- #1

ait.abd

- 26

- 0

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Let [itex]X(t)[/itex] be a random process such that:

$$

X(t) = \begin{cases}

t & \text{with probability } \frac{1}{2} \\

2-at & \text{with probability } \frac{1}{2} \\

\end{cases},

$$

where [itex]a[/itex] is a constant.

I need to find the value of [itex]a[/itex] for which [itex]X(t)[/itex] is a wide-sense stationary process. I have made the following definition of the random process:

$$

\begin{equation}

X(t) = \alpha t + (1-\alpha)(2-at),

\end{equation}

$$

where [itex]\alpha[/itex] is a Bernoulli random variable with [itex]p=q=0.5[/itex].

For mean, we have

$$

E[X(t)] = \frac{t + 2-at}{2}.

$$

For the autocorrelation function,

$$

\begin{align*}

R_X(t_1,t_2) &= E[X(t_1)X(t_2)]\\

&=E[(\alpha t_1 + (1-\alpha)(2-at_1))\times(\alpha t_2 + (1-\alpha)(2-at_2))]\\

&=E[\alpha^2 t_1 t_2 + (1-\alpha)^2(2-at_1)(2-at_2)]\\

&=\frac{t_1 t_2}{2} + \frac{4-a(t_1+at_2)+a^2 t_1 t_2}{2}\\

\end{align*}

$$

As clear from the above equation, there is no value of [itex]a[/itex] for which autocorrelation function is a function of time difference [itex]t_2-t_1[/itex].

My confusion starts from the way we define autocorrelation function as above in so many textbooks. The above-mentioned definition shows that we sample the ensemble at two time instants [itex]t_1[/itex] and [itex]t_2[/itex] to get two random variables. The two random variables [itex]X(t_1)[/itex] and [itex]X(t_2)[/itex] are (possibly different) functions of the same random variable [itex]\alpha[/itex]. My question is why do we need to take the same random variable [itex]\alpha[/itex] at two time instants? It is like saying that if we know [itex]X(t_1)[/itex], we can figure out [itex]X(t_2)[/itex] right away. Shouldn't it be like that at [itex]t_1[/itex] we should take [itex]\alpha_1[/itex] and at [itex]t_2[/itex] we should take [itex]\alpha_2[/itex], where both [itex]\alpha_1[/itex] and [itex]\alpha_2[/itex] are Bernoulli with [itex]p=0.5[/itex].

I can describe the confusion like the following as well. When we sample the random process at two time instants, we get two random variables [itex]A = X(t_1)[/itex] and [itex]B = X(t_2)[/itex], where

$$

A = \begin{cases}

t_1 & \text{ with probability 0.5} \\

2-at_1 & \text{ with probability 0.5} \\

\end{cases}

$$

and

$$

B = \begin{cases}

t_2 & \text{ with probability 0.5} \\

2-at_2 & \text{ with probability 0.5} \\

\end{cases}.

$$

To calculate [itex]E[AB][/itex], we need to take the cases into account where [itex]A=t_1[/itex] and [itex]B=2-at_2[/itex], and [itex]A=2-at_1[/itex] and [itex]B=t_2[/itex]. Both of these cases do not appear in the calculation of ensemble autocorrelation function [itex]R_X(t_1,t_2)[/itex]. Why do I get to take the two cases into account when I use the formulation in terms of [itex]A[/itex] and [itex]B[/itex], and these two cases do not appear when I calculated [itex]R_X(t_1, t_2)[/itex] using the formulation with [itex]\alpha[/itex] as discussed in the start of the problem?