Inverse Fourier-Laplace transformation

The inverse Fourier-Laplace transformation of l/(z — E(z)) provides the autocorrelation function (survival probability) represented in Figure 1.4. The... 

The eigenenergies Si and S2 are complex (< i and (021 are the duals of 0i) and 102) [8]. The inverse Fourier-Laplace transformation (Appendix A) and the Cauchy theorem lead to the time-dependent wavefunction... 

The complex energies k (k = 1,2,..., 5) are the roots of the polynomial Q(z) and the/j s are generalized oscillator strengths. The inverse Fourier-Laplace transformation provides the survival probability P t) of the initial state Is... 

The inverse Fourier-Laplace transformation ot the Green tunction in the case ot a weak coupling (A T) leads to... 

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. 

Using the Fourier-Laplace transforms (x, t) (k, s), we derive from the balance equations (8.40) and (8.42) expressions for sp ik, s) - pf k). Using the Fourier-Laplace inversion formula, we obtain a system of integro-differential equations. 

Previous sections dealt with the analytical development of Laplace transform and the inversion process. The method of residues is popular in the inversion of Laplace transforms for many applications in chemical engineering. However, there are cases where the Laplace transform functions are very complicated and for these cases the inversion of Laplace transforms can be more effectively done via a numerical procedure. This section will deal with two numerical methods of inverting Laplace transforms. One was developed by Zakian (1969), and the other method uses a Fourier series approximation (Crump 1976). Interested readers may also wish to perform transforms using a symbolic algebra language such as Maple (Heck 1993). 

Crump, K. S. Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation, J. Assoc. Comput. Machinery, 23, 89-96 (1976). 

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. 

An important technical development of the PFG and STD experiments was introduced at the beginning of the 1990s the Diffusion Ordered Spectroscopy, that is DOSY.69 70 It provides a convenient way of displaying the molecular self-diffusion information in a bi-dimensional array, with the NMR spectrum in one dimension and the self-diffusion coefficient in the other. While the chemical-shift information is obtained by Fast Fourier Transformation (FFT) of the time domain data, the diffusion information is obtained by an Inverse Laplace Transformation (ILT) of the signal decay data. The goal of DOSY experiment is to separate species spectroscopically (not physically) present in a mixture of compounds for this reason, DOSY is also known as "NMR chromatography."... 

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... 

Hence using the Bromwich integral (complex inversion formula for the Laplace transform) and the inverse Fourier transform we have... 

If the real part of p is zero, the Laplace Transformation becomes the Fourier transformation. The inversion formula of (4.4.1) consequently resembles the inverse Fourier transformation (4.1.2)... 

Integral transforms were discussed, including Fourier and Laplace transforms. Fourier transforms are the result of allowing the period of the function to be represented by a Fourier series to become larger and larger, so that the series approaches an integral in the limit. Fourier transforms are usually written with complex exponential basis functions, but sine and cosine transforms also occur. Laplace transforms are related to Fourier transforms, with real exponential basis functions. We presented several theorems that allow the determination of some kinds of inverse Laplace transforms and that allow later applications to the solution of differential equations. 

Substitution of (3.146) into (3.150) and inversion of the Fourier and Laplace transforms leads to the RD equation (2.3) with D = j l (t>- If we substitute (3.147) into (3.150) and invert the Fourier and Laplace dansforms, we obtain the reaction-telegraph equation ... 

At the beginning of this chapter, we quoted the Mellin-Fourier Inversion theorem for Laplace transforms, worth repeating here... 

Another method of getting the numerical inverse is by the Fourier series approximation. The Laplace transform pairs are given as in Eqs. 9.1 and 9.3, written again for convenience... 

These are the two building blocks to prove the Fourier-Mellin inversion theorem for Laplace transforms. 

This form is now suitable for deducing the Fourier-MeUin inversion formula for Laplace transforms. In terms of real time as the independent variable, we can write the Fourier integral representation of any arbitrary function of time, with the provision that fit) = 0 when / < 0, so Eq. C39 becomes... 

By replacing p by yro, the definition of is changed from time domain (inverse Laplace transform) to frequency domain (Fourier transform). One then characterizes the electrieal system by a representation in the Bode diagram of the modulus of the transfer function cf. Figure 3) ... 

Based on the MT model a method has been developed to calculate the distribution of capture rates from the measured photocurrent [68, 69]. In contrast to Ref [34], the method is based on a Fourier transform tecbnique with better numerical stability as compared to the inverse Laplace transform used in Ref [34]. 

Comparison with a Fourier or Laplace transform (see appendix) suggest that H(t) can be found using the inversion integral,... 

To go down a level, one applies either an inverse Laplace transform or an inverse Fourier... 

---