Nyquist-Shannon Sampling Theorem
In the field of digital signal processing, the sampling theorem is a fundamental bridge between continuous-time signals (often called ‘analog signals’) and discrete-time signals (often called ‘digital signals’). It establishes a sufficient condition between a signal’s bandwidth and the sample rate that permits a discrete sequence of samples to capture all the information from the continuous-time signal. Strictly speaking, the theorem only applies to a class of mathematical functions having a Fourier transform that is zero outside of a finite region of frequencies. Intuitively we expect that when one reduces a continuous function to a discrete sequence and interpolates back to a continuous function, the fidelity of the result depends on the density (or sample rate) of the original samples. The sampling theorem introduces the concept of a sample rate that is sufficient for perfect fidelity for the class of functions that are bandlimited to a given bandwidth, such that no actual information is lost in the sampling process. It expresses the sufficient sample rate in terms of the bandwidth for the class of functions. The theorem also leads to a formula for perfectly reconstructing the original continuous-time function from the samples. Perfect reconstruction may still be possible when the sample-rate criterion is not satisfied, provided other constraints on the signal are known. (See § Sampling of non-baseband signals below, and Compressed sensing.) The name Nyquist-Shannon sampling theorem honors Harry Nyquist and Claude Shannon. The theorem was also discovered independently by E. T. Whittaker, by Vladimir Kotelnikov, and by others. So it is also known by the names Nyquist-Shannon-Kotelnikov, Whittaker-Shannon-Kotelnikov, Whittaker-Nyquist-Kotelnikov-Shannon, and cardinal theorem of interpolation.
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Maximal Label Search (MLS)
Many graph search algorithms use a vertex labeling to compute an ordering of the vertices. We examine such algorithms which compute a peo (perfect elimination ordering) of a chordal graph and corresponding algorithms which compute an meo (minimal elimination ordering) of a non-chordal graph, an ordering used to compute a minimal triangulation of the input graph. We express all known peo-computing search algorithms as instances of a generic algorithm called MLS (maximal label search) and generalize Algorithm MLS into CompMLS, which can compute any peo. We then extend these algorithms to versions which compute an meo and likewise generalize all known meo-computing search algorithms. We show that not all minimal triangulations can be computed by such a graph search, and, more surprisingly, that all these search algorithms compute the same set of minimal triangulations, even though the computed meos are different. Finally, we present a complexity analysis of these algorithms. An extended abstract of part of this paper was published in WG 2005.
Computing a clique tree with algorithm MLS (Maximal Label Search)