Theoretical and experimental results for the distribution of a certain nonlinear functional of the Ornstein-Uhlenbeck process

Let
x(t)
be the Ornstein-Uhlenbeck process and
y(t)
the result of low-pass
RC
filtering of sgn
x(t)
. This paper considers the problem of determining the first-order probability density function of
y(t)
. The approach is to apply the
\nu
th-order Fokker-Planck-Kolmogorov type equations. Based upon an assumption as to the linearity of a coefficient of the resulting differential equation, a closed-form solution is obtained for
p(y)
. The result agrees with the previous work of Doyle, McFadden and Marx who solved the special case when the bandwidth of the
RC
filter is twice the bandwidth of the input noise. The result also agrees, to within experimental error, with a Monte Carlo simulation over four orders of magnitude of variation of the ratio of the bandwidths of the
RC
filter and the input process.