Transforms convolution

If the term in brackets, which represents a particular (convolutive) transformation of the experimental i(t) data, is defined as /(t), then equation 6.7.1 becomes (20)... 

For the sake of completeness, the convolution transform shall also be mentioned because it is so closely related to cross-correlation ... 

Convolutive linear sweep voltammetry (CLSV) The convolutive transformation of a LSV curve yields the expression for surface concentration of the electroactive species A (in a reversible reduction process) as... 

Fig. 54. Convolutive transformations. Linear sweep voltammogram of 1 mM in 0.1 M KNO3 electrode of 4.7 mm area, sweep rate v = 0.1 Vs . 1, capped potential-time dependence (dotted line) 2, current-time transient (full line) 3, convoluted (semiintegrated vs. time) current (dashed line). Adapted according to [119].

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... 

Inversion of equation 22, using the convolution transform, gives... 

The term inside the square brackets is the convolution transform of the i f) data which we will denote as F t) (/(f) is frequently used but tends to be confused with current density), and therefore Equation (6.50) can be rewritten as... 

The convolution transformation described by Equation (6.50) is readily performed on a small computer using numerical techniques, and Fig. 6.26 shows a typical cyclic voltammogram for a reversible system and the corresponding convolution voltammogram. h can be seen immediately that the convolution voltammogram retraces itself on the back sweep, and this is one test of a reversible process, the other being that a plot of the left-hand side of Equation (6.56) varies linearly with E. 

The essential property that we use, is the transformation of the product of convolution in a sum. 

Applying to Eq. (4) an integral transform (usually, a Fourier transform) <., one derives by (integral) convolution, symbolized by the expression... 

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... 

The integrals in Eqs. (17) and (18) are called convolution integrals. In Fourier space they are products of the Fourier transforms of c r). Thus, Eq. (18) is a geometric series in Fourier space, which can be summed. Performing this summation and returning to direct space, we have the OZ equation... 

This result can also be established by making use of Eq. (3-130) and the well-known formula for the Fourier transform of the convolution of two functions. 

By means of Laplace transforms of the foregoing three equations mating use of the convolution theorem and the assumptions Pf(t) — Pt a constant which is the ratio of the in use time (t the total operating time of the 4th component), Gt(t) si — exp ( — t/dj (note that a double transform is applied to Ff(t,x)), we obtain an expression in terms of the lifetime distribution, i.e.,... 

In the discrete case, the convolution by the PSF is diagonalized by using the discrete Fourier transform (DFT) ... 

Noticing the fact that the formula for determining surface deformation takes the form of convolution, the fast Fourier transform (FFT) technique has been applied in recent years to the calculations of deformation [35,36]. The FFT-based approach would give exact results if the convolution functions, i.e., pressure and surface topography take periodic form. For the concentrated contact problems, however. 

To use the DFT properly for evaluating normal surface deformation, the linear convolution appearing in Eq (27) has to be transformed to the circular convolution. This requires a pretreatment for the influence coefficient Kj and pressure pj so that the convolution theorem for circle convolution can be applied. The pretreatment can be performed in two steps ... 

Another resolution-enhancement procedure used is convolution difference (Campbell et ai, 1973). This suppresses the ridges from the cross-peaks and weakens the peaks on the diagonal. Alternatively, we can use a shaping function that involves production of pseudoechoes. This makes the envelope of the time-domain signal symmetrical about its midpoint, so the dispersionmode contributions in both halves are equal and opposite in sign (Bax et ai, 1979,1981). Fourier transformation of the pseudoecho produces signals... 

It is a known property of Fourier transforms that given a convolution product in the reciprocal space, it becomes a simple product of the Fourier transforms of each term in the real space. Then, as the peak broadening is due to the convolution of size and strains (and instrumental) effects, the Fourier transform A 1) of the peak profile I s) is [36] ... 

If X (0 and Xjit) are the input and output functions in the time domain (for example, the contents in the reservoir and in the plasma compartment), then XJj) is the convolution of Xj(r) with G(t), the inverse Laplace transform of the transfer function between input and output ... 

In the i-domain, convolution is simply the product of the Laplace transform ... 

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. 

These four steps are illustrated in Fig. 40.17 where two triangles (array of 32 data points) are convoluted via the Fourier domain. Because one should multiply Fourier coefficients at corresponding frequencies, the signal and the point-spread function should be digitized with the same time interval. Special precautions are needed to avoid numerical errors, of which the discussion is beyond the scope of this text. However, one should know that when J(t) and h(t) are digitized into sampled arrays of the size A and B respectively, both J(t) and h(t) should be extended with zeros to a size of at least A + 5. If (A -i- B) is not a power of two, more zeros should be appended in order to use the fast Fourier transform. 

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