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Decibel

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The decibel (dB) is a measure of the ratio between two quantities, and is used in a wide variety of measurements in acoustics, physics and electronics. While originally only used for power and intensity ratios, it has come to be used more generally in engineering. The decibel is widely used as a measure of the loudness of sound. It is a "dimensionless unit" like percent. Decibels are useful because they allow even very large or small ratios to be represented with a conveniently small number. This is achieved by using a logarithm.

Definition

An intensity I or power P can be expressed in decibels with the standard equation

where I0 and P0 are a specified reference intensity and power.

If PdB is 10 dB greater than PdB0 then P is ten times P0. If PdB is 3 dB greater, the power ratio is very close to a factor of two.

For sound intensity, I0 is typically chosen to be 10−12 W/m2, which is roughly the threshold of hearing. When this choice is made, the units are said to be "dB SIL". For sound power, P0 is typically chosen to be 10−12 W, and the units are then "dB SWL".

In engineering, voltage V or pressure p can be expressed in decibels with the standard equation

where V0 and p0 are a specified reference voltage and pressure. Note that in physics, these equations are considered to give power in decibels, and it is then incorrect to use them if the electrical or acoustic impedence is not the same at the two points where the voltage or pressure are measured. In this formalism, decibels are always a measure of relative power or intensity, and the value is the same regardless whether power or voltage/pressure measurements are used.

If VdB is 20 dB greater than VdB0 then V is ten times V0. If VdB is 6 dB greater, the voltage ratio is very close to a factor of two.

For sound pressure, p0 is typically chosen to be 10−5 N/m2, or pascals (Pa) which is roughly the threshold of hearing. When this choice is made, the units are said to be "dB SPL".

Standards

The decibel is not an SI unit, although the International Committee for Weights and Measures (BIPM) has recommended its inclusion in the SI system. Following the SI convention, the d is lowercase, as it is the SI prefix deci-, and the B is capitalized, as it is an abbreviation of a name-derived unit, the bel, named for Alexander Graham Bell. Written out it becomes decibel. This is standard English capitalization.

Merits

The use of decibels has three different merits:

  • It is more convenient to add the decibel values of, for instance, two consecutive amplifiers rather than to multiply their amplification factors.
  • A very large range of ratios can be expressed with decibel values in a range of moderate size, allowing one to clearly visualize huge changes of some quantity.
  • In acoustics, the decibel as a logarithmic measure of ratios fits well to the logarithmic dependence of perceived loudness on sound intensity. In other words, at all levels of loudness, increasing the decibel level by the same amount creates approximately the same increase in perceived loudness — humans perceive the increase from 20 dB to 25 dB as being about the same as the increase from 90 dB to 95 dB, for example. This is known as Stevens' power law.

History of bels and decibels

A bel (symbol B) is a unit of measure of ratios, such as power levels and voltage levels. It is mostly used in telecommunication, electronics, and acoustics. Invented by engineers of the Bell Telephone Laboratory to quantify the reduction in audio level over a 1 mile length of standard telephone cable, it was originally called the transmission unit or TU, but was renamed in 1923 or 1924 in honor of the laboratory's founder and telecommunications pioneer Alexander Graham Bell.

The bel was too large for everyday use, so the decibel (dB), equal to 0.1 bel (B), became more commonly used. The bel is still used to represent noise power levels in hard drive specifications.

The neper is a similar unit which uses the natural logarithm. The Richter scale uses numbers expressed in bels as well, though this is implied by definition rather than explicitly stated. In spectrometry and optics, the absorbance unit used to measure optical density is equivalent to −1 B. In astronomy, the apparent magnitude measures the brightness of stars logarithmically, since just as the ear responds logarithmically to acoustic power, the eye responds logarithmically to brightness.

Uses

Acoustics

The decibel unit is often used in acoustics to quantify sound levels relative to some 0 dB reference. The reference may be defined as a sound pressure level (SPL), commonly 20 micropascals (20 μPa). To avoid confusion with other decibel measures, the term dB(SPL) is used for this. The reference sound pressure (corresponding to a sound pressure level of 0 dB) can also be defined as the sound pressure at the threshold of human hearing, which is conventionally taken to be 2×10−5 newtons per square metre, or 20 micropascals. That is roughly the sound of a mosquito flying 3 m away.

The reason for using the decibel is that the ear is capable of detecting a very large range of sound pressures. The ratio of the sound pressure that causes permanent damage from short exposure to the limit that (undamaged) ears can hear is above a million. Because the power in a sound wave is proportional to the square of the pressure, the ratio of the maximum power to the minimum power is above one (short scale) trillion. To deal with such a range, logarithmic units are useful: the log of a trillion is 12, so this ratio represents a difference of 120 dB.

Psychologists have found that our perception of loudness is roughly logarithmic — see the Weber-Fechner law. In other words, you have to multiply the sound pressure by the same factor to have the same increase in loudness. This is why the numbers around the volume control dial on a typical audio amplifier are related not to the voltage amplification, but to its logarithm.

Various frequency weightings are used to allow the result of an acoustical measurement to be expressed as a single sound level. The weightings approximate the changes in sensitivity of the ear to different frequencies at different levels. The two most commonly used weightings are the A and C weightings; other examples are the B and Z weightings.

Sound levels above 85 dB are considered harmful, while 120 dB is unsafe and 150 dB causes physical damage to the human body. Windows break at about 163 dB. Jet airplanes cause A-weighted levels of about 133 dB at 33 m, or 100 dB at 170 m. Eardrums rupture at 190 dB to 198 dB. Shock waves and sonic booms cause levels of about 200 dB at 330 m. Sound levels of around 200 dB can cause death to humans and are generated near bomb explosions (e.g., 23 kg of TNT detonated 3 m away). The space shuttle generates levels of around 215 dB (or an A-weighted level of about 175 dB at a distance of 17 m). Even louder are nuclear bombs, earthquakes, tornadoes, hurricanes and volcanoes, all capable of exceeding 240 dB. A more detailed chart can be found at makeitlouder.com.


Some other values:

dB(SPL) Source (with distance)
250 Inside of tornado; conventional or nuclear bomb explosion at 5 m.
180 Rocket engine at 30 m; blue whale humming at 1 m;
Krakatoa explosion at 100 miles (160 km)[1]
150 Jet engine at 30 m
140 Rifle being fired at 1 m
130 Threshold of pain; train horn at 10 m
120 Rock concert; jet aircraft taking off at 100 m
110 Accelerating motorcycle at 5 m; chainsaw at 1 m
100 Pneumatic hammer at 2 m; inside disco
90 Loud factory, heavy truck at 1 m
80 Vacuum cleaner at 1 m, curbside of busy street
70 Busy traffic at 5 m
60 Office or restaurant inside
50 Quiet restaurant inside
40 Residential area at night
30 Theatre, no talking
10 Human breathing at 3 m
0 Threshold of human hearing (with good ears)

Note that the SPL emitted by an object changes with distance from the object. Commonly-quoted measurements of objects like jet engines or jackhammers are meaningless without distance information. The measurement is not of the object's noise, but of the noise at a point in space near that object. For instance, it is intuitively obvious that the noise level of a volcanic eruption will be much higher standing inside the crater than it would be measured from 5 kilometers away.

Measurements of ambient noise do not need a distance, since the noise level will be relatively constant at any point in the area (and are usually only rough approximations anyway).

Measurements that refer to the "threshold of pain" or the threshold at which ear damage occurs are measuring the SPL at a point near the ear itself.

Under controlled conditions, in an acoustical laboratory, the trained healthy human ear is able to discern changes in sound levels of 1 dB, when exposed to steady, single frequency ("pure tone") signals in the mid-frequency range. It is widely accepted that the average healthy ear, however, can barely perceive noise level changes of 3 dB.

On this scale, the normal range of human hearing extends from about 0 dB to about 140 dB. 0 dB is the threshold of hearing in healthy, undamaged human ears; 0 dB is not an absence of sound, and it is possible for people with exceptionally good hearing to hear sounds at −10 dB. A 3 dB increase in the level of continuous noise doubles the sound power, however experimentation has determined that the frequency response of the human ear results in a perceived doubling of loudness with every 10 dB increase; a 5 dB increase is a readily noticeable change, while a 3 dB increase is barely noticeable to most people.

Sound pressure levels are applicable to the specific position at which they are measured. The levels change with the distance from the source of the sound; in general, the level decreases as the distance from the source increases. If the distance from the source is unknown, it is difficult to estimate the sound pressure level at the source.

Frequency weighting

Main article: Frequency weighting

Since the human ear is not equally sensitive to all the frequencies of sound within the entire spectrum, noise levels at maximum human sensitivity — middle A and its higher harmonics (between 2,000 and 4,000 hertz) — are factored more heavily into sound descriptions using a process called frequency weighting.

The most widely used frequency weighting is the "A-weighting", which roughly corresponds to the inverse of the 40 dB (at 1 kHz) equal-loudness curve. Using this filter, the sound level meter is less sensitive to very high and very low frequencies. The A weighting parallels the sensitivity of the human ear when it is exposed to normal levels, and frequency weighting C is suitable for use when the ear is exposed to higher sound levels. Other defined frequency weightings, such as B and Z, are rarely used.

Frequency weighted sound levels are still expressed in decibels (with unit symbol dB), although it is common to see the incorrect unit symbols dBA or dB(A) used for A-weighted sound levels.

Electronics

The decibel is used rather than arithmetic ratios or percentages because when certain types of circuits, such as amplifiers and attenuators, are connected in series, expressions of power level in decibels may be arithmetically added and subtracted. It is also common in disciplines such as audio, in which the properties of the signal are best expressed in logarithms due to the response of the ear.

In radio electronics, the decibel is used to describe the ratio between two measurements of electrical power. It can also be combined with a suffix to create an absolute unit of electrical power. For example, it can be combined with "m" for "milliwatt" to produce the "dBm". Zero dBm is one milliwatt, and 1 dBm is one decibel greater than 0 dBm, or about 1.259 mW.

Although decibels were originally used for power ratios, they are commonly used in electronics to describe voltage or current ratios. In a constant resistive load, power is proportional to the square of the voltage or current in the circuit. Therefore, the decibel ratio of two voltages V1 and V2 is defined as 20 log10(V1/V2), and similarly for current ratios. Thus, for example, a factor of 2.0 in voltage is equivalent to 6.02 dB (not 3.01 dB!). Similarly, a ratio of 10 times gives 20 dB, and one tenth gives −20 dB.

This practice is fully consistent with power-based decibels, provided the circuit resistance remains constant. However, voltage-based decibels are frequently used to express such quantities as the voltage gain of an amplifier, where the two voltages are measured in different circuits which may have very different resistances. For example, a unity-gain buffer amplifier with a high input resistance and a low output resistance may be said to have a "voltage gain of 0 dB", even though it is actually providing a considerable power gain when driving a low-resistance load.

In professional audio, a popular unit is the dBu (see below for all the units). The "u" stands for "unloaded", and was probably chosen to be similar to lowercase "v", as dBv was the older name for the same thing. It was changed to avoid confusion with dBV. This unit (dBu) is an RMS measurement of voltage which uses as its reference 0.775 VRMS. Chosen for historical reasons, it is the voltage level at which you get 1 mW of power in a 600 ohm resistor, which used to be the standard impedance in almost all professional audio circuits.

Since there may be many different bases for a measurement expressed in decibels, a dB value is meaningless unless the reference value (equivalent to 0 dB) is clearly stated. For example, the gain of an antenna system can only be given with respect to a reference antenna (generally a perfect isotropic antenna); if the reference is not stated, the dB gain value is not usable.

Optics

In an optical link, if a known amount of optical power, in dBm (referenced to 1 mW), is launched into a fibre, and the losses, in dB (decibels), of each component (e.g., connectors, splices, and lengths of fibre) are known, the overall link loss may be quickly calculated by simple addition and subtraction of decibel quantities.

Telecommunications

In telecommunications, decibels are commonly used to measure signal-to-noise ratios and other ratio measurements.

Decibels are used to account for the gains and losses of a signal from a transmitter to a receiver through some medium (free space, wave guides, coax, fiber optics, etc.) using a Link Budget.

Seismology

Earthquakes were formerly measured on the Richter scale, which is expressed in bels. (The units in this case are always assumed, rather than explicit.) The more modern moment magnitude scale is designed to produce values comparable to those of the Richter scale.

Typical abbreviations

Absolute measurements

Electric power

dBm or dBmW
dB(1 mW) — power measurement relative to 1 milliwatt.
dBW
dB(1 W) — same as dBm, with reference level of 1 watt.

Electric voltage

dBu or dBv
dB(0.775 V) — (usually RMS) voltage amplitude referenced to 0.775 volt. Although dBu can be used with any impedance, dBu = dBm when the load is 600Ω. dBu is preferable, since dBv is easily confused with dBV. The "u" comes from "unloaded".
dBV
dB(1 V) — (usually RMS) voltage amplitude of a signal in a wire, relative to 1 volt, not related to any impedance.

Acoustics

dB(SPL)
dB(Sound Pressure Level) — relative to 20 micropascals (μPa) = 2×10−5 Pa, the quietest sound a human can hear. This is roughly the sound of a mosquito flying 3 metres away. This is often abbreviated to just "dB", which gives some the erroneous notion that a dB is an absolute unit by itself.

Radio power

dBm
dB(mW) — power relative to 1 milliwatt.
dBμ or dBu
dB(μV/m) — electric field strength relative to 1 microvolt per metre.
dBf
dB(fW) — power relative to 1 femtowatt.
dBW
dB(W) — power relative to 1 watt.
dBk
dB(kW) — power relative to 1 kilowatt.

Note regarding absolute measurements

The term "measurement relative to" means so many dB greater, or smaller, than the quantity specified.

Examples:

  •  3 dBm means 3 dB greater than 1 mW.
  • −6 dBm means 6 dB less than 1 mW.
  •  0 dBm means no change from 1 mW, in other words 0 dBm is 1 mW.

Relative measurements

dB(A), dB(B), and dB(C) weighting
These symbols are often used to denote the use of different frequency weightings, used to approximate the human ear's response to sound, although the measurement is still in dB (SPL). Other variations that may be seen are dBA or dBA. According to ANSI standards, the preferred usage is to write LA = x dB, as dBA implies a reference to an "A" unit, not an A-weighting. They are still used commonly as a shorthand for A-weighted measurements, however.
dBd
dB(dipole) — the forward gain of an antenna compared to a half-wave dipole antenna.
dBi
dB(isotropic) — the forward gain of an antenna compared to an idealized isotropic antenna.
dBFS or dBfs
dB(full scale) — the amplitude of a signal (usually audio) compared to the maximum which a device can handle before clipping occurs. In digital systems, 0 dBFS would equal the highest level (number) the processor is capable of representing. (Measured values are negative, since they are less than the maximum.)
dBr
dB(relative) — simply a relative difference to something else, which is made apparent in context. The difference of a filter's response to nominal levels, for instance.
dBrn
dB above reference noise.
dBC
dB relative to carrier — in fiberoptic telecommunications, this indicates the relative levels of noise or sideband peak power, compared to the optical carrier power.

Reckoning

Decibels are handy for mental calculation, because adding them is easier than multiplying ratios. First, however, one has to be able to convert easily between ratios and decibels. The most obvious way is to memorize the logs of small primes, but there are a few other tricks that can help.

Round numbers

The values of coins and banknotes are round numbers. The rules are:

  1. One is a round number
  2. Twice a round number is a round number: 2, 4, 8, 16, 32, 64
  3. Ten times a round number is a round number: 10, 100
  4. Half a round number is a round number: 50, 25, 12.5, 6.25
  5. The tenth of a round number is a round number: 5, 2.5, 1.25, 1.6, 3.2, 6.4

Now 6.25 and 6.4 are approximately equal to 6.3, so we don't care. Thus the round numbers between 1 and 10 are these: 

Ratio  1    1.25 1.6  2    2.5  3.2  4    5    6.3  8   10
dB     0    1    2    3    4    5    6    7    8    9   10

This useful approximate table of logarithms is easily reconstructed or memorized.

The 4 → 6 energy rule

To one decimal place of precision, 4.x is 6.x in dB (energy).

Examples:

  • 4.0 → 6.0 dB
  • 4.3 → 6.3 dB
  • 4.7 → 6.7 dB

The "789" rule

To one decimal place of precision, x → (½ x + 5.0 dB) for 7.0 ≤ x ≤ 10.

Examples:

  • 7.0 → ½ 7.0 + 5.0 dB = 3.5 + 5.0 dB = 8.5 dB
  • 7.5 → ½ 7.5 + 5.0 dB = 3.75 + 5.0 dB = 8.75 dB
  • 8.2 → ½ 8.2 + 5.0 dB = 4.1 + 5.0 dB = 9.1 dB
  • 9.9 → ½ 9.9 + 5.0 dB = 4.95 + 5.0 dB = 9.95 dB
  • 10.0 → ½ 10.0 + 5.0 dB = 5.0 + 5.0 dB = 10 dB

−3 dB ≈ ½ power

A level difference of ±3 dB is roughly double/half power (equal to a ratio of 1.995). That is why it is commonly used as a marking on sound equipment and the like.

Another common sequence is 1, 2, 5, 10, 20, 50 ... . These preferred numbers are very close to being equally spaced in terms of their logarithms. The actual values would be 1, 2.15, 4.64, 10 ... .

The conversion for decibels is often simplified to: "+3 dB means two times the power and 1.414 times the voltage", and "+6 dB means four times the power and two times the voltage ".

While this is accurate for many situations, it is not exact. As stated above, decibels are defined so that +10 dB means "ten times the power". From this, we calculate that +3 dB actually multiplies the power by 103/10. This is a power ratio of 1.9953 or about 0.25% different from the "times 2" power ratio that is sometimes assumed. A level difference of +6 dB is 3.9811, about 0.5% different from 4.

To contrive a more serious example, consider converting a large decibel figure into its linear ratio, for example 120 dB. The power ratio is correctly calculated as a ratio of 1012 or one trillion. But if we use the assumption that 3 dB means "times 2", we would calculate a power ratio of 2120/3 = 240 = 1.0995 × 1012, giving a 10% error.

6 dB per bit

In digital audio, each bit offered by the system doubles the (voltage) resolution, corresponding to a 6 dB ratio. For instance, a 16-bit (linear) audio format offers an approximate theoretical maximum of (16 × 6) = 96 dB, meaning that the maximum signal (see 0 dBFS, above) is 96 dB above the quantization noise.

See also

Converters

Reference

  • Martin, W. H., "DeciBel – The New Name for the Transmission Unit", Bell System Technical Journal, January 1929.
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