# Decibel

The bel (symbol B) expresses the ratio of two powers by the decimal logarithm of this ratio. This unit is not often used, having been replaced by the decibel (symbol dB) which is one-tenth of a bel.

For example, if P1 and P2 are two powers, their ratio expressed in decibels is:

10 10log (P1 / P2)

Originally the dB is used to express the ratio of two powers, such as:

• the signal-to-noise ratio,
• the ratio of received power to the transmitted power
• the ratio of output and input power, e.g.:
• the gain of an amplifier,
• antenna gain.

The decibel is also used to express the signal power itself. The signal power is then given as a ratio to 1 Watt. Sometimes the symbol dBW is used to denote this ratio. The signal power can also be given as a ratio to 1 mW (1/1000 Watt). In that case the symbol dBm is used. A power in dBm is equal to a power in dBW + 30.

The decibel is also used to express the ratio of two field quantities, such as voltage, current, sound pressure, electric field, charge velocity or density, the square of which in linear systems is proportional to power. To obtain the same numerical value as a power ratio, the logarithm of the field quantity ratio is multiplied by the factor 20, assuming that the impedances are equal.

For example, if V1 and V2 are two voltages, their ratio expressed in decibels is:

20 10log (V1 / V2)

The relationship between a current or voltage ratio and that of the corresponding power ratio is impedance dependent. Use of the decibel when the impedances are not equal is not appropriate unless adequate information is given concerning the impedances involved.

## Calculating in dB's

In the following table a number of ratios and their equivalent in decibels are given.

Other values can be calculated with the use of the table and the following rules:

log (a · b) = log(a) + log(b)   and   log (ax) = x · log (a)

For example, an power ratio of 42 (6 · 7) is equal to 7.8 + 8.5 = 16.3 dB and a power ratio of 1000 (103) is equal to 3 · 10 = 30 dB.