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Michael Greenfield SpectrochimicaActa, Vol. Received 12 July ; in final form and accepted 23 March Abstract--A novel spectral analysis technique is presented for the determination of peak areas from spectra in which there are multiple overlapping peaks. The technique involves a combination of Fourier spectral analysis and profile modeling. Although the technique presented could be applied to many different types of problems analysis of UV spectra or chromatographic results it is applied here to the problem of determining, from Fourier transform infrared FI'-IR spectra, the peak areas which correspond to monomeric and hydrogen- bonded species. Use of the Voigt profile leads to more accurate and descriptive spectral peak parameters and, in turn, to more accurate determinations of concentrations of monomeric and hydrogen-bonded species. IL O : Lorentzian modeling function.

If the spectral peaks monomeric and hydrogen- bonded did not overlap, the area of each peak could be determined directly by integrating the spectrum. However, since the peaks do overlap, a method of separating them must be included in the data analysis.

If the bands are only slightly overlapped it may be possible to use the conventional method of difference spectrometry [13]. However, this method does not explicitly allow for the shift of spectral bands which, as 0. When a shift occurs, it gives rise to derivative-like bands [14, 15] which make a finite contribution to the area but which cannot be resolved or assigned unambiguously to a particular peak.

Additionally, difference spectrometry requires that a calibration curve be established which is based on a knowledge of the concentrations of monomeric and associated species. Such concentrations are not known a priori and are in fact the desired outcome of the measurements. In effect, to use difference spectrometry to determine these concentrations requires that one knows them to begin with. A more reliable approach for the determination of area of partially overlapped peaks is to use spectral analysis in combination with profile modeling also referred to as curve fitting.

Fourier self- deconvolution [13, 18] is a technique used to narrow and therefore partially resolve overlapping peaks. For peaks that are only slightly overlapped either profile modeling or deconvolution can be used separately. For peaks that are strongly overlapped neither technique is useful by itself and their combined application becomes necessary.

Our technique involves a combination of spectral analysis and profile modeling. The spectra are first analyzed using derivatives and Fourier self-deconvolution to establish the number and position of peaks that are present. Profile modeling is then used to determine the spectral peak parameters h, o0, 2, y that characterize each peak in the spectrum.

These spectral peak parameters are determined by fitting not only the raw data, but also by fitting the derivative and the Fourier self-deconvolution of the data. The parameters obtained can be used to reproduce the spectra and therefore to accurately calculate the peak areas. A more detailed description of the spectral analysis technique is given below. Spectrophotometric grade chemicals were used and stored under nitrogen. Purity was checked using gas chromatography and by IR spectroscopy. All spectra were measured at K.

To overcome this limitation, we developed a tech- nique that is as efficient and as simple to use as the Lorentzian profile. One technique would be to find an analytical solution to the Voigt profile. However, such a solution has not been found. A reliable method of evaluation is to recognize that the Voigt profile is proportional to the real part of the complex error function [20, 21], and to evaluate this function instead.

The properties of the complex error function have been the subject of numerous investigations [], and are discussed in Appendix A. However, Armstrong found that a 20 term Gauss-Hermite integration is needed to obtain an accuracy of one digit in the sixth significant figure. However, for x and y less than 3 we simplify the evaluation of the complex error function by interpolation of the tabulated values found in Ref.

Since we are working with a limited range of x and y values and the table itself is not excessively large elements , interpolation is fast and reliable. The accuracy of these approximations was tested using Simpson's rule and good agreement was found.

The spectral analysis technique As discussed above, our technique involves a combination of spectral analysis and profile modeling. The first step in spectral analysis is to determine the number and positions of the peaks that are present.

This involves partially resolving the experimental spectra to the extent that we do not introduce distortion. This is accomplished by differentiation and Fourier self-deconvolution of the raw spectral data. Spectral analysis is displayed pictorially in Fig. Fourier self-deconvolution [18] has the effect of decreasing the width of the individual peaks, which decreases their overlap and makes it easier to determine peak positions. It involves three steps: transforming the spectral data to the Fourier domain to obtain the interferogram, smoothing the data, and taking the inverse transform to obtain the deconvoluted spectrum.

The first derivative is the slope of the data, helping to better understand the shapes of the individual peaks. The numerical method used to differentiate a spectrum is a three- point formula [22, 26]. In this work, differentiated data are smoothed using a "boxcar truncation" technique. The differentiated data is first converted to the Fourier domain using a fast Fourier transform [27].

Next, all frequencies greater than some optimal frequency are set equal to zero. Transformation back to the time domain results in a smooth curve. The optimal frequency is found by trial and error. A potential problem with this method is that the derivative can become distorted if the boxcar truncation eliminates an important part of the spectrum in addition to the noise.

To avoid this, a comparison of the smoothed and original data should show that they have the same features. A comparison of an unfiltered and a filtered derivative is shown in Fig.

The profile modeling technique involves several steps in order to obtain a set of spectral parameters that best represent the spectral curve. Each type of data raw data, differentiated data, self-deconvoluted data is curve-fit modeled using, respectively, the profile modeling function, the derivative of the profile modeling function and the JOHN T.

Spectral analysis is displayed pictorially, in which a simulated spectral curve is shown before and after differentiation and self-deconvolution.

Each type of data raw data, differen- tiated data, and self-deconvoluted data is then curve-fit using, respectively, the profile modeling function, the derivative of the profile modeling function, and, the self-deconvolution of the modeling function.

An external optimization is used to find a single set of spectral parameters which gives the best overall curve-fit shown by the final spectrum. In general, a slightly different set of spectral peak parameters is obtained from modeling each type of data.

In order to obtain a single set of spectral peak parameters which gives the best overall curve-fit, it is necessary to further regress the parameters determined from the self-deconvolution and differentiation routines. The spectral peak parameters obtained from the self-deconvolution and differentiation routines are used as first guesses in the raw data regression routine. At first, none of these spectral peak parameters are regressed.

The regression routine is set up such that, a choice of which parameters are regressed, and the order in which they are regressed, is allowed. The reason the spectral peak parameters are not regressed is to determine a Analysis of FT-IR spectroscopic data: the Voigt profile value for the sum of the errors squared for each set of parameters.

This gives a basis to which further regression of any of the peak parameters may be compared. After minimizing the error we are again left with three sets of peak parameters, one from the original attempt at regressing the raw data and two from regression of the raw data beginning with the parameters determined from self-deconvolution and from differentiation.

From these three sets of spectral peak parameters the set with the minimum error is selected as the potential best-fit set. We used the term "potential" because the process of determining the best-fit of the spectral curve is not over at this point. The "best-fit" parameters for each peak are plotted as a function of the concentration.

As an example, the heights of the monomeric and associated peaks of acetone plotted as a function of the methanol concentration are shown in Fig. A smooth transition of the heights from zero methanol concentration to the final concentration of methanol is found. If, however, a smooth transition was not found, the regression process would be reiterated until a smooth transition occurs.

It is at this point, when all of the spectral peak parameters give smooth transitions as a function of the methanol concentration, that we take the peak parameters to be the best-fit parameters. From these best-fit parameters the area of each peak is determined and the individual peaks can be simulated. An example of individual peaks determined from overlapping peaks is represented by the final spectrum in Fig. By curve-fitting single peaks we found the uncertainty of the area of the spectral band to be 0.

An additional problem when fitting overlapped peaks, that will be addressed later in the paper, is that the shapes of individual peaks may be different. That is, for two or more overlapped peaks, one may have either a pure Lorentzian or Gaussian shape and the other a Voigt shape. Therefore, when curve-fitting the peaks with say, a Lorentzian profile a reasonable fit of the overall band may be found but, the spectral peak parameters for each peak may not be the best-fit parameters.

Analysis of synthetic spectra Before applying the spectral analysis technique to actual spectra, we tested it on synthetic spectra. Synthetic Voigt spectra were generated numerically by specifying Voigt profile parameter values and calculating absorbance vs wavenumber.

Next, we curve-fit the synthetic Voigt spectra using the Lorentzian profile.

This gives an estimate of the errors to be expected when real spectra are fit with a Lorentzian. The advantage of 0. Smoothed and unsmoothed spectral data of the first derivative of the carbonyl peak of a methanollCCIJacetonemixture.

In curve-fitting we did not use the actual Voigt parameter values to influence our search for the best-fit Lorentzian parameter values. The values of the spectral peak parameters are listed in Table 3 and a plot of one synthetic curve and its best fit is shown in Fig. Note that the Lorentzian profile does not fit near the peak maximum nor in the "wings" of the spectra. As lambda approaches zero, the shape index approaches zero and the Voigt profile becomes Gaussian in character.

In addition, as gamma approaches zero, the shape index approaches infinity and the Voigt profile approaches the pure Lorentzian profile. As expected, we found for small values of y the Lorentzian profile is a poor model of a Voigt spectrum, while for large values of y the Lorentzian profile is very good.

This is in keeping with the capability of the Voigt profile to go through a smooth transition from Gaussian to Lorentzian shape. A sensitivity analysis was performed to test the error introduced by the use of an incorrect value of the shape index. A synthetic Voigt spectrum was generated and is shown in Fig. We then curve-fit the synthetic spectrum with the Voigt profile and obtained, as expected, the original spectral peak parameters used to generate the spectrum. Next, we changed the value of y and keeping it, along with h and o0, constant Table 3.

Voigt 0. Resultsof syntheticspectra modeledwith the Lorentzianprofile. Syntheticspectra were generated numericallyby specifyingVoigtprofileparametervaluesand calculatingabsorbancevs wavenumber. This yielded a new value for 2 and a new value for the sum of the errors squared.

A plot of the sum of the errors squared vs the reduced spectral peak index is shown in Fig. There are several important points to be made concerning this figure.

The first is that the minimum error occurs between the Lorentzian and Gaussian profiles. In fact, as expected, it occurs at the values of the spectral peak parameters that were used to generate the synthetic data. Second, based on the values of the sum of the errors squared, the plot indicates that neither the Lorentzian profile nor the Gaussian profile can accurately model synthetic spectral data that has a Voigt lineshape. This suggests that actual spectra which exhibit both Gaussian and Lorentzian broadening can be fit better with a Voigt profile.

Analysis of actual spectral data As a first attempt at modeling spectral data, the Lorentzian profile was used. The height, width and position parameters were determined by the combined spectral analysis and profile modeling technique outlined in the introduction and described in detail above. As an example of the results obtained, in Fig. One is near the peak maximum, the other is in the "wings".

Any attempt to improve the modeling results for either of these two regions resulted in a deterioration in the overall accuracy of the curve-fit. The spectral data shown in Fig. The procedure used to obtain the Voigt parameters was the same as that used to obtain the Lorentzian parameters.

As a quantitative measure of the improved curve- fitting results obtained with the Voigt profile, the sum of the errors squared and the calculated areas are shown in Table 4, and compared with those obtained using the Lorentzian profile. There is a difference in areas of roughly seven per cent which is an indication that spectral analysis results do indeed depend on the type of modeling function used.

Presumably the areas obtained using the Voigt profile are more accurate. Additionally, listed in Table 4 are the sum of the errors squared and the calculated area for the carbonyl peak of methyl isobutyl ketone MIBK in carbon tetrachloride.

The difference in calculated areas between the Lorentzian profile and Voigt profile is approximately five per cent. Having analyzed the spectra of non-hydrogen bonding mixtures, we turn now to the goal of our work which is to analyze the spectra of hydrogen bonding mixtures.

One sees they are highly overlapped. In this spectrum there are two additional peaks: one is a combination band of acetone [29] and the other is the tail of several vibrational peaks of methanol between and cm -1 [30]. We approximate this last tail with a small but broad peak around cm-L As shown in Fig. However, as in the spectra above, the peak maximum and the "wings" are not fit very well.

All attempts to improve the fit of these two regions resulted in a deterioration of the overall accuracy. In Fig. Though the two profiles appear to give similar results the Voigt profile gives a better fit. The important difference between the two profile modeling functions appears in the spectral parameters and areas obtained. Using the Voigt profile, the monomeric and hydrogen-bonded carbonyl peaks were found to have quite different shapes.

As shown in Table 5, the reduced shape index for the monomeric peak is 0.

A pure 2. Results of the sensitivity analysis for synthetic spectral data shown in Fig. The sum of the errors squared between the modeled synthetic Voigt spectrum and the modeling profile Voigt profile with y fixed at an incorrect value is plotted vs the reduced shape index s. The minimum error occurs between a pure Lorentzian lineshape and a pure Gaussian lineshape. Table 4. Lorentzian peak has a reduced shape index of 1.

Comparison of the reduced shape indices suggests that the Voigt profile gives a much better representation than the Lorentzian profile for the hydrogen-bonded peak. A sensitivity analysis was performed on the monomeric and hydrogen-bonded carbo- nyl bands of acetone. We determined for the experimental spectrum shown in Fig.

Next, we changed the value of?

This yielded a new value of 2 for the monomeric peak and a new value for the sum of the errors squared. This was done several times using different values of? Figure 12 displays the sum of the errors squared between the experimental spectrum and the modeling profile Voigt 0.

Analysis of Flr-IR spectroscopic data: the Voigt profile 0. There are several points to be made concerning this figure. For the hydrogen-bonded peak the minimum error occurs between the pure Lorentzian and the pure Gaussian profiles. In fact, the minimum error occurs at a value of 0. We consider this peak to have a substantial Gaussian component and conclude that in order to accurately model this peak the Voigt profile should be used.

Although it is difficult to determine from Fig. There is relatively little difference in values of the sum of the errors squared between the Voigt profile and the pure Lorentzian profile. However, in order to minimize the error the Voigt profile should be used as the modeling profile.

Table 5. A full discussion is found in Ref. The sum of the errors squared between the actual spectra and the modeling profile Voigt profile with y fixed at an incorrect value is plotted vs the reduced shape index s. The minimum error for each peak occurs between a pure Lorentzian lineshape and a pure Gaussian lineshape. The areas obtained from the two modeling functions are the most important result since they are related to the concentrations of m o n o m e r i c and hydrogen-bonded species.

These are given in Table 5. There is no significant difference in area for the m o n o m e r i c p e a k whereas for the hydrogen-bonded p e a k there is a difference of roughly three per cent. We found this to be the case for all of the spectra analyzed in this study.

That is, for the m o n o m e r i c carbonyl peaks we found differences in areas of approximately one per cent and for hydrogen-bonded carbonyl peaks the differences ranged from three to five per cent. Figure 13 is a plot of the areas for the m o n o m e r i c and hydrogen-bonded peaks vs methanol concentration for the two modeling functions. Again, there are slight differences of the areas for the m o n o m e r i c peaks. However, for the hydrogen-bonded p e a k the areas calculated using the Voigt profile are, in general, lower than the areas calculated using the Lorentzian profile.

An additional item to be discussed is the apparent discrepancy of the areas for the combination band of acetone listed in Table 5. We first used the Lorentzian profile to curve-fit the overlapping spectral peaks, i.

Next, we used the Voigt profile to curve-fit each of the peaks. The shape of each p e a k is 6. Click the icon to activate the drawing tool. Draw what you consider to be the first phase of the current trend. There is no need to draw the full trend, only draw what you consider to be the beginning of the trend. The platform will continue to draw the trend zigzag trendline until the trend reverses.

Drawing the first phase of the trend in a logical way is crucial it determines all other parameters of the strategy. The dedicated section below, "Drawing the zigzag trendline", explains this action in more detail.

The drawing tool indicates 3 zones based on the zigzag trendline and the current market price. These 3 levels, which Voigt considers crucial, are: The buy or short sell zone. Around this price level positions are opened.

The level is drawn as a small coloured zone. Green for the buy zone, red for the short sell zone. The zone will only appear in the chart when the market closes below above the sideways zone. The sideways zone. The sideways zone is indicated by two dotted blue lines. The end-of-the-trend level. When a candle closes below this level, a bullish trend ends. When a candle closes above this level, a bearish trend ends.

This example shows a bullish trend. The market closed below the sideways zone 2. As a consequence the green buy zone 1 appeared in the chart. This example shows a bearish trend. The market closed above the sideways zone 2. As a consequence the red short sell zone 1 appeared in the chart. These trading techniques can be used individually or in parallel. All three techniques use the same buy and short sell zones as indicated in the charts.

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