Level (logarithmic quantity)

"Level quantity" redirects here. For other uses, see Level measurement.

A logarithmic ratio quantity is the logarithm of the ratio of the value of a physical quantity to a constant reference value of the same quantity. These quantities may be referred to as types of level depending on the types of the initial quantities and the scale used:

• power level for power, power density or sometimes energy, with commonly used unit decibel (dB),
• field level (or root-power level) for voltage, sound pressure, etc., with commonly used units neper (Np) or decibel (dB),
• frequency level for frequency, with commonly used units octave (oct) and decade (dec),

with units generally used and indicate the selected base of the logarithm and may depend on the nature of the physical quantity, even though a logarithm may be considered to be a dimensionless quantity. The reference values are often specified by international standards.

Examples in acoustics of power levels are sound power level (SWL), sound exposure level (SEL). An example of field level is sound pressure level (SPL).

Field (or root-power) level

The level of a root-power quantity (also known as a field quantity), denoted L_F, is defined by

$L_{F}=\log _{\mathrm {e} }\!\left({\frac {F}{F_{0}}}\right)\!~\mathrm {Np} =2\log _{10}\!\left({\frac {F}{F_{0}}}\right)\!~\mathrm {B} =20\log _{10}\!\left({\frac {F}{F_{0}}}\right)\!~\mathrm {dB} .$

where

• F is the root-power quantity, proportional to the square root of power quantity;
• F₀ is the reference value of F.

See also: Decibel § Conversions, and Neper § Units

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. The neper, bel, and decibel are defined by

• Np = 1;
• B = ⁠1/2⁠ log_e10 Np;
• dB = 0.1 B = ⁠1/20⁠ log_e10 Np.

Power level

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

$L_{P}={\frac {1}{2}}\log _{\mathrm {e} }\!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {Np} =\log _{10}\!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {B} =10\log _{10}\!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {dB} .$

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.

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

Frequency level

Frequency level is the logarithm of a ratio of frequencies, though the use of the term frequency level is seldom seen outside of the ANSI standard that defines it.

In electronics, the octave (oct) is used as a unit with logarithm base 2, and the decade (dec) is used as a unit with logarithm base 10:

$L_{f_{2}/f_{1}}=\log _{2}\!\left({\frac {f_{2}}{f_{1}}}\right)~{\text{oct}}=\log _{10}\!\left({\frac {f_{2}}{f_{1}}}\right)~{\text{dec}},$

where f₂ and f₁ are the frequencies of the ratio.

In music theory, the octave is a unit used with logarithm base 2 (called interval). A semitone is one twelfth of an octave. A cent is one hundredth of a semitone.

Logarithmic ratio quantity generalization

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 quantity;
• Q₀ is the reference value of Q.

Standards

Level and its units are defined in ISO 80000-3.

See also

• 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)

References

• Fletcher, H (1934), "Loudness, pitch and the timbre of musical tones and their relation to the intensity, the frequency and the overtone structure", Journal of the Acoustical Society of America, 6 (2)
• Taylor, Barry (1995), Guide for the Use of the International System of Units (SI): The Metric System, Diane Publishing Co., p. 28
• ISO 80000-3 (2006), Quantities and units, vol. Part 3: Space and Time, International Organization for Standardization{{citation}}: CS1 maint: numeric names: authors list (link)
• Carey, W. M. (2006), "Sound Sources and Levels in the Ocean", IEEE Journal of Oceanic Engineering, 31
• ISO 80000-8 (2007), Quantities and units, vol. Part 8: Acoustics, International Organization for Standardization{{citation}}: CS1 maint: numeric names: authors list (link)
• ANSI/ASA S1.1 (2013), Acoustical Terminology, vol. ANSI/ASA S1.1-2013, Acoustical Society of America{{citation}}: CS1 maint: numeric names: authors list (link)
• Ainslie, M. A. (2015), "A Century of Sonar: Planetary Oceanography, Underwater Noise Monitoring, and the Terminology of Underwater Sound", Acoustics Today, 11 (1)
• D'Amore, F. (2015), Effect of moisturizer and lubricant on the finger‒surface sliding contact: tribological and dynamical analysis
• IEEE/ASTM SI 10 (2016), American National Standard for Metric Practice, IEEE Standards Association{{citation}}: CS1 maint: numeric names: authors list (link)