Editing Signal-to-noise ratio (section)

==Definition==
{{refimprove section|date=February 2022}}
One definition of signal-to-noise ratio is the ratio of the [[power (physics)|power]] of a [[signal]] (meaningful input) to the power of background [[noise (electronic)|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]].

The signal-to-noise ratio of a random variable ({{mvar|S}}) to random noise {{mvar|N}} is:<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 display="block">
\mathrm{SNR} = \frac{\mathrm{E}[S^2]}{\mathrm{E}[N^2]} \, ,
</math>

where E refers to the [[expected value]], which in this case is the [[mean square]] of {{mvar|N}}.

If the signal is simply a constant value of ''{{mvar|s}}'', this equation simplifies to:

<math display="block">
\mathrm{SNR} = \frac{s^2}{\mathrm{E}[N^2]} \, .
</math>

If the noise has [[expected value]] of zero, as is common, the denominator is its [[variance]], the square of its [[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 [[Electrical impedance|impedance]]. Their [[root mean square]]s can alternatively be used according to:
:<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]], signals are often expressed using the [[logarithm]]ic [[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 [[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|''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 )  = {A_\mathrm{signal,dB} - A_\mathrm{noise,dB}} .
</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-[[distortion|distorted]] signal on a [[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 [[audio engineering]], the reference signal is usually a [[sine wave]] at a standardized [[nominal level|nominal]] or [[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 [[power (physics)|power]] of an AC signal is defined as the average value of voltage times current; for [[resistive]] (non-[[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>