Decibel
The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity) relative to a specified or implied reference level. Since it expresses a ratio of two (same unit) quantities, it is a dimensionless unit. A decibel is one tenth of a bel (B).
The decibel is useful for a wide variety of measurements in science and engineering (e.g., acoustics and electronics) and other disciplines. It confers a number of advantages, such as the ability to conveniently represent very large or small numbers, a logarithmic scaling that roughly corresponds to the human perception of, for example, sound and light, and the ability to carry out multiplication of ratios by simple addition and subtraction.
The decibel is not an SI unit. However, following the SI convention, the d is lowercase, as it represents the SI prefix deci-, and the B is capitalized, as it is an abbreviation of a name-derived unit (the bel). The full name decibel follows the usual English capitalization rules for a common noun.
The decibel symbol is often qualified with a suffix, which indicates which reference quantity or frequency weighting function has been used. For example, "dBm" indicates that the reference quantity is one milliwatt, while "dBu" is referenced to 0.775 volts. The practice of attaching a suffix in this way, though not permitted by SI, is widely followed.
The definitions of the decibel and bel use base-10 logarithms. For a similar unit using natural logarithms to base e, see neper.

# History
The bel was originally devised by engineers of the Bell Telephone Laboratories to quantify the reduction in audio level over a 1 mile (approximately 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 Bell System's founder and telecommunications pioneer Alexander Graham Bell. In many situations, however, the bel proved inconveniently large, so the decibel has become more common.
In April 2003, the International Committee for Weights and Measures (CIPM) considered a recommendation for the decibel's inclusion in the SI system, but decided not to adopt the decibel as an SI unit.

# Definitions

## Power
When referring to measurements of power or intensity, a ratio can be expressed in decibels by evaluating ten times the base-10 logarithm of the ratio of the measured quantity to the reference level.  Thus, if L represents the ratio of a power value P1 to another power value P0, then LdB represents that ratio expressed in decibels and is calculated using the formula:
L_\mathrm{dB} = 10 \log_{10} \bigg(\frac{P_1}{P_0}\bigg) \,
Naturally, P1 and P0 must have the same dimension (that is, must measure the same type of quantity), and must as necessary be converted to the same units before calculating the ratio of their numerical values.  Note that if P1 = P0 in the above equation, then LdB = 0.  If P1 is greater than P0 then LdB is positive; if P1 is less than P0 then LdB is negative.
Rearranging the above equation gives the following formula for P1 in terms of P0 and LdB:
P_1 = 10^\frac{L_\mathrm{dB}}{10} P_0 \,
Since a bel is equal to ten decibels, the corresponding formulae for measurement in bels (LB) are
L_\mathrm{B} = \log_{10} \bigg(\frac{P_1}{P_0}\bigg) \,
P_1 = 10^{L_\mathrm{B}} P_0 \,

## Amplitude, voltage and current
When referring to measurements of amplitude it is usual to consider the ratio of the squares of A1 (measured amplitude) and A0 (reference amplitude). This is because in most applications power is proportional to the square of amplitude. Thus the following definition is used:
L_\mathrm{dB} = 10 \log_{10} \bigg(\frac{A_1^2}{A_0^2}\bigg) = 20 \log_{10} \bigg(\frac{A_1}{A_0}\bigg) \,
The formula may be rearranged to give
A_1 = 10^\frac{L_\mathrm{dB}}{20} A_0 \,
Similarly, in electrical circuits, dissipated power is typically proportional to the square of voltage or current when the impedance is held constant. Taking voltage as an example, this leads to the equation:
G_\mathrm{dB} =20 \log_{10} \left (\frac{V_1}{V_0} \right ) \quad \mathrm \quad  
where V1 is the voltage being measured, V0 is a specified reference voltage, and GdB is the power gain expressed in decibels. A similar formula holds for current.

## Examples
Note that all of these examples yield dimensionless answers in dB because they are relative ratios expressed in decibels.
- To calculate the ratio of 1 kW (one kilowatt, or 1000 watts) to 1 W in decibels, use the formula
G_\mathrm{dB} = 10 \log_{10} \bigg(\frac{1000 \mathrm{W}}{1 \mathrm{W}}\bigg) = 30 \mathrm{dB} \,
- To calculate the ratio of 1 mW (one milliwatt) to 10 W in decibels, use the formula
G_\mathrm{dB} = 10 \log_{10} \bigg(\frac{.001 \mathrm{W}}{10 \mathrm{W}}\bigg) = -40 \mathrm{dB} \,
- To find the power ratio corresponding to a 3 dB change in level, use the formula
G = 10^\frac{3}{10} \times 1\ = 1.99526...  \approx 2 \,
It is seen that there is a 10 dB increase (decrease) for each factor 10 increase (decrease) in the ratio of the two power levels, and approximately a 3 dB increase (decrease) for every factor 2 increase (decrease). In exact terms, the factor is 103/10, or 1.9953, about 0.24% different from exactly 2. Similarly, an increase of 3 dB implies an increase in voltage by a factor of approximately 2, or about 1.41, an increase of 6 dB corresponds to approximately four times the power and twice the voltage, and so on. (In exact terms the power factor is 106/10, or about 3.9811, a relative error of about 0.5%.)

# Merits
The use of the decibel has a number of merits:
- The decibel's logarithmic nature means that a very large range of ratios can be represented by a convenient number, in a similar manner to scientific notation. This allows one to clearly visualize huge changes of some quantity. (See Bode Plot and half logarithm graph.)
- The mathematical properties of logarithms mean that the overall decibel gain of a multi-component system (such as consecutive amplifiers) can be calculated simply by summing the decibel gains of the individual components, rather than needing to multiply amplification factors. Essentially this is because log(A  B  C  ...) = log(A) + log(B) + log(C) + ...
- The human perception of, for example, sound or light, is, roughly speaking, such that a doubling of actual intensity causes perceived intensity to always increase by the same amount, irrespective of the original level. The decibel's logarithmic scale, in which a doubling of power or intensity always causes an increase of approximately 3 dB, corresponds to this perception.

# Uses

## Acoustics
The decibel is commonly used in acoustics to quantify sound levels relative to some 0 dB reference. The reference level is typically set at the threshold of perception of an average human and there are common comparisons used to illustrate different levels of sound pressure.
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. Since the human ear is not equally sensitive to all the frequencies of sound within the entire spectrum, noise levels at maximum human sensitivity  for example, the higher harmonics of middle A (between 2 and 4 kHz)  are factored more heavily into sound descriptions using a process called frequency weighting.

## Electronics
In electronics, the decibel is often used to express power or amplitude ratios (gains), in preference to arithmetic ratios or percentages. One advantage is that the total decibel gain of a series of components (such as amplifiers and attenuators) can be calculated simply by summing the decibel gains of the individual components. Similarly, in telecommunications, 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.
The decibel unit can also be combined with a suffix to create an absolute unit of electric power. For example, it can be combined with "m" for "milliwatt" to produce the "dBm".  Zero dBm is the power level corresponding to a power of one milliwatt, and 1 dBm is one decibel greater (about 1.259 mW).
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 which delivers 1 mW of power in a 600 ohm resistor, which used to be the standard reference impedance in almost all professional low-impedance audio circuits.
The bel is used to represent noise power levels in hard drive specifications. It shares the same symbol (B) as the byte.

## Optics
In an optical link, if a known amount of optical power, in dBm (referenced to 1 mW), is launched into a fiber, and the losses, in dB (decibels), of each electronic component (e.g., connectors, splices, and lengths of fiber) are known, the overall link loss may be quickly calculated by addition and subtraction of decibel quantities.
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 a star logarithmically, since, just as the ear responds logarithmically to acoustic power, the eye responds logarithmically to brightness; however astronomical magnitudes reverse the sign with respect to the bel, so that the brightest stars have the lowest magnitudes, and the magnitude increases for fainter stars.

# Common reference levels and corresponding units

## "Absolute" and "relative" decibel measurements
Although decibel measurements are always relative to a reference level, if the numerical value of that reference is explicitly and exactly stated, then the decibel measurement is called an "absolute" measurement, in the sense that the exact value of the measured quantity can be recovered using the formula given earlier. For example, since dBm indicates power measurement relative to 1 milliwatt,
- 0 dBm means no change from 1 mW. Thus, 0 dBm is the power level corresponding to a power of exactly 1 mW.
- 3 dBm means 3 dB greater than 0 dBm. Thus, 3 dBm is the power level corresponding to 103/10  1 mW, or approximately 2 mW.
- 6 dBm means 6 dB less than 0 dBm. Thus, 6 dBm is the power level corresponding to 106/10  1 mW, or approximately 250 W (0.25 mW).
If the numerical value of the reference is not explicitly stated, as in the dB gain of an amplifier, then the decibel measurement is purely relative.  The practice of attaching a suffix to the basic dB unit, forming compound units such as dBm, dBu, dBA, etc, is not permitted by SI.   However, outside of documents adhering to SI units, the practice is very common as illustrated by the following examples.

## Absolute measurements

### Electric power
dBm or dBmW
dBW

### Voltage
Note that the decibel has a different definition when applied to voltage (as contrasted with power). See the "Definitions" section above.
dBV
dBu or dBv
dBmV
dBV or dBuV

### Acoustics
dB(SPL)
dB SIL
dB SWL
dB(A), dB(B), and dB(C)
dB HL or dB hearing level is used in Audiograms as a measure of hearing loss. The reference level varies with frequency according to a Minimum audibility curve as defined in ANSI and other standards, such that the resulting audiogram shows deviation from what is regarded as 'normal' hearing.
dB Q is sometimes used to denote weighted noise level, commonly using the ITU-R 468 noise weighting

### Radar
dBZ

### Radio power, energy, and field strength
dBc
dBJ
dBm
dB or dBu
dBf
dBW
dBk

## Relative measurements
dBd
dBFS or dBfs
dB-Hz
dBi
dBiC
dBov or dBO
dBr
dBrn
dBc