Methods of numerical integration of sampled data are compared in terms of their frequency responses and resolving power. Compared, theoretically and by numerical experiments, are trapezoidal, Simpson, Simpson-3/8 methods, method based on cubic spline data interpolation and Discrete Fourier Transform (DFT) based method. Boundary effects associated with DFT- based and spline-based methods are investigated and an improved Discrete Cosine Transform based method is suggested and shown to be superior to all other methods both in terms of approximation to the ideal continuous integrator and of the level of the boundary effects.
In some measurement techniques the profile, f ͑x͒, of a function should be obtained from the data on measured slope f 0 ͑x͒ by integration. The slope is measured in a given set of points, and from these data we should obtain the profile with the highest possible accuracy. Most frequently, the integration is carried out by numerical integration methods [Press et al., Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987)] that assume different kinds of polynomial approximation of data between sampling points. We propose the integration of the function in the Fourier domain, by which the most-accurate interpolation is automatically carried out. Analysis of the integration methods in the Fourier domain permits us to easily study and compare the methods' behavior.
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