Signal:noise

Signal-to-noise ratio (abbreviated SNR, S/N, or S:N) is a measure in science and engineering that compares the level of a desired signal relative to the background noise. It is defined as the ratio of signal power to noise power. A ratio greater to one indicates that there is more signal than noise. Signal-to-noise ratio is commonly used to describe electrical signals and time series, but can be used to describe any measurement with a blend of useful information and unwanted noise.

Definition
Signal-to-noise ratio is defined as the power ratio between a signal (meaningful information) and the background noise (unwanted signal):
 * $$\mathrm{S:N} = \frac{P_\mathrm{signal}}{P_\mathrm{noise}},$$

where P is average power. Both signal and noise power must be measured at the same or equivalent points in a time series. The SNR can be obtained by calculating the square of the amplitude ratio:
 * $$\mathrm{S:N} = \frac{P_\mathrm{signal}}{P_\mathrm{noise}} = \left ( \frac{A_\mathrm{signal}}{A_\mathrm{noise} } \right )^2,$$

where A is root mean square (RMS) amplitude. Because many signals have a very wide dynamic range, SNRs are often expressed using the logarithmic decibel scale. In decibels, the SNR is defined as
 * $$\mathrm{S:N_{dB}} = 10 \log_{10} \left ( \frac{P_\mathrm{signal}}{P_\mathrm{noise}} \right ) = {P_\mathrm{signal,dB} - P_\mathrm{noise,dB}},$$

which may equivalently be written using amplitude ratios as
 * $$\mathrm{S:N_{dB}} = 10 \log_{10} \left ( \frac{A_\mathrm{signal}}{A_\mathrm{noise}} \right )^2 = 20 \log_{10} \left ( \frac{A_\mathrm{signal}}{A_\mathrm{noise}} \right ).$$