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The decibel (dB) is a dimensionless unit of ratio which is used to express the relationship between a variable quantity and a known reference quantity. Since decibels express a relationship between a variable and a known reference, they are useful for a wide variety of measurements in acoustics, physics, electronics and other disciplines. The calculation of decibels uses a logarithm to allow very large or very small relations to be represented with a conveniently small number (similar to scientific notation).
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.
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 blocking 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.
A decibel is defined in two equivalent ways.
When referring to measurements of power or intensity it is:
But when referring to measurements of amplitude it is:
where X0 is a specified reference with the same units as X. Which reference is used 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.
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 .
For sound intensity, I0 is typically chosen to be 10−12 W/m2, which is roughly the threshold of human hearing in air. 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 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.
Substituting a measured voltage and a reference voltage 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:
where V0 is a specified reference voltage. This means a 20 dB increase for every factor 10 increase in the voltage ratio, or approximately 6 dB increase for every factor 2.
The use of decibels has a number of merits:
The decibel unit is commonly used in acoustics to quantify sound levels relative to some 0 dB reference. See Sound#Examples of sound pressure and sound pressure levels.
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.
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.
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.
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.
For the same source pressure at 1 m, the underwater sound pressure level will be higher by 26 dB, due to the difference in reference levels (20 µPa vs 1 µPa = 26.0 dB difference). Additional confusion is sometimes caused by the difference in characteristic acoustic impedance, which is a factor of 3600 higher in water than in air, due to the higher speed of sound and density in water. This difference results in a sound wave in water having an intensity level 36 dB lower than a sound wave in air of the same pressure amplitude.
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 and telecommunications, 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.
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.
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 reference impedance in almost all professional low impedance audio circuit.
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 (often a theoretical a perfect isotropic antenna); if the reference is not stated, the dB value is a relative measurement, such as the gain of an amplifier.
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.
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.
The term "measurement relative to" means so many dB greater than or less than the quantity specified. Some examples:
dBm or dBmW
dBu or dBv
dBμ or dBu
dB(A), dB(B), and dB(C)
dBFS or dBfs
dBov or dBO
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 values of coins and banknotes are round numbers. The rules are:
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 10dB 0 1 2 3 4 5 6 7 8 9 10
This useful approximate table of logarithms is easily reconstructed or memorized.
To one decimal place of precision, 4.x is 6.x in dB (energy).
To one decimal place of precision, x → (½ x + 5.0 dB) for 7.0 ≤ x ≤ 10.
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.
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 (power) 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.
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.