Decibel

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 in measurements 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 (similar to scientific notation). This is achieved by using a logarithm.

History
The bel (symbol B) 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 (1.6 km) 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, for instance. The Richter scale uses numbers expressed in bels as well, though they are not labeled with a unit. 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.

Definition
A decibel is defined in two common ways.

When referring to measurements of power or intensity it is:
 * $$ X_\mathrm{dB} = 10 \log_{10} \bigg(\frac{X}{X_0}\bigg) \ $$

But when referring to measurements of amplitude it is:
 * $$ X_\mathrm{dB} = 20 \log_{10} \bigg(\frac{X}{X_0}\bigg) \ $$

where X0 is a specified reference with the same units as X. In many cases, the reference is 1 and so is ignored. Which one people use depends on convention and context. When the impedance is held constant, the power is proportional to the square of the amplitude of either voltage or current, and so the above two definitions become consistent.

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

I_\mathrm{dB} = 10 \log_{10} \left(\frac{I}{I_0} \right) \quad \mathrm{or} \quad P_\mathrm{dB} = 10 \log_{10} \left(\frac{P}{P_0} \right)\ , $$ where I0 and P0 are a specified reference intensity and power.

Examples
As examples, 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 $$(10^{3 \over 10} = 1.99526)$$.

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".

decibels in electrical circuits
In electrical circuits, the dissipated power is typically proportional to the square of the voltage V, and for sound waves, the transmitted power is similarly proportional to the square of the pressure amplitude p. Effective sound pressure is related to sound intensity I, density ρ and speed of sound c by the following equation:
 * $$I = p_e^2 / \rho_0 c$$

Substituting a measured voltage or pressure and a reference voltage or pressure and rearranging terms leads to the following equations and accounts for the difference between the multiplier of 10 for intensity or power and 20 for voltage or pressure:

V_\mathrm{dB} =20 \log_{10} \left (\frac{V_1}{V_0} \right ) \quad \mathrm{or} \quad p_\mathrm{dB} = 20 \log_{10} \left (\frac{p_1}{p_0} \right )\ , $$ where V0 and p0 are a specified reference voltage and pressure. This means a 20 dB increase for every factor 10 increase in the voltage or pressure ratio, or approximately 6 dB increase for every factor 2. Note that in physics, decibels refer to power ratios only; it is incorrect to use them if the electrical or acoustic impedances are not the same at the two points where the voltage or pressure are measured, though this usage is very common in engineering. For example, the power carried by a sound wave at atmospheric pressure is only proportional to the squared pressure amplitude as long as the latter is much smaller than 1 atmosphere.

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 a number of 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. (See Bode Plot and half logarithm graph.)
 * In acoustics, the decibel scale was adopted for measuring sound intensity, which, according to Fechner's Law is a good fit to loudness perception. However, "not long after they had adopted the decibel scale for measuring sound intensities, the engineers noted that equal steps on the logarithmic (decibel) scale do not behave like equal steps. A level 50 dB positive threshold does not sound at all like half of 100 dB, as Fechner's Law implies it should.” (Stevens, 1957: 163). This led to the development of Stevens' Power Law which is generally found to be a better fit to data. Stevens (1957) suggested replacing the decibel scale with the Sone Scale, but it did not seem to take root.

Difficulties
The use of decibels frequently causes confusion:
 * It is unclear to many users whether any unit requires the 20&middot;log10 or 10&middot;log10 formulation.
 * The 'deci' formulation causes confusion - understanding that this is merely bels divided by ten, and that a one bel increase means an increase of 10 to the power 1, i.e. a factor of ten increase, may add clarity.

Acoustics
The decibel unit is commonly used in acoustics to quantify sound levels relative to some 0 dB reference. Commonly, sound intensities are specified as a sound pressure level (SPL) relative to 20 micropascals (20 µPa) in gases and 1 µPa in other media (standardized in ANSI S1.1-1994). 20 µPa corresponds to the threshold of human hearing (roughly the sound of a mosquito flying 3 m away). Often, the unit dB(SPL) is used, implying the standard reference, though this is discouraged by the Acoustical Society of America, which recommends explicitly stating the reference level for each measurement; "100 dB re 20 µPa". . In the remainder of this section, the reference level of 20 µPa is implied.

Rationale
A 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.

Psychology
Psychologists have debated whether loudness perception is better described as roughly logarithmic (see the Weber-Fechner law) or as a power law (see Stevens' power law), where the latter is now generally more accepted. A consequence of either model is that a volume control dial on a typical audio amplifier that is labeled linearly in voltage amplification will affect the loudness much more for lower numbers than higher ones. This is why some are labeled in relation to decibels, i.e. the numbers are related to the logarithm of intensity.

Weightings
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.

Safety
In air, sound pressure levels above 85 dB are considered harmful, while 95 dB is considered unsafe for prolonged periods and 120 dB causes an immediate perforation of the ear drum (tympanic membrane). Windows break at about 163 dB. Jet aircraft cause A-weighted levels of about 133 dB at 33 m, or 100 dB at 170 m. In air at atmospheric pressure, the simple relationship between pressure and power of a sound wave breaks down for pressures on the order of or greater than 1 atmosphere, which corresponds to an SPL of 194 dB re 20 µPa (i.e. $$20\log_{10}(1\ \mathrm{atm}/20\ \mu\mathrm{Pa})=194.09$$). Waves with higher pressures are more properly called shock waves rather than sound waves; their properties are very different from those of normal sound waves. One could extend the meaning of sound pressure level in order to describe the pressure waves emitted by processes such as earthquakes and explosions, and get numbers exceeding 194 dB, but these numbers should only be used if it is clear how the measurable quantities are converted into SPL. An extensive list can be found at makeitlouder.com.

Note that the SPL emitted by an object changes with distance d from the object with 1/d. 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 the air near that object; sound pressure levels are applicable only to the specific position at which they are measured. The levels change with the distance from the source of the sound; generally decreasing as the distance from the source increases. For instance, there is no single number to describe the sound level of a volcanic explosion; 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 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. 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).

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(SPL) to about 140 dB(SPL). 0 dB(SPL) is the threshold of hearing in healthy, undamaged human ears at 1 kHz; 0 dB(SPL) is not an absence of sound, and it is possible for people with exceptionally good hearing to hear sounds at −10 dB(SPL). A 3 dB increase in the level of continuous noise doubles the sound power, however experimentation has determined that the response of the human ear results in a perceived doubling of loudness for approximately every 10 dB increase (part of Stevens' power law).

Relation to Loudspeakers
Speaker sensitivity is usually given in dBSPL @ 1 Watt @ 1 meter.

The equation for dBSPL is :$$ X_\mathrm{dB} = 20 \log_{10} \bigg(\frac{X}{X_0}\bigg) \ $$. This means that a doubling in sound pressure output from a speaker relates to a 6 dBSPL increase.

A practical example
A fictional 2 way speaker (A box with separate driver for high("Treble") and low("Bass") ) has the following specs:

High driver: 92 dBSPL @ 1W @ 1m. A Low driver: 86 dBSPL @ 1W @ 1m. B

Now if we want to match the output of the two speakers so the sound is "equally loud" we need to do the following:

Get the difference between the two by subtracting the sensitivity: Difference in sensitivity = A-B = 92 dBSPL - 86 dBSPL = 6 dBSPL As we concluded earlier this 6dB difference requires that we double the power delivered to the low driver. Since a doubling in [power] relates to 3 dB, we need to adjust the cross-over unit in this system so that the [gain] of the Low signal is 3dB more than the Highs. If there is no crossover you can always adjust the Amplifier's output to be 3dB more.

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 and 4 kHz) — 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. Performance characteristics for professional and consumer audio products are commonly measured with A-weighted filtering.

In water
For the same source pressure at 1 m, the underwater sound pressure level will be higher by 62 dB, due to the difference in reference levels (20 µPa vs 1 µPa = 26.0 dB difference), and the difference in acoustic impedance between air and water (3600 times = 35.6 dB difference).

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 also 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 considered an absolute measurement only if 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 value is a relative measurement, such as the gain of an amplifier.

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 electronic 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.

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 600 Ω. 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 "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 than or less than the quantity specified.

Some 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. This is an instantaneous (sample) value as compared to the dBm/dBu/dBv which are typically RMS.(Measured values are usually negative, since they should be 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 See also dBrnC.
 * dBc : dB relative to carrier — in telecommunications, this indicates the relative levels of noise or sideband peak power, compared to the 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 linear pulse-code modulation, the first bit (least significant bit, or LSB) produces residual quantization noise (bearing little resemblance to the source signal) and each subsequent bit offered by the system doubles the (voltage) resolution, corresponding to a 6 dB ratio. So for instance, a 16-bit (linear) audio format offers 15 bits beyond the first, for a dynamic range (between quantization noise and clipping) of (15 × 6) = 90 dB, meaning that the maximum signal (see 0 dBFS, above) is 90 dB above the theoretical peak(s) of quantization noise. The negative impacts of quantization noise can be reduced by implementing dither.

dB chart
As is clear from the above description, the dB level is a logarithmic way of expressing not only power ratios, but also voltage ratios The following tables are cheat-sheets that provide values for various dB power ratios and also "voltage" ratios.

Converters

 * Vpeak, VRMS, Power, dBm, dBu, dBV converter
 * Conversion: dBu to volts, dBV to volts, and volts to dBu, and dBV
 * Conversion of sound level units: dBSPL or dBA to sound pressure p and sound intensity J
 * Conversion: Voltage V to dB, dBu, dBV, and dBm
 * Only Power: dBm to mW conversion
 * Decibel - Description and calculations

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