Adding levels correctly: coherent and incoherent

Two loudspeakers each producing 80 dB do not produce 160 dB. Combining levels takes more than adding the numbers, and the correct method depends on whether the signals are related.

Incoherent sources

Two independent sources, such as two instruments, add their powers. The energy quantities are summed, and the result converted back to decibels:

L = 10 · log (10^(L₁/10) + 10^(L₂/10))

Two equal sources add 3 dB. Four add 6 dB, ten add 10 dB. For n equal sources the increase is 10 · log n.

When the levels differ, the quieter source matters less:

For example, 80 dB and 70 dB make 80.4 dB. A source more than 10 dB below another hardly counts, which is useful when judging noise.

Coherent signals

Two identical or related signals add their amplitudes, and the result depends on their phase difference:

In phase, the amplitudes double and the level rises by 6 dB. Out of phase, the signals cancel. This is why two loudspeakers playing the same bass note at the same level can give 6 dB at one place and almost nothing at another, see Comb filtering.

Real situations lie between the two cases: low frequencies from nearby loudspeakers tend to add coherently, whereas high frequencies and random noise add by power.

Subtracting a level

To remove background noise from a measurement, subtract the powers:

L_source = 10 · log (10^(L_total/10) − 10^(L_noise/10))

If a measurement reads 85 dB with the source and 80 dB without it, the source alone is about 83.3 dB. If the two readings differ by less than 3 dB, the result is unreliable, because small errors in the measurement produce large changes.

A note on hearing

Adding levels in this way describes sound pressure level. How loud the result sounds depends on the loudness perception, see Equal-loudness contours.

Try it

The adding levels tool handles lists of levels, two coherent signals with a chosen phase, and background-noise correction.

Draft: written by an AI assistant, awaiting technical review.