The deconvolution of detector signals in correlation chromatography

W Original Research Paper

155

Chemome~ricsand Intelligent L&oratory Systems, 12 (1991) 155-168 Elsevier Science Publishers B.V., Amsterdam

The deconvolution

of detector signals in correlation chromatography

Recovering response functions from fine-sampled convolutions with pseudo-random binary sequences R. Mulder, C. Mars and H.C. Smit * Laboratory for Analytical Chemistry, University of Amsterabm, Nieuwe Achtergracht 1018 WV Amsterdam (Netherlandr) (Received 22

166,

October 1990; accepted 2 September 1991)

Mulder, R., Mars, C. and Smit, H.C., 1991. The deconvolution of detector signals in correlation chromatography. Recovering response functions from fine-sampled convolutions with pseudo-random binary sequences. Chetnomelti’csand Intelligent bboratov Sysrems, 12: 155-168. In most applications of correlation chromatography the output signal is the response of a chromatographic system to a sometimes modified - pseudo-random binary sequence of injections. To obtain an interpretable chromatogram, the output can be deconvoluted by cross-correlating with the input sequence. It can easily be overlooked that straightforward correlation may result in unnecessary and unwanted smoothing of the result. A modified sequence that is generally applicable in deumvoluting the output signal without supplementary peak-broadening is described. The theory is of interest in those cases where peak-width is small compared to the width of the (virtual) injection.

INTRODUCTlON

The best known applications of multiplex or correlation chromatography (CC) [l-7] are in the field of trace analysis. Another interesting feature is, for instance, a design in which individual chromatograms are obtained from different samples although they are actually analyzed simultaneously [81. The most widely used injection pattern in correlation chromatography is a pseudo-random binary sequence (PRBS) of single injections: ideally, plugs of solely reference compound (gas in GC, liquid in LC) and solely sample compound are fed into the column. As CC techniques become increasingly refined, there is a growing interest in details on a time scale of the (single shot) injection. During the development of some CC applications [8,91 in our laboratory we noticed that the usual straightforward correlation of the PRBS and detector signal may lead to unexpected and sometimes undesirable smoothing of the correlogram. This effect can easily be avoided. 0169-7439/91/$03.50

0 1991 - Elsevier Science Publishers B.V. All rights reserved

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Similar correlation techniques are used in some fields of spectrometry where the mathematics are commonly described in terms of Hadamard matrices 17,101,although, from a somewhat different point of view, an analogous solution was reported by Fenimore [lO,ll] in imaging spectroscopy.

THEORY

~l~sical ~conuolut~n

of detector signals in co~e~t~n

aunts

Assuming linearity and stationarity of the system the resulting detector signal may be described as a convolution of the input signal (p, the PRBS) and the impulse response function (h, the chromatogram obtained from a single i~~tion experiment in which the ~ntribution of the injection time to peak broadening is negligible): y(t)

=k*p(t

-T’)~(T’)

dr’

(1)

where the integration time T is chosen to be equal to the repetition length of p. Notice that, apart from independent noise contributions, the detector signal therefore has the same periodic&y if h(r’) = 0 for T’ > T. In Eqn. 1 and throughout this paper a steady state of the detector signal is assumed, that is, the injection pattern started not later than at t = -T. Estimation of the impulse response is generally performed by straightforward cross-correlation of the detector signal with the input signal:

R&l

=&t-W)

dt

in which a scaling factor is omitted for convenience.

R&)

d+

=kTp(t--‘)(/Oa(t+(r’) =

p( t - #)p(

T’ - ~)h(r’)

Inserting Eqn. 1 into Eqn. 2:

t - T)

dt h( 7’) dr’

dr’

One may conclude that, apart from a scaling factor and disregarding noise and nonlinearities, the resulting cross-correlogram R,, is identical to a chromatogram obtained from an injection with a profile equal to the autocorrelation of the input sequence. For this reason, R, is sometimes referred to as the ‘virtual injection’ profile. Any basic PRBS pattern consists of a combination of l/204 - 1) ‘low’ and l/204 + 1) ‘high’ levels. If the values of these levels are set to 0 and 1, the autocorrelation is:

Rpp(r) =

l/2( M+ 1)(2 - T/AT)

for

l/2( M + 1)

for

i 1/2(M+

1)(2+7/AT-M)

for

0 G T/AT mod(T)

< 1

lgr/ATmod(T)
1
mod(T)

(4)
Where AT (the so-called ‘clock period’) is the longest period of time in which the PRBS does not change levels. The PRBS pattern repeats after a time MAT = 1”. The cross-correlogram thus resembles a

n

Original Research Paper

correlation

157

of:

hl-n_YMLu-itti

Fig. 1. Cross-correlation of sequences based on a seven-clock period PRBS with ten sample points per clock period. Simplified detector signals (x): (a) PRBS delayed for 15 sample points; (b) sum of two PRBSs delayed for 15 and 26 sample points; (c) PRBS delayed for 15 points with a ‘1st order’ disturbance; (d) 5 : 10 duty cycle PRBS delayed for 15 sample points. The signals are correlated with: (A) a full PRBS, levels - 1 and + 1 (p*); (B) a two-level 1: 10 duty cycle PRBS, levels - 1 and + 1; (C) a three-level 1: 10 duty cycle PRBS, levels - 1, 0 and + 1 (q*).

chromatogram obtained from a triangular shaped injection profile. This simple result is symbolically depicted in Fig. 1Aa. Some authors [4] use a ‘true’ random binary sequence. In that case Eqn. 4 does not hold for finite T and deconvolution according to Eqn. 3 is not appropriate and other methods are necessary, such as deconvolution in the Fourier domain. Origin of correlation broadening

In conventional chromatography it is common practice to choose the injection time in such a way that its contribution to the total peak broadening is negligible. Increasing the injection time to some extent may be allowable, as long as the resolution of peaks of interest is not seriously affected. In many CC designs the sample is fed into the column during approximately 50% of the total analysis time, regardless of whether this is accomplished with a PRBS with many short clock periods or one with a relatively low number of long clock periods. If the signal to noise ratio is poor it may be helpful in conventional chromatography to increase the injection time at the expense of peak shape. Increasing the ‘virtual injection time’, i.e. the PRBS clock period in CC, has fewer advantages in this respect because the net sample throughput remains 50%. Of course the spectral density of the input signal should be chosen with respect to the impulse response. In both methods extremely low frequencies in the input signal will not contribute to the analytical information because of loss of separation. Extreme high frequencies in the input signal in CC will not contribute to the power of the output signal. An important drawback of CC is the necessity of some kind of modulation device at the column inlet which is capable of injecting reproducible quantities without disturbing the chromatographic process already in progress. In most cases injection disturbances play a bigger role if the number of transitions in

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an input sequence (given an analysis time) increase, i.e. if the clock period of a PRBS decreases [12]. From this point of view the clock period in CC should be chosen to be as large as may be allowable with respect to the frequency response of the system. Moreover, minimizing the number of input transitions may reduce wear and tear of mechanical input devices. After sampling the detector signal with some kind of analog-to-digital converter, the calculations are carried out on a digital computer. One sample point per clock period is sufficient to describe the peaks in the correlogram if the (virtual> injection time is small compared to peak broadening in the column. (Of course, proper filtering of the analog signal before sampling is assumed.) More sample points are required per clock period if, for whatever reason, a clock period is chosen which is not significantly smaller than the chromatographic peaks. In that case, however, the resolution of the peaks in the deconvoluted signal will be less than that obtained in a conventional recorded chromatogram with an injection time AT, because the triangular-shaped virtual injection in CC has a base which is twice the clock period (= 2AT). This effect can be illustrated by considering a hypothetical impulse response (chromatogram) consisting of two rectangular peaks separated by only 10% of the clock period. Regardless of the number of sample points per clock period, almost all separation in the deconvoluted signal will be lost. Fig. 1Ab gives an impression of the result that will be obtained after correlation of the hypothetical detector signal (Fig. lb) with the input PRBS (Fig. 1A). A method to prevent broadening in the ideal standard experiment The rectangular input pattern is a consequence of the experimental design, but the choice of deconvoluting the detector signal by correlating with this pattern is in fact based on historical grounds. The triangular shape of the autocorrelation function of the PRBS (and therefore of the virtual injection) is essentially the result of the correlation function between two rectangular patterns. This was recognized by Fenimore [ll] in imaging spectroscopy. At first sight the avoidance of correlation broadening seems to be simple if one correlates the detector signal with a PRBS-like profile in which, in each ‘high’ clock period, only one sample point is left at the upper level and is reset to the lower level everywhere else (Fig. 1B):

4(t) =

p(t) 0

for t=O, AT,2AT,... for every other t

(5)

The virtual injection defined this way (Fig. 1Ba) does indeed look satisfactory. Any other signal that may be regarded as the sum of (weighted) time shifted versions of the PRBS (i.e. any convolution with the full PRBS) will therefore give a neat result after correlation with the modified PRBS (see for example Fig. 1Bb). In practice detector signals in CC are sometimes seriously afflicted by nonlinearities (a nonlinear behavior of the column, for instance) as well as by PRBS dependent injection errors. These reproducible errors give rise to specific disturbances in the correlogram (‘ghost’ peaks [121). In Fig. lc, a strongly simplified disturbance is added to the profile of Fig. la. Correlation with a PRBS that is modified according to Eqn. 5 leads to serious baseline disturbances (Fig. lBc), while these disturbances are less distinct in the correlogram obtained with a full PRBS (Fig. 1Ac). In some applications of CC, e.g. simultaneous [S] CC, the basic input pattern is a PRBS in which the duration of the physical ‘high’ level is less than the ‘logic’ high level. Fig. Id shows a 50% duty cycle PRBS; that is, a maximum of 50% of each clock period is actually at the high level. This signal cannot be considered as a convolution of some function with a full PRBS, and again heavy disturbances occur in the correlogram (Fig. 1Bd).

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Original Research Paper

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The 1: K duty cycle PRBS as the basic input pattern The detector signal in CC (Eqn. 1) can be rewritten in digitized form as: N-l y(i6T) = c p(i~T-j~T)b( j&T) j=O

briefly: y(i) = &(i j

-W(j)

(6)

6T denotes the distance between two consecutive sample points. Consequently the number of points in a clock period is AT/GT =K and the total number of points per repetition length is KM=N. The digitized version of the modified PRBS in Eqn. 5 may be written: p(i) 0

for i=O, K, 2K,... for every other i

(7) { This signal will be called a 1: K duty cycle PRBS. The full K : K PRBS can be described as the convolution of a 1: K duty cycle PRBS with a simple rectangular window, W, with width K6T = AT: X-l p(i) = [II0 4(i - 041) (8)

4(i) =

where: 1 0

w(l) =

for l=O,...,K-1 for l=K,...,N-

(9)

1

The input signal in Eqn. 8 may be generalized w(l): x(i) = C4(i I

using an arbitrary function f(l) instead of the window

-W(l)

(10)

The output signal in Eqn. 6 is redefined accordingly: y(i) = cx
-j)h(j)

Define the response of the chromatographic the injection profile) by: c(i)

= Ch(j

(11) system M j), the chromatogr~)

to the input unction

- l)f(l)

I

Rearranging y(i) = &(i i

(f(l), (12)

the detector signal in Eqn. 11 using the associativity property of convolutions: -#r(j)

C(C4(i-j-l)f(l))h(j) j f = C4(i -j)( Ch(j - Qf(r)) =f4(i - j)c(:) =

i

(13)

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The output signal may thus be considered chromatogr~. In matrix form:

4(N- 1)

4(l) q(2)

4(O) 41)

4(j)

4t.i - 1) .

I

*

.

1..

.

.

.

of the 1: K duty cycle PRBS with the

4(-k)

q(l)

q(N-k+l)

@I

q(N-k+2)

q(3) .. .

q(N-k+j)

..,

4(j+l) .

I..

c?(N‘1) q( fv*- 2)

11,

,Yw-

40)

as a convolution

’

I

49 41) 42) ** C(J)

4(b),\ c(N- - 1)

q(N-k-l)

...

n

In short:

(14)

y=Q.c Finding the response c is tantamount

to finding the inverse of Q. This will be our next aim.

One sample point per clock period

In the classical routine case K = 1 thus matrix formed by cyclic rotation of a PRBS multiplication) simply the transpose of that levels + 1 and - 1 (denoted by + and -). [+

+

+

0

0

+

o+++oo+ + o+++oo

o\+

o+o+++o +

0

+

+

+

+004-o++ f 0 ,+

0

+

0

+

0

0

1

+

0

=40

0 oooo+oo 0 0 \o 0

+ + + + -

+ + + + - + - + - +

0

0

0

0’

0

+

0

0

0

0 0

0 0

0 0

+ 0

0 +I

0

o+ooooo oo+oooo

x(i) I= q(i)=p(i). It is easily verified that the inverse of a with one point a clock period is (apart from an addition and matrix. The addition factor is avoided by using a PRBS with For example 7 clock periods a sequence: -

-

+ -

+ + + +

+ + + -

+ + + +

+ +

+ +

Summarizing: 1st row of the input matrix = PRBS values: 1st calm of the inverted matrix ( * 4) :

+

+

+

0

+

+

+

-

0

+

0

+

-

Briefly:

(15)

n

Original Research Paper

161

where J is a matrix with all elements equal to 1. Resolving c: c = Q-‘y = (2.Q - J)r/4y A PRBS with levels - 1 and + 1 will be denoted by p*:

(16)

p*=2p-1 Writing the entries of c as a summation and using Eqn. 16:

49 = C [2P(j - i) - 1]/4y(i) j =

Cp*(i - i)/4yfj) i

= Cz
(17)

i

The sequence z(O), . . . , z(N - 1) in the above equation is short for the first column of Q-‘. Because Q-’ is cyclic the other columns are defined by shifting the index of z(j). For a PRBS with a repetition-length of M clock periods and one sample point a clock period the general form of z( j> is analogous: z(j)

1)/2)1

=p*(j)/[(M+

(18)

Thus:

R,,(i)=2/(M+l)Cp*(j_i)p(j)=(~ ~:it~,._.,,_, i The same results may be obtained using the theory of Had~ard Example

of theinverse of a

(19)

matrices 12,7,10,131.

1: 2 duty cycle PRBS

Taking a seven-clock period PRBS with two sample points a clock period and setting every second sample point in each clock period to the lower level leads to the following matrix:

+o+o+ooooo+ooo o+o+o+ooooo+oo

oo+o+o+ooooo+o ooo+o+o+ooooo+

Q=

+ooo+o+o+ooooo o+ooo+o+o+oooo oo+ooo+o+o+ooo 0

0

0

+

0

0

0

+

0

+

0

+

oooo+ooo+o+o+o

ooooo+ooo+o+o+ +00000+000+0+0

o+ooooo+ooo+o+

o+ooooo+ooo+o ,~+o+ooooo+ooo+

0

0

(20)

Chemometrics and Intelligent Laboratory Systems w

162

Straigh~o~ard

Q-1

=i

matrix inversion results in:

0

0 -

0 0 +

+ 0 -

0 + 0 -

-

0

0 -

-

0

0 0

-

0 0 + 0 -

-

-

0

0

0

0

0

0 +

-

0

0

-

+

-

-

+

+

0

0

+ 0 +

0

0 + 0 + 0 -

f

0 0 + 0 + 0 + 0

+

0

+

+

0

0

+

+ 0

0 f 0

0 +

0

0

+

-

0

0

-

+

0

0

+

+

0

0 +

0

-

0 + 0

0 + 0 + 0 0

+

0 + 0 -

0 + 0 + 0 + 0

0 0 0 +

+ 0 0 0 + 0 0

0 0 +

0 + 0 +

+ 0

+ 0

0 + 0 0 0 + 0 0 + 0 +

+

0

0 +

+ 0

0 0 0 + 0 0 + 0

+ 0 0 0 + 0 0 +

* l/4

(21)

The inverse is a three-level, but still cyclic, matrix. Summarizing:

fu&PRBS values 1st row of the input matrix 1st column of the invertedmatrix

(*4)

.. :

++ +o

++ +o

++ +o

:

+0

+0

+O

00 00 -0

00

++

00

00

t-0

00

-0

t-0

-0

llte inverse of a i : K duty cycle PRBS

From the inverse of the 1: 2 input cycle in the previous paragraph, the inverse of the 1: K input cycle can be expected to be described by: z(i)

p*(i)/[(M+ o

=

1)/2]

for i=O, K, 2K,... for every other i

(22)

Analogously to Eqn. 7 we define:

4*(i) =

p*(i)

for i=O, K,2K,... for every other i

o

Thus: z(i)

=4*(i)A(M+

Correlation K,,(i)

of this sequence with the 1: K duty cycle PRBS:

= Cr(j-i)q(j) i =

I)/21

2/(M+

= C2/(M+ j l)p*(j-i)p(j)

l)s*(j-i)q(j) for j, (j -i)

= 0, K, 2K,. . .

for every other j, (j-i) =

1 0

for i = 0 for every other i

H

Original Research Paper

Correlating

163

z with the general detector signal of Eqn. 13 using Eqn. 25:

R,,(k) = Cz
k)y(i)

i

=

Cz(i i

=

’ i

C( j

=

k)(C4(i -_i)c(i)) J

Cz(i-k)*(i--j))c(jf i

CR,,(k -j)c(.i) =c(k)

(26)

i

Apart from noise contributions, the cross-correlogram equals the classical chromatogram obtained with a single injection with the shape of the arbitrary function f. Correlating the signals in Fig. la-d with the three-level duty cycle PRBS (Fig. 10 does indeed give satisfactory results.

for the signal

Ca~eq~nces

to noise ratio

If, for whatever reason, a correlogram with more than one sample point per clock period is preferred, it is advantageous to correlate the detector signal with a three-level 1: K duty cycle PRBS to maintain maximum resolution although a poorer signal to noise ratio will be obtained. However, smoothing this correlogram afterwards with a moving average of K sample points leads to the same result as obtained from at once using correlation with a full W: K) PRBS. The relation between the full PRBS and the inverse of the (1: K) sequence is obtained from Eqn. 16 and the equivalence of Eqn. 8: K-1 p(i)

=

l/2 + l/Zp*(i)

= l/2 + l/2

C

q*(i

-l)w(f)

I=0 K-l

=1/2+1/4(M+l)

C

(27)

z(i-l)w(l)

I=0

Correlation

of the detector signal with the full PRBS yields: K-l

Rp,(k)

=

Cp(i-k)y(i) i

= C l/2+

1/4(M+

i

= 1/2Cy(i)

+ 1/4(&f+

1) C

z(i-k-Z)w(l)

I-O

1)~~~~~z(i-k-r)y(i))w(I)

i

i K-l

=

1/2Cy(i) i

+ 1/4(M+

1) C c(l+k)w(l) I=0

(28)

Because w is a rectangular window the second term in Eqn. 28 is merely a moving summation over K sample points (the first term is only an offset value). Indeed smoothing the correlograms in Fig. lCa-Cd results in the correlograms in Fig. lAa-Ad. It must be emphasized that any digital filtering procedure can be applied to the correlogram Rzy, not necessariIy the brute force smoothing incorporated in correIation with a full PRBS. If the actual input pattern is not a perfect PRBS and is known, for instance, by the use of a pre-column detector, it may be tempting to use this information in the deconvolution by cross-correlating

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n

the output with the ‘real’ input. If the input satisfies Eqn. 10 the result will be analogous to Eqn. 28 with the window w replaced by the single injection pattern f:

The result is again unnecessarily distorted due to the supplementary correlation. Aspects of using the actual injection profile in the deconvolution of detector signals in CC will be discussed in a forthcoming paper of the series by Engelsma et al. [9].

RESULTS AND DISCUSSION

As mentioned in the theoretical section the method merely prevents unnecessary correlogram and illustrations are therefore rather trivial.

I

4

0

smoothing of the

120

30

150

time""/ ~IockpeE]

Fig. 2. Correlograms of a headspace analysis of methanol,. . . , n-hexanol in water at 46 o C (approximately 1 x lo-’ mol l- ’ each in the liquid phase) on a wall-coated 50 mX0.23 mm open tubular column (Chrompack CP-SIL-SCB, chemically bonded dimethylpolysiloxanel at 110 o C with FID detection. Nitrogen at 300 kPa (3 bar) was used as carrier. See ref. 12 for further details on the experimental setup. Clock period = 4 s (the full PRBS was used as injection pattern); sequence length = 127 clock periods (508 s); detector sampling rate 2.5 Hz. The detector signal is correlated with (above) the full PRBS, (middle) the two-level duty cycle PRBS and (below) the 1: 10 three-level duty cycle PRBS. Part of the baselines enlarged ten-fold.

n

t 0

Original Research Paper

165

I

7 tim:,

pcxm2i,

28

35

Fig. 3. Correlograms of an injection sequence generated by electrochemical modulation of hydroquinone (10W6 mol 1-l) with UV detection. No analytical column was used. Clock period = 5 s; modulation duty cycle, 2 s:S s (only during 2 s of each high PRBS clock period 0.9 V is applied, during the other parts of the high periods as well as the low periods, -0.3 V, see ref. 9 for further details); sequence length = 31 clock periods (155 s); detector sampling rate 4 m The detector signal is correlated with sequences analogous to Fig. 2 Gitll, two-level, three-level 1: 20 duty cycle PRBS).

Fig. 2 shows correlograms of a headspace analysis of a mixture of alcohols on a capillary column. A relatively large clock period was used and the resolution of the first two peaks is unsatisfactory if the detector signal is correlated with the full PRBS but is almost complete if correlated with a three-level 1: K PRBS like the one in Fig. 1C. Correlation of the detector signal with a two-level 1: K sequence results in a baseline disturbance repeating each clock period (Fig. 2, middle). As the injection pattern was a full PRBS this phenomenon reveals the presence of ghost peaks [12] (compare to Fig. 1Bc). The algorithm may be of help in (self-)checking procedures for CC systems. One may consider for instance a simplex algorithm acting on the tuning of inlet valves or inlet pressures thereby enduing the variance of the baseline. If only a non-retained test component is present, this procedure could minimiie imbalances in the injection system. With retained components ghost peaks due to column nonlinearities may occur and the impact of minimizing these by modulation of the input pattern is not yet known but may be an interesting feature. The performance of an electrochemical concentration modulator as an injection device depends strongly on the electrode potential, tune constant and clock period [9]. Correlation of the detector signal (which is the actual injection pattern since no column was used) with the full PRBS only reveals a broad virtual injection due to memory effects (Fig. 3, upper part). Optimization of the modulator is a lot easier if the indi~dual ghost peaks can be identified in the properly de~nvoluted signal (Fig. 3, lower part). As

Chemometrics and Intelligent Laboratory Systems

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, 0

n

3

60

30

time/

90

120

150

[clockperiod]

Fig. 4. Correlograms of simultaneous HPLCC analyses of 3 different samples on a 75 mmX4.6 mm Hypersil 5 pm MOS column with UV detection at 2.54 nm (eluent methanol/water, 84% v/v, 1 ml/min). The sample corresponding to the first part of each correlogram contains both phenanthrene (0.30 mg l-*1 and anthracene (0.06 mg 1-t). The second sample contains only phenantrene (0.28 mg l-r), the third only anthracene (0.07 mg I-‘). Clock period = 6 s; duty cycle, 2 s:6 s for each sample; sequence length = 127 clock periods (762 s); detector sampling rate 1 &, the detector signal is correlated with sequences analogous to Fig. 2 (full, two-level, three-level 1: 6 duty cycle PRBS). Experimental setup as in ref. 8.

a duty cycle was applied in the modulation signal the strong periodic disturbances may be expected after correlation with a two-level 1: I( sequence (compare to Fig. 1Bd). The results of simultaneous analyses [S] of three different samples are depicted in Fig. 4. Two compounds of the same sample are hardly separated and the peak top of the second ~rn~und is only observable if deconvoluted with an 1: K sequence. Although a duty cycle is intrinsic to simultaneous CC the effect of deconvolution with the two-level 1: K sequence is less abundant because the chromatographic dispersion is large compared to the clock period.

LIST OF SYMBOLS

~nt~uous t, 7, 7’

T

variables (arbitrary) time repetition time (sequence length) of the PRBS

n

Original Research Paper

AT 6T Discrete variables 4 i, k, 1, m N M K Periodic functions P P* 4 4* x Y ; Tran:ent 3 W

167

basic clock period of the PRBS sampling period time

(arbitrary) indices = T/6T, the number of samples per PRBS sequence = T/AT, the number of clock periods per PRBS sequence = AT/6T, the number of samples per PRBS clock period (repetition time T) PRBS with levels 0 and 1 PRBS with levels. - 1 and 1 at the start of each clock period q =p, else q = 0 at the start of each clock period q * = p *, else q * = 0 input sequence output sequence inverse sequence correlation of u and u functions (0 after time T) impulse-response of the chromatographic system arbitrary function specified weighing function or window response to an input f (chromatogram)

ACKNOWLEDGEMENTS

The authors are indebted to one of the referees for directing our attention to similar techniques in imaging spectroscopy reported by Fenimore and co-workers [lO,ll]. This work was supported by The Netherlands Foundation for Technical Research (STW), Future Technical Science Division of The Netherlands Organization for the Advancement of Pure Research (NWO).

REFERENCES 1 H.C. Smit, Random input and correlation methods to improve the signal-to-noise ratio in chromatographic trace analysis, Chromatographia, 3 (1970) 515-518. 2 M. Kaljurand and E. Kiillik, Application of the Hadamard transform to gas chromatograms of continuously sampled mixtures, Chromatographia, 11 (1978) 328-330. 3 H.C. Smit, Correlation chromatography; A semi-continuous trace analysis method with potential possibilities in process monitoring and process control, Joumal A, Journal of Automatic Control, 24 (1983) 145-148. 4 J.R. Valentin, G.C. Carle and J.B. Phillips, Determination of methane in ambient air by multiplex gas chromatography, Analytical Chemistry, 57 (1985) 1035-1039. 5 E. Kiillik and M. Kaljurand, Correlation gas chromatography of thermal degradation products of thermostable polymers, Analytica Chimica Acta, 181 (1986) 51-56. 6 J.M. Laeven, H.C. Smit and J.C. Kraak, Differential cross-correlation HPLC a method of establishing small concentration differences in samples of similar origin, Analytica Chimica Acta, 194 (1987) l&-24. 7 M. Kaljurand and E. Kiillik, Computerized Multiple Input Chromatography, Ellis Honvood Ltd., West Sussex, 1989. 8 H.C. Smit, C. Mars and J.C. Kraak, Simultaneous correlation chromatography, a new technique applied to calibration in high-performance liquid chromatography, Anafytica Chimica Acta, 181 (1986) 37-49.

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