Monitoring of randomly varying chemical processes by correlation chromatography

Analytica Chimica Acta 373 (1998) 175±187

Monitoring of randomly varying chemical processes by correlation chromatography M. Kaljuranda,*, H.C. Smitb a

b

Institute of Chemistry, Estonian Academy of Sciences, Akadeemia Road 15, Tallinn EE0026, Estonia Laboratory for Analytical Chemistry, University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands Received 8 May 1997; received in revised form 15 May 1998; accepted 26 May 1998

Abstract Correlation chromatography (CC) is a well known method for reducing detection limits in chromatographic analysis. Its usefulness for the monitoring of varying substance ¯ows is however more ambiguous. Although generally believed as a useful tool for this purpose, very little work has been done to prove it. In this paper CC and common, single injection chromatography (SIC) were compared with respect to their restoration capabilities of a stochastic ®rst order process, which is assumed to be the varying concentration input function of a chromatograph. It is demonstrated that CC can restore the input function with a smaller error than SIC when the process rate and input sequence length are related by a certain condition. The effect was found to be more profound when measurement noise is present due to the multiplex advantage of the CC method. # 1998 Elsevier Science B.V. All rights reserved. Keywords: Process monitoring; Multiple input/correlation chromatography; Measurement noise

1. Introduction Monitoring can be de®ned as the measurement of the changing properties of an object in order to detect too large deviations from a pre-set value [1]. The need for such an operation frequently appears in the chemical industry, where the input/output ¯ow of materials have to be controlled in order to manufacture a product of a desired quality. Chromatography, being a simple and straightforward analytical method, is a popular monitoring tool. In common chromatography the sample is introduced by a single injection and after *Corresponding author. Tel.: +372-2-536438; fax: +372-2536371; e-mail: [email protected] 0003-2670/98/$19.00 # 1998 Elsevier Science B.V. All rights reserved. PII S0003-2670(98)00391-2

that a chromatogram is recorded. The variations in time of the process components are reconstructed by using the values of the chromatographic peak areas obtained after a certain time interval. We call this interval the time resolution of the process. The chromatographic method, however, appears to be slow when the separation time of the components of the injected sample is comparable or larger than the characteristic time of the monitored process. In this case the process course cannot be reconstructed (in the sense of the sampling theorem) using the concentration values obtained at discrete time moments. In most cases the separation time in contemporary gas chromatography (GC) is (depending on the type of the column) hardly less than 1 min. Processes that could

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be monitored by GC are in the same time scale. In high performance liquid chromatography (HPLC) the separation time is even longer. Evidently, for the analysis of more rapid processes new sampling strategies are needed to reduce the required time resolution. One of the possibilities that might provide such a result is random sample injection with time intervals much less than the separation time of the sample components. The method is known as correlation chromatography (CC). CC is a chromatographic signal enhancement method. In CC, unlike in common single injection chromatography (SIC), the sample is introduced after short intervals, with a duration much less than the length of the chromatogram. The time moments for sample introduction are chosen randomly or according to a pseudo random binary sequence (PRBS). The resulting chromatogram ± correlogram ± is obtained by correlating the input sequence with the detector signal. CC is a multiplex measurement method and just as one of the aims of its spectroscopic analogies (FT-IR, FT-NMR, Hadamard spectroscopy), CC is intended to lower the detection limits of the objects of analysis by improving the signal-to-noise ratio. Another important aim in the spectroscopic multiplex methods is to improve the spectroscopic resolution. In that respect it might be interesting to note that the S/N ratio in separation methods is also decisive for the demanded minimum (peak) resolution if a certain predetermined estimation precision is required [2]. It has been demonstrated that enhancing the S/N ratio can be attained successfully in CC [3]. Because in CC the sample is introduced after short time intervals (typically 0.5±2 s), there exists an hypothesis among the users of CC that the technique could be useful for monitoring of time varying sample ¯ows, e.g. those appearing in chemical reactions or in process control. Compared to common chromatography, where the next analysis is not possible before the previous one is completed, CC enables a semi-continuous update of the current chromatogram, thus seemingly reducing the time resolution of recording input concentration curves by a factor of thousand and more. The result, however, is not straightforward because in theory the input to CC must be stationary. Thus the aim of the usage of CC for studying time varying substance ¯ows violates a basic assumption of CC, and if the time varying samples have been ana-

lysed by CC, the peculiarities of the CC process must be thoroughly considered to obtain meaningful results. For a time varying input the CC method calculates an average value over a certain period of time, which depends on the input sequence length. Also, the resulting correlogram baselines are in¯uenced by a speci®c disturbance: correlation noise, which disturbs the real concentration values obtained from the peak area measurements. Thus the essential question arises: how will this disturbance affect the analysis results of the time varying concentration ¯ows, or equivalently, does the analysis of the time varying ¯ows by CC has an advantage compared to common, SIC? In the present paper the study of this problem was undertaken using both theoretical means and simulations on a computer. 2. Theory In general, much what can be said on the nature of processes to be monitored depends on the pre-information of the investigator. It is well known that the more general and the less speci®c assumptions over the phenomena to be studied are, the more general and less speci®c are the conclusions the theory can predict. Talking about monitoring of time varying processes by chromatography, one encounters four principally different situations: 1. The sample is known in advance (i.e. it is known how many peaks there are in the chromatogram and also their location and shape) and the process describing the change of the concentrations at the beginning of the column is known in advance (i.e. the functional form or underlying statistics of the process are known). The task is to determine the process parameters. 2. The chromatogram is known, but the process nature is not. 3. The chromatogram is unknown but the process is known. 4. Neither chromatogram nor process are known. Examples of these situations are given in Table 1. The meaning of the `unknown' sample or process should not be taken too literally. The nature of chromatography itself restricts severely the possible set of the compounds that can be analysed. For example, if we are using GC with an FID the sample consists of

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177

Table 1 Examples of processes encountered in monitoring time varying concentrations by chromatography Process/sample

Known

Unknown

Known

Catalytic reactions with known kinetics and reactants; control in chemical plants Monitoring in chemical plants

Monitoring specific compounds in environment

Unknown

organic compounds with a molecular weight less than 400, and a good chromatographer takes precautions to protect the column from interferences of compounds which cannot be detected by this system. For different situations different sampling strategies can be considered. When sample and process are known there seems to be no need for implementing sophisticated input sequences, because common regular interval sampling with curve ®tting might work well. The chromatogram runtime can be reduced to be short enough to meet the sampling theorem requirements. Possible peak overlap can be resolved since peak number positions and shape are known. Much less can be done if the sample and process are both unknown. Sophisticated data analysis such as maximum entropy may be useful [4]. Maximum entropy has been used successfully for image processing in astronomy [5], but no reports are available of using this technique in chromatography. Nevertheless, the use of maximum entropy seems to be attractive since no pre-knowledge of the sample is necessary. The peak shape must be known, however, together with the statistical properties of the chromatogram noise. The case where the process nature is known but the chromatogram is not is also of interest. In this paper we consider a situation where a sample amount in front of the column changes according to a ®rst order random process with known parameters, but the chromatogram is not known. This is an important situation in chemical plant monitoring. 2.1. First order processes It is well known [6,7] that many chemical processes behave like ®rst order stochastic processes, which can be described by a differential equation: T

dx ‡ x…t† ˆ u…t† dt

(1)

Environmental processes; space studies; polymer thermal degradation; chemical reactions with unknown kinetics

where T is the process time constant, t is time, x(t) is the output concentration of a compound participating in the process, and u(t) is the input concentration of the compound. When u(t) is a stochastic varying concentration with `white noise' properties, the output of the ®rst order system of Eq. (1) will be a ®rst order stochastic process. The task of monitoring is to detect e.g. the process deviations from the given (e.g. safety) limits. During monitoring the samples of x are injected into the chromatograph. Assuming that the chromatograms have been recorded digitally with a certain interval, say
t, which has been determined by the narrowest peak width in the chromatogram, it will be convenient to write Eq. (1) as a recursive difference equation. By replacing dt!
t and dx !
x ˆ x…i
t†ÿ x……i ÿ 1†
t† one obtains  
t
t x…i
t† ˆ 1 ÿ u……i ÿ 1†
t†; x……i ÿ 1†
t† ‡ T T i ˆ 1; 2 . . .

(2)

or using the common notation x…i† ˆ ax…i ÿ 1† ‡ w…i ÿ 1† thus a ˆ 1 ÿ
t=T and w…i† ˆ …1 ÿ a†u…i†

(3)

Also assuming that w is white noise with a standard deviation sw, it is well pknown  that the standard deviation of x is sx ˆ sw = 1 ÿ a2 [8]. 2.2. The maximum frequency that can be monitored by SIC and CC As already mentioned, comparison of CC and SIC for monitoring purposes involves the determination of the highest frequency that can be monitored in the time varying process with acceptable accuracy. In SIC the samples are taken with a time interval n
t, where n is the number of points in the chroma-

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togram. The highest frequency, fcut, that can be monitored must satisfy the sampling theorem [9], which says that a time varying process can be completely restored from its discrete values if 1 …SIC† > 2fcut n
t

(4)

In CC the samples are measured after an interval
t, but due to the averaging property of the decorrelation procedure, the CC process acts as a moving average ®lter. Recalling that the shape of the power spectrum of the moving average ®lter is proportional to ‰sin…fn
t†=…fn
t†Š2 , it follows that the boxcar ®lter has in its power spectrum the ®rst zero intensity value at the frequency fˆ1/n
t which for our purposes can be considered as the ®lter cut-off frequency, i.e. 1 …CC† > fcut n
t

(5)

Comparing Eqs. (4) and (5) it follows that …CC†

…SIC†

fcut ˆ 2fcut . The result is general and does not depend on the functional form of the time varying process spectra as far as it is bandlimited. Unfortunately, the models of all processes of practical interest are theoretically not bandlimited. A solution to overcome this dif®culty is the introduction of an effective signal frequency bandwidth. Usually, the motivation behind the use of an effective value of a parameter is to replace an unusable or poorly de®ned function by a simple function which lends itself to the required mathematical manipulations. The effective parameter has to be chosen such, that the equivalent calculations give results that would have been obtained when the true function had been used. An effective power bandwidth can be de®ned in several ways; there can be no unique de®nition because no single parameter can pretend to completely characterise the spectral properties of a random process. In our case the effective or equivalent power bandwidth was de®ned by choosing a rectangular power spectral density (PSD) function, which has the same area as the curve below the original PSD. It is customary to choose P(0) as the height of the rectangle, where P(0) is the PSD value at zero frequency. The power spectrum of the ®rst order process has a shape proportional to the function [10]

P…f † ˆ

P…0† 1 ‡ …2fT†2

where f is the frequency. The equivalent bandwidth for a ®rst order process, fFOP can be calculated as solution of the equation Z1 fFOP
P…0† ˆ 0

P…0†df 1 ‡ …2fT†2

The integral is equal to 1/4T. Thus fFOPˆ1/4T. It follows from Eq. (4) that for SIC 1 1 >2 ; n
t 4T

n
t < 2T and a > 1 ÿ

2 n

…40 †

and for CC 1 1 > ; n
t 4T

n
t < 4T and a > 1 ÿ

4 n

…50 †

The simple conclusion that can be made from Eqs. (40 ) and (50 ) is that TCC  TSIC =2, which re¯ects the general conclusion that CC enables monitoring two times more rapid processes than SIC. An example clari®es this conclusion: taking a value typical for GC,
tˆ1 s, and assuming that the last peak in a chromatogram requires nˆ600, one gets that the chromatogram length n
t is 10 min. Thus T …CC† ˆ2.5 min and T …SIC† ˆ5 min. Similar calculations can be made for other processes of interest. For example, Gaussian input reveals p to the equivalent cut off frequency equal to 1=… 2†, where  is the standard deviation of the Gaussian, and the chromatogram length required for monitoring the Gaussian function input is equal to 2.5 for CC and 1.25 for SIC, correspondingly. 2.3. Process reconstruction error by SIC The sampling theorem establishes the correct digitising interval for the restoration of the continuous function from its digitised values after the measurements are performed. On the contrary, monitoring means obtaining information on-line when the process is in progress. Thus, having measurements x(0), x(1), . . ., x(i
t) by SIC the further process must be somehow extrapolated from these measurements before a new measurement becomes available at moment i‡n
t. The process values between x(i)

M. Kaljurand, H.C. Smit / Analytica Chimica Acta 373 (1998) 175±187

and x(i‡n
t) at time moment  can be predicted by the autocorrelation function as follows [8] x…i
t ‡ t† ˆ x ‡ …x…i
t† ÿ x†eÿt=T

(6)

where x is the process mean value and  is runtime between the interval i
t and (i‡n)
t. The prediction error variance is given by the formula [8] s2 ˆ s2x …1 ÿ eÿ2t=T †

(7)

The mean variance, s2SIC , over the input sequence length interval, n
t, would be a useful integral parameter for the latter comparisons. It can be calculated as follows nZ
t

s2SIC

s2CC ˆ 2 n…b
t†2

(9)

where b is the input function slope and  is a dimensionless coef®cient between 0.5 and 2.5, which value depends on the peak retention time. By de®nition the input slope value over the interval n
t is in 95% of the cases less than 2D…
xc †=…n
t†2 . Then the standard deviation of the correlation noise is in 95% of the cases less than s2CC ˆ 2 n

2D…
xc † …n
t†

2


t2 ˆ

22 2 s …1 ÿ eÿn…1ÿa† †2 n x (10)

s2CC ˆ 22 ns2x …1 ÿ a†2

(11)

It follows from Eq. (10) that s2CC does not depend on , thus averaging over n
t is not necessary.

For small
t/T values the latter expression can be developed to the Taylor series up to the second term as follows n
t ˆ s2x n…1 ÿ a† ˆ s2x T

depends on the linear input slope as follows

or for a linear approximation of the exponent

s2x …1 ÿ eÿ2=T †d n
t 0   T 2 …eÿ2n
t=T ÿ 1† ˆ sx 1 ‡ 2n
t

s2SIC ˆ

179

(8)

2.4. Process reconstruction error by CC and the influence of correlation noise For a linearly changing input, the correlation noise standard deviation is proportional to the amplitude of the input change over the interval n
t [3]. This result can be implemented for monitoring ®rst order processes also since if n
t<4T (Eq. (50 )) the input can be considered approximately linear (see e.g. Figs. 4 and 5). The amplitude can be estimated using the autocorrelation prediction of the process values. Thus, using Eq. (6) one can calculate
xc ˆ x…i
t† ÿ x…i
t ‡ …i ‡ n†
t† ˆ …x…t† ÿ x†…1 ÿ eÿn…1ÿa† † The mean of the
xc is zero and the variance is D‰
xc Š ˆ s2x …1 ÿ eÿn…1ÿa† †2 It can be shown [11] that correlation noise variance

2.5. Comparing prediction errors of CC and SIC To compare the single injection method and the CC method it would be convenient to investigate the ratio of prediction error standard deviations of both methods. From Eqs. (8) and (11) follows s2SIC s2x n…1 ÿ a† 1 ˆ ˆ 2 2 2 2 2 sCC 2 n…1 ÿ a† sx 2 …1 ÿ a†

(12)

When a!1 the ratio of both errors approaches to zero, but the CC method error does it more rapidly than the single injection prediction error because the ratio …sSIC =s2CC † ! 1. It means that when a!1, the CC method predicts ®rst order process values more precisely than the single injection method, because the prediction error for the latter is larger than that for CC. 3. Simulation methods and means Both chromatographic methods of reconstruction of ®rst order processes, SIC and CC, were investigated by using computer simulations of the reconstruction methods. First an ensemble of the process with given parameters sw and a was generated. Members of this ensemble were considered as input functions to the chromatograph. Correlograms were updated after

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every
t, and x(t) values were measured from the chromatogram peak area measurements. Those values were compared with the real process values at point i to measure the estimation error. Using common chromatography for restoring the process, the input concentration values were ®rst measured at time moments 0, n
t, 2n
t, 3n
t,. . . Concentration values within time interval n
t were estimated according to Eq. (6). Prediction by CC was done for every measured point by decorrelating the corresponding detector output window. The whole sequence of actions was the following: 1. Generation of the ®rst order ensembles of processes with given statistics. 2. Generation of the PRBSs and corresponding circulant input matrices. 3. Generation of the detector signals of the chromatograph for the case of PRBS inputs and with a given number of peaks. The input signal variation was assumed to correspond to the first order process, generated in Eq. (1). 4. Decorrelation of the window of the detector signals with length n points by inversion of the input matrices and calculation of peak areas. The input signal reconstruction was done by sliding the window over the whole process run. 5. Reconstruction of the first order processes by SIC applying Eq. (6), using the first order process values registered with interval n
t. 6. Calculating the standard deviation of the squared error between generated (ideal) process values and predicted process values for both the SIC and the CC method. The simulations were performed on a `286' and a `486' type personal computer. All simulations were done using MATLAB software (MathWorks). MATLAB is a special purpose language specially designed for matrix calculations (which form the significant part in CC theory). Also MATLAB contains an extensive set of means for spectral and statistical analysis of signals. 4. Results and discussion During this study several ensembles of ®rst order processes were generated with given parameters. Cal-

culation of one reconstructed ensemble requires 2±4 h on a `286' computer, depending on the PRBS length and number of PRBSs generated. Typical parameters were nˆ6 and 7, number of PRBSsˆ6±10, and number of process realisations in ensembleˆ1±10. 4.1. CC as a boxcar filter By simulating a ®rst order process with given a and sw values (keeping the sx constant) and using different PRBSs with different lengths, it was established that if a<1±4/n (i.e. it violates the condition in Eq. (50 )) then the CC noise standard deviation value is constant and the input can be considered as white noise averaged by the CC process. If a>1±4/u, the CC noise power starts to decrease (Fig. 1). Theoretically, the a value where the CC noise standard deviation begins to decrease is around aˆ1±4/63ˆ0.93 (nˆ63) and aˆ1±4/127ˆ0.97 (nˆ127), correspondingly. It follows from Fig. 1 that those values are in good agreement with the results of the simulations. CC noise power drops to zero when aˆ1 and sxˆ0, since the process then has a constant value. 4.2. Comparison of CC and common chromatography Some typical mean standard deviation values of the difference between generated and reconstructed process standard deviation values are given in Table 2. In Table 2 scc represents the error standard deviation between input signals simulated as well as reconstructed by CC, calculated over the ensemble as a function of time. The mean standard deviation is calculated over the time. sSIC has the similar meaning for single injection reconstruction. It follows from Table 2 that the CC and SIC reconstructions give almost the same mean error for the processes with a<0.95. The advantage of the CC method, although statistically existing, is not very signi®cant. When a increases, the error standard deviations for both methods decrease, since the overall change of the input function over the interval n
t decreases. However, comparing the ratios of the average standard deviations of the reconstruction errors, it follows from Table 2 that the input function can be better reconstructed by CC than by SIC. In this sense we can talk about the veri®cation of the theoretically

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181

Fig. 1. Standard deviation of CC noise as a function of the first order process rate parameter a, open squares nˆ127, triangles nˆ63. Table 2 Mean reconstruction errors for selected ensembles n

No. of sequences

Number of realisations

a

sw

Mean (sCC)

Mean (sSIC)

sSIC/sCC

63 127 127 127 127 127 127

40 9 6 6 6 6 6

5 5 5 1 5 1 1

0.9990 0.9500 0.9500 0.9950 0.9990 0.9990 0.9999

0.007 0.100 0.005 0.050 0.007 0.022 0.0008

0.0180.001 0.2200.016 0.1310.052 0.1490.000 0.0240.009 0.0580.000 0.00170.00

0.0300.005 0.2730.045 0.1530.049 0.2780.000 0.0590.027 0.1820.000 0.00580.00

1.66 1.24 1.16 1.86 2.37 3.14 3.41

predicted advantage (by Eq. (12)) of CC for process monitoring. Comparisons with the theory are however dif®cult, since Eq. (12) holds only when a is near to one. This advantage is increasing when a approaches to unity (Fig. 2). There is a remarkable difference in the distribution of the reconstruction error over the process runtime (Fig. 3). In this ®gure the standard deviation of the

reconstruction error as a function of time is given. The standard deviations were calculated over the ensemble for every time moment. Indeed, according to Eq. (7), the reconstruction error using SIC must be zero in the measurement point, increasing rapidly to the ®rst order process standard deviation sx. It follows from Fig. 3 that simulation results con®rm this prediction. The standard deviation of the reconstruction error in

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Fig. 2. Ratio or reconstruction error mean standard deviation as a function of the first order rate parameter a.

4.3. Effect of the measurement noise

to the generated ®rst order process. A typical result is given in Fig. 5. From this ®gure it follows that the single injection method fails to reconstruct the simulated process when the CC results are not much affected by the measurement noise. This is the result of the well known multiplex advantage that the CC method has. For SIC there is no multiplex advantage, so the in¯uence of the detector noise with the level easy to deal with by CC is disastrous for the reconstruction process by SIC. In process control the measurability, m, is an important way to express the quality of the measurement in¯uencing the control. It is de®ned as s s2x ÿ s2x;opt mˆ s2x

Because the measured chromatographic peak area values are always affected by detector noise, it is of interest to study the effect of high frequency white noise on the reconstruction process. Thus, to make the model more realistic, a zero mean Gaussian noise record with given standard deviation sd was added

where sx,opt is the output variance of the optimal controlled process. It can be proven that m is linearly dependent of the standard deviation of the measurement noise, but exponentially dependent of the sampling frequency. Therefore the measurability in case of CC, compared with SIC, is considerably higher,

case of the CC method appeared to be distributed uniformly over the process runtime. Results presented in this ®gure demonstrate the fact that was already discussed above: CC reconstructs the ®rst order process better than SIC only if a>1±4/n (i.e. a>0.95) in the present case. In that case the CC method has a de®nite advantage compared with the single injection approach for reconstructing the process course for a time period, when a long interval has passed from the previous single injection measurement moment and the next injection cannot be done because the column is still busy with the analysis of the sample. In Fig. 4 an example of the reconstruction is given for a typical member of the ensemble.

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Fig. 3. Reconstruction error standard deviation. Thin line: single injection reconstruction; thick line: CC reconstruction; sxˆ0.16.

183

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M. Kaljurand, H.C. Smit / Analytica Chimica Acta 373 (1998) 175±187

Fig. 4. Simulated and reconstructed first order processes. Solid thin line: original process course; solid thick line: CC reconstruction; dashed line: reconstruction by Eq. (7); sxˆ0.1. CC does not make predictions for the first n points.

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185

Fig. 5. Simulated and reconstructed first order processes. Solid thin line: original process course; solid thick line: CC reconstruction; dashed line: reconstruction by Eq. (7); sxˆ0.05 measurement noise standard deviation 0.2 units.

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particularly in case of long retention times. Of course, one should note that in any case the retention time has to be as small as possible, the increase in m is always relative compared to SIC. An extra but in general minor increase comes from the reduction of the measurement noise in CC. The results of this paper indicate that the CC method might require further modi®cation and improvement. Its direct application to the identi®cation of ®rst order processes has advantages over the common approach by SIC. A potential user of CC for process monitoring purposes should weigh these advantages with respect to the necessary investments of the chromatographic soft- and hardware necessary for CC implementation. One promising way for improving the CC performance is to use the available a priori information about the sample: e.g. chromatogram shape. Analysts usually know the composition of the sample and the aim of chromatography is to establish the kinetics of the sample components variation. The pre-knowledge can be used for modi®cation of the de-correlation equations introducing the chromatogram shape and developing the peak intensity values into the series of polynomials to better ®t the course of the ®rst order process. Such work is already submitted for publication. 5. Conclusions Correlation chromatography has an advantage over SIC in a ®rst order process reconstruction, despite the in¯uence of the correlation noise disturbing the peak area measurements. The critical aspect for applying the CC technique lies in the fact that the input sequence length and process time constant must be related via the inequality a>1±4/n. The better the inequality is ful®lled the more clearly the CC method advantage is expressed, particularly in the controllability of a controlled process. When detector noise can not be neglected, the CC method advantage is very signi®cant due to the multiplex advantage. Using CC, the standard deviation of the reconstruction error is distributed uniformly over the time scale. However, in case of single injection the error distribution is non-uniform. It is zero at the measurement point and approaches to the sx value at the end of the interpolation interval.

6. List of abbreviations CC PRBS SIC a b D[] f fcut fFOP i m n P(f) sw sx sx,opt s2() s2SIC s2CC T TCC TSIC t u(t) w(i) x(t) x(i) 
t 

correlation chromatography pseudo random binary sequence single injection chromatography first order process constant mean slope of the first order process over interval variance operation frequency cut-off frequency equivalent bandwidth of the first order process runtime index measurability number of points in a chromatogram power spectral density input concentration standard deviation output concentration standard deviation output concentration standard deviation for an optimal process variance of the process prediction error mean of the prediction error variance over the interval n
t measured by SIC prediction error variance measured by CC, correlation noise first order process time constant first order process time constant measured by CC first order process time constant measured by SIC time input concentration input concentration (digitised) output concentration (continuous) output concentration (digitised) numerical factor digitising interval runtime

Acknowledgements This work was supported by European Community Grants `Go West' (CIPA3510PL926561) and `Go East' (CIPA3510CT921274).

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