Autocorrelation function

Correlation measures how predictable a random variable is from another one. The autocorrelation function is a function which depends on the time instants values defining the random variables and measures the predictability degree of two random variables defined in different time instants in a random signal. It is defined with the following expression:

R_{x x} (t_{1}, t_{2}) = E [X (t_{1}) X (t_{2})] = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} x_{1} x_{2} p_{x} (x_{1}, x_{2}, t_{1}, t_{2}) d x_{1} d x_{2}

When time instants are equal, we define the Mean Squared Value:

R_{x x} (t, t) = E [X {(t)}^{2}] = \int_{- \infty}^{\infty} x^{2} p_{x} (x, t) d x

Review of Basic Concepts of Probability

Autocorrelation function