# Level (logarithmic quantity)

(Redirected from Frequency level)

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]

## Definitions

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

### Level

Level of a quantity Q, denoted LQ, is defined by[4]

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

where

• r is the base of the logarithm;
• Q is the quantity;
• Q0 is the reference value of Q.

### Level of a root-power quantity

The level of a root-power quantity (also known as a field quantity), denoted LF, is defined by [5]

${\displaystyle L_{F}=\log _{\mathrm {e} }\!\left({\frac {F}{F_{0}}}\right)\!,}$

where

• F is the root-power quantity, proportional to the square root of 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 LP, is defined by

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

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.[6]

## 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.[7] The neper, bel, and decibel are defined by

• Np = 1;
• B = 1/2 loge10 Np;
• dB = 0.1 B = 1/20 loge10 Np.

If F is a root-power quantity:

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

If P is a power quantity:

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

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 LF and LP are equal.

### Frequency level

The octave is a unit of level (specifically "frequency level",[8] for r = 2) though that concept is seldom seen outside of the standard.[9] A semitone is one twelfth of an octave. A cent is one hundredth of a semitone.