Zero-Crossings of Random Processes with Application to Estimation and Detection

The origin of Rice’s formula for the average level-crossing rate of a general class of random processes can be traced back to his 1936 notes on “Singing Transmission Lines,” [35]. For a stationary, ergodic, random process {X(t)} , -∞ < t < ∞, with sufficiently smooth sample paths, the average number of crossings about the level u, per unit time, is given by Rice’s formula [36] (1)$$ E\left[ {D_u } \right] = \int_{ - \infty }^\infty {\left| {x'} \right|p\left( {u,x'} \right)dx'} $$ where p(x, x’) is the joint probability density of the X(t) and its mean-square derivative X’(t), and Du is the number of u level-crossings of {X(t)} for t in the unit interval [0,1]. Rice’s formula is quite general in nature, and as we shall see, has a simplified form when the process is Gaussian.