Definition

Digital random signal

• A random signal is a vector of random variables: $\boldsymbol{X} = (X[0],X[1],\ldots,X[N-1])$.
• Each $X[n]$ is a random variables
• In practice we observe one trajectory, that is one realization $\boldsymbol{x}$ : $$ \boldsymbol{x} = (x[0],x[1],\ldots,x[N-1]). $$
• $x[n]$ is one realization of the random variables $X[n]$

Useful assumptions

• The random signal is assumed to be ergodic and stationary
• We can work with only one realization
• The autocorrelation function is deterministic