Theorem convolution

The derivative of a convolution is the convolution of either of the functions with the derivative of the other, that is, 

This can be shown as follows. According to (B.18) and the convolution theorem, 

Associating the factor Hits with F(s) we then have 

Occasionally, products of factors occur wherein each factor has a known inversion, and it is desired to find the product of the two. The mechanics for doing this is accomplished by the method of convolution, the derivation of which is given in standard operational mathematics textbooks (Churchill 1958). Thus, if the product occurs 

The two forms shown suggest that the convolution integral is synunetrical. 

We illustrate the application of convolution in the next series of e unples. 

This could be performed directly, since it is recognized that l/s implies integration. Thus, recalling Eq. 9.111, which stated 

Some people are confused by the difference between Fourier filters and linear smoothing and resolution functions. In fact, both methods are equivalent and are related 

The principles of convolution have been discussion in Section 3.3.3. Two functions, / and g, are said to be convoluted to give h if 

Filtering a time series, using Fourier time domain filters, however, involves multiplying die entire time series by a single function, so diat 

The convolution theorem states diat /, g and h are Fourier transforms of F, G and H. Hence linear filters as applied directly to spectroscopic data have their equivalence as Fourier filters in die time domain in other words, convolution in one domain is equivalent to multiplication in die other domain. Which approach is best depends largely on computational complexity and convenience. For example, bodi moving averages and exponential Fourier filters are easy to apply, and so are simple approaches, one applied direct to die frequency spectrum and die other to die raw time series. Convoluting a spectrum widi die Fourier transform of an exponential decay is a difficult procedure and so die choice of domain is made according to how easy the calculations are. 

Here we are in the area of advanced mechanics. All the equations must fidiill boundary conditions, for example, convergence of sequence. For the simple operations to be valid, they must be understood to satisfy all conditions by limiting the domains in which the variables x,y, t,. .. belong. 

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The last relation in equation (Al.6.107) follows from the Fourier convolution theorem and tlie property of the Fourier transfonn of a derivative we have also assumed that E(a) = (-w). The absorption spectmm is defined as the total energy absorbed at frequency to, nonnalized by the energy of the incident field at that frequency. Identifying the integrand on the right-hand side of equation (Al.6.107) with the total energy absorbed at frequency oi, we have... 

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.,... 

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

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 ... 

An important aspect of convolution is its translation into the frequency domain and vice versa. This translation is known as the convolution theorem [7], which states that ... 

From the convolution theorem it follows that the convolution of the two triangles in our example can also be calculated in the Fourier domain, according to the following scheme ... 

In order to deduce Scherrer s equation first an infinite crystal is considered that is, second, restricted (i.e multiplied) by a shape function (cf. p. 17). Thus from the Fourier convolution theorem (Sect. 2.7.8) it follows that in reciprocal space each reflection is convolved by the Fourier transform of the square of the shape function - and Scherrer s equation is readily established. 

Desmearing. In practice, there are two pathways to desmear the measured image. The first is a simple result of the convolution theorem (cf. Sect. 2.7.8) which permits to carry out desmearing by means of Fourier transform, division and back-transformation (Stokes [27])... 

This fact has been discussed in Sect. 2.7.5, p. 24 on the basis of the Fourier convolution theorem (Sect. 2.7.8). 

The convolution theorem, item 8 of Table 1.3, is of value when the transform is a product of two factors whose inverses are known individually. Take the case of the equation,... 

The usefulness of the Fourier transforms lies in the fact that the following convolution theorem can be established.4 The sum over all configurations of n defects in a chain ... 

If we then notice that the bracketed expression in the first integral of (65 ) is nothing else but the Laplace transform of PoKP> t)>we obtain, applying the well-known convolution theorem of Laplace transforms together with the definitions (64) and (65),... 

Using the convolution theorem for Laplace transforms, we rewrite Eq. (336) as ... 

The last two terms of Eq. (121) are evaluated by observing that they are products of Laplace transforms. By applying the convolution theorem, we obtain... 

By applying a variant of the extremely powerful convolution theorem stated above, computing the overlap integral of one scalar field (e.g., an electron density), translated by t relative to another scalar field for all possible translations t, simplifies to computing the product of the two Fourier-transformed scalar fields. Furthermore, if periodic boundary conditions can be imposed (artificially), the computation simplifies further to the evaluation of these products at only a discrete set of integral points (Laue vectors) in Fourier space. 

Making use of equation (15) and (22), together with the Fourier convolution theorem, we can see that... 

To obtain the corresponding structure factor expression, the Fourier convolution theorem is applied, or... 

According to the Fourier convolution theorem, further discussed in section 5.1.3, the Fourier transform of the convolution in expression (2.14) is the product of the Fourier transforms of the individual functions, or... 

It is of importance that expression (5.12) holds even when /(x) is known only in part of space, as is the case in a crystallography experiment at finite resolution determined by Hmax. Using the Fourier convolution theorem, we may write (Dunitz and Seiler 1973)... 

The derivation of the electrostatic properties from the multipole coefficients given below follows the method of Su and Coppens (1992). It employs the Fourier convolution theorem used by Epstein and Swanton (1982) to evaluate the electric field gradient at the atomic nuclei. A direct-space method based on the Laplace expansion of 1/ RP — r has been described by Bentley (1981). 

In Eq. (8.18), we wrote the potential as a convolution of the total density and the operator 1 /r. Similarly, the integrals encountered in the evaluation of the peripheral electronic contributions to Eqs. (8.35) (8.37) are convolutions of the electron density p(r) and the pertinent operator. They can be evaluated with the Fourier convolution theorem (Prosser and Blanchard 1962), which implies that the convolution of /(r) and p(r) is the inverse transform of the product of their... 

If we apply the convolution theorem to the constitutive relation (2.23), for example, we obtain... 

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