Resolution Enhancement in Second-Derivative Spectra Mirosław A. Czarnecki Faculty of Chemistry, University of Wrocł aw, F. Joliot-Curie 14, 50-383 Wrocł aw, Poland
Derivative spectroscopy is a powerful tool for the resolution enhancement in infrared, near-infrared, Raman, ultraviolet-visible, nuclear magnetic resonance, electron paramagnetic resonance, and fluorescence spectroscopy. Despite its great significance in analytical chemistry, not all aspects of the applications of this method have been explored as yet. This is the first systematic study of the parameters that influence the resolution enhancement in the second derivative spectra. The derivative spectra were calculated with the Savitzky–Golay method with different window size (5, 15, 25) and polynomial order (2, 4). The results obtained in this work show that the resolution enhancement in the second derivative spectra strongly depends on the data spacing in the original spectra, window size, polynomial order, and peak profile. As shown, the resolution enhancement is related to variations in the width of the peaks upon the differentiation. The present study reveals that in order to maximize the separation of the peaks in the second derivative spectra, the original spectra should be recorded at high resolution and differentiated using a small window size and high polynomial order. However, working with the real spectra one has to compromise between the noise reduction and optimization of the resolution enhancement in the second derivative spectra. Index Headings: Derivative spectroscopy; Second derivative; Resolution enhancement; Savitzky–Golay derivatization; Peak positions; Computer-aided spectroscopy.
INTRODUCTION The spectra obtained by different techniques including infrared (IR), near-infrared, Raman, ultravioletvisible, nuclear magnetic resonance, electron paramagnetic resonance, and fluorescence spectroscopy often consist of broad and heavily overlapped features. To extract useful information from these spectra, one first has to separate the individual peaks. Derivative spectroscopy is a routine method for resolution enhancement and has been implemented in many commercial instruments and software for data analysis. Differentiation often allows the detection of trace elements in the presence of strongly absorbing components and improves the accuracy of quantitative analyses. The first and second derivatives are frequently used as the preprocessing procedures in the multivariate analysis.1–3 In the literature one finds numerous available examples of successful applications of derivative spectroscopy for quantitative and qualitative analysis of various kinds of samples measured by diverse techniques.4-26 Numerical calculation of the derivatives is usually based on the Savitzky–Golay method. 27,28 In this
approach, a small portion of data (window) is fitted by a low-degree polynomial (2–4) using simplified least squares procedures. The calculation of the derivative spectra by this method is accompanied by simultaneous data smoothing. An increase in the window size improves the degree of smoothing. On the other hand, too broad of a window reduces the resolution enhancement effect and distorts the derivative spectra. A lower degree of the polynomial leads to more effective noise suppression, while a higher polynomial degree provides more accurate reproduction of the peak profiles. The best parameters for the Savitzky–Golay routine are usually selected by a trial-and-error method. Recently, Zimmermann and Koehler published a paper on optimization of Savitzky–Golay parameters for improving spectral resolution and quantification in IR spectroscopy.29 The authors studied the influence of spectral anomalies like baseline variations, noise, and scattering on the second derivative spectra. It was concluded that a particular window size is associated with the specific component and frequently does not apply for the other components. Hence, the best results can be obtained by combining data sets preprocessed with different parameters. Previous works have focused on removing the baseline drift1,29 or noise30,31 in the derivative spectra. Several papers have been dedicated to studies of the relationship between the width of the original peaks and the amplitude of the second derivative peaks. 32,33 However, little attention has been paid to an examination of the parameters determining the resolution enhancement in the second derivative spectra. In addition, most studies of derivative spectroscopy have been performed only for two limiting cases: pure Lorentzian or pure Gaussian peak profiles. The real peak shapes are neither Lorentzian nor Gaussian and in most cases are well approximated by a combination (sum or product) of these two functions. This study is focused on the effect of the full width half-maximum (FWHM), the separation between the peaks (Dm0 = jm02m01j, where m01 and m02 are the peak positions), the fraction of a Lorentzian function, and the data spacing in the original spectra on the resolution enhancement in the second derivative spectra. In addition, this work provides arguments for selection of the best parameters for the Savitzky–Golay method (window size, polynomial order) in terms of the resolution enhancement in the second derivative spectra.
SIMULATED DATA Received 16 April 2014; accepted 9 July 2014. E-mail: [email protected]. DOI: 10.1366/14-07568
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The peak shapes were modeled by a sum of Lorentzian and Gaussian functions: 0003-7028/15/6901-0067/0 Q 2015 Society for Applied Spectroscopy
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AðmÞ ¼ A0 a
e
Dm21=2
þ ð1 aÞ Dm21=2 þ 4 ðm m0 Þ2
4lnð2Þðmm0 Þ2 =Dm21=2
Þ
ð1Þ
where A0 is a peak height at m = m0 (default value of A0 = 1), Dm1/2 is FWHM, m0 is a peak position, and a is a fraction of a Lorentzian function (1a is a fraction of a Gaussian function). The noise-free spectra were simulated in the range 800–1200 cm1 with a data spacing (Dm = jmiþ1mij) of 0.1, 0.5, 1, and 2 cm1. The changes in FWHM and the separation between the peaks were linear with a step of 1 or 0.1 cm1, while those in the fraction of a Lorentzian function were calculated with a step of 0.1. The second derivative spectra were computed with the savgol function from PLS-toolbox 6.2 (Eigenvector Research) for Matlab. This function is based on the Savitzky–Golay method of derivatization.27 The window size was 5, 15, or 25, and the polynomial degree was 2 or 4. To reduce the number of figures, in this paper are presented only the results obtained with a polynomial degree of 2. The corresponding data for a polynomial degree of 4 are shown in the Supplemental Material (Figs. 1S–8S). All calculations were performed with software written in Matlab 7 (The MathWorks).
RESULTS AND DISCUSSION Effect of Full Width Half-Maximum and Fraction of a Lorentzian Function on Resolution Enhancement in the Second Derivative Spectra. Figures 1 and 2 display the effect of FWHM on positions of two Lorentzian and Gaussian peaks (at 995 and 1005 cm1), respectively, in the second derivative spectra obtained with a polynomial degree of 2. Corresponding figures for a polynomial order of 4 are displayed in the Supplemental Material (Figs. S1 and S2). For reference, in each figure are plotted the peak positions determined from the original spectra. If the bands are broad, the positions of the highand low-frequency peaks determined from the original or second derivative spectra are the same (1000 cm1). This means that the peaks are not resolved yet. Upon reducing the band width, the peak positions are differentiated and gradually approach the true values. From Figs. 1–2 and S1–S2 it is evident that in going from the original to the second derivative spectra a significant resolution enhancement appears, and this effect is more pronounced for Lorentzian peaks. As can be seen (Figs. 1a and 2a), for data separation of 0.1 cm1 the Lorentzian peaks are resolved at FWHM = 30.8 cm1, while the Gaussian peaks are separated at FWHM = 15.9 cm1. The separation of the original Lorentzian and Gaussian peaks starts at FWHM = 17.4 and 11.8 cm1, respectively. The smaller window size and the higher polynomial order, the better resolution in the second derivative spectra. If the data spacing is small as compared with FWHM, the window size does not influence the peak positions determined from the second derivative spectra (Fig. 1a). An increase in the data spacing leads to discrimination of the peak positions obtained from the second derivative spectra calculated with different
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values of the window size (Figs. 1b and 1c). If the window size is too large, even relatively narrow peaks are not resolved in the derivative spectra (Fig. 1d). The same trends are observed for Gaussian peaks (Figs. 2a– 2d). The real positions of the original peaks are 995 and 1005 cm1; hence it is clear that the peak positions obtained from the second derivative spectra may significantly depart from the true values. Figure 3 displays the effect of a fraction of a Lorentzian function on limiting the value of FWHM at which two peaks (at 995 and 1005 cm1) start to merge in the original and second derivative spectra. These data were obtained at a polynomial order of 2, while the analogous results for a polynomial order of 4 are shown in the Supplemental Material (Fig. S3). From Figs. 3a and 3b it results that if the original spectra have small data spacing, the peaks in the second derivative spectra start to merge at higher values of FWHM as compared with the original spectra (regardless of the fraction of a Lorentzian function). An increase in the fraction of a Lorentzian function increases the limiting value of FWHM both in the original (from 11.8 to 17.4 cm1) and in the second derivative from (15.9 to 30.6 cm1) spectra. This means that Lorentzian peaks are better resolved than Gaussian peaks. If the data spacing is small, Lorentzian peaks are merged in the second derivative spectra at approximately twice higher FWHM as compared with Gaussian peaks, and the shape of this relationship does not depend on the window size (Figs. 3a and 3aS). For larger data spacing an increase in the window size leads to weaker dependence of the limiting FWHM on the fraction of a Lorentzian function. An increase in the polynomial order reduces the effect of the window size on this relationship (Fig. S3). Sometimes the limiting value of FWHM in the second derivative spectra may decrease with the increase in fraction of a Lorentzian function (Fig. S3d), demonstrating that Gaussian peaks are better resolved than Lorentzian ones. In addition, the separation of overlapped peaks in the normal spectra may be better as compared with that in the second derivative spectra. However, this situation occurs only for the spectra recorded with a large data spacing and differentiated by using a broad window. Effect of Peaks Separation and Fraction of a Lorentzian Function on Resolution Enhancement in the Second Derivative Spectra. Figures 4–5 and S4–S5 display the relationships between the peak separation and the peak positions in the normal and second derivative spectra. The original spectra consist of two Lorentzian (Figs. 4 and S4) or Gaussian (Figs. 5 and S5) peaks with FWHM = 10 cm1. The initial positions of both peaks were set to 1000 cm1 (Dm0 = 0), and next the peaks were gradually shifted in the opposite directions. At first, the positions of both peaks remain unchanged (1000 cm1), indicating that the peaks are not yet resolved. When the distance between the peaks still further increases, the positions of both peaks start to differentiate. This means that the peaks are resolved in the original or second derivative spectra. Initially the changes in the peak positions are nonlinear, whereas the real peak positions are changing in a linear manner. Hence, it is clear that for heavily overlapped peaks the positions determined from the original and second
FIG. 1. Effect of FWHM on the peak positions in the original (*) and second derivative (*) spectra for two Lorentzian peaks at 995 and 1005 cm1. Figures (a), (b), (c), and (d) show the relationships for the data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. The peak positions obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red, and green lines, respectively, while the values determined from the original spectra are drawn in magenta.
FIG. 2. Effect of FWHM on the peak positions in the original (*) and second derivative (*) spectra for two Gaussian peaks at 995 and 1005 cm1. Figures (a), (b), (c), and (d) show the relationships for data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. The peak positions obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red, and green lines, respectively, while the values determined from the original spectra are drawn in magenta.
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FIG. 3. Effect of fraction of Lorentzian function on limiting FWHM at which two peaks at 995 and 1005 cm1 start to merge in the original (*) and second derivative spectra (*). Figures (a), (b), (c), and (d) show the relationships for data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. The limiting values of FWHM obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red, and green lines, respectively, while the values determined from the original spectra are drawn in magenta.
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derivative spectra deviate from the real positions. Finally, the variations are linear, and the positions obtained from the normal and derivative spectra are the same as the real ones. Comparison of Figs. 4–5 and S4–S5 reveals that for small data spacing (0.1 or 0.5 cm1), the peaks are resolved in the second derivative spectra earlier (at smaller peaks separation) as compared with the original spectra, and the peak positions do not depend on the window size. Obviously, Lorentzian peaks are resolved earlier than Gaussian peaks in both the normal and second derivative spectra. The limiting values of the peak separation in the original spectra for Lorentzian and Gaussian peaks are 5.8 and 8.5 cm1, respectively, while the corresponding values in the second derivative spectra are 3.3 and 6.3 cm1. If the data spacing in the original spectra is relatively small, an increase in the polynomial order does not enhance the resolution in the second derivative spectra. In contrast, for the data spacing higher than 1 cm1 an increase in the polynomial order improves the peaks’ separation in the second derivative spectra. Figures 6 and S6 show the effect of the fraction of a Lorentzian function on limiting the value of the separation at which two peaks with FWHM = 10 cm1 are resolved in the original and second derivative spectra. As can be seen, at a small data spacing the separation of the peaks in the second derivative spectra starts earlier (at smaller peaks separation) as compared with that in the original spectra (Fig. 6a). In most cases, an increase in the fraction of a Lorentzian function leads to better separation of the peaks in both the original and second derivative spectra. In other words, Lorentzian peaks are resolved at smaller values of the separation than Gaussian peaks. Exceptions from this rule may appear for the spectra with larger data spacing (.1 cm1) differentiated with a broad window (Figs. 6c and 6d). An increase in the window size reduces the resolution enhancement in the second derivative spectra, and this effect is more prominent for smaller polynomial order. Additional calculations (not shown) reveal that the magnitude of the changes displayed in Figs. 6 and S6 is proportional to FWHM. Effect of Full Width Half-Maximum and Fraction of Lorentzian Function in the Original Spectra on Full Width Half-Maximum in the Second Derivative Spectra. The resolution enhancement in the second derivative spectra results from variations of FWHM upon differentiation, and this effect has a different extent for Lorentzian and Gaussian peaks. However, an exact relationship between FWHM in the original and second derivative spectra has not been reported as yet. In Figs. 7 and S7 are displayed these relationships for a polynomial order of 2 and 4, respectively. The solid black line indicates FWHM of the original peak. If the relationships are below this black line, it means that FWHM of the second derivative peak is smaller than that of the original peak (Figs. 7a and S7a). As a result, in the second derivative spectrum occurs the resolution enhancement. In the other case, the derivative peaks are broader than the original ones, and the peaks are better resolved in the original spectra. This situation takes place only for the spectra with a large data spacing differentiated with a broad window (Figs. 7c, 7d, and
FIG. 4. Effect of the peak separation on the peak positions in the original (*) and second derivative (*) spectra for two Lorentzian peaks at 995 and 1005 cm1 (FWHM = 10 cm1). Figures (a), (b), (c), and (d) show the relationships for data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. The peak positions obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red, and green lines, respectively, while the values determined from the original spectra are drawn in magenta.
FIG. 5. Effect of the peak separation on the peak positions in the original (*) and second derivative (*) spectra for two Gaussian peaks at 995 and 1005 cm1 (FWHM = 10 cm1). Figures (a), (b), (c), and (d) show the relationships for data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. The peak positions obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red, and green lines, respectively, while the values determined from the original spectra are drawn in magenta.
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FIG. 6. Effect of the fraction of a Lorentzian function on limiting peak separation at which two peaks of FWHM = 10 cm1 start to separate in the original (*) and second derivative spectra (*). Figures (a), (b), (c), and (d) show the relationships for data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. The limiting peaks separation obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red and green lines, respectively, while the values determined from the original spectra are drawn in magenta.
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FIG. 7. Effect of FWHM in the original spectra on FWHM in the second derivative spectra for Lorentzian (*) and Gaussian (u) peaks. Figures (a), (b), (c), and (d) show the relationships for data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. FWHM obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red, and green lines, respectively. The black solid line shows FWHM of the original peak.
S7d). If the data spacing is relatively small, FWHM in the second derivative spectra does not depend on the window size or the polynomial power (Figs. 7a and S7a). It is also clear that FWHM of Lorentzian peaks in the second derivative spectra is significantly smaller than that of Gaussian peaks. As can seen from Figs. 7a and S7a, if FWHM of the original peak is 30 cm1, then FWHM of the Lorentzian and Gaussian peaks in the second derivative spectrum is 9.8 and 15.9 cm1, respectively. Upon an increase in the data spacing this difference becomes less apparent. FWHM obtained at different values of the window size starts to differentiate, and this effect is more evident for lower polynomial power (Fig. 7). In addition, depending on the window size, FWHM of Lorentzian peaks may be greater than that of Gaussian peaks (Figs. 7c, 7d, S7c, and S7d). Comparison of Figs. 7 and S7 shows that using the higher polynomial order provides better resolution enhancement regardless of the data spacing and the window size. Figures 8 and S8 display the effect of fraction of Lorentzian function on FWHM of the second derivative peaks. The original spectra consist of a single peak of FWHM = 10 cm1. As can be seen (Figs. 8a, 8b, S8a, and S8b), at small and moderate values of the data spacing an increase in the fraction of a Lorentzian function reduces FWHM of the second derivative peak. For a data spacing of 0.1 cm1 the FWHM of the original peak is reduced upon differentiation from 10 cm1 to 5.3 and 3.3 cm1 for Gaussian and Lorentzian peaks, respectively. This observation confirms that Lorentzian peaks are better resolved in the second derivative spectra as compared with the Gaussian peaks. An opposite situation may occur only for the spectra with large data spacing differentiated with a broad window (Figs. 8c, 8d, and S8d). In the worst situation (data spacing of 2 cm1 and window size of 15 or 25), the second derivative peaks are broader than the original ones. This effect is more evident for the derivative spectra calculated with the low polynomial order (Fig. 8d). Additional calculations (not shown) reveal that the magnitude of the changes shown in Figs. 8 and S8 is related to FWHM of the original peak.
CONCLUSIONS In the second derivative spectra the resolution enhancement appears, and this effect is more evident for the spectra with small data spacing. An increase in the data spacing and the window size significantly reduces the extent of the resolution enhancement effect. If the spectra with a large data spacing are differentiated with a broad window, the separation of the peaks in the original spectra may be better than those in the second derivative spectra, regardless of the fraction of a Lorentzian function. The degree of the resolution enhancement increases in going from Gaussian to Lorentzian peak profiles, and this difference is more pronounced for the spectra with small data spacing. An increase in the window size reduces the difference between Gaussian and Lorentzian peaks. In general, the higher the polynomial order, the better the resolution of the peaks in the second derivative spectra. As shown,
FIG. 8. Effect of the fraction of a Lorentzian function on FWHM of the peak in the derivative spectrum (*). Figures (a), (b), (c), and (d) show the relationships for data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. FWHM obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red and green lines, respectively. The original spectrum comprise of a single peak of FWHM = 10 cm1.
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the resolution enhancement effect simply results from variations in FWHM upon differentiation. This study clearly shows that in order to optimize the resolution enhancement in the second derivative spectra, the original spectra should be recorded with a high resolution and differentiated with a small window size and high polynomial order. However, working with a real spectra one has to compromise between maximizing the spectral resolution and the noise suppression. In addition, this study reveals that the positions of overlapped peaks may significantly deviate from the true values for both the normal and second derivative spectra. SUPPLEMENTAL MATERIAL All supplemental material mentioned in the text, including Figs. S1– S8 and figure captions, is available in the online version of the journal at http://www.s-a-s.org. 1. C.D. Brown, L. Vega-Montoto, P.D. Wentzell. ‘‘Derivative Preprocessing and Optimal Corrections for Baseline Drift in Multivariate Calibration’’. Appl. Spectrosc. 2000. 54(7): 1055-1068. 2. R.S. Delwiche, J.B. Reeves III. ‘‘A Graphical Method to Evaluate Spectral Preprocessing in Multivariate Regression Calibrations: Example with Savitzky-Golay Filters and Partial Least Squares Regression’’. Appl. Spectrosc. 2010. 64(1): 73-82. 3. A˚. Rinnan, F. van den Berg, S.B. Engelsen. ‘‘Review of the Most Common Pre-Processing Techniques for Near-Infrared Spectra’’. TRAC-Trend Anal. Chem. 2009. 28(10): 1201-1222. 4. T.C. O’Haver, G.L. Green. ‘‘Numerical Error Analysis of Derivative Spectrometry for the Quantitative Analysis of Mixtures’’. Anal. Chem. 1976. 48(2): 312-318. 5. C. Balestrieri, G. Colonna, A. Giovane, G. Irace, L. Servillo. ‘‘Second-Derivative Spectroscopy of Proteins’’. Eur. J. Biochem. 1978. 90(3): 433-440. 6. K.G. Jones, D.G. Sweeney. ‘‘Quantitation of Urinary of Porphyrins by Use by Second-Derivative Spectroscopy’’. Clinical Chem. 1979. 25(1-2): 71-74. 7. M.R. Whitbeck. ‘‘Second Derivative Infrared Spectroscopy’’. Appl. Spectrosc. 1981. 35(1): 93-95. 8. L. Servillo, G. Colonna, C. Balestrieri, R. Ragone, G. Irace. ‘‘Simultaneous Determination of Tyrosine and Tryptophan Residues in Proteins by Second-Derivative Spectroscopy’’. Anal. Biochem. 1982. 126(2): 251-257. 9. H. Susi, D.M. Byler. ‘‘Protein Structure by Fourier Transform Infrared Spectroscopy: Second Derivative Spectra’’. Biochem. Biophys. Res. Comm. 1983. 115(1): 391-397. 10. R. Ragone, G. Colonna, C. Balestrieri, L. Servillo, G. Irace. ‘‘Determination of Tyrosine Exposure in Proteins by SecondDerivative Spectroscopy’’. Biochemistry. 1984. 23(8): 1871-1875. 11. D.C. Lee, J.A. Hayward, C.J. Restall, D. Chapman. ‘‘SecondDerivative Infrared Spectroscopic Studies of the Secondary Structures of Bacteriorhodopsin and Ca2þ-ATPase’’. Biochemistry. 1985. 24(16): 4364-4373. 12. N. Suzuki, R. Kuroda. ‘‘Direct Simultaneous Determination of Nitrate and Nitrite by Ultraviolet Second-Derivative Spectrophotometry’’. Analyst. 1987. 112(7): 1077-1079. 13. A. Dong, P. Huang, W.S. Caughey. ‘‘Protein Secondary Structures in Water from Second-Derivative Amide I Infrared Spectra’’. Biochemistry. 1990. 29(13): 3303-3308. 14. H. Mach, C.R. Middaugh, R.V. Lewis. ‘‘Detection of Proteins and Phenol in DNA Samples with Second-Derivative Absorption Spectroscopy’’. Anal. Biochem. 1992. 200(1): 20-26.
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15. M. Galligani, S. Garrigues, M. Guardia. ‘‘Direct Determination of Ethanol in All Types of Alcoholic Beverages by Near-Infrared Derivative Spectrometry’’. Analyst. 1993. 118(9): 1167-1173. 16. H. Mach, C.R. Middaugh. ‘‘Simultaneous Monitoring of the Environment of Tryptophan, Tyrosine, and Phenylalanine Residues in Proteins by Near-Ultraviolet Second-Derivative Spectroscopy’’’. Anal. Biochem. 1994. 222(2): 323-331. 17. S. Torrado, S. Torrado, R.J. Cado´rniga. ‘‘Comparison of Assay Methods by Second-Derivative Spectroscopy, Colorimetry and Fluorescence Spectroscopy of Salicylic Acid in Aspirin Preparations with a High-Performance Liquid Chromatographic Method’’. J. Pharm. Biomed. Anal. 1994. 12(3): 383-387. 18. J.A. Kornblatt, M.J. Kornblatt, G.H.B. Hoe. ‘‘Second Derivative Spectroscopy of Enolase at High Hydrostatic Pressure—An Approach to the Study of Macromolecular Interactions’’. Biochemistry. 1995. 34(4): 1218-1223. 19. D.G. Konstantianos, R.C. Ioannou. ‘‘Second Derivative Synchronous Fluorescence Spectroscopy for the Simultaneous Determination of Naproxen and Salicylic-Acid in Human Serum’’. Analyst. 1996. 121(7): 909-912. 20. B.S. Kendrick, A. Dong, S.D. Allison, M.C. Manning, J.F. Carpenter. ‘‘Quantitation of the Area of Overlap Between Second-Derivative Amide I Infrared Spectra to Determine the Structural Similarity of a Protein in Different States’’. J. Pharm. Sci. 1996. 85(2): 155-158. 21. Y.C. Micalizzi, N.B. Poppano, N.B. Debattista. ‘‘First and Second Order Derivative Spectrophotometric Determination of Benzyl Alcohol and Diclofenac in Pharmaceutical Forms’’. Talanta. 1998. 47(3): 525-530. 22. J. Zhang, Y.B. Yan. ‘‘Probing Conformational Changes of Proteins by Quantitative Second-Derivative Infrared Spectroscopy’’. Anal. Biochem. 2005. 340(1): 89-98. 23. Y. Kim, D.S. Himmelsbach, S.E. Kays. ‘‘ATR-Fourier Transform MidInfrared Spectroscopy for Determination of trans Fatty Acids in Ground Cereal Products Without Oil Extraction’’. J. Agric. Food Chem. 2007. 55(11): 4327-4333. 24. A. Ikehata, K. Ohara, H. Shinzawa, Y. Ozaki. ‘‘Sensitive Detection and Identification of Organic Liquids Using the Second Derivative of Surface Plasmon Resonance Near-Infrared Spectra’’. Appl. Spectrosc. 2008. 62(5): 517-524. 25. S. Kasemsumran, W. Thanapase, V. Punsuvon, Y. Ozaki. ‘‘A Feasibility Study on Nondestructive Determination of Oil Content in Palm Fruits by Visible–Near Infrared Spectroscopy’’. J. NIR Spectrosc. 2012. 20(6): 687-694. 26. J. Cielecka-Piontek. ‘‘Derivative Spectrophotometry for the Determination of Faropenem in the Presence of Degradation Products: An Application for Kinetic Studies’’. Appl. Spectrosc. 2013. 67(7): 703-708. 27. A. Savitzky, M.J.E. Golay. ‘‘Smoothing and Differentiation of Data by Simplified Least Squares Procedures’’. Anal. Chem. 1964. 36(8): 1627-1639. 28. J. Steinier, Y. Termonia, J. Deltour. ‘‘Comments on Smoothing and Differentiation of Data by Simplified Least Squares Procedures’’. Anal. Chem. 1972. 44(11): 1906-1909. 29. B. Zimmermann, A. Koehler. ‘‘Optimizing Savitzky-Golay Parameters for Improving Spectral Resolution and Quantification in Infrared Spectroscopy’’. Appl. Spectrosc. 2013. 67(8): 892-902. 30. L. Antonov, S. Stoyanov. ‘‘Approach for Increased Information from the Second-Derivative Spectra in UV-Vis Absorption Spectroscopy’’. Appl. Spectrosc. 1993. 47(10): 1712-1715. 31. L. Antonov, S. Stoyanov. ‘‘Step by Step Filter–An Approach for Noise Reduction in the Derivative UV-Visible Spectra’’. Anal. Chim. Acta. 1996. 324(1): 77-83. 32. H. Mark, J. Workman Jr. ‘‘Derivatives in Spectroscopy, Part I: The Behavior of Derivative’’. Spectroscopy. 2003. 18(4): 32-37. 33. H. Mark, J. Workman Jr. ‘‘Derivatives in Spectroscopy, Part II: The True Derivative’’. Spectroscopy. 2003. 18(9): 25-28.
Derivative spectroscopy is a powerful tool for the resolution enhancement in infrared, near-infrared, Raman, ultraviolet-visible, nuclear magnetic resonance, electron paramagnetic resonance, and fluorescence spectroscopy. Despite its great significance in analytical chemistry, not all aspects of the applications of this method have been explored as yet. This is the first systematic study of the parameters that influence the resolution enhancement in the second derivative spectra. The derivative spectra were calculated with the Savitzky–Golay method with different window size (5, 15, 25) and polynomial order (2, 4). The results obtained in this work show that the resolution enhancement in the second derivative spectra strongly depends on the data spacing in the original spectra, window size, polynomial order, and peak profile. As shown, the resolution enhancement is related to variations in the width of the peaks upon the differentiation. The present study reveals that in order to maximize the separation of the peaks in the second derivative spectra, the original spectra should be recorded at high resolution and differentiated using a small window size and high polynomial order. However, working with the real spectra one has to compromise between the noise reduction and optimization of the resolution enhancement in the second derivative spectra. Index Headings: Derivative spectroscopy; Second derivative; Resolution enhancement; Savitzky–Golay derivatization; Peak positions; Computer-aided spectroscopy.
INTRODUCTION The spectra obtained by different techniques including infrared (IR), near-infrared, Raman, ultravioletvisible, nuclear magnetic resonance, electron paramagnetic resonance, and fluorescence spectroscopy often consist of broad and heavily overlapped features. To extract useful information from these spectra, one first has to separate the individual peaks. Derivative spectroscopy is a routine method for resolution enhancement and has been implemented in many commercial instruments and software for data analysis. Differentiation often allows the detection of trace elements in the presence of strongly absorbing components and improves the accuracy of quantitative analyses. The first and second derivatives are frequently used as the preprocessing procedures in the multivariate analysis.1–3 In the literature one finds numerous available examples of successful applications of derivative spectroscopy for quantitative and qualitative analysis of various kinds of samples measured by diverse techniques.4-26 Numerical calculation of the derivatives is usually based on the Savitzky–Golay method. 27,28 In this
approach, a small portion of data (window) is fitted by a low-degree polynomial (2–4) using simplified least squares procedures. The calculation of the derivative spectra by this method is accompanied by simultaneous data smoothing. An increase in the window size improves the degree of smoothing. On the other hand, too broad of a window reduces the resolution enhancement effect and distorts the derivative spectra. A lower degree of the polynomial leads to more effective noise suppression, while a higher polynomial degree provides more accurate reproduction of the peak profiles. The best parameters for the Savitzky–Golay routine are usually selected by a trial-and-error method. Recently, Zimmermann and Koehler published a paper on optimization of Savitzky–Golay parameters for improving spectral resolution and quantification in IR spectroscopy.29 The authors studied the influence of spectral anomalies like baseline variations, noise, and scattering on the second derivative spectra. It was concluded that a particular window size is associated with the specific component and frequently does not apply for the other components. Hence, the best results can be obtained by combining data sets preprocessed with different parameters. Previous works have focused on removing the baseline drift1,29 or noise30,31 in the derivative spectra. Several papers have been dedicated to studies of the relationship between the width of the original peaks and the amplitude of the second derivative peaks. 32,33 However, little attention has been paid to an examination of the parameters determining the resolution enhancement in the second derivative spectra. In addition, most studies of derivative spectroscopy have been performed only for two limiting cases: pure Lorentzian or pure Gaussian peak profiles. The real peak shapes are neither Lorentzian nor Gaussian and in most cases are well approximated by a combination (sum or product) of these two functions. This study is focused on the effect of the full width half-maximum (FWHM), the separation between the peaks (Dm0 = jm02m01j, where m01 and m02 are the peak positions), the fraction of a Lorentzian function, and the data spacing in the original spectra on the resolution enhancement in the second derivative spectra. In addition, this work provides arguments for selection of the best parameters for the Savitzky–Golay method (window size, polynomial order) in terms of the resolution enhancement in the second derivative spectra.
SIMULATED DATA Received 16 April 2014; accepted 9 July 2014. E-mail: [email protected]. DOI: 10.1366/14-07568
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The peak shapes were modeled by a sum of Lorentzian and Gaussian functions: 0003-7028/15/6901-0067/0 Q 2015 Society for Applied Spectroscopy
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AðmÞ ¼ A0 a
e
Dm21=2
þ ð1 aÞ Dm21=2 þ 4 ðm m0 Þ2
4lnð2Þðmm0 Þ2 =Dm21=2
Þ
ð1Þ
where A0 is a peak height at m = m0 (default value of A0 = 1), Dm1/2 is FWHM, m0 is a peak position, and a is a fraction of a Lorentzian function (1a is a fraction of a Gaussian function). The noise-free spectra were simulated in the range 800–1200 cm1 with a data spacing (Dm = jmiþ1mij) of 0.1, 0.5, 1, and 2 cm1. The changes in FWHM and the separation between the peaks were linear with a step of 1 or 0.1 cm1, while those in the fraction of a Lorentzian function were calculated with a step of 0.1. The second derivative spectra were computed with the savgol function from PLS-toolbox 6.2 (Eigenvector Research) for Matlab. This function is based on the Savitzky–Golay method of derivatization.27 The window size was 5, 15, or 25, and the polynomial degree was 2 or 4. To reduce the number of figures, in this paper are presented only the results obtained with a polynomial degree of 2. The corresponding data for a polynomial degree of 4 are shown in the Supplemental Material (Figs. 1S–8S). All calculations were performed with software written in Matlab 7 (The MathWorks).
RESULTS AND DISCUSSION Effect of Full Width Half-Maximum and Fraction of a Lorentzian Function on Resolution Enhancement in the Second Derivative Spectra. Figures 1 and 2 display the effect of FWHM on positions of two Lorentzian and Gaussian peaks (at 995 and 1005 cm1), respectively, in the second derivative spectra obtained with a polynomial degree of 2. Corresponding figures for a polynomial order of 4 are displayed in the Supplemental Material (Figs. S1 and S2). For reference, in each figure are plotted the peak positions determined from the original spectra. If the bands are broad, the positions of the highand low-frequency peaks determined from the original or second derivative spectra are the same (1000 cm1). This means that the peaks are not resolved yet. Upon reducing the band width, the peak positions are differentiated and gradually approach the true values. From Figs. 1–2 and S1–S2 it is evident that in going from the original to the second derivative spectra a significant resolution enhancement appears, and this effect is more pronounced for Lorentzian peaks. As can be seen (Figs. 1a and 2a), for data separation of 0.1 cm1 the Lorentzian peaks are resolved at FWHM = 30.8 cm1, while the Gaussian peaks are separated at FWHM = 15.9 cm1. The separation of the original Lorentzian and Gaussian peaks starts at FWHM = 17.4 and 11.8 cm1, respectively. The smaller window size and the higher polynomial order, the better resolution in the second derivative spectra. If the data spacing is small as compared with FWHM, the window size does not influence the peak positions determined from the second derivative spectra (Fig. 1a). An increase in the data spacing leads to discrimination of the peak positions obtained from the second derivative spectra calculated with different
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values of the window size (Figs. 1b and 1c). If the window size is too large, even relatively narrow peaks are not resolved in the derivative spectra (Fig. 1d). The same trends are observed for Gaussian peaks (Figs. 2a– 2d). The real positions of the original peaks are 995 and 1005 cm1; hence it is clear that the peak positions obtained from the second derivative spectra may significantly depart from the true values. Figure 3 displays the effect of a fraction of a Lorentzian function on limiting the value of FWHM at which two peaks (at 995 and 1005 cm1) start to merge in the original and second derivative spectra. These data were obtained at a polynomial order of 2, while the analogous results for a polynomial order of 4 are shown in the Supplemental Material (Fig. S3). From Figs. 3a and 3b it results that if the original spectra have small data spacing, the peaks in the second derivative spectra start to merge at higher values of FWHM as compared with the original spectra (regardless of the fraction of a Lorentzian function). An increase in the fraction of a Lorentzian function increases the limiting value of FWHM both in the original (from 11.8 to 17.4 cm1) and in the second derivative from (15.9 to 30.6 cm1) spectra. This means that Lorentzian peaks are better resolved than Gaussian peaks. If the data spacing is small, Lorentzian peaks are merged in the second derivative spectra at approximately twice higher FWHM as compared with Gaussian peaks, and the shape of this relationship does not depend on the window size (Figs. 3a and 3aS). For larger data spacing an increase in the window size leads to weaker dependence of the limiting FWHM on the fraction of a Lorentzian function. An increase in the polynomial order reduces the effect of the window size on this relationship (Fig. S3). Sometimes the limiting value of FWHM in the second derivative spectra may decrease with the increase in fraction of a Lorentzian function (Fig. S3d), demonstrating that Gaussian peaks are better resolved than Lorentzian ones. In addition, the separation of overlapped peaks in the normal spectra may be better as compared with that in the second derivative spectra. However, this situation occurs only for the spectra recorded with a large data spacing and differentiated by using a broad window. Effect of Peaks Separation and Fraction of a Lorentzian Function on Resolution Enhancement in the Second Derivative Spectra. Figures 4–5 and S4–S5 display the relationships between the peak separation and the peak positions in the normal and second derivative spectra. The original spectra consist of two Lorentzian (Figs. 4 and S4) or Gaussian (Figs. 5 and S5) peaks with FWHM = 10 cm1. The initial positions of both peaks were set to 1000 cm1 (Dm0 = 0), and next the peaks were gradually shifted in the opposite directions. At first, the positions of both peaks remain unchanged (1000 cm1), indicating that the peaks are not yet resolved. When the distance between the peaks still further increases, the positions of both peaks start to differentiate. This means that the peaks are resolved in the original or second derivative spectra. Initially the changes in the peak positions are nonlinear, whereas the real peak positions are changing in a linear manner. Hence, it is clear that for heavily overlapped peaks the positions determined from the original and second
FIG. 1. Effect of FWHM on the peak positions in the original (*) and second derivative (*) spectra for two Lorentzian peaks at 995 and 1005 cm1. Figures (a), (b), (c), and (d) show the relationships for the data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. The peak positions obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red, and green lines, respectively, while the values determined from the original spectra are drawn in magenta.
FIG. 2. Effect of FWHM on the peak positions in the original (*) and second derivative (*) spectra for two Gaussian peaks at 995 and 1005 cm1. Figures (a), (b), (c), and (d) show the relationships for data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. The peak positions obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red, and green lines, respectively, while the values determined from the original spectra are drawn in magenta.
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FIG. 3. Effect of fraction of Lorentzian function on limiting FWHM at which two peaks at 995 and 1005 cm1 start to merge in the original (*) and second derivative spectra (*). Figures (a), (b), (c), and (d) show the relationships for data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. The limiting values of FWHM obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red, and green lines, respectively, while the values determined from the original spectra are drawn in magenta.
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derivative spectra deviate from the real positions. Finally, the variations are linear, and the positions obtained from the normal and derivative spectra are the same as the real ones. Comparison of Figs. 4–5 and S4–S5 reveals that for small data spacing (0.1 or 0.5 cm1), the peaks are resolved in the second derivative spectra earlier (at smaller peaks separation) as compared with the original spectra, and the peak positions do not depend on the window size. Obviously, Lorentzian peaks are resolved earlier than Gaussian peaks in both the normal and second derivative spectra. The limiting values of the peak separation in the original spectra for Lorentzian and Gaussian peaks are 5.8 and 8.5 cm1, respectively, while the corresponding values in the second derivative spectra are 3.3 and 6.3 cm1. If the data spacing in the original spectra is relatively small, an increase in the polynomial order does not enhance the resolution in the second derivative spectra. In contrast, for the data spacing higher than 1 cm1 an increase in the polynomial order improves the peaks’ separation in the second derivative spectra. Figures 6 and S6 show the effect of the fraction of a Lorentzian function on limiting the value of the separation at which two peaks with FWHM = 10 cm1 are resolved in the original and second derivative spectra. As can be seen, at a small data spacing the separation of the peaks in the second derivative spectra starts earlier (at smaller peaks separation) as compared with that in the original spectra (Fig. 6a). In most cases, an increase in the fraction of a Lorentzian function leads to better separation of the peaks in both the original and second derivative spectra. In other words, Lorentzian peaks are resolved at smaller values of the separation than Gaussian peaks. Exceptions from this rule may appear for the spectra with larger data spacing (.1 cm1) differentiated with a broad window (Figs. 6c and 6d). An increase in the window size reduces the resolution enhancement in the second derivative spectra, and this effect is more prominent for smaller polynomial order. Additional calculations (not shown) reveal that the magnitude of the changes displayed in Figs. 6 and S6 is proportional to FWHM. Effect of Full Width Half-Maximum and Fraction of Lorentzian Function in the Original Spectra on Full Width Half-Maximum in the Second Derivative Spectra. The resolution enhancement in the second derivative spectra results from variations of FWHM upon differentiation, and this effect has a different extent for Lorentzian and Gaussian peaks. However, an exact relationship between FWHM in the original and second derivative spectra has not been reported as yet. In Figs. 7 and S7 are displayed these relationships for a polynomial order of 2 and 4, respectively. The solid black line indicates FWHM of the original peak. If the relationships are below this black line, it means that FWHM of the second derivative peak is smaller than that of the original peak (Figs. 7a and S7a). As a result, in the second derivative spectrum occurs the resolution enhancement. In the other case, the derivative peaks are broader than the original ones, and the peaks are better resolved in the original spectra. This situation takes place only for the spectra with a large data spacing differentiated with a broad window (Figs. 7c, 7d, and
FIG. 4. Effect of the peak separation on the peak positions in the original (*) and second derivative (*) spectra for two Lorentzian peaks at 995 and 1005 cm1 (FWHM = 10 cm1). Figures (a), (b), (c), and (d) show the relationships for data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. The peak positions obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red, and green lines, respectively, while the values determined from the original spectra are drawn in magenta.
FIG. 5. Effect of the peak separation on the peak positions in the original (*) and second derivative (*) spectra for two Gaussian peaks at 995 and 1005 cm1 (FWHM = 10 cm1). Figures (a), (b), (c), and (d) show the relationships for data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. The peak positions obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red, and green lines, respectively, while the values determined from the original spectra are drawn in magenta.
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FIG. 6. Effect of the fraction of a Lorentzian function on limiting peak separation at which two peaks of FWHM = 10 cm1 start to separate in the original (*) and second derivative spectra (*). Figures (a), (b), (c), and (d) show the relationships for data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. The limiting peaks separation obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red and green lines, respectively, while the values determined from the original spectra are drawn in magenta.
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FIG. 7. Effect of FWHM in the original spectra on FWHM in the second derivative spectra for Lorentzian (*) and Gaussian (u) peaks. Figures (a), (b), (c), and (d) show the relationships for data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. FWHM obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red, and green lines, respectively. The black solid line shows FWHM of the original peak.
S7d). If the data spacing is relatively small, FWHM in the second derivative spectra does not depend on the window size or the polynomial power (Figs. 7a and S7a). It is also clear that FWHM of Lorentzian peaks in the second derivative spectra is significantly smaller than that of Gaussian peaks. As can seen from Figs. 7a and S7a, if FWHM of the original peak is 30 cm1, then FWHM of the Lorentzian and Gaussian peaks in the second derivative spectrum is 9.8 and 15.9 cm1, respectively. Upon an increase in the data spacing this difference becomes less apparent. FWHM obtained at different values of the window size starts to differentiate, and this effect is more evident for lower polynomial power (Fig. 7). In addition, depending on the window size, FWHM of Lorentzian peaks may be greater than that of Gaussian peaks (Figs. 7c, 7d, S7c, and S7d). Comparison of Figs. 7 and S7 shows that using the higher polynomial order provides better resolution enhancement regardless of the data spacing and the window size. Figures 8 and S8 display the effect of fraction of Lorentzian function on FWHM of the second derivative peaks. The original spectra consist of a single peak of FWHM = 10 cm1. As can be seen (Figs. 8a, 8b, S8a, and S8b), at small and moderate values of the data spacing an increase in the fraction of a Lorentzian function reduces FWHM of the second derivative peak. For a data spacing of 0.1 cm1 the FWHM of the original peak is reduced upon differentiation from 10 cm1 to 5.3 and 3.3 cm1 for Gaussian and Lorentzian peaks, respectively. This observation confirms that Lorentzian peaks are better resolved in the second derivative spectra as compared with the Gaussian peaks. An opposite situation may occur only for the spectra with large data spacing differentiated with a broad window (Figs. 8c, 8d, and S8d). In the worst situation (data spacing of 2 cm1 and window size of 15 or 25), the second derivative peaks are broader than the original ones. This effect is more evident for the derivative spectra calculated with the low polynomial order (Fig. 8d). Additional calculations (not shown) reveal that the magnitude of the changes shown in Figs. 8 and S8 is related to FWHM of the original peak.
CONCLUSIONS In the second derivative spectra the resolution enhancement appears, and this effect is more evident for the spectra with small data spacing. An increase in the data spacing and the window size significantly reduces the extent of the resolution enhancement effect. If the spectra with a large data spacing are differentiated with a broad window, the separation of the peaks in the original spectra may be better than those in the second derivative spectra, regardless of the fraction of a Lorentzian function. The degree of the resolution enhancement increases in going from Gaussian to Lorentzian peak profiles, and this difference is more pronounced for the spectra with small data spacing. An increase in the window size reduces the difference between Gaussian and Lorentzian peaks. In general, the higher the polynomial order, the better the resolution of the peaks in the second derivative spectra. As shown,
FIG. 8. Effect of the fraction of a Lorentzian function on FWHM of the peak in the derivative spectrum (*). Figures (a), (b), (c), and (d) show the relationships for data spacing of 0.1, 0.5, 1, and 2 cm1, respectively. FWHM obtained from the second derivative spectra calculated with a window size of 5, 15, and 25 are plotted with blue, red and green lines, respectively. The original spectrum comprise of a single peak of FWHM = 10 cm1.
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the resolution enhancement effect simply results from variations in FWHM upon differentiation. This study clearly shows that in order to optimize the resolution enhancement in the second derivative spectra, the original spectra should be recorded with a high resolution and differentiated with a small window size and high polynomial order. However, working with a real spectra one has to compromise between maximizing the spectral resolution and the noise suppression. In addition, this study reveals that the positions of overlapped peaks may significantly deviate from the true values for both the normal and second derivative spectra. SUPPLEMENTAL MATERIAL All supplemental material mentioned in the text, including Figs. S1– S8 and figure captions, is available in the online version of the journal at http://www.s-a-s.org. 1. C.D. Brown, L. Vega-Montoto, P.D. Wentzell. ‘‘Derivative Preprocessing and Optimal Corrections for Baseline Drift in Multivariate Calibration’’. Appl. Spectrosc. 2000. 54(7): 1055-1068. 2. R.S. Delwiche, J.B. Reeves III. ‘‘A Graphical Method to Evaluate Spectral Preprocessing in Multivariate Regression Calibrations: Example with Savitzky-Golay Filters and Partial Least Squares Regression’’. Appl. Spectrosc. 2010. 64(1): 73-82. 3. A˚. Rinnan, F. van den Berg, S.B. Engelsen. ‘‘Review of the Most Common Pre-Processing Techniques for Near-Infrared Spectra’’. TRAC-Trend Anal. Chem. 2009. 28(10): 1201-1222. 4. T.C. O’Haver, G.L. Green. ‘‘Numerical Error Analysis of Derivative Spectrometry for the Quantitative Analysis of Mixtures’’. Anal. Chem. 1976. 48(2): 312-318. 5. C. Balestrieri, G. Colonna, A. Giovane, G. Irace, L. Servillo. ‘‘Second-Derivative Spectroscopy of Proteins’’. Eur. J. Biochem. 1978. 90(3): 433-440. 6. K.G. Jones, D.G. Sweeney. ‘‘Quantitation of Urinary of Porphyrins by Use by Second-Derivative Spectroscopy’’. Clinical Chem. 1979. 25(1-2): 71-74. 7. M.R. Whitbeck. ‘‘Second Derivative Infrared Spectroscopy’’. Appl. Spectrosc. 1981. 35(1): 93-95. 8. L. Servillo, G. Colonna, C. Balestrieri, R. Ragone, G. Irace. ‘‘Simultaneous Determination of Tyrosine and Tryptophan Residues in Proteins by Second-Derivative Spectroscopy’’. Anal. Biochem. 1982. 126(2): 251-257. 9. H. Susi, D.M. Byler. ‘‘Protein Structure by Fourier Transform Infrared Spectroscopy: Second Derivative Spectra’’. Biochem. Biophys. Res. Comm. 1983. 115(1): 391-397. 10. R. Ragone, G. Colonna, C. Balestrieri, L. Servillo, G. Irace. ‘‘Determination of Tyrosine Exposure in Proteins by SecondDerivative Spectroscopy’’. Biochemistry. 1984. 23(8): 1871-1875. 11. D.C. Lee, J.A. Hayward, C.J. Restall, D. Chapman. ‘‘SecondDerivative Infrared Spectroscopic Studies of the Secondary Structures of Bacteriorhodopsin and Ca2þ-ATPase’’. Biochemistry. 1985. 24(16): 4364-4373. 12. N. Suzuki, R. Kuroda. ‘‘Direct Simultaneous Determination of Nitrate and Nitrite by Ultraviolet Second-Derivative Spectrophotometry’’. Analyst. 1987. 112(7): 1077-1079. 13. A. Dong, P. Huang, W.S. Caughey. ‘‘Protein Secondary Structures in Water from Second-Derivative Amide I Infrared Spectra’’. Biochemistry. 1990. 29(13): 3303-3308. 14. H. Mach, C.R. Middaugh, R.V. Lewis. ‘‘Detection of Proteins and Phenol in DNA Samples with Second-Derivative Absorption Spectroscopy’’. Anal. Biochem. 1992. 200(1): 20-26.
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15. M. Galligani, S. Garrigues, M. Guardia. ‘‘Direct Determination of Ethanol in All Types of Alcoholic Beverages by Near-Infrared Derivative Spectrometry’’. Analyst. 1993. 118(9): 1167-1173. 16. H. Mach, C.R. Middaugh. ‘‘Simultaneous Monitoring of the Environment of Tryptophan, Tyrosine, and Phenylalanine Residues in Proteins by Near-Ultraviolet Second-Derivative Spectroscopy’’’. Anal. Biochem. 1994. 222(2): 323-331. 17. S. Torrado, S. Torrado, R.J. Cado´rniga. ‘‘Comparison of Assay Methods by Second-Derivative Spectroscopy, Colorimetry and Fluorescence Spectroscopy of Salicylic Acid in Aspirin Preparations with a High-Performance Liquid Chromatographic Method’’. J. Pharm. Biomed. Anal. 1994. 12(3): 383-387. 18. J.A. Kornblatt, M.J. Kornblatt, G.H.B. Hoe. ‘‘Second Derivative Spectroscopy of Enolase at High Hydrostatic Pressure—An Approach to the Study of Macromolecular Interactions’’. Biochemistry. 1995. 34(4): 1218-1223. 19. D.G. Konstantianos, R.C. Ioannou. ‘‘Second Derivative Synchronous Fluorescence Spectroscopy for the Simultaneous Determination of Naproxen and Salicylic-Acid in Human Serum’’. Analyst. 1996. 121(7): 909-912. 20. B.S. Kendrick, A. Dong, S.D. Allison, M.C. Manning, J.F. Carpenter. ‘‘Quantitation of the Area of Overlap Between Second-Derivative Amide I Infrared Spectra to Determine the Structural Similarity of a Protein in Different States’’. J. Pharm. Sci. 1996. 85(2): 155-158. 21. Y.C. Micalizzi, N.B. Poppano, N.B. Debattista. ‘‘First and Second Order Derivative Spectrophotometric Determination of Benzyl Alcohol and Diclofenac in Pharmaceutical Forms’’. Talanta. 1998. 47(3): 525-530. 22. J. Zhang, Y.B. Yan. ‘‘Probing Conformational Changes of Proteins by Quantitative Second-Derivative Infrared Spectroscopy’’. Anal. Biochem. 2005. 340(1): 89-98. 23. Y. Kim, D.S. Himmelsbach, S.E. Kays. ‘‘ATR-Fourier Transform MidInfrared Spectroscopy for Determination of trans Fatty Acids in Ground Cereal Products Without Oil Extraction’’. J. Agric. Food Chem. 2007. 55(11): 4327-4333. 24. A. Ikehata, K. Ohara, H. Shinzawa, Y. Ozaki. ‘‘Sensitive Detection and Identification of Organic Liquids Using the Second Derivative of Surface Plasmon Resonance Near-Infrared Spectra’’. Appl. Spectrosc. 2008. 62(5): 517-524. 25. S. Kasemsumran, W. Thanapase, V. Punsuvon, Y. Ozaki. ‘‘A Feasibility Study on Nondestructive Determination of Oil Content in Palm Fruits by Visible–Near Infrared Spectroscopy’’. J. NIR Spectrosc. 2012. 20(6): 687-694. 26. J. Cielecka-Piontek. ‘‘Derivative Spectrophotometry for the Determination of Faropenem in the Presence of Degradation Products: An Application for Kinetic Studies’’. Appl. Spectrosc. 2013. 67(7): 703-708. 27. A. Savitzky, M.J.E. Golay. ‘‘Smoothing and Differentiation of Data by Simplified Least Squares Procedures’’. Anal. Chem. 1964. 36(8): 1627-1639. 28. J. Steinier, Y. Termonia, J. Deltour. ‘‘Comments on Smoothing and Differentiation of Data by Simplified Least Squares Procedures’’. Anal. Chem. 1972. 44(11): 1906-1909. 29. B. Zimmermann, A. Koehler. ‘‘Optimizing Savitzky-Golay Parameters for Improving Spectral Resolution and Quantification in Infrared Spectroscopy’’. Appl. Spectrosc. 2013. 67(8): 892-902. 30. L. Antonov, S. Stoyanov. ‘‘Approach for Increased Information from the Second-Derivative Spectra in UV-Vis Absorption Spectroscopy’’. Appl. Spectrosc. 1993. 47(10): 1712-1715. 31. L. Antonov, S. Stoyanov. ‘‘Step by Step Filter–An Approach for Noise Reduction in the Derivative UV-Visible Spectra’’. Anal. Chim. Acta. 1996. 324(1): 77-83. 32. H. Mark, J. Workman Jr. ‘‘Derivatives in Spectroscopy, Part I: The Behavior of Derivative’’. Spectroscopy. 2003. 18(4): 32-37. 33. H. Mark, J. Workman Jr. ‘‘Derivatives in Spectroscopy, Part II: The True Derivative’’. Spectroscopy. 2003. 18(9): 25-28.
