The human ear does not react to changes in sound in a simple, linear way. To double the loudness of a sound, say by turning up the volume control on an audio amplifier, most people find that they need to increase the power produced by the loudspeaker by about ten times. If we were to increase the power by a further factor of ten, i.e. 100 times the power that we started with, the sound would only appear to double in loudness again – so subjectively, it would be about four times as loud as the original sound.
A tenfold ratio of power is defined as 1 bel (from Alexander Graham Bell (1847 – 1922)). The bel scale is thus logarithmic to the base 10. So the ratio of power expressed in bels (called the “level difference”) is found by taking the log (base 10) of the ratio of the power levels. This unit is too large for practical purposes, so we use one tenth of a bel – the decibel:
Level difference = 10log(10)( P2/P1) dB.
Notice the correct notation: small b for bel when written in full (as it is an ordinary word), large B for the abbreviation (to honour Mr. Bell) and small d for deci.
Under good listening conditions, the minimum change in sound level which can be detected by the human ear is one decibel (1 dB).
From now on all logarithms are assumed to be base 10.
dB, Power and Voltages
To use the decibel system for voltage levels, we must use Ohm’s Law to convert power to voltage. Power W (in Watts) is voltage V (in Volts) multiplied by current I (in Amps): W = VI and Ohm’s Law gives I = V/R where R is the resistance (in Ohms). Simple substitution shows that the power varies as the square of the voltage. Squaring a number corresponds to doubling its logarithm, so:
Level difference = 20log(V1/V2) dB
The statement that the power varies as the square of the voltage is only true if the voltages being compared are across the same impedance. In audio we are mainly interested in voltage, not power (leaving aside loudspeaker amplifiers), so it is very convenient to forget the impedances altogether, and compare voltages using the above formula regardless; and that is what we do.
This does mean that we cannot convert back to a power ratio a level difference that has been derived from a voltage ratio, unless we happen to know that the voltages are across the same impedance.
Calculating with decibels
When you add the logarithms of two numbers together, the result is the logarithm of the product of the two numbers: logA + logB = log(AB)
So the effect of multiplying a voltage by a series of ratios can be calculated by adding the corresponding decibels. For example, a power ratio of 2:1 is a level difference of 3 dB, and a power ratio of 3:1 is a level difference of 4.8 dB, so a power ratio of 6:1 (2 x 3) is a level difference of (3+4.8) = 7.8 dB.
Similarly, dividing by a factor corresponds to subtracting the corresponding level difference in decibels.
To convert a level difference back to a power or voltage ratio: divide by 10 to change it to bels; (only if converting to a voltage ratio, divide by two;) take 10 to the power of the resulting number (also called the anti-logarithm or antilog).
Any form of power ratio can be expressed in decibels; for example, the ratio of the acoustic power produced by a loudspeaker to its electrical power input. In practice decibels find most application with sound and signals in telecommunications and broadcasting.
As well as power, they are commonly used with many quantities that have a direct connection with power such as voltage, sound pressure and radio frequency field strength. As the decibel describe ratios, it can also be used to specify such factors as the gain of electronic amplifiers and the loss of attenuators, whether electronic or acoustic.