Why does the matrix contain only \(R(n)\) intead of X(n)?
The unknowns in the Yule–Walker equations are the AR coefficients \(\phi_1, \phi_2, \ldots, \phi_p\). To estimate these coefficients, we need equations that contain known quantities.
The known quantities are the autocorrelation values \(R[0], R[1], \ldots, R[p]\), which are computed from the signal. Therefore, the Yule–Walker equations can be written as:
$$ \underbrace{ \begin{bmatrix} R[0] & R[1] & \cdots & R[p-1] \\ R[1] & R[0] & \cdots & R[p-2] \\ \vdots & \vdots & \ddots & \vdots \\ R[p-1] & R[p-2] & \cdots & R[0] \end{bmatrix} }_{\text{Known autocorrelation matrix}} \; \underbrace{ \begin{bmatrix} \phi_1\\ \phi_2\\ \vdots\\ \phi_p \end{bmatrix} }_{\text{Unknown AR coefficients}} = \underbrace{ \begin{bmatrix} R[1]\\ R[2]\\ \vdots\\ R[p] \end{bmatrix} }_{\text{Known autocorrelation vector}} $$
The only unknowns are the coefficients \(\phi_1, \phi_2, \ldots, \phi_p\), while every \(R[k]\) is already known (or estimated from the data).
If we instead used the signal samples \(x[0], x[1], x[2], \ldots\), the equations would depend on one specific realization of the signal rather than its statistical properties. Individual samples vary randomly and generally do not satisfy the Yule–Walker equations exactly.
Autocorrelation values \(R[k]\) measure the average relationship between samples separated by a lag of \(k\). By averaging over many observations, they reduce the effect of random noise and provide a more reliable description of the underlying process. As a result, the Yule–Walker equations form a stable linear system whose coefficient matrix is symmetric and Toeplitz, allowing the AR coefficients to be estimated efficiently.