Numerical deconvolution

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. 

The deconvolution is the numerical solution of this convolution integral. The theory of the inverse problem that we exposed in the previous paragraph shows an idealistic character because it doesn t integrate the frequency restrictions introduced by the electro-acoustic set-up and the mechanical system. To attenuate the effect of filtering, we must deconvolve the emitted signal and received signal. 

As expected from the design of the experiment, the HPLC column packed with CSP 14 containing all 36 members of the library with tt-basic substituents separated 7t-acid substituted amino acid amides. Although encouraging since it suggested the presence of at least one useful selector, this result did not reveal which of the numerous selectors on CSP 14 was the most powerful one. Therefore, a deconvolution process involving the preparation of series of beads with smaller numbers of attached selectors was used. The approach is schematically outlined in Fig. 3-17. 

Fig. 17 FTIR absorbance spectrum of two-phonon processes in single crystalline a- Sg in the range 550-1000 cm, after [109], The strong bands in the range 800-950 cm result from combinations of components of the stretching vibrations. The insert shows a numerical deconvolution of the prominent spectral feature between 750-950 cm ...

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. 

This key paper was followed by a flurry of activity in this area, spanning several years." " "" A variety of workers reported attempts to deconvolute the temperature dependence of carbene singlet/triplet equilibria and relative reactivities from the influence of solid matrices. Invariably, in low-temperature solids, H-abstraction reactions were found to predominate over other processes. Somewhat similar results were obtained in studies of the temperature and phase dependency of the selectivity of C-H insertion reactions in alkanes. While, for example, primary versus tertiary C-H abstraction became increasingly selective as the temperature was lowered in solution, the reactions became dramatically less selective in the solid phase as temperatures were lowered further. Similar work of Tomioka and co-workers explored variations of OH (singlet reaction) versus C-H (triplet reaction) carbene insertions with alcohols as a function of temperature and medium. Numerous attempts were made in these reports to explain the results based on increases in triplet carbene population... 

As noted earlier, osmotic systems have been shown to provide good in vitro-in vivo correlations between the observed release rates. This has been shown explicitly for the core I devices described above [33], The data are shown in Figure 16, where the in vitro data are plotted along with the release curves from six devices administered to dogs. The animals were in the fed state at the time of administration and maintained that way throughout the duration of the experiment via the administration of —50 g of dog chow before device administration and every hour thereafter. The individual release curves shown in Figure 16 were obtained by numerical deconvolution of the plasma data with an oral solution dose given to the same dog. Clearly the in vivo and in vitro data are... 

After H0bs has properly been extracted (cf. Sect. 2.2.2), the effect of instrumental broadening can be eliminated by numerical deconvolution (see p. 38). If the peaks shall be modeled by analytical functions (Sects. 8.2.5.7-8.2.5.8), the consideration... 

In the presented form Eq. (8.13) is only valid, if Hj (s) is, indeed, constant over the whole angular range required for analysis. If this is not the case and numerical deconvolution is aimed at, the standard algorithm may be adapted by consideration of the fact that, in any case, the broadening is a slowly varying function of 29. 

According to the RG calculations, vahd for relatively low functionalities, the mean contribution of EV in the numerator and denominator of Eq. (20) should cancel for any branched structure in a good solvent. Therefore, ratio g for a star in a good solvent should be very close to gg, Eq. (21). Different experimental data included in [49] seem to support this conclusion. Croxton [50] carried out iterative deconvolution theoretical calculations for uniform stars with up to six arms of model lengths that yielded, g=f a result that is not in agreement with Eq. (21) for the considered range of low functionalities. On the other hand, Eq. (14) shows that the Daoud and Cotton theory gives or, approxi-... 

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. 

These properties carry back to the discrete formulation. We shall use both discrete and continuous formulations in this volume, changing back and forth as needs require. The continuous regime allows us to avoid consideration of sampling effects when such consideration is not of immediate concern. Deconvolution algorithms, on the other hand, are numerically implemented on sampled data, and we find the discrete representation indispensable in such cases. 

Even considering our brief treatment of this subject thus far, we may have enough information to foresee that difficulties could arise when we attempt to compute values of G from given values of A and B. Such difficulties do indeed occur. The problem of deconvolution has therefore been the subject of a vast literature spanning the numerous special fields of science. 

In preceding chapters we laid a foundation for the study of deconvolution. We presented several linear methods that exemplify the groundwork available before recent developments revolutionized the deconvolution field. Why, in their simplicity and elegance, did the linear methods fail to stimulate the wide adoption of deconvolution methods in spectroscopy After all, available instrumental resolution is limiting in many applications, and the simplicity of the microcomputer makes numerical processing attractive. 

To grasp the flavor of this caveat, consider this the adjustment range of an instrument s physical parameters is limited. Furthermore, many modifications, because they are time consuming and expensive, are carefully considered before implementation and fully tested afterwards. On the other hand, because deconvolution is numerical, radical changes are readily made and often not as thoroughly tested as are physical changes to the instrument. Very subtle effects are possible. 

Today questions such as, Would you rather use deconvolution or build a higher-resolving-power instrument are not uncommon. In fact, today they may actually be justified, considering what we have not learned about de-convolution. These same questions cannot be justified on the basis of a negative reaction to a numerical process that is no more mystical than the operation of any state-of-the-art instrument. 

The literature on deconvolution is rich with the contributions of many investigators. These contributions are, however, scattered among journals devoted to numerous specialties. No single volume has been available that provides both an overview and the detail needed by a newcomer to this field. When a specific need arises, a recent journal article or the advice of a colleague often initiates a considerable, but not always successful, search. The lack of a suitable volume has fostered this understandable approach. Although it may be sorely needed, deconvolution is a distraction for the researcher in pursuit of personal scientific goals. The present work conveys an understanding of the field and presents under one cover a selection of the most effective, practical techniques. 

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. 

Consider now the problem of identifying a linear system in the form of its weighting function h(t), using the relationship (5.66). This problem is called deconvolution. Discrete Fourier transformation offers a standard technique performing numerical deconvolution as mentioned in Section 4.3.3. It... 

DETERMINING THE INPUT CF A LINEAR SYSTEM BY NUMERICAL DECONVOLUTION... 

Since the convolution integral is symmetrical in u(t) and hit), this problem is similar to the one of system identification considered in the previous section. Nevertheless, it is usually easier to find the weighting function h(t) since its form is more - or - less known (e.g., as a sum of polyexponentials), and hence parametric methods apply, whereas the input function u(t) is a priori arbitrary. Therefore, the non - parametric point -area method is a popular way of performing numerical deconvolution. It is really simple evaluating the integral means h, f, . .., fy, of the weighting function over the subinterval [t. ., t ] we can easily solve the set (6.68)... 

D.J. Cutler, Numerical deconvolution by least squares Use of prescribed input functions, J. Pharmacokinetics and Biopharm., 6 (1978) 227-242. 

F. Langenbucher, Numerical convolution/deconvolution as a tool for correlating in vitro with in vivo drug availability, Pharm. Ind.,... 

S. Vajda, K.R. Godfrey and P. Valkd, Numerical deconvolution using system identification methods, J. Pharmacokinetics and Biopharm., 16 (1988) 85-107. 

A study of the expanded plot of the dppp complex resulted in deconvoluting the ABCX coupling pattern in terms of P-Rh and P-P couplings as indicated by Figure 12. The numerical values for the assigned chemical shifts and coupling constants are given in Table II. 

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