Shannons sampling theorem

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

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 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. 

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. 

Jerry AJ (1997) The Shannon sampling theorem - its various extensions and applications a tutorial review. Proceedings of the Institute of Electrical and Electronic Engineers 65 1565-1595. 

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. 

It is rare in praetise to work near to the limit given by Shannon s theorem. A useful rule of thumb is to sample the signal at about ten times higher than the highest frequeney thought to be present. 

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 ... 

The basis for pulse modulation applications is representing analog signals as properly spaced samples. The theoretical justification for this is Shannon s sampling theorem (Ziemer and Tranter, 2002), which may be stated succinctly as follows ... 

Shannon s sampling theorem, 318 Shewhart control chart, 415 signals... 

For synchronized cultivations, it is further inevitable to satisfy the Nyquist-Shannon theorem, that is, the sampling rate must be at least twice the oscillation frequency - the higher, the better. In the case of CHO cells with an oscillation rate of approximately 15-20 h, this means a sampling rate of 7 h or (much) less, covering multiple cell cycles. This drives the need for (semi)automated, possibly noninvasive, sampling and analysis methods covering population diversity (Figure 4.2). 

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