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,/spl tau/), we allow a class of probability distributions /spl phi/ for the time constant /spl tau/ and consider the following question: Given frequency /spl omega//spl ges/0 and unknown probability distribution f /spl isin/ F, to what extent can the expected filter gain g(/spl omega/,/spl tau/)=|G(j/spl omega/,/spl tau/)| deviate from some desired nominal value, g(/spl omega/, /spl tau//sub 0/)? It turns out that the deviations of concern are surprisingly low. For example, with 20% variation in /spl tau/, the expected filter gain deviates from g(/spl omega/,/spl tau//sub 0/) 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 /spl tau//sub 0/. The frequency /spl omega//spl ges/0 and the admissible probability distributions d/spl isin/F.