Adding decibels: +3, +6, or nothing at all

Does adding a second identical source give +3 dB or +6 dB? Both answers are correct, which is why the question keeps coming up. The deciding factor is not the equipment and not the levels: it is whether the two signals are correlated, that is, whether their waveforms are locked in phase with each other.

One formula covers every case, from +6 dB through +3 dB to complete cancellation, and the calculator below runs it in both directions — adding sources, and removing a noise floor from a measurement that already contains one.

Powers add, amplitudes add as vectors

Two sound waves arriving at the same point do not add as powers; the pressures add, and then the power is the square of the total pressure. If the two waves are in phase the pressures add directly and the power goes up by a factor of four — +6.02 dB. If they are unrelated, their relative phase wanders over the measurement interval, the cross term averages to zero, and the powers add — +3.01 dB. Giving the correlation coefficient k the value cos φ covers both, and the whole range in between:

For a list of sources with a single pairwise correlation it generalises to ΣP_i + k ( (Σ√P_i)² − ΣP_i ), which is what the calculator evaluates. The three cases worth remembering:

• k = 0, unrelated: two equal sources give +3.01 dB.
• k = 1, identical and in phase: two equal sources give +6.02 dB.
• k = −1, identical and inverted: they cancel completely. Two equal sources give nothing at all, and the arithmetic returns −∞ dB.

Between the extremes the numbers are less familiar but just as real: two equal sources with k = 0.5 give +4.77 dB, which is the answer for a pair of microphones that hear mostly the same source with a little decorrelation between them. Unequal levels behave as you would expect — a source 6 dB down contributes only +0.97 dB uncorrelated, or +3.53 dB if it is identical and in phase.

Stacking N identical sources

With every source at the same level and a common pairwise correlation, the sum collapses to a single factor: N(1 + (N−1)k).

Sources	Uncorrelated (k = 0)	Half correlated (k = 0.5)	Coherent (k = 1)
1	0.00 dB	0.00 dB	0.00 dB
2	+3.01 dB	+4.77 dB	+6.02 dB
4	+6.02 dB	+10.00 dB	+12.04 dB
8	+9.03 dB	+15.56 dB	+18.06 dB
16	+12.04 dB	+21.34 dB	+24.08 dB

The uncorrelated column is the familiar 3 dB per doubling and 10 log₁₀N overall. The coherent column doubles that in decibels — 6 dB per doubling, and 20 log₁₀N. Sixteen uncorrelated sources buy 12 dB; sixteen coherent ones buy 24 dB. That difference is the entire reason subwoofer arrays are stacked tightly and why the coupling only holds below the frequency where the boxes stop being acoustically close: once the path difference between them approaches a wavelength, the correlation falls and the gain slides from +6 towards +3 dB per doubling.

Where each case turns up in audio

• Two speakers on a mono signal. On the centre line at low frequency they are within a fraction of a wavelength and add coherently, so the level is +6 dB over one speaker. Well off axis, or at high frequency where the path difference exceeds a wavelength, the sum falls towards +3 dB.
• Panning. Two identical tracks hard-panned left and right carry +3.01 dB of total acoustic power compared with one, and +0 dB at each ear — because each ear hears one speaker. A centre-panned mono signal, by contrast, is coherent at the listening position and sums to +6.02 dB on the centre line. Pan laws exist to paper over that difference: a −3 dB centre keeps the total power constant, a −6 dB centre keeps the level at each ear constant, and neither is wrong.
• Two noise sources. Unrelated by definition, so +3.01 dB per doubling, and that is why a second identical fan is not twice as loud.
• Two microphones on one source. Partially correlated, so somewhere between +3 and +6 dB, moving towards +6 as you put them closer together.
• Doubling a track in a mix. Two copies of the same recording are coherent and add at +6 dB, not +3 dB — which is why a duplicated track is not a subtle effect.

Removing a noise floor

The reverse calculation is just as useful: a measurement contains the signal and the floor, and you want the signal alone. Subtract the powers:

The correction is small when the measurement is well clear of the floor and large when it is not:

Total above the floor	Correction to apply	Practical reading
3 dB	−3.02 dB	Half the power is noise; the uncertainty is as large as the correction
6 dB	−1.26 dB	Usable, with care
10 dB	−0.46 dB	Comfortable
20 dB	−0.04 dB	The floor is irrelevant

The row to take seriously is the first. When the total is only 3 dB above the floor, the signal and the noise are nearly equal in power, and any error in either measurement is amplified by the subtraction — a 0.5 dB error in the total becomes a 1.5 dB error in the answer. Below about 3 dB the result is not small, it is undefined: the calculator refuses the calculation rather than returning a number that looks plausible.

Averaging levels over time

Average the power, not the decibels. Averaging dB values understates the true level because the logarithm is concave, and the size of the error depends on how spread out the levels are: averaging 90 dB and 80 dB as numbers gives 85 dB, while the correct power average is 87.40 dB — an error of 2.40 dB, and the naive number always falls on the low side.

For a signal whose power is exponentially distributed — which is what a noise floor looks like in a spectrum, and what a random signal looks like in a level meter — the bias has a closed form: averaging the dB values sits 10γ/ln 10 = 2.51 dB low, where γ is Euler's constant. That is the same figure the FFT article measured on a noise record, from the same cause.

Five mistakes worth avoiding

1. Assuming +6 dB for two sources. In a diffuse field, or for two unrelated sources, the answer is +3.01 dB.
2. Assuming +3 dB for two identical signals. Two copies of the same recording, or two speakers within a quarter wavelength, add at +6.02 dB.
3. Averaging decibels. It reads low, always, and the error grows with the spread of the levels.
4. Subtracting a floor from a measurement only 2 or 3 dB above it. The correction is as large as the uncertainty in the inputs.
5. Expecting a hard-panned doubled track to be twice as loud. It is +3 dB of total power, and 0 dB in each ear.

Summary

Sum powers for unrelated sources, sum amplitudes for identical ones, and use the correlation coefficient when the answer is somewhere in between. Remember that the result is per point in space: two sources can be coherent at one listening position and incoherent at another, which is exactly why loudspeaker summation is a measurement and not a calculation.

Values shown are engineering aids rather than measurements; see the disclaimer and the tool index.