## Correlation Chromatography

flow of the column is rapidly switched between the sample and the eluent, according to a (pseudo) random pattern. A cross-correlation function (correlo-gram) of the random input signal and the resulting very complex detector signal is also identical to the impulse response (chromatogram) of the chromatographic system, if the input satisfies certain conditions. In other words, the correlogram is identical to a conventional chromatogram obtained by a pulse-shaped injection.

In CC the sample is usually injected according to a pseudo random binary sequence (PRBS) pattern p(i), where i is the discrete time. A PRBS is a binary noise with a specific length M, the sequence length, of 2n — 1 periods (n is a positive integer) controlled by a clock; the only levels are + 1 and — 1, or 1 and 0. The M clock periods correspond to I = 2n — 1 injections. The periodic nature of the PRBS input signals yields low estimate variance of the estimation of statistical quantities such as correlation functions if taken over an integer number of sequences.

The signal power of a PRBS, determining the final intensity of the detector response, is much higher than that of an impulse-like injection function with similar amplitude, and it is equally spread over the frequency range of the chromatographic system. This 'white noise' property is essential for the application of CC. In addition the levels can be used to control simple on/off valves, corresponding to injection of sample or mobile phase.

The detector signal yi is built up of noise-free chromatograms h(i) shifted in time, according to the PRBS pattern, plus detector noise n(i):

The PRBS length is chosen to be equal to or longer than the time duration of the comparable chromato-gram obtained from a single injection. The detector signal becomes circular after one PRBS sequence, the so-called presequence. The calculation of a correlo-gram comparable with a similar chromatogram requires the inverse of the PRBS, defined as the function p-i(i), producing a Kronecker delta function A(i) after circularly cross-correlating with p(i):

Owing to the special properties of a PRBS, the inverse calculated from a PRBS with one point per period and levels 1 and 0 gives the same PRBS, but with levels + M/I and — M/I instead of +1 and 0. Cross-correlating the detector signal y(i) with the inverse p^1(i) results in a correlogram with a reduced noise level. In the calculations, non-correlated (white) noise is assumed:

= ™ X- \p-i(i + k) ■ I X [h(i) ■ p(i-i)]+n(i)

Considering the levels + M/I and — M/I for p ^(i # k), p~ 1(i + k) ■ n(i) can be replaced by

Adding M noncorrelated points for every k results in noise with a standard deviation (SD) of M1/2 times the original SD of the noise:

With one point per period, the detector signal can also be cross-correlated with the original PRBS. This produces a comparable correlogram multiplied by a factor I/M.

A similar derivation can be made in the continuous time domain. It has been shown that the resulting cross-correlogram is identical to a chromatogram obtained from an injection with a profile equal to the autocorrelogram of the input sequence. For this reason this autocorrelogram is sometimes referred to as the 'virtual injection' profile. Sometimes a 'true' random binary sequence is used. In that case other de-convolution methods are necessary, such as decon-volution in the Fourier domain.

The correlation procedure can be continued for an arbitrary integer number of sequences. Theoretically, noise not correlated with the input pattern can be reduced to any desired level - but at the cost of time -assuming that the chromatographic system is stationary and that enough sample is available. The noise reduction in only one sequence is about a factor of 10 to 20.

Correlation techniques can be applied in different column separation methods, applications in gas chromatography (GC), liquid chromatography (LC) and capillary zone electrophoresis (CZE) are known. Particularly in LC and CZE, the detection limit can be a problem and correlation techniques in principle offer possibilities to increase the signal-to-noise ratio considerably without preconcentration of the sample. Another feature is the possibility of using CC for continuous monitoring; CC allows a fast and almost continuous updating of the value of the varying concentrations to be monitored. However, the moving average effect, typical for the correlation procedure, limits the highest frequency that can be monitored. CC permits to monitor a frequency about a factor of 2 higher than conventional chromatography. A considerable improvement is possible, if the multiple injection (PRBS) input is maintained, but the correlation procedure is replaced by non-linear fitting. The time-varying concentra tions are described as functions of the time and a number of parameters are optimized in the fitting procedure. A detector signal is calculated using the parameters, the known PRBS and known peak shapes; the squared differences with the real detector signal are minimized. The maximum frequency is not determined by the chromatogram length but by the peak width, orders of magnitude better.

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