In Exercise 7.4, consider \(a=0.8\) and plot the minimum-variance solution for window lengths equal to \(L=4, L=10\), and \(L=50\). Compare the estimated PSD in each case with the actual PSD and comment on the results.

Exercise 7.4

An AR process is generated by applying white Gaussian noise, with variance \(\sigma_{X}^{2}\), to a first-order filter with transfer function

\[H(z)=\frac{z}{z-a} .\]

This process has the autocorrelation matrix

\[\mathbf{R}_{Y}=\frac{\sigma_{X}^{2}}{1-a^{2}}\left[\begin{array}{cccc}1 & a & \cdots & a^{7} \\a & 1 & \cdots & a^{6} \\\vdots & \vdots & \ddots & \vdots \\a^{7} & a^{6} & \cdots & 1\end{array}\right]\]

whose inverse can be shown to be \[\mathbf{R}_{Y}^{-1}=\frac{1}{\sigma_{X}^{2}}\left[\begin{array}{ccccc}
1 & -a & \cdots & 0 & 0 \\
-a & 1+a^{2} & \cdots & 0 & 0 \\
0 & -a & \cdots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & 1+a^{2} & -a \\
0 & 0 & \cdots & -a & 1 \end{array}\right]\]
For the signal above, calculate a closed-form solution for its minimum-variance estimate and comment on this solution when \(L\) approaches infinity.