Laplace transformation excitation

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... 

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 presence of the h(z, P) factor makes Eq. (7.44) different from a Laplace transform of C(z). If the z dependence of h(z, P) is ignored,(34 36) then calculated concentrations of fluorophore near an interface derived from collected fluorescence are approximations. Also, the P dependence in the tf1,11 causes the integral in Eq. (7.44) to differ from the form of a Laplace transform even after the excitation term is factored out. 

If the excitation electric field is an s-polanzed evanescent field instead of the above p-polarized example, then wH 11 [ = wHJI(z)] does not depend upon p. Therefore, an approximate C(z) can be calculated from the observed fluorescence (P) (obtained experimentally by varying 0) by ignoring the z dependence in the bracketed term in Eq. (7.45) and by inverse Laplace transforming Eq. (7.44) after the ,(0, /J) 2 term has been factored out.,37 39)... 

Furthermore, it is also not necessary to discuss different excitations in detail as long as we restrict ourselves to the linear response regime. There it holds that the response to any excitation allows the calculation of the response to other excitations via the convolution theorem of cybernetics.213 In the galvanostatic mode, e.g., we switch the current on from zero to /p (or switch it off from 7p to zero) and follow IKj) as a response to the current step. The response to a sinusoidal excitation then is determined through the complex impedance which is given by the Laplace transform of the response to the step function multiplied with jm (j = V-I,w = angular frequency). 

After instantaneous excitation of A, the normalized decay of A is given by R(t) = NA(t)/NA0). According to (3.10), its Laplace transformation, related to the lifetime xA, defines the relative quantum yield of luminescence ... 

Here the kernels R and R are different from their analogs in (3.266a) and (3.266b) because they account not only for forward (Wi) but also for backward transfer to the excited state WB [in line with recombination to the ground state with a rate WR(r). Both kernels are given by their Laplace transformations ... 

The Laplace transformations of the kernels, representing the bimolecular recombination to the ground and excited states, are... 

Whatever the excitation, the transformation of the response from the frequency to the time domain (Fig. 11.21) is done with the inverse Fourier transform, normally as the FFT (fast Fourier transform) algorithm, just as for spectra of electromagnetic radiation. Remembering that the Fourier transform is a special case of the Laplace transform with... 

Thus, application of delta function is equivalent to exciting all the circular frequencies with equal emphasis. This is the basis of finding the natural frequency of any oscillator via impulse response. When the oscillator is subjected to an impulse, all frequencies are equally excited and the system dynamics picks out the natural frequency of vibration, leaving others to decay in due course of time. It is noted that this result also applies to Laplace transform and we are going to use it often by replacing time by space and circular frequency by wave numbers. 

The description of the local solution and other details of selecting the Bromwich contours are given in Sengupta et al. (1994) and Sengupta Rao (2006). The near-held response created due to wall excitation is shown in these references as due to the essential singularity of the bilateral Laplace transform of the disturbance stream function. While the experiments of Schubauer Skramstad (1947) verihed the instability theory, the instability theory is incapable of explaining all the aspects of the experiments or... 

Following the above, let us explore the relation between the original and the image of bilateral Laplace transforms in relation to the evolution of small disturbances in boundary layers. In particular, one would be interested in the behaviour of the original in the neighbourhood of the exciter (x = 0) as determined by the image 4> y, a) for a —> oo according to the Tauber s theorem. 

In Fig. 2.32, the bilateral Laplace transform, (f>onormal derivative, have been shown that correspond to the imposed velocity boundary conditions of Fig.2.31. It is clearly evident that oo is much larger than cf) and this information is used here to compare different types of free stream excitations in the next subsection. 

Let us consider now the more complicated case of an arbitrary excitation. The approach to be followed in this case uses the Laplace transform. From Eqs. (16.249) through (16.251), the transformed equations for the displacement and stress can be written as... 

Free vibration, the motion that persists after the excitation is removed, is governed by Eq. (17.85), in which the applied transverse force has been made zero. Let us assume a solution of the form Uy(x, t) = /(x)exp(io)0 where f x) specifies the lateral displacement and is the angular frequency of the motion. For low loss viscoelastic materials, the free vibrations can be assumed to be quasi-harmonic, and therefore the complex modulus in the equation of motion can be used. The Laplace transform of Eq. (17.85) gives... 

In other words, the dynamics is equivalent to two successive static evolutions, the second one starting from initial conditions (aA(r), BdA(T). (3W>A(r) ). Using the Laplace transform of the system (4) with the initial condition (6), it is possible to express the dynamic amplitude of the excited state after the sudden change as... 

A useful technique for treating reversal methods in chronopotentiometry (and other techniques in electrochemistry) is based on the response function principle (2, 17). This method, which is also used to treat electrical circuits, considers the system s response to a perturbation or excitation signal, as applied in Laplace transform space. One can write the general equation (2)... 

The maximum entropy method can handle Laplace transforms such as those found in pulse fluorometry, without restricting the validity of the solution or suffering from any instability. It allows the recovery of the distribution of exponentials describing the decay of the fluorescence (i.e., inverting the Laplace transform), which is, in turn, convolved by the shape of the excitation flash. Also, it can determine the background level and amount of parasitically scattered radiation. 

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... 

Z(w) = 1/wC, and in the s-domain Z(s) = 1/sC. The Laplace transforms of some very important excitation waveforms are very simple for example, for a unit impulse it is 1, a unit step function 1/s, a ramp 1/s, etc. That is why the excitation with, for example, a unit impulse is of special interest examining the response of a system. In the extended immittance definition, calculations with some nonsinusoidal waveforms become very simple. Even so, Laplace transforms are beyond the scope of this book. 

It is seen that the last term of this equation causes mixing of different modes. Since the equations have been linearized they are soluble by an exponential solution or by Laplace transform methods. For the present discussion we shall assume that the modes with v 0 are much less excited than the fundamental mode, and we then obtain the equations ... 

Up to this point we have described methods in which impedance is measured in terms of a transfer function of the form given by Eq. (56). For frequency domain methods, the transfer function is determined as the ratio of frequency domain voltage and current, and for time domain methods as the ratio of the Fourier or Laplace transforms of the time-dependent variables. We will now describe methods by which the transfer function can be determined from the power spectra of the excitation and response. 

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, ... 

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