# Understanding the signal to noise ratio (SNR)

In my work, I came across the following expression of signal to noise ratio (SNR) which contains expectation operator.

$$\textbf{y}(n) = \textbf{h}s(n)+\textbf{u}(n)\tag{1}$$

where $$\textbf{u}(n)$$ is additive white Gaussian noise vector, $$s(n)$$ is transmitted signal, $$\textbf{h}$$ is Rayleigh channel vector.

The SNR of (1) is expressed as

SNR = $$E(||\textbf{h}s(n)||^2_2)$$/$$E(||\textbf{u}(n)||^2_2)\tag{2}$$

where $$||.||_2$$ is Euclidean norm of a vector and $$E(.)$$ denotes statistical expectation.

My query is why statistical expectation appears in the expression number (2).

Any help in this regard will be highly appreciated.

• See, e.g. this previous answer. Jan 11 at 3:35
• Thank you so much sir for your detailed answer .....Very well written answer... Jan 11 at 9:18

Typically signal to noise ratio is quoted over a time average. Remember that \begin{align} \textbf{u}(n) \end{align} is additive white noise. This means that the values $$\textbf{u}(n)$$ are random and could be very big or very small based on the probability distribution of the noise. If desired, the SNR could be calculated for each sample but would vary wildly with the noise in that sample. To remove this time dependence and create a "useful" metric the expectation value (aka Average) across samples is taken