Linear convolution method

The peak area errors for the two most studied de-convolution methods (i.e., perpendicular drop and linear tangential skim) are dependent on a complex combination of resolution, relative peak width, relative peak height, and asymmetry ratio [1]. Exponential skimming assumes that the tailing of the first peak can be described by an exponential decay and that the peaks are sufficiently resolved to determine the decay parameters. Nonetheless, some broad generalizations can be made ... 

Another advanced analysis method applicable to the SSITKA studies makes use of convolution, well known for linear systems. Making use of linear convolution in parametric and nonparametric kinetic analyses, provides increased accuracy for the determination of the kinetic parameters... 

Equations (11) and (12) enable the generation of the total isotopic transient responses of a product species given (a) the transient response that characterises hypothesized catalyst-surface behaviour and (b) an inert-tracer transient response that characterises the gas-phase behaviour of the reactor system. Use of the linear-convolution relationships has been suggested as an iterative means to verify a model of the catalyst surface reaction pathway and kinetics. I This is attractive since the direct determination of the catalyst-surface transient response is especially problematic for non-ideal PFRs, since a method of complete gas-phase behaviour correction to obtain the catalyst-surface transient response is presently unavailable for such reactor systems.1 1 Unfortunately, there are also no corresponding analytical relationships to Eqs. (11) and (12) which permit explicit determination of the catalyst-surface transient response from the measured isotopic and inert-tracer transient responses, and hence, a model has to be assumed and tested. The better the model of the surface reaction pathway, the better the fit of the generated transient to the measured transient. 

These rescaling operations, which operate on individual pixels, are limited in their applications. More powerful image enhancement methods take into account the entire image or at least neighboring pixels. Linear convolution operations... 

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... 

Our new method of determining nonlinearity (or showing linearity) is also related to our discussion of derivatives, particularly when using the Savitzky-Golay method of convolution functions, as we discussed recently [6], This last is not very surprising, once you consider that the Savitzky-Golay convolution functions are also (ultimately) derived from considerations of numerical analysis. 

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-... 

Equation (25) is general in that it does not depend on the electrochemical method employed to obtain the i-E data. Moreover, unlike conventional electrochemical methods such as cyclic or linear scan voltammetry, all of the experimental i-E data are used in kinetic analysis (as opposed to using limited information such as the peak potentials and half-widths when using cyclic voltammetry). Finally, and of particular importance, the convolution analysis has the great advantage that the heterogeneous ET kinetics can be analyzed without the need of defining a priori the ET rate law. By contrast, in conventional voltammetric analyses, a specific ET rate law (as a rule, the Butler-Volmer rate law) must be used to extract the relevant kinetic information. 

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. 

Note now that the present iterative method relies on sequential application of convolution by s. All these convolutions could be linearly combined into a... 

Since the value of bounds has come to be widely accepted, numerous other effective bounded methods have appeared. Linear programming has provided the basis for a method presented by Mammone and Eichmann (1982a, 1982b). In a method loosely related to linear programming, MacAdam (1970) exploited the relationship between polynomial multiplication and convolution. His method is particularly suited to human interactive adjustment of constraints. 

Linear sweep voltammetry Ep measurements have not been applied extensively for the study of heterogeneous charge transfer kinetics. A serious problem with the use of this method is that Ep in itself is not significant in this respect but rather Ep — Etev is the quantity of interest. While AEP in CV is readily measured, this cannot be said for Etev using only LSV as a measurement technique. Therefore, there does not appear to be any advantage in LSV for the study of electrode kinetics. A more detailed analysis of the LSV wave, by convolution potential sweep or normalized potential sweep voltammetry (both to be discussed later) can provide both a and k°. 

The extension of the formulas to degenerate accepting modes which occur when a Jahn-Teller effect in the excited state is present is relatively easy. In this case the products of distribution functions can be rewritten by convolution into fundamental distributions which does not change the overall expression of Eq. (29) [38, 66]. Also, it is possible to consider an intermixing of modes in the excited state by virtue of the bi-linear term in Eq. (1) (Duschinsky effect [67]). Since it is difficult to decide from most of the spectra if this effect is really observed in the case of the present complex compounds, we will not consider it here and refer to the literature [68,69]. This is justified as long as we can explain the experimental spectra satisfactorily applying the parallel mode approximation leading to the line shape function of Eq. (29) as it has been described in the method above. 

An answer to this lies in the transformation of the linear sweep response into a form which is readily analysable, i.e. the form of a steady-state voltammetric wave. Two independent methods of achieving this goal have been described the convolution technique by Saveant and co-workers11,12, and semi-integration by Oldham13. In this section we describe the convolution technique, and demonstrate the equivalence of the two approaches at the end. 

Potential or current step transients seem to be more appropriate for kinetic studies since the initial and boundary conditions of the experiment are better defined unlike linear scan or cyclic voltammetry where time and potential are convoluted. The time resolution of the EQCM is limited in this case by the measurement of the resonant frequency. There are different methods to measure the crystal resonance frequency. In the simplest approach, the Miller oscillator or similar circuit tuned to one of the crystal resonance frequencies may be used and the frequency can be measured directly with a frequency meter [18]. This simple experimental device can be easily built, but has a poor resolution which is inversely proportional to the measurement time for instance for an accuracy of 1 Hz, a gate time of 1 second is needed, and for 0.1 Hz the measurement lasts as long as 10 seconds minimum to achieve the same accuracy. An advantage of the Miller oscillator is that the crystal electrode is grounded and can be used as the working electrode with a hard ground potentiostat with no conflict between the high ac circuit and the dc electrochemical circuit. 

In addition to obtaining correlograms, a large battery of methods are available to smooth time series, many based on so-called windows , whereby data are smoothed over a number of points in time. A simple method is to take the average reading over five points in time, but sometimes this could miss out important information about cyclicity especially for a process that is sampled slowly compared to the rate of oscillation. A number of linear filters have been developed which are apphcable to this time of data (Section 3.3), this procedure often being described as convolution. 

Fourier filters can be related to linear methods in Section 3.3 by an important principle called the convolution theorem as discussed in Section 3.5.3. 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

Each E (l,2) in Eq. (17) is an integrand whose structure can be represented by an "elementary diagram with base points 1,2, say. For the precise definition of these diagrams the reader is referred to Reference 30. Because of the form of E r), Eq. (16), while exact, apart from some questions of convergence of the series for (r), hardly provides a practicable method of determining g r), since many if not all of the terms of E(r) have to be computed. E (r) has so far had to be drastically approximated. The simplest procedure is to set E(r) = 0 in Eq. (16) [E[r) contains no term linear in p this yields the so-called hyper-netted chain or convolution approximation. In a second approximation, the E[i) approximation, the first "elementary diagram having two... 

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