The convolution theorem

Convolution integrals can readily be calculated in Fourier space. To this end, the Volterra series is Fourier transformed [Blal, Bliil], 

The functions written in capital letters in (4.2.13)-(4.2.16) are the Fourier transforms of the functions written in small letters in (4.2.5)-(4.2.8). The superscript s indicates that the nonlinear transfer functions K (o)i,. .., o ) in (4.2.15) and (4.2.16) are the Fourier transforms of impulse-response functions with indistinguishable time arguments, where the causal time order ti r is not respected. These transfer functions are invariant against permutation of frequency arguments. Equivalent expressions for the Fourier transforms of impulse-response functions with time-ordered arguments cannot readily be derived. 

Equation (4.2.14) is also referred to as the convolution theorem [Blal]. According to it, a linear convolution of the functions k (t) and x(t) in the time domain can be evaluated by complex multiplication of the Fourier transforms in the frequency domain 

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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 ... 

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])... 

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,... 

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... 

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

Its transform is just (l/ /ln) sinc2(co/27r). This may easily be shown with the aid of the convolution theorem discussed in Section IV.B.10. 

We know that the variance of a gaussian is the reciprocal of the variance of its transform. We apply the convolution theorem to obtain... 

Normally, discrete convolution involves shifting, adding, and multiplying —a laborious and time-consuming process, even in a large digital computer. The convolution theorem presents us with an alternative. It reveals the possibility of computing in the Fourier domain. What are the trade-offs between the two methods ... 

The convolution theorem plays a valuable role in both exact and approximate descriptions of functions useful for analyzing resolution distortion and in helping us understand the effects of these functions in Fourier space. Functions of interest and their transforms can be constructed from our directory in Fig. 2 by forming their sums, products, and convolutions. This technique adds immeasurably to our intuitive grasp of resolution limitations imposed by instrumentation. 

By applying the convolution theorem, we see that replication in the x domain has produced a sampling effect in the frequency domain. The wider the replication interval, the finer is the frequency sampling. Sampling in the x domain, on the other hand, appears in Fourier space as replication. Fine sampling in x produces wide spacing between cycles in co. The area under each scaled Dirac function of co may be taken as the numerical value of a sample. 

The convolution theorem aided us in understanding a property of the spurious component 0(x). It also hints at a method of deconvolution when applied to Eq. (86). This method is developed in Section IV of Chapter 3. 

The entire discussion of relaxation methods was conducted without examining Fourier space consequences. Van Cittert s method is easy to study this way and has been treated by Burger and Van Cittert (1933), Bracewell and Roberts (1954), Sakai (1962), and Frieden (1975). By applying the convolution theorem to Eq. (14), we may write... 

The convolution operation is a way of describing the product of two overlapping functions, integrated over the whole of their overlap, for a given value of their relative displacement (Bracewell 1978 Hecht 2002). The symbol is often used to denote the operation of convolution. The convolution theorem states that the Fourier transform of the product of two functions is equal to the convolution of their separate Fourier transforms... 

Fourier analysis is used to find the velocity and attenuation of surface waves. Let the range in z over which data is available be (. If there were no attenuation, then by the convolution theorem the Fourier transform F ( ) would be a sine function centred at a spatial frequency... 

This rule is known as the convolution theorem and the integrals in eqn. (92) are called convolution integrals [74]. 

After dividing both members of eqns. (166a) and (166b) by s, the convolution theorem (see Sect. 2.5.1) is applied to find their counterpart in the time domain. 

The Convolution Theorem. Let wt(x) and w2(y) be two probability distributions. Often one is interested in the distribution of finding the sum z = x + y. Such a case occurs in polymer science, for instance, in the radical polymerization with termination by radical combination126-1281. The resulting probability distribution is given by... 

Now multiplying Eq. (C.74) on both sides by s1 and summing over all i, we obtain after application of the convolution theorem... 

The monomer unit chose as a root, however, bears f functional groups, and therefore the distribution of offspring is the f-fold convolution of the functional-group probability or applying the convolution theorem (C.IV.c)... 

For the units in the first generation, the result is different when poup A is similar to the preceding generation, or poup B, or group C. Applying the convolution theorem, one finds by inspection of Fig. 9... 

To determine the function x(t), it is necessary to find the solution of the integral of Eq. (3.6) for a known h(t) and measured y(t) functions. One proposed method of calculating x(t) assumes that the shape of the function x(t) is known beforehand and that the transfer function h(t) can be experimentally determined.158 A more general solution uses Fourier transforms. If Eq. (3.6) is rewritten with the Fourier transforms of the functions x(t), y(t), and h(t - t), which are denoted by Y, X, and H, then, using the convolution theorem ... 

The convolution theorem reduces Eq. (57) to a simple product in frequency space Y = XH (58)... 

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