A theory of 1/f noise

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.