Level (logarithmic quantity)

In the International System of Quantities, the level of a quantity is the logarithm of the ratio of the value of that quantity to a reference value of the same quantity.^[1]^[2] Examples are the various types of sound level: sound power level (literally, the level of the sound power, abbreviated SWL), sound exposure level (SEL), sound pressure level (SPL) and particle velocity level (SVL).^[3]

Mathematical definitions

The level and its units are defined in ISO 80000-3.

Level

Level of a quantity Q, denoted L_Q, is defined by

L_{Q}=\log _{r}\!\left({\frac {Q}{Q_{0}}}\right)\!,

where

r is the base of the logarithm;
Q is the root-power quantity;
Q₀ is the reference value of Q.

Level of a field quantity

Level of a field quantity, denoted L_F, is defined by

L_{F}=\ln \!\left({\frac {F}{F_{0}}}\right)\!,

where

F is the field quantity;
F₀ is the reference value of F.

For the level of a field quantity, the base of the logarithm is r = e.

Level of a root-power quantity

A root-power quantity is a field quantity. The Level of a root-power quantity, denoted L_F, is therefore

L_{F}=\ln \!\left({\frac {F}{F_{0}}}\right)\!,

where

F is the root-power quantity;
F₀ is the reference value of F.

For the level of a root-power quantity, the base of the logarithm is r = e.

Level of a power quantity

Level of a power quantity, denoted L_P, is defined by

L_{P}=\log _{{e^{2}}}\!\left({\frac {P}{P_{0}}}\right)={\frac {1}{2}}\ln \!\left({\frac {P}{P_{0}}}\right)\!,

where

P is the power quantity;
P₀ is the reference value of P.

For the level of a power quantity, the base of the logarithm is r = e².^[4]

Units of level

Power level

The neper, bel, and decibel (one tenth of a bel) are units of level that are often applied to such quantities as power, intensity, or gain.^[5] The neper, bel, and decibel are defined by

Np = 1;
B = 1/2 ln 10 Np;
dB = 0.1 B = 1/20 ln 10 Np.

If F is a root-power quantity:

L_{F}=\ln \!\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}}.

If P is a power quantity:

L_{P}={\frac {1}{2}}\ln \!\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}}.

If the power quantity P is equal to F², and if the reference value of the power quantity, P₀, is equal to F₀², the levels L_F and L_P are equal.

Frequency level

The octave is a unit of level (specifically "frequency level", for r = 2) though that concept is seldom seen outside of the standard.^[6] A semitone is one twelfth of an octave.

References

↑ ISO 80000-3:2006, Quantities and units, Part 2: Space and Time
↑ W. M. Carey, Sound Sources and Levels in the Ocean, IEEE J Oceanic Eng 31:61-75(2006)
↑ ISO 80000-8:2007, Quantities and units, Part 8: Acoustics
↑ Ainslie, M. A. A Century of Sonar: Planetary Oceanography, Underwater Noise Monitoring, and the Terminology of Underwater Sound. Acoustics Today, 23 February 2015
↑ Barry Taylor (1995). Guide for the Use of the International System of Units (SI): The Metric System. Diane Publishing Co. p. 28.
↑ ANSI/ASA S1.1-2013, Acoustical Terminology

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Level (logarithmic quantity)

Mathematical definitions

Level

Level of a field quantity

Level of a root-power quantity

Level of a power quantity

Units of level

Power level

Frequency level

See also

References