Digital filters for improved resolution enhancement in spectral analysis

3098

Anal. C h m . 1993, 65, 3098-31 12

Digital Filters for Improved Resolution Enhancement in Spectral Analysis Fabian Janssens and Jean-Pierre Franqois' Limburgs Universitair Centrum, Department SBG, Research Group Analytical Chemistry, Universitaire Campus, B-3590 Diepenbeek, Belgium

A new set of digital filters for the analysis of complex spectral data is introduced. These filters are based on the consecutive application of two zero-area square-wave or Gaussian filters and are, in fact, convolutions of the individual filters. It is shown that these filters are also even and of the zero-area type. Analytical expressions for both zero-area square-wave and Gaussian filters combined are presented. It is found that the signals, obtained with the individual and combined filters as well, are characterized by a strong positive central peak, situated at the centroid of the spectral line, and two negative side lobes. A remarkable difference is the presence of two positive sidebands when the combined filters are used. Criteria to exclude detection of these sidebands, using both individual and combined filters, have been developed. For numerical work, it has been found that it is more efficient to use two consecutive zero-area filters rather than the corresponding convolved filters. In the first part of the present work, the technique has been tested on simulated spectral data, where statistical noise has been introduced by using a Gaussian pseudorandom generator. The technique proposed here (consecutive application of two individual zero-area filters or use of the convolution expressions) has a number of advantages over a single filtering operation: (1) the constant, linear, quadratic, and cubic terms in a polynomial background contribution to the signal are eliminated (this property has been generally proved); (2) the resolution enhancement as well as the signal-tonoise ratio can be improved by using two identical filters and the following working interval for the filter widths M, 0.25 fwhm, IM I fwhm, (fwhm, is the full width at half-maximum of the signal); (3) the broadening of the convolution signal with the combined filters for Voigt profiles is less pronounced with increasing damping constants a; and (4) the use of combined filters or the application of two consecutive zero-area filters is a clear improvement over a single filtering operation, and even over Zimmermann's method, for the unraveling of multiplets composed of strongly overlapping lines. In the second part of this paper, successful applications to the following spectroscopic techniques are described inductively coupled argon plasma optical emission spectrometry, solid-state NMR, IR, and UV-vis absorption spectrometry. 0003-2700/93/0365-3098$04.00/0

INTRODUCTION In recent years, digital filtering techniques have gained increased interest in analytical chemistry; for recent reviews, see refs 1-3. Digital filtering allows the extraction of important information from complex signals. An important problem in spectral analysis, for instance, is the unraveling of multiplets. It has already been shownu that digital filtering techniques can give important information about the number of lines and their positions and about the presence of spectral interference. In our previous paper? we studied the performance of zeroarea square-wave, triangular, and Gaussian filters for peak recognition and interference detection in automated spectral data analysis. It has been found that zero-area Gaussian filters give better resolution enhancements and signal-to-noise ratios than the others. Generally, zero-area filters perform better than non-zero-area ones and have the great advantage of filtering out constant and first-order terms of polynomial background contributions.6 In the present work, the performance of a new set of digital filters, based on the consecutive application of two zero-area filters of the same functional form, is investigated. General expressions for the combination of two zero-areasquare-wave and Gaussian filters are presented. It will be proved, in general, that the combined filters themselves are even and of the zero-area type. Someimportant properties of the resulting signal are formulated by means of an analytical expression when a combined square-wave filter acta upon a single Gaussian line. Expressions describing the convolution of discrete data with discrete combined square-wave and Gaussian filters are presented. Very important properties, such as peak position,resolution enhancement, signal-to-noiseratio, and elimination of background components, are studied. In the subsequent study, combined Gaussian filters are used, rather than square-wave filters, since the former give an enhanced resolution. It will also be proved, in general, that the new filters remove the constant, linear, quadratic, and cubic terms from a polynomial background component of the signal. In the last part of the work, the performance of the new digital filters will be illustrated for ICP-OE spectra, UV-vis absorption, IR, and solid-state NMR spectra. THEORY General Outline. In one of our previous papers? hereafter indicated as paper 1,the following general definition for the convolution of a continuous signal D ( t ) and a filter function (1)Bialkowski, S. E. Anal. Chem. 1988,60,355A. (2)Bialkowski, S.E.Anal. Chem. 1988,60,403A. (3)Brown, S.D.;Bear, R. S., Jr.; Blank, T.B. Anal. Chem. 1992,64, 22R. (4) Op de Beeck, J. At. Energy Rev. 1975,13, (4),743. (5)Taylor, P.; Schutyser, P. Spectrochim. Acta 1988,41B,81. (6)Janssens, F.; FranCois, J.-P. A d . Chem. 1991, 63,320.

0 1993 American Chemical Society

ANALYTICAL CHEMISTRY, VOL. 65, NO. 21, NOVEMBER 1, 1993

C1(f - t ) has been given:

Table I. Expressions for the Combined Filter C(y) of Two Zero-Area Square-Wave Filters (y = 7 - t ) case

R ( f ) = JmCl(f -m - t)D(t)dt where D ( t ) = S ( t ) + B ( t ) and R(f is the convolution signal at position f . S(t) representa the net signal and B ( t ) the background, both at position t. The convolution S(T)of R ( f ) with a second filter function cZ(7- f ) , is expressed as S(7) = S-mm C 2 ( r -f ) R ( f ) d f = JmC2(7 -m - f)dr'JmCl(f -m

3099

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d e

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The integral in bracketa in eq 2 can also be written as follows

J-:Cz(r

- t - u)C,(u)du E C(T- t)

e-h a b

(3)

which shows that the integral is, in fact, the convolution C(7 - t ) of the functions Cl(u) and CZ(T- t - u). Equation 2 can thus be written as (4) Thus, S(7) is the convolution signal at position 7 of the continuous signal D ( t ) and the filter function C(7 - t ) . Further on, the filter function C(T - t ) will be called the combined filter function of the even zero-area functions C1 and CZ. The existence of the filter function C(T- t ) can also be proved by using the well-known Fourier transform convolution theorem. For continuous functions, integrated over finite intervals, the convolution integral becomes

C

d

e-h a b C

d e

f g, h a

b C

d e

f g

h

a-c

MI and MZare the filter widths of the first (C1)and second (CZ)filter functions, respectively. In the present study, the same functional form for both filter functions will be taken. The convolution of the fiiter functions C1 and C2 in eq 5 can also be obtained for finite intervals, however, the detailed integration limits are dependent upon the functional form of both filters. Equation 5 can be written in a general form as

In the following, expressionsfor the combined filters are given for zero-area square-wave and Gaussian filters. For two continuous zero-area square-wavefilters (SWFs), the widths M1 and MZsuffice for the complete characterization of C1and CZ(see Figure 1A in paper I). When two zero-area Gaussian fiters (GFs) are used, both the width, M, and the full width at half-maximum, f w k , for each filter should be specified (see Figure 1B and eqs 10,ll in paper I). Combined Filter of Two Continuous Zero-Area Square-WaveFilters. The expressions for C(T- t ) can be obtained from eq 3 with

d e f g, h

following, it is assumed (with no loss of generality) that M2 IMI. A detailed analysis shows that the following six cases should be considered: (I) (3M1/4) I MZ I MI, (11) (2M1/3) 5 Mz I (3Mi/4), (111) (Mi/2) I M2 I(m1/3), (IV) (Mi/3) I MZI (M1/2), (V) (Mi/4) 5 Mz I (Mi/3), and (VI) M2 I(Mi/ 4). The expressions for C(y) are summarized in Table I and are shown graphically in Figures la,b. It can be seen that the filter operates over 2(M1 + M2) channels. Combined Filter of Two Zero-Area Gaussian Filters. The expressions for the combined filters C(y), with y 7 t, can be obtained from eq 3 with

C,(u) = +1 for -(M1/2) IIuI I (M1/2)

-1 for (M1/2) IJuII Ml 0 for lul1 Ml (7) and similar expressions for Cz(y - u ) with y = r - t. In the

and a similar expression for Cl(u). 1 0 , ~stands z for the intensity of the Gaussian filter, and uc,z is its standard deviation. It is also assumed that Mz IMI. The following expressions are

I " " l " " 1 " " .

3101


SWFs on a single Gaussian line and background. In most cases, spectral lime profiles can be approximated quite well by Gaussian distributions, although it is well known that, in general, this is not the real functional form. In emission spectrometry, for instance, the data should be represented by a Voigt profile. It has been shown frequently6J that the central part of an emission line can be approximated by a Gaussiandistribution; for an accurate description, especially in the line wings, the Voigt profile must be used. Here, the following expression is used for D(t):

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where S(t) is the Gaussian model function, a, is the standard deviation given in units of channels or distance between data points, tor is the position of the centroid, and lo, is the intensity of the Gaussian distribution at tor. When the expressions listed in Table I and eq 11 are substituted in eq 4, the following expression for S(7 ) is finally obtained

Figure 3. Qaussian profile with lor= 1000 counts/channeland fwhm, = 10 channels superimposed on a fourthorder background characterlred by bo = 2500, b, = h = 2.5 X lo-*, 9 = b, = -2.75 X lo-'. The F(Ir)(O)and@ T )(0)convolutionsignalsare obtained with Gaussian filters, where M,,2 = (fWhr&)l,2 = 10 channels.

In paper 1,the following expression(after the notation was slightly changed) has been obtained for the signalR(f) when a zero-area square-wave filter acts upon a single Gaussian line

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t = m1 ~ + m , f s m l - -m2 B = m l + -m2 2 2 B'U' (b4, ba, ...; M I ; Mz;7 ) represents the background component of the second-ordersignal S(r)and can be obtained from eqs AII-8 and AII-9-11 in Appendix 11. From these equations it can be seen that S(7) is enhanced by the background term ELn& when all b, are positive. It should be remarked that eq 12 can be obtained in an easier and more straightforward way by calculating the convolution of R ( f ) with the second filter function C2(7- f ). (8) Francpis, J.-P.; Janesens, F. Spectrochim. Acta 1990,45B,177.

(16)

When arg cosh z is developed into a well-known series expansion valid for bl> 1, the following very good approximation for the position of the side lobes is obtained: f = to,

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The general forms of R ( f )and S(7)are illustrated in Figure 3. A single Gaussian line ( l o , = 1000 countdchannel, fwhm, = 10channels)is superimposed on a fourth-order background B ( t ) = 2500 (2.5 X 10-2)t- (2.75 X lW)t2 (2.5 X 10-2)ts

+

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+

Two zero-area Gaussian filters have been

3102


used here (vide infra), but the results for square-wave filters are nearly the same. From Figure 3 it appears that S(T)has approximatelythe same shape as R(T'): there is a very strong positive central peak and two negative minima, and the only difference is the occurrence of two positive maxima, which will be hereafter referred to as sidebands. The position of the central maximum, the two minima, and the sidebands can be found by solving eq 18: d -S(7) =0 dr When ml and m2 equal m, and eq 19 should be solved:

bq,

(18)

bg, ...,are set equal to 0,

erf(x + 2m) + erf(x - 2m) + 4 erf(x + m) + 4 erf(n - m) +

) : + 4 erfk -) :

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= IOerf x

+ 4 erf(x + 3 / z ) + 4 erf(x - '12) (19)

It is readily seen by inspection that x = 0 is a solution corresponding to the positive central peak. Thus, S(7)and R(7) reach their maximum values at T = to,,. For the two sidebands, the following approximate solution can be obtained: T

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(21)

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1

1 1 I- R(7 - 2) + R ( r - 1)+ R ( T )

(36a) and

R ( 7 - 1) + R(7) + R(7 + 1)> R(7) + R(7 + 1) + R(7 + 2) (36b) The R(7)convolution signal and the background corrected original spectrum are then compared. Using additional criteria, a selection is made between real and ghost peaks. Since the peak detection criteria are rather involved, their detailed description will be given in a forthcoming paper.12 For a single (original) line, the S(7)signal will differ from R(7) only by the presence of two sidebands. However, multiplets may not yet be resolved in the R(7) convolution spectrum, and new lines may appear in S(r)due to the better resolution enhancement. These lines appear with or without their respective sidebands, depending, among others, upon the intensity ratios of the individual lines and their exact separations. Thus, sidebands have to be differentiated from the new real maxima by means of a criterion. Based on eqs 17 and 20 and the working interval, (eq 21), which has been found in the RE*SNR study, the following criterion can be formulated. A maximal RE could be obtained for X = CY = 0.4,while an optimal RE*SNR is realized for X = CY = 0.8 (see Figure 5) and y = S = 1.0 (see Figure 6). The condition expressed by eq 21, 0.25 fwhm, I M Ifwhm,, is thus always satisfied. It is then proved (eq 24) that the sidebands (with positive intensities) in the S(T)convolution spectrum are always further removed from the centroid tO,,,i than the side lobes (with negative intensities) in the R(T) (12)Janssens, F.; Franqois, J.-P., to be published.

convolution signal. This property enables us to exclude most sidebands in the S(T)signal from being detected as real lines. Indeed, we found that the new real peaks, appearing in the S(7)convolution spectrum, will rarely be single lines but will be heavily interfered ones, close to the position of a peak already found in the R(7) signal. Thus, a new maximum in the S(T)spectrum will be considered as a real line if its position is closer to to,,,i than that of the side lobe in the R(7)signal. On the other hand, a new maximum in the S(7)signal which is situated further away from to,,,i than the side lobe in R ( T ) will be rejected as a real peak. In practical situations, due to statistical fluctuations and small shifts of the positions in complex spectra, a flexibility of M-1 channels is allowed for the left and right positions of the side lobes and sidebands, respectively. An illustration of this criterion is given in Figure 7. Figure 7a depicts an original signal (upper part) consisting of two Gaussian lines characterized by IO,,^ = 1000 counts/channel, t0,,,1 = 40, Ioa2 = 500 counts/channel, t0,,,2= 70, fwhmal,2= 10 channels. The R(7) and S(7)spectra are calculated for M = fwhm, = 4 channels. It can be seen that the two lines are well resolved in the original spectrum as well as in R(T). In S(T) the sidebands between the two peaks coincide and give rise to one positive maximum. This maximum is rejected as an additional line since it is situated outside the side lobes of the first and second R(7) lines. The outermost sidebands are also rejected, as they are more than M-1 channels away from the side lobes. Shifting the lines toward each other (to+,l= 50 and tor,2 = 70) gives the situation depicted in Figure 7b. In the R(7) spectrum, the lines can no longer be considered as fully resolved since the two central side lobes coincide. Use of a combined filter gives a better resolution, but no extra maximum is found between the two lines. The outermost sidebands are rejected, as they are farther away from the centroid than the side lobes in the R ( T )signal. Intermediate situations (between 7a and 7b) sometimes result in a false detection of an additional line. Depending upon the exact separation of the two lines in the original spectrum and their intensity ratio, they may remain unresolved in the R ( T )spectrum, i.e., the central side lobes still coincide. An option in our peak detection routine is to consider such lines as a doublet (multiplet). If the lines are, however, fully separated in S(T),the coinciding central sidebands, as in Figure 7a, give rise to an additional maximum. The position of this maximum is then compared to the side lobes of the multiplet, i.e., the outermost side lobes in the R(7)spectrum. It is obvious that, in this case, the additional maximum is recognized as a peak. The lines in Figure 7c are positioned at tor,1 = 50 and tos,2 = 57. In this situation, the R ( T )convolution signal reveals only one line, although asymmetric. The S(T)signal clearly shows two peaks. The additional maximum is considered a real peak since its position is closer to tori than the right side lobe in the R(7) signal. The outermost sidebands are eliminated for the same reasons as those stated previously. In Figure 7d, both use of a single and use of a combined = 50 and torz = filter are unable to resolve the lines at t0,~,1 56. The sidebands in the S(7) signal are situated farther away from the centroid than the side lobes in the R ( T )signal and are thus not accepted as real peaks. Nevertheless, especially the S ( r )signal is strongly asymmetric, indicating the presence of an interfering line. It can also be seen here that the positions of the maxima in R(7) and S(7)are shifted by one channel with respect to each other. (B) Study of Interferences. In this part of the study, the ability to detect heavily interfered lines will be treated. Simulated data are used, and their parameters are presented

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Flgwo 7. (a) Two Gausslan lines characterired by Io,,,, = 1000 and ZO,,,2 = 500 counts/channel, b ,,1 = 40, b,r,2= 70, f ~ h m ,=~10 , ?channels, and the corresponding M T ) (0)and S(T) (0)signals, obtalned for M = fvh" = 4 channels and y = b = 1.0. (b) Two Gaussian lines characterired by I o , l= 1000 and = 500 countslchannel, b,8,1= 50, b,1,2= 70, f ~ h m , , ~=, ~10 channels, and the corresponding MT)(0)and q r ) (0) slgnals, obtalned for M = fwhm, = 4 channels and y = 6 = 1.0. (c) Two Gausslan lines characterlzed by = 1000 and = 500 countslchannel, b,,,1 = 50, b,8,2= 57, fwhm,l,2= 10 channels, and the corresponding R(T) (0)and g ~(0) ) slgnals, obtalned for M = fwhm, = 4 channels and y = 6 = 1.0. (d) Two Gausslan llnes characterlzed by IO,^,^ = 1000 and = 500 countslchannel, = 50, b,r.2= 56, f ~ h m , , ~=, ~10 channels, and the correspondlng MT)(0) and g ~(0) ) signals, obtalned for M = fwhm, = 4 channels and y = 6 = 1.0.

in Table VII. Each of the situations has been treated with a zero-area Gaussian filter with M = fwhm, = 3 channels, iq the presence of a 3 7% noise level, as already discussed in one of our previous papers.6 The positions found in the R ( T ) spectrum are presented in the fourth column of Table VII. In only a few cases could all lines be detected. Zimmermann's method18 (see also refs 6 and 8) sometimes enabled the detection of the missing line@),which is indicated with an asterisk. Failures in the peak detection with the Gaussian filter often happen when a small line is interfered by more intense ones. In order to realize the best possible RE, small filters have been used, for both operations with a single filter and those with combined filters. The second filter is therefore characterized by the same widths as the first one. It is obvious from the data in the last column of Table VI1 that the combined fiiter gives a far better resolution, even for intensity ratios of 10/1 (nos. 19-22). The situations indicated with two asterisks concern S(T)convolution signals where the correct number of lines can be detected visually but not by the peak criteria. This can be, for instance, a situation where the S(T) signal is asymmetric, indicating a composed signal, as in Figure 7d. However, it is obvious from the results in Table VI1 that the use of combined filters is an improvement over both the previous ones and Zimmermann's method.698 Use of Combined Digital Filters on Experimental Spectral Data. Combined digital filters will be applied now tovarious spectroscopic data to demonstrate the applicability of the technique as a general detection method. The spectroscopic techniques considered here are inductively (13)Zimmermann, W.Rev. Sci. Instrum. 1961,32, 1063.

coupled argon plasma optical emission spectrometry, solidstate NMR, IR, and UV-vis absorption spectroscopy. It should be remarked that the NMR spectral data and the UV-vis absorption spectra were not available in a digital form; the original spectra were digitized by means of a computer SCBn.

(A) Inductively Coupled Argon Plasma Optical Emission Spectrometry. ICP-OE spectra are often rich in lines, and the analytical line is more or less interfered by matrix lines. Besides these, molecular bands are also often present, such as OH bands originating from aqueous solutions. Since it is not always possible to choose another (often less sensitive) analyte line, unravelingthe multiplets is the only way to solve the problem. Figure 8a depicts the 3092.71-Aline (indicated with *) of a 0.5-ppm Al solution (0). The quantification of A1 is often hampered by the presence of molecular OH bands. Dieke and Crosswhite14published an extensive study of OH bands in the spectral region 2800-3550 A. The OH bands in the neighborhood of the A1 line, together with their relative intensities (bH), are given in Table VIII. The S(T ) convolution signal in Figure 8a (0,lower part) has been obtained with M1.2 = (fwhmJ1.p = 2 channels. The lines found with MI,^ = (fwhm,)I,z = 2 and 3 channels are also listed in Table VIII. The line at channel number 54 or 55 (M= 2 or M = 3 channels) resembles a sideband escaping from the peak criteria, especially since the original signal is almost resolved at its position. However, this line is not a sideband and is not detected as such, since it also appears in the R(T)convolution (14)Dieke, G.H.;Crosswhite, H.M.J. Quantum Spectrosc. Radiat. Transfer 1962,2, 97.

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Table VII. Digital Filtering on Complex Spectral Data* positions t and intensities peaks found in the peaks found in the I in the simulated multbleta R(7) sDectnunb S(7) smctrumc no. tlJ--/tru I+/I,,, tlJ...ltm tlJ...ltnc 1 2 3 4 5 6 7 8

9 10 11

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

40148156 40148156 40148156 40/48/58 40/48/58 40148158 40/48/58 40/48/58 40148158 40/48/60 40/50/60 40/52/60 40/54/60 40/56/60 40/48/60 40/50/60 40/52/60 40/54/60 40/48/60 40/50/60 40/52/60 40/54/60 30/40/60/80 40/50/60/80 45/55/60/80 55/60/65/80 60/65/75/80

1/0.50/1 1/0.50/0.50 0.50/1/0.50 1/0.75/1 1/0.50/1 1/0.25/1 l/O.lO/l 0.50/0.50/1 0.25/0.25/1 1/0.50/0.75 1/0.50/0.75 1/0.50/0.75 1/0.50/0.75 1/0.50/0.75 1/0.25/0.75 U0.2510.75 1/0.25/0.75 1/0.25/0.75 1/0.10/0.75 1/0.10/0.75 1/0.10/0.75 1/0.10/0.75 0.50/0.50/1/1 0.50/0.60/1/1 0.50/0.50/1/1 0.50/1/0.50/1 1/0.50/0.50/1

40/*-/56 40/-/56 39/48/...* 40/-/58* 40/.../58* 40/-/58 40/-*/58 40/.../58* 39/.../5a* 40/-/60 40/50/60 40/-**/60 40/.../59 40/.../59 40/*-/60 40/50/60 40/*-/60 40/.-/60 40/-*/60 40/*-/60 40/-/60 40/*-/60 29/40/60/80 39/49/60/80 44/*-/59/80 *-/60/--/80* 61/-/.-*/80*

40148156 40149156 38/48/58 40/48/59 40/48/59 40/48/58 40/48/58 39/48/58 40/47/58 40/49/60 40/50/60 40/52/60 40/54/60 40/.../59 40/49/60 40/50/60 40/51/60 40/*.*/60 40/49/60 40/50/60 40/50/60 40/-/60** 29/40/60/80 39/49/60/80 45/*-/60/80 -/60/65/80 60/-*/-*/80**

afwhm, = 10 channels, Gaussian filter with M = fwhm, = 3 channels, and combined Gaussian filter with y = 6 = 1.0. A noise level of 3% has been introduced on the original data. b*Zimmermann'a plot indicates the presence of additional line(s).B 0 **An additional peak can be recognized visually but is not detected by the peak detection routine, or an important asymmetry is noticed.

spectrum. An assignment, based on the data of Dieke and Crosswhite or by consulting tables of spectral lines, could not be made. On the other hand, the line at 3092.88 A (channel number 84) can be assigned to another Al line,ls namely one at 3092.84 A, with an intensity ratio of approximately 1/10 compared to the 3092.71-A line, which has been displaced by the intense OH band. An example of matrix interference is given in Figure 8b, where the 2286.16-A Co(I1) line (indicated with *) is surrounded by several matrix lines (AISI 4340A Steel, Standard Reference Material 361) (0). The correspondingS(T)signal (lower part, 0)has been obtained with MI,^ = (fwhm,)l,z = 2 channels. Table IX contains the lines found from S(T)and, where possible, the assignments to matrix elements based upon the Tables of Spectral Lines.'= The analyte line is indicated with *. (B)Solid-stateNMR. Solid-state NMR spectra are often characterized by rather broad lines, separated by small shifts. Thus, assignment of chemical shifts to certain molecular structures becomes difficult. If no better resolution can be achieved by changing pulse sequencesor other modifications, the S(T) signal might be a helpful tool for determining the number of lines hidden under a spectral envelope. Two 13C CP/MAS NMR spectra are shown in Figure 9 obtained with a Varian XL-200 spectrometer, equipped with an auxiliary high-power amplifier and a solid-state probe with magic angle capability. The sampleswere placed in SisNl rotors and spun at 6 kHz. Figure 9a represents the spectrum of the low bandgap conjugated polymer poly(isothi0naphthene),l8J7recorded with (15) Zaidel, A. N.; Prokofev, V. K.; Raiskii, S. M.; Slavnyi, V. A.; Shreider,E.Y. TablesofSpectralLines; IFYPlenum: New York-London, 1970. (16) Voelkel, R. Angew. Chem., Znt. Ed. Engl. 1988,27, 1468.

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-5000

-500

-loo00 - 15000

-1500

0

-E

$

.-g

3

100

120 450

6000

350

5000

b


40

7000 ~ " ' " " " " ' " ' ' l h

.-u

20

4000

I' i:i

s5

250

2

r

9'

150

3000

5

50

yi

2000

e.

G

C

-50

1000 0 0

20


80

-150 100

Figure 8. (a) ICP-OE spectrum (0)of 0.5-ppm Ai where the Ai (I) 3092.71-A line is Interfered by molecular OH bands and the corresponding g ~spectrum ) (lower part, 0)obtalned for M = fwhm, = 2 channels and 7 = 6 = 1.0. (b) ICP-OE spectrum (0)of the AISI 4340A steel (Reference Meterlal 381) where the Co(I1) 2286.16-A line is surroundedby matrix lines and the corresponding 9 ( ~spectrum ) (lower part, 0)obtained for M = fwhm, = 2 channels and 7 = 6 =

1.0. Table VIII. AI(I) 3092.71-A Line Surrounded by Molecular OH Bands MI, = 3 Mia 2 channel assign- channel assign~ H ( A PIo# number A & ) ment number A(A) ment 3092.394 lo00 3092.577 3092.650

50 70

3092.786 800

55 61

3092.40 OH 3092.49 3092.58 OH

70 77

3092.70 Al,OH 3092.79 OH 3092.88 Al

48

84 a

48 54 61 66 71 77 84

3092.40 3092.48 3092.58 3092.64 3092.71 3092.79 3092.88

OH OH OH

AI OH

Al

Values taken from ref 14.

the pulse sequence for measuring the spin-locking relaxation time, a contact time of 400 ps, and a delay time of 1200 c(s. The signal at about 140 ppm could be assigned to a nonprotonated carbon,17 while in the region from 125 to 135 ppm, lines from two protonated carbons can be expected. Applicationof a combined digital filter characterized by MI,^ = (fwhm,)1,2 = 5 channels clearly reveals the presence of these three lines at approximately 138,126,and 119 ppm (channel numbers 45, 54, and 63, respectively). After this peak detection procedure, regular deconvolution (fitting) procedures can be applied on the basis of the foregoing results. Figure 9b shows the 1W CP/MAS NMR spectrum of 1,3diphenylisothionaphthene recorded with the proton dephasing pulse sequence;18a contact time and a delayed decoupling (17) Hoogmartens,1.; Vanderzande,D.; Martens, H.; Gelan, J. Synth. Met. 1991,41, 513. (18) Opella, S. J.; Frey, M. H. J. Am. Chem. SOC.1979,101, 5854.

ANALYTICAL CHEMISTRY, VOL. 65,NO. 21,NOVEMBER 1, 1993

3109

Table IX. Lines Found in the Steel Sample (AIS1 4340A Steel, Standard Reference Material 361) in the Neighborhood of the Co(I1) 2286.16-A Analytical Line

channel

number

wave1ength (A)

16.3 22.4 25.7 31.3 36.2 43.6 53.9 57.4 63.9 71.2 74.6 78.3 a

2285.81 2285.88 2285.92 2285.99 2286.06 2286.16 2286.29 2286.34 2286.43 2286.53 2286.57 2286.62

c

I

+

2

v

10

5

-10

100

K '1 '

'

'

'

40

'

' ' ' ' ' 160 Channel number '

'

80

'

'

120

'

-300

200

Table X. UV-Visible Absorption Maxima (nm) of Polyenes as a Function of the Number (n)of Conjugated Double Bonds.

n 3 4 5 6 7 8

250 200 150 100

50

0 -50 -100

100 150 200 Channel number Flguro 0. (a) l% CPIMAS NMR spectrum (0)of the low bandgap conJugatedpolymer poly(isothionaphthene). The Q(T) (lower part, 0) has been obtained for M = fwhm, = 5 channels and y = 6 = 1.0. (b) '% CPIMAS NMR spectrum (0)of 1,3dlphenyllsothlonaphthene and the conespondlng Q(T) signal (lower part, 0) obtained for M = 10 channels, fwhm, = 8 channels, and y = 6 = 1.0.

0

50

time of 400 and 50 FS, respectively, have been used here. Under these conditions, only the nonprotonated carbons can be seen, correspondingto the limes 1,2,and 3 (channelnumbers 99,114, and 134, respectively) in the S(T)convolution signal. The characteristics of the filters used here are M1.z = 10 and (fwhm&,Z = 8 channels. A satisfactory assignment of the lines at the positions 69 and 156 could not be given." (C) IR Spectroscopy. Figure 10 shows the IR spectrum of C, species in an Ar matrix annealed at 35 K19 from 1870 to 1970 cm-l. This complex spectrum was analyzed by using the S(T)signal characterized by MIS= (fwhm&l,z= 4 channels (19)KrHtschmer, W.;Nachtigall, K. In Polycyclic Aromatic Hydrocarbons and Astrophysics; LBger, A., D'Hendecourt, L. B., Bocarra, N., E%; Reidel: Dordrecht, 1987.

A1

A2

x9

xl

ha

279 300 316 332

240 267 290 313 332 349

248 278 303 328 350 367

257 290 317 344 368 386

268 304 334 364 390 410

Taken from ref 25.

a

300

g

1' 0

Flguro 10. IR spectrum In the 1870-1970-~m-~range of carbon clusters (C,) In an Ar matrix annealed at 35 K.

-30

5

-1000

30

I

.-2.

B

Fe(1,II) CO(1I)O W(I1) Nb(I1) Mo(II),Fe(I,II) W(I1) PtiII),V(I)

I

50

.-x

-B

6oo 200

.-2.

Refers to the anale line in Figure 8b (indicated with *). 70

.-C

awignment Co(1,II) W(1,II)

and y = 6 = 1.0. The first IR band at 1895 cm-1 (channel number 52) is a single line and has been assigned to linear C, by Martin, Frangois, and Gijbels20321 on the basis of quantum chemical computations. This assignment has been proved experimentally with high-resolution diode laser spectroscopy of a supersonic cluster beam by Heath and coworkers.22 The band at 1952 cm-l (channel number 168) appears to be broadened and was shown by Martin et al.'-Q21 to be caused by linear CS. The assignmenthas been confirmed experimentallywith FTIR spectroscopy by Vala et al.29 The band around 1915 cm-l appears to be a doublet (channel numbers 89-90 and 103). The feature at 1915 cm-1 has been tentatively assigned to cyclic C, (an alongated species with CzUsymmetry) by Slanina et al.u (D) UV-Visible Absorption Spectroscopy. Degradation of poly(viny1 chloride) (PVC) results, among others, in a discoloration of the sample, caused by the formation of conjugated polyenes after dehydrochlorination.% Several studies of this PVC degradation have been published.%@ Wavelengths of the absorption maxima of polyenes are given in Table X26 for various numbers of conjugated double bonds n (n = 3-8). It is obvious that the UV-vis spectrum of degraded PVC consists of an overlapof the individual spectra of polyenes. In Figure 11the UV-vis absorption spectrum of degraded poly(viny1chloride) with 0.12% Zn stearate as a stabilizer is (20)Martin,J. M.L.; Franwis, J.-P.; Gijbels, R. J. Comput. Chem.

1991,12, 52. (21)Martin,J. M.L.; Franwie, J.-P.; Gijbels, R. J. Chem. Phys. 1990,

93,8850. (22)Heath, J. R.:Van Orden,. A.:. Kuo,. E.:. Savkallv, - . R. J. Chem. Phvs. Lett. 1991,182, 17.' (23)Vala, M.;Chandrasekhar, T. M.;Szczepaneki, J.; Pellow, R. In MateriaZs Chemistry at High Temperatures; Hastie, J., Ed.; Humana: Clifton. NJ. 1990,Vol. 11. (24)Sladna, Zi;Kurtz,J.;Adamowicz, L. Chem.Phys. Lett. 1992,196,

208. (25)Daniels, V. D.; Rees, N. H. Polym. Sci. 1974,12, 2115. (26)Braun, D. &re Appl. Chem. 1971,26,173. (27)Crompton, T.R. The Analysis of Plastics; Pergamon Press: Oxford.. UK. 1984. ~~, (28) Marks,G. C.; Benton,J. L.; Thomas, C. M.SOC.Chem.2nd.Monogr. 1967,26,204. ~

~~~

~

~

~

~

~~

~

~

3110


!, 0

20

, ,


,

, 80

,

,%: -3 100

Flgure 11. UV-vls absorption spectrum (0)of degraded poly(vlny1 &&de) with 0.12% Zn stearate and the corresponding a ~signal ) (lower part, 0)obtained for M = fwhm, = 2 channels and y = 6 = 1.o.

Table XI. Lines Found in the UV-Visible Spectrum of PVC

with a Combined Gaussian Filter channel number wavelength (nm) 6 10 14 19 21 26 30 35 40 49 56 63 70 77

80 85

90

241.0 255.5 269.0 286.5 293.5 311.0 325.0 342.5 360.0 391.5 416.0 440.5 465.0 489.5 500.0 517.5 535.0

n 3 3 394 4,5 495 4,5,6,7 4,5,6,7 5,6,7,8,9,10 5,6,7,8,9,10 7 , 8 , 9 , 10, 11,12,13

depicted. The spectral range is from 220 to 570 nm. Due to the scanning procedure (vide supra), the resolution of the spectrum is as poor as 3.5 nm per channel. Our technique, based upon the S(T)signal, will thus be limited in its possibilities. The convolution signal in Figure 11(lower part, 0)has been obtained with M1,2 = (fwhmJl,2 = 2 channels, and y = 6 = 1.0. Table XI presents the wavelengths of the maxima in the S(r) convolution spectrum and the n values of the polyenes possibly contributing to these lines. The availability of more data points over the same spectral region (for instance, via better scanning possibilities) would result in a much higher resolution in the S(7)signal, allowing a more detailed analysis.

been developed. For numerical work, it has been found that it is more efficient to use two consecutive zero-area filters than to use the corresponding convolved filters. The technique (consecutive application of two individual zero-area filters or the use of the convolution expressions) has a number of advantages over a single filtering operation. These can be summarized as follows. (1)The constant, linear, quadratic, and cubic terms in a polynomial background contribution to the signal are eliminated. (2) The resolution enhancement as well as the signal-to-noise ratio can be improved by using two identical filters (C1 = C2) and the following working interval for the filter widths M0.25 fwhm, IM I f w h q (fwhm, represents the full width at halfmaximum of the signal). (3) The broadening of the convolution signal S(T) for Voigt profiles is less pronounced with increasing damping constants a. (4) The use of combined filters or the application of two consecutive zero-area filters is a clear improvement over a single filtering operation and even over Zimmermann’s method for the unraveling of multiplets, composed of strongly overlapping lines. The new method has been successfully applied to data from various spectroscopictechniques, such as ICP-OES, solid-state NMR, IR, and UV-vis absorption spectrometry.

ACKNOWLEDGMENT One of us (J.P.F.)wishes to thank the National Fund for ScientificResearch of Belgium (NFWO/FNRS)for a research grant. The authors are indebted to Prof. Dr. W. Kriitschmer (Max-Planck-Institut ftir Kernphysik, Heidelberg, BRD) for supplying the argon matrix IR spectra of the carbon clusters. Further thanks go to Dr. I. Hoogmartens and M. Scheepers for providing us with the 13C CP/MAS NMR and UV-vis absorption spectra, respectively, and to R. Garner for the introductioninto the DISSPLA routines. Last, but not least, the authors wish to thank a refereefor making very stimulating remarks.

APPENDIX I Theorem. A combined filter C(T - t ) is an even and zeroarea filter. Proof. (a) The combined filter can, for a finite window (2M being the window width), be written as

Now,

u’)Cl(-u’)du’ CONCLUSIONS

A new set of digital filters for the analysis of complex spectral data has been introduced. These filters are based on the consecutive application of two zero-area square-wave and Gaussian fiiters and are, in fact, convolutions of the individual filters. It has been shown that these fiiters are even and of the zero-area type. It is found that the signals obtained with the individual and combined filters are characterized by a strong positive central peak, situated at the centroid of the spectral line, and two negative side lobes. A remarkable difference is the presence of two positive sidebands when the combined filters are used. Criteria to exclude the detection of these sidebands for complex spectral data, using both the individual and combined filters, have

(AI-2) which proves the even property of C(T - t ) since the filters C1 and Cz are both even. (b) In order to prove that C(r - t) is a zero-area filter, we have to show that (AI-3)

with M Ml + M2 (2Ml and 2 M z are the window widths of the filters C1and C2, respectively). Now,

An infinite integration interval for u has been taken here. The finite integration limits are dependent upon the choice


of the fiiter function: however, the exact values of the integration limits for u are d no importance here. Further,

9111

[n/21 represents the integer not exceeding 4 2 . If the terms up to n = 7 are worked out explicitly, the following expression for R ( f ) is obtained

since C1 and C2 are both zero-area filters, and

from which eq AI-3 follows.

APPENDIX I1 The expression for S(T)can be transformed as

Theorem. A combined even zero-area filter C(T - t ) or, equivalently, the consecutiveoperation of two zero-area filters Cl(f - t ) and CZ(T- f ) ,eliminates the constant (bo), linear (blt),quadratic (b&, and cubic (b3t3)terms in the background component of the signal D(t). Proof. In order to prove this theorem generally, the consecutive operation of two zero-area filters C1(f - t ) and c2(7 - f ) will be used. The general expression for the convolution spectrum R ( f 1 can be stated as follows:

If 0, + d W 2 p is developed according to the binomium of Newton, the following result is finally obtained: where eq AIL2 has been used, which holds for an even zeroarea filter.

It should be remarked that, according to a theorem proved in paper 1, the constant (bo) and linear (bit) terme of the background distribution do not contribute to the convolution spectrum R(f). Equation AII-1 can be transformed 88 follows:

where

and (AII-7~) It is readily verified that the terms for n = 2 and 3 vanish; non-zero contributions start to come for n 14. Thiscompletes the proof of the theorem. If the terms up to n = 7 are worked out explicitly, the following expression is then obtained

where

G: = l:yzPCl(y)dy

p = 1,2,

...

6b4G1)G2) + 30b,~G')&~) + 15b,[6~~&"G~' + g1)fi2) + (AII-3b)

and (AII-3~) The integrals

G:+l = s-Zy2P+'C,(y)dy = 0

G1)fi2)]+

vanish since y2p+K'1(y) is an uneven function in y.

+ fi1'd2']+ ... (AII-8)

For zero-area.squarewave filters, the following values for the {-integrals are found: $1*2)

p = 0, 1,2, ... (AI1-3d)

+

= -'/&,2

fila

and

= #12)

= -B/3&,2

(AII-O,lO, 11)

so that the background contributions to the R ( f ) and S(T) signals can be obtained from eqs AII-4 and AII-8, respectively.

3112


The f-integrals for zero-area Gaussianfilters are as follows: = 2 ~ c ( 1 , 2 ) Q ~ ( 1 , 2 ) ~ - m c ( 1exP[-m,2(1,2)1 ,2) +6

1 1 / 2

-

1

/3m~(1,2)1 erf mc(1,2)j(A11-12,13)

AII-4 and AII-8, respectively. It can be remarked here that the two last terms of eq 12b in paper 1 correspond to eqs AII-12,13 when the following properties of the incomplete yfunction are used

RE” The background contributions in R(#) and 8 ( ~can ) be obtained by combining eqs AII-12, 13 and AII-14, 15 with

for review March 29, 1993. Accepted July 13,

1993.’ Abstract published in Advance ACS Abetractcr, September 1,1993.

Digital filters for improved resolution enhancement in spectral analysis

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