Why does downsampling increase the phase angle of poles in autoregressive models?

Question

When reading about autoregressive models (AR), people often "downsample to increase the phase angle of the poles of the corresponding transfer function".

How exactly does downsampling affect the phase angle of the poles?

Answer 1

Decimating a signal $x[n]$ by an integer factor $M$ results in the following $\mathcal{Z}$-transform:

$$Y(z)=\frac{1}{M}\sum_{k=0}^{M-1}X\left(e^{-j2\pi k/M}z^{1/M}\right)\tag{1}$$

If you assume that $x[n]$ was filtered by an anti-aliasing filter before decimation, the transfer function of the decimated filter can be approximated by the term for $k=0$ in (1):

$$Y(z)\approx\frac{1}{M}X(z^{1/M})\tag{2}$$

So if $z_{\infty}$ is a pole of $X(z)$ then $z_{\infty}^{M}$ is a pole of $Y(z)$. Writing the pole in polar coordinates $z_{\infty}=re^{j\phi}$ gives

$$z_{\infty}^M=r^Me^{jM\phi}$$

So the angle of the pole is increased by a factor $M$.