# How to characterize this whitening filter?

A little bit of background: I am interested in whitening in the least-square estimation.

In this sense, consider a univariate signal $$y$$. Assuming the covariance of $$y$$ is $$\mathbf{A}\sigma^2$$. If we knew matrix $$\bf A$$, then, whitenning of this signal could be done by: $$\mathbf{A}^{-1/2}\bf y$$. Since the $$\mathbf{A}^{-1/2}\bf y$$ has a covariance matrix of $$\mathbf{A}^{-1/2} \bf A \mathbf{A}^{-1/2}\sigma^2 = I\sigma^2$$. Thus, $$\mathbf{A}^{-1/2}\bf y$$ is whitened.

In practice, the $$A$$ is unknown but can be estimated using different methods such maximum likelihood or Yule-Walker. Here is what I am interested to figure out.

To be more specific, for this purpose one can use Yule-Walker equation in particular and assume an AR(p) model, say order $$p=1$$ and estimate the model parameter of this autoregressive.

This is indeed a filter, but my question is how to characterize this filter in frequency domain? How can I find the $$H(z)$$ of this filter? Is there a way to find its zeros and poles and further know the characteristics of this filter?

I would appreciate if you could help me.