Convolution-deconvolution techniques

Such a treatment, within its mechanistic restrictions, is general across the full range of experimental techniques. This convolution-deconvolution approach is the basis of the E G G Condecon system. 

The convolution-deconvolution voltammetry, combined with digital simulation techniques, was applied [36] to determine the electrochemical and chemical parameters for the Cd(II)/Cd(Hg) system in aqueous NaNOs solution. The agreement between experimental and theoretical data indicated that the reduction mechanism at the mercury electrode proceeds via consisting in chemical step (C) followed by charge transfer step (E)-so-called CE mechanism [37]. 

The traditional approach to treat the IP is based on deconvolution techniques. In fact, it can be shown that the PD profile is a convolution ((8>) of profile components produced by different sources [IP, size (S), strain (D), faulting (F), etc.] ... 

The complex convolution integral is reduced to a simple product and it is obvious that an excitation at a frequency f only results in a response at the same frequency. If, as in the case of gas—surface studies, the overall response is determined by a number of processes, then the one of interest may be extracted by deconvolution techniques, which are particularly simple in the frequency domain [53]. In particular, attenuation and phase shift of the signal produced by the flight time of molecules from the modulator to the detector and any non-ideal response of the detector may be taken into account [55]. 

The BE in APS is obtained directly from the recorder plots by applying the correction for the work function of the thermionic electron soiuce. To avoid the imcertainty introduced due to this correction in BE measiuements, Fukuda et al. [34] have used a field-emission soiurce. In earlier measurements in APS, the BE was determined in a simple way by the intersection of the extrapolated projection of the bac ound and positive going low energy slope of the peak. Since the APS yield is proportional to the self-convolution of the density of the final electron states broadened by the finite lifetime of the core hole and other effects stated earlier, precise knowledge about BE can be obtained by using deconvolution techniques. Successful deconvolution techniques have been developed by Fukuda et al. [34], Dose et al. [35,36], and Schulz et al. [37]. 

In addition, the time-dependence of these concentrations also contains (albeit in encoded form) the homogeneous parameters of the particular mechanism being considered. These latter techniques are termed convolutions. Convolution (and its reverse, i.e. deconvolution) are ideal for the electroanalyst because the theoretical calculation of current, and direct comparison with experimental data, is often not viable. This alternative of testing experimental currents via convolutions results in expressions for concentrations at the electrode which arise directly from the data rather than requiring iterations(s). The electrode concentrations thus estimated for a particular mechanism then allow for correlations to be drawn between potential and time, thereby assessing the fit between the data and the model. 

Vardya and Hester used their OEX model in a constrained linear optimi2ation procedure, based on the Box-Complex method which is essentially a constrained simplex minimization technique. The method does not require derivatives of the object function and is not subject to scaling problems. As an example. Fig. 8 shows a de-convolution of a chromatogram of Dextran T-2000 with water as the mobile phase in a controlled porous glass column. The badly fused peaks are successfully deconvoluted. 

The matrix notation serves to stress that the technique is applicable to shift-variant spread functions, that is, where sjk slm for j — k = l — m. Many of the deconvolution methods described here with the convolution notation may thus be generalized. In the convolution notation, the present method may be expressed by the equation... 

Iterative deconvolution is, with positional scanning, the most popular de-convolution technique. It was introduced by Geysen et al. [29-31] and exploited by Houghten et al [32,33] and Ecker et al. [34,35] for the deconvolution of peptide and oligonucleotide libraries, but it has been applied to the deconvolution of many small-molecule libraries. 

Other procedures applied to the LSV technique Semidifferentiating (deconvolutive) procedure may be considered as a counterpart to semi-integrating (convolutive) transformation. The replacement of Pick s laws by formulations involving semidifferentiation was proposed 25 years ago [122]. Five years later [123] the deconvolutive transformation of recordered (sampled) currents represented by the equation... 

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