Dirac impulse

The function g(x) is named impulse response of the system, because it is the response to an unit pulse 5(x) applied at =0 [2]. This unit impulse 5(x), also called Dirac impulse or delta-function, is defined as... 

The superposition integral (1) corresponds to a division of the input signal u(x) into a lot of Dirac impulses 5 x). which are scaled to the belonging value of the input. The output of each impulse 5fx) is known as the impulse response g(x). That means, the output y(x) is got by addition of a lot of local shifted and scaled impulse responses. 

Fig. 1 (right) shows upside an example of an input. The marked points are some of the scaled Dirac impulses. The belonging scaled impulse responses are shown downside. 

If the true value can be considered to be error-free (otrue —> 0), r(x) degenerates into a Dirac impulse N(ptrue,Q). Considering real samples and the bias 8 = ptrue — x, the estimate of Eq. (9.18) is given by... 

For the case of a Dirac impulse injection, the solution of the Thomas model was given by Wade et al. [10] ... 

In the special case the input consists of a single Dirac impulse, the first sampling time can be different from zero. Then the resulting weighting function must be appropriately adjusted (ref. 27). In any other case, however, the method applies only, if the first time point is t = 0. 

Here, the response functions of the diffusion equation for a number of discrete input signals were calculated based upon the solution for a Dirac impulse input signal. 

Fig. 4.33 Shift of concentration pulses (Dirac impulses) for an aspect ratio of 0.1 in the water-acetone system (arrows indicate the individual peak maximum the hatched line connects the peak maxima).

Here, the response functions of the diffusion equation for a number of discrete input signals were calculated based on the solution for a Dirac impulse input signal. A set of transformation relations for the injected mass M and the axial coordinate was used to obtain the solutions for a reacting gas directly from the solutions of a non-reacting gas ... 

Cavazzini et at. showed that the above Monte Carlo model of nonlinear chromatography is equivalent to the Thomas kinetic model of second order Langmuir kinetics [70]. The solution of the Thomas model for a Dirac impulse injection is given by Eq. 14.65. When the chromatographic process is modeled at the molecular level with the stochastic model, the Thomas model becomes [70] ... 

The application in [24] is to celestial mechanics, in which the reduced problem for consists of the Keplerian motion of planets around the sun and in which the impulses account for interplanetary interactions. Application to MD is explored in [14]. It is not easy to find a reduced problem that can be integrated analytically however. The choice /f = 0 is always possible and this yields the simple but effective leapfrog/Stormer/Verlet method, whose use according to [22] dates back to at least 1793 [5]. This connection should allay fears concerning the quality of an approximation using Dirac delta functions. 

Mathematically,/(l) can be determined from F t) or W t) by differentiation according to Equation (15.7). This is the easiest method when working in the time domain. It can also be determined as the response of a dynamic model to a unit impulse or Dirac delta function. The delta function is a convenient mathematical artifact that is usually defined as... 

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 Dirac delta function represents an intense impulse of very short time duration. An example is the hit1 of a baseball by the bat From a mathematical point of view this function can be defined by the relations... 

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) ... 

Impulse. The impulse is defined as the Dirac delta function, an infinitely high pulse whose width is zero and whose area is unity. This kind of disturbance is, of course, a pure mathematical fiction, but we will find it a useful tool. 

Theoretically, the best possible input pulse would be an impulse or a Dirac function S, y. The Fourier transformation of is equal to unity at all frequencies. 

Let us now define an infinite sequence of unit impulses or Dirac delta functions whose strengths are all equal to unity. One unit impulse occurs at every sampling time. We will call this series of unit impulses, shown in Fig. 18.4, the function /, . 

Since all tracer entered the system at the same time, t = 0, the response gives the distribution or range of residence times the tracer has spent in the system. Thus, by definition, eqn. (8) is the RTD of the tracer because the tracer behaves identically to the process fluid, it is also the system RTD. This was depicted previously in Fig. 3. Furthermore, eqn. (8) is general in that it shows that the inverse of a system transfer function is equal to the RTD of that system. To create a pulse of tracer which approximates to a dirac delta function may be difficult to achieve in practice, but the simplicity of the test and ease of interpreting results is a strong incentive for using impulse response testing methods. 

Such an operator is indeed the first derivative of the familiar impulse or Dirac S function. It can, like the S function, be represented as the limiting... 

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