Anal. Chem. 1992, 64, 2164-2174
2164
Fourier Analysis of Multicomponent Chromatograms. Recognition of Retention Patterns Attila Felingert Department of Analytical Chemistry, University of Veszprhm, P.O. Box 158, H-8201 Veszpr6m, Hungary
Luisa Pasti' and Francesco Dondi' Department of Chemistry, University of Ferrara, via Luigi Borsari, 46, I-44100 Ferrara, Italy
ita Fourier Stiltjes transform (Le. the characteristic function2), A procedure based on flttlng the experimentally computed O(o). 6 ( w ) is the Dirac function. Equation 1was derived by autocovariance functlon (ACVF) of multicomponent chromatcb the fair general hypotheses of lack of correlation between grams to theoretical models is introduced by which both the peak height and peak position (uncorrelatsd chromatograms) single component interdistance model (IM) of the retention and constancy of the shape properties of the SC peak along times Is tested and the statistkal attributes of the multicomthe chromatogram. The usefulness of the PS approach hae ponent chromatogram (Le. number m of angle components, been shown in a previous papel.3 where a numerical method peak width, and parametersof the IM) are determined. Four was presented for evaluating the statistical properties of a different IM-exponential, uniform, normal, and gamma-are multicomponent chromatogram under the hypothesis of a considered. I n essence, when Wed to these theoretical Poissonian-type IM. The method which consisted of a modek, the experimental ACVF-exprerrlng the chromatoprocedure for the nonlinear fitting of the theoretical expresgraphic response correlation on the t h e distance-provides sion of the PS derived from eq 1to the PS of the experimental ~ h r f o r m a t k n n ~ r y t o ~ k d r b o t h t h e t y p e o f r ~ ~ chromatogram kn was validated by numerical simulation. In pattem and gives the necessary parameter estimatlon. The general the results were excellent and superior to those procedure Ir tested by using computergeneratedchromatoobtained by the so-called statistical model of overlapping grams with different IMs and uncorreiaed peak heights, in (SMO)originally presented by Davis and Giddings.4-s Recently, the PS method was extended to handle multicomwhich deMny and m are varied. I t lo shown that the ponent chromatograms with non constant SC peak width? chromatographk attrlbutes m and peak wldth derived from The PS approach appears in principle to be much more the best flttlng I M are unbiased. Moreover, even if the best powerful than the method derived from the SMOPs6 since fffllng IMs do not always coincide wlth the true model, because it is potentially able not only to take into account the peak of their flexiblitty and approxlmatlng properties they always shape and the peak height dispersion but also to represent give a correct description of the retention pattern provided specific retention patterns through the term in 9(w) of eq 1. that the results are correctly interpreted.
INTRODUCTION In a previous paper, the statistical properties of a complex multicomponent chmabgram-obtained under programmed elution technique conditions-were represented in terms of ita power spectrum (PS, see Glossary at the end).' The general form of the PS is
where w is the frequency. In essence, the PS expresses the correlation existing in the chromatographic response on the time distance. According to this equation the PS of the total multicomponent chromatogram, F ( o ) , is the function of following quantities: (1)the PS of the normalized peak of a single detectable component (SC),lg(o)I2(the same for all SC peaks); (2) the mean interdistance between subsequent SC peaks, (3) the sc peak height dispersion ratio, d a h (bh being the peak height standard deviation and a h the mean peak height); (4) the distribution of the interdistances between subsequent SC peaks along the chromatogram (called here for the sake of simplicitythe interdistance model, IM) through t Present address: Department of Chemistry, University of Tennessee, Knoxville, T N 37996. Present address: Enichem, 12, pl Donegani, 1-44100 Ferrara, Italy. (1) Felinger, A., Pasti, L.; Dondi, F. Anal. Chem. 1990, 62, 1846.
*
0003-2700/92/0364-2164$03.00/0
With respect to this treatment, the SMO was originally developed by assuming a Gaussian peak shape of the SC peak, constant SC peak height, and Poisson interdistance distribution. In any case the SMO approach retains ita general validity not only because of ita intrinsic simplicity and elegance but also because of its completeness. In fact it also contains a full description of how to evaluate the overlapping probabilities and the requirements for attaining a given separation god4 The choice of assuming Poissonian-likedistribution of SC retention times along the chromatogram, made by Davis and Giddings4and by other authors who either previously faced@ or further developed the topic,1s13 was well founded. A Poissonian pattern means completelyrandom peak positions, and it is the most likely consequence of the many slight differences existing in molecular structure among componenta of a complex mixture. According to the molecular theory of
(2) Cram&, H. Mathematical Methods of Statistics; Princeton University Press: Princeton, NJ, 1974. (3) Felinger, A.; Pasti, L.; Dondi, F. Anal. Chem. 1990, 62, 1864. (4) Davis, J. M.; Giddings, J. C. Anal. Chem. 1983,55, 418. (5) Davis, J. M.; Giddings, J. C. J. Chromatogr. 1984,289, 277. (6) Davis, J. M.; Giddings, J. C. Anal. Chem. 1986, 57, 2168. (7) Feliier, A.; Pasti, L.; Dondi, F. Anal. Chem. 1991,63,2627. (8) Rosenthal, D. Anal. Chem. 1982,54,63. (9) Nagels, L. J.; Cretan, W. L.; Vanpeperstraete, P. M. Anal. Chem. 1983, 55, 216. (10) Herman, D. P.; Gonnord, M. F.; Guiochon, G. Anal. Chem. 1984, 56, 995. (11) Martin, M.; Guiochon, G. Anal. Chem. 1986,57,289. (12) Martin, M.; Herman, D. P.; Guiochon, G. Anal. Chem. 1986,58, 2200. (13) Cretan, W. L.; Nagels, L. J. Anal. Chem. 1987,59,822.
0 1992 Amerlcan Chemical Society
ANALYTICAL CHEMISTRY, VOL. 64, NO. 18, SEPTEMBER 15, 1992
2165
multicomponent chromatogramand to determine ita unbiased solutions, the total partition free energy of a compound is a statistical attributes. It has in fact been shown that if the PS sum of many intermolecular force contributions. In a mulmethod based on the hypothesis of chromatogram Poieeoticomponent mixture the partition free energies of the nian character is applied when this is not the caee, the different compounds should result in an uncorrelated, completely random retention time distribution.4J4 It has been estimated SC number m could be completely ~r0ng.3More over, establishing the correct IM for a chromatogram is a shown that the effect of a lack of correlation in peak positions necessary requirement for correct evaluation of other imis reached even in the limit case of a great number of ordered portant statistical features of a multicomponent chromatoretention time sequences when they randomly superimpose gram such as the mutual SC overlapping probabilities? the one another.’ Thus, any very complex mixture, composed number of resolvable peaks and resolvable componenta,12 and either of several homologous series of chemical compounds the determination limita,13 which have been analyzed but or of a large enough number of different chemicalcompounds, only under the hypothesis of Poisson IM. would always produce Poissonian chromatograms. All these arguments were rightly advocated in previous ~ o r k s l as ~ ~ - ~ ~ The detection of the correct IM in an unknown chromatophysical justification for their approach. However Poisson gram will be made on the basis of the attained degree of fitting, and the method will be validated by numerical distribution is a “limit distribution” and it should hold true in practice only in the limit case of a maximum complexity simulation by using computer-generated multicomponent chromatograms. of the mixture.’ How closely a real caee should meet this asymptotic condition is still an unexploited question and the THEORY need to explore alternative IMs is fully justified. The critical role played by the Poisson assumption in SMO General Properties of the SC Peak Interdistance was well presented by many authors who tried either to prove Models. As far as we know, no generaland extensive handling the Poissonian character of an experimental multicompoof the mathematical expressions of IM (which should repnent chr~matogram,’~J~ or to obtain a meaningful and resent the elution patter of complexmulticomponent misturea consistent SC number by using different columns for the same in programmed elution techniques) has been presented. An complex unknown mixture,17J8or to check the theory by an example was previously reported on how a mixture of several extensive analysis of gas chromatograms of synthetic mixtures homologous series with constant retention time incrementa containing a known number of detectable c o m p o n e r ~ t a . ~ ~ ~in~ ~programmed elution techniques became close to a poisIn all these cases the Poissonian hypothesis could not be sonian pattern when they simultaneously elute,’ provided rejected. However, it is well-known that this does not mean that both the number of the homologousseriesis great enough that the Poissonian character was proved nor that the Poisand the retention time incrementa in each homologous series sonian model will hold elsewhere. It was only shown that are random. However, it can also be shown (e.g. by numergeneral real multicomponent Chromatograms cannot be ical simulation) that the resulting distribution of interdiestatistically distinguished from the theoretical uncorrelated tances between subsequent peak positions can assume a wide Poissonian chromatogram model. The strength of this variety of shapes-more or less Gaussian and in certain cases conclusion obviously relies on the ypower*of the employed uniform as well-depending on to what extent the above statistical testa,e.g. the x2test,applied in verifying the random mentioned limiting conditions are respected: e.g. if the character of peak positions over the ~hromatogram.’~J~ retention time increment of the different homologous series However when applied to a chromatogram where the deis more or less constant and the total number of SCs is limited tectable SC number does not usually exceed 100-200, these (about 100-200).22 It is not the specific aim of the present statistical testa have limited power. It is not our intent to paper to fully develop the topic; however the main result to criticize these studies; rather, setting up alternative IMs retain in the present context is that the SC interdistance should have a positive impact even in further confirming the distribution in any real multicomponent chromatogram can basic role of the Poissonian hypothesis. assume a great variety of shapes. In order to be of general In this paper the ability of the PS method to handle chroutility, the IMs to be assumed should be both flexible and matograms with SC peak position patterns other than Poisvaried. Expressions and properties of the four chosen IM are sonian are exploited. However, instead of followingthe same reported in Table I. The skewness S and the excess E are approach as previously described3 which consisted of fitting defined as the PS of the chromatogram to a theoretical model derived from eq 1,an alternative method is presented which consists of fitting the autocovariancefunction (ACVF). This approach should overcome some troubles deriving from the need to set up a filtering step in the experimental PS c o m p u t a t i ~ nIn .~~~~ addition to the Poiasonian case three other types of IM will be considered here. The case of a Poissonian chromatogram where p3 and p4 are respectively the third and the fourth is characterized by the exponential (E) 1M.l The three other central momenta of the distribution and UT is ita standard IMs considered here are the normal (N), the uniform (U), deviati0n.~,~3 S and E characterize a given distribution with and the gamma (r)distributions. respect to the normal distribution, for which both S and E The aim of this study is to set up a method able to correctly are zero? a positive Svalue is related to atailing and anegative recognize the peak position pattern type of an experimental E value to flattening shape compared to the normal distribution. Another significantparameter isthe relative standard (14) Giddings, J. C. The Unified Separation Science; John Wiley and deviation:
Sons: New York, 1991. (15) Davis, J. M.; Giddings, J. C. Anal. Chem. 1985,60,2178. (16) Dondi, F.; Kahie, Y. D.; Lodi, G.; Remelli, M.; Reschiglian, P.; Bighi, C. Anal. Chim. Acta 1986,191, 261. (17) Coppi, S.; Betti, A.; Dondi, F. Anal. Chim. Acta 1988,212, 165. (18)Oros, F. J.; Davis, J. M. J. Chromatogr. 1991,550, 135. (19) Davis, J. M. J. Chromatogr. 1988,449, 41. (20) Delinger, S. L.; Davis, J. M. Anal. Chem. 1990,62, 436. (21) Jenkins, G. M.; Watts, D. G. Spectral Analysis and Its Applications; Holden-Day: San Francisco, 1968.
RSD = uT/T (4) It can be seen that the four chosen distributions are able to represent a wide variety of conditions (see Table I). In (22) Dondi, F.; Felinger, A. Unpublished results. (23) Abramowitz, M.; Segun, I. A. Handbook ofhfathematical hcnctiom; Dover Publications: New York, 1965.
2166
ANALYTICAL CHEMISTRY, VOL. 64, NO. 18, SEPTEMBER 15, 1992
Table I. Some Theoretical Expressions of SC Peak Interdistance Frequency Functions, qt),and Their Statistical Parameters f(t) T UT RSD S Exponential
method (ACVF Method) will be validated by numerical simulation with computer-generated chromatograms. The following function
~~
(1/7)e+ (05 t )
7
-e-(t-T)'/Z+/&q
T
1/2T (0 < t < 2T)
Uniform T TI&
7
Normal
E
*-' TACVF(i)-EACVF(i)
2
6
d T
0
0
l/&
0
-1.2
&(
(9) TACVF(0) is minimized with respect to the quantities m and ug,in addition to UT or p, respectively, for the N or the r IM cases. One of the most important attributes of a multicomponent chromatogram (either experimental or simulated) containing m SCs is the saturation factor CY:^
"=
1
a = m/N,
(10)
where N , is the peak capacity a chromatogram spanning over a range X particular, the I? and N distributions, by appropriate choice of the p parameter value or the RSD value, respectively, can represent more or less tailed and more or less sharp distributions. In the Appendix the theoretical PS (TPS) expressions for the four considered models are derived. The expressions are reported in Table 11. Theoretical Autocovariance Models. The corresponding expressions for the theoretical autocovariance functions (TACVF) can be obtained from TPS expressions by applying the Wienel-Khinchin theorem:24 1 C(t)= 47r j"F(w)e"' --
dw = L27rj m0F ( w ) cos (ut) dw
(5)
where t is the time axis. Only in selected cases can the Fourier integral be analytically solved. One case is the E model:'
In the other cases (U,N, and r models) the TACVF can be evaluated from PS by numerical inversion by using the fast Fourier transform algorithm.
PROCEDURE The TACVF computed as discussed above can be compared with the "experimental" ACVF (EACVF)which is computed over a digitized chromatogram according to the well-known expresion^^^^
where pis the mean computed from the sampled chromatogram:
N, = X / r , (11) x o is the interdistance between two SC peaks separated at a given resolution R,: x g = 4ups (12) The mean interdistance between two subsequent SC peaks in the chromatogram, T, is
T =X/m (13) and from eqs 10-13 the following relationship between T, u,, a and R, is obtained:
T = 4upJa
In a simulation run,once a,, R,, a and m values are fixed, T and X are consequently defined; thereafter, only the SC peak interdistance model (IM) is to be specified. For the N and r case, the value of the third distribution parameter-respectively uTand p-must be in addition assigned. For the N model, instead of UT, the relative standard deviation,RSD (eql), can be specified. The simulated chromatogram is thus generated by fixing the first SC peak position and then determining the positions of all the subsequent peaks according to the chosen IM. The peak heights are determined according to an independent specific distribution. Under given conditions, a characteristic random overlapping pattern among SC peaks appears over the chromatogram. Consequently only "separated bands" (i.e. the detected 'peaks" in the chromatogram), containing even more than one SC peak (singlet,doublet, triplet peak), can be singled out (for a definition of peak or of the 'separated band" see ref 4). The number of peaks in a given chromatogram is identified by p and, to be fully defied, the assumed resolution value must be assigned. Since in the present work R. = 0.5, p corresponds to the number of maxima. The extent of separation is defined as91 Y =p/m
N p is the number of points in the digitized chromatogram. In eq 7 N pis a norming factorz1and M- 1 is the maximum extension over which the EACVF is computed (here M = 64, see Computation). C,(k) is called the experimental autocovariancefunction (EACVF), in order to distinguish it from the TACVF which derives from a model, since it is calculated from an experimental chromatogram (either real or simulated). By nonlinear least squares fitting of TACVF to EACVF, all the parameters of the TPS models of Table 11 (with the exclusion of AT and X which are experimental quantities) can be estimated, provided that the peak height dispersion ratio term d a h is evaluated. This will be made by the corresponding peak maxima disperison ratio &UM, as previously des~ribed.~ Note that this term can be evaluated provided that maxima in the chromatogram are identified. Since an approximation was introduced, the new ~
~~
(24) Margenau, H.;Murphy, G. M. The Mathematics of Physics and Chemistry;van Nostrandand Reinhold Co.: New York, 1966; Vol. II, p
183.
(14)
(15)
COMPUTATION All the programs were written in Basic and run on an IBM PS/2 Model 50 computer. All the simulated chromatogramswere generated as previously des~ribed.~ Using different random sequences for each parameter combination, 25 runs were performed. In order to have enough points in the EACVF computation, a frequency of 4/u, and 6/u,, respectively, for nonnoisy and noisy peaks was used and the experimental autocovariance function was calculated in the interval 0 I t I 16u,. The white noise and the interdistances distributed accordingto the different IM (exponential, uniform, normal, and gamma) were generated by using STATGRAPHICS routines.26 The same peak sensing algorithm as described in ref 3 was used for peak maximum identification. The TACVF was computed by using the fast Fourier transform routine of ref 26. Saturation values a,ranging from 0.333 to 0.667, were considered, under conditions of R, = (26) STATGWHYCS, Version 1.2;StatisticalGraphyce Corp.,Copyright 1985 STSC, Inc. (26)Amino, R.; Driver,R. D.Scientific andEngineeringApplications with Personal Computers; John Wiley & Sons: New York, 1988.
ANALYTICAL CHEMISTRY, VOL. 64, NO. 18, SEPTEMBER
15, 1992 2167
Table 11. Some Theoretical Expressions of SC Peak Interdistance Frequency Function, qt),and Their Corresponding Power Smctrum. F(o)
(0 It )
Normal u:/a:
mX
- 2 cos (Tu)e3+ cos (Tw)e"ldJ2 -
+ 1 + 2edd
(b)
Uniform 1 (0< t < 2T) 3 COB
2 - 2 COB (2To) - TO sin (2Tw) (2Tw) + 2Tw sin (2Tw) - 2T2u2- 1
T-Pt'le-t/'
up)
(0 It )
0.5. A standard value of ug = 4 and an exponential distribution
for peak heighta were chosen for all runs. The signal-to-noise (S/N) ratio reported in Table VI1 was computed by dividing the maximum peak by 4 times the noise standard deviation. The nonlinear parameter estimation was performed by the SIMPLEX method as described in ref 3.
RESULTS AND DISCUSSION In Figure l b examples of IM plots are reported. As can be seen in Figure la, the four models (E, U, N, and I') may represent quite different SC interdistance distributions. However, as can be seen in Figure lb, for selected values of their parameters they may represent quite similar patterns. In this instance it must be observed that the RSD values calculated according to Table I are close to each other (for U case RSD = 0.577; for the N case RSD = 0.5; for I' case RSD = 0.447). The point shown in Figure lb, and above described, is of basic importance to understanding some findings which will be reported below. It will,in fact, be observed that the ACVF derived from the IMs of type N and r are flexible as the original IM distributions because these are threeparameter functions. Consequently, both the IMs and the corresponding ACVFs can reprocically approximate one another. The E and the U models are, instead, "rigid" but can nonetheless approximate sufficiently well the N and the I' cases for selected values of their additional parameter (UT and p, respectively, for N and;'l see Table 11). These characteristics will be referred to below as the "mutual approximating properties" of the IMs. It must also be observed that RSD values greater than 0.5 cannot be considered for the N model, since otherwise negative interdistancevalues would have significantprobabilities (see Figure Ib). In Figures 2-6b TPSs and TACVFs of multicomponent chromatograms for different IM types and parameters and for different a values are reported. The r cases (Figures 2 and 3) exhibit a strong dependence on p and a values. Remember that when p = 1, the I'model becomes identical to the E model. Aa far as the PS shape is concerned (Figures 3a and 4a), a maximum appears for p > 2. The onset of a maximum in the TPS plot is related to the onset of a maximum in the corresponding plot of the IM (see Figure 1). In fact,
a /
om 0.07
f
om0.08
0
I
.
-l a
,
,
I
\
I
,
I I
1 ,
......_......__..... 2a
, , , I,
;
,' ,
I
II
,,
__..._..._ 3a _.._-__ 4 a
Flgwo 1. Examples of SC peak lnterdlstancedlstrlbution models (IMs) wkh T = 24: (a) (la) exponentlal, (2a) uniform, (3a)normal wkh UT = 0.2T, (4)gamma, p = 2; (b) (lb) uniform (RSD = 0.577), (2b) normal with uT= O.BT(RSD = 0.5), (3b) gamma, p = 5 (RSD = 0.447).
a peak in PS at a given frequency 00 means a reproducible repetition of peak position with an average distance between adjacent peaks that is reciprocal of 00. This is more distinctly
2168
ANALYTICAL CHEMISTRY, VOL. 84, NO. 18, SEPTEMBER 15, 1992 1.6 1
i
I .
-P= 0.5
F
a
__-_._. p. 1
1.2 1.1
......,, ..... p- 2 ---. p= 5
,
\
'.
0.7
\
0.8 0.5
0.4 0.3 0.2
. \
1
0.1 0
0
1
2
3
4
0 og 1
0.24 0.22
C
-P = 0.5
b
- - - _P_=.1 ... , ..... .. P = 2 ---_p= 5
0.18
C
0.14 0.12 0.1
0.1
I I
;'TI '.
I
0.08
0.08
0.06
0.08
0.04
0 04
0.02
0.02
0
0
__-----
\
.0.02
.0.02 0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
t
12
14
18
18
20
t /ag
t /og Flgure 2.
Power spectrum (a) and autocovariance functlon (b). SC peak interdistance dlsbibution model (IM): gamma with different p values. a = 0.333 (Rs= 0.5).
Flgure 3. Power spectrum (a)and
autocovarlance functlon (b). SC peak interdistance distribution model (IM): gamma, with different p values. a = 0.867 (RB= 0.5).
singled out at lower a saturation values when peak overlapping is less effective in obscuring peak position repetitions (see Figures 2 and 3). N distribution cases (Figures 4 and 5) can likewise be interpreted. When the RSD is low, there is a onset of a maximum in TPS plots, as in the 'I case for p > 2. Moreover at a = 0.333, pronounced periodic oscillations are observed in the TACVF plot (see Figures 4)provided that RSD < 0.5. The effect is practically canceled out when a is increased to 0.667 (cf. Figure 4a vs Figure 5a). Note that when a > 0.5 and R, = 0.5, the average distance between adjacent peaks, T, is less than the baseline width of the SC peaks (4ug,see eq 14). In such a cramped chromatogram, the long-distance correlations, that where observed at low saturations, no longer exist. The U TPS plots are more jagged (see Figure 6), but details in both the TPS and the TACVF plots are lost when a is increased. When different IM plots are compared, some features are distinctly singled out especially when the IM is sharp and the saturation factor low (see the N case for RSD = 0.1 and a = 0.333 reported in Figure 4). In the opposite case-that is when the IM is not sharp and the saturation factor is moderately high-TACVF plots look quite similar especially in the first part of the plot, for t / u g< 8 (see Figure 5). It is interesting to compare the three cases previously drawn in Figure lb: the r case of Figure 2b at p = 5; the N case of Figure 4b at RSD = 0.5; the U case of Figure 6b. All these cases are to be considered at the same a value, i.e. a = 0.333. It can be seen that the ACVF plots look very similar with a more or less pronounced concavity in the negative ACVF region, for t / u , < 8. The peculiarities are, instead, more pronounced in the high t value region (8 < t / u g< 16). The general conclusion is that TACVFs or TPSs are sensitive to
the IMs but they also exhibit properties of reciprocal approximation when the parent IMs are similar. Let us now see how multicomponent chromatograms look with the different IMs. Figures 7-11 report five examples of simulated multicomponent chromatograms all having the same number of components (m= 2001, the same saturation value (a = 0.333, at R, = 0.5), the same ug value (=4), and consequently the same mean interdistance value (2' = 24, see eq 141,but different IMs. By careful inspection one can verify that the number of peaks is not the same (the maximum is for the case of Figure 10 which corresponds to the N case with RSD = 0.2). However, it is absolutely impossible by simple inspection either to prove that they have the same number of components or to express a precise estimation about overlapping attributes like the separation extent y or the number of singlet, doublet, triplet peaks. For example, it is impossibleto discover whether the chromatogramsin Figures 8 and 9 have the same type of interdistance distribution (the normal) with the same mean SC interdistance value (2' = 24) but with different SC interdistance standard deviations (respectivelyUT = 12 and UT = 4.8). Likewise, it is impossible to prove that the chromatograms in Figures 7,9, and 10 have roughly the same T, UT, and RSD values. Finally, since the chromatogram in Figure 7 has the greatest amount of peakfree baseline, it becomes difficult even to imagine that it is, instead, the one with the worst y value. How the EACVF is able to single out the SC position pattern is shown in Figures 12-16. In each figure the EACVF and the best fitting TACVFs obtained with the four IMs, in the range 0 < t/ug< 16,are reported (note that the correct model is that underlined). It can be seen that the EACVF are quite different for the different cases. For example, the I' and the two N
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 18, SEPTEMBER 15, 1992 1 2.8 2.6 2.4
1.6
RSD
...... .. .. ....
RSD
-
0.9
=1
RSD = 0.5
=0.1
F
1
0.8
_....
0.7
.... ...
RSD-
1.
i
RSD- 0.5 RSD-
0.1
0.6
0.5 0.4 0.3
.. .
0.2
“b.
0.2
J
0
1
0
0
4
-.,
0.18 C0.160.14 0.12 0.1
0.1
*i
b
-
RSD = 1
......
RSD = 0.5 RSD
2
00
::j
4
3
9
0.161
:x
c
-0.1
J
.....
0.1 1 0.1
0.08
b RSD=1. RSD -0.5 RSD
=0,1
0.09 0.08 .
0.06 0.02
0.07. 0.06 . 0.05.
0
0.04.
0.04
-0.02 -0.04 -0.08
1
0
0.03
4 0
2
4
6
8
10
12
14
18
18
1
0 .
-0.01
20
cases exhibit the expected oscillatorypattern, which is instead almost absent in the E case. The U case exhibits a spread concavityaround t = 20 @ / a g= 5 ) as expected for this IM (see Figure 6b and compare in Figure 16 the EACVF, solid line, with the corresponding best fitting TACVF for the same U model, dotted line). However one can see that in general the best fitting TACVF obtained by employing the correct IM is not always the one that fits most perfectly. This finding is a direct consequence of the above mentioned properties of mutual approximation for the different models. Moreover, it must be remembered that at high t values, in the EACVF, a random component is present3p21which partially obscures the features of the deterministic part making imperfect the degree of fitting even with the correct model. This random component comes from the fact that, when computing the EACVF (eq 7), only a finite extension of the chromatogram-considered a stochastic process-is available. This effect was also shown in Figure 3 of ref 3. One must recall that just in this part-that obscured by these random components-the IM peculiarities are singled out. Since the random components are different from case to case the fitting results can be different. This behavior is shown in Tables I11 and IV where the ACVF fitting results of five different simulated chromatograms of type E and N (RSD = 0.2), respectively, are reported. The corresponding sample chromatograms are those of Figures 7 and 9 respectively, whereas the corresponding sample fittings are those of Figures 12 and 14, respectively. It can be observed that the beet fitting IM (the one underlined in the Tables I11and IV) does not always coincide with the true model. Moreover, the estimation of m and ug obtained by using the different IMs can be significantlybiased. This is more evident in the cases of Table IV where the interdistance distribution is of type N and its RSD value is 0.2. In this case the IM distribution is sharp
v
.
0
t /a, Flgure 4. Power spectrum (a) and autocovariance function (b). SC peak interdistance distribution model (IM): normal, at different RSD values. cy = 0.333 (Rs= 0.5).
.
0.02. 0.01 .
. 2
.
. .
.
4
6
. . . . 8 10 12
1 14
16
18
20
t /ag Flgure 5. Power spectrum (a) and autocovariance function (b). SC peak Interdistance distribution Model (IM): normal, at different RSD values. cy = 0.667 (Rs = 0.5).
Table 111. Selected Cases of Best Fitting Results. case no. 1
no. of peaks,p 128
fitting
IM
E U N
r
2
128
E U N
r
3
133
E U N
r
4
130
E U N
r
5
136
E U N
r
m est 177 149 181 164 192 152 126 176 207 160 161 194 177 151 246 287 199 160 179 203
ug
est 4.01 4.39 3.83 4.06 3.67 4.20 5.15 3.81 3.54 4.07 4.05 3.38 4.11 4.46 3.09 3.74 3.75 4.24 3.97 3.74
thirdparam (mor p)
29.9 1.19 17.8 1.23 19.9 1.15 25.8 0.43 23.7 0.98
52 0.042 0.208
0.026
0.049 0.085 0.132 0.254 0.081 0.089 0.029 0.026 0.070 0.129 0.161 0.417 0.182 0.057 0.072 0.035 0.057
-
Multicomponent chromatogram with exponentialinterdistance model. m = 200; T = 24; ug = 4;a = 0.333.
(see Figure la) which the other models can only poorly approximate, as above mentioned. Because the fitting behavior is so spread, the problem will be consequently to statistically check these fitting properties. For a given set of m, a,and IM, 25 different simulated chromatograms were processed, and the synthesis of the results from five sets all having m = 200 and a = 0.333 is reported in Table V. It can be seen that the mean number
2170
ANALYTICAL CHEMISTRY, VOL. 64, NO. 18, SEPTEMBER 15, 1992 1.0 1.5 1.4
F
1.3 1.2 1.1 1 0.9 0.a 0.7
0.0 0.5 0.4 0.3
0 200 t 400 Flgurr 8. Simulated multicomponent chromatogram: m = 200; a = 0.333 (R. = 0.5); peak height distribution, exponentlai; SC peak interdistance distribution model (IM), uniform (RSD = 0.577).
0.2 0.1 0
Flguro 9. Simulated multicomponent chromatogram: m = 200; a = 0.333(Rs = 0.5); peak height distribution, exponential: SC peak interdistance distribution model (IM), normal, wlth uT = 0.2T (RSD = 0.2). 2
0
4
6
8
12
10
14
16
18
20
t /a, Flguro 6. Power spectrum (a) and autocovariancefunction (b). SC peak interdistancedistributionmodel (IM): uniform, at different a values (R. = 0.5).
6.
Y 4.
Flgurr 10. Simulated multicomponent chromatogram: m = 200; a = 0.333 (R. = 0.5); peak height distribution, exponentlab SC peak interdistance distribution model (IM), normal, wlth uT= 0.5T(RSD = 0.5).
0
200
t
400
Flgure 7. Simulated multicomponent chromatogram: m = 200; a = 0.333 (R. = 0.5): peak height distribution, exponential: SC peak lnterdistance distribution model (IM), exponential (RSD = 1). 4
of peaks @) detected in the different types of chromatograms are quite different: a minimum value of 130 and a maximum value of 198-practically equal to the S C number m-are observed respectively for E type and for N type (RSD = 0.2) chromatograms. The retention pattern as expressed by the separation extent y, the number of singlet, doublets, triplets, etc., should be strongly dependent on the IM type, whose correct detection thus proves to be of basic interest. The first information searched for in these tests was to see whether the ACVF method is as powerful as the PS methods previously set up for only the E model.3 As one can see in Table V, if the correct model is fitted, that is if, e.g., the exponential, the uniform, etc. models are applied in fitting the EACVF of chromatograms having interdistance distributions respectively exponential, uniform, etc., good parameter estimation is always attained. As for as the m and ug estimation, similar results were obtained in studying Pois-
Flguro 11. Simulated multicomponent chromatogram: m = 200; a = 0.333 (R. = 0.5); peak height distribution, exponential; SC peak lnterdistance distribution model (IM), gamma, p = 5 (RSD = 0.447).
sonian chromatogram cases.3 In the present study the estimation of the additional parameters uTand p, respectively, in the case of N and I' distributions, is equallygood. However, the results obtained from the correct model application are not of particular usefulness since,when an experimental chromatogram is processed, one does not have any a priori knowledge of the true IM. It willbe instead mostly important to verify how correct is the mean estimation of chromato-
ANALYTICAL CHEMISTRY, VOL. 64, NO. 18, SEPTEMBER 15, 1992
0.009 -I
2171
0.01
i
0.009
EACVF .............. E model
0.006 0.005
_______N
........ ...
u
”
*I
0.001 . 0-0.00 1
20
0 Flgure 12. Experimentalautocovariancefunctbn (EACVF) of the chromatogram in Figure 7 and best fMng theoretical autocovariance functions. E, U, N, SC peak Interdistancedistributionmodeis(IMs), exponential,uniform, normal, and gamma, respectively. Correct model:
r:
E.
v’wwo 0.007
C 0.006
I,
0.005
0.004 0.003 0.002 0.00 1
1
EACVF ..___..___.... E model 0.008
\
0.006
,.
0.004 0.002
20
0
0.0 12
EACVF _ _ _ _ _ _E_model __.___. - *’ _ _ _ _ _ _ _ N ..
u r
40
60
t
Flgure 13. Exp@nmtai autocovariancefunctbn (EACVF) of the chromatogram in Figure 8 and best fitting theoretical autocovariance functions. E, U, N, SC peak Interdistancedistributionmodels (IMs), exponential, uniform, normal, and gamma, respectively. Correct model: U.
r:
60
t
Flgwe 15. Experimentalautocovariancefunction(EACVF)of thechromatogram In Figure 10 and best fitting theoretical autocovariance functions. E, U, N, I’: SC peak Interdistancedistributionmodels (IMs), exponentiei,uniform, normal, and gamma, respectively. Correct model: N with UT = O.ST(RSD = 0.5).
I
n nna
40
’
1
-0.002 0
20
40
60
t
Flgure 16. Experimental autocovariance function (EACVF) of the chromatogramin Figure 11and best ftttlng theoreticalautocovarlance functions. E, U, N, I’: SC peak Interdistancedistributionmodels (IMs), exponential, uniform, normal, and gamma, respectively. Correct modeif with p = 5.
Table IV. Selected Cases of Best Fitting Results.
EACVF _ _ _ _ _ ~ _ model _.____.E .........................
0.008
u
case no.
1
no.of peake,p 195
*’
_______ N r ’.
m
ug
IM
est
est
E U N
355 290 206 380 377 303 206 306 387 307 206 315 367 304 203 200 363 321 209 342
3.05 3.25 3.92 2.94 2.91 3.16 3.90 3.06 2.89 3.14 3.88 3.07 2.95 3.14 3.84 3.96 3.26 3.07 3.79 2.94
fitting
r
2
198
0.004-
E U N
r
3
199
E U N
r
4 Flgure 14. Experimentalautocovariancefunction(EACVF)of the chromatogram in Figure 9 and best fitting theoretlcai autocovariance functions. E, U, N, SC peak interdistancedistributionmodels (IMs), exponential, uniform, normal, and gamma, respectively. Correct model: N with UT = 0.2T(RSD = 0.2).
r:
gram attributes, obtained from the best fitting model. In
fact it is only the last one which can be unambigously identified when fitting an unknown case. These data are reported in the central part of Table V under the “best fitting model results”. The success frequency is the number of times a given model gave the best fit of the four IMs: e.g. in the E case (first row), 11 times the N model fit best (over a total number of cases of 25), being thus more powerful than the
195
E U N
r
5
199
E U N
r
thirdparam (mor p)
5.94 0.87 5.18 1.74 4.71 1.57 5.89 24.4 5.13 1.47
82
0.183 0.145 0.025 0.189 0.282 0.181 0.033 0.264 0.457 0.341 0.175 0.435 0.330 0.260 0.048 0.022 0.548 0.369 0.136 0.471
-
Multicomponent chromatogram with normal interdistance model. T = 24; q = 4.8; RSD = 0.2. m = 200, u8 = 4; a = 0.333.
true model-the E model-which instead gives the best fitting in only 7 cases. The U set (second row) behaves regularly. In fact, the true model is largely the most powerful one (20 times over 25 cases). The N model pattern strongly depends
ANALYTICAL CHEMISTRY, VOL. 04, NO. 18, SEPTEMBER 15, 1992
2172
Table V. Parameter Estimation and Recognition of the Retention Pattern by the ACVF Method.
set no.
-
interno. of peaks distance model (IM) P
E U N (RSD = 0.2) N (RSD = 0.5)
1 2 3 4 5
est params from the correct model third
r (P = 5)
m (200)
130 f 5 156 f 5 198 f 1 171 f 3 182 f 3
185 f 14 192 f 13 202 f 4 197 f 27 194 f 11
P”
uc (4)
(UT or
success frequency
E U
u)
third params for N and r success cases
best fitting model results
N
r
7 1 11 6 1 20 3 1 0 0 23 2 1 6 7 11 0 2 10 13
3.92 f 0.22 3.97 f 0.24 3.90 f 0.10 4.95 f 0.84 (4.8) 4.03 f 0.38 12.3 f 3.1 (12) 3.98 f 0.18 5.09 f 1.3 (5)
params
m (200) 179 f 19 193 f 15 202 f 4 197 f 22 200 f 15
u,(4) 4.02 f 0.22 3.94 i 0.19 3.91 fO.10 3.95 f 0.17 3.97 f 0.13
pr
$
success) success) 26.3 i 2.8 (24) 0.98 f 0.29 (1) 18.2 i 1.8 (13.9) 4.70 (3) 4.95 f 0.84 (4.8) 26.2i 2.4 (25) 12.8 f 3.6 (12) 3.92 i 0.58 (4) 10.8f 2.7 (10.7) 5.27 f 1.4 (5)
0 m = 200; T = 24; (Y = 0.333 (R,= 0.5). Comparison of the results obtained by using correct and best fitting model. Data reported in parentheses are true or reference values.
nent chromatogram is always close to the true value (seeTable V). Instead, ugestimations obtained by using and IMs which are not the best fitting one can be significantly biased (see the cases reported in Table IV). Thus, in an experimental application, an agreement between the a, value estimated from the ACVF procedure by using the results of the best fitting IM and that determined from separated SC peaks should confirm the goodness of the m estimation. When either the N or the r distribution model results in being the best fitting one, a third parameter-uT and p, respectively-in addition to m and ug is obtained. These cases are considered in Table V in the last two columns. First, it must be observed that in cases where N or I’ is both the true and the best fitting model, agreement is observed in the third additional parameter and the case does not require further comments. It proves interesting to analyze the estimated value and the meaning of the third parameter in the cases where these models give the best fitting for another IM in the place of the true one. One can see that, in this case, the third parameter always closely meets its “expected value”, which is reported in parentheses (see the last two columns in Table V). These expected values were computed by using relationships among m, UT,RSD, p, etc., reported in Table VI. These relationships were derived from the basic parameters reported in Table I. For example, in fitting E chromatograms (first row of Table V), in 11cases the one which fits best was the N model. In this case, the third parameter is UT and its true value is 24, since for the E model UT = T (see Table I) and T = 24. This true value falls within the
Table VI. Correspondence Table of the Third Additional Parameter of the Auuroximatina IM approximated IM (=true)
third param of the approximating IM (=best fitting) UT (N case) P (r case)
on its RSD value (see Table V third and fourth rows). In fact, for RSD = 0.2, it behaves like the U case. On the contrary, for RSD = 0.5, the N model (fourth row) exhibits a spread pattern: the best fitting frequency is 1,6,7,11, respectively, for E, U, N, and r models. These differences in the behavior of N cases are not unexpected since only for RSD close to 0.5 can the N model be well approximated by U or by a r distribution with a p value approximately equal to 5, as previously discussed (see Figure la,b and the comments at the beginning of this section). Let us now consider the best fitting parameters (see Table V). One can see that the agreement of the m estimate with the true value (200) is as good as that found by making use of the true model (compare the two m estimations of Table V). The a, estimate is another important control since it can be estimated from both a complex multicomponent chromatogram and from well-separated peaks of SCs obtained under the same experimental conditions. It can be seen that the ug estimation from the best fitting over a multicompo-
Table VII. Parameter Estimation and Recognition of the Retention Pattern by the ACVF Method on Additional Case Sets. best fitting model results success frequency set no.
IM
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
E E E E E N(RSD50.2) N(RSD=0.2) N(RSD 10.5) N(RSD =0.5) N(RSD=0.2) N(RSD=0.2) N(RSD-0.5) N(RSD=0.5) r ( p = 5) r (P = 5) r(p=5) r (p = 5) N(RSD=O.2) N(RSD =0.2) N ( R S D =0.2)
a
S / N m
m
100 50 200 100 50
m
100
m
50 100 50 100 50 100 50 100 50 100 50 200 200 200
m 01
m m
m m
m m W
m
100 50 5
( 0.333 0.333 0.667 0.667 0.667 0.333 0.333 0.333 0.333 0.667 0.667 0.667 0.667 0.333 0.333 0.667 0.667 0.333 0.333 0.333
Y 4 0 4 6 9 0 0 1 1 7 8 8 8 1 1 5 4 0 0 0
params
E
U
N
1 0 4 10 8 0 1 6 8 12 13 6 11 9 8 7 11 0 0 0
10
10 8 11
17 6
4 3 15 10 10 10 0 0 0 1 5 7 4 2 15 20 17
Data reported in parentheses are true or reference values.
5 5
10 14 8 6 6 4 11 5 10 9 9 8 10 5 8
~
third params for N and r success cases UT
98 f 11 50 f 9 189 f 27 97 f 14 49 f 7 100f3 50 2 93 f 11 48 f 7 103 f 13 51 f 7 94 f 15 48 f 5 99 f 13 50 f 5 95 f 15 46 i 6 203 f 5 207 f 7 198 f 13
*
4.08 f 0.22 4.12 f 0.42 4.17 f 0.30 4.17 f 0.49 4.10 f 0.47 3.95 f 0.11 3.93 f 0.16 4.08 f 0.21 4.04 f 0.33 4.30 f 0.40 4.31 i 0.46 4.16 i 0.76 3.98 f 0.38 4.11 f 0.37 4.00 f 0.30 4.12 f 0.29 4.13 0.48 3.94 f 0.12 4.15 f 0.14 3.90 f 0.15
(N success)
22.7 f 4.0 (24) 25.3 f 9.9 (24) 13.2 f 2.5 (12) 13.8 f 4.3 (12) 12.6 f 4.7 (12) 4.8 f 0.4 (4.8) 4.3 f 1.4 (4.8) 10.0 f 3.0 (12) 11.3 f 1.8 (12)
6.67 (6) 11.6 f 1.7 (10.7) 9.1 f 1.9 (10.7) 10.7 f 1.3 (5.4) 11.8 f 3.0 (5.4) 4.78 f 0.83 (4.8) 3.66 i 0.45 (4.8) 5.00 i 0.95 (4.8)
p
(r success)
1.31 f 0.39 (1) 1.81 f 0.67 (1) 1.21 i 0.37 (1) 1.08 f 0.61 (1) 1.10 i 0.35 (1) 25.6 f 8.0 (25) 26.2 f 7.4 (25) 6.5 f 3.7 (4) 11.7 i 4.8 (4) 3.23 & 0.64 (25) 2.6 f 1.3 (25) 3.6 f 0.5 (4) 3.6 f 0.4 (4) 5.1 i 3.5 (5) 8.0 f 7.1 ( 5 ) 3.1 i 1.0 (5) 3.6 f 1.1 (5) 32 i 12 (25) 35 f 10 (25) 28 f 21 (25)
ANALYTICAL CHEMISTRY, VOL. 64, NO. 18, SEPTEMBER 15, 1992
range of those derived from the fitting (26.3 f 2.8, see Table V). Likewise for the six cases of success obtained with the l? model, a p value of 0.98 f 0.28 is obtained which is close to 1 and which is just what would be expected when the I’ distribution becomes the E distribution. The other cases should have similar explanations. From this close correspondence between expected and determined values, one can infer that even if the best fitting distribution type does not coincide with the true model, nonetheless one always obtains correct information regarding the SC peak interdistance pattern, provided a correct interpretation is made of the numerical values of the additional fitting parameters. In order to do this one can make reference to the correspondence table reported in Table VI, with the insight to always calculate the T value by using eq 14. If, in this analysis, an inverse correspondence is found between the N or the I’ best fitting model and another model, one can make correct inferences about the SC position pattem. As a general conclusion it would appear that the ACVF method is substantially able to give the two first momenta of the interdistance distribution between the subsequent SC peaks. It must be stressed that the usefulness of the N or the I? models is derived from their flexibility. The potentialities of the ACVF method to work at lower SC numbers ( m = 100,50), at higher saturation factor values (a= V 3 ) , and in the presence of noise are exploited in the case seta of Table VII. For a = l/3, the ACVF method is very powerful even a t low m values, since the m and a, estimations are equally good. Even if the method fails in recognizing the true IM, as above for m = 200, the estimation of the third additional parameter (UT for the N success and p for the I’ success) is, nonetheless, generally correct for a proper interpretation of the distribution type. With a = 2/3 and low m values, the performance of the ACVF method in recognizing the correct retention pattern weakens (see Table VII, set nos. 17,18,21, and 22) since the long-distance correlations in the ACVF plots are smoothed out and obscured by random components. The consequence is that almost any IM can give a good fitting. However, the m and u, estimates are still unbiased. This last finding is very important since, from a correct estimation of both m and ug estimation, a correct a value is also obtained (see eqs 1Ck14). If this last value proves too high (e.g. a = 0.666) and a full description of the IM is required, one knows that this can be attained only by lowering the saturation factor, that is by applying better separation conditions. Since a = m/N,, this is attained by increasing the peak capacity with a more efficient column or by lowering the SC value m by e.g. selective preseparation. This last requirement is more stringent if the analyzed chromatogram span is short and thus the searched details are increasingly fine. For the sake of completeness some case seta with white noise a t different S/N levels are reported in Table VII. It can be seen that the ACVF method is insensitive to this noise type. This result was not, however, unexpected since this type of noise, which contributes to the EACVF with a spike a t t = 0, is easily filtered, by simply excluding from the fitting the very first part of the EACVF.3 It would be more interesting to analyze different types of noise, but this would require a specific numerical and theoretical handling which lies beyond the aims of the present work. The conclusion is that by this approach not only is correct evaluation of the SC number m obtained, but significant insights into the interdistance distribution type are attained as well. Thus one of the major constraintsin the interpretation of complex multicomponent chromatograms-the underlying interdistance distribution hypothesis-has been overcome. The point to face now is to apply all these numerical
2173
approaches based on PS and ACVF to real cases. This will be the subject of a forthcoming paper.
ACKNOWLEDGMENT This work was made possible by the financial support of the Italian Ministry of the University and the Scientific Research (MURST), the Italian Research Council (CNR). ACVF A
AT ah aM
C(t) C&) E E EACVF f(t)
F(4 g(0)
h IM
m M N NP NC P P
PS
RSD Re
R, S S2
sc SMO T
T t TACVF TPS U xo
X
P yi (Y
Y
r
GLOSSARY autocovariance function SC peak area total area of the multicomponent chromatogram mean value of SC peak height mean value of peak maxima in the multicomponent chromatogram autocovariance function value at time t numerically computed ACVF at point k , eq 7 exponential (distributionfunction of the interdistance between subsequent SC peaks) excess experimental autocovariance function, eq 7 peak interdistance frequency function PS value at frequency Fourier transform of the unitary peak shape located at the origin single-component peak height single-component interdistance model, i.e. distribution type of the interdistance between subsequent SC peaks number of SCs maximum time point in EACVF computation, eq 7 normal (distribution of the interdistance between subsequent SC peaks) number of points in the digitized chromatogram peak capacity computed at a given resolution parameter of the gamma IM number of separated bands computed at a given R, value. Since R, = 0.5, p = number of maxima. power spectrum relative standard deviation, eq 4 real part chromatographic resolution skewness total squared deviations between EACVF and TACVF,eq 9 single component statistical model of overlapping mean value in the U and N type distributions (Table 11) mean value of interdistance between subsequent SC peaks time axis theoretically computed autocovariance function theoretical power spectrum uniform (distribution of the interdistance between subsequent SC peaks) SC peak interdistance at a given resolution, eq 12 time span of the considered multicomponent chromatogram mean value of the chromatographic response, eq 8 chromatographic response at point j saturation factor (=m/N,) Dirac function separation extent ( = p / m ) gamma (distribution of the interdistance between subsequent SC peaks)
2174 a, Qh
m UT T
w
ANALYTICAL CHEMISTRY, VOL. 64, NO. 18, SEPTEMBER 15, 1992
standard deviation of the Gaussian form of the SC peak (=peak width) standard deviation of SC peak height distribution standard deviation of peak maximum distribution standard deviation of the SC peak interdistance distribution function decay constant on the E and r IMs (Table 11) frequency
(e’ = cos ( 2 ) + i sin ( 2 ) ) eq 23 is written cos (Tu) e(w) = sin(Tw)Tw
sin2(Tu) - sin (2To) Tu 2Tw 1- COS (2Tw) i 2Tw +
+i
and
APPENDIX Mathematical Derivation of the Theoretical PS Models. In eq 1the term lg(o)I2is the PS of the SC peak shape function at the origin.’ Under the hypothesis of a Gaussian shape function ita value (g(w)I2 = 2 r u t exp(-w2ut)
(16) where ug is the peak standard deviation. The area of component peak i is Ai = &uJzi (17) where hi is the peak height. The total area of the chromatogram is in
AT = d G o , C h i
where i is the imaginary unit. Using Euler’s formula
(18)
R=-
1- cos (2Tw) - T w sin (2To) 1- B(w) cos (2Tw) + 2Tw sin (2To) - 2T202- 1 ( c ) Normal IM. The characteristic function is23
e(w) = e3d/2+iTw = [cos ( T U )
+ i sin ( ~ w ) l e ” d / ~
and
eWzd - cos (Tw)e-w2a/2 2 cos (Tw)eW’df 2 -e-3d - 1 ( d ) Gamma type IM. Ita characteristic function is23
R=-
1-
1 (1- iwdP since the absolute value and the phase of
e(w) =
(29)
e(@) is
I=’
As ah is the mean height of the component peaks
(30) (19)
and
4 = arctan ( T O ) (31) moreover applying de Moivre’s rule for the power of complex numbers
eq 18 can be written as
By combination of eqs 13,16, and 20, the multiplying factor in eq 1 is expressed as a function of chromatographic quantities:
+
(cos 4 i sin 4)fl = cos (n4)+ i sin (nq5) the characteristic function can be written as tvw)=
(21) The delta function term in eq 1 can be neglected if the chromatogram is centered around ita mean value.’ The expression ReB(w)l[l - O(w)l of eq 1 describes the IM contribution to the PS of a multicomponent chromatogram. This term is calculated by using some properties of the complex numbers. (a) Poissonian chromatogram-IM of type E. It was previously shown’ that
Re=0 (22) 1- tqw) ( b ) Uniform IM. When peak positions are uniformly distributed, the characteristic function is23 =
sin (To)iTw Tw e
(23)
cos[p arctan (TU)]
(32)
+ i sin [p arctan ( T U ) ] +
(T2W2
(33)
For this reason R= -
B(W)
1-
+ 2(T2W2+ (TZU2
- (T2W2 + COS
[p arctan
COS
[p arctan
(TO)]
- (T2W2 +
(TW)]
- 1 (34)
When the pertinent ReB(w)/[l- e(w)l terms are introduced for each IM (eqs 22,26,28, and 34))with the aid of eq 21 and the delta Dirac term is neglected as explained above, theoretical PS (TPS) expressions are obtained for the four models considered. These expressions are reported in Table 11.
RECEIVED for review January 22,1992. Accepted June 22, 1992.
