fG 1 :0.32
vola
(38)
We can now define an effective response speed, S,, derived from Equation 38 and for which Equation 37 will apply, thus S, = a/uo 1:0.32/fa
(39)
and in terms of the measurable critical frequency, A, this gives us S,
‘v
0.32/0.55fc N 0.58/f,
This means that Figure 15 and Equation 37 can be used for any recorder provided that the effective full scale response time defined by Equation 40 is employed. In Table 111, values of S, are compared with the full scale response times previously obtained from Equation 13, and they are seen to be about 2 to 3 times the latter. Also the values of S, are about twice the manufacturers’ span step response times, and it is suggested that a minimum factor of 2 should be adopted in the absence of relevant frequency response data. The last two sets of figures in the Table were obtained from published chromatographic results (12) for which we know that d&/21: 16. The relevant minimum retention times found were 11 and 30 sec, respectively, giving the ratio figures shown. These results are also in reasonable agreement with the suggested factor of 2.0. (12) S. Dal Nogare and J. Chiu, ANAL.CHEM., 34, 890 (1962).
It should perhaps be emphasized at this point that our discussion has been concerned with a chromatographic peak of height equal to the full scale deflection of the recorder. For smaller peaks, a smaller value of tmwould be obtained, but not in direct relationship to the peak heights. This is because the settling time effect becomes more pronounced for the smaller peaks, as can be seen from the two sets of frequency response results (50% and 10% full scale) for the Honeywell recorder in Figure 4. Finally, we must again consider the effect of the delay time. As outlined in the previous Section (Figure 13), provided that the delay time does not exceed one fifth of the response time, the error introduced is negligible. It might be expected that the introduction of the effective response time, which makes allowance for the settling time effect, would also compensate to some extent for delay time effects. By redefining the allowable maximum delay time in terms of S,, thus d,,,
‘v
0.2 S,
(41)
we retain the correct factor for use when the delay time dominates and s = S, but relax the specification for most practical cases for which the values of S, will be about twice the manufacturers’ span step response times. On this basis, the delay time figures given in Table I are therefore acceptable. RECEIVED for review May 19, 1969. Accepted July 15, 1969. This work was made possible by the award of a Research Fellowship to one of the authors (1.G.M.) by the Shell Group of Companies in Australia.
Detection and Resolution of Overlapped Peaks for An On-Line Computer System for Gas Chromatographs A. W. Westerberg’ Control Data Corp., 4455 Eastgate Mall, LaJolla, Calif. 92037 Peak detection and resolution methods are discussed for an existing computer system which handles several concurrently operating on-line gas chromatographs. The use of digital filtering to smooth the input data and then of heuristic criteria to distinguish peaks from noise solves the detection problem; the design criteria for the required digital filter are given together with a minimum sampling rate. After a detailed study of two commonly used resolution techniques of triangulation and perpendicular drop, the paper concludes these methods are too inaccurate and are poorly reproducible. It then presents an alternate method of resolving peaks which uses model curve fitting and covers the details for a Gaussian, a modified Gaussian, and a general tabular data model which may be used with this approach. The paper concludes with a sample trace analyzed by the system.
THISPAPER is the second of a series presenting a software system developed on the CDC 1700 computer for on-line gas chromatographic data input and analysis. An earlier paper (1) presented a real-time periodic sampling algorithm. This paper will discuss the detection and resolution of peaks in the nonbaseline segments of a trace. (1) A. W. Westerberg, ANAL..CHEM., 40, 1595 (1969). 1770
COMPONENT DETECTION
One problem for the computer in gas chromatography data analysis is the detection of the number of components contributing peaks within a nonbaseline segment of a trace. A human interpreter can manage to detect peaks from noise quite successfully because he can view the whole trace segment and make a judgement based on this perspective. Unfortunately this problem is much more difficult for the computer. Two approaches will be discussed here. The first technique, used successfully for NMR (2, 3), assumes a precise peak shape model. One first guesses the number of components by finding only the obvious major peaks in the trace. Using this number of components, one easily least squares fits the data to the model (see Peak Resolution, Curve Fitting Section). If the fit is poor, as evidenced by the sum of squares of differences between the data and fitted model, one assumes added components in those local Present address, Dept. of Chemical Engineering, University of Florida, Gainesville, Fla. 32601
(2) W. D. Gwinn,A. C. Luntz, C. H. Sederholm, and R. Millikan, J . Cornp~ifafional Phys., 2, 439 (1968). (3) W. Keller, T. Lusebrink, and C. H. Sederholm, J. Chem. Phys., 44, 782 (1966).
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
tions where the fitted model and data differ excessively. One then refits the expanded model to the data and repeats until the fit is satisfactory. In this manner one solves both the detection and resolution problem at the same time. Another advantage is that component peaks can be found which were not evident to the eye. Unfortunately the technique has the requirement of a well defined peak shape model, and it therefore cannot be used in gas chromatography for detection as successfully as in NMR. The second method which can be used to detect the number of peaks in a nonbaseline curve segment is to clean up the data by filtering and then count inflection point pairs; one pair occurs for each visually detectable peak or shoulder in a trace. A limit for detectability results in that two peaks sufficiently overlapped cease to produce two pairs of inflection points and will be detected as one. (See Detectability Limit Section.) Applying the method requires therefore two steps-data filtering and then the counting of true inflection points. First filtering will be considered. Filtering Chromatograph Data. The initial problem in filtering experimental data is the determination of a suitable cut-off frequency for any filter (digital or analog) to be used. Only digital filtering will be considered here. Nyquist criteria (4) state the sampling rate for a signal must be greater than twice the highest frequency contained in the signal if one wishes to reconstruct completely the original signal. It is not the desire here to reconstruct the signal but only to obtain sufficiently smoothed sampled data so inflection points may be found. Also the data are then to be used to provide area estimates for quantitative analysis as discussed later so some minor distortions can be tolerated here too. The following analysis produced a rule of thumb for properly sampling and filtering chromatographic data without significantly distorting it. Approximate a chromatographic peak by the Gaussian model : y
=
H exp(-(t - i)2/2u2)
(1 1
where H = peak height i = time of peak maximum u 2 = variance of peak t = time y = detection output voltage Letting
7 = YIH x = t - i
(2)
q = exp(-x2/2u2),
(3)
Equation 1 becomes :
and the Fourier Transform of Equation 3 is ( 5 , 6 ) : C ( U ) = 5(q(x))=
r
o
