Ratio of the desired signal to the background noise
Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in decibels. A ratio higher than 1:1 (greater than 0 dB) indicates more signal than noise.
Because many signals have a very wide dynamic range, signals are often expressed using the logarithmicdecibel scale. Based upon the definition of decibel, signal and noise may be expressed in decibels (dB) as
In a similar manner, SNR may be expressed in decibels as
Using the definition of SNR
Using the quotient rule for logarithms
Substituting the definitions of SNR, signal, and noise in decibels into the above equation results in an important formula for calculating the signal to noise ratio in decibels, when the signal and noise are also in decibels:
In the above formula, P is measured in units of power, such as watts (W) or milliwatts (mW), and the signal-to-noise ratio is a pure number.
However, when the signal and noise are measured in volts (V) or amperes (A), which are measures of amplitude,[note 1] they must first be squared to obtain a quantity proportional to power, as shown below:
The concepts of signal-to-noise ratio and dynamic range are closely related. Dynamic range measures the ratio between the strongest un-distorted signal on a channel and the minimum discernible 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 VRMS).
SNR is usually taken to indicate an average signal-to-noise ratio, as it is possible that 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'.
Difference from conventional power
In physics, the average power of an AC signal is defined as the average value of voltage times current; for resistive (non-reactive) circuits, where voltage and current are in phase, this is equivalent to the product of the rms voltage and current:
But in signal processing and communication, one usually assumes that  so that factor is usually not included while measuring power or energy of a signal. This may cause some confusion among readers, but the resistance factor is not significant for typical operations performed in signal processing, or for computing power ratios. For most cases, the power of a signal would be considered to be simply
where is the signal mean or expected value and is the standard deviation of the noise, or an estimate thereof.[note 2] Notice that such an alternative definition is only useful for variables that are always non-negative (such as photon counts and luminance), and it is only an approximation since . It is commonly used in image processing, where the SNR of an image is usually calculated as the ratio of the mean pixel value to the standard deviation of the pixel values over a given neighborhood.
This definition is closely related to the sensitivity index or d', when assuming that the signal has two states separated by signal amplitude , and the noise standard deviation does not change between the two states.
The Rose criterion (named after Albert Rose) states that an SNR of at least 5 is needed to be able to distinguish image features with certainty. An SNR less than 5 means less than 100% certainty in identifying image details.
Output signal-to-noise ratio (of AM receiver) is given by
Channel signal-to-noise ratio is given by
Output signal-to-noise ratio is given by
Recording of the noise of a thermogravimetric analysis device that is poorly isolated from a mechanical point of view; the middle of the curve shows a lower noise, due to a lesser surrounding human activity at night.
All real measurements are disturbed by noise. This includes electronic noise, but can also include external events that affect the measured phenomenon — wind, vibrations, the 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.
Internal electronic noise of measurement systems can be reduced through the use of low-noise amplifiers.
When the characteristics of the noise are known and are different from the signal, it is possible to use a filter to reduce the noise. For example, a lock-in amplifier can extract a narrow bandwidth signal from broadband noise a million times stronger.
When the signal is constant or periodic and the noise is random, it is possible to enhance the SNR by averaging the measurements. In this case the noise goes down as the square root of the number of averaged samples.
When a measurement is digitized, 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 already noisy (as is usually the case), the signal's 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, including the intentional addition of dither.
Although noise levels in a digital system can be expressed using SNR, it is more common to use Eb/No, the energy per bit per noise power spectral density.
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:
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-scalesine 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 and uniform distribution. In this case, the SNR is approximately
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:
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.
Optical signals have a carrier frequency (about 200 THz and more) that is much higher than the modulation frequency. This way the noise covers a bandwidth that is much wider than the signal itself. The resulting signal influence relies mainly on the filtering of the noise. To describe the signal quality without taking the receiver into account, the optical SNR (OSNR) is used. The OSNR is the ratio between the signal power and the noise power in a given bandwidth. Most commonly a reference bandwidth of 0.1 nm is used. This bandwidth is independent of the modulation format, the frequency and the receiver. For instance an OSNR of 20 dB/0.1 nm could be given, even the signal of 40 GBit DPSK would not fit in this bandwidth. OSNR is measured with an optical spectrum analyzer.
^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.
^The exact methods may vary between fields. For example, if the signal data are known to be constant, then can be calculated using the standard deviation of the signal. If the signal data are not constant, then can be calculated from data where the signal is zero or relatively constant.
^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.
^Charles Sherman and John Butler (2007). Transducers and Arrays for Underwater Sound. Springer Science & Business Media. p. 276. ISBN 9780387331393.CS1 maint: uses authors parameter (link)
^Michael A. Choma, Marinko V. Sarunic, Changhuei Yang, Joseph A. Izatt. Sensitivity advantage of swept source and Fourier domain optical coherence tomography. Optics Express, 11(18). Sept 2003.
^Gabriel L. A. de Sousa and George C. Cardoso (18 June 2018). "A battery-resistor analogy for further insights on measurement uncertainties". Physics Education. IOP Publishing. 53: 055001. arXiv:1611.03425. doi:10.1088/1361-6552/aac84b. Retrieved 5 May 2021.CS1 maint: uses authors parameter (link)
D. J. Schroeder (1999). Astronomical optics (2nd ed.). Academic Press. p. 278. ISBN 978-0-12-629810-9., p.278
^ abBushberg, J. T., et al., The Essential Physics of Medical Imaging, (2e). Philadelphia: Lippincott Williams & Wilkins, 2006, p. 280.
^Rafael C. González, Richard Eugene Woods (2008). Digital image processing. Prentice Hall. p. 354. ISBN 978-0-13-168728-8.
^Tania Stathaki (2008). Image fusion: algorithms and applications. Academic Press. p. 471. ISBN 978-0-12-372529-5.
^Jitendra R. Raol (2009). Multi-Sensor Data Fusion: Theory and Practice. CRC Press. ISBN 978-1-4398-0003-4.
^John C. Russ (2007). The image processing handbook. CRC Press. ISBN 978-0-8493-7254-4.
Rose, Albert (1973). Vision – Human and Electronic. Plenum Press. p. 10. ISBN 9780306307324. [...] to reduce the number of false alarms to below unity, we will need [...] a signal whose amplitude is 4–5 times larger than the rms noise.
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
Interactive webdemo of visualization of SNR in a QAM constellation diagram Institute of Telecommunicatons, University of Stuttgart
Bernard Widrow,István Kollár (2008-07-03), Quantization Noise: Roundoff Error in Digital Computation, Signal Processing, Control, and Communications, Cambridge University Press, Cambridge, UK, 2008. 778 p., ISBN 9780521886710
Quantization Noise Widrow & Kollár Quantization book page with sample chapters and additional material
Signal-to-noise ratio online audio demonstrator - Virtual Communications Lab