derive optlmum cross-correlatlon parameters for minlmlzlng signal dlstortlon while enhanolng S I N in Gaussian peak shapes. The optimal parameters 81'. then applied to Gram-Schmidt chromatograms reconstructed from interferometrlc data In GC/ FTIR. The cross-correiatlon technlque Increases the S I N of the peaks while retairilng quantitative Informatlon. The precislon of Gram-Schmltlt quantltatlve GC/FTIR is improved by a factor of 3 In somt~cases uslng cross-correlation. The cross-correlation technique Is also shown to be superior to muitlbasls vector Gram-Schmidt recoristructlons as a S I N enhancement method.
The gas chromatography/Fourier transform infrared spectrometry (GC/FT.IR) experiment has suffered since its inception from a lack of sensitivity in detecting microgram and submicrogram amounts of material. The poorer sensitivity of interferometric measurements compared to mass spectrometry effectively limits the concentration ranges that may be easily handled by current GC/FTIR hardware. Improvements in light pipe design and detectors and faster interferometric scanning modes coupled with signal averaging do lower the detection limit of GC/F'TIR. Beyond these measures, however, a drastic improvement in instrumentation hardware or corresponding improvements in data handling software are necessary if GC/FTIR and the companion technique of GC/IR/MS are to become weful in routine trace analysis. Our laboratory has concentrated on software development to improve detection limits and sensitivity of GC/FTIR measurements ( I , 2). Hieftje has reviewed several signal to noise ( S I N ) enhancement methods applied to various instrumental techniques (3, 4 ) . Correlation analysis of wave forms in communicatioins theory has also been studied by Lee (5,6)and applied to S / W improvement in flame spectrometry (7) and chromatography (8). Horlick has reviewed the detection of spectral information using autocorrelation (9) and cross-correlation (10). This paper describer3 the application of cross-correlation to S I N enhancement of Gram-Schmidt reconstructed gas chromatograms in GC/FTIR. Cross-correlation is shown to improve the SIN of chromatographic peaks while maintaining quantitative information about the peak areas. Experiments on synthetic data are rleported to demonstrate the effect of changing various cross-correlation parameters on S I N and peak distortion. Optimum parameters are obtained for the cross-correlation of Gaussian peaks and these parameters are applied to quantitative GC/FTIR data collected on submicrogram amounts of material. Cross-correlation S I N enFoxboro Analytical, 140 Water St., P.O.B. 5449, Norwalk, C'T 06856.
hancement increases the precision of quantitative GC/FTIR measurements. Cross-correlation is also compared to the use of additional basis vectors in Gram-Schmidt reconstructions for improving SIN. The cross-correlation technique is demonstrably superior to multibasis vector reconstructions in terms of sensitivity and detection limit.
THEORY Cross-correlation is a technique which was developed in communications theory to enable the detection of periodic signals in noise. Both periodic and aperiodic signals arise in many chemical measurement systems and both types of signals are detectable by cross-correlation. Equation 1 gives the expression for the cross-correlation of two functions gl(x)and g2(3c). This function may be more easily evaluated by using Fourier transforms. The details of this procedure including the proper scaling of the result using the fast Fourier transform algorithm are given elsewhere (11).
Basically, cross-correlation S I N enhancement operates by finding the frequency information common to g, and gz. Typically, g, is a wave form containing both a signal and random noise. A reference signal g2 containing little or no noise components is shifted in time relative to g, and the two functions are multiplied and averaged over the abscissa. The average value i s the cross-correlation of the two wave forms at a single delay or shift point. The complete cross-correlation result is represented by a plot of the averages obtained vs. the delay between the two functions. The cross-correlation result will be a maximum when the two signals are in phase (i.e., zero delay). The random noise in gl will be incoherent with the signal in g, at all delay times and will average to zero. The cross-correlation result therefore exhibits an increased
SIN. As cross-correlation is essentially a form of digital smoothing, some distortion of the original peak shape is expected. This distortion is usually a broadening of the peak which decreases resolution of the chromatographic measurement. To minimize broadening and still obtain a significant improvement in S / N , we carried out a preliminary study of how Gaussian functions behave under cross-correlation. Gaussian functions as defined in eq 2 were used for all synthetic data experiments. H , 2,and s are the height, mean, and standard deviation of the Gaussian, respectively. Random noise was added to the signal using a FORTRAN random number generator which approximates white noise as determined from an examination of the noise power spectrum. The noisy Gaussian functions were synthesized with H = 10, i= 4,and s = 0.5 using 256 points over an abscissa range from 0 to 10. S I N was computed by dividing the maximum value of the function by the root mean square of the noise on either side of the peak. All numbers presented in the following tables represent averages of a set of simulations on synthetic data. g ( x ) = He-(X-zP/2s2
0003-2700/82/0354-1927$01.25/0 0 1982 American Chemical Society
(2)
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ANALYTICAL CHEMISTRY, VOL. 54, NO. 12, OCTOBER 1982
Table I. SIN Values and SIN Enhancement for Synthetic Gaussian Data Before and After Cross-Correlation (CC) SIN
before CC
after CC
38 20
250
6.2 3.5 2.3
SIN enhancement
61 26
6.6 5.5 5.5 4.2
11
3.1
110
11
3.7
Table 11. SIN Enhancement as a Function of the Number of Points, N , Describing the Abscissa Range of the Synthetic Gaussian Peak Data
s IN
1.6 a
w/CCb
N
w / o CCa
32 64 128 256 512
21 20 21 20
110
19
184
SIN enhancement
47 55 67
Without cross-correlation.
2.5 2.8 3.2 5.5 8.8
With cross-correlation.
Table 111. Tabulated Values of the Standard Deviation of a Test Gaussian vs. the SIN Enhancement Resulting from the Cross-Correlation of a Gaussian Peak with a Standard Deviation of 0.5 and an Original SIN of 10
B
A
St est
SIN
%est
SIN
0.05 0.1
1.4 1.5
0.2 0.3
1.9
0.6 0.7 0.8
2.1 2.1 2.0
0.9 1.0
1.9 1.8
2.1 2.2 2.2
0.4
0.5
i
Figure 1. (A) A synthetic Gaussian function with a SINof
T
11. (6) The
result of cross-correlation of 1A with a noiseless Gaussian peak.
?i
1
T
T
P Ozi
'0 100
'0 200
'0 300
'0 100
I b 500
'0 600
'0 1 0 0
'0 800
'0 900
'1 PO
STEN
Figure 3. A plot of the distortion measure vs. values of the standard deviation of the test Gaussian, stest.
synthetic Gaussian with a S I N of 0.97. (B) The cross-correlation result.
Flgure 2. (A) A
Table I shows some SIN values resulting from cross-correlating the noisy Gaussians with s = 0.5 with a noise-free Gaussian which has s = 0.5. Varying the height of the noise-free Gaussian, hereafter referred to as the test Gaussian, has no effect on the S I N of the cross-correlation result after proper scaling using the sampling parameters (11). Varying the mean of the test Gaussian simply shifts the cross-comelation peak aIong the abscissa of the cross-correlation result. This shift in the mean of the cross-correlation peak has no effect on the qualitative or quantitative information content of the cross-correlation result. Furthermore, the shift can be totally eliminated as discussed later in this section. Dividing the S I N after cross-correlation by the original S I N gives the S I N enhancement listed in Table I, which ranges from 2 to 7 depending on the original SIN. As expected, the lower S I N cases show correspondingly less S I N improvement as the limiting case of no signal present is approached. Figures 1 and 2 show the results of cross-correlating noisy Gaussian
functions with SIN ratios of 11 and 0.97, respectively. These figures illustrate the power of cross-correlation in recovering a signal buried in noise. Table I1 shows the effect of varying the number of points, N , describing the abscissa range of the synthetic Gaussian peaks. Using Gaussians with s = 0.5, the range spanned by N = 64 or 128 corresponds to a GC/FTIR scan rate of approximately 1interferogram15 with packed column GC separations. The expected S I N enhancement is therefore about a factor of 3 using cross-correlation on data collected as described in the Experimental Section. As N increases to 512, the S I N enhancement approaches an order of magnitude improvement. This point may be realized with state-of-the-art FTIR spectrometers, where scan rates are 10-20 interferogramsls. In a test of the effect of varying the standard deviation of the test Gaussian on the cross-correlation result, a Gaussian with a S I N of 10 and s = 0.5 was cross-correlated with Gaussian functions where the standard deviation of the test Gaussian, stest,ranged from 0.05 to 1.0. Table 111shows the S I N after cross-correlation as a function of stest.Note that the S I N steadily increases with increased values of stest However, broadening of the result also occurs to a significant extent a t higher values of the standard deviation, as deter-
ANALYTICAL CHEMISTRY, VOL. 54, NO. 12, OCTOBER 1982
mined from examining plots of the cross-correlation results. A measure of the distortion in the cross-correlation peak shape is the sum of the squared deviations, S , between the cross-correlation result and the original Gaussian before any noise is added. A plot of S vs. skst is sihown in Figure 3. As predicted, the distortion is greatest at high values of stest. At very low values of stent the distortion is also large. Low stest values describe relatively narrow test Gaussian functions which have correspondingly broad frequency or Fourier domain representations. Therefore, higher frequency noise cornponents of the original noisy signal are not zeroed out when multiplied by the Fourier transform of the narrow test Gaussian. The cross-correlation peak thus contains more higher frequency coinponents and this noise contributes to the distortion measuirement. This is confirmed by noting the low S I N for low values of stest in Table 111. The minimum in Figure 3 occurs a t skst = 0.3, or 60% of the original standard deviation of the noisy Gaussian. From Table 111, this poinit. still shows a near optimum S I N enhancement while minimizing the distortion of the peak shape. For this work, the optimum standard deviation of the test Gaussian for cross-correlation with noisy Gaussian functions was therefore taken to be 60% of the standard deviation of the noisy Gaussian. 'This point apparently represents a good trade-off between S I N enhancement and peak distortion or resolution. Another feature of cross-correlation which has not been extensively implemented is the property of retaining quantitative information. This is the primary reason for choosing cross-correlation ovei*other digital smoothing techniques. If two Gaussian functions gl(x) and gz(x) are cross-correlated, the result is a Gaussiian function g,,(x) given by eq 3, where H,,, x,,, and s,, are defined by eq 4, 5, and 6 in terms of the heights, means, and standard deviations of the original Gaussians. As shown by eq 5, choosing f 2= 0 for the test Gaussian eliminates the mean shift in the cross-correlation peak.
(3) (4)
Equation 7 gives the expression for the area A of a Gaussian peak. Substituting eq 4 and 6 into eq 7 yields an expression for the area of the cross-correlation result in terms of the heights and standard deviations of the original functions. Simplification of thici expression gives eq 8, showing the relationship between the area of the cross-correlation peak A,, and the areas A1 and Az of the original Gaussian peaks.
A =Hs(~T)~/~ A,, = AlA2
(7)
The area of the test Gaussian is known from eq 7 and the area of the cross-corrielation result may be computed by numerical integration after base line correction. The area of the original function ma!? thus be determined using eq 8 if the peak shape is assumed to be Gaussian. That is the approach used in this paper for cross-correlation quantitative GCIFTIR. Theory and experimental results for quantitative GC/FTIR are presented in ref 112.
EXPERIMENTAL SECTION The GC/FTIR quantitative analysis was performed on a series of six standards consisting of pentyl propionate as the analyte, acetophenone as the internal standard, and benzene as solvent. The standards were prepared as described previously (12) with a concentration range ifor pentyl propionate listed in Table IV. The absolute analyte concentrations in Table IV correspond to
1929
Table IV. Average SIN Values for the Pentyl Propionate (PP) Chromatographic Peak at Six Different Concentratiansa SIN WICC
concn of PP,ng/3fiL 288.7 481.2 673.7 769.9 866.1 962.4
(fwhm = w/o CC 6) 4.8 7.9 11 11 11
10
9.8 22
29 25 31 20
WICC (fwhm = 10)
12 27 33
25 26 19
a SIN values are given for the original data without crosscorrelation ( w / o CC) and with cross-correlation (wlCC) using a fwhm of 6 and 10 for the test Gaussian.
a molar concentration range of approximately 0.7-2 mM. Injections of each solution were made in triplicate and all quantitative data represent the average of three measurements. Each standard contained 13.87 fig of acetophenone in 3 fiL of solution. Injection size for each standard was 3 fiL. Chromatography was carried out on a 6 f t X ' / 4 in. 0.d. column packed with 15% SE-30 on 80-100 mesh Chromosorb P AW-DMCS. Injection and column temperatures were 210 and 139 "C, respectively. Flow rate measured with a soap bubble flowmeter was 46.5 mL/min. GC/FTIR transfer lines and light pipe temperatures were 220 "C and 198 "C, respectively. A Digilab FTS-14 was used to collect interferogramswith 8 cm-' resolution at the rate of 7 interferograms/6 s. The GC/FTIR interface is described elsewhere (12,13). Data were written onto magnetic tape and transferred to a Nova 3/12 minicomputer for further data processing using the GIFTS software package (14). The International Mathematical and Statistical Library (IMSL) FFT package was used for all Fourier transform cross-correlation computations. Base line corrections for the Gram-Schmidt chromatographic peaks and area computations were calculated as described previously (12).
RESULTS AND DISCUSSION A comparison of the results obtained from Gram-Schmidt quantitation of GC/FTIR data before and after cross-correlating the chromatographic peaks is presented. Figure 4B is a plot of a peak representing 481.2 ng of pentyl propionate. Cross-correlation of this peak with a Gaussian peak produces the lower noise signal in Figure 4A. Integration of the cross-correlation signal should be more precise as less noise is present and base line corrections and integration limits for the peak are easier to determine. Examination of higher SIN analyte peaks under the same chromatographic conditions determined the full width at half-maximum (fwhm) as 10 interferograms or approximately 8.6 s. Cross-correlation computations were then carried out using fwhm values for the test Gaussian of 6 and 10 interferograms, respectively. A fwhm = 6 corresponds to the optimum 60% value determined from the study of synthetic data. Table IV shows the S I N calculated from the original chromatographic peaks and the S I N after cross-correlating the peaks using the two fwhm values for the test Gaussian. All SIN values are averages over three chromatographic peaks. Using a fwhm = 6 improves the S I N values by a factor of 2 to 3 as predicted from the synthetic data study. The fwhm = 10 data show no substantial improvement over the fwhm = 6 data. Furthermore, using a fwhm = 10 causes a 25% increase in peak width compared to the fwhm = 6 data. The optimum value for the test Gaussian width determined from the synthetic data thus appears to work well with actual data. After background correction and area computation for the pentyl propionate and acetophenone peaks, the ratio of the two peak areas was computed to correct for differences in injection volume among replicate injections. The 95% con-
1930
ANALYTICAL CHEMISTRY, VOL. 54, NO. 12, OCTOBER 1982
Table VI. Tabulated Goodness of Fit Values from a Linear Least-Squares Analysis of the Four Different Data Sets Described in the Table A
I
description of data
goodness of fit
original data 1 7 basis vector reconstruction original data 30 basis vector reconstruction cross-correlated data fwhm = 6 1 7 basis vector reconstruction cross-correlated data fwhm = 1 0 1 7 basis vector reconstruction
1.77 0.971 1.24 1.33
A
il
Ill
INTERFEROGRRM NUMER
Figure 4. (A) The result of cross-correlating a portion of the Gram-
Schmidt reconstruction containing a pentyl propionate peak, 4B.
Table V. 95% Confidence Intervals for the Six Amounts of Pentyl Propionate Analyzed, Computed as its/N”2, Where t Is the Student’s t Value and N is the Number of Determinations for Each Standard concn of PP, ngI3 P L 288.7 481.2 673.7 769.9 866.1 962.4
cc
103.3
65.3 74.2 76.2 94.8 92.6
b.2
‘0.9
b.+
‘0.5
b.6
’0.7
‘0.8
0.9
\.I
PP P P / 3 P L
Figure 5. A comparison of anaiytlcal Calibration curves. The area
95% confidence limits
wlo
‘0.1
WICC 150.4 60.8 29.5 24.6 67.3 87.9
fidence limits for each concentration were computed before and after cross-correlation using a fwhm = 6 for the test Gaussian. The acetophenone internal standard peak was not cross-correlated as the original S I N for this peak was very high in each chromatogram. Thus area measurements on the internal standard peak do not reduce the precision of the quantitative data. The cross-correlation areas were corrected using eq 8 by dividing by the area of the test Gaussian. The confidence limits are presented in Table V. Cross-correlation improved the precision of the area ratios in every case but the lowest concentration. This concentration value was too close to the detection limit under the present experimental conditions to yield precise estimates. Note that as only three points were used to determine standard deviations for confidence interval estimates, some fluctuation is observed in the limits presented in Table V. Table VI shows the goodness of fits computed as overall standard deviations for the linear least-squares fits to the quantitative data. All computations on the original data and cross-correlated data were done on a 17 basis vector GramSchmidt reconstructed chromatogram. The goodness of fit for the cross-correlated data using a fwhm = 6 for the test Gaussian represents a 30% improvement over the goodness of fit for the original data. The cross-correlation fwhm = 10 data are given for comparison, again exhibiting no improvement over the fwhm = 6 data and in fact showing a slightly worse fit. A plot in Figure 5A of the area ratios after crosscorrelation using fwhm = 6 vs. concentration shows the ex-
ratios of the analyte to the internal standard are plotted vs. concentration of analyte. Figure 5A represents the cross-correlated data with a fwhm = 6 for the test Gaussian. Figure 58 represents the data obtained with the 30 basis vector reconstruction.
cellent linearity for Gram-Schmidt quantitative GC/FTIR with cross-correlation S I N enhancement. As additional basis vectors are added in the Gram-Schmidt calculation, the S I N of the resulting chromatogram is improved. In a comparison of this method of SIN enhancement to cross-correlation, a 30 basis vector Gram-Schmidt reconstruction was computed on the original data. An examination of Table VI indicates a better goodness of fit for the 30 basis vector reconstruction than for either the original or crosscorrelated data. Figure 5 shows a comparison of the calibration plots for the fwhm = 6 cross-correlation data and the 30 basis vector reconstruction data. Note that the slope of the calibration curve for the 30 basis vector data shows the poorer sensitivity of this S I N enhancement technique compared to cross-correlation. In addition, the line extrapolates to the x axis at a larger concentration, thereby showing a worse detection limit. The decrease in sensitivity (slope) obtained by using more basis vectors to increase S I N may be appreciated by examining eq 9 used for computing the Gram-Schmidt reconstruction. The bulk of the analyte signal is found by computing the dot product of the sample vector 7 with itself. Adding an additional basis vector Bi subtracts a portion of this signal equal to the square of the dot product of the sample vector with the new reference. As more reference information is subtracted from the signal, the S I N increases but only at the cost of decreased absolute magnitude of signal. Crosscorrelation SIN enhancement maintains the greater sensitivity of using fewer basis vectors for chromatogram reconstructions while increasing S I N and overall quantitative precision. Gram-Schmidt remonse =
ANALYTICAL CHEMISTRY, VOL. 54, NO. 12, OCTOBER 1982
d
A
B p
0.
i
,
0.1
I
,
0.2
0.3
,
1
0.1
0.5
I
G.6
1931
1
,
0.1
0.8
0.0
1.0
/4 P P / 3 P L
Figure 6. The relative errors vs. concentrations for three different linear fits: 6A represents the original 17 basis vector reconstruction data, 6B represents the cross-correlated version of the 6A data, and 6C represents the 30 b'asis vector reconstruction data.
Figure 6 summarizes the comparisons among the quantitative GC/FTIR data. The relative error for each linear fit is plotted over the concentration range spanned by the data. The relative error is computed by dividing the goodness of fit by the predicted value of the response at each concentration, thus giving a measure of how the relative error in the quantitative response changes with concentration. Curve 6B represents the relative error for the cross-correlated data, which is lower over the entire range of concentrations than the original data using either 17 (curve 6A) or 30 (curve 6C) basis vector reconstructions. Note that at high concentrations, the relative error in the quantitative response reaches an asymptotic limit corresponding to the error inherent in preparing the standard solutions (12). At low concentrations approaching the detection limit, the relative error increases rapidly but, using cross-correlation, a smaller amount of material can be detected. The fact that the 30 basis vector response shows a higher detection limit than the 17 basis vedor response despite better SIN is the result of subtracting the additional basis vector components from the low magnitude signal as discussed for eq 9 above. As a final example, work is proceeding in this laboratory using cross-correlation on more complex chromatographic signals. Figure 7 illuntrates the use of cross-correlation on a portion of a GCIFTIRL chromatogram from a shale oil analysis. In addition to improving quantitative precision, the S I N enhancement methold may prove to be beneficial from a qualitative standp0in.t. in locating low S I N peaks in complicated chromatogramia or other signals.
CONCLUSIONS The cross-correlation S I N enhancement technique described here is not confined in its appllication to GC/FTIR. Quantitative cross-correlation results may be related back to original signals arising from any source if the signal has a Gaussian shape. For other peak shapes such as Lorentzian or exponentially modiified Gaussians, the appropriate relationships must be derived from the cross-correlation equation. However, eq 6 is a general relation irrespective of peak shape if s2 is taken to be the statistical second moment of the peak (15). A problem with crloss-correlation in general is some uncertainty in constructing base lines through the cross-correlation peak which still! contains a substantial quantity of low frequency noise. Sevleral peaks in this study exhibited this difficulty and work is progressing to minimize this effect. This
P
500
'
~
600
,
~
,
100
~
.
~
I
800 900 INTERFERODRRH NUmER
~
~
~
1000
~
Figure 7. The cross-correlation of a portion of a GCIFTIR chromatogram of a shale oil mixture (78).
problem would be significantly worse for those instrumental techniques having a substantial l l f noise or drift component which is an inherent property of electronic amplifiers. Another problem with cross-correlation is the necessity of matching the reference and noisy peak shapes. This may be overcome in some cases if the functional form of the peaks is approximately known (as in the application described in this paper). Alternatively, a measured peak having a high S I N may be used as the reference function. As additional chemical analysis problems arise, software data handling techniques become increasingly important for efficient data evaluation. However, the advantages and limitations of data manipulation algorithms must be well understood. Cross-correlation S / N enhancement should prove beneficial in future studies requiring both qualitative and quantitative capabilities.
LITERATURE CITED (1) Hanna, D. A.; Hangac. G.; Hohne, B. A,; Small, G. W.; Wieboldt, R. C.; Isenhour, T. L. J . Chromatogr. Sci. 1979, 77, 423. (2) Lam, R. B.; Isenhour, T. L. Anal. Chem. 1981, 53, 1179. (3) Hieltje, G. M. Anal. Chem. 1972, 44 (6), 81A. (4) Hieftje, G. M. Anal. Chem. 1972, 44 (7), 69A. (5) Lee, Y. W.; Cheatham, T. P.; Wiesner, J. B. R o c . IRE 1950, 38, 1165. (6) Lee, Y. W. "Statlstlcal Theory of Communication"; Wiley: New York, 1960. (7) Hieftje, G. M.; Bystroff, R. I.; Lim, R. Anal. Chem. 1973, 45, 253. (8) Annlno, R. J . Chromatogr. Sci. 1976, 74, 265. (9) Betty, K. R.; Horlick, G. Anal. Chem. 1978, 45, 1899. (10) Horlick, G. Anal. Chem. 1973, 45,319. (11) Lam, R. B.; Wieboldt, R. C.; Isenhour, T. L. Anal. Chem. 1981, 53, 889A. (12) Sparks, D. T.; Lam R. B.; Isenhour, T. L. Anal. Chem. 1982, 5 4 , 0000. (13) Wieboldt, R. C.; Hohne, B. A.; Isenhour, T. L. Appl. Spectrosc. 1980, 34,7. (14) Hanna, A.; Marshall, J. C.; Isenhour, T. L. J . Chromatogr. Sci. 1979, 77, 434. (15) Bracewell, R. "The Fourier Transform and Its Applications", 2nd. ed., New York: McGraw-Hill: New York, 1978.
RECEIVED for review January 11,1982. Accepted June 4,1982. The financial support of the National Science Foundation, Grant No. CHE 8026747, is gratefully acknowledged.
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