Product convolution

The diffusion stage, characterized by H(t), will be discussed in detail with respect to the instantaneous wetting condition. However, in the presence of a time dependent wetting function (p(t), Eq. 2.1, we see from Eig. 2 that diffusion will have progressed to different extents in different areas of the interface. If the intrinsic diffusion function, H(t) as given by Eq. 1.1 does not change its nature with time due to the other stages, then the net diffusion, H"(t) can be expressed as the convolution product... 

The observed profile P(20) of a Bragg peak at do = dhu is a convolution product between the instrumental profile r(20) and the physical profile p(20) ... 

The Voigt function is a convolution product ( ) between L and G. As the convolution is expensive from a computational point of view, the pseudo-Voigt form is more often used. The pseudo-Voigt is characterized by a mixing parameter r], representing the fraction of Lorentzian contribution, i.e. r] = 1(0) means pure Lorentzian (Gaussian) profile shape. Gaussian and Lorentzian breadths can be treated as independent parameters in some expressions. 

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] ... 

In Eq. (A.4), the convolution products that appear are the inverted Laplace transforms of the operators and we get the results ... 

The principles of pulse and phase-modulation fluorometries are illustrated in Figures 6.5 and 6.6. The d-pulse response I(t) of the fluorescent sample is, in the simplest case, a single exponential whose time constant is the excited-state lifetime, but more often it is a sum of discrete exponentials, or a more complicated function sometimes the system is characterized by a distribution of decay times. For any excitation function E(t), the response R(t) of the sample is the convolution product of this function by the d-pulse response ... 

Phase-modulation fluorometry The sample is excited by a sinusoidally modulated light at high frequency. The fluorescence response, which is the convolution product (Eq. 6.9) of the pulse response by the sinusoidal excitation function, is sinusoidally... 

Figure 2. Space-scale representation of the GC content of a 10-Mbp-long fragment of human chromosome 22 when using a Gaussian smoothing filter (x) [Eq. (6)]. (a) GC content flucmations computed in adjacent 1 kbp intervals, (b) Color coding of the convolution product Wg(o)[GC](n,a) = (GC /a))(n) using 256 colors from black (0) to red (max) superimposed...

I> = 8c0ifi, and where I is the time spent at site i. When a random variable is defined as the sum of several independent random variables, its probability distribution is the convolution product of the distributions of the terms of the... 

If b and g are peaked functions (such as in a spectral line), the area under their convolution product is the product of their individual areas. Thus, if b represents instrumental spreading, the area under the spectral line is preserved through the convolution operation. In spectroscopy, we know this phenomenon as the invariance of the equivalent width of a spectral line when it is subjected to instrumental distortion. This property is again referred to in Section II.F of Chapter 2 and used in our discussion of a method to determine the instrument response function (Chapter 2, Section II.G). 

Convolution has an interesting property with respect to differentiation. The first derivative of the convolution product of two functions may be given by the convolution of either function with the derivative of the other. Thus, if... 

If two Gaussian functions are convolved, the result is a gaussian with variance equal to the sum of the variances of the components. Even when two functions are not Gaussian, their convolution product will have variance equal to the sum of the variances of the component functions. Furthermore, the second moment of the convolution product is given by the sum of the second moments of the components. The horizontal displacement of the centroid is given by the sum of the component centroid displacements. Kendall and Stuart (1963) and Martin (1971) provide helpful additional discussions of the central-limit theorem and attendant considerations. 

The fast Fourier transform (FFT) requires 5N log N (Bergland, 1969) elementary arithmetic operations for an array of N samples. Computation of the convolution product requires three such transforms plus 4N elementary operations (N complex products) in the transform domain. 

In Eq. (45), TM represents the true transmittance only if instrumental spreading is negligible. In previous sections, however, we learned that instrumental spreading may be described by the convolution product of the flux and the spectrometer response function, here called r(x), that incorporates all the instrumental spreading phenomena ... 

For successive overrelaxation, we understand Eq. (26) to incorporate the use of o(k+1 values in place of o k) values in the convolution product as soon as they are formed for preceding x values. This adaptation can be explicitly displayed by the appropriate use of the Heaviside step function in a modified version of Eq. (26). The method of Van Cittert is a special case of simultaneous relaxation in which C = 1. 

The specimen intensity transform X is a type of convolution product of the particle intensity transform Ip and the particle orientation density function ( 1,2). The procedure that we have used to simulate Ip involves firstly the calculation of the intensity transform for an infinite particle, with appropriate allowances for random fluctuations in atomic positions and for matrix scattering. A mapping of Xp is then carried out which includes the effects of finite particle dimensions and of intraparticle lattice disorder, if this is present. A mapping of Is is then obtained from Tp by incorporating the effects of imperfect particle orientation. 

Further manipulations can be performed on the convolution products in the integral... 

The retention-time model expressed by the convolution (9.10). The derivation of this convolution product leads to (cf. Appendix E)... 

Equation (41) can be viewed as the reciprocal of Eq. (37). It can be rewritten, using convolution product notations, as... 

Exactly like its classical analog Eq. (94), Eq. (125) allows one to express the displacement response function in terms of the time-dependent diffusion coefficient. However, contrary to the classical case in which Xxx( 0 is directly proportional to D(t — t ), in the quantum formulation Xxv( 0 is a convolution product, for the value t — t of the argument, of the functions D(t ) and logcoth(7i fi /2 h). Inverting the convolution equation (125) yields an expression for D(t) in terms of the dissipative part of the displacement response function ... 

Assuming for a molecular system the vibrational, rotational, and translational motions as uncoupled, the incoherent scattering law can be expressed as a convolution product of the individual scattering laws for each motion, which can be examined separately ... 

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