Impulse function

Common time domain input functions 3.4.1 The impulse function... 

Fig. 3.11 Response of a first-order system to an impulse function of area A.

Figure 2.3. Depiction of unit step, time delay, rectangular, and impulse functions.

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... 

The (unit) impulse function is called the Dirac (or simply delta) function in mathematics.1 If we suddenly dump a bucket of water into a bigger tank, the impulse function is how we describe the action mathematically. We can consider the impulse function as the unit rectangular function in Eq. (2-20) as T shrinks to zero while the height 1/T goes to infinity ... 

The impulse function is rarely defined in the conventional sense, but rather via its important property in an integral ... 

The Laplace transform of the impulse function is obtained easily by taking the limit of the unit rectangular function transform (2-20) with the use of L Hospital s rule ... 

In mathematics, the unit rectangular function is defined with a height of 1/2T and a width of 2T from -T to T. We simply begin at t = 0 in control problems. Furthermore, the impulse function is the time derivative of the unit step function. 

Example 2.10 What is the time domain response C (t) in Eq. (2-27) if the change in inlet concentration is (a) a unit step function, and (b) an impulse function ... 

The time variations of the effluent tracer concentration in response to step and pulse inputs and the frequency-response diagram all contain essentially the same information. In principle, any one can be mathematically transformed into the other two. However, since it is easier experimentally to effect a change in input tracer concentration that approximates a step change or an impulse function, and since the measurements associated with sinusoidal variations are much more time consuming and require special equipment, the latter are used much less often in simple reactor studies. Even in the first two cases, one can obtain good experimental results only if the average residence time in the system is relatively long. 

An impulse function (F) is also useful in some problems where the force exerted on bounding surfaces is desired ... 

To answer this question, away from the context of PF, consider a characteristic function / ( ) that, at t = b, is suddenly increased from 0 to 1/C, where C is a relatively small, but nonzero, interval of time, and is then suddenly reduced to 0 at t = b + C, as illustrated in Figure 13.6. The shaded area of C(l/C) represents a unit amount of a pulse disturbance of a constant value (1/C) for a short period of time (C). As C - 0 for unit pulse, the height of the pulse increases, and its width decreases. The limit of this behavior is indicated by the vertical line with an arrow (meaning goes to infinity ) and defines a mathematical expression for an instantaneous (C - 0) unit pulse, called the Dirac delta function (or unit impulse function) ... 

From a well-known result of calculus, the definite integral on the right-hand side is s/n so M is just equal to the quantity of diffusing substance. The present solution is therefore applicable to the case where M grams (or moles) per unit surface is deposited on the plane x=x at t=0. In terms of concentration, the initial distribution is an impulse function (point source) centered at x=x which evolves with time towards a gaussian distribution with standard deviation JlQit (Figure 8. 13). Since the standard deviation is the square-root of the second moment, it is often stated that the mean squared distance traveled by the diffusion species is 22t. 

The impulse function is an infinitely high spike that has zero width and an area of one (see Fig. 9.1a). It is a function that cannot occur in any real system, but it is a useful mathematical function that will be used in several spots in this book. 

F. UNIT IMPULSE FUNCTION. By definition, the z transformation of an impulse-sampled function is... 

Defining in this way permits us to use transfer functions in the z domain [Eq. (18.57)] just as we use transfer functions in the Laplace domain. G,, is the z transform of the impulse-sampled response of the process to a unit impulse function <5( . In z-transforming functions, we used the notation =... 

AF = AF/Ii, where = P + fm ) hw is the axial impulse function at the combustor entrance. It is seen that the magnitude of AF increases with increasing values of ER and is clearly higher for elliptic nozzles than for round nozzles. 

The points along a Rayleigh-Mikhel son Line are characterized by constancy of impulse function (P+mq)A as well as of mass velocity but not, in general, of temperature (Ref 2, p 101). Movement along such a line can represent flow without losses in a constant-area duct, with heat addition or removal and consequent changes rn temperature and hence in entropy. The Rayleigh-Mikhel son line is curved when plotted in the s,T (temperature vs entropy), the s,h (enthalpy vs entropy), and the dimensionless temperature vs density planes (Ref 2, p 173)... 

Examining a given concentration profile in greater detail, we may construct the model illustrated in Figure 2.3, which approximates the bell shape by many hypothetical impulse functions of different heights located Ax distance increments apart. Simple diffusion is a linear process that fits the two criteria discussed earlier (Chap. 1). According to the superposition principle, the impulse functions may be treated completely independently if there is no solute-solute interaction (e.g., dimerization). The future of the total profile will be described accurately by a summation of the behavior of each independent impulse. 

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 ... 

Table 32 Hypothetical dissolution data for XYZ1234 obtained by deconvolution using Treatment A(NF) as impulse function. Medians.

The mismatch of surpassing 100% absorption of the active metabolite is probably due to method constraints in combination with the immediate release data, as the deconvolution method requires data from a formulation with zero order absorption for the impulse function, e.g. an oral solution (oral bolus input) the immediate release formulation only provides an approximation to the required properties. 

The true ideal impulse function is difficult to introduce into the system. In most practical cases, the.ttacer inputs are pulsed. Careful precautions should be taken to ensure good pulse injections. If the pulse is imperfect, the analysis of the RTD curve becomes much more difficult. Quite often the RTD curve obtained from an arbitrary input is transported to the one which could correspond to a pulse input, using the convolution integral... 

Observations of metabolic cross-correction provided the rationale for cellular replacement, achieved primarily through allogeneic hematopoietic stem cell or bone transplantation (HSCT) (Prasad and Kurtzberg, 2008). More recently, the use of neural stem cells (NSC) implanted in the brain of patients with late-infantile neuronal ceroid lipofuscinosis has been contemplated (Pierret et al., 2008) but there are no reports as yet of its potential efficacy. Within the central nervous system there must be proper integration of donor cells, and differentiation into appropriate cell types. As specialized cell types within the nervous system elaborate neurotransmitters and are involved with conducting electrical impulses, functional differentiation may be a major hurdle for the neurodegenerative LSDs. 

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