Although on-lme digital-computer analysis of reactor dynamics is possible73 and perhaps will be prevalent in the future, it has been the practice until now to perform digital analysis off-line. As shown in Table 6 11, the digitizing process may be on-line (creating the proper magnetic-tape format for a computer) or off-line The off-line digitizing may be automatic from an f-m tape or semiautomatic, as in the case of manually operated strip — chart readers whose electrical output punches cards In any event the result is that one or more sequences (x,, у,, etc ) of variables at time spacings At are generated m a form suitable for input to a digital computer

The selection of a digitizing interval, At, and of a total duration of the data collection, T, is discussed in Sec 6-7 4 It will suffice here to note that the digitizing interval determines the upper frequency limit, fmax = l/(2At), of the analysis and the total duration is associated with the frequency resolution (1 e, minimum frequency interval between independently determined spectral values) and accuracy of results The quotient, T/At, is the number of digital values per signal and may be 103 to 10s in typical experiments

A number of versatile programs are available to users of the various commercial computers for statistical analysis of large quantities of data Typical of these are the Biomedical Computer Programs,115 a series of 42 programs that are useful not only in biomedical research but also in any field requiring analysis of data for frequency counts, variances, correlations, and related functions Table 6 14 lists the

Ckn(T) = iGlNkg cos (сот — в) (6 36)

iGl and 0 may be determined from as few as two values, Ckn(0) and Скп(яУ2со)

Whether the digital approach discussed here or the continuous-signal approach discussed in previous sections should be used depends on a variety of factors, some of which are mentioned in Table 6 15 The digital approach has been more common in recent years as digitizing costs and computer rental costs per data point decrease and as demands for computer versatility (see Table 6.14) increase

available computer outputs from just one of these 42 programs (BMD-02T) if one has, for example, three related system variables All possible time-correlation functions and their Fourier transforms are computed with x(t) regarded as an input signal Evidently there is sufficient versatility and generality to permit adaption to almost any type of transfer function experiment

Even more versatile than the Biomedical Computer Programs series is the BOMM system of programs 1 1 6 Here the user describes in few-word control statements the step by-step data-handhng operations to be performed on a time series, such as finding the mean, doing a cross correlation, or plotting an answer These control statements call in standard subprograms to the computer that perform all the detailed calculations Thus individualized data — processing needs can be satisfied with BOMM, although more effort is required to list the control statements than in the Biomedical Computer Programs

Although a computer is a virtual necessity for per­forming the required analysis on random fluctuations, it is not necessarily required in the special case of obtaining transfer functions from strip-chart recordings of sinusoidal oscillations where the signal-to-noise ratio is good 15 76 For example, the transfer function of the BORAX-4 reactor1 could be obtained to an accuracy of ±5% by chart reading and simple hand calculations, even though the root-mean-square oscillatory amplitude, lGlNk0/(2)^, ex­cited by k0 sin cot was only about twice the root-mean — square boiling noise In this technique the digitally de­termined (Eq. 6.9) normalized cross-correlation function, Ckn, of the reactivity and the power [which is lGlNk0 sin (cot + в) + noise] is equated to its theoretical expectation

However, for applications requiring many repetitive de terminations of a single function, the special-purpose continuous analyzer is strongly entrenched The consider­ations of Table 6 15 also apply to frequency domain analysis, as discussed in the following sections

As the use of on-line digital computers becomes more prevalent and accepted in reactor operation, on-line digital analysis of noise may be expected to be used competitively with other methods Cohn683’73 has demonstrated the ability to sample noise as often as every 0 5 Rsec and to do on-line correlations with a digital computer. Polarity correlating (ie., replacing the noise amplitude value by +1 or —1 for its fluctuation about an average of zero in computing correlation functions) was found useful in this application.