8-function pulses

This is a prelude to the important impulse function. We can define a rectangular pulse such that the area is unity. The Laplace transform follows that of a rectangular pulse function... 

Array programming is used here which allows the graphing of the axial profile. Closed-end boundary conditions are used for the first and last segments. Also included is a PULSE function for simulating tracer experiments. Thus it should be possible to calculate the E curve and from that the reaction conversion obtained on the basis of tracer experiment. The example CSTRPULSE should be consulted for this. 

The full listing of the MADONNA program is shown here both to illustrate the use of array programming and also the use of the PULSE function. 

PULSE functions could be used for studying the axial dispersion of heat or, alternatively, the axial dispersion of mass (see DISRE). 

The program includes a Pulse function, which has been added to the X phase. This is used to represent the effect of sudden addition of solute to this phase at TIME=Tswitch. 

This can be done in two ways Either use the pre-programmed PULSE function (see the program CSTRPULSE for an example) or use an IF-THEN-ELSE statement to turn a stream on and off (see example CHROM). 

Adjust the pulse/function generator with the aid of the oscilloscope to create a sine wave as shown in Figure S-4 with an amplitude accuracy of +5 percent or better. 

Switch the pulse/function generator from the sine wave to a square wave signal as shown in Figure S-5. 

An example is the relatively simple moving average filter. In case of a digitized signal, the values of a fixed (odd) number of data points (a window) are added and divided by the number of points. The result is a new value of the center point. Then the window shifts one point and the procedure, which can be considered as a convolution of the sipal with a rectangular pulse function, repeats. Of course, other functions like a triangle, an exponential and a Gaussian, can be used. 

Note the unit pulse function is used for modelling NMR spectra. 

The impulse is a special case of the pulse function in which to - 0 but the area A/ under the impulse function remains constant and finite (Figure 7.26). Thus, for the impulse function ... 

Figure 1. Time distribution of the number of photons observed by BATSE in channels 1 and 3 for GRB 970508, compared with the following fitting functions (a) Gaussion, (b) Lorentzian, (c) tail function, and (d) pulse function. We list below each panel the positions tp and widths ap (with statistical errors) found for each peak in each fit. We recall that the BATSE data are binned in periods of 1.024 s.

The response to the step-function for a non-reacting gas is given by integrating the response of the pulse function as Eq. (10). 

A graphical representation is shown in Fig. 4.31. For the extended pulse function of width X, the solution given by Eq. (12) is obtained in the same manner. 

Fig. 4.31 Response to the extended pulse function for t=0 s (broken line) and t=0.01 s for a non-reacting gas (upper solid line) and a reacting gas ( =0.3, lower solid line).

The kinetics captured in disordered systems like polymers, glasses and poly-cristalline structures has been often described in terms of continuous relaxation times and exciton diffusion at recombination centers [10]. Assuming a <5— pulse function, the temporal data are best fitted by a monomolecular kinetic equation,... 

In the methods previously described, the signal imposed on the concentration of the steady inlet flow to the open reactor w as a step f unction or a pulse function (step up followed by step down). These signals enter the analysis only through the initial conditions imposed on Eq. (4), for example. Another interesting input signal is the sinusoidal variation of c,f,... 

Commonly encountered forcing functions (or input variables) in process control are step inputs (positive or negative), pulse functions, impulse functions, and ramp functions (refer to Figure 44). 

Consider an impulse function. The impulse function is also known as a Dirac delta function and is represented by S(t). The function has a magnitude oo and an area equal to unity at time t = 0. The Laplace transform of an impulse function is obtained by taking the limit of a pulse function of unit area ast 0. Thus, the area of pulse function HT = 1. The Laplace transform is given by... 

Fig. 4.1.2 The rectangular pulse function (a) and the sine function (b) form a Fourier pair, transformation,...

The Fourier transform of an infinite short pulse function h(t) = Kb(t), where 5(f) is Dirac s delta function, equals//(jco) = K, that is, it contains all the frequencies with the same amplitude K. Such a function caimot be realized in practice and must be substituted by a pulse of a short duration At. However, such a function does not have uniform response in the Fourier (i.e., frequency) space. The Fourier transform of such a function, defined as h(t) = 1 for r = 0 to To and h(t) = 0 elsewhere, equals... 

For use in interpreting infinitesimal pulse-function distributions, or chromatograms (cf. Section IV, C, 2), with longitudinal dispersion controlling, differentiation of Eq. (113d) yields... 

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