The Probability Distributions of the Outputs of First-Order and Tuned Second-Order Filters with Coin-Toss Square Wave Inputs

A method is presented for constructing the probability distribution functions of the outputs of an RC low-pass filter and a tuned second-order filter for coin-toss square wave inputs. The input is a binary process in which axis crossings can occur only on a set of equally spaced sample points, with
p = 1 - q
being the probability that a crossing does not occur at a given sample point. The method is valid only when
exp (-\tau_0/T) < 1/2
, where
T
is the time constant of the filter and
\tau_0
is the minimum time between axis crossings. The method involves construction of the point sets which represent the allowed and unallowed values of the output of the RC filter at a sample point. These sets are used to solve an integral equation involving the distribution functions of the output of the RC filter at an axis crossing. It is shown that at a sample point the output of a tuned higher-order filter obeys the same distribution as the output of the RC filter at a sample point. These results are then extended to give the distributions at an arbitrary point.