I am trying to understand the meaning of the term Stationary Process. For example, I was told that $\sin(t)$ is a stationary process.
Could someone try to explain, in simple words, why is $\sin(t)$ (for example) is a stationary process?
Answer
$\sin(t)$ is no random process, because there's nothing random about it. You could add a random amplitude to get a random process:
$$x(t)=A\sin(t)\tag{1}$$
This is a random process because $A$ is a random variable. However, $x(t)$ is not stationary, but it is cyclostationary, i.e., its statistical properties vary periodically. You can make the process $x(t)$ stationary by adding a random phase:
$$\tilde{x}(t)=A\sin(t+\phi)\tag{2}$$
The phase $\phi\in [0,2\pi]$ is a uniformly distributed random variable that is independent of $A$. It can be shown that the statistical properties of $\tilde{x}(t)$ given by $(2)$ are independent of $t$, and hence, the process is stationary.
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