Fourier-Laplace transform, response

We denote the fluctuations of the number density of the monomers of component j at a point r and at a time t as pj r,t). With this definition we have pj(r,t))=0. In linear response theory, the Fourier-Laplace transform of the time-dependent mean density response to an external time dependent potential U r,t) is expressed as ... 

A wave may be viewed as a unit of the response of the system to applied input or disturbances. These responses could be in terms of physical deflections, pressure, velocity, vorticity, temperature etc., those physical properties relevant to the dynamics, showing up in general, as function of space and time. Any arbitrary function of space and time can be written in terms of Fourier-Laplace transform as given by,... 

Just as the longitudinal response, Eq. (A2.7) can be solved exactly for the Fourier-Laplace transform gi(i ) in terms of ordinary continued fractions to yield... 

There exist powerful simulation tools such as the EMTP [35]. These tools, however, involve a number of complex assumptions and application limits that are not easily understood by the user, and often lead to incorrect results. Quite often, a simulation result is not correct due to the user s misunderstanding of the application limits related to the assumptions of the tools. The best way to avoid this type of incorrect simulation is to develop a custom simulation tool. For this purpose, the FD method of transient simulations is recommended, because the method is entirely based on the theory explained in Section 2.5, and requires only numerical transformation of a frequency response into a time response using the inverse Fourier/Laplace transform [2,6,36, 37, 38, 39, 40, 41-42]. The theory of a distributed parameter circuit, transient analysis in a lumped parameter circuit, and the Fourier/Laplace transform are included in undergraduate course curricula in the electrical engineering department of most universities throughout the world. This section explains how to develop a computer code of the FD transient simulations. 

A numerical calculation code of the Fourier/Laplace transform is prepared in commercial software such as MATLAB, MAPLE, or even Excel. Therefore, it is easy to carry out an inverse transform provided that all frequency responses are given by the user. Similarly, if the user can prepare the time response of a transient voltage, for example, as digital data of a measured result, then the user can easily obtain its frequency response using the software. However, it is better to first understand the basic theory of the Fourier/Laplace transform. 

Considerable effort has gone into solving the difficult problem of deconvolution and curve fitting to a theoretical decay that is often a sum of exponentials. Many methods have been examined (O Connor et al., 1979) methods of least squares, moments, Fourier transforms, Laplace transforms, phase-plane plot, modulating functions, and more recently maximum entropy. The most widely used method is based on nonlinear least squares. The basic principle of this method is to minimize a quantity that expresses the mismatch between data and fitted function. This quantity /2 is defined as the weighted sum of the squares of the deviations of the experimental response R(ti) from the calculated ones Rc(ti) ... 

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... 

The traditional way is to measure the impedance curve, Z(co), point-after-point, i.e., by measuring the response to each individual sinusoidal perturbation with a frequency, to. Recently, nonconventional approaches to measure the impedance function, Z(a>), have been developed based on the simultaneous imposition of a set of various sinusoidal harmonics, or noise, or a small-amplitude potential step etc, with subsequent Fourier- and Laplace transform data analysis. The self-consistency of the measured spectra is tested with the use of the Kramers-Kronig transformations [iii, iv] whose violation testifies in favor of a non-steady state character of the studied system (e.g., in corrosion). An alternative development is in the area of impedance spectroscopy for nonstationary systems in which the properties of the system change with time. 

Thus, it becomes apparent the output and the impulse response are one-sided in the time domain and this property can be exploited in such studies. Solving linear system problems by Fourier transform is a convenient method. Unfortunately, there are many instances of input/ output functions for which the Fourier transform does not exist. This necessitates developing a general transform procedure that would apply to a wider class of functions than the Fourier transform does. This is the subject area of one-sided Laplace transform that is being discussed here as well. The idea used here is to multiply the function by an exponentially convergent factor and then using Fourier transform technique on this altered function. For causal functions that are zero for t < 0, an appropriate factor turns out to be where a > 0. This is how Laplace transform is constructed and is discussed. However, there is another reason for which we use another variant of Laplace transform, namely the bi-lateral Laplace transform. 

Gangwall et al. [47] were the first to apply Fourier analysis for the evaluation of the transport parameters of the Kubin-Kucera model. Gunn et al. applied the frequency response [80] and the pulse response method [83] in order to determine the coefficients of axial dispersion and internal diffusion in packed beds from experiments performed at various Reynolds numbers. Bashi and Gunn [83] compared the methods based on the analytical properties of the Fourier and the Laplace transforms for the calculation of transport coefficients. MacDonnald et al. [84] discussed the applications of the method of moments to the analysis of the profiles of skewed chromatographic peaks. When more than two parameters have to be determined from one single run, the moment analysis method is less suitable, because only the first and second moments are reliable (see Figure 6.9). Therefore, only two parameters can be determined accurately. 

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. 

When the system responds in a linear range, however, the same information can be obtained from steady-state frequency response measurements . On the other hand, in the nonlinear response region this is no longer the case, and it is inappropriate to derive frequency response results from a Fourier or Laplace transform of transient response results. 

Laplace transform becomes a Fourier transform g 0 ) we can convert the transform domain model to a frequency-response model as shown in Table 5.1 for both g (5) and g (5). Therefore, the frequency-domain model becomes... 

Few articles on receptivity present a qualitative view of particular transition routes created by not so well-defined excitation field (see e.g. Saric et al. (1999)). Such approaches do not demonstrate complete theoretical and /or experimental evidence connecting the cause (excitation field) and its effect(s) (response field). Here, a model based on linearized Navier- Stokes equation is presented to show the receptivity route for excitation applied at the wall. This requires a dynamical system approach to explain the response of the system with the help of Laplace-Fourier transform. 

The function F(ico)=F f(t) is the image function of f(t). After rearrangement of Eq. (6A.2), using the laws of the Laplace-Fourier transformation (cf. Appendix 4E), the measurable surface tension response Ay(t) of the system to the area disturbance Ain A(t) is obtained. 

From the definitions given in Section 4.4.2, it is apparent that the interfacial impedance can be calculated from the perturbation and response in the time domain, in which the excitation can be any arbitrary function of time. In principle, any one of several linear integral transforms can be used (Macdonald and McKubre [1981]) to convert from the time domain into the frequency domain, but the two most commonly used are the Laplace and Fourier transforms ... 

The time response of the frequency dependence explained in Section 1.5.2 is calculated by a numerical Fourier or Laplace inverse transform in the following form [1,20] ... 

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