concept

Decibels (dB)

The strongest sound a healthy ear tolerates carries about a trillion times the power of the quietest one it can detect. Numbers that span twelve zeros are unreadable, and perception tracks ratios: a doubling of power registers as roughly the same step whether it starts from a whisper or from a shout. The decibel is built on both facts. It is a logarithm of a ratio, so equal ratios become equal, readable steps.

This concept builds on logarithms, in particular the fact that equal ratios take equal steps on a log scale.

A logarithm of a power ratio

A decibel value always compares two signal powers, the rates at which they deliver energy. The level in decibels is ten times the base-10 logarithm logarithms ↗ of their ratio:

level in dB  =  10 log⁡10 ⁣PPref\text{level in dB} \;=\; 10 \,\log_{10}\!\frac{P}{P_{\text{ref}}}

PP is the power being measured and PrefP_{\text{ref}} is the reference it is compared against. A decibel value is therefore always relative: quoting one fixes a reference, explicitly or by convention. The factor 10 sizes the unit conveniently (the “deci” in decibel): a tenfold power ratio corresponds to 10 dB, a hundredfold to 20 dB, and a trillion-fold to 120 dB, so equal ratios take equal steps, exactly as the logarithm guarantees. A level of 0 dB means the ratio is 1, that is, equal to the reference; it does not mean silence.

Amplitude on the same scale, with a factor of 20

Audio work usually measures a wave's amplitude, the size of its swing. Amplitude and power are linked: power grows as the square of amplitude, because a wave with twice the swing pushes the air twice as far and twice as hard, and so delivers 2×2=42 \times 2 = 4 times the power. Squaring the input of a logarithm doubles its output, log⁡(a2)=2log⁡a\log(a^2) = 2\log a, so for amplitudes the formula's factor of 10 becomes a 20:

level in dB  =  20 log⁡10 ⁣aaref\text{level in dB} \;=\; 20 \,\log_{10}\!\frac{a}{a_{\text{ref}}}

This is the same scale with the same steps, entered through amplitude instead of power. Two landmark values follow from it. Doubling the amplitude adds 20log⁡102≈6.0220\log_{10} 2 \approx 6.02 dB, and since power is amplitude squared, +6 dB means twice the amplitude and four times the power. Doubling the power adds 10log⁡102≈3.0110\log_{10} 2 \approx 3.01 dB. When a dB figure is quoted, the choice between 10 and 20 is already settled by which quantity was measured; the scale itself is one and the same.

A worked example from digital audio: 16-bit samples can express amplitudes from 1 unit up to 216=65,5362^{16} = 65{,}536 units, a span of sixteen doublings of amplitude. At ≈6.02\approx 6.02 dB per doubling, that span is 16×6.02≈9616 \times 6.02 \approx 96 dB, which is why “16-bit audio” and “about 96 dB of range” describe the same format. Each extra bit allows one more doubling of amplitude and so adds another 6 dB to the range.

In the widget below, press ×2 amplitude repeatedly: the marker advances by the same 6.02 dB step each time, while the power ratio quadruples. Press ×10 amplitude to add exactly 20 dB. Press the 16-bit span chip to jump to the 96 dB range computed above.

The ruler is the dB scale itself, a log-scale ruler for ratios. Equal ratios occupy equal lengths on it, and the dB readout stays small and legible long after the raw ratios have run to six digits.

Reading dB figures in practice

Reading a quoted dB figure requires three pieces of information. The first is the reference: “dBFS” references digital full scale (0 dBFS is the largest representable sample, so levels are negative), while “dB SPL” references the threshold of hearing (so levels are positive). The second is whether the quantity measured is power or amplitude, since the measured quantity decides the factor of 10 or 20. The third is the conversion back to ordinary ratios, which a few landmark values handle without a calculator: +6 dB is approximately a doubling of amplitude, +20 dB is exactly amplitude ×10, and −3 dB is approximately a halving of power. Chains of gains add in dB because the underlying ratios multiply; the logarithm turns multiplication of ratios into addition of levels, the same property that gives equal ratios equal steps.

Check your understanding

In the web

Builds on Logarithms and exponents.