Decibel (dB) Calculator

The decibel is one of the most widely used and most widely misunderstood units in engineering. It is not a quantity of sound, power, or signal, it is a logarithmic ratio, a compressed way of writing "this compared to that" when the comparison spans many orders of magnitude. Human hearing covers roughly a trillion-to-one range in acoustic intensity from the faintest audible sound to the threshold of pain; radio receivers routinely handle signals spanning a similar range. Writing those numbers out in linear form is unmanageable, so we take a base-10 logarithm and multiply by ten. That is the entire idea behind the decibel.

This calculator handles the four operations that actually come up: converting a ratio into a level and back, adding several sound levels together correctly, subtracting a background level from a measurement, and moving between the absolute power scales dBm, dBW and watts.

How to Use the Decibel Calculator

1. Ratio ⇄ decibels. First choose whether your quantity is power-like (watts, acoustic intensity, energy) or amplitude-like (voltage, current, sound pressure). Then type either a ratio or a level; the other updates instantly. The calculator also shows the alternative interpretation of the same ratio alongside it, so you can see immediately what a wrong 10-versus-20 choice would have cost you.
2. Add sound levels. Enter up to six individual levels in decibels. Blank fields are ignored. The combined level appears, along with how much the combination raises the loudest single source, usually a much smaller number than people expect.
3. Subtract background. Enter the total measured level and the background level. The calculator recovers the source level alone by energy subtraction and warns you when the gap between the two is too small for the correction to be meaningful.
4. dBm, dBW, watts. Type into any of the four fields and the others follow. Preset buttons load the canonical anchor points: 1 µW, 1 mW, 100 mW, 1 W and 100 W.

Every input is validated. A ratio of zero or a negative ratio has no decibel value at all, and the calculator refuses to produce one rather than showing you a NaN or an infinity.

The Formulas

Power ratio to level. The bel is defined on power. One bel is a factor of ten in power, and the decibel is one tenth of a bel:

L = 10 × log₁₀(P₂ / P₁)

Amplitude (root-power) ratio to level. Voltage, current and sound pressure are field quantities. Power goes as the square of a field quantity under the standard assumption of equal reference impedance, and log₁₀(V²) = 2 × log₁₀(V), so the factor of ten becomes a factor of twenty:

L = 20 × log₁₀(V₂ / V₁)

ISO 80000-3 calls these "root-power quantities"; the older term "amplitude quantities" is still in universal use among engineers. The distinction matters: the same ratio of 2 is +3.0103 dB as a power ratio and +6.0206 dB as an amplitude ratio.

Level back to ratio. Both relations invert straightforwardly:

P₂ / P₁ = 10^(L / 10) and V₂ / V₁ = 10^(L / 20)

Adding levels (incoherent sources). Decibels do not add. To combine independent sound sources you convert each level back into a relative energy, add the energies, then convert the total back into a level:

L_total = 10 × log₁₀( Σ 10^(Lᵢ / 10) )

This is the standard incoherent (uncorrelated) sum assumed throughout environmental and occupational noise work, including ISO 9613 outdoor propagation modelling.

Subtracting a background level. The same idea in reverse:

L_source = 10 × log₁₀( 10^(L_total / 10) − 10^(L_bg / 10) )

This is only defined when the total exceeds the background. When the gap is small, the subtraction amplifies measurement error dramatically, which is why standardised measurement practice treats a total that is less than 3 dB above the background as too close to correct reliably.

Absolute power levels. dBm and dBW are levels with an implied reference:

L(dBm) = 10 × log₁₀(P / 1 mW) and L(dBW) = 10 × log₁₀(P / 1 W)

Because 1 W = 1000 mW exactly and 10 × log₁₀(1000) = 30, the two scales differ by exactly 30: dBm = dBW + 30.

Sound pressure level. The acoustic reference pressure is p₀ = 20 µPa (ISO 80000-8), and because pressure is a root-power quantity:

L_p = 20 × log₁₀(p / 20 µPa)

Worked Examples

Example 1, doubling a power. An amplifier stage takes 5 W in and puts 10 W out. The power ratio is 10 / 5 = 2, so the gain is 10 × log₁₀(2) = 3.0103 dB. This is the origin of the engineer's rule of thumb that "3 dB is double". It is not exact, the exact figure is 3.0103, but for almost every practical purpose 3 dB is close enough.

Example 2, doubling a voltage. A preamp raises a signal from 0.5 V to 1.0 V. The voltage ratio is also 2, but voltage is a root-power quantity, so the gain is 20 × log₁₀(2) = 6.0206 dB. Same ratio, different quantity type, double the decibel figure. If you had entered this as a power ratio you would have reported 3 dB and understated the gain by a factor of two in dB, a factor of two in power.

Example 3, two identical machines. Two ventilation fans each produce 60 dB at the operator position. Naively adding gives 120 dB, which would be near the threshold of pain. The correct calculation converts to energy: 10^(60/10) + 10^(60/10) = 2 × 10⁶, and 10 × log₁₀(2 × 10⁶) = 63.0103 dB. Doubling the number of identical sources always adds exactly 10 × log₁₀(2) = 3.0103 dB, whatever the starting level. Turning off one fan buys you 3 dB, barely a perceptible difference.

Example 4, three unequal sources. A workshop has three noise sources at 85 dB, 82 dB and 78 dB. The energy sum is 10^8.5 + 10^8.2 + 10^7.8 = 3.1623×10⁸ + 1.5849×10⁸ + 6.3096×10⁷ = 5.3781×10⁸, giving 10 × log₁₀(5.3781×10⁸) = 87.31 dB. Note that the 78 dB source contributes only about 0.5 dB to the total. This is the practical lesson of logarithmic addition: quieting the loudest source is almost always where the improvement lies, and eliminating anything more than about 10 dB below the loudest source is close to pointless.

Example 5, dBm and watts. A transmitter is rated at 30 dBm. Converting: P = 10^(30/10) mW = 1000 mW = 1 W. Its dBW figure is 30 − 30 = 0 dBW, which is another way of saying 1 W. At the other end of the scale, 0 dBm is exactly 1 mW and −30 dBm is exactly 1 µW. A Wi-Fi client reporting −70 dBm is receiving 10^(−70/10) mW = 10⁻⁷ mW = 100 pW, which is a perfectly usable signal, negative dBm values are normal and carry no implication of anything being wrong.

Where It Shows Up

Audio engineering. Mixing consoles, compressors and level meters are calibrated in decibels throughout. Fader movements, gain staging, and headroom are all discussed in dB because the ear responds roughly logarithmically to intensity. A 10 dB increase is often described as "roughly twice as loud" subjectively, even though it represents ten times the acoustic power, the gap between those two statements is the difference between physics and perception.

Radio and telecommunications. Transmit power, antenna gain, cable loss, receiver sensitivity and link budgets are all in decibels, which turns a chain of multiplications into a chain of additions. A link budget is literally a column of dB figures added and subtracted: transmit power in dBm, plus antenna gain in dBi, minus path loss in dB, minus cable loss in dB, gives received power in dBm.

Occupational and environmental noise. Workplace exposure limits are written in A-weighted decibels. OSHA's permissible exposure limit under 29 CFR 1910.95 is 90 dBA over an 8-hour time-weighted average, while NIOSH recommends a more protective 85 dBA over 8 hours. Environmental noise assessments combine contributions from roads, plant and machinery using exactly the logarithmic addition this calculator performs.

Electronics and instrumentation. Signal-to-noise ratio, dynamic range, common-mode rejection ratio and filter roll-off (the familiar "−3 dB point" and "20 dB per decade") are all decibel quantities. The −3 dB point of a filter is the frequency at which output power has fallen to half its passband value.

Ultrasound and medical imaging. Attenuation coefficients in tissue are quoted in dB per centimetre per megahertz, and gain compensation in an ultrasound machine is applied in decibels to offset depth-dependent attenuation.

Common Mistakes

Using 10·log₁₀ for a voltage ratio (or 20·log₁₀ for a power ratio). This is by far the most common decibel error, and it silently doubles or halves every answer. The rule is simple: if the quantity is power, intensity or energy, use 10. If it is voltage, current or pressure, use 20. When in doubt, ask what the quantity would do to power, if doubling it quadruples the power, it is a root-power quantity.

Adding decibels arithmetically. Two 60 dB sources are 63 dB, not 120 dB. Levels only add arithmetically when they represent gains in a chain, where the underlying ratios multiply. Sound levels from separate sources represent energies that add, so the levels must be combined logarithmically.

Treating dB as an absolute unit. A bare "dB" is always a ratio and requires a stated reference to be an absolute value. dBm implies 1 mW, dBW implies 1 W, and dB SPL implies 20 µPa. Writing "the signal is 40 dB" without a reference is incomplete.

Ignoring frequency weighting in acoustics. A-weighting, C-weighting and Z-weighting produce different numbers for the same sound. Occupational limits are specified in dBA. Quoting a level without the weighting makes it ambiguous, and comparing a dBA figure with a dBC figure compares two different measurements.

Subtracting a background that is too close. If the total is only 1 dB above the background, the energy subtraction produces a source level with an uncertainty of many decibels. The arithmetic runs; the answer means nothing. Standard practice is to discard corrections when the gap falls below about 3 dB.

Expecting the ratio of a negative or zero value. The logarithm of zero is undefined and the logarithm of a negative number is not real. A power of zero has no decibel level, and a "negative power" is not a physical quantity. This is different from a negative level, which is perfectly ordinary and just means the ratio is less than one.

Frequently Asked Questions

Why is doubling the power only 3 dB?

Because the decibel is logarithmic and defined so that a factor of ten in power is 10 dB. A factor of two is 10 × log₁₀(2) = 3.0103 dB. The "3 dB rule" that engineers quote is this figure rounded. It follows that four times the power is about 6 dB, eight times is about 9 dB, and ten times is exactly 10 dB.

When do I use 20·log₁₀ instead of 10·log₁₀?

Use 20·log₁₀ whenever the ratio is of a field or root-power quantity, voltage, current, sound pressure, particle velocity, electric field strength. Use 10·log₁₀ for power, intensity and energy. The factor of two arises because power is proportional to the square of a field quantity under equal reference impedance, and taking the logarithm of a square brings the exponent out front.

How do I add two sound levels together?

Convert each level back to relative energy with 10^(L/10), add the energies, then convert back with 10 × log₁₀ of the sum. Two 60 dB sources give 63.01 dB. Two levels 10 dB apart give essentially the louder one plus 0.4 dB. This calculator does the arithmetic for up to six sources at once.

What is the difference between dBm and dBW?

Only the reference power. dBm is referenced to 1 milliwatt and dBW to 1 watt. Since 1 W = 1000 mW and 10 × log₁₀(1000) = 30, the two scales differ by exactly 30 dB: dBm = dBW + 30. A 1 W transmitter is 30 dBm and 0 dBW.

Can a decibel value be negative?

Yes, and it is completely normal. A negative level simply means the ratio is less than one, an attenuation rather than a gain. A filter with −20 dB of attenuation passes one hundredth of the power. A received signal of −70 dBm is 100 picowatts. Nothing about a negative dB figure implies an error.

Is 10 dB louder really twice as loud?

Perceptually, roughly, but this is psychoacoustics, not physics. A 10 dB increase corresponds to ten times the acoustic power and is commonly reported as sounding about twice as loud to a typical listener at moderate levels. The relationship depends on frequency, level and the individual, so treat "10 dB ≈ twice as loud" as a rule of thumb rather than a measurement.

Why do sound level meters report dBA rather than dB?

Because human hearing is much less sensitive to low and very high frequencies than to the midrange, an unweighted measurement over-represents parts of the spectrum that people barely hear. A-weighting applies a filter that approximates the ear's response at moderate levels, which is why occupational exposure limits are written in dBA. C-weighting is flatter and used for peak and low-frequency work.

can the Decibel (dB) Calculator be used for professional or commercial purposes?

The mathematics is exact and suitable for professional, commercial and educational use. However, the reference sound-level table gives typical ranges only, and no calculator substitutes for a calibrated sound level meter when compliance, hearing conservation or legal exposure limits are at stake. For regulated work, consult the applicable standard and a qualified acoustician.

References

• IEC 60027-3, Letter symbols to be used in electrical technology, Part 3: Logarithmic and related quantities, and their units. The international standard defining the bel, the decibel and the neper, and the distinction between power and field quantity levels.
• ISO 80000-3, Quantities and units, Part 3: Space and time. Establishes the modern terminology of "root-power quantities" for the family previously called amplitude quantities, and the associated 20·log₁₀ convention.
• ISO 80000-8, Quantities and units, Part 8: Acoustics. Defines sound pressure level, the reference sound pressure of 20 µPa, and the related acoustic level quantities.
• NIST Special Publication 811, Section 8.7, Guide for the Use of the International System of Units (SI): Logarithmic quantities and units, level, neper, bel. The authoritative US guidance on how logarithmic quantities are written and used alongside SI units.
• ISO 9613, Acoustics, Attenuation of sound during propagation outdoors. The standard modelling framework for outdoor sound propagation, which uses the incoherent logarithmic sum of source contributions implemented in this calculator.
• OSHA 29 CFR 1910.95, Occupational noise exposure. The US regulation establishing the 90 dBA 8-hour permissible exposure limit and the associated hearing conservation requirements.
• NIOSH Publication No. 98-126, Criteria for a Recommended Standard: Occupational Noise Exposure. The NIOSH recommended exposure limit of 85 dBA as an 8-hour time-weighted average, with a 3 dB exchange rate.