Pulse function Laplace transform

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

The rectangular pulse can be generated by subtracting a step function with dead time T from a step function. We can derive the Laplace transform using the formal definition... 

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... 

For the design of the actively compensated RF pulse, experimental and numerical determination of the response function h(t) of the circuit is necessary. We should also keep in mind that modification to the circuit, such as probe timing, insertion or removal of RF filters, and so on, can alter h(t). In practice, it is convenient to measure the response y t) to a step excitation u(t) instead of that to the impulse excitation. By performing Laplace transformation to... 

Figure 22C summarizes the procedure for calculating the programming pulse shape v(t) from the target pulse shape bi(t) and the step response u(t). In addition to the measurement of y t), we need to perform a number of data operations such as multiplication, division, Laplace and inverse Laplace transformations. All of these functions can be performed on the software. 

Figure 23 Calculation of the shape of the actively compensated pulse can be carried out on the software. (A) shows the real (red line) and the imaginary (green line) component of an example of the target pulse shape t>,(f). Its leading and the trailing edges have a cosine shape with a transition time of 1.25 xs in 50 steps, and the width of the plateau is 5 ps. (B) Laplace transformation B(s) multiplied by the Laplace transformed step function U(s). (C) It was then divided by the Laplace transformation Y(s) of the measured step response y(t) of the proton channel of a 3.2-mm Varian T3 probe tuned at 400.244 MHz to obtain V(s). (D) Finally, inverse Laplace transformation was performed on V(s) to obtain the compensated pulse that results in the RF pulse with the target shape. Time resolution was 25 ns, and o = 20 was used for the Laplace and inverse Laplace transformations.

Next, bi(t) was Laplace transformed into B(s), and then multiplied by the Laplace transformation U(s) of the step function u(t). The result B(s)U(s) is displayed in Figure 23B. In this example, the step response y(t) was measured for the 1H channel of a Varian 3.2 mm T3 probe tuned at 400.244 MHz with a time resolution of 25 ns, and Laplace transformed into Y(s). By dividing B(s)U(s) by Y(s), the function plotted in Figure 23C was obtained, from which, by performing inverse Laplace transformation, the programming pulse shape v(t) was finally obtained, as shown in Figure 23D. The amplitude and the phase of the complex function v(t) give the intensity and the phase of the transient-compensated shaped pulse. 

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... 

When the boundary condition is a Dirac S t) function at z = 0 [which is different from the Dirac S z) pulse at z = 0 of the open-open boundary condition], the solution obtained is different. It can be derived using the inverse Laplace transform [21,22] ... 

The Laplace transform of the unit pulse function of duration A is... 

The stochastic Liouville equations are readily solved for the time-dependent density matrix elements pu (e.g., through Laplace transforms) the latter may then be used in turn to develop expressions for the polarized fluorescence or absorption difference signals. The initial values of the density matrix elements under 5-function pulsed excitation are given by... 

One can envision three types of perturbation an infinitesimally narrow light pulse (a Dirac or S-functional), a rectangular pulse (characteristic of chopped or interrupted irradiation), or periodic (usually sinusoidal) excitation. All three types of excitation and the corresponding responses have been treated on a common platform using the Laplace transform approach and transfer functions [170]. These perturbations refer to the temporal behavior adopted for the excitation light. However, classical AC impedance spectroscopy methods employing periodic potential excitation can be combined with steady state irradiation (the so-called PEIS experiment). In the extreme case, both the light intensity and potential can be modulated (at different frequencies) and the (nonlinear) response can be measured at sum and difference frequencies. The response parameters measured in all these cases are many but include... 

Electron Drift in a Constant Electric Field. As an example, let us consider the system discussed in the time-of-flight section. In this system, charge carriers are generated close to the injecting contact and drift to the collecting contact under the force of a constant electric field. As discussed above, the current response on a laser pulse has a constant value of Jph = AQIr for 0 < t < t, and drops instantly to zero at t=r. The input signal is a delta function and the output response is a step function. Linear-response theory shows that the system function H s) is the Laplace transform of the impulse response function h(t). In our example ... 

Now, to find the current/voltage relationship in the time domain, we only need to substitute V(5) for the particular excitation of interest and carry out the inverse Laplace transform for the entire equation. However, first we need to know how the particular excitation we want to apply looks in the Laplace domain. Here is an example of how to do it for a voltage step. In the time domain it can be expressed as v(0 = Heaviside(t) Vi where the function Heaviside(/) takes a value 1 if i > 0 and 0 otherwise. Vi is the value of voltage applied during the step. The Laplace transform of this function, V s) =. [Heaviside(i) Vj] = VJs. To find the Laplace current of the above mentioned network to pulse excitation, we substitute this V(x) into Eq. (4) and perform inverse Laplace transformation of the resulting equation. We obtain, ... 

The various terms in the equation are as follows i is the current, n is the number of electrons transferred, F is the Faraday, A is the electrode area, C is the concentration, D is the diffusion coefficient, and t is the time. Thus the technique may be used to estimate, among other things, the charge transport parameter. The reader interested in the solution of the Fick s law equation using the Laplace transformation should consult Ref. 6. A closely related technique is chronocoulometry, in which the excitation function is still the potential pulse, but instead of monitoring the current, the integrated charge is monitored as a function of time. This entails less error as a cumulative measurement is made. The equation for chronocoulometry is... 

The unit step function corresponds to the integral of the unit pulse function with respect to time. The Laplace transform of the unit step function is y(j = 1/. ... 

In the case of a dispersion model, for instance, transfer function G(p) is given as Eq. (6-59). Transfer function is defined as the ratio of the Laplace transform of the elution concentration curve and that of the input concentration curve, the latter of which is a constant in the case of impulse input. Laplace parameter, p, is a complex variable but if a response curve, C(r), is transformed by using Eq. (6-8) by assuming p as a real parameter, then the resultant C(p) gives a transfer function G(p) by dividing by the size of pulse, M, in a real plane. C(p) is then compared with the solution of basic equations obtained in a Laplace domain. 

For several types of time function, such as pulse, step, ramp, wave changes, the Laplace transformations are shown in Table 5.1. The time functions belonging to a particular Laplace transform are given in Table 5.2. 

Using Laplace transform tables the function can be transformed back to the time domain, the result for the response to a pulse in the input is the so-called impulse transfer function ... 

Impulse Function. A limiting case of the unit rectangular pulse is the impulse or Dirac delta function, which has the symbol 8(r). This function is obtained when ty while keeping the area under the pulse equal to unity. A pulse of infinite height and infinitesimal width results. Mathematically, this can be accomplished by substituting h = lltyy into (3-22) the Laplace transform of b(i) is... 

We know how to find the z transformations of functions. Let us now turn to the problem of expressing input-output transfer-function relationships in the z domain. Figure 18.9a shows a system with samplers on the input and on the output of the process. Time, Laplace, and z-domain representations are shown. G(2, is called a pulse transfer function. It will be defined below. 

The pulse response function is the output function/j(t) caused by the action of the input impulse function (Dirac function). It is applied for determination of the particular forms of the Laplace transmittance. It can be obtained by applying the Laplace inverse transformation to the transmittance Eq. (2.41) ... 

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