Question

A Random Telegraph Signal with rate λ > 0 is a random process X(t) (where for each t, X(t) ∈ {±1}) defined on [0,∞) with the following properties: X(0) = ±1 with probability 0.5 each, and X(t) switches between the two values ±1 at the points of arrival of a Poisson process with rate λ i.e., the probability of k changes in a time interval of length T is

P(k sign changes in an interval of length T) = e −λT (λT)^k/k! 

(a) Find the first-order PMF of the process X(t). 

(b) Find E[X(t)] 

(c) Find the autocorrelation function of X. 

(d) Find the power spectral density of X.

Answer 1

(a) First-order PMF of the process X(t): The first-order PMF (Probability Mass Function) of the process X(t) can be found by considering the number of sign changes within a given time interval.

Let's denote the number of sign changes within a time interval of length T as N. According to the given information, the number of sign changes follows a Poisson distribution with rate λ.

Therefore, the first-order PMF of X(t) is given by:

P(X(t) = ±1) = P(N is even) = e^(-λT) * (1 + (λT)^2/2! + (λT)^4/4! + ...)

This represents the probability that there is an even number of sign changes within the time interval T.

(b) Expected value of X(t) (E[X(t)]): To find the expected value of X(t), we need to calculate the average of X(t) over all possible values.

E[X(t)] = (1) * P(X(t) = 1) + (-1) * P(X(t) = -1)

From part (a), we know the probabilities P(X(t) = ±1) based on the number of sign changes within the time interval.

(c) Autocorrelation function of X: The autocorrelation function of X(t) measures the correlation between X(t) and X(s), where s and t are time points.

The autocorrelation function can be defined as:

R_X(t,s) = E[X(t) * X(s)]

To calculate this, we need to consider the probabilities of X(t) and X(s) being ±1 and account for the sign changes within the time interval.

(d) Power spectral density of X: The power spectral density (PSD) of X(t) is a measure of the distribution of power with respect to frequency in the random process.

To find the PSD, we need to take the Fourier Transform of the autocorrelation function obtained in part (c). The Fourier Transform will give us the PSD as a function of frequency.

Please note that the details for parts (b), (c), and (d) will depend on the specific values of t and s that you want to analyze.