Signal to noise ratio¶
Compares the level of a desired signal to the level of background noise. A ratio higher than 1:1 indicates more signal than noise.
Given a Linear Gaussian system:
\[x \sim \mathcal{N}(\mu_0, \Sigma_0)\]
System $\(y = x + \epsilon\)$
where
\[\epsilon \sim \mathcal{N}(0, \Sigma_y)\]
is the noise term:
Then the signal to noise ratio is:
\[
SNR \triangleq \frac{E[X^2]}{E[\epsilon^2]} = \frac{
\Sigma_0 + \mu_0^2}{
\Sigma_y}
\]