To derive the Yule–Walker equations, we multiply the AR model by $x[n-m]$ and take expectations: (m represents the lag (or time delay))
$$ E[x[n]x[n-m]] = \sum_{k=1}^{p} \phi_k E[x[n-k]x[n-m]] + E[e[n]x[n-m]] $$
Since $e[n]$ is white noise and uncorrelated with past samples:
$$ E[e[n]x[n-m]] = 0 \quad \text{for } m \ge 1 $$
Using the definition of autocorrelation, we obtain:
$$ R[m] = \sum_{k=1}^{p} \phi_k R[m-k], \quad m = 1,2,\dots,p $$
These equations can be written in matrix form 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 matrix is Toeplitz because each element depends only on the lag difference.
Spectral Estimation (AR-Modeled Yule-Walker)
Follow the steps to simulate (1. Generate input signal 2. Simulate)
Parameters
Input Signal
AWGN Noise
Operations
PSD Estimation
Real Data Input
Upload CSV
Input Signal
Output
Noise Variance
Once the AR coefficients are estimated, the driving noise variance is given by:
$$ \sigma_e^2 = R[0] - \sum_{k=1}^{p} \phi_k R[k] $$
This represents the portion of the signal power that cannot be explained by the AR model. In other words, even after using past samples to estimate the current value, there is still some remaining randomness or error. That leftover uncertainty is what the noise variance measures.
Power Spectral Density (PSD)
The PSD of the AR process is:
$$ P_x(f) = \frac{\sigma_e^2}{\left|1 - \sum_{k=1}^{p} \phi_k e^{-j 2 \pi f k}\right|^2} $$
The denominator corresponds to the frequency response of the all-pole system.
Frequencies where the denominator is small correspond to peaks in the spectrum.