# Decibel (dB)

Decibel (dB) is a measurement of the ratio of two values V1 and V2 given by: 10 log10(V2/V1) or, in some cases, by 20 log10(V2/V1) as described below.

A decibel is one tenth of a bel.

The decibel is useful to measure very large ratios. The dynamic range of human hearing, for example, includes barely audible sounds as well as very loud sounds, the ratio of which can go up to 10,000,000,000. The same number represented in decibels is 10 log10(Loudest audible / Quietest audible) = 10 log10(10,000,000,000)=100 dB, which is an easier number to work with. Thus, the range of human hearing is about 100 dB.

When decibels are used to measure voltage it helps to consider the fact that power is proportional to the square of voltage (P = V2 / R; power is equal to the square of voltage divided by resistance). In order to get a decibel measurement that is consistent between power and voltage, the first formula above is used for ratios of power and the following formula is used for ratios of voltage:

10 log10(V22/V12) = 20 log10(V2/V1)

Similarly, this last formula is used to measure sound amplitude ratios. The power of a sound wave per unit of area is P = (1/2) ω2 A2 ρ c, where ω is the angular frequency, ρ is the material density, c is the speed of sound in a particular material, and A is the displacement amplitude. Also, A = Δp0 / ω ρ c, where Δp0 is the pressure amplitude. Thus, the power of sound is proportional to the square of the displacement amplitude and to the square of the pressure amplitudes. To ensure that the decibel values are consistent 10 log10(V2/V1) is used to compare measures of sound power and 20 log10(V2/V1) is used to compare measures of amplitude.

The dynamic range of CD audio, for example, can also be presented in decibels. CD audio uses 16-bit sampling resolution. The ratio of loudest to quietest sound that can be represented with 16 bits is 215:1, which in decibels is 20 log10(215/1) = 90 dB approximately. The 16-bit sampling resolution of CDs thus is close to the dynamic range of human hearing.

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