Differentiability fast Fourier transform

In fig. 2 an ideal profile across a pipe is simulated. The unsharpness of the exposure rounds the edges. To detect these edges normally a differentiation is used. Edges are extrema in the second derivative. But a twofold numerical differentiation reduces the signal to noise ratio (SNR) of experimental data considerably. To avoid this a special filter procedure is used as known from Computerised Tomography (CT) /4/. This filter based on Fast Fourier transforms (1 dimensional FFT s) calculates a function like a second derivative based on the first derivative of the profile P (r) ... 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements. One simply combines the methods for ordinary differential equations (see Ordinary Differential Equations—Boundary Value Problems ) with the methods for initial-value problems (see Numerical Solution of Ordinary Differential Equations as Initial Value Problems ). Fast Fourier transforms can also be used on regular grids (see Fast Fourier Transform ). 

The spectral method is used for direct numerical simulation (DNS) of turbulence. The Fourier transform is taken of the differential equation, and the resulting equation is solved. Then the inverse transformation gives the solution. When there are nonlinear terms, they are calculated at each node in physical space, and the Fourier transform is taken of the result. This technique is especially suited to time-dependent problems, and the major computational effort is in the fast Fourier transform. 

In addition to the statistical response, the effectiveness of the active control system is further demonstrated using the method of Monte Carlo simulation. Sample functions of the components of the buffeting loads in the normal coordinates are simulated using the Fast Fourier transform (FFT) technique [25]. Then, a system of simultaneous coupled differential equations is solved using a 4 order Runge-Kutta numerical integration method to obtain the sample function of bridge response quantities [11]. 

We already know that PM-IRRAS combines Fourier transform mid-IR reflection spectroscopy with fast PM of the incident beam (ideally between p- and s-linear states) and with two-channel electronic and mathematical processing of the detected signal in order to get a differential reflectivity spectrum AR/... 

It is possible, for example, to use the method of Fourier transformations to solve the boundary problems for the system of differential equations (5.198), (5.199). In this case we have jmax linear differential equations of the second order. Because jmax is usually of the order of hundred, a further analytical investigation of Eqs. (5.198), (5.199) is senseless. However, the problem can be essentially simplified if the surfactant diffusion is treated separately for time scales comparable with the relaxation times of the fast and slow steps of micellisation, respectively. 

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