View source for Physics:Signal-to-noise ratio

{{short description|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 (signal processing)|noise]]. SNR is defined as the ratio of signal [[Physics:Power|power]] to [[Noise power|noise power]], often expressed in [[Physics:Decibel|decibel]]s. A ratio higher than 1:1 (greater than 0&nbsp;dB) indicates more signal than noise.

SNR is an important parameter that affects the performance and quality of systems that process or transmit signals, such as communication systems, audio systems, radar systems, imaging systems, and [[Data acquisition|data acquisition]] systems. A high SNR means that the signal is clear and easy to detect or interpret, while a low SNR means that the signal is corrupted or obscured by noise and may be difficult to distinguish or recover. SNR can be improved by various methods, such as increasing the signal strength, reducing the noise level, filtering out unwanted noise, or using error correction techniques.

SNR also determines the maximum possible amount of data that can be transmitted reliably over a given channel, which depends on its bandwidth and SNR. This relationship is described by the [[Shannon–Hartley theorem]], which is a fundamental law of information theory.

SNR can be calculated using different formulas depending on how the signal and noise are measured and defined. The most common way to express SNR is in decibels, which is a logarithmic scale that makes it easier to compare large or small values. Other definitions of SNR may use different factors or bases for the logarithm, depending on the context and application.

==Definition==
Signal-to-noise ratio is defined as the ratio of the [[Physics:Power|power]] of a signal (meaningful input) to the power of background noise (meaningless or unwanted input):
:<math>
\mathrm{SNR} = \frac{P_\mathrm{signal}}{P_\mathrm{noise}},
</math>
where {{mvar|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 (signal processing)|bandwidth]].

Depending on whether the signal is a constant ({{mvar|s}}) or a random variable ({{mvar|S}}), the signal-to-noise ratio for random noise {{mvar|N}} becomes:<ref>{{cite book |author1=Charles Sherman |author2=John Butler |title=Transducers and Arrays for Underwater Sound |date=2007 |publisher=Springer Science & Business Media |isbn=9780387331393 |page=276 |url=https://books.google.com/books?id=srREi-ScbFcC&q=%22signal+to+noise+ratio%22+define+mean-square&pg=PA276}}</ref>
:<math>
\mathrm{SNR} = \frac{s^2}{\mathrm{E}[N^2]}
</math>
where E refers to the [[Expected value|expected value]], i.e. in this case the [[Mean square|mean square]] of {{mvar|N}},
or
:<math>
\mathrm{SNR} = \frac{\mathrm{E}[S^2]}{\mathrm{E}[N^2]}
</math>

If the noise has [[Expected value|expected value]] of zero, as is common, the denominator is its [[Variance|variance]], the square of its [[Standard deviation|standard deviation]] {{math|''σ''<sub>N</sub>}}.


The signal and the noise must be measured the same way, for example as voltages across the same [[Physics:Electrical impedance|impedance]]. The [[Root mean square|root mean square]]s can alternatively be used in the ratio:
:<math>
\mathrm{SNR} = \frac{P_\mathrm{signal}}{P_\mathrm{noise}} = \left ( \frac{A_\mathrm{signal}}{A_\mathrm{noise} } \right )^2,
</math>
where {{mvar|A}} is root mean square (RMS) amplitude (for example, RMS voltage).

===Decibels===
Because many signals have a very wide [[Dynamic range|dynamic range]], signals are often expressed using the [[Logarithm|logarithm]]ic [[Physics:Decibel|decibel]] scale. Based upon the definition of decibel, signal and noise may be expressed in decibels (dB) as

:<math>P_\mathrm{signal,dB} = 10 \log_{10} \left ( P_\mathrm{signal} \right ) </math>

and

:<math>P_\mathrm{noise,dB} = 10 \log_{10} \left ( P_\mathrm{noise} \right ). </math>

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

:<math>
\mathrm{SNR_{dB}} = 10 \log_{10} \left ( \mathrm{SNR} \right ).
</math>

Using the definition of SNR

:<math>
\mathrm{SNR_{dB}} = 10 \log_{10} \left ( \frac{P_\mathrm{signal}}{P_\mathrm{noise}} \right ).
</math>

Using the quotient rule for logarithms

:<math>
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 ).
</math>

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:
:<math>
\mathrm{SNR_{dB}} = {P_\mathrm{signal,dB} - P_\mathrm{noise,dB}}.
</math>

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,{{#tag:ref|The connection between [[Physics:Optical power|optical power]] and [[Physics:Voltage|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.<ref>Michael A. Choma, Marinko V. Sarunic, Changhuei Yang, Joseph A. Izatt. [https://www.osapublishing.org/oe/fulltext.cfm?uri=oe-11-18-2183 Sensitivity advantage of swept source and Fourier domain optical coherence tomography]. Optics Express, 11(18). Sept 2003.</ref>|group="note"}} they must first be squared to obtain a quantity proportional to power, as shown below:

:<math>
\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 )  = 2 \left ( {A_\mathrm{signal,dB} - A_\mathrm{noise,dB}}\right ).
</math>

===Dynamic range===
The concepts of signal-to-noise ratio and dynamic range are closely related. Dynamic range measures the ratio between the strongest un-[[Physics:Distortion|distorted]] signal on a [[Engineering:Communication channel|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 [[Engineering:Audio engineering|audio engineering]], the reference signal is usually a [[Physics:Sine wave|sine wave]] at a standardized [[Nominal level|nominal]] or [[Physics:Alignment level|alignment level]], such as 1&nbsp;kHz at +4 dBu (1.228 V<sub>RMS</sub>).

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&nbsp;dB) and measuring how far the signal 'stands out'.

===Difference from conventional power===
In physics, the average [[Physics:Power|power]] of an AC signal is defined as the average value of voltage times current; for resistive (non-[[Engineering:Reactance (electronics)|reactive]]) circuits, where voltage and current are in phase, this is equivalent to the product of the [[Root mean square|rms]] voltage and current:
:<math>  \mathrm{P} = V_\mathrm{rms}I_\mathrm{rms}
</math>

:<math>
\mathrm{P}= \frac{V_\mathrm{rms}^{2}}{R} = I_\mathrm{rms}^{2} R
</math>
But in signal processing and communication, one usually assumes that <math>R=1 \Omega</math> <ref>{{cite journal |author1=Gabriel L. A. de Sousa |author2=George C. Cardoso |title= A battery-resistor analogy for further insights on measurement uncertainties |date= 18 June 2018 |url= https://doi.org/10.1088/1361-6552/aac84b |journal= Physics Education |volume= 53 |issue= 5 |pages= 055001 |doi= 10.1088/1361-6552/aac84b | publisher= IOP Publishing |access-date= 5 May 2021|arxiv= 1611.03425 |bibcode= 2018PhyEd..53e5001D |s2cid= 125414987 }}</ref> 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
:<math>
\mathrm{P}= V_\mathrm{rms}^{2}
</math>

== Alternative definition ==
An alternative definition of SNR is as the reciprocal of the [[Coefficient of variation|coefficient of variation]], i.e., the ratio of [[Mean|mean]] to [[Standard deviation|standard deviation]] of a signal or measurement:<ref>
{{cite book | title = Astronomical optics | author = D. J. Schroeder | edition = 2nd | publisher = Academic Press | year = 1999 | isbn = 978-0-12-629810-9 | page = 278 | url = https://books.google.com/books?id=v7E25646wz0C&pg=PA433}}, [https://books.google.com/books?id=VZvqqaQ5DvoC&q=signal%20to%20noise&pg=PA278 p.278]</ref><ref name=b1>Bushberg, J. T., et al., ''[https://books.google.com/books?id=VZvqqaQ5DvoC&pg=PA280 The Essential Physics of Medical Imaging],'' (2e). Philadelphia: Lippincott Williams & Wilkins, 2006, p. 280.</ref>
:<math>
 \mathrm{SNR} = \frac{\mu}{\sigma}
</math>
where <math>\mu</math> is the signal mean or [[Expected value|expected value]] and <math>\sigma</math> is the standard deviation of the noise, or an estimate thereof.<ref group="note">The exact methods may vary between fields. For example, if the signal data are known to be constant, then <math>\sigma</math> can be calculated using the standard deviation of the signal. If the signal data are not constant, then <math>\sigma</math> can be calculated from data where the signal is zero or relatively constant.</ref> Notice that such an alternative definition is only useful for variables that are always non-negative (such as photon counts and [[Physics:Luminance|luminance]]), and it is only an approximation since <math>\operatorname{E}\left[X^2 \right] = \sigma^2 + \mu^2 </math>. It is commonly used in [[Image processing|image processing]],<ref>{{cite book|url=https://books.google.com/books?id=8uGOnjRGEzoC&pg=PA354|title=Digital image processing|page=354|author=Rafael C. González, Richard Eugene Woods|publisher=Prentice Hall|year=2008|isbn=978-0-13-168728-8}}</ref><ref>{{cite book|url=https://books.google.com/books?id=VmvY4MTMFTwC&pg=PA471|title=Image fusion: algorithms and applications|page=471|author=Tania Stathaki|publisher=Academic Press|year=2008|isbn=978-0-12-372529-5}}</ref><ref>{{cite book|url=https://books.google.com/books?id=7s8xpR-5rOUC&pg=PA471 |title=Multi-Sensor Data Fusion: Theory and Practice|author=Jitendra R. Raol|publisher=CRC Press|year=2009|isbn=978-1-4398-0003-4}}</ref><ref>{{cite book|url=https://books.google.com/books?id=Vs2AM2cWl1AC&pg=PA26 |title=The image processing handbook|author=John C. Russ|publisher=CRC Press|year=2007|isbn=978-0-8493-7254-4}}</ref> where the SNR of an [[Image|image]] is usually calculated as the ratio of the [[Mean|mean]] pixel value to the [[Standard deviation|standard deviation]] of the pixel values over a given neighborhood.

Sometimes SNR is defined as the square of the alternative definition above, in which case it is equivalent to the more common definition:
:<math>
 \mathrm{SNR} = \frac{\mu^2}{\sigma^2}
</math>

This definition is closely related to the [[Sensitivity index|sensitivity index]] or ''d{{'}}'', when assuming that the signal has two states separated by signal amplitude <math>\mu</math>, and the noise standard deviation <math>\sigma</math> does not change between the two states.

The ''Rose criterion'' (named after [[Biography:Albert Rose (physicist)|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.<ref name=b1/><ref>
{{cite book |last= Rose |first= Albert |title= Vision – Human and Electronic |publisher=  Plenum Press  |isbn= 9780306307324| year = 1973 | page=[https://archive.org/details/visionhumanelect01rose/page/10 10] | url=https://archive.org/details/visionhumanelect01rose |url-access= registration |quote= [...] 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.}}</ref>

Yet another alternative, very specific, and distinct definition of SNR is employed to characterize [[Physics:Film speed|sensitivity]] of imaging systems; see [[Physics:Signal-to-noise ratio (imaging)|Signal-to-noise ratio (imaging)]].

Related measures are the "[[Engineering:Contrast ratio|contrast ratio]]" and the "[[Contrast-to-noise ratio|contrast-to-noise ratio]]".

==Modulation system measurements==

===Amplitude modulation===

Channel signal-to-noise ratio is given by
:<math>\mathrm{(SNR)_{C,AM}} = \frac{A_C^2 (1 + k_a^2 P)} {2 W N_0}
</math>
where W is the bandwidth and <math>k_a</math> is modulation index

Output signal-to-noise ratio (of AM receiver) is given by
:<math>\mathrm{(SNR)_{O,AM}} = \frac{A_c^2 k_a^2 P} {2 W N_0}
</math>

===Frequency modulation===

Channel signal-to-noise ratio is given by
:<math>\mathrm{(SNR)_{C,FM}} = \frac{A_c^2} {2 W N_0}
</math>

Output signal-to-noise ratio is given by
:<math>\mathrm{(SNR)_{O,FM}} = \frac{A_c^2 k_f^2 P} {2 N_0 W^3}
</math>

==Noise reduction==
[[Image:Analyse thermo gravimetrique bruit.png|thumb|Recording from a [[Chemistry:Thermogravimetric analysis|thermogravimetric analysis]] device with poor mechanical isolation; the middle of the plot shows lower noise due to reduced 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 [[Engineering:Low-noise amplifier|low-noise amplifier]]s.

When the characteristics of the noise are known and are different from the signal, it is possible to use a [[Filter (signal processing)|filter]] to reduce the noise. For example, a [[Engineering:Lock-in amplifier|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 [[Signal averaging|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 [[Physics:Noise|noise]] level is the [[Error|error]] caused by the [[Quantization (signal processing)|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 converter|analog-to-digital converter]]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 [[Dither|dither]].

Although noise levels in a digital system can be expressed using SNR, it is more common to use E<sub>b</sub>/N<sub>o</sub>, the energy per bit per noise power spectral density.

The [[Engineering:Modulation error ratio|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 ([[Quantization (signal processing)|uniform quantization]]) the [[Dynamic range|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 2<sup>''n''</sup>/1. The formula is then:
:<math>
\mathrm{DR_{dB}} = \mathrm{SNR_{dB}} = 20 \log_{10}(2^n) \approx 6.02 \cdot n
</math>

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

Assuming a full-scale [[Physics:Sine wave|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|sawtooth wave]] with peak-to-peak amplitude of one quantization level<ref name="maxim 728">[http://www.maxim-ic.com/appnotes.cfm/appnote_number/728 Defining and Testing Dynamic Parameters in High-Speed ADCs] — Maxim Integrated Products Application note 728</ref> and uniform distribution. In this case, the SNR is approximately
:<math>
\mathrm{SNR_{dB}} \approx 20 \log_{10} (2^n {\textstyle\sqrt {3/2}}) \approx 6.02 \cdot n + 1.761
</math>

===Floating point===
[[Floating-point arithmetic|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 [[Logarithm|mantissa]] and m bits in the exponent:
:<math>
\mathrm{DR_{dB}} = 6.02 \cdot 2^m
</math>

:<math>
\mathrm{SNR_{dB}} = 6.02 \cdot (n-m)
</math>

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.<ref name="rane fixed vs floating">[https://web.archive.org/web/20060515074349/http://www.rane.com/note153.html Fixed-Point vs. Floating-Point DSP for Superior Audio] — [[Company:Rane Corporation|Rane Corporation]] technical library</ref><ref group="note">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|comb filter]]s may be used.</ref><ref group="note">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.</ref>

==Optical signals==
Optical signals have a [[Carrier frequency|carrier frequency]] (about {{val|200|u=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&nbsp;nm is used. This bandwidth is independent of the modulation format, the frequency and the receiver. For instance an OSNR of 20&nbsp;dB/0.1&nbsp;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|peak signal-to-noise ratio]]. GSNR stands for geometric signal-to-noise ratio.<ref>{{cite book |author1=Tomasz Pander |title=Man-Machine Interactions 3 |chapter=An Application of Myriad M-Estimator for Robust Weighted Averaging |series=Advances in Intelligent Systems and Computing |date=2013 |volume=242 |publisher=ICMMI |isbn=9783319023090 |page=265--272 |doi=10.1007/978-3-319-02309-0_28 |chapter-url=https://link.springer.com/chapter/10.1007/978-3-319-02309-0_28}}</ref> SINR is the [[Signal-to-interference-plus-noise ratio|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 [[Physics:Isotope|isotope]] levels in an [[Earth:Ice core|ice core]], biochemical signaling between cells, or [[Finance:Financial signal processing|financial trading signals]]. The term is sometimes used metaphorically to refer to the ratio of useful [[Information|information]] to false or irrelevant data in a conversation or exchange. For example, in [[Internet forum|online discussion forums]] and other online communities, off-topic posts and [[Spamming|spam]] are regarded as {{em|noise}} that interferes with the {{em|signal}} of appropriate discussion.<ref>{{cite book |url=https://books.google.com/books?id=W5bAcxc2TcgC&pg=PA128 |page=128 |title=The Music Internet Untangled: Using Online Services to Expand Your Musical Horizons |last=Breeding |first=Andy |publisher=Giant Path |year=2004 |isbn=9781932340020}}</ref>

SNR can also be applied in marketing and how business professionals manage information overload. Managing a healthy signal to noise ratio can help business executives improve their KPIs (Key Performance Indicators).<ref>{{Cite web |title=What Is Signal To Noise Ratio? |url=https://www.thruways.co/blog/what-is-signal-to-noise-ratio |access-date=2023-11-09 |website=www.thruways.co |language=en}}</ref>

==See also==
{{Div col|colwidth=20em}}
* [[Engineering:Audio system measurements|Audio system measurements]]
* [[Generation loss]]
* [[Matched filter]]
* [[Near–far problem]]
* [[Engineering:Noise margin|Noise margin]]
* [[Omega ratio]]
* [[Physics:Pareidolia|Pareidolia]]
* [[Peak signal-to-noise ratio]]
* [[Signal-to-noise statistic]]
* [[Signal-to-interference-plus-noise ratio]]
* [[Engineering:SINAD|SINAD]]
* [[Subjective video quality]]
* [[Physics:Total harmonic distortion|Total harmonic distortion]]
* [[Video quality]]
{{Div col end}}

==Notes==
{{Reflist|group="note"|1}}

==References==
{{Reflist}}

==External links==
* {{citation |title=Taking the Mystery out of the Infamous Formula,"SNR = 6.02N + 1.76dB," and Why You Should Care |author=Walt Kester |url=http://www.analog.com/static/imported-files/tutorials/MT-001.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.analog.com/static/imported-files/tutorials/MT-001.pdf |archive-date=2022-10-09 |url-status=live |publisher=Analog Devices |access-date=2019-04-10}}
* [http://www.maxim-ic.com/appnotes.cfm/appnote_number/641 ADC and DAC Glossary] – Maxim Integrated Products
* [http://www.analog.com/static/imported-files/tutorials/MT-003.pdf Understand SINAD, ENOB, SNR, THD, THD + N, and SFDR so you don't get lost in the noise floor] – Analog Devices
* [https://web.archive.org/web/20060522134626/http://www.techonline.com/community/related_content/20771 The Relationship of dynamic range to data word size in digital audio processing]
* [http://www.sengpielaudio.com/calculator-noise.htm Calculation of signal-to-noise ratio, noise voltage, and noise level]
* [http://www.vias.org/simulations/simusoft_spectaccu.html Learning by simulations – a simulation showing the improvement of the SNR by time averaging]
* [http://focus.ti.com/lit/an/sbaa055/sbaa055.pdf Dynamic Performance Testing of Digital Audio D/A Converters]
* [http://www.circuitdesign.info/blog/2008/11/fundamentals-of-analogrf-design-noise-signal-power/ Fundamental theorem of analog circuits: a minimum level of power must be dissipated to maintain a level of SNR]
* [http://webdemo.inue.uni-stuttgart.de/webdemos/02_lectures/uebertragungstechnik_1/qam_constellation_diagram_from_snr Interactive webdemo of visualization of SNR in a QAM constellation diagram] Institute of Telecommunicatons, University of Stuttgart
* {{citation |title=Quantization Noise: Roundoff Error in Digital Computation, Signal Processing, Control, and Communications|author=Bernard Widrow,István Kollár|date=2008-07-03|url=http://www.cambridge.org/9780521886710 |publisher=Cambridge University Press, Cambridge, UK, 2008. 778 p. |isbn=9780521886710}}
* [http://oldweb.mit.bme.hu/books/quantization/ Quantization Noise] Widrow & Kollár Quantization book page with sample chapters and additional material
* [https://www.etti.unibw.de/labalive/experiment/snr/ Signal-to-noise ratio online audio demonstrator - Virtual Communications Lab]

{{Noise}}

[[Category:Engineering ratios]]
[[Category:Measurement]]
[[Category:Electrical parameters]]
[[Category:Noise (electronics)]]
[[Category:Statistical ratios]]
[[Category:Acoustics]]
[[Category:Sound]]

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