Aliasing errors

The resolution of the analog I/O channels of the controller vaiy somewhat, with 12-bit and 14-bit conversions quite common. Sample rates for the majority of the constant sample rate controllers range from I to 10 samples/second. Hard-wired single-pole, low-pass filters are installed on the analog inputs to the controller to protect the sampler from aliasing errors. 

Not only is the choice of a uniform prior-prejudice distribution not sensible it also exposes the calculation to two main sources of computational errors, both connected with the functional form of the MaxEnt distribution of scatterers, and with its numerical evaluation namely series termination ripples and aliasing errors in the numerical sampling of the exponential modulation of mix). The next two paragraphs will illustrate these issues in some detail. 

Within the computational scheme described in the course of this work, the available information about the atomic substructure (core+valence) can be taken into account explicitly. In the simplest possible calculation, a fragment of atomic cores is used, and a MaxEnt distribution for valence electrons is computed by modulation of a uniform prior prejudice. As we have shown in the noise-free calculations on l-alanine described in Section 3.1.1, the method will yield a better representation of bonding and non-bonding valence charge concentration regions, but bias will still be present because of Fourier truncation ripples and aliasing errors ... 

Sample times for the microprocessor-based SLCs vary from 0.1 to 0.4 seconds. Low-pass analog electronic filters are installed on the process inputs to stop aliasing errors caused by fast changes in the process signal. Input filter time constants are typically in the range from 0.1 to 1 s. Microprocessor-based SLCs may be made part of a DCS by using the communication port (RS-488 is common) on the controller or may be operated in a standalone mode independent of the DCS. 

Here, the first sum gives the errors introduced by the finite cutoff in k of (1), while the second sum represents the aliasing errors due to the interpolation of the charges onto a mesh. Admittedly, (13) looks rather complicated. Still, in combination with (11) it gives the rms force error of the Fourier space part of P M, and therefore allows to tune the P M algorithm a priori. In addition, the computations of k and Gopt are quite similar, and have to be done only once in the code, and play a negligible role in the overall computation time. 

The early meteorological finite difference studies of long-term numerical time integrations of the equations of fluid motion, which involve non-linear convection terms, revealed the presence of non-linear instabilities due to aliasing errors [143, 144, 7,145, 210]. To avoid the occurrence of these non-linear instabilities, Arakawa [7] was the first to recognize the importance of the use of numerical schemes which conserve kinetic energy. 

We have already remarked that the pulse waveforms in TDS equipment have small amplitudes at and above f = 15 GHz, and the appropriate A would be no greater than 30 ps by this criterion if aliasing errors are to be avoided. For arbitrarily chosen frequencies rather than harmonics of the base frequency fo = 1/NAthe situation is less simple, but in practice values of A smaller than given by this criterion are usually appropriate for a reasonable simulation of the pulse itself. As a rule of thumb, it is prudent to take A less than l/4fm, where f is the highest frequency of interest. 

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