Yule-Walker equations

A way of solving the linear relationship between filter parameters and the autocorrelation sequence for an Autoregressive (AR) Process.

$$\begin{array}{rl}\tilde{\Gamma }{\tilde{a}}_k& ={\tilde{\sigma }}_w^2\\ & \\ \left[\begin{array}{ccccc}{\gamma }_{xx}(0)& {\gamma }_{xx}(-1)& {\gamma }_{xx}(-2)& \dots & {\gamma }_{xx}(-p)\\ {\gamma }_{xx}(1)& {\gamma }_{xx}(0)& {\gamma }_{xx}(-1)& \dots & {\gamma }_{xx}(-p+1)\\ ⋮& ⋮& ⋮& & ⋮\\ {\gamma }_{xx}(p)& {\gamma }_{xx}(p-1)& {\gamma }_{xx}(p-2)& \dots & {\gamma }_{xx}(0)\end{array}\right]\left[\begin{array}{c}1\\ a_1\\ ⋮\\ a_p\end{array}\right]& =\left[\begin{array}{c}{\gamma }_{xx}(1)\\ {\gamma }_{xx}(2)\\ ⋮\\ {\gamma }_{xx}(p)\end{array}\right]\end{array}$$