# Signal-to-noise ratio

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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.

## Contents

SNR, bandwidth, and channel capacity of a communication channel are connected by the Shannon–Hartley theorem.

## Definition

Signal-to-noise ratio is defined as the ratio of the power of a signal (meaningful input) to the power of background noise (meaningless or unwanted input):

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

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.

Depending on whether the signal is a constant (s) or a random variable (S), the signal-to-noise ratio for random noise N becomes: [1]

${\displaystyle \mathrm {SNR} ={\frac {s^{2}}{\mathrm {E} [N^{2}]}}}$

where E refers to the expected value, i.e. in this case the mean square of N, or

${\displaystyle \mathrm {SNR} ={\frac {\mathrm {E} [S^{2}]}{\mathrm {E} [N^{2}]}}}$

If the noise has expected value of zero, as is common, the denominator is its variance, the square of its standard deviation σN.

The signal and the noise must be measured the same way, for example as voltages across the same impedance. The root mean squares can alternatively be used in the 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).

### Decibels

Because many signals have a very wide dynamic range, signals are often expressed using the logarithmic decibel scale. Based upon the definition of decibel, signal and noise may be expressed in decibels (dB) as

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

and

${\displaystyle P_{\mathrm {noise,dB} }=10\log _{10}\left(P_{\mathrm {noise} }\right).}$

In a similar manner, SNR may be expressed in decibels as

${\displaystyle \mathrm {SNR_{dB}} =10\log _{10}\left(\mathrm {SNR} \right).}$

Using the definition of SNR

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

Using the quotient rule for logarithms

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

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:

${\displaystyle \mathrm {SNR_{dB}} ={P_{\mathrm {signal,dB} }-P_{\mathrm {noise,dB} }}.}$

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:

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

### Dynamic range

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:

${\displaystyle \mathrm {P} =V_{\mathrm {rms} }I_{\mathrm {rms} }}$
${\displaystyle \mathrm {P} ={\frac {V_{\mathrm {rms} }^{2}}{R}}=I_{\mathrm {rms} }^{2}R}$

But in signal processing and communication, one usually assumes that ${\displaystyle R=1\Omega }$ [3] 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

${\displaystyle \mathrm {P} =V_{\mathrm {rms} }^{2}}$

## Alternative definition

An alternative definition of SNR is as the reciprocal of the coefficient of variation, i.e., the ratio of mean to standard deviation of a signal or measurement: [4] [5]

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

where ${\displaystyle \mu }$ is the signal mean or expected value and ${\displaystyle \sigma }$ is the standard deviation of the noise, or an estimate thereof. [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 ${\displaystyle \operatorname {E} \left[X^{2}\right]=\sigma ^{2}+\mu ^{2}}$. It is commonly used in image processing, [6] [7] [8] [9] 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.

Sometimes[ further explanation needed ] SNR is defined as the square of the alternative definition above, in which case it is equivalent to the more common definition:

${\displaystyle \mathrm {SNR} ={\frac {\mu ^{2}}{\sigma ^{2}}}}$

This definition is closely related to the sensitivity index or d', when assuming that the signal has two states separated by signal amplitude ${\displaystyle \mu }$, and the noise standard deviation ${\displaystyle \sigma }$ 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. [5] [10]

Yet another alternative, very specific, and distinct definition of SNR is employed to characterize sensitivity of imaging systems; see Signal-to-noise ratio (imaging).

Related measures are the "contrast ratio" and the "contrast-to-noise ratio".

## Modulation system measurements

### Amplitude modulation

Channel signal-to-noise ratio is given by

${\displaystyle \mathrm {(SNR)_{C,AM}} ={\frac {A_{C}^{2}(1+k_{a}^{2}P)}{2WN_{0}}}}$

where W is the bandwidth and ${\displaystyle k_{a}}$ is modulation index

Output signal-to-noise ratio (of AM receiver) is given by

${\displaystyle \mathrm {(SNR)_{O,AM}} ={\frac {A_{c}^{2}k_{a}^{2}P}{2WN_{0}}}}$

### Frequency modulation

Channel signal-to-noise ratio is given by

${\displaystyle \mathrm {(SNR)_{C,FM}} ={\frac {A_{c}^{2}}{2WN_{0}}}}$

Output signal-to-noise ratio is given by

${\displaystyle \mathrm {(SNR)_{O,FM}} ={\frac {A_{c}^{2}k_{f}^{2}P}{2N_{0}W^{3}}}}$

## Noise reduction

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.

## Digital signals

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.

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

### Fixed point

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 [11] 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

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. [12]

## Optical signals

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.

## Types and abbreviations

Signal to noise ratio may be abbreviated as SNR and less commonly as S/N. PSNR stands for peak signal-to-noise ratio. GSNR stands for geometric signal-to-noise ratio. SINR is the signal-to-interference-plus-noise ratio.

## Other uses

While SNR is commonly quoted for electrical signals, it can be applied to any form of signal, for example isotope levels in an ice core, biochemical signaling between cells, or financial trading signals. SNR is sometimes used metaphorically 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. [13]

## Notes

1. 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. [2]
2. 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.
3. 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.
4. 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.

## Related Research Articles

The decibel is a relative unit of measurement corresponding to one tenth of a bel (B). It is used to express the ratio of one value of a power or root-power quantity to another, on a logarithmic scale. A logarithmic quantity in decibels is called a level. Two signals whose levels differ by one decibel have a power ratio of 101/10 or root-power ratio of 10120.

In electronics, an analog-to-digital converter is a system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a digital signal. An ADC may also provide an isolated measurement such as an electronic device that converts an input analog voltage or current to a digital number representing the magnitude of the voltage or current. Typically the digital output is a two's complement binary number that is proportional to the input, but there are other possibilities.

Noise figure (NF) and noise factor (F) are measures of degradation of the signal-to-noise ratio (SNR), caused by components in a signal chain. It is a number by which the performance of an amplifier or a radio receiver can be specified, with lower values indicating better performance.

In electronics, noise temperature is one way of expressing the level of available noise power introduced by a component or source. The power spectral density of the noise is expressed in terms of the temperature that would produce that level of Johnson–Nyquist noise, thus:

Shot noise or Poisson noise is a type of noise which can be modeled by a Poisson process. In electronics shot noise originates from the discrete nature of electric charge. Shot noise also occurs in photon counting in optical devices, where shot noise is associated with the particle nature of light.

The total harmonic distortion is a measurement of the harmonic distortion present in a signal and is defined as the ratio of the sum of the powers of all harmonic components to the power of the fundamental frequency. Distortion factor, a closely related term, is sometimes used as a synonym.

In information theory, the Shannon–Hartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise. It is an application of the noisy-channel coding theorem to the archetypal case of a continuous-time analog communications channel subject to Gaussian noise. The theorem establishes Shannon's channel capacity for such a communication link, a bound on the maximum amount of error-free information per time unit that can be transmitted with a specified bandwidth in the presence of the noise interference, assuming that the signal power is bounded, and that the Gaussian noise process is characterized by a known power or power spectral density. The law is named after Claude Shannon and Ralph Hartley.

Johnson–Nyquist noise is the electronic noise generated by the thermal agitation of the charge carriers inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. Thermal noise is present in all electrical circuits, and in sensitive electronic equipment such as radio receivers can drown out weak signals, and can be the limiting factor on sensitivity of an electrical measuring instrument. Thermal noise increases with temperature. Some sensitive electronic equipment such as radio telescope receivers are cooled to cryogenic temperatures to reduce thermal noise in their circuits. The generic, statistical physical derivation of this noise is called the fluctuation-dissipation theorem, where generalized impedance or generalized susceptibility is used to characterize the medium.

Quantization, in mathematics and digital signal processing, is the process of mapping input values from a large set to output values in a (countable) smaller set, often with a finite number of elements. Rounding and truncation are typical examples of quantization processes. Quantization is involved to some degree in nearly all digital signal processing, as the process of representing a signal in digital form ordinarily involves rounding. Quantization also forms the core of essentially all lossy compression algorithms.

Crest factor is a parameter of a waveform, such as alternating current or sound, showing the ratio of peak values to the effective value. In other words, crest factor indicates how extreme the peaks are in a waveform. Crest factor 1 indicates no peaks, such as direct current or a square wave. Higher crest factors indicate peaks, for example sound waves tend to have high crest factors.

Delta-sigma modulation is a method for encoding analog signals into digital signals as found in an analog-to-digital converter (ADC). It is also used to convert high bit-count, low-frequency digital signals into lower bit-count, higher-frequency digital signals as part of the process to convert digital signals into analog as part of a digital-to-analog converter (DAC).

In digital communication or data transmission, Eb/N0 is a normalized signal-to-noise ratio (SNR) measure, also known as the "SNR per bit". It is especially useful when comparing the bit error rate (BER) performance of different digital modulation schemes without taking bandwidth into account.

Decibels relative to full scale is a unit of measurement for amplitude levels in digital systems, such as pulse-code modulation (PCM), which have a defined maximum peak level. The unit is similar to the units dBov and decibels relative to overload (dBO).

dBc is the power ratio of a signal to a carrier signal, expressed in decibels. For example, phase noise is expressed in dBc/Hz at a given frequency offset from the carrier. dBc can also be used as a measurement of Spurious-Free Dynamic Range (SFDR) between the desired signal and unwanted spurious outputs resulting from the use of signal converters such as a digital-to-analog converter or a frequency mixer.

Signal-to-quantization-noise ratio is widely used quality measure in analysing digitizing schemes such as pulse-code modulation (PCM). The SQNR reflects the relationship between the maximum nominal signal strength and the quantization error introduced in the analog-to-digital conversion.

In telecommunications, the carrier-to-noise ratio, often written CNR or C/N, is the signal-to-noise ratio (SNR) of a modulated signal. The term is used to distinguish the CNR of the radio frequency passband signal from the SNR of an analog base band message signal after demodulation, for example an audio frequency analog message signal. If this distinction is not necessary, the term SNR is often used instead of CNR, with the same definition.

In digital audio using pulse-code modulation (PCM), bit depth is the number of bits of information in each sample, and it directly corresponds to the resolution of each sample. Examples of bit depth include Compact Disc Digital Audio, which uses 16 bits per sample, and DVD-Audio and Blu-ray Disc which can support up to 24 bits per sample.

Effective number of bits (ENOB) is a measure of the dynamic range of an analog-to-digital converter (ADC), digital-to-analog converter, or their associated circuitry. The resolution of an ADC is specified by the number of bits used to represent the analog value. Ideally, a 12-bit ADC will have an effective number of bits of almost 12. However, real signals have noise, and real circuits are imperfect and introduce additional noise and distortion. Those imperfections reduce the number of bits of accuracy in the ADC. The ENOB describes the effective resolution of the system in bits. An ADC may have 12-bit resolution, but the effective number of bits when used in a system may be 9.5.

Signal-to-noise ratio (SNR) is used in imaging to characterize image quality. The sensitivity of a imaging system is typically described in the terms of the signal level that yields a threshold level of SNR.

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 (SNR) will be increased, ideally in proportion to the square root of the number of measurements.

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