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Mathematical Analysis of Random Noise-Conclusion
About: This article is published in Bell System Technical Journal.The article was published on 1945-01-01 and is currently open access. It has received 807 citations till now.
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TL;DR: A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media as discussed by the authors, where a geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.
Abstract: A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media. Statistical topography involves the geometrical properties of the isosets (contour lines or surfaces) of a random potential $\ensuremath{\psi}(\mathrm{x})$. For rapidly decaying correlations of $\ensuremath{\psi}$, the isopotentials fall into the same universality class as the perimeters of percolation clusters. The topography of long-range correlated potentials involves many length scales and is associated either with the correlated percolation problem or with Mandelbrot's fractional Brownian reliefs. In all cases, the concept of fractal dimension is particularly fruitful in characterizing the geometry of random fields. The physical applications of statistical topography include diffusion in random velocity fields, heat and particle transport in turbulent plasmas, quantum Hall effect, magnetoresistance in inhomogeneous conductors with the classical Hall effect, and many others where random isopotentials are relevant. A geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.
1,059 citations
TL;DR: In this paper, a review of coherence properties of electromagnetic fields and their measurements, with special emphasis on the optical region of the spectrum, is presented, based on both the classical and quantum theories.
Abstract: This article presents a review of coherence properties of electromagnetic fields and their measurements, with special emphasis on the optical region of the spectrum. Analyses based on both the classical and quantum theories are described. After a brief historical introduction, the elementary concepts which are frequently employed in the discussion of interference phenomena are summarized. The measure of second-order coherence is then introduced in connection with the analysis of a simple interference experiment and some of the more important second-order coherence effects are studied. Their uses in stellar interferometry and interference spectroscopy are described. Analysis of partial polarization from the standpoint of correlation theory is also outlined. The general statistical description of the field is discussed in some detail. The recently discovered universal "diagonal" representation of the density operator for free fields is also considered and it is shown how, with the help of the associated generalized phase-space distribution function, the quantum-mechanical correlation functions may be expressed in the same form as the classical ones. The sections which follow deal with the statistical properties of thermal and nonthermal light, and with the temporal and spatial coherence of blackbody radiation. Later sections, dealing with fourth- and higher-order coherence effects include a discussion of the photoelectric detection process. Among the fourth-order effects described in detail are bunching phenomena, the Hanbury Brown-Twiss effect and its application to astronomy. The article concludes with a discussion of various transient superposition effects, such as light beats and interference fringes produced by independent light beams.
889 citations
TL;DR: The performance of collaborative beamforming is analyzed using the theory of random arrays and it is shown that with N sensor nodes uniformly distributed over a disk, the directivity can approach N, provided that the nodes are located sparsely enough.
Abstract: The performance of collaborative beamforming is analyzed using the theory of random arrays. The statistical average and distribution of the beampattern of randomly generated phased arrays is derived in the framework of wireless ad hoc sensor networks. Each sensor node is assumed to have a single isotropic antenna and nodes in the cluster collaboratively transmit the signal such that the signal in the target direction is coherently added in the far-field region. It is shown that with N sensor nodes uniformly distributed over a disk, the directivity can approach N, provided that the nodes are located sparsely enough. The distribution of the maximum sidelobe peak is also studied. With the application to ad hoc networks in mind, two scenarios (closed-loop and open-loop) are considered. Associated with these scenarios, the effects of phase jitter and location estimation errors on the average beampattern are also analyzed.
611 citations
TL;DR: In this paper, the outer intermittent region of a fully developed turbulent boundary layer with zero pressure gradient was extensively explored in the hope of shedding some light on the shape and motion of the interface separating the turbulent and non-turbulent regions as well as on the nature of the related large-scale eddies within the turbulent regime.
Abstract: The outer intermittent region of a fully developed turbulent boundary layer with zero pressure gradient was extensively explored in the hope of shedding some light on the shape and motion of the interface separating the turbulent and non-turbulent regions as well as on the nature of the related large-scale eddies within the turbulent regime. Novel measuring techniques were devised, such as conditional sampling and conditional averaging, and others were turned to new uses, such as reorganizing in map form the space-time auto- and cross-correlation data involving both the U and V velocity components as well as I, the intermittency function. On the basis of the new experimental results, a conceptual model for the development of the interface and for the entrainment of new fluid is proposed.
579 citations
TL;DR: In this paper, the authors discuss the morphologies of caustics and their diffraction patterns in catastrophe optics, and discuss the diffraction catastrophes that both clothe and underlie caustic structures.
Abstract: Publisher Summary This chapter discusses the morphologies of caustics and their diffraction patterns. In catastrophe optics, wave motion is viewed in terms of the contrast and interplay among the morphologies of three extreme regimes. Firstly, if the wavelength λ is small in comparison with scales of variation of diffracting objects or refracting media, the wavefield is dominated by the caustics and associated diffraction patterns. Secondly, when waves propagate in environments which can be modeled by a hierarchy of scales extending to the infinitely small, caustics cannot occur and the limit λ → 0 is not geometrical optics. And thirdly, when waves are explored on the scale of λ, the principal features are wavefronts, which are dominated by their singularities in the form of lines in space. The chapter also discusses the diffraction catastrophes that both clothe and underlie caustics. Each structurally stable caustic has its characteristic diffraction pattern, whose wave function has an integral representation in terms of the standard polynomial describing the corresponding catastrophe. The diffraction catastrophes constitute a new hierarchy of functions, different from the special functions of analysis. The newest application of catastrophe optics is to random short waves, whose statistical properties are determined by the random caustic structure.
509 citations