Decibel
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The '''decibel''' ('''dB''') is a [[logarithmic units|logarithmic unit]] of measurement that expresses the magnitude of a physical quantity (usually [[power (physics)|power]] or intensity) relative to a specified or implied ''reference level''. Since it expresses a ratio of two quantities with the same unit, 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 language|English]] capitalization rules for a [[common noun]].
The decibel symbol is often qualified with a suffix, which indicates which reference quantity or [[A-weighting|frequency weighting function]] has been used. For example, "dBm" indicates that the reference quantity is one milliwatt, while "dBu" is referenced to 0.7746 volts.{{Fact|date=July 2008}} The practice of attaching a suffix in this way, though not permitted by SI,<ref>[http://physics.nist.gov/Pubs/pdf.html Taylor 1995, Guide for the Use of the International System of Units (SI), NIST Special Publication SP811]</ref> 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 Labs|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.<ref>[http://www.bipm.org/utils/common/pdf/CCU15.pdf Consultative Committee for Units, Meeting minutes], Section 3</ref>
== 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 ''P''<sub>1</sub> to another power value ''P''<sub>0</sub>, then ''L''<sub>dB</sub> represents that ratio expressed in decibels and is calculated using the formula:
<!-- L means level and lambda means wavelength -->
:<math>
L_\mathrm{dB} = 10 \log_{10} \bigg(\frac{P_1}{P_0}\bigg) \,
</math>
Naturally, ''P''<sub>1</sub> and ''P''<sub>0</sub> 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 ''P''<sub>1</sub> = ''P''<sub>0</sub> in the above equation, then ''L''<sub>dB</sub> = 0. If ''P''<sub>1</sub> is greater than ''P''<sub>0</sub> then ''L''<sub>dB</sub> is positive; if ''P''<sub>1</sub> is less than ''P''<sub>0</sub> then ''L''<sub>dB</sub> is negative.
Rearranging the above equation gives the following formula for ''P''<sub>1</sub> in terms of ''P''<sub>0</sub> and ''L''<sub>dB</sub>:
:<math>
P_1 = 10^\frac{L_\mathrm{dB}}{10} P_0 \,
</math>.
Since a bel is equal to ten decibels, the corresponding formulae for measurement in bels (''L''<sub>B</sub>) are
:<math>
L_\mathrm{B} = \log_{10} \bigg(\frac{P_1}{P_0}\bigg) \,
</math>
:<math>
P_1 = 10^{L_\mathrm{B}} P_0 \,
</math>.
===Amplitude, voltage and current===
When referring to measurements of ''amplitude'' it is usual to consider the ratio of the squares of ''A''<sub>1</sub> (measured amplitude) and ''A''<sub>0</sub> (reference amplitude). This is because in most applications power is proportional to the square of amplitude. Thus the following definition is used:
:<math>
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) \,
</math>
The formula may be rearranged to give
:<math>
A_1 = 10^\frac{L_\mathrm{dB}}{20} A_0 \,
</math>
Similarly, in [[Electronic circuit|electrical circuits]], dissipated power is typically proportional to the square of [[voltage]] or [[Electric current|current]] when the [[Electrical impedance|impedance]] is held constant. Taking voltage as an example, this leads to the equation:
:<math>
G_\mathrm{dB} =20 \log_{10} \left (\frac{V_1}{V_0} \right ) \quad \mathrm \quad
</math>
where ''V''<sub>1</sub> is the voltage being measured, ''V''<sub>0</sub> is a specified reference voltage, and ''G''<sub>dB</sub> 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
:<math>
G_\mathrm{dB} = 10 \log_{10} \bigg(\frac{1000 \mathrm{W}}{1 \mathrm{W}}\bigg) = 30 \mathrm{dB} \,
</math>
*To calculate the ratio of 1 mW (one milliwatt) to 10 W in decibels, use the formula
:<math>
G_\mathrm{dB} = 10 \log_{10} \bigg(\frac{.001 \mathrm{W}}{10 \mathrm{W}}\bigg) = -40 \mathrm{dB} \,
</math>
*To find the power ratio corresponding to a 3 dB change in level, use the formula
:<math>
G = 10^\frac{3}{10} \times 1\ = 1.99526... \approx 2 \,
</math>
<br>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 10<sup>3/10</sup>, 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 10<sup>6/10</sup>, 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 [[logarithm]]ic 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|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 ===
{{main article|Sound pressure}}
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 [[Sound pressure#Examples of sound pressure and sound pressure levels|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 pressure]]s. 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 ([[long and short scales|short scale]]) [[1000000000000 (number)|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 [[harmonic]]s of middle [[A (musical note)|A]] (between 2 and 4 [[hertz|kHz]]) — are factored more heavily into sound descriptions using a process called [[frequency weighting]].
{{further|[[Sound_pressure_level#Examples_of_sound_pressure_and_sound_pressure_levels|Examples of sound pressure and sound pressure levels]]}}
=== Electronics ===
In electronics, the decibel is often used to express power or amplitude ratios ([[gain]]s), in preference to [[arithmetic]] ratios or [[percent]]ages. One advantage is that the total decibel gain of a series of components (such as [[amplifier]]s and [[attenuator]]s) 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 guide]]s, [[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 [[root mean square|RMS]] measurement of voltage which uses as its reference 0.775 V<sub>RMS</sub>. 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.{{Fact|date=December 2007}}
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 [[Optics|optical]] power, in [[dBm]] (referenced to 1 mW), is launched into a [[Optical fiber|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 10<sup>3/10</sup> × 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<sup>−6/10</sup> × 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.<ref>[http://physics.nist.gov/Pubs/pdf.html Taylor 1995, SP811]</ref> 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. X<sub>dBm</sub> = X<sub>dBW</sub> + 30.
'''[[dBW]]'''
:dB(1 W) — similar to dBm, except the reference level is 1 [[watt]]. 0 dBW = +30 dBm; −30 dBW = 0 dBm; X<sub>dBW</sub> = X<sub>dBm</sub> − 30.
==== Voltage ====
Note that the decibel has a different definition when applied to voltage (as contrasted with power). See the "Definitions" section above.
[[Image:Relationship between dBu and dBm.svg|thumb|300px|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 V<sub>[[root mean square|RMS]]</sub>) — [[volt]]age relative to 1 volt, regardless of impedance.<ref name = "clqgmk">[http://www.analog.com/Analog_Root/static/techSupport/designTools/interactiveTools/dbconvert/dbconvert.html Analog Devices : Virtual Design Center : Interactive Design Tools : Utilities : V<sub>RMS</sub> / dBm / dBu / dBV calculator<!-- Bot generated title -->]</ref>
'''dBu''' or '''dBv'''
:dB(0.775 V<sub>[[root mean square|RMS]]</sub>) — [[volt]]age relative to 0.775 volts.<ref name = "clqgmk"/> Originally dBv, it was changed to dBu to avoid confusion with dBV.<ref>[http://stason.org/TULARC/entertainment/audio/pro/3-3-What-is-the-difference-between-dBv-dBu-dBV-dBm-dB.html 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]]</ref> 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 mV<sub>[[root mean square|RMS]]</sub>) — [[volt]]age 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 μV<sub>[[root mean square|RMS]]</sub>) — [[volt]]age relative to 1 microvolt. Widely used in television and aerial amplifier specifications. 60 dBμV = 0 dBmV.
==== Acoustics ====
'''dB(SPL)'''
:dB ([[Sound pressure level|Sound Pressure Level]]) — for sound in air and other gases, relative to 20 micropascals (μPa) = 2×10<sup>−5</sup> 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 [[Underwater acoustics|sound in water]] and other liquids, a reference pressure of 1 μPa is used.<Ref>Morfey, C. L. (2001). Dictionary of Acoustics. Academic Press, San Diego.</Ref>
'''dB SIL'''
:dB [[Sound intensity level|Sound Intensity Level]] — relative to 10<sup>−12</sup> W/m<sup>2</sup>, which is roughly the [[threshold of human hearing]] in air.
'''dB SWL'''
:dB [[Sound power level|Sound Power Level]] — relative to 10<sup>−12</sup> W.
'''dB(A)''', '''dB(B)''', and '''dB(C)'''
:These symbols are often used to denote the use of different [[weighting filter]]s, 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 dB<sub>A</sub> or [[A-weighting|dBA]]. According to ANSI standards, the preferred usage is to write L<sub>A</sub> = 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 [[audiogram]]s 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.{{Fact|date=March 2008}}
'''dB Q''' is sometimes used to denote weighted noise level, commonly using the [[ITU-R 468 noise weighting]]{{Fact|date=March 2008}}
==== Radar ====
'''[[DBZ (meteorology)|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.<ref>{{cite web |url=http://www.srh.noaa.gov/tsa/radar_faq.html |title=Radar FAQ from WSI|accessdate=2008-03-18 |format= |work= }}</ref>
==== Radio power, energy, and field strength ====
[[DBc|'''dBc''']]
: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 (electronics)|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 (signal processing)|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 (signal processing)|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.
'''[[dBrn]]'''
:dB above [[reference noise]]. See also [[dBrnC]].
'''[[dBc]]'''
:dB relative to carrier — in [[telecommunication]]s, this indicates the relative levels of noise or sideband peak power, compared to the carrier power. Compare dBC, used in acoustics.
== See also ==
*[[Cent (music)|Cent]] in music
*[[dB drag racing]]
*[[Equal-loudness contour]]
*[[ITU-R 468 noise weighting]]
*[[Neper]]
*[[Noise (environmental)]]
*[[Richter magnitude scale]]
*[[Signal noise]]
*[[Weighting filter]] — discussion of '''dBA'''
==Footnotes==
{{reflist}}
== References ==
*{{cite journal | author = Martin, W.H. | year = 1929 | title = DeciBel--The New Name for the Transmission Unit | journal = Bell System Technical Journal | volume = January}}
*{{cite journal |author=STEVENS SS |title=On the psychophysical law |journal=Psychol Rev |volume=64 |issue=3 |pages=153–81 |year=1957 |pmid=13441853 |doi=}}
== External links ==
*[http://www.phys.unsw.edu.au/jw/dB.html What is a decibel? With sound files and animations]
*[http://www.sengpielaudio.com/calculator-db-volt.htm Conversion of dBu to volts, dBV to volts, and volts to dBu, and dBV]
*[http://www.cvel.clemson.edu/emc/tutorials/DB_Notes/DB_Notes.html Working with decibels - a tutorial ]
*[http://www.sengpielaudio.com/calculator-soundlevel.htm Conversion of sound level units: dBSPL or dBA to sound pressure p and sound intensity J]
*[http://www.sengpielaudio.com/calculator-volt.htm Conversion of voltage V to dB, dBu, dBV, and dBm]
*[http://www.osha.gov/pls/oshaweb/owadisp.show_document?p_table=STANDARDS&p_id=9735 OSHA Regulations on Occupational Noise Exposure]
*[http://www.analog.com/Analog_Root/static/techSupport/designTools/interactiveTools/dbconvert/dbconvert.html V<sub>peak</sub>, V<sub>RMS</sub>, Power, dBm, dBu, dBV online converter at Analog Devices]
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[[Category:Units of measure]]
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