The convolution method

it is required to predict the time course of the plasma concentration from a model with oral administration or with continuous infusion, when only data from a single intravenous injection are available. In this case, the Laplace transform can be very useful, as will be shown from the following illustration. 

In the catenary model of Fig. 39.14a we have a reservoir, absorption and plasma compartments and an elimination pool. The time-dependent contents in these compartments are labelled X, X, and X, respectively. Such a model can be transformed in the 5-domain in the form of a diagram in which each node represents a compartment, and where each connecting block contains the transfer function of the passage from one node to another. As shown in Fig. 39.14b, the 

This model can now be solved for various inputs to the absorption compartment. 

In the case of rapid administration of a dose D to the absorption compartment (such as the gut, skin, muscle, etc.), the Laplace transform of the reservoir function is given by  

In this case we obtain the simplest possible expression for the plasma function in the 5-domain  

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An example of the use of the convolution method to detect the change in mechanism is shown in Figure 3.15. A more complete scanning of the passage between the two mechanisms is obtained thanks to the use of convolution and placing data obtained at two temperatures on the same diagram.20... 

Figure 5 Range distribution in nitrogen for 200-eV electrons obtained via the convolution method at 110 eV [38]. The final energy is about twice the ionization potential. See text for details.

A further advantage of the convolution method is that iR drop is very easily accounted for. In the case of conventional cyclic voltammetry, the nonlinearity of the sweep makes data analysis difficult however, in convolution voltammetry all that is necessary is to replace E by E + when plotting the voltammogram [43]. 

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. 

Equations of Convolution Type The equation u x) = f x) + X K(x — t)u(t) dt is a special case of the linear integral equation of the second land of Volterra type. The integral part is the convolution integral discussed under Integral Transforms (Operational Methods) so the solution can be accomplished by Laplace transforms L[u x)] = E[f x)] + XL[u x)]LIK x)] or... 

Madden, H. H., Comments on the Savitzky-Golay Convolution Method for Least-Squares Fit Smoothing and Differentiation of Digital Data, Anal. Chem. 50, 1978, 1383-1386. 

Gorry, P. A., General Least-Squares Smoothing and Differentiation by the Convolution (Savitzky-Golay) Method, Anal. Chem. 62, 1990, 570-573. 

In order to compress the measured data through a wavelet-based technique, it is necessary to perform a series of convolutions on the data Becau.se of the finite size of the convolution filters, the data may be decomposed only after enough data has been collected so as to allow convolution and decomposition on a wavelet basis. Therefore, point-bypoint data compression as done by the boxcar or backward slope methods is not possible using wavelets. Usually, a window of data of length 2" m e Z, is collected before decomposition and selection of the appropriate... 

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. 

Early studies of ET dynamics at externally biased interfaces were based on conventional cyclic voltammetry employing four-electrode potentiostats [62,67 70,79]. The formal pseudo-first-order electron-transfer rate constants [ket(cms )] were measured on the basis of the Nicholson method [99] and convolution potential sweep voltammetry [79,100] in the presence of an excess of one of the reactant species. The constant composition approximation allows expression of the ET rate constant with the same units as in heterogeneous reaction on solid electrodes. However, any comparison with the expression described in Section II.B requires the transformation to bimolecular units, i.e., M cms . Values of of the order of 1-2 x lO cms (0.05 to O.IM cms ) were reported for Fe(CN)g in the aqueous phase and the redox species Lu(PC)2, Sn(PC)2, TCNQ, and RuTPP(Py)2 in DCE [62,70]. Despite the fact that large potential perturbations across the interface introduce interferences in kinetic analysis [101], these early estimations allowed some preliminary comparisons to established ET models in heterogeneous media. 

A transformation of the peak voltammogram to the sigmoidal shape shown in the preceding section, Fig. 5.13, is achieved by the convolution analysis method proposed by K. Oldham. The experimental function j(t) = j[T(E — Ej)/v] is transformed by convolution integration... 

The heterogeneous rates of electron transfer in eq 7 were measured by two independent electrochemical methods cyclic voltammetry (CV) and convolutive potential sweep voltammetry (CPSV). The utility of the cyclic voltammetric method stems from its simplicity, while that of the CPSV method derives from its rigor. 

We deemed it necessary to confirm the CV results by the alternate method using convolutive potential sweep voltammetry, which requires no assumptions as to the form of the free energy relationship and is ideally suited for an independent analysis of curvature revealed in Figure 7. In convolutive linear sweep voltammetry, the heterogeneous rate constant ke is obtained from the cur-... 

The convolution integral in (1.19) and (1.20) can be solved by the method of numerical integration proposed by Nicholson and Olmstead [47], The time t is divided into m time increments t = md. It is assumed that within each time increment the function 1 can be replaced by the average value Ij ... 

Stonehouse and Keeler developed an intriguing method for the accurate determination of scalar couplings even in multiplets with partially convoluted peaks (one- or two-dimensional). They recognized that the time domain signal is completely resolved and that convolution of the frequency domain spectrum is a consequence of the Fourier transform of the signal decay. The method requires that the multiplet be centred about zero frequency and this was achieved by the following method ... 

Due to the convolution with the instrumental resolution, an accurate determination of the spectral shape is not possible. Therefore, the value of has to be determined by other methods or from coherent scattering measurements. 

For PSDs measured by GPC, we expect a greater degree of success with the simple model for retention (eq. 5). Halasz noted that the PSDs he measured were always broader than corresponding PSDs from porisimetry and capillary condensation. This is in keeping with the convolution model (eq. 7) and indicates that the PSDs measured by GPC already contain the convolution between Kqpq and the classical PSD. If this is the case, then the "effective PSDs" provided by the GPC method should be useful for the direct prediction of calibration curves. 

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