**Signal-to-noise ratio** is a measure used in science and engineering that compares the level of a desired signal to the level of background

noiseIn common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...

. It is defined as the ratio of signal power to the noise power. 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 coreAn ice core is a core sample that is typically removed from an ice sheet, most commonly from the polar ice caps of Antarctica, Greenland or from high mountain glaciers elsewhere. As the ice forms from the incremental build up of annual layers of snow, lower layers are older than upper, and an ice...

or biochemical signaling between cells).

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 forumsAn Internet forum, or message board, is an online discussion site where people can hold conversations in the form of posted messages. They differ from chat rooms in that messages are at least temporarily archived...

and other online communities, off-topic posts and spam are regarded as "noise" that interferes with the "signal" of appropriate discussion.

## Definition

Signal-to-noise ratio is defined as the

powerIn physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...

ratio between a signal (meaningful information) and the background noise (unwanted signal):

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

impedanceElectrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...

, then the SNR can be obtained by calculating the square of the

amplitudeAmplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

ratio:

where

*A* is

root mean squareIn mathematics, the root mean square , also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids...

(RMS)

amplitudeAmplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

(for example, RMS voltage). Because many signals have a very wide

dynamic rangeDynamic range, abbreviated DR or DNR, is the ratio between the largest and smallest possible values of a changeable quantity, such as in sound and light. It is measured as a ratio, or as a base-10 or base-2 logarithmic value.-Dynamic range and human perception:The human senses of sight and...

, SNRs are often expressed using the

logarithmThe logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

ic

decibelThe decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

scale. In decibels, the SNR is defined as

which may equivalently be written using amplitude ratios as

The concepts of signal-to-noise ratio and dynamic range are closely related. Dynamic range measures the ratio between the strongest un-

distortedA distortion is the alteration of the original shape of an object, image, sound, waveform or other form of information or representation. Distortion is usually unwanted, and often many methods are employed to minimize it in practice...

signal on a

channelIn telecommunications and computer networking, a communication channel, or channel, refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel...

and the minimum discernable 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 engineeringAn audio engineer, also called audio technician, audio technologist or sound technician, is a specialist in a skilled trade that deals with the use of machinery and equipment for the recording, mixing and reproduction of sounds. The field draws on many artistic and vocational areas, including...

, the reference signal is usually a

sine waveThe sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

at a standardized

nominalNominal level is the operating level at which an electronic signal processing device is designed to operate. The electronic circuits that make up such equipment are limited in the maximum signal they can output and the low-level internally-generated electronic noise they add to the signal...

or

alignment levelThe alignment level in an audio signal chain or on an audio recording is a defined anchor point that represents a reasonable or typical level...

, such as 1 kHz at +4

dBuDBU may refer to:* Dansk Boldspil-Union, in English known as the Danish Football Association* dBu, a decibel measurement of voltage* 1,8-Diazabicyclo[5.4.0]undec-7-ene...

(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'.

## Alternative definition

An alternative definition of SNR is as the reciprocal of the

coefficient of variationIn probability theory and statistics, the coefficient of variation is a normalized measure of dispersion of a probability distribution. It is also known as unitized risk or the variation coefficient. The absolute value of the CV is sometimes known as relative standard deviation , which is...

, i.e., the ratio of

meanIn statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

to

standard deviationStandard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

of a signal or measurement:

where

is the signal mean or

expected valueIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

and

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 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.] Notice that such an alternative definition is only useful for variables that are always positive (such as photon counts and

luminanceLuminance is a photometric measure of the luminous intensity per unit area of light travelling in a given direction. It describes the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle. The SI unit for luminance is candela per square...

). Thus it is commonly used in

image processingIn electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...

, where the SNR of an

imageAn image is an artifact, for example a two-dimensional picture, that has a similar appearance to some subject—usually a physical object or a person.-Characteristics:...

is usually calculated as the ratio of the

meanIn statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

pixel value to the

standard deviationStandard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

of the pixel values over a given neighborhood. Sometimes SNR is defined as the square of the alternative definition above.

The

*Rose criterion* (named after

Albert RoseAlbert Rose was an American physicist, who made major contributions to TV video camera tubes such as the orthicon, image orthicon, and vidicon....

) 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.

Yet another alternative, very specific and distinct definition of SNR is employed to characterize

sensitivityFilm speed is the measure of a photographic film's sensitivity to light, determined by sensitometry and measured on various numerical scales, the most recent being the ISO system....

of imaging systems; see signal to noise ratio (imaging).

Related measures are the "

contrast ratioThe contrast ratio is a property of a display system, defined as the ratio of the luminance of the brightest color to that of the darkest color that the system is capable of producing...

" and the "contrast-to-noise ratio".

### Amplitude Modulation

Channel Signal-to-Noise Ratio is given by

where W is the bandwidth and ka is modulation index

Output Signal-to-Noise Ratio (of AM receiver) is given by

### Frequency Modulation

Channel Signal-to-Noise Ratio is given by

Output Signal-to-Noise Ratio is given by

## Improving SNR in practice

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

filterIn signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...

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.

## Digital signals

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

noiseIn common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...

level is the

errorThe word error entails different meanings and usages relative to how it is conceptually applied. The concrete meaning of the Latin word "error" is "wandering" or "straying". Unlike an illusion, an error or a mistake can sometimes be dispelled through knowledge...

caused by the

quantizationQuantization, in mathematics and digital signal processing, is the process of mapping a large set of input values to a smaller set – such as rounding values to some unit of precision. A device or algorithmic function that performs quantization is called a quantizer. The error introduced by...

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 converterAn analog-to-digital converter is a device that converts a continuous quantity to a discrete time digital representation. An ADC may also provide an isolated measurement...

s 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

ditherDither is an intentionally applied form of noise used to randomize quantization error, preventing large-scale patterns such as color banding in images...

.

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 ratioThe modulation error ratio or MER is a measure used to quantify the performance of a digital radio transmitter or receiver in a communications system using digital modulation...

(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 quantizationQuantization, in mathematics and digital signal processing, is the process of mapping a large set of input values to a smaller set – such as rounding values to some unit of precision. A device or algorithmic function that performs quantization is called a quantizer. The error introduced by...

) the

dynamic rangeDynamic range, abbreviated DR or DNR, is the ratio between the largest and smallest possible values of a changeable quantity, such as in sound and light. It is measured as a ratio, or as a base-10 or base-2 logarithmic value.-Dynamic range and human perception:The human senses of sight and...

(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 2

^{n}/1. The formula is then:

This relationship is the origin of statements like "

16-bit-16-bit architecture:The HP BPC, introduced in 1975, was the world's first 16-bit microprocessor. Prominent 16-bit processors include the PDP-11, Intel 8086, Intel 80286 and the WDC 65C816. The Intel 8088 was program-compatible with the Intel 8086, and was 16-bit in that its registers were 16...

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 waveThe sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

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 waveThe sawtooth wave is a kind of non-sinusoidal waveform. It is named a sawtooth based on its resemblance to the teeth on the blade of a saw....

with peak-to-peak amplitude of one quantization level and uniform distribution. In this case, the SNR is approximately

### Floating point

Floating-pointIn computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...

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* The mantissa is the significand in a common logarithm or floating-point number.* Metaphorically, it is the part of the self that eludes conscious awareness or self-understanding.* An addition of little importance.Mantissa may also refer to:...

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.

[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 filter]In signal processing, a comb filter adds a delayed version of a signal to itself, causing constructive and destructive interference. The frequency response of a comb filter consists of a series of regularly spaced spikes, giving the appearance of a comb....

s 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.]
## Optical SNR

Optical signals have a carrier frequency that is much higher than the modulation frequency (about 200 THz and more). 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 20dB/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

## See also

- Audio system measurements
Audio system measurements are made for several purposes. Designers take measurements so that they can specify the performance of a piece of equipment. Maintenance engineers make them to ensure equipment is still working to specification, or to ensure that the cumulative defects of an audio path are...

- Generation loss
Generation loss refers to the loss of quality between subsequent copies or transcodes of data. Anything that reduces the quality of the representation when copying, and would cause further reduction in quality on making a copy of the copy, can be considered a form of generation loss...

- Near-far problem
The near-far problem or hearability problem is a situation that is common in wireless communication systems, in particular, CDMA. In some signal jamming techniques, the near-far problem is exploited to disrupt communications.-Technical explanation:...

- Noise margin
In electrical engineering, noise margin is the amount by which a signal exceeds the minimum amount for proper operation. It is commonly used in at least two contexts:...

- Peak signal-to-noise ratio
The phrase peak signal-to-noise ratio, often abbreviated PSNR, is an engineering term for the ratio between the maximum possible power of a signal and the power of corrupting noise that affects the fidelity of its representation...

- Signal-to-noise statistic
- Subjective video quality
Subjective video quality is a subjective characteristic of video quality. It is concerned with how video is perceived by a viewer and designates his or her opinion on a particular video sequence...

- Total harmonic distortion
The total harmonic distortion, or THD, of a signal is a measurement of the harmonic distortion present and is defined as the ratio of the sum of the powers of all harmonic components to the power of the fundamental frequency...

- Video quality
Video quality is a characteristic of a video passed through a video transmission/processing system, a formal or informal measure of perceived video degradation...

## External links

- Taking the Mystery out of the Infamous Formula,"SNR = 6.02N + 1.76dB," and Why You Should Care. http://www.analog.com/static/imported-files/tutorials/MT-001.pdf Analog Devices
- ADC and DAC Glossary - Maxim Integrated Products
Maxim Integrated Products is a publicly traded company that designs, manufactures, and sells analog and mixed-signal semiconductor products. Maxim develops integrated circuits for the industrial, communications, consumer, and computing markets....

- Understand SINAD, ENOB, SNR, THD, THD + N, and SFDR so you don't get lost in the noise floor - Analog Devices
Analog Devices, Inc. , known as ADI, is an American multinational semiconductor company specializing in data conversion and signal conditioning technology, headquartered in Norwood, Massachusetts...

- 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