Deconvolution procedure, numerical

Typically, in measurements of time-resolved luminescence in the time regime of tens of picoseconds, data obtained from 10 to 20 laser shots are averaged to improve the signal-to-noise ratio and to minimize the effects of shot-to-shot variations in the laser pulse energy and shape. Once the reliability of the data has been ensured by application of the corrections described above and made necessary by detector-induced distortions, the time-resolved fluorescence data is analyzed in terms of a kinetic model which assumes that the emitting state is formed with a risetime, xR, and a decay time, Tp. Deconvolution of the excitation pulse from the observed molecular fluorescence is performed numerically. The shape of the excitation pulse to be removed from the streak camera data is assumed to be the same as the prepulse shape, and therefore the prepulse is generally used for the deconvolution procedure. Figure 6 illustrates the quality of the fit of the time-dependent fluorescence data which can be achieved. 

With a numerical deconvolution procedure, corresponding to the convolution method demonstrated above, Voigt widths for some measured analytical lines were determined. The results are collected in Table 2.1 and compared with calculated values corresponding to Equations 2.7, 2.11, and 2.13. 

Itagaki et al [76] have proposed that the decay kinetics observed for fluid solutions of poly(l-methoxy-4-vinylnaphthalene) (PMVN) may be explained in terms of two excimer species in addition to excited monomer. The proposal is based on variations in steady-state l,3-di(4-methoxy-1-naphthyl)propane (BMP) [77,78], presumes that the lower energy excimer comprises the normal structure with fully overlapped aromatic rings, whereas the second excimer is thought to comprise a partially overlapped structure. Steady state spectroscopy indicated a reasonable degree of evidence that three excited state species do indeed exist in the sterically hindered naphthalene systems. However the spectral and numerical deconvolution procedures adopted make the separation of intensity components used in the kinetic analysis, necessary for excited state assignment, rather optimistic. 

By means of numerical convolution one can obtain Xg t) directly from sampled values of G t) and Xj(t) at regular intervals of time t. Similarly, numerical deconvolution yields Xj(t) from sampled values of G(t) and Xg(t). The numerical method of convolution and deconvolution has been worked out in detail by Rescigno and Segre [1]. These procedures are discussed more generally in Chapter 40 on signal processing in the context of the Fourier transform. 

The above algorithm works well for pure compounds and simple mixtures, but it becomes increasingly difficult to assign all peaks properly when complex mixtures are to be addressed. Additional problems arise from the simultaneous presence of peaks due to protonation and alkali ion attachment etc. Therefore, numerous refined procedures have been developed to cope with these requirements. [102] Modem ESI instrumentation is normally equipped with elaborate software for charge deconvolution. 

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

Before comparing with experiment, however, the theoretical results at an energy of 40 meV (equivalent to room temperature) were convoluted with the energy resolution function of the detector used for the measurement. This procedure was adopted because deconvolution of the experimental data was found to be numerically unstable. The convoluted theoretical data were then normalized to the experimental data at zero Doppler shift to yield the results shown. The agreement between the convoluted theoretical results and experiment is extraordinarily good, extending as it does over more than three orders of magnitude. These results also reveal... 

Various procedures have been proposed for solving this equation. For example the pore size distribution may be assumed to be of log-normal form (Seaton et al., 1989), or alternatively integration may be replaced by summation and a numerical deconvolution approach adopted (Olivier et al., 1994). 

Although deconvolution is a wdl defined mathematical procedure, its a lication to fluorescence decay curves is attended with numerous difficulties owing to the counting enors and instmmental distortions that accompany sin e photon countii data. It is now generally accepted that least squares iterative reconvdution is the most satisfactory method of analymg nano cond decay data In its amplest... 

SE7 Mathematically inexact deconvolution. Numerical procedures such as numerical integration, numerical solution of differential equations, and some matrix-vector formulations of linear systems are numerical approximations and as such contain errors. This type of error is largely eliminated in the direct deconvolution method where the deconvolution is based on a mathematical exact deconvolution formula (see above). Similarly, the prescribed input function method ( deconvolution through convolution ) wiU largely eliminate this numerical type of error if the convolution can be done analytically so that numerical convolution is avoided. 

Various analytical procedures are applied to extract valence numbers from Lm spectra. They are based on a deconvolution of two superimposed and shifted single-peaked sub-spectra taken either from experimental or from numerical reference spectra. Within Av = +0.1 the different techniques yield the same results on identical spectra. An estimate of the absolute and relative uncertainties may be obtained from table 3. It lists the valence numbers from most of the Lnui spectra published from 1975 until 1986. These numbers have been worked out by more than ten different laboratories. Obviously the numbers extracted from identical systems agree fairly well in spite of the fact that they have been obtained with the use of quite different deconvolution techniques. Large systematic deviations (>0.1) from the average numbers (where available) should be attributed to different experimental results rather than to the specific valence determination procedure. 

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