Estimate the PSD of the ARMA system output in Exercise 7.11 to a unit-variance white-noise input. Use the standard periodogram method and the autocorrelation method with \(N=1,2,3,4\) and compare the results in each case.

Exercise 7.11

Find an \(N\) th-order AR approximation for the ARMA system

\[H(z)=\frac{\left(1-0.8 z^{-1}\right)\left(1-0.9 z^{-1}\right)}{\left(1-0.5 z^{-1}\right)\left(1+0.5 z^{-1}\right)}\]

and compare the resulting magnitude response with the original ARMA system.