## Abstract

Let u(θ) be an absolutely integrable function and define the random process[Figure not available: see fulltext.] where the t_{i} are Poisson arrivals and the s_{i}, are identically distributed nonnegative random variables. Under routine independence assumptions, one may then calculate a formula for the spectrum of n(t), S_{n}(ω), in terms of the probability density of s, p_{s}(α). If any probability density p_{s}(α) having the property p_{s}(α) ∼ I for small α is substituted into this formula, the calculated S_{n}(ω) is such that S_{n}(ω)∼ 1 ω for small ω. However, this is not a spectrum of a well-defined random process; here, it is termed a limit spectrum. If a probability density having the property p_{s}(α) ∼α^{δ} for small α, where δ > 0, is substituted into the formula instead, a spectrum is calculated which is indeed the spectrum of a well-defined random process. Also, if the latter p_{s} is suitably close to the former p_{s}, then the spectrum in the second case approximates, to an arbitrary, degree of accuracy, the limit spectrum. It is shown how one may thereby have 1/f noise with low-frequency turnover, and also strict 1/f^{1-δ} noise (the latter spectrum being integrable for δ > 0). Suitable examples are given. Actually, u(θ) may be itself a random process, and the theory is developed on this basis.

Original language | English (US) |
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Pages (from-to) | 199-216 |

Number of pages | 18 |

Journal | Journal of Statistical Physics |

Volume | 4 |

Issue number | 2-3 |

DOIs | |

State | Published - Mar 1 1972 |

## Keywords

- 1/f noise
- Flicker effect
- Poisson process

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics