ACKNOWLEDGMENTS

The authors acknowledge the work of Mark Grandy for development of the computer-controlled automated microdensitometer system. Atomic absorption analysis of high-purity aluminum samples was car- ried out by Maria Hultenheim, visiting CHUST scholar from the Swed- ish Royal Academy. We are further indebted to Lee Peterson, Milt Shopniek, Albert Wilson, and Eugene Snowden of the Wayne State University Liberal Arts Instrumentation Shop, for fabrication of spe- cialized components. Gifts-in-kind (electrical and optical components) from the Jarreli-Ash Corp. (Waltham, MA) and from the Baird Corp. (Bedford, MA) are appreciated. Financial assistance from the Depart- ment of Chemistry, Wayne State University, and the award of two Wayne State University Faculty Research Fellowships are acknowl- edged and appreciated. Acknowledgment is made to the Donors of the Petroleum Research Fund, administered by the American Chemical Society, for their partial support of this research. Portions of this re- search were funded by the National Science Foundation under Grant CHE-80-16148.

1. D. M. Coleman, M. A. Sainz, and H. T. Butler, Anal. Chem. 53, 746 (1980).

2. J. W. Hosch, Ph.D. Thesis, University of Wisconsin (1975), Uni- versity Microfilms #75-12903.

3. R. J. Klueppel, D. M. Coleman, J. W. Hosch, and J. P. Walters, Spectrochim. Acta 33B, 741 (1978).

4. D. M. Coleman and J. P. Waiters, Appl. Phys. 48, 3297 (1977). 5. W. H. Hsu and D. M. Coleman, Anal. Chem. 58, 2106 (1986). 6. I. Kleinmann and V. Svoboda, Anal. Chem. 41, 1029 (1969). 7. D. E. Nixon, V. E. Fassel, and R. N. Kniseley, Anal. Chem. 46, 210

(1974). 8. A. B. Whitehead and H. H. Heady, Appl. Spectrosc. 22, 7 (1968). 9. F. Brech, Current Status and the Potentials of Laser Excited

Spectrochemical Analysis, Reprint No. 2 (Jarrell-Ash Company, Waltham, Massachusetts, 1966).

10. R. L. Watters, presented at the Institut fiir Spektrochemie und Angewandte Spektroskopie, Dortmund, Germany (1978).

11. J. S. Beaty, paper presented at the 9th Inter. Conf. on Atomic Spectrosc. and XXII Colloq. Spectrosc. Internat., Tokyo, Japan (1981).

12. Jarrell-Ash Corp., Waltham, Massachusetts (now Thermo Jarrell- Ash).

13. J. W. Chris and L. S. Birks, Anal. Chem. 40, 1080 (1968). 14. G. M. Allen and D. M. Coleman, Anal. Chem. 56, 2981 (1984). 15. G. M. Allen and D. M. Coleman, Appl. Spectrosc. 41, 381 (1987). 16. W. H. Hsu, M. A. Sainz, and D. M. Coleman, Spectrochim. Acta

44B, 109 (1989). 17. Cordin, Inc., Salt Lake City, Utah. 18. D. M. Coleman and M. A. Sainz, Anal. Chem. Acta 34, 219 (1986). 19. Microlab/FXR, Livingston, New Jersey. 20. A. L. Lance, Introduction to Microwave Theory and Measure-

ments (McGraw-Hill, New York, 1964). 21. Bird Electronics Corporation, 30303 Aurora Road, Cleveland, Ohio. 22. Princeton Applied Research, Princeton, New Jersey. 23. S. E. Mathews, Ph.D. Thesis, University of Wisconsin (1982). 24. A. Scheeline, personal communication. 25. A. Scheeline, J. Norris, J. L. Travis, J. R. DeVoe, and J. P. Walters,

Spectrochim. Acta 36B, 373 (1981). 26. A. Scheeline, J. L. Travis, J. R. DeVoe, and J. P. Waiters, Spec-

trochim. Acta 36B, 153 (1981). 27. W. Hsu, V. Majidi, and D. M. Coleman, Appl. Spectrosc. 41, 739

(1987). 28. V. Majidi, W. Hsu, and D. M. Coleman, Spectrochim. Acta 43B,

561 (1988). 29. A. Scheeline and M. J. Zoellner, Appl. Spectrosc. 38, 245 (1984). 30. J. Sheffield, Plasma Scattering of Electromagnetic Radiation (Ac-

ademic Press, New York, 1975).

A Mixed Linear/Nonlinear Least-Squares Method for Determining Line Parameters from Noisy Spectra

A M E D E O P R E M O L I and M A R I A L U I S A R A S T E L L O * Dipartimento di Elettrotecnica, Elettronica ed Informatica (DEED, Universitd di Trieste, via A. Valerio I0, 34127 Trieste, Italy (A.P.); and Istituto Elettrotecnico Nazionale Galileo Ferraris (IENGF), Strada delle Cacce 91,10135 Torino, Italy (M.L.R.)

The determination of the parameters characterizing line profiles embed- ded in a noisy spectrum is an important task in IR and Raman spec- troscopy. A direct analysis from an inspection of the spectrum is often difficult in the presence of overlapped profiles and/or considerable back- ground. This paper presents an algorithm based on alternate stages of linear and nonlinear optimization for extracting Lorentzians or Gauss- ians from the measured data, taking into account the presence of a frequency-dependent background superimposed onto white noise. The analysis of several experimental spectra shows the capability of the algorithm in discriminating noisy and partially overlapped profiles. Index Headings: Line fitting; Spectroscopic techniques; Computer ap- plications.

Received 28 July 1988. * Author to whom correspondence should be sent.

INTRODUCTION

S e ns i t i v i t y is a n essen t i a l factor in spec t roscopy, b u t i t is u sua l l y l i m i t e d by the noise assoc ia ted wi th t he spec t romete r . The r e f o r e the ana lys i s of l o w - i n t e n s i t y l ine profi les is o f t en a diff icul t task.

S ince the s igna l is cha rac te r i zed by l ine b r o a d e n i n g , the s p e c t r u m r e p r e s e n t i n g the l ine profi le shows a Lo- r e n t z i a n or G a u s s i a n l i ne sha pe accord ing to the d i f f e ren t phys i ca l m e c h a n i s m s occur r ing in the b r o a d e n i n g 2 B o t h L o r e n t z i a n s a n d G a u s s i a n s are cha rac te r i zed b y a max- i m u m in tens i ty , a ha l f -wid th a t h a l f - m a x i m u m ( H W H M ) , a n d a c e n t r a l f r equency .

T h e a im of th i s p a p e r is to p r e s e n t a n a lgo r i t hm de- vo ted to the d e t e r m i n a t i o n of these pa r ame te r s . T h e

558 Volume 43, Number 3, 1989 0003-7028/89/4303-055852.00/9 APPLIED SPECTROSCOPY © 1989 Society for Applied Spectroscopy

method here proposed is able to take into account several overlapped profiles by using an appropriate model for analyzing the measured spectrum, and an ad hoc opti- mization algorithm.

The ideal spectrum y(w) should take the analytical form of a sum of m spectral lines, that is

y(c0) = 2 aih(co, as, Fj) (la) j= l

where the function

~1/{[(o~ - ~2)/F] 2 + 1 } h(co, a, r) = [exp{-[(co - a)/r] 2} (lb)

represents a normalized (Gaussian or Lorentzian) profile; ai is the maximum intensity, a i is the central frequency, and F s is the half-width at half-maximum.

Unfortunately, the measured samples of y(c0) are af- fected by a random noise r(co) superimposed on a back- ground p (co) slowly varying with oa, so that the measured spectrum s (co) takes the form

s(co) = y(co) + r(co) + p(co). (2)

Random noise can be classified into two groups; one group is the so-called white noise, which is statistically distributed around the true signal and arises partly from the electronic equipment and partly from the dark noise of the photomultiplier. Moreover, there exists a second type of noise which is caused by random external events and is called single spike outlierY Since the algorithm proposed here does not require any presmoothing of the data, the latter is not dealt with explicitly.

As for the background, it is likely to be present in all types of experimental data, and it is necessary to take it into account in any curve-fitting method. In the case of infrared spectra, a parabolic function a-5 seems to provide the degree of flexibility required to deal with more dif- ficult situations. There are greater problems in the case of Raman spectroscopy, where curvature of the baseline may occur because of overlap by the tails of remote lines.

On the basis of these remarks, the background p(co) is assumed to be modeled by an (n - 1)th-degree poly- nomial:

P(co) = 2 Pi wi-1 (3)

where n denotes the number of unknown coefficients. Since the measured signal exhibits a rather regular be- havior, n is assumed not to exceed 4. In other words, the background is assumed to be constant (n = 1), linear (n = 2), quadratic (n = 3), or cubic (n = 4). Therefore it is not necessary to express the background in terms of or- thogonal polynomials. 6

The fitting procedure involves the minimization of the random noise variance over the whole measured spec- trum. The minimization should be repeated by assuming a different number of profiles and degrees of background. The comparison of the resulting variances allows one to check whether the chosen number m of spectral lines and the chosen degree n of the background are suitable.

An ad hoc optimization algorithm has been realized on the basis of the following motivations:

(1) The number of unknowns may be relevant. The CPU time required by a generic nonlinear optimization algorithm, such as those available in the literature, increases rapidly with the number of unknowns.

(2) The unknowns involved in the model can be sepa- rated into two groups: those appearing linearly in the expression of s (co) and those appearing nonlinearly.

Therefore, the numerical procedure consists of alter- nating linear and nonlinear optimization stages. So the linearly appearing unknowns are treated in a CPU-time- sparing way, since the linear stage reduces to the solution of a linear system of equations; the more expensive non- linear stage is limited to a part of the unknowns 6,7 and is based on the Davidon-Fletcher-Powell numerical method. This method does not require any analytical calculation of partial derivatives; hence it is suitable for any line shape.

THE OPTIMIZATION METHOD

Our aim is to estimate the parameters Pi (i = 1, . . . , n) characterizing the background and the parameters aj, tl i, Fj (1" = 1 . . . . , m) characterizing the unknown (Gaus- sian or Lorentzian) profiles.

This estimation is essentially based on the fitting of the measured data by the mathematical model consid- ered in the introduction. The residual error in this fitting corresponds to the random noise. Then one should min- imize the following quadratic functional

p 1 £~b = - - [r(co)] 2 dco (4)

COb COa a

where

r(co) = s(co) - ~ ajh(w, as, Y i) - 2 Pi coi-' (5) j= l i=1

where co b and coo are the limits of the spectral range within which s (co) is measured. Note that the minimum of the functional P represents the variance of the random noise in the range COa -+ COb; it will be denoted by ~ in the following discussion.

From a mathematical point of view, the problem is of a nonlinear least-squares type; however, some unknowns appear linearly in expression 5. This peculiarity is exploited in order to realize an algorithm that both im- proves the performance of a generic nonlinear least- squares method and spares CPU t imeY

According to the above considerations, the unknown parameters are grouped into two different vectors:

V = [ P l , " ' ' , P n , OQ . . . . . O[m] T ( 6 a )

o r

u = [al , . . . , am, r , . . . . . r m ] r (6b)

where the dimensions of v and u are n + m and 2m, respectively, and the superscript T denotes transposi- tion.

By means of this notation, functional P can be ex- pressed in terms of u and v as follows:

P(u , v) = Po + vTA(u)v -- 2vTb(u) (7)

APPLIED SPECTROSCOPY 559

where

1 f f ~ Po = bob boo ~ [s(bo)] 2 dbo. (8)

A(u) is a symmetrical (n + m) x (n + m) matrix partitioned as follows:

A' A+(u)] A(u) = A+(u)T A"(u)J (9)

where A' is a symmetric n x n matrix, independent of u, with entries

, 1 f ~ b = _ _ boi+j--2 dbo a ij bob boa a

i = 1 , 2 . . . . , n j = l , 2 . . . . . n. (10a)

A+(u) is an n × m matrix, dependent on u, with entries

1 ~ a + . . . . boi-lh(bo, a t , rj) dbo

tJ bob boa a

i = l , 2 , . . . , n j = l , 2 . . . . , m . (10b)

A"(u) is a symmetric m × m matrix, dependent on u, with entries

1 fff~ a t t i j : --60b - - boa a h ( b o , ~ i , ri)h(bo, t2j, Fj) dbo

i = 1 , 2 , . . . , m j = 1 , 2 . . . . . m (10c)

and b(u) is an (n + m) vector partitioned as follows:

b(u) T = [b '~, b"(u) T] (11) where b' is an n-vector, independent of u, with entries

1 fff~ ' : - - boi--ls(bo) dbo b i bob boa a

i = 1, 2 . . . . . n; (12a)

and b"(u) is an m-vector, dependent on u, with entries

1 j~ffb " = ~ h(bo, fli, ri)s(bo) dbo b i bob - - boa a

i = 1, 2 . . . . . m . (12b)

Appendix A reports the formulas used in the computer code implementing this algorithm for Lorentzian pro- files. Although there have been some examples of the use of Gaussian profiles to fit IR, NIR, and Raman peaks, the evidence in favor of the rather general applicability of the Lorentzian shape is very strong) Formulas in Ap- pendix A, obtained by analytical integrations, drastically reduce CPU time and improve the accuracy.

The optimization consists of solving the problem

~2 = minimum [P(u, v)]. (13) <u,v)

From Eq. 7, note that the minimization reduces to a linear least-squares problem with respect to v, if sub- vector u is assumed to be assigned. So, first let us consider the linear optimization stage obtained by keeping u in- variant. In other words, let us solve

/~(u) = minimum [P(u, v)]. (14) <v)

As this is a linear least-squares problem, the value of v minimizing P(u, v) is given by

v = C ( u ) b ( u ) (15)

where C(u) is the inverse of A(u). Appendix B gives the partitioned expression of C(u) vs. the submatrices of A(u). The partitioned form of C(u) allows one to save CPU time because the inverse of submatrix A', independent of u, is calculated only once. By substituting in Eq. 7, one has

P(u) = Po - b(u)TC(u)b(u) • (16)

To solve the problem formulated in Eq. 13, it is necessary to also consider the nonlinear optimization stage, con- sisting of minimizing/5(u) with respect to u, that is

~2 = minimum [/5(u)]. (17) (u)

This nonlinear optimization stage is solved by a numer- ical algorithm based on the Davidon-Fletcher-Powell method. 9

NUMERICAL PROCEDURE AND REMARKS

The Davidon-Fletcher-Powell method is a particular variant of the conjugate directions method and consists of successive cycles: each cycle is composed of a number of descent line steps equal to the dimensions of the un- known vector plus 1 (2m + 1 in our case). Each descent line step moves along a line to obtain even better partial solutions. The method is characterized by a good con- vergence rate without requiring any analytical calcula- tion of partial derivatives, which is rather troublesome in the case of Lorentzians and Gaussians.

As this method is detailed in Ref. 9, we will not go into it in this paper. For the user, it is sufficient to know that the algorithm requires a suitable starting value for u and iterated calculations of P(u). Some intervals, within which ~ and Fy (j = 1 , . . . , m) are confined, are generally known and allow the user to choose good starting values for them. In this context, the value of visual inspection should not be underestimated as a means for locating individual parameters in composite profiles. Convergence is reached when all the absolute cycle-to-cycle variations I~u~l (j = 1 . . . . , 2m) of the entries of u are less than some prefixed small values.

Since the measured signal s(bo) is given by K equi- spaced samples sl, s2 . . . . , st, the (continuous) frequency range of measurement boa -~ bob has to be replaced by a discrete domain bok = kAbo characterizing the samples

sh = s(boh) k = 1, 2, . . . , K. (18)

Consequently, the integrals appearing in Eqs. 8, 10, and 12 are replaced by summations. Those terms indepen- dent of u need be evaluated only once, whereas those terms depending on u have to be calculated in each itera- tion of the nonlinear optimization stage.

Some remarks on the mathematical performances of the algorithm are needed. The starting values of central frequencies and HWHMs can be directly derived from a straightforward inspection of the recorded spectrum.

Parameters controlling the convergence of the algo- rithm are chosen so that they are equal to one tenth of

560 Volume 43, Number 3, 1989

. . . . . . . . . . . . . . . . . , . . . . . . . . . ..

I

1 0 0

5 . 0 0 0 -

4 . 0 0 0 -

3 . 0 0 0 -

2 . 0 0 0 -

1 . O O O -

FIG. 1.

I I

2 0 0 3 0 0 L O

Measured spec t rum (K = 361; do t ted line) conta in ing a single Lorentz ian and its fit for case n = 3 (solid line).

the error expected on the unknowns ~i and F i (j = 1, 2, . . . , m), to ensure significant results. Under the above circumstances, the typical number of iterations needed in the nonlinear stage is in the range of 100 to 500, de- pending on m.

For the experimental measurements reported below, a Lorentzian line shape has been assumed. The opti- mization has been repeated by using different starting values for ~2~ and F i (j = 1, 2, . . . , m). If the number of Lorentzians (m) and the starting values of their param- eters are chosen reasonably, according to the shape of the measured spectrum, the final values obtained in dif- ferent cases are coincident, apart from an error of less than the allowed uncertainty. This means that the same local minimum is reached from different but reasonable starting points.

On the other hand, different final values are reached if the starting points are not reasonably related to the measured spectrum. In any case, the inspection of the values of the parameters and variances in the different local minima allows the discrimination of an effective solution, as will be shown in the next section.

PROCESSING OF EXPERIMENTAL DATA

The experimental test of the proposed algorithm is related to the determination of H W H M and the central frequency of Lorentzian line shapes, since, at this mo- ment, applications are oriented to the study of molecular dynamics in liquid crystals.

More specifically, Raman spectroscopy is used to study the reorientational diffusion of complex molecules in an- isotropic fluids, such as liquid crystalline phases} ° In principle, both vibrational and rotational relaxations of the single molecules contribute concurrently to the band- width, but, for anisotropic fluids, the two contributions can be separated. If both the intrinsic vibrational band shape and the instrumental line shape are Lorentzian,

the reorientational diffusion coefficients can be obtained directly by a proper analysis of the HWHMs of bands corresponding to particular scattering geometries, n

Usually, vibrational modes are selected with band shapes sufficiently well separated from other peaks. Fu- jiyama and Crawford 12 suggested truncating the analyzed band by at least ten times the full width at half-maxi- mum; unfortunately, Raman spectra of mesogenes con- sist of many bands relatively close to each other, and this fact makes it impractical to follow the Fujiyama- Crawford suggestion.

The sets of experimental data on which the proposed algorithm has been checked were obtained at the Physics Department, University of Parma (Italy). Raman spectra were taken with a microcomputer-driven and -controlled spectrometric system, the computer also acting as a data buffer. Spectra were obtained mainly with the 488-nm

T A B L E I. Parameters supplied by the processing of the measured spec- trum of Fig. 1 for various m and n.

m n j a i ~i Fj f i Pi

S ta r t ing 1 185.0 10.00 For the four poin t cases below

1 1 1 5128 186.8 11.12 131.6 1 302.1

1 2 1 5127 186.7 11.09 121.1 1 198.9 2 0.498

1 3 1 5215 186.7 12.04 85.3 1 419.9 2 -3 .6 4 9 3 0.0114

1 4 1 5209 186.7 11.98 75.2 1 527.1 2 -7 .1 7 6 3 0.0361 4 0.461E-4

Star t ing 1 180.0 10.00 po in t 2 190.0 10.00

2 2 1 2648 182.4 8.47 108.6 1 234.3 2 3138 190.5 10.12 2 0.491

APPLIED SPECTROSCOPY 561

TABLE II. Parameters supplied by the processing of the measured spectrum of Fig. 1 for n = 2, m = 2, and distinct starting points.

Case rn n j % ~2~ Fj F

Star t ing 1 180.0 10.00 poin t 2 190.0 10.00

2 2 1 2648 182.4 8.47 108.6 2 3138 190.5 10.12

Star t ing 1 184.0 10.00 po in t 2 192.0 10.00

2 2 1 3492 183.5 9.08 109.1 2 2313 192.1 9.68

S tar t ing 1 50.0 10.00 poin t 2 190.0 10.00

2 2 1 403.7 24.2 17.63 69.1 2 5190 186.7 11.87

S tar t ing 1 100.0 10.00 po in t 2 190.0 12.00

2 2 1 - 1 5 3 3 106.8 258.1 73.7 2 5219 186.7 12.07

argon laser line, which was cylindrically focused onto the sample cell in a back-scattering geometry.

In all the tests, the (discrete) output signal has been sampled at equispaced intervals; so, the frequency unit can be chosen to make the sampling interval (Ao~) uni- tary. By convention, ~% is assumed to be equal to zero. On the contrary, an arbitrary unit is adopted for the measured signals; so, the reported numerical values of ~, ~2~. r i (3" = 1 . . . . . m), Pl (i = 1 . . . . , n), and F have to be considered in this sense.

The first test concerns the influence of the degree of the polynomial modeling of the background. The test consisted of a single noisy Lorentzian band with a back- ground that significantly varied with frequency. The overall spectrum is shown as the dotted line in Fig. 1. The spectrum was fitted with the use of backgrounds represented by polynomials of different degree (n = 1, 2, 3, and 4).

Results are reported in Table I. The expected decrease in the noise variance for n = 4 is insignificant; this case is then not further considered, and the case of n = 3 has to be chosen as a final result (see solid line in Fig. 1). Note that the case of n = 1 and the case of n = 2 give

TABLE IV. Parameters supplied by the processing of the measured spectrum of Fig. 2 for n = 2, m = 2, and distinct starting points.

Case m n j a t ~# r j r

a S ta r t ing 1 250.0 20.00 po in t 2 250.0 20.00

2 2 1 1412 236.3 18.01 62.51 2 4226 258.1 16.92

b Star t ing 1 245.0 20.00 po in t 2 255.0 20.00

2 2 1 1323 236.2 17.82 62.29 2 4252 257.9 17.28

c S ta r t ing 1 240.0 20.00 po in t 2 260.0 20.00

2 2 1 1323 236.2 17.82 62.29 2 4252 257.9 17.28

d S ta r t ing 1 220.0 20.00 poin t 2 280.0 20.00

2 2 1 4873 255.3 23.20 62.18 2 - 5 2 4 . 6 294.8 21.10

e S ta r t ing 1 240.0 20.00 poin t 2 350.0 20.00

2 2 1 4888 254.0 22.37 92.40 2 3284 822.2 412.0

f S ta r t ing 1 50.00 20.00 poin t 2 250.0 20.00

2 2 1 1366 34.10 199.9 88.80 2 4932 253.9 22.91

close values of HWHM, whereas a significant difference in HWHM results from the use of a quadratic back- ground (n = 3).

Table I also reports the results of a check concerning the possible existence of a second Lorentzian that is par- tially responsible for the varying background. This as- sumption can be reasonably excluded because of the noise variance values showing that the curve fitting is better when one is considering a single curve with a quadratic background. For a closer inspection, the same measured data have been analyzed by fixing n = 2 and m -- 2 and varying the optimization starting point. Table II sum- marizes the results for four distinct starting points.

When the starting frequency of a Lorentzian is out of the range of the peak envelope, the final related curve

TABLE IlL Parameters supplied by the processing of the measured spectrum of Fig. 2 for various m and n.

m n j a t ~j r j ? i p~

S tar t ing 1 250.0 20.00 po in t

1 1 1 4823 254.1 21.34

1 2 1 4821 254.1 21.31

1 3 1 4883 254.0 22.31

1 4 1 4889 254.0 22.36

S tar t ing 1 240.0 20.00 poin t 2 260.0 20.00

2 2 1 2006 236.2 17.82 2 3451 257.9 17.28

For the four cases below

102.9 1 51.50

101.3 1 20.18 2 0.1340

92.1 1 122.4 2 - 1 . 5 4 6 3 0.344E-2

91.4 1 91.49 2 -0 .7740 3 -0 .717E-3 4 0.590E-5

62.29 1 28.98 2 0.1944

TABLE V. Parameters supplied by the processing of the measured spectrum of Fig. 3 for m = 3 and different n.

m n j a i fli r j ~ i p~

1 63.00 5.000 Star t ing 2 113.0 11.00

po in t 3 174.0 8.000

3 1 1 189.8 62.10 8.622 2 413.3 112.8 14.13 3 175.3 173.1 8.803

3 2 1 189.2 62.17 8.377 2 414.0 112.8 14.17 3 177.6 173.1 9.234

3 3 1 212.9 62.04 10.27 2 458.2 113.0 16.32 3 200.6 172.7 11.34

3 4 1 208.5 62.16 9.764 2 463.4 113.2 16.61 3 211.9 172.7 12.41

10.57

For the four cases below

1 - 1 5 . 5 0

10.34 1 - 11.05 2 - 0 . 0 5 0

8.31 1 0.9119 2 -1 .2 5 2 3 0.528E-2

8.12 1 - 5 . 2 4 5 2 -0 .7 1 7 5 3 0.219E-2 4 0.236E-4

562 Volume 43, Number 3, 1989

3 ,000

2 .000 "

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . J

5 . 000 -

J 300 400 eO 100 200

5 .000 -

3 . 000 "

2 . 000 -

1 . 000 -

o,2h(~,,2.r2)

r i b i 100 200 300 400 (,O

FIG. 2. (A) Measured spectrum (K = 481; dotted line) containing two Lorentzians and its fit for case n = 2 (solid line). (B) Reconstructed Lorentzi~ms and background.

gives mainly a contribution to a fit of the random noise (cases c and d). On the other hand, when both starting frequencies are within the range of the peak envelope, final results related to different starting points do not converge to the same final values (cases a and b). This consideration strongly supports the speculation that the peak effectively contains only one Lorentzian profile.

The second test has been focused on the capability of the algorithm in dealing with a line shape characterized by two Lorentzians so close to each other that they look like a unique peak (dotted line in Fig. 2A). Table III

summarizes the results, with the same notation as for Table I. Note that one does not significantly decrease the noise variance by increasing n when considering a single Lorentzian in the optimization, whereas a signif- icant improvement is reached by taking into account two Lorentzians (solid line in Fig. 2A). Figure 2B shows the reconstructed Lorentzians and background.

As previously, the same measured data have been ana- lyzed by fixing n = 2 and m = 2 and varying the opti- mization starting point. Table IV summarizes the results for six distinct starting points. Again, when one starting

APPLIED SPECTROSCOPY 563

T A B L E VI. Parameters supplied by the processing of the measured spectrum of Fig. 3 for n = 3, m = 3, and distinct starting points.

Case m n j a t fit Ft F

a 1 63.00 5.000 Star t ing 2 113.0 11.00

po in t 3 174.0 8.000

3 3 1 212.9 62.05 10.27 8.31 2 458.2 113.0 16.32 3 200.6 172.7 11.34

b 1 60.00 10.00 Star t ing 2 110.0 15.00

po in t 3 176.0 10.00

3 3 1 212.9 62.05 10.27 8.31 2 458.2 113.0 16.32 3 200.6 172.7 11.34

c 1 55.00 10.00 S tar t ing 2 100.0 15.00

poin t 3 165.0 10.00

3 3 1 212.7 62.02 10.27 8.33 2 459.0 113.0 16.35 3 201.3 172.8 11.42

d 1 45.00 10.00 S tar t ing 2 90.00 15.00

poin t 3 150.0 10.00

3 3 1 - 3 6 6 . 9 36.15 34.80 34.97 2 - 3 5 0 . 0 84.34 13.01 3 - 3 7 4 . 0 147.1 22.48

frequency is out of the range of the peak envelope, it gives a contribution to the fit of the random noise (cases c, d, and e), while, when both starting frequencies are within the range of the peak envelope, final results do converge to the same point (cases a, b, and c). These results, together with those in Table II, show the good capability of the algorithm in discriminating overlapped line shapes.

The third and last test relates to the analysis of a spectrum characterized by three Lorentzians. As ex-

pected, no significant change is obtained by varying the polynomial approximating the background (see Table V). Figure 3 shows the goodness of fit for n = 3. The influence of the starting point on final results has been studied also in this case, by fixing n = 3 and m = 3. Table VI summarizes the results for four distinct starting points.

The algorithm is characterized by a very good con- vergence to the same final point when the starting central frequencies are chosen within the range of the peak en- velope (cases a, b, and c). Otherwise, completely incorrect results are obtained, with no convergence (case d).

As an overall conclusion from the performed tests, it should be noted that an effective Lorentzian in the ana- lyzed spectrum is characterized by a maximum whose value is positive and is significantly higher than the noise standard deviation. Moreover, optimization results for different starting central frequencies within the width at half-maximum do converge to the same final values. On the contrary, a lack of convergence is associated with erroneous Lorentzians in the final results. Finally note that the choice of the starting values of Fj (j = 1, 2 . . . . , m) is generally not crucial from the convergence point of view.

CONCLUSIONS

A new algorithm has been developed for determining the parameters characterizing the lines embedded in a measured spectrum. This algorithm fits the measured data by a mathematical model based on several theo- retical (overlapped) profiles and an additive smoothly frequency-varying background. The least-squares for- mulation is used as the fitting criterion: this corresponds to the minimization of the variance due to superimposed random noise on the measured data.

The algorithm is based on alternating linear and non- linear stages according to the way in which the unknowns appear in the functional to be minimized. So the required

FIG. 3.

o ~o ,oo ,;o ~oo Measured spectrum ( K = 222; dot ted l ine) containing three Lorentzians and its f i t for case n = 3 (solid l ine).

564 V o l u m e 43, Number 3, 1 9 8 9

C P U t ime decreases and the convergence is i m p r o v e d wi th respec t t o the classical non l inea r leas t -squares me thods .

T h e a lgo r i thm is appl ied to the analysis o f R a m a n spec t ra o f l iquid crystals . I t allows one to d i sc r imina te be tween two s t rong ly ove r l apped Loren t z i ans a n d / o r th ree ove r l apped Loren tz ians . A sui table s ta r t ing po in t for the a lgor i thm can be chosen by a d i rec t inspec t ion of the m e a s u r e d spec t rum.

Final ly , t he behav io r of the a lgor i thm wi th respec t to d i f fe ren t s t a r t ing poin ts allows one to d is t inguish an ef- fect ive line f rom e r roneous ones and to recognize two d i f fe ren t l ines i m b e d d e d in the same peak.

1. R. G. Breene, Jr., Theory of Spectral Lineshapes (Wiley, New York, 1981).

2. B. M. Bussian and W. Hardle, Appl. Spectrosc. 38, 309 (1984). 3. D. Audo, Y. Armand, and D. Arnaud, J. Mol. Struct. 2, 287 (1968). 4. H.V. Drushel, J. S. Ellerbe, R. C. Cox, and L. H. Lane, Anal. Chem.

40, 370 (1968). 5. D. Papousek and J. Pliva, Collect. Czech. Chem. Commun. 30, 3007

(1965). 6. R. J. Noll and A. Pires, Appl. Spectrosc. 34, 351 (1980). 7. Y. S. Chang and J. H. Shaw, Appl. Spectrosc. 31, 213 (1977). 8. W. F. Maddams, Appl. Spectrosc. 34, 245 (1980). 9. S. L. S. Jacoby, J. S. Kowalik, and J. T. Pizzo, Iterative Methods

for Nonlinear Optimization Problems (Prentice Hall, Englewood Cliffs, New Jersey, 1972).

10. S. Jen, N. A. Clark, and P. S. Pershan, J. Chem. Phys. 66, 4635 (1977); see also these authors' earlier publications cited therein.

11. N. Kirov, I. Dozov, and M. P. Fontana, J. Chem. Phys. 83, 5267 (1985).

12. T. Fujiyama and B. Crawford, J. Chem. Phys. 73, 4040 (1969).

APPENDIX A

In the case of L o r e n t z i a n profiles, t he e lements o f submat r i ces A' , A÷(u), and A"(u) r educe to the fol lowing expressions:

where

with A i

a'ij-~ [60b i+j-1 - - wai+j-l]/[(i A - j - 1)(O~b -- 0~)]

a+lj = Fj (arc tg[(¢o b - ~j)/F~] - arc tg[(wa - ~j)/Fj]) 0~ b - - 0~ a

rJ2 ln([(wb -- ~j)2 + rj2]/[(wo _ 9y)2 + r2 ] ) + ~ya+lj a+2~ = 2(o~ b -- o~o)

a+3i = r j 2 + 2~ja+2i _ (~j2 + Fj2)a+lj

a+4j = Fj2(Wb + ¢oo)/2 + 2~sa+aj - ( ~ j 2 + Fj2)a+2j

(W b -- ~j)112(60 b -- 60:)] (W a -- ~ j ) I [2 (W b -- ¢0=)1 1 a " = - + ~a+l j z [(wb - f~y)/Fj] 2 -4- 1 [(w. - ~ ) /F j ] 2 + 1

a"i~ = olija+2i + ~i ja+v + %ja+li Jr- 5qa+lj

i = l , . . . , n j = l , . . . , n

j = l . . . . , m

j = l , . . . , m

j = l . . . . . m

j = l . . . . . m

j = l , . . . , m

i = l , . . . , m j = l . . . . . m i ¢ j

o~ij = _F/2/~.. r l 2 t-t/

2Fi2( 9, - ~i)

~ij = (A i _ Aj)2 + 4(~j -- ~ , ) ( ~ j A i -- ~ iA j )

A~ = _ .

/ F i 2 [ A , - A j - - 4~/(t2 , - - i ~ ) ]

~'J = ( A / - - t,~)~ + a(u~ - u , ) ( e jA , - u/A~)

= u,~ + r , ~ a n d Zs~. = aj~ + r / .

APPLIED SPECTROSCOPY .565

APPENDIX B

This appendix supplies matrix C(u), the inverse of A(u), in the partitioned form. By definition of the inverse matrix it follows that:

[A+(u)T A+(u)] [c+(C:)T C+(u) 1 A, ,u, =[o o] A"(u)[ where I and 0 denote identity and null matrices of ap-

propriate dimensions. The submatrices of C(u) have the same dimensions of the corresponding submatrices of A ( u ) .

It is easy to verify that the submatrices of C(u) are given by

C"(u) = [A"(u) - A+(u)TA'-~A+(u)] -~ C+(u) = -A'-IA+(u)C"(u) C'(u ) = A ' - I [ I - A + ( u ) C + ( u ) r ] .

566 Volume 43, Number 3, 1989