On the Theory of Random Noise. Phenomenological Models. I

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Abstract
Three phenomenological models are considered from which can be constructed a macroscopic statistical description of varieties of electronic noise, of which shot, thermal, and Barkhausen noise, electron multiplier and precipitation noise, clutter, ignition, and impulsive random noise in general are representative examples. The models examined are (I) non‐overlapping, periodic noise waves, common in pulse‐time modulation and other communication schemes, where the amplitude, phase, duration, and epoch in a period interval are subject to statistical variations; (II), non‐overlapping, nonperiodic disturbances, encountered in servo‐mechanism operation and keyed‐carrier communication techniques, for instance, which are like (I), but lack the basic periodic structure; and (III), poisson noise, consisting of the superposition of independent, randomly occurring elementary impulses. Much of the electronic noise mentioned above belongs to this more comprehensive type, where overlapping of the basic pulses is the characteristic feature. Because all (second‐order) moments are required in general for the analysis of noise in nonlinear systems, the attempt is made here to determine explicitly on the basis of the appropriate model the second‐order probability density W2 in the important stationary cases. For noise of types (I) and (II) this appears impractical except in the simplest cases: only the lower order moments prove tractable. However, for poisson noise (III) an explicit treatment is possible for impulsive random noise, nearly normal random noise, and for the limiting, normal random cases (of which shot and thermal noise are examples). In Part I the main features of the models (I–III) are discussed, and the general probability density Wl(X1, t1; …; Xs, ts) in the nonstationary instances is formally constructed. In Part II, the distribution density for nearly normal random noise is given, the first and second (second‐order) moments of the various distributions are determined, and from these in turn are found the spectral distribution of the energy in the random waves.