Solve Exercise 7.21 by estimating the AR parameters using the Levinson-Durbin recursions with the Burg reflection coefficients.

Exercise 7.21

Use the covariance method to estimate the PSD of \(L=1024\) samples for the following signal:

\[x(n)=\cos \frac{2 \pi}{L} n+x_{1}(n)\]
where \[x_{1}(n)=a x_{1}(n-1)+x_{2}(n),\]
with \(x_{2}(n)\) being a white Gaussian noise with variance \(1-a^{2}\).

(a) Choose \(a=-0.9\) and \(N=4\).

(b) Choose \(a=-0.9\) and \(N=20\).

(c) Choose \(a=0.9\) and \(N=4\).
Comment on your results.