Estimation of AR Parameters

We estimate the AR parameters with the sinusoidal parameters fixed. The derivatives for the AR parameters may be found in Section 10.2.7 (assume $B_k=1$). But, we do not need to use iterative methods to find the ML parameters that maximize (9.13) since we can obtain them in one step using Levinson algorithm. Treating the residual error ${\bf e} = {\bf x}-{\bf y}$ as AR noise, we compute the auto-correlation function using the frequency-domain method by taking the inverse FFT of the magnitude-squared of the FFT of ${\bf e}$. The Levinson algorithm is then used to find the AR coefficients and $\sigma^2$.

The Fisher's information matrix for AR modeling was given in Section 5.2.8 in terms of the CR bound matrix ${\bf C}$ which is the inverse of the FIM. We have

$${\bf C}_{\bf a} = \frac{1}{\sigma^2 N}{\bf R},$$ (9.16)

where ${\bf R}$ is the $P\times P$ auto-correlation matrix. The CR bound for $\sigma^2$ is independent of ${\bf I}_{\bf a}$ and given by

$${\bf C}_{\sigma^2}=\frac{2\sigma^4}{N}.$$ (9.17)