Signal averaging

Signal averaging is a signal processing technique applied in the time domain, intended to increase the strength of a signal relative to noise that is obscuring it. By averaging a set of replicate measurements, the signal-to-noise ratio, S/N, will be increased, ideally in proportion to the square root of the number of measurements.

Assumed that

Assuming we sample the noise, we get a per-sample variance of

${\displaystyle \mathrm {Var} (z)=E[z^{2}]=\sigma ^{2}}$.

Averaging a random variable leads to the following variance:

${\displaystyle \mathrm {Var} \left({\frac {1}{n}}\sum _{i=1}^{n}z_{i}\right)={\frac {1}{n^{2}}}\mathrm {Var} \left(\sum _{i=1}^{n}z_{i}\right)={\frac {1}{n^{2}}}\sum _{i=1}^{n}\mathrm {Var} \left(z_{i}\right)}$.

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