In signal processing, signal averaging is a time domain technique applied to repetitive signals, used to recover waveforms that are obscured by random noise. By averaging a set of repeated measurements of the same signal, the signal-to-noise ratio (SNR) increases, ideally in proportion to the square root of the number of repetitions.
Deriving the SNR for averaged signals Assumed that
Signal s ( t ) {\displaystyle s(t)} is uncorrelated to noise, and noise z ( t ) {\displaystyle z(t)} is uncorrelated : E [ z ( t ) z ( t − τ ) ] = 0 = E [ z ( t ) s ( t − τ ) ] ∀ t , τ {\displaystyle E[z(t)z(t-\tau )]=0=E[z(t)s(t-\tau )]\forall t,\tau } . Signal power P s i g n a l = E [ s 2 ] {\displaystyle P_{signal}=E[s^{2}]} is constant in the replicate measurements. Noise is random, with a mean of zero and constant variance in the replicate measurements: E [ z ] = 0 = μ {\displaystyle E[z]=0=\mu } and 0 < E [ ( z − μ ) 2 ] = E [ z 2 ] = P n o i s e = σ 2 {\displaystyle 0<E[\left(z-\mu \right)^{2}]=E[z^{2}]=P_{noise}=\sigma ^{2}} . We (canonically) define Signal-to-Noise ratio as S N R = P s i g n a l P n o i s e = E [ s 2 ] σ 2 {\displaystyle SNR={\frac {P_{signal}}{P_{noise}}}={\frac {E[s^{2}]}{\sigma ^{2}}}} .
Noise power for sampled signals Assuming we sample the noise, we get a per-sample variance of
V a r ( z ) = E [ z 2 ] = σ 2 {\displaystyle \mathrm {Var} (z)=E[z^{2}]=\sigma ^{2}} . Averaging a random variable leads to the following variance:
V a r ( 1 n ∑ i = 1 n z i ) = 1 n 2 V a r ( ∑ i = 1 n z i ) = 1 n 2 ∑ i = 1 n V a r ( z i ) {\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)} . Since noise variance is constant σ 2 {\displaystyle \sigma ^{2}} :
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