Sampling, discrete theorem

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... 

Selected entries from Methods in Enzymology [vol, page(s)] Application in fluorescence, 240, 734, 736, 757 convolution, 240, 490-491 in NMR [discrete transform, 239, 319-322 inverse transform, 239, 208, 259 multinuclear multidimensional NMR, 239, 71-73 shift theorem, 239, 210 time-domain shape functions, 239, 208-209] FT infrared spectroscopy [iron-coordinated CO, in difference spectrum of photolyzed carbonmonoxymyo-globin, 232, 186-187 for fatty acyl ester determination in small cell samples, 233, 311-313 myoglobin conformational substrates, 232, 186-187]. 

As with the continuous Fourier transform, we could treat the equations of the discrete Fourier transform (DFT) completely independently, derive all the required theorems for them, and work entirely within this closed system. However, because the data from which the discrete samples are taken are usually continuous, some discussion of sampling error is warranted. Further, the DFT is inherently periodic, and the limitations and possible error associated with a periodic function should be discussed. 

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. 

Note that this has nothing to do with the Fourier transform per se, but everything with the more general problem of representing continuous functions by discrete samples of such functions. The Nyquist theorem specifies that the underlying, continuous, repetitive function cannot be defined properly unless one samples it more than twice per its repeat period. 

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 ... 

A function that is compact in momentum space is equivalent to the band-limited Fourier transform of the function. Confinement of such a function to a finite volume in phase space is equivalent to a band-limited function with finite support. (The support of a function is the set for which the function is nonzero.) The accuracy of a representation of this function is assured by the Whittaker-Kotel nikov-Shannon sampling theorem (29-31). It states that a band-limited function with finite support is fully specified, if the functional values are given by a discrete, sufficiently dense set of equally spaced sampling points. The number of points is determined by Eq. (26). This implies that a value of the function at an intermediate point can be interpolated with any desired accuracy. This theorem also implies a faithful representation of the nth derivative of the function inside the interval of support. In other words, a finite set of well-chosen points yields arbitrary accuracy. 

The trouble with practical applications of Fourier s integral theorem is that it requires continuous functions for an infinite length of time. These conditions are clearly impossible, so the case of a finite number of observations must be considered. Consider sampling the data every At for 2N equally spaced points from — N to N—1 with tj = jAt-, j= —N, — N-bl,..., 0,..., N—1. The FT, eqn [1], then becomes the discrete FT ... 

Compared to discrete variables, ctmtinuous variables have a large information content so that small or moderate sample sizes are sufficient for a powerful test The probability distribution of the sample mean should be known and preferably normal, either because the population distribution is normal, or the sample is large enough to let the distribution of the sample mean approach normality (central limit theorem). 

One critical mathematical preHminary for the processing of discrete-time waveforms, which originate from sampling continous-time waveforms, is the issue of sampUngrate and aliasing. The Nyquist sampling theorem can be summarized as follows ... 

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