I was going through AR modeling.
The AR model of a covariance stationary process can be expressed as:
$$x[n]=\sum\limits_{i=1}^{p} \alpha_i x[n-i] + \epsilon[n]$$
where $p$ is the model order and $\epsilon[n]$ is the residual
1) Why is it that the residual $\epsilon[n]$ is a white noise? Is it by definition or can it be shown to be a white noise?
2) Where is the assumption on the stationarity of $x[n]$ used? In other words, if $x[n]$ is not stationary, what would have changed?
Thanks,
it's a hand-wavy argument, but the idea is to derive the coefficients $\alpha_i$ so that the norm of the error $\Big\||\epsilon[n]|^2\Big\|$ is minimized. we assume that $x[n]$ is a stationary "random" process, we know all of the previous samples $x[n-i]$ and we want to make a good guess at $x[n]$ with a linear combination of the previous samples. we can only make some guess of the next $x[n]$ if it is not a white random process. if it is white, knowing the previous samples is not going to help us at all. if it is not white, then the autocorrelation of $x[n]$ is not zero for non-zero lags and from that we can compute $\alpha_i$.
now if $\epsilon[n]$ was not white, we could do further LPC to make a good guess at $\epsilon[n]$ and make a better guess for $x[n]$, but if the coefficients $\epsilon[n]$ are optimal, then we can't make a better guess for $x[n]$. so then the error of our guess for $x[n]$ and what we really get has to be white.
