So suppose I have the following signal:
$$ x(t) = s(t) + n(t) $$
where $n(t) \sim \mathcal{N}(0, \sigma_{n}^{2})$ and $s(t)$ is an autoregressive process of order $P$. My question -- If I model $x(t)$ with an autoregressive process of order $P$, how are the roots of the estimated AR process related to the original?
I can think of some ideas:
From the paper Bounding the Roots of Polynomials by Holly P. Hirst and Wade T. Macey, the authors claim that the radius of the roots can be bounded by:
$$ | z| \leq \sum_{p=0}^{P-1} |a_{p}| $$
So then if $\hat{a}_{p}$ are the estimated AR coefficients, and we could say that $\hat{a}_{p} \sim \mathcal{N}(a_{p}, \sigma^{2}_{n})$, then we could estimate the pdf of $|z|$ and get some information that way.
However -- is that claim true? Can we simply say $\hat{a}_{p} \sim \mathcal{N}(a_{p}, \sigma^{2}_{n})$?
In the paper The Effects of Noise on the Autoregressive Spectral Estimator by Steven Kay, the author shows that noise flattens the autoregressive spectrum, suggesting that the angle of the roots stays the same, but the magnitude moves towards the origin.
However, it isn't clear how the estimated coefficients $\hat{a}_{p}$ relate to the true coefficients $a_{p}$ due to noise.
TLDR Questions:
If I model a noisy signal with an autoregressive process of order $P$, how are the roots of the estimated AR process related to the AR process of the true underlying signal? Can we simply say $\hat{a}_{p} \sim \mathcal{N}(a_{p}, \sigma^{2}_{n})$?
Everything I see about bounding the roots of polynomials seems to care only about the radius, ie the magnitude of the roots. But what about the angle? Can we bound that at all via the coefficients?