Finite inverse Fourier transform

The temperature-dependent coupling spectrum is the Fourier transform of the bath response function in Eq. (4.202), and it usually has a certain width proportional to the inverse of the correlation time. The time-dependent modulation spectrum is the finite-time Fourier transform of the modulation function, eft). 

The intensity of scattering at finite q, on the other hand, reflects the concentration fluctuations that exist on a more local scale. In the case of a single-component system, as discussed in Section 4.1, the finite-angle intensity data can be converted, through an inverse Fourier transform, to a radial distribution function g(r). With a two-component system a comparable general procedure is not available, and information on the structure is derived usually by comparing the observed intensity data, on the q plane, with expressions derived from theoretical models. 

In practice, the sampling in each projection is limited, so that f(x, y) is bounded according to -AP. < X < A/2 and -AH < y < AH. Also, physical limitations ensure that only a finite number of Fourier components are known, so that F(u, v) is bounded according to —NH < u < NH and —NH < v < N/2. With reference to the discrete inverse Fourier transform (26.47), and in view of the limiting bounds, Eq. (26.64) becomes... 

Then, to analyze the obtained current, a Fourier transform is applied and the responses at the fundamental, co, and harmonic, 2m, 3m, 4m,..., frequencies are obtained. Next, the current responses at the fundamental and harmonic frequencies are extracted by an inverse Fourier transform. Harmonics up to the eighth order were obtained. Analysis of the kinetic parameters is carried out by comparison of the experimental and simulated data. Theoretical ac voltammograms were simulated using classical numerical simulations of the diffusion-kinetic process using an implicit finite-difference method [658, 659] with a subsequent Fourier analysis of the simulated data. An example of the comparison of the experimental and simulated data is shown in Fig. 15.5. In this case, oxidation of ferrocenmethanol appeared reversible, and a good agreement was found with the simulated data for the reversible process. 

If fit) is defined everywhere in the interval —c interval converges (i.e., is finite), the Fourier transform F(v) also will converge. In addition, an inverse Fourier transform will regenerate the original function ... 

Let the discrete spectrum, which consists of the coefficients of u(k) and v(k), be denoted by U(n) and V(n), respectively. The low-frequency spectral components U(n) are most often given by the most noise-free Fourier spectral components that have undergone inverse filtering. For these cases V(n) would then be the restored spectrum. However, for Fourier transform spectroscopy data, U(n) would be the finite number of samples that make up the interferogram. For these cases V(n) would then represent the interferogram extension. 

Tool RF burst waveforms serve as a convenient means for determining the ejection resolution limits that are due to effects intrinsic in the operation of the cubic trap. More general computed waveforms (59) are obtained via inverse discrete Fourier transformation. These computed waveforms are just a linear superposition of a finite number of RF bursts. As a result, it is proposed that the best performance obtained with RF bursts anticipate the best performance obtained with computed waveforms,... 

An expression for solute concentration versus angular displacement at the column outlet requires inversion of this solution back to the 0 domain, a procedure which cannot be performed analytically. A fast Fourier transform algorithm was used to perform the inversion numerically (21). Equations 1 and 2 were also solved using a finite difference algorithm. 

Deconvolution of the observed decay pulse was necessary because of the finite width (approx. 1.9ns FWHM) of the excitation pulse. This was carried out using a Fourier transform method (4). Despite the non-trlvlal computational requirements and the need to filter out the high frequency components of the Fourier division before taking the Inverse transform (5) It was felt that this method was the most direct way of generating the true Impulse response function of the system. The resulting fluorescence decay curves were fit to a single (or double) exponential. The phase plane plot method of Demas (6) and others (2), was also used for deconvolution but with less success. 

To determine the convolution for a discrete sample we follow Eq. (26.43) and find the product of two finite Fourier transforms Z = X T and take the inverse of the result, according to... 

Fig. 6. Paramagnetic critical scattering in La2Cu04. Energy-integrating scans across the two-dimensional rod of scattering measure the Fourier transform of the instantaneous spin-spin correlation function in La2Cu04 above T. The peak width (corrected for finite instrumental resolution) is the inverse correlation length for antiferromagnetic spin fluctuations. The dashed lines show the experimental resolution function, and the solid lines are the results of fits to a Lorentzian lineshape convolved with the resolution for three different temperatures. From Kcimer et al. (1992).

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