The sample function of a stochastic process X(t) is shown in Figure P4.18a, where we see that the sample function x(t) assumes the values ±1 in a random manner. It is assumed that at time t = 0, the values X(0) = -1 and X(1) = +1 are equiprobable.
From there on, the changes in X(t) occur in accordance with a Poisson process of average rate l. The process X(t), described herein, is sometimes referred to as a telegraph signal.
a. Show that, for any time t > 0, the values X(t) = -1 and X(t) = +1 are equiprobable.
b. Building on the result of part a, show that the mean of X(t) is zero and its variance is unity.
c. Show that the autocorrelation function of X(t) is given by
d. The process X(t) is applied to the simple low-pass filter of Figure P4.18b. Determine the power spectral density of the process Y(t) produced at the filter output.
