Signal-to-noise ratio (often abbreviated SNR or S/N) is a measure used in science and engineering to quantify how much a signal has been corrupted by noise. It is defined as the ratio of signal power to the noise power corrupting the signal. A ratio higher than 1:1 indicates more signal than noise. While SNR is commonly quoted for electrical signals, it can be applied to any form of signal (such as isotope levels in an ice core or biochemical signaling between cells).

In less technical terms, signal-to-noise ratio compares the level of a desired signal (such as music) to the level of background noise. The higher the ratio, the less obtrusive the background noise is. "Signal-to-noise ratio" is sometimes used informally to refer to the ratio of useful information to false or irrelevant data in a conversation or exchange. For example, in online discussion forums and other online communities, off-topic posts and spam are regarded as "noise" that interferes with the "signal" of appropriate discussion.

Signal-to-noise ratio is defined as the power ratio between a signal (meaningful information) and the background noise (unwanted signal):

${\displaystyle \mathrm {SNR} ={\frac {P_{\mathrm {signal} )){P_{\mathrm {noise} ))},}$

where P is average power. Both signal and noise power must be measured at the same or equivalent points in a system, and within the same system bandwidth. If the signal and the noise are measured across the same impedance, then the SNR can be obtained by calculating the square of the amplitude ratio:

${\displaystyle \mathrm {SNR} ={\frac {P_{\mathrm {signal} )){P_{\mathrm {noise} ))}=\left({\frac {A_{\mathrm {signal} )){A_{\mathrm {noise} ))}\right)^{2},}$

where A is root mean square (RMS) amplitude (for example, RMS voltage). 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

${\displaystyle \mathrm {SNR_{dB)) =10\log _{10}\left({\frac {P_{\mathrm {signal} )){P_{\mathrm {noise} ))}\right)={P_{\mathrm {signal,dB} }-P_{\mathrm {noise,dB} )),}$

which may equivalently be written as

${\displaystyle \mathrm {SNR_{dB)) =10\log _{10}\left({\frac {A_{\mathrm {signal} )){A_{\mathrm {noise} ))}\right)^{2}=20\log _{10}\left({\frac {A_{\mathrm {signal} )){A_{\mathrm {noise} ))}\right).}$

The concepts of signal-to-noise ratio and dynamic range are closely related. Dynamic range measures the ratio between the greatest un-distorted signal on a channel and the smallest detectable signal, which for most purposes is the noise level. SNR measures the ratio between an arbitrary signal level (not necessarily the most powerful signal possible) and noise. Measuring signal-to-noise ratios requires the selection of a representative or reference signal. In audio engineering, the reference signal is usually a sine wave at a standardized nominal or alignment level, such as 1 kHz at +4 dBu (1.228 V_RMS).

SNR is usually taken to indicate an average signal-to-noise ratio, as it is possible that (near) instantaneous signal-to-noise ratios will be considerably different. The concept can be understood as normalizing the noise level to 1 (0 dB) and measuring how far the signal 'stands out'.

A common alternative definition of SNR is the ratio of mean to standard deviation of a signal or measurement:^[1][2]

${\displaystyle \mathrm {SNR} ={\frac {\mu }{\sigma ))}$

where ${\displaystyle \mu }$ is the signal mean or expected value and ${\displaystyle \sigma }$ is the standard deviation of the noise, or an estimate thereof. The exact methods may vary between fields. For example, if the signal data are known to be constant, then ${\displaystyle \sigma }$ can be calculated using the standard deviation of the signal. If the signal data are not constant, then ${\displaystyle \sigma }$ can be calculated from data where the signal is zero or relatively constant.

All real measurements are disturbed by noise. This includes electronic noise, but can also include external events that affect the measured phenomenon — wind, vibrations, gravitational attraction of the moon, variations of temperature, variations of humidity, etc. depending on what is measured and of the sensitivity of the device. It is often possible to reduce the noise by controlling the environment. Otherwise, when the characteristics of the noise are known and are different from the signals, it is possible to filter it or to process the signal. When the signal is constant or periodic and the noise is random, it is possible to enhance the SNR by averaging the measurement.

In image processing, the SNR of an image is usually defined as the ratio of the mean pixel value to the standard deviation of the pixel values, that is, SNR${\displaystyle =\mu /\sigma }$ (the inverse of the coefficient of variation). Related measures are the "contrast ratio" and the "contrast-to-noise ratio".

The connection between optical power and voltage in an imaging system is linear. This usually means that the SNR of the electrical signal is calculated by the 10 log rule. With an interferometric system, however, where interest lies in the signal from one arm only, the field of the electromagnetic wave is proportional to the voltage (assuming that the intensity in the second, the reference arm is constant). Therefore the optical power of the measurement arm is directly proportional to the electrical power and electrical signals from optical interferometry are following the 20 log rule.^[3]

The Rose criterion (named after Albert Rose) states that an SNR of at least 5 is needed to be able to distinguish image features at 100% certainty. An SNR less than 5 means less than 100% certainty in identifying image details.^[4]

When a measurement is digitised, the number of bits used to represent the measurement determines the maximum possible signal-to-noise ratio. This is because the minimum possible noise level is the error caused by the quantization of the signal, sometimes called Quantization noise. This noise level is non-linear and signal-dependent; different calculations exist for different signal models. Quantization noise is modeled as an analog error signal summed with the signal before quantization ("additive noise").

This theoretical maximum SNR assumes a perfect input signal. If the input signal is noisy (as is usually the case), the measurement noise may be larger than the quantization noise. Real analog-to-digital converters also have other sources of noise that further decrease the SNR compared to the theoretical maximum from the idealized quantization noise.

Although noise levels in a digital system can be expressed using SNR, it is more common to use E_b/N_o, the energy per bit per noise power spectral density.

The modulation error ratio (MER) is a measure of the SNR in a digitally modulated signal.

For n-bit integers with equal distance between quantization levels (uniform quantization) the dynamic range (DR) is also determined.

Assuming a uniform distribution of input signal values, the quantization noise is a uniformly-distributed random signal with a peak-to-peak amplitude of one quantization level, making the amplitude ratio 2ⁿ/1. The formula is then:

${\displaystyle \mathrm {DR_{dB)) =\mathrm {SNR_{dB)) =20\log _{10}(2^{n})\approx 6.02\cdot n}$

This relationship is the origin of statements like "16-bit audio has a dynamic range of 96 dB". Each extra quantization bit increases the dynamic range by roughly 6 dB.

Assuming a full-scale sine wave signal (that is, the quantizer is designed such that it has the same minimum and maximum values as the input signal), the quantization noise approximates a sawtooth wave with peak-to-peak amplitude of one quantization level^[5] and uniform distribution. In this case, the SNR is approximately

${\displaystyle \mathrm {SNR_{dB)) \approx 20\log _{10}(2^{n}{\sqrt {3/2)))\approx 6.02\cdot n+1.761}$

Floating-point numbers provide a way to trade off signal-to-noise ratio for an increase in dynamic range. For n bit floating-point numbers, with n-m bits in the mantissa and m bits in the exponent:

${\displaystyle \mathrm {DR_{dB)) =6.02\cdot 2^{m))$

${\displaystyle \mathrm {SNR_{dB)) =6.02\cdot (n-m)}$

Note that the dynamic range is much larger than fixed-point, but at a cost of a worse signal-to-noise ratio. This makes floating-point preferable in situations where the dynamic range is large or unpredictable. Fixed-point's simpler implementations can be used with no signal quality disadvantage in systems where dynamic range is less than 6.02m. The very large dynamic range of floating-point can be a disadvantage, since it requires more forethought in designing algorithms.^[6]

• Often special filters are used to weight the noise: DIN-A, DIN-B, DIN-C, DIN-D, CCIR-601; for video, special filters such as comb filters may be used.
• Maximum possible full scale signal can be charged as peak-to-peak or as RMS. Audio uses RMS, Video P-P, which gave +9 dB more SNR for video.

• Audio system measurements
• Video quality
• Subjective video quality
• Near-far problem
• Peak signal-to-noise ratio
• Total harmonic distortion
• Noise margin

• ADC and DAC Glossary - Maxim Integrated Products
• Understand SINAD, ENOB, SNR, THD, THD + N, and SFDR so you don't get lost in the noise floor - Analog Devices
• The Relationship of dynamic range to data word size in digital audio processing
• Calculation of signal-to-noise ratio, noise voltage, and noise level
• Learning by simulations - a simulation showing the improvement of the SNR by time averaging
• Dynamic Performance Testing of Digital Audio D/A Converters
• Fundamental theorem of analog circuits: a minimum level of power must be dissipated to maintain a level of SNR