View source for Level (logarithmic quantity)

{{redirect|Level quantity||Level measurement}}

In [[science and engineering]], a '''power level''' and a '''field level''' (also called a '''root-power level''') are [[logarithmic scale|logarithmic measures]] of certain quantities referenced to a standard reference value of the same type.

* A ''power level'' is a logarithmic quantity used to measure power, power density or sometimes energy, with commonly used unit [[decibel]] (dB).
* A ''field level'' (or ''root-power level'') is a logarithmic quantity used to measure quantities of which the square is typically proportional to power, etc., with commonly used units [[neper]] (Np) or [[decibel]] (dB).

The type of level and choice of units indicate the scaling of the logarithm of the ratio between the quantity and it reference value, though a logarithm may be considered to be a dimensionless quantity.{{sfn|IEEE/ASTM SI 10|2016|pp=26–27}}{{sfn|ISO 80000-3|2006}}{{sfn|Carey|2006|pp=61–75}} The reference values for each type of quantity are often specified by international standards.

Power and field levels are used in [[electronic engineering]], [[telecommunications]], [[acoustics]] and related disciplines.  Power levels are used for signal power, noise power, sound power, sound exposure, etc.  Field levels are used for voltage, current, [[sound pressure level|sound pressure]].{{sfn|ISO 80000-8|2007}}{{clarify|date=February 2019}}

==Power level{{anchor|Power quantities}}==
Level of a ''power'' quantity, denoted ''L''<sub>''P''</sub>, is defined by
:<math>L_P = \frac{1}{2} \log_{\mathrm{e}}\!\left(\frac{P}{P_0}\right)\!~\mathrm{Np} = \log_{10}\!\left(\frac{P}{P_0}\right)\!~\mathrm{B} = 10 \log_{10}\!\left(\frac{P}{P_0}\right)\!~\mathrm{dB}.</math>
where
*''P'' is the power quantity;
*''P''<sub>0</sub> is the reference value of ''P''.

==Field (or root-power) level{{anchor|Root-power quantities|Field quantities}}==
The level of a ''root-power'' quantity (also known as a ''field'' quantity), denoted ''L''<sub>''F''</sub>, is defined by{{sfn|D'Amore|2015}}
:<math>L_F = \log_{\mathrm{e}}\!\left(\frac{F}{F_0}\right)\!~\mathrm{Np} = 2 \log_{10}\!\left(\frac{F}{F_0}\right)\!~\mathrm{B} = 20 \log_{10}\!\left(\frac{F}{F_0}\right)\!~\mathrm{dB}.</math>
where
*''F'' is the root-power quantity, proportional to the square root of power quantity;
*''F''<sub>0</sub> is the reference value of ''F''.

If the power quantity ''P'' is proportional to ''F''<sup>2</sup>, and if the reference value of the power quantity, ''P''<sub>0</sub>, is in the same proportion to ''F''<sub>0</sub><sup>2</sup>, the levels ''L''<sub>''F''</sub> and ''L''<sub>''P''</sub> are equal.

The [[neper]], [[Decibel|bel]], and [[decibel]] (one tenth of a bel) are units of level that are often applied to such quantities as power, intensity, or gain.{{sfn|Taylor|1995}}  The neper, bel, and decibel are related by
*{{nowrap|1=1 B = {{sfrac|1|2}} log<sub>e</sub>10 Np}};
*{{nowrap|1=1 dB = 0.1 B = {{sfrac|1|20}} log<sub>e</sub>10 Np}}.

{{see also|Decibel#Conversions|Neper#Units}}

==Standards==
Level and its units are defined in [[ISO 80000-3#Logarithmic quantities and units|ISO 80000-3]].

The ISO standard defines each of the quantities power level and field level to be dimensionless, with {{nowrap|1=1 Np = 1}}.  This is motivated by simplifying the expressions involved, as in systems of [[natural units]].

==Related quantities==

===Logarithmic ratio quantity===
Power and field quantities are part of a larger class, logarithmic ratio quantities.

[[ANSI/ASA S1.1-2013]] defines a class of quantities it calls ''levels''. It defines a level of a quantity ''Q'', denoted ''L''<sub>''Q''</sub>, as{{sfn|ANSI/ASA S1.1|2013|loc=entry 3.01}}
:<math>L_Q = \log_r\!\left(\frac{Q}{Q_0}\right)\!,</math>
where
*''r'' is the base of the logarithm;
*''Q'' is the quantity;
*''Q''<sub>0</sub> is the reference value of ''Q''.
For the level of a root-power quantity, the base of the logarithm is {{nowrap|1=''r'' = [[e (mathematical constant)|e]]}}.
For the level of a power quantity, the base of the logarithm is {{nowrap|1=''r'' = e<sup>2</sup>}}.{{sfn|Ainslie|2015}}

===Frequency level===
Frequency level of a frequency ''f'' is the logarithm of a ratio of the frequency ''f'' to a reference frequency ''f''{{sub|0}}. The reference frequency is [[C_(musical_note)#Designation_by_octave|C{{sub|0}}]], four octaves below [[middle C]]. {{sfn|ANSI/ASA S1.1|2013}}

In [[electronics]], the [[Octave (electronics)|octave]] (oct) is used as a unit with logarithm base 2, and the [[Decade (log scale)|decade]] (dec) is used as a unit with logarithm base 10:
: <math>L_f = \log_2 \!\left( \frac{f}{f_0} \right) ~\text{oct} = \log_{10} \!\left( \frac{f}{f_0} \right) ~\text{dec}. </math>

In [[music theory]], the [[octave]] is a unit used with logarithm base 2 (called ''[[Interval (music)|interval]]'').{{sfn|Fletcher|1934|pp=59–69}} A [[semitone]] is one twelfth of an octave. A [[Cent (music)|cent]] is one hundredth of a semitone.

==See also==
* {{section link|Decibel|Definition}}
* [[Field, power, and root-power quantities]]
* [[Logarithmic scale]]
* [[Sound level (disambiguation)]]
* [[Leveling (tapered floating point)]]
* [[Level-index arithmetic]] (LI) and [[symmetric level-index arithmetic]] (SLI)

==Notes==
{{reflist}}

==References==
* {{citation |year=1934 |author=Fletcher |first=H |title=Loudness, pitch and the timbre of musical tones and their relation to the intensity, the frequency and the overtone structure |journal=[[Journal of the Acoustical Society of America]] |volume=6 |issue=2 }}
* {{citation |year=1995 |author=Taylor |first=Barry |title=Guide for the Use of the International System of Units (SI): The Metric System |publisher=Diane Publishing Co. |url=https://books.google.com/books?id=I3HRRrDd1n8C&pg=PA28 |page=28}}
* {{citation |year=2006 |author=ISO 80000-3 |title=Quantities and units |publisher=[[International Organization for Standardization]] |volume=[[ISO 80000-3|Part 3: Space and Time]] }}
* {{citation |year=2006 |author=Carey |first=W. M. |title=Sound Sources and Levels in the Ocean |journal=[[IEEE Journal of Oceanic Engineering]] |volume=31 }}
* {{citation |year=2007 |author=ISO 80000-8 |title=Quantities and units |publisher=[[International Organization for Standardization]] |volume=[[ISO 80000-8|Part 8: Acoustics]] }}
* {{citation |year=2013 |author=ANSI/ASA S1.1 |title=Acoustical Terminology |volume=[[ANSI/ASA S1.1-2013]] |publisher=[[Acoustical Society of America]] }}
* {{citation |year=2015 |author=Ainslie |first=M. A. |title=A Century of Sonar: Planetary Oceanography, Underwater Noise Monitoring, and the Terminology of Underwater Sound |url=http://acousticstoday.org/a-century-of-sonar-planetary-oceanography-underwater-noise-monitoring-and-the-terminology-of-underwater-sound-michael-a-ainslie/ |journal=[[Acoustics Today]] |volume=11 |issue=1 }}
* {{citation |year=2015 |author=D'Amore |first=F. |title=Effect of moisturizer and lubricant on the finger‒surface sliding contact: tribological and dynamical analysis |url=https://www.researchgate.net/publication/316104462_Effect_of_moisturizer_and_lubricant_on_the_finger-surface_sliding_contact_tribological_and_dynamical_analysis }}
* {{citation |year=2016 |author=IEEE/ASTM SI 10 |title=American National Standard for Metric Practice |publisher=[[IEEE Standards Association]] }}

[[Category:Mathematical terminology]]
[[Category:Logarithmic scales of measurement]]