# Spectral flatness or Wiener Entropy for AR(1) and AR(2)

I'm sudiying compressibility of random processes by using Spectral flatness aka Wiener Entropy

I would like to know if there is any reference which derives this quantity, for autoregressive processes AR(1) and AR(2) in terms of their coeficients.

The AR(1) Gaussian process is defined as:

$$x(n) = a x(n-1) + \sqrt{1-a^2}\epsilon (n)$$

Where $\epsilon (n)$ is zero-mean stationary white Gaussian noise.

It has PSD: $$\gamma_{xx}(f)=\frac{\sigma^2_{x}(1-a^2)}{1-2a\cos(2\pi f)+a^2} \ \ \ |f|<1/2$$

Where $$\frac{\sigma^2_{x}}{\sigma^2_{\epsilon}}=\frac{1}{1-a^2}$$

The AR(2) is defined as:

$$x(n) = a_2 x(n-2)+a_1 x(n-1) + \sqrt{G}\epsilon (n)$$

The definition of $\epsilon (n)$ is the same as above.

Its PSD reads: $$\gamma_{xx}(f)=\frac{\sigma^2_{x}G}{|1-a_1 e^{-j2\pi f}-a_2 e^{-j4\pi f}|^2} \ \ \ |f|<1/2$$

Here: $$\frac{\sigma^2_{x}}{\sigma^2_{\epsilon}}=\frac{1}{G}$$

According to this this Paper for an AR(p) Gaussian, the Spectral Flatness Measure (SFM) is very easy to compute:

$$\hbox{SFM=}e^{-2\rho}\ \hbox{ where } \rho=\frac{1}{2}\log_2\left(\frac{\sigma^2_{x}}{\sigma^2_{\epsilon}}\right)$$

Their SFM is respectively:

$$e^{\log_2(1-a^2)}$$ and $$e^{\log_2(G)}$$