Digital differentiation

We are accustomed to computation in the decimal system, but this need not be so. Computers, for instance, are based on a binary system, the binary numbers being 0 and 1. Digitalization of an analog signal is nothing more than the conversion of an electric potential to such a number. In Table 3-3, some decimal numbers are converted into digital numbers. 

One of the advantages of analog differentiation is the continuous computing of the signal. No information is lost. In contrast to this, digitalization requires to be the signals quantized, that is, data is only measured at certain time intervals. In the space between two data points no information is available, and it is assumed that the two points can be connected by a straight line. But this is not always the case (Fig. 3-41). Therefore, 

The steps necessary for digital (or numerical) differentiation are similar to those for analog differentiation. After AD conversion the effective signals have to be smoothed - or the data separated from the noise in another way -, then differentiated, and finally printed, or plotted, or analog DA converted and smoothed again (Fig. 3-42) for recording. 

Depending on the circumstances, either all or only some of the modules will be necessary. 

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The composition of permeates was deduced refractomet-rically by digital differential refractometer (Atago DD-5, Atago Co. Ltd., Japan). 

To calculate the rates of styrene polymerization with the help of experimental data on conversion-time the method of the digital differentiation using five points (16) was applied. In table V the values of Kp/Kt for different conversions are represented, the values have been obtained during the styrene polymerization in the presence of different initial concentrations of BzjOa at temperature of 70° C, Up to conversion of - 40% the ratio of Kp/Kt somewhat increases smd does not depend on the initial concentration of initiator. But at more high stages of polymerization the difference of MW of polymer being formed reveals itself and the ratio of Kp/Kt increases as quickly as the initial concentration of initiator decreases. Dependences of Kp/Kt on the con-... 

Figure 3-42. Flow sheet of a first-order digital differentiator. To get higher-order derivatives the data must be stored, and differentiated again.

The core of a digital differentiator is the computer. Prices vary considerably depending on quality. Nowadays, at least a 16- or 32-bit computer with a minimum of 512 kB, or preferably 1 MB, internal memory (RAM), two floppy drives, and, if possible, a Winchester drive (hard disk) is required. This equipment usually provides derivatives of high quality. 

There are different reasons for the numerical manipulation of data. One of these is data reduction others are filtering and smoothing, magnification of the amplitude of the spectra or of certain spectral features, and standardization. After the preceding computations, different algorithms for digital differentiation can be applied. 

There are many proposals in the literature for curve fitting and digital differentiation, and the most important ones are listed in Table 3-9. Morrey [119], for example, fitted the digital data of a segment of a spectrum to a quartic equation and differentiated this convolute. A least-squares analysis of this equation gives rise to a set of linear equations. Peak positions were obtained without consideration of the type of distribution function representing the peak. 

As already described, there are many methods of digital differentiation. Each method has its advantages and disadvantages - nevertheless all algorithms are based on approximating polynomials. Therefore, deviations of the true signals may occur and a compromise between SNR and resolution is necessary. In order to estimate the results, it is important to know which method to use for differentiation (see Table 3-9). The possibility of artifacts can then be minimized. 

MD D (S-G 11) corn means for example digital differentiation according Savitzky-Golay 11 point polynomial shifting of the signal (AA) corrected. 

In Fig. 4-44 c, the result of digital differentiation using the 11-point polynomial according to the Savitzky-Golay method is demonstrated. It is in good agreement with... 

For measuring pressure differences at elevated pressures, when the pressure is varying or when the two pressure points are far apart, a portable digital differential-pressure indicator is required. 

The differential pressure test is used for the vacuum test and is also used to test gas pipelines for leaks. In this test, the leakproofness of the sewer pipeline is compared with the absolute leakproofness of a heat-insulated control bottle (reference vessel). This is carried out with digital differential pressure measuring equipment. The pre-requisite for this is an absolutely leakproof pipe system. Both the pipe material and the couplings must be gastight. 

An additional problem is the usual introduction of scatter when one tried to digitally differentiate data as implied in Eq. (126). 

In the anuran, Xenopus, the prechondrogenic pattern shows features of the Shubin-Alberch bauplan, although digits differentiate early in relation to the digital arch . Amphibian patterns are briefly compared with amniotes (bird). 

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