# Bel

dB

The decibel (abbreviated to dB) is a logarithmical unit used to express relative difference in intensity or power between two signals, usually acoustic.

It is equal to ten times the common logarithm of the ratio of the two levels.

${di\! f\!\! f} = 10 \cdot \log_{10}\left(\frac{P_2}{P_1}\right )$

The 10 factor comes from the fact the Bel unit is too large for everyday use; decibel is just more convenient.

There is a second definition of the decibel. In acoustics, it is used to express the ratio in sound pressure, because power is roughly proportional to the square of effective sound pressure.

Let 'p' be pressure, 'P' power.

${di\! f\!\! f} = 10 \cdot \log_{10}\left(\frac{P_2}{P_1}\right )$
${di\! f\!\! f} = 10 \cdot \log_{10}\left(\frac{p_2^2}{p_1^2}\right )$
${di\! f\!\! f} = 20 \cdot \log_{10}\left(\frac{p_2}{p_1}\right )$ because $\log_{10}(a^2) = 2 \cdot \log_{10}(a)$

In this scale, it means an increase of 20 dB is equal to a ten-fold ratio, or 6 dB a two-fold ratio ($20 \cdot \log_{10}(2)$ is approximately equal to 6).