Co-indicator functions and related 1/ƒ noises

  • Benoit B. Mandelbrot


In thin metallic films, semiconductors, nerve tissues, and many other media, the measured spectral density of noise is proportional to ƒ D − 2, where ƒ is the frequency and D a constant 0 < D < 2. The energy of these “ƒ D − 2 noises” behaves more “erratically” in time than is expected from functions subject to the Wiener-Khinchin spectral theory. Moreover, when D < 1, blind extrapolation of the expression ƒ D − 2 to ƒ = 0 suggests that the total energy is infinite. This divergence, called “infrared catastrophe” raised problems of great theoretical interest and of great practical importance in the design of electronic devices.

This paper interprets these spectral measurements without paradox, by introducing a new concept, “conditional spectrum.” Examples are given of random functions that have both the “erratic” behavior and the conditional spectral density observed for ƒ D − 2 noise.

A conditional spectrum is obtained when a procedure that was meant to measure a sample Wiener-Khinchin spectrum is used beyond its domain of applicability. The conditional spectrum is defined for nonconstant samples from all random functions of the Wiener-Khinchin theory and in addition 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 has a ƒ− 2 conditional spectral density. It is a direct current broken by a single discontinuity uniformly distributed over the whole time axis. White noise is the D = 2 limit of ƒ D − 2 noise. The other ƒ D − 2 noises to be described partake both of direct current and of white 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 “direct current” component.

Empirical studies are suggested, and the descriptive value of the concepts of direct current component and of spectrum are discussed.


IEEE Transaction Spectral Density Independent Random Variable Random Function Renewal Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Benoit B. Mandelbrot 1999

Authors and Affiliations

  • Benoit B. Mandelbrot
    • 1
    • 2
  1. 1.Mathematics DepartmentYale UniversityNew HavenUSA
  2. 2.IBM T.J. Watson Research CenterYorktown HeightsUSA

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