Sampling theorem

Shannon s sampling theorem states that A funetion f t) that has a bandwidth is uniquely determined by a diserete set of sample values provided that the sampling frequeney is greater than 2uj, . The sampling frequeney 2tJb is ealled the Nyquist frequeney. 

Equation (4-210) is known as the sampling theorem. It states that any time function limited to a frequency W can be uniquely determined by its values at sample points spaced 1/2W apart. Thus if both the input and output from a channel are limited to a frequency W, then we can represent the input and output as sequences of real numbers spaced at intervals of 1/2 W. If each output in this sequence depends on only one input, then the results for continuous input, continuous output channels can be applied. One particularly important case in which ... 

Sales response to advertising, 265 Salpeter, E. E641 Sample, adequacy of, 319 Sampling theorem, 245 Scalar product of two wave functions, 549,553... 

For image slicers, contiguous slices of the sky are re-arranged end-to-end to form the pseudo-slit. In that case it is obvious that the sky can be correctly sampled (according to the Nyquist sampling theorem) by the detector pixels on which the slices are projected in the same way as required for direct imaging. 

Let us return for the moment to Eq. (2.2). In atmospheric problems it is impossible to solve the equations of motion analytically. Under these conditions information about the instantaneous velocity field u is available only from direct measurements or from numerical simulations of the fluid flow. In either case we are confronted with the problem of reconstructing the complete, continuous velocity field from observations at discrete points in space, namely the measuring sites or the grid points of the numerical model. The sampling theorem tells us that from a set of discrete values, only those features of the field with scales larger than the discretization interval can be reproduced in their entirety (Papoulis, 1%5). Therefore, we decompose the wind velocity in the form... 

This basic sampling theorem has profound implications. It says that any high-frequency components in the signal (for example, 60-cycle-per-second electrical noise) can necessitate very fast sampling, even if the basic process is quite slow. It is, therefore, always recommended that signals be analog-filtered before they are sampled. This eliminates the unimportant high-frequency components. 

To prove the sampling theorem let us consider a continuous that is a sine wave with a frequency coq and an amplitude Aq. ... 

Brief reflection on the sampling theorem (Chapter 1, Section IV.C) with the aid of the Fourier transform directory (Chapter 1, Fig. 2) leads to the conclusion that the Rayleigh distance is precisely two times the Nyquist interval. We may therefore easily specify the sample density required to recover all the information in a spectrum obtained from a band-limiting instrument with a sine-squared spread function evenly spaced samples must be selected so that four data points would cover the interval between the first zeros on either side of the spread function s central maximum. In practice, it is often advantageous to place samples somewhat closer together. 

The sampling theorem tells us that we must sample the signal at a rate equal to or greater than twice the highest-frequency component in the Fourier transform of the signal. For an actual spectrum of absorption lines, this maximum frequency depends on the selected scanning rate in a direct way. In addition, the bandpass of the electronics must be established in such a way that the maximum signal frequency will be minimally attenuated. 

One procedure for recovering the continuous (band-limited) function exactly is provided by the Whittaker-Shannon sampling theorem, which is expressed by the equation... 

Anti-aliasing filters (considered part of Analog Conditioning ) are needed at the input to remove out of band energy that might alias down into baseband. The antialiasing filter at the output removes the aliases that result from the sampling theorem. 

As demonstrated for a simple sine wave, if the sampling (the dotted sequence in Figure 10.2a and Figure 10.2b) is done with at least twice the frequency of the highest frequency present in the sampled wave, perfect reconstruction of the continuous sequence from the discrete sequence of samples (the starred points) is possible. This remarkable fact is known as the sampling theorem. 

The matter of sampling and limited representation of frequencies requires a second look at the representation of data in the time and frequency domains, as well as the transformation between those domains. Specifically, we need to consider the Fourier transformation of bandwidth-limited, finite sequences of data so that the S/N enhancement and signal distortion of physically significant data can be explored. We begin with an evaluation of the effect of sampling, and the sampling theorem, on the range of frequencies at our disposal for some set of time-domain data. 

G(cu) is thus defined for all frequencies contdned in the incident pulse, I.e, it is defined for all oi where Fi(o>) 0. The only major restriction in the use of this Fourier transform approach is that the pulse form must be known for all values of t or, in other words, the decay of any relaxation process must be monitored effectivdy to completion. The Samulon modification of the Shannon sampling theorem has been shown to be an accurate and convenient method of Fourier-transforming step pulses. The relevant relation is... 

In order to compare with the experimentally measured spectrum, one would ideally like to have the spectra simulated with equal or better energy resolution and with a good statistical reliability for the predicted intensity. Therefore, a suitable estimate of these quantities is important, in order to minimise the output to a manageable level (i.e. to output the trajectories and velocities as less as possible). According to the sampling theorem [23], the smallest time step for the Z(co) calculations is a factor of n times larger than MD simulation step which is determined by the maximum frequency, (Omox, of the system to be simulated. Hence the appropriate time step n At is given by the Nyquist sample rate, as ... 

In both of these examples there is a maximum of curvature near each control point and a minimum in between. There is no systematic movement of the control points which can remove this variation. If proof of this is required, we can appeal to the Shannon sampling theorem. Although this was originally conceived in the temporal domain it applies equally to the spatial. 

Equation (2.3.13) states the Nyquist sampling theorem. If At is set too high, the spectral width is too low, and signals at higher frequencies will appear at false positions (Fig. 2.3.9). This phenomenon is called signal aliasing [Deri]. 

There exist also methods which do not require a whole series of swelling experiments to evaluate the continuous structure factors. Shannon s sampling theorem from information theory implies that only two sets of discrete structure factors are sufficient. More specifically, this theory states that the complete continuous structure factor F(s) can be reconstructed by Fourier interpolation from just one set of observed F ... 

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