Some noises with spectrum, a bridge between direct current and white noise

• Authors: B. Mandelbrot
• Venue: IEEE Transactions on Information Theory
• Year: 2006

Noises in thin metallic fills, semiconductors, nerve tissues, and many other media, have measured spectral densities proportional tof^theta - 2}withfthe frequency andthetaa constant0 leq theta < 2. The energy of these "f^{theta-2}noises" behaves more "erratically" in time, than expected from functions subject to the Wiener-Khinchin spectral theory. Moreover, blind extrapolation of the "f^{theta -2}law" tof=0incorrectly suggests, when0 leq 1, that the total energy is infinite ("infrared catastrophe"). The problems thus raised are of the greatest theoretical interest, and of the greatest practical importance in the design of electronic devices. The present paper reinterprets these spectral measurements without paradox, by introducing a concept to be called "conditional spectrum." Examples are given of functions ruled by chance, that have the observed "erratic" behavior and conditional spectral density. A conditional spectrum is obtained when a procedure, meant to measure a sample Wiener-Khinchin spectrum, is applied to a sample conditioned to be nonconstant. The conditional spectrum is defined, not only for nonconstant samples from all random functions of the Wiener-Khinchin theory, but also for nonconstant samples from certain nonstationary random functions, and for nonconstant samples from a new generalization of random functions, called "sporadic functions." The simplest sporadic functions, having af^{-2}conditional spectral density, is a "direct current" with a single discontinuity uniformly distributed over- infty < t < infty. The otherf^{theta -2}noises to be described partake both of direct current and of white noise (thetheta =2limit off^{theta - 2}noise), and continuously span the gap between these limits. In many cases, their noise energy can be said to be proportional to the square of their "dc" component. Empirical studies will be suggested, and the descriptive value of the concepts of dc component and of spectrum will be discussed.