If the ACF estimates are replaced by the true ACF, the relationship holds exactly and is called the Yule-Walker equations. The Yule-Walker equation relates the AR coefficients to the ACF coefficients as follows (example of shown):
Now we have come full-circle. From the original AR coefficients, we can write the theoretical power spectrum (9.8), which through the inverse FT becomes the theoretical ACF (9.1), which through the Yule-Walker equations and the Levinson algorithm becomes the AR coefficients again. Note that the Levinson algorithm also produces the reflection coefficients (RC) as a by-product . If estimates of the ACF are used in place of the theoretical ACF, we obtain estimates of the AR coefs. These AR coefficient estimates can then be transformed into an AR power spectrum estimate. Note that the AR model has the ``ACF-matching" property that the theoretical ACF corresponding to the AR coefs will match the ACF estimates up to lag . Note that MATLAB examples in this section can be found in software/ar_example.m.