# Decibel

Although it is widely used as a measure of the loudness of sound, the decibel is more generally a measure of the ratio between two quantities, and can be used to express a wide variety of measurements in acoustics and electronics. The decibel is in fact a "dimensionless unit" like percent.

 Contents

### Definition

The ratio between two power values [itex]P_1[itex] and [itex]P_0[itex] in decibels is given by the formula:

[itex]10\log_{10} \left (\frac{P_1}{P_0} \right )[itex]

Usually, [itex]P_1[itex] is the quantity being measured and [itex]P_0[itex] is a reference level.

Where the ratio is between two field strength or voltage values (so that the power being transmitted is proportional to the square of this value), the formula is:

[itex]20\log_{10} \left (\frac{V_1}{V_0} \right )[itex]

It can therefore be seen that a ratio expressed in decibels is independent of whether the measurements are made as field strength or power values.

### 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. The d is therefore 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 convenient to add the decibel values e.g. of two subsequent 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 e.g. 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.

## 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, 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 B, became more commonly used.

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.

## 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 newton per square metre, 2×10−5 N/m² or 20 micropascals. That is roughly the sound of a mosquito flying 3 m away. The ears are only sensitive to sound pressure deviations.

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 more than 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 more than 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.

Some other values:

 dBSPL Source (with distance!) 180 Rocket engine at 30 m 150 Jet engine at 30 m 130 Threshold of pain 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 20 Only Rustling of leaves 10 Human breathing at 3 m 0 Threshold of hearing (human with good ears)

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

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 nowadays 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!).

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 (dBv) 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; if the reference is not stated, the dB gain value is not useable.

### 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.

### Seismology

Earthquakes are measured on the Richter scale, which is expressed in bels. (The units in this case are always assumed, rather than explicit.)

## Typical abbreviations

### Absolute measurements

#### Electric power

dBm or dBmW
dB(1 mW@600 Ω) — in analog audio, power measurement relative to 1 milliwatt into a 600 ohm load
dBW
dB(1 W@600 Ω) — 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 volts, not related to any impedance. dBu is preferable, since dBv is easily confused with dBV. The "u" comes from "unloaded".
dBV
dB(1 V) — (usually RMS) voltage amplitude of an audio 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.

### 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 imaginary 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.

## 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.

### 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 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, for 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 a theoretical maximum of (16 x 6) = 96 dB, meaning that the maximum signal (see 0dBFS, above) is 96 dB above the quantization noise.

## Reference

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