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:

$SNR \triangleq \frac{E[X^2]}{E[\epsilon^2]} = \frac{ \Sigma_0 + \mu_0^2}{ \Sigma_y}$

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