Operators convolution

To illustrate the self-convolution operation, draw two identical boxcars and evaluate the area in common as a function of their relative separation along the abscissa. 

Note that in equations (1) and (4), (V + Q is a multiplicative operator in r representation, and a convolution operator in p representation. F E) is neither convolution nor multiplicative-like in either representation hecause... 

Before we can imderstand why Fourier cycling works, we have to deepen our understanding of the Fourier transform. In particular, we need to imderstand the effects of modifying density on the structure factor amplitudes and phases. The mathematical tool that describes this is the convolution operator. 

Figure 10.4 Vector diagram of the effect of convolution of many structure factors in reciprocal space. In (a) is shown the true structure factor (thick arrow) and the initial, unmodified structure factor (thin arrow). In (b) the convolution operator applied to the initial structure factor (thin arrow) results in a closer estimate (dashed arrow) of the true structure factor (thick arrow).

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). 

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... 

To obtain a complete drug concentration profile, both the input and disposition kinetics must be known or assumed. If the input is an intravenous bolus, zero or first order, and disposition is first order, then the input and disposition can be combined mathematically through the convolution operation, represented by the symbol. Mathematically, this is represented as... 

Patel, D., Davies, E.R., and Hannah, I. 1996. The use of convolution operators for detecting contaminants in food images. Pattern Recognit. 29(6), 1019-1029. 

Taking the discrete Laplace transform of this equation and noting the relation mentioned in the preceding text regarding the convolution operator, gives d(p /dx = (p)2, which has solution... 

IVote In this table, a and b are arbitrary constants. Also, the symbol used between variables denotes a convolution operation, such that the notation a b = l a(k)b(r - k). When used as a superscript, the symbol denotes complex conjugation, such that (a + ib) = (a - ib). 

Peaks departing from Gaussians are often described by exponentially modified Gaussian (EMG) functions, a combination (via a convolution operation) of a Gaussian and an exponential function [27,28]. Other empirical functions have been used as well [29]. None of the above functions has a very sound theoretical footing, but they are nonetheless important for descriptive purposes. 

Starting with AT/T = kit (A x Ip), where and x are the correlation and convolution operators, respectively, we apply the Fourier transform, keeping in mind the fundamental identities ... 

Sometimes this operation is written using a convolution operator denoted by an asterisk, so that... 

Leonard [97] was apparently the first to use the term Large Eddy Simulation. He also introduced the idea of filtering as a formal convolution operation on the velocity field and gave the first general formulation of the method. Since Leonard s approach form the basis for application of LES to chemical reactor modeling, we discuss this approach in further details. 

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