Editing Signal-to-noise ratio (section)

===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>