A universal figure of merit for stochastic first order filters

The focal point of this paper is a new result on the probabilistic robustness of a stochastic first order filter. For a first order filter transfer function, G(s,τ), we allow a class of probability distributions φ for the time constant τ and consider the following question: Given frequency ω⩾0 and unknown probability distribution f ∈ F, to what extent can the expected filter gain g(ω,τ)=|G(jω,τ)| deviate from some desired nominal value, g(ω, τ₀)? It turns out that the deviations of concern are surprisingly low. For example, with 20% variation in τ, the expected filter gain deviates from g(ω,τ₀) by no more than 0.4% of the zero frequency gain. In addition to performance bounds such as this, we also provide a so-called universal figure of merit. The word “universal” is used because the performance bound attained holds independently of the nominal τ₀. The frequency ω⩾0 and the admissible probability distributions d∈F