dB Converter

dB = 10 log₁₀(P₂ / P₁) for power, or dB = 20 log₁₀(A₂ / A₁) for amplitude, so the ratio is 10^(dB/10) or 10^(dB/20)

dB

The level, in decibels, which is a logarithmic way of writing a ratio rather than a quantity in its own right. A positive number is a gain and a negative one an attenuation, and the scale is symmetric: +10 dB and −10 dB are a factor of ten up and a factor of ten down. It carries no unit and no dimension, which is the whole difficulty, because a bare number cannot say what is being compared with what

P₂ / P₁

The power ratio, recovered as 10 raised to the power of dB divided by 10. Power here means anything proportional to energy per second: watts, milliwatts, acoustic intensity, the energy in a pulse. A doubling of power is very nearly 3 dB, which is where that rule of thumb comes from — the exact value is 10 log₁₀2 = 3.010, and the 0.01 is what accumulates when you chain ten of them

A₂ / A₁

The amplitude ratio, recovered as 10 raised to the power of dB divided by 20, where amplitude means volts, amperes, sound pressure, field strength — anything whose square is proportional to power. The divisor is twice the power one because log(x²) is 2 log x, so a decibel figure applied to an amplitude always means the square root of the ratio it would mean applied to the corresponding power

R₀

The reference value, which is the quantity that 0 dB stands for. It is a plain number here with no unit attached, so the same 20 dB against a reference of 1 mW is 100 mW and against 20 micropascals is a hundred times that pressure. Nothing in the arithmetic depends on which of those it is, which is precisely why the page asks you rather than assuming on your behalf

Reach for it when a level is quoted in decibels and you need the factor behind it — how many times more power, how much voltage gain, how many times the sound pressure — or the other way round, when you have measured a ratio and want to write it as a level. It is also the page for checking which convention a figure is in, and that is the mistake that costs the most: the same −6 dB means half a voltage but a quarter of a power, so an answer that is out by a factor of two is usually a convention error rather than an arithmetic one. Use it with the reference value whenever the number carries a suffix such as dBm or dB SPL.

1. The defaults: 20 dB as a power ratio

   1. Level 20 dB, convention: power, reference 1
   2. Power ratio is 10^(20 / 10) = 10² = 100
   3. Under the amplitude convention the same 20 dB would be 10^(20 / 20) = 10
   4. Absolute level is 1 × 100 = 100, in whatever unit the reference was in

   Read the first and second results against each other: 100 and 10, from one input, and neither is wrong. That is the whole of the confusion this page exists to remove. If you were measuring power and got 100, you are done. If you were measuring a voltage and got 100 while expecting 10, your reading is not wrong either — the convention was. Notice that the third result cannot be interpreted at all without knowing what the reference of 1 was: 100 mW, 100 W and 100 micropascals are all consistent with these three inputs.

2. The same 20 dB as an amplitude ratio

   1. Level 20 dB, convention: amplitude, reference 1
   2. Amplitude ratio is 10^(20 / 20) = 10¹ = 10
   3. Under the power convention the same 20 dB would be 10^(20 / 10) = 100
   4. Absolute level is 1 × 10 = 10

   This is the previous example with the selector moved one notch, and the two ratios have swapped places. 20 dB is a decade in voltage and two decades in power, and both are quotable as 20 dB. The practical case is a preamplifier: a gain of 10 specified in dB is 20 dB if the specification is about voltage, and 10 dB if it is about power, which is why audio specifications state which one they mean.

3. A small gain, 3 dB, against a reference of 2

   1. Level 3 dB, convention: power, reference 2
   2. Power ratio is 10^(3 / 10) = 1.995262
   3. Under the amplitude convention 3 dB would be 10^(3 / 20) = 1.412538
   4. Absolute level is 2 × 1.995262 = 3.990525

   The famous doubling is not 3 dB, it is 3.010 dB, and the difference is worth knowing rather than memorising: at 3 dB exactly you are 0.24% short of double. That is harmless once, and it is not harmless ten times over — a chain of ten 3 dB stages claiming a factor of 1024 actually delivers 1000, a 2.4% shortfall. The other half of this example is the reference: the ratio does not care that it is 2, but the absolute level does, and 3.991 is only meaningful if you know what the 2 was.

4. An attenuation of −3 dB on a half-watt signal

   1. Level −3 dB, convention: power, reference 0.5
   2. Power ratio is 10^(−3 / 10) = 0.501187
   3. Under the amplitude convention −3 dB would be 10^(−3 / 20) = 0.707946
   4. Absolute level is 0.5 × 0.501187 = 0.250594

   Halving a 0.5 W signal gives 0.25 W, and the arithmetic obliges: the ratio is 0.501 and half of 0.5 is 0.2506, which rounds to 0.251. This is the one place where the two conventions are often confused in the opposite direction, because the familiar 0.707 — the half-power point of a filter — is the amplitude figure, and a filter described as 3 dB down halves the power while reducing the voltage to 70.7% of what it was.

A decibel figure means nothing until you say what 0 dB refers to, and this page deliberately does not guess. The reference value is a plain number with no unit attached, so the absolute level it returns is in whatever unit you had in mind — milliwatts, pascals, volts, or none at all. The two conventions are the only thing modelled here, and the page assumes a single load impedance throughout, which is what makes the power form and the amplitude form two statements about the same circuit rather than two different measurements. Frequency-weighted scales such as dBA are a different quantity and are not the same number as unweighted dB SPL.

How many times is 20 dB?

A hundred times if it is a power figure, and ten times if it is an amplitude figure, and the same 20 dB is both. The two come from the same definition read two ways: power uses 10 log₁₀, so 20 dB gives 10² = 100, while amplitude uses 20 log₁₀, so 20 dB gives 10¹ = 10. They are not in conflict, because power goes as the square of amplitude and squaring 10 gives 100. So the answer to "how many times" always needs the second question answered first: how many times what?

Why does −6 dB mean half?

Because it halves an amplitude, which is the same thing as quartering a power, and −6 dB is the rounded value of both. Halving an amplitude is 20 log₁₀(0.5) = −6.0206 dB, and quartering a power is 10 log₁₀(0.25) = −6.0206 dB — the same number, because quartering a power is exactly what halving an amplitude does. The 3 dB and 6 dB figures people quote are the rounded versions; the calculator returns the exact ones, which is why −6 dB comes out as 0.501 rather than 0.5.

What is the difference between dB and dBm?

dBm is a decibel figure with its reference stated: it is the power relative to 1 milliwatt, so 0 dBm is 1 mW, 20 dBm is 100 mW and −30 dBm is a microwatt. Plain dB is a bare ratio with no reference at all and cannot be converted to watts. This page handles the ratio and leaves the reference to you, which is why the reference field is a plain number: put 1 in it for dBW, 0.001 for dBm in watts, or any other 0 dB point you are working from.

Is 3 dB exactly double?

No, it is 1.995 times — doubling is 3.0103 dB — and the gap only matters when the stages stack. One 3 dB stage is 0.24% short of double, which is invisible; ten of them in a chain multiply the error to 2.4%, so a chain advertised as 1024 times delivers 1000. The forward direction is worth remembering too: doubling a power is 3 dB, doubling an amplitude is 6 dB, and neither is exactly 3 or 6.

How do I convert dB to a voltage ratio?

Use the amplitude convention, which divides by 20 rather than 10: the ratio is 10 raised to the power of dB divided by 20, so 40 dB of voltage gain is 100 times. Voltage, current and sound pressure all belong to this convention because their squares are proportional to power. If you have a dB figure for power and want the voltage ratio, halve the level first or take the square root of the power ratio — 20 dB of power is 100 times the power and 10 times the voltage, and 10 is the square root of 100.

What does 0 dB mean?

It means the ratio is 1: the quantity is unchanged, and 10^0 is 1 in either convention. It does not mean silence and it does not mean nothing, which is where the phrase "0 dB" causes trouble — 0 dB SPL is a real sound pressure of 20 micropascals, the quietest a healthy young ear can detect, and 0 dBm is a real milliwatt. A level of 0 dB is the reference itself, so the only question it leaves open is what you chose as the reference.

• NIST Guide for the Use of the International System of Units (SI), §8.7 — the bel and the decibel as dimensionless logarithmic units for expressing ratios — NIST
• Decibel — the two conventions and their 10 and 20 divisors, the reference suffixes such as dBm, dBW and dB SPL, and the difference between field and power quantities — Wikipedia
• NOISE AND HEARING LOSS PREVENTION — the recommended exposure limit of 85 dBA over eight hours, the standard behind the noise figure most often quoted in decibels — NIOSH