# 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,[1] 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.[2]

## 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:

${\displaystyle L_{\mathrm {dB} }=10\log _{10}{\bigg (}{\frac {P_{1}}{P_{0}}}{\bigg )}\,}$

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:

${\displaystyle P_{1}=10^{\frac {L_{\mathrm {dB} }}{10}}P_{0}\,}$.

Since a bel is equal to ten decibels, the corresponding formulae for measurement in bels (LB) are

${\displaystyle L_{\mathrm {B} }=\log _{10}{\bigg (}{\frac {P_{1}}{P_{0}}}{\bigg )}\,}$
${\displaystyle P_{1}=10^{L_{\mathrm {B} }}P_{0}\,}$.

### 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:

${\displaystyle L_{\mathrm {dB} }=10\log _{10}{\bigg (}{\frac {A_{1}^{2}}{A_{0}^{2}}}{\bigg )}=20\log _{10}{\bigg (}{\frac {A_{1}}{A_{0}}}{\bigg )}\,}$

The formula may be rearranged to give

${\displaystyle A_{1}=10^{\frac {L_{\mathrm {dB} }}{20}}A_{0}\,}$

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:

${\displaystyle G_{\mathrm {dB} }=20\log _{10}\left({\frac {V_{1}}{V_{0}}}\right)\quad \mathrm {\quad } }$

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
${\displaystyle G_{\mathrm {dB} }=10\log _{10}{\bigg (}{\frac {1000\mathrm {W} }{1\mathrm {W} }}{\bigg )}=30\mathrm {dB} \,}$
• To calculate the ratio of 1 mW (one milliwatt) to 10 W in decibels, use the formula
${\displaystyle G_{\mathrm {dB} }=10\log _{10}{\bigg (}{\frac {.001\mathrm {W} }{10\mathrm {W} }}{\bigg )}=-40\mathrm {dB} \,}$
• To find the power ratio corresponding to a 3 dB change in level, use the formula
${\displaystyle G=10^{\frac {3}{10}}\times 1\ =1.99526...\approx 2\,}$

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.[citation needed]

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 10−6/10 × 1 mW, or approximately 250 μW (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.[3] 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

dB(1 mW) — power measurement relative to 1 milliwatt. XdBm = XdBW + 30.
dB(1 W) — similar to dBm, except the reference level is 1 watt. 0 dBW = +30 dBm; −30 dBW = 0 dBm; XdBW = XdBm − 30.

#### Voltage

Note that the decibel has a different definition when applied to voltage (as contrasted with power). See the "Definitions" section above.

File:Relationship between dBu and dBm.svg
A schematic showing the relationship between dBu (the voltage source) and dBm (the power dissipated as heat by the 600 Ω resistor)

dBV

dB(1 VRMS) — voltage relative to 1 volt, regardless of impedance.[4]

dBu or dBv

dB(0.775 VRMS) — voltage relative to 0.775 volts.[4] Originally dBv, it was changed to dBu to avoid confusion with dBV.[5] The "v" comes from "volt", while "u" comes from "unloaded". dBu can be used regardless of impedance, but is derived from a 600 Ω load dissipating 0 dBm (1 mW). Compare ambiguous use of dBu in radio engineering.

dBmV

dB(1 mVRMS) — voltage relative to 1 millivolt, regardless of impedance. Widely used in cable television networks, where the nominal strength of a single TV signal at the receiver terminals is about 0 dBmV. Cable TV uses 75 Ω coaxial cable, so 0 dBmV corresponds to −48.75 dBm or ~13 nW.

dBμV or dBuV

dB(1 μVRMS) — voltage relative to 1 microvolt. Widely used in television and aerial amplifier specifications. 60 dBμV = 0 dBmV.

#### Acoustics

dB(SPL)

dB (Sound Pressure Level) — for sound in air and other gases, 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. For sound in water and other liquids, a reference pressure of 1 μPa is used.[6]

dB SIL

dB Sound Intensity Level — relative to 10−12 W/m2, which is roughly the threshold of human hearing in air.

dB SWL

dB Sound Power Level — relative to 10−12 W.

dB(A), dB(B), and dB(C)

These symbols are often used to denote the use of different weighting filters, 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. Nevertheless, the units dBA and dB(A) are still commonly used as a shorthand for A-weighted measurements. Compare dBc, used in telecommunications.

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.[citation needed]

dB Q is sometimes used to denote weighted noise level, commonly using the ITU-R 468 noise weighting[citation needed]

dBZ

dB(Z) - energy of reflectivity (weather radar), or the amount of transmitted power returned to the radar receiver. Values above 15-20 dBZ usually indicate falling precipitation.[7]

#### Radio power, energy, and field strength

dBc — power relative to the power of the main carrier frequency; typically used to describe spurs, noise, channel crosstalk, and intermodal signals which may interfere with the carrier. Compare dB(C), used in acoustics.

dBJ

dB(J) — energy relative to 1 joule. 1 joule = 1 watt per hertz, so power spectral density can be expressed in dBJ.

dBm

dB(mW) — power relative to 1 milliwatt.

dBμ or dBu

dB(μV/m) — electric field strength relative to 1 microvolt per meter. Compare the ambiguous use of dBu as a unit of voltage level.

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

dBd

dB(dipole) — the forward gain of an antenna compared to a half-wave dipole 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 (peak) would equal the highest level (number) the processor is capable of representing. Measured values are usually negative, since they should be less than the maximum.

dB-Hz

dB(hertz) — bandwidth relative to 1 Hz. E.g., 20 dB-Hz corresponds to a bandwidth of 100 Hz. Commonly used in link budget calculations.

dBi

dB(isotropic) — the forward gain of an antenna compared to the hypothetical isotropic antenna, which uniformly distributes energy in all directions.

dBiC

dB(isometric circular) — power measurement relative to a circularly polarized isometric antenna.

dBov or dBO

dB(overload) — the amplitude of a signal (usually audio) compared to the maximum which a device can handle before clipping occurs. Similar to dBFS, but also applicable to analog systems.

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.
dB relative to carrier — in telecommunications, this indicates the relative levels of noise or sideband peak power, compared to the carrier power. Compare dBC, used in acoustics.

## Footnotes

1. Taylor 1995, Guide for the Use of the International System of Units (SI), NIST Special Publication SP811
2. Taylor 1995, SP811
3. Analog Devices : Virtual Design Center : Interactive Design Tools : Utilities : VRMS / dBm / dBu / dBV calculator
4. What is the difference between dBv, dBu, dBV, dBm, dB SPL, and plain old dB? Why not just use regular voltage and power measurements? - rec.audio.pro Audio Professional FAQ
5. Morfey, C. L. (2001). Dictionary of Acoustics. Academic Press, San Diego.
6. "Radar FAQ from WSI". Retrieved 2008-03-18.

## References

• Martin, W.H. (1929). "DeciBel--The New Name for the Transmission Unit". Bell System Technical Journal. January.
• STEVENS SS (1957). "On the psychophysical law". Psychol Rev. 64 (3): 153–81. PMID 13441853.