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.


I found the answer:

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 $$e^{\log_2(G)} $$

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