Linear convolution integral

In the case of our linear, stationary and causal device, input and output are linked by the convolution integral ... 

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... 

A somewhat more complex case is that of an integral equation that cannot be formulated in any closed form. This is a frequently encountered situation, the integral in the equation often being a convolution integral involving the linear diffusion function 1 / fwz, while the other side contains a function, F, of the function sought, ij/ 9... 

The star in this formula represents a convolution integral with e being the linear response function corresponding to the frequency dependent e(u),x,y). The non-linear polarization is an arbitrary function of the electric field P = P(E). We will also include a current density that is driven by the optical field... 

In a linear response regime the solvent polarization at a given time due to a TD electric field can be expressed as a convolution integral on previous times as [43] ... 

The measurable linear viscoelastic functions are defined either in the time domain or in the frequency domain. The interrelations between functions in the firequenpy domain are pxirely algebraic. The interrelations between functions in the time domain are convolution integrals. The interrelations between functions in the time and frequency domain are Carson-Laplace or inverse Carson-Laplace transforms. Some of these interrelations will be given below, and a general scheme of these interrelations may be found in [1]. These interrelations derive directly from the mathematical theory of linear viscoelasticity and do not imply any molecular or continuum mechanics modelling. 

Upon evaluating the convolution integral from the experimental current-potential (time) curve and its limiting values (Eq. 77), kinetic analysis can be performed with the help of Eq. (76). Conversely, Eq. (76) or similar equations can be used to calculate the theoretical current-potential curve, e.g., for the linear potential sweep voltammogram, provided that the values of all the parameters are known. Some illustrative examples were provided by Girault and coworkers [183]. 

If the deformations are small enough, the functional can be written in terms of linear differential equations with constant coefficients or, equivalently, in terms of convolution integrals with difference kernels. 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

A linear system is fully characterized by its pulse response, since the reaction y t) to any excitation x(t) can be derived from the pulse response by the convolution integral... 

For a linear system the input signal, e.g., a voltage, may be a function of time. The same is true for the output signal fo t). The input signal and the output signal are related by the convolution integral as... 

Let us consider a stable system with a single input x and single output y. Dynamic response of a linear system to an arbitrary input x(t) can be defined using a convolution integral... 

The FT has many useful mathematical properties including linearity, and using eqn [2], the derivative d/( )/d has the transform (i2nv)P v). The convolution theorem is particularly useful because it relates the product of two functions, F(v) G(v), in the frequency domain to the convolution integral... 

Experimental decay curves measured after a pulsed excitation are affected by the limited frequency response of the detection system and the width of the exciting light pulse. Assuming a linear response for all contributions, the measured decay curve / t) is given by the convolution integral... 

Models in which the damping force is afunction of past history of motion via convolution integrals over a suitable Kernel function constitutes non-viscous damping. They are called non-viscous because the force depends on state variables other than just the instantaneous velocity (Adhikari et al 2003). The most generic form of linear non-viscous damping given in the form of modified dissipation function is as follows (Woodhouse 1998, Adhikari 2000) ... 

If the apparatus operates in a linear and repeatable manner, desmearing according to the convolution integral is an exact method capable of producing the true heat flow rate of the respective sample. Figure 6.23 illustrates this in the case of the melting curve of octadecane on the basis of a measured curve obtained by means of a DSC. The increase of the resolution in temperature is obvious and shows the pretransition more clearly, whereas the fluctuations (noise Ag) of the desmeared heat flow rate function increase accordingly . 

An autonomous linear system is one in which all system parameters are independent of time, and integrals that exist in the system description are of convolution type (a convolution integral has the form/o° f t- t )a t )dt ). It is readily... 

In particular, the LS-Dyna finite viscoelastic relationship [175] takes into accotmt rate effects through linear viscoelasticity by a convolution integral. The model corresponds to a Maxwell fluid consisting of dampers and springs in series. The Abaqus FEA model is reminiscent of, and similar to, a well-established model of finite viscoelasticity, namely the Pipkin-Rogers model [161]. This model, with an appropriate choice of the constitutive parameters, reduces to the Fung (QLV) model [173, 177]. 

In Chapter 6 it was shown that linear viscoelastic materials could be represented by the hereditary convolution integrals. 

From a mathematical point of view, the reaction force experienced by a linear viscoelastic device at rest for f < 0 can be expressed in the time domain through the following convolution integral ... 

The conventional filtered white-noise process f(t, k) is the stationary response of a linear time-invariant filter subjected to a white-noise process. White-noise w t) is a stationary random process in time that has a zero mean and a constant spectral density for all frequencies. The response of a linear filter to a white-noise process may be calculated by using the Duhamel convolution integral, and hence the general formulation of a filtered white-noise process can be written in the following form ... 

The general approach to discussing linear viscoelasticity comes from the Boltzmann superposition principle represented as a convolution integral. For the shear stress as a function of shear strain, one obtains... 

Figure 7b illustrates the way in which the responses add for a two-step history in which each step has the same magnitude. Equation 20a is the general form and is the discrete form of the linear superposition principle cast as a simple shear. It shows the simple linear additivity of the responses. A similar equation could be written for the strain response in terms of the stresses for a creep history. The equations can easily be generalized to include the full range of strains and stresses discussed in the next section. Furthermore, the responses can he written in terms of convolution integrals ... 

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