Xvi. Signal Processing Academic and Research Staff A. First-passage Time Statistics of a First-order Markov Process with a Linear Ramp 1. Statement of the Problem Y(t)

where 6(T) is the unit impulse function. If the coefficients of Eq. 1 and the noise spectral density were arbitrary rather than as specified above, the problem could easily be reduced to that described with suitable scaling of x and t. The problem is illustrated in Fig. XVI-1. Solution for the first zero of y(t) = Et + x(t) is equivalent to determining the first time instant that x(t) = -Et. As the slope, E, decreases, we would expect the mean FPT to become more negative, and the standard deviation of the FPT to increase. In this report, we have obtained asymptotic expressions for the probability distribution w(O), the mean value 0, and the standard deviation cr6 02 of the FPT. The results are valid when E << 1, which turns out to be the region where computer simulation of the process is most difficult.