discrete signals - Linear Prediction of AR Process

A discrete signal x is generated by the recursive process $$ x_n = x_{n-1} - 0.2 x_{n-2} + w_n $$

where $w_n$ is white noise with zero mean and unit variance. What is the optimum order of a linear predictor for this signal? What are the values of the prediction coefficients? What is the average power of the residual?

I would really appreciate if someone could help me with this question.

It's a past paper question not homework.

Thanks

Answer

From the definition of the process you know that

$$x_{n+1}=x_n-0.2x_{n-1}+w_{n+1}\tag{1}$$

Since $w_n$ is white you can't predict it, so the best linear predictor for the given process is the filter

$$P(z)=1-0.2z^{-1}\tag{2}$$

which is a first order filter. It estimates the future sample $x_{n+1}$ by computing

$$\hat{x}_{n+1}=x_n-0.2x_{n-1}\tag{3}$$

The residual error is equal to $w_n$, which has an average power of $1$.