Decibel

This article is about the ratio of measures. For other uses, see Decibel (disambiguation).

Attenuation chart of an RG-6 Coaxial cable, measured in decibels per 100 feet of cable

The decibel (dB) is a logarithmic unit that indicates the ratio of a physical quantity (usually power or intensity) relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities. Being a ratio of two measurements of a physical quantity in the same units, it is a dimensionless unit. A decibel is one tenth of a bel, a seldom-used unit.

The decibel is widely known as a measure of sound pressure level, but is also used for a wide variety of other measurements in science and engineering, most prominently in acoustics, electronics, and control theory. In electronics, the gain of amplifiers, attenuation of signals, and signal to noise ratios are often expressed in decibels. 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 sound and light, and the ability to carry out multiplication of ratios by simple addition and subtraction.

The decibel symbol is often qualified with a suffix, that 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 RMS. and dBμV/m referenced to microvolts per metre for radio frequency signal strength.

The definitions of the decibel and bel use logarithms to base 10. The neper, used in electronics, uses natural logarithm to base (e).

History

The decibel originates from methods used to quantify reductions in audio levels in telephone circuits. These losses were originally measured in units of Miles of Standard Cable (MSC), where 1 MSC corresponded to the loss of power over a 1 mile (approximately 1.6 km) length of standard telephone cable at a frequency of 5000 radians per second (795.8 Hz), and roughly matched the smallest attenuation detectable to an average listener. Standard telephone cable was defined as "a cable having uniformly distributed resistances of 88 ohms per loop mile and uniformly distributed shunt capacitance of .054 microfarad per mile" (approximately 19 gauge).

The transmission unit (TU) was devised by engineers of the Bell Telephone Laboratories in the 1920s to replace the MSC. 1 TU was defined as ten times the base-10 logarithm of the ratio of measured power to a reference power level. The definitions were conveniently chosen such that 1 TU approximately equaled 1 MSC (specifically, 1.056 TU = 1 MSC). Eventually, international standards bodies adopted the base-10 logarithm of the power ratio as a standard unit, named the bel in honor of the Bell System's founder and telecommunications pioneer Alexander Graham Bell. The bel was larger by a factor of ten than the TU, such that 1 TU equaled 1 decibel. For many measurements, the bel proved inconveniently large, giving way to the decibel becoming the common unit of choice.

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. However, the decibel is recognized by other international bodies such as the International Electrotechnical Commission (IEC). The IEC permits the use of the decibel with field quantities as well as power and this recommendation is followed by many national standards bodies, such as NIST, which justifies the use of the decibel for voltage ratios.

Definition

A decibel (dB) is one tenth of a bel (B), i.e. 1B = 10dB. The bel is the logarithm of the ratio of two power quantities of 10:1, and for two field quantities in the ratio ${\sqrt {10}}:1$. A field quantity is a quantity such as voltage, current, sound pressure, electric field strength, velocity and charge density, the square of which in linear systems is proportional to power. A power quantity is a power or a quantity directly proportional to power, e.g. energy density, acoustic intensity and luminous intensity.

The calculation of the ratio in decibels varies depending on whether the quantity being measured is a power quantity or a field quantity.

Power quantities

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, the ratio of a power value P₁ to another power value P₀ is represented by L_dB, that ratio expressed in decibels, which is calculated using the formula:

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

P₁ and P₀ must measure the same type of quantity, and have the same units before calculating the ratio. If P₁ = P₀ in the above equation, then L_dB = 0. If P₁ is greater than P₀ then L_dB is positive; if P₁ is less than P₀ then L_dB is negative.

Rearranging the above equation gives the following formula for P₁ in terms of P₀ and L_dB:

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

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

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

$P_{1}=10^{L_{\mathrm {B} }}P_{0}\,$.

Field quantities

When referring to measurements of field 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, and it is desirable for the two decibel formulations to give the same result in such typical cases. Thus the following definition is used:

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

This formula is sometimes called the 20 log rule, and similarly the formula for ratios of powers is the 10 log rule, and similarly for other factors. The equivalence of $10\log _{10}{\frac {a^{2}}{b^{2}}}$ and $20\log _{10}{\frac {a}{b}}$ is one of the standard properties of logarithms.

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 G_dB is the power gain expressed in decibels. A similar formula holds for current.

Examples

File:X and log x.svg

An example scale showing x and 10 log x. It is easier to grasp and compare 2 or 3 digit numbers than to compare up to 10 digits.

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

$G_{\mathrm {dB} }=10\log _{10}{\bigg (}{\frac {1000~\mathrm {W} }{1~\mathrm {W} }}{\bigg )}\equiv 30~\mathrm {dB} \,$

• To calculate the ratio of ${\sqrt {1000}}~\mathrm {V} \approx 31.62~\mathrm {V}$ to $1~\mathrm {V}$ in decibels, use the formula

$G_{\mathrm {dB} }=20\log _{10}{\bigg (}{\frac {31.62~\mathrm {V} }{1~\mathrm {V} }}{\bigg )}\equiv 30~\mathrm {dB} \,$

Notice that $({31.62\,\mathrm {V} }/{1\,\mathrm {V} })^{2}\approx {1\,\mathrm {kW} }/{1\,\mathrm {W} }$, illustrating the consequence from the definitions above that $G_{\mathrm {dB} }$ has the same value, $30~\mathrm {dB}$, regardless of whether it is obtained with the 10-log or 20-log rules; provided that in the specific system being considered power ratios are equal to amplitude ratios squared.

• To calculate the ratio of 1 mW (one milliwatt) to 10 W in decibels, use the formula

$G_{\mathrm {dB} }=10\log _{10}{\bigg (}{\frac {0.001~\mathrm {W} }{10~\mathrm {W} }}{\bigg )}\equiv -40~\mathrm {dB} \,$

• To find the power ratio corresponding to a 3 dB change in level, use the formula

$G=10^{\frac {3}{10}}\times 1\ =1.99526...\approx 2\,$

A change in power ratio by a factor of 10 is a 10 dB change. A change in power ratio by a factor of two is approximately a 3 dB change. More precisely, the factor is 10^3/10, 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 $\scriptstyle {\sqrt {2}}$, 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 ratio is 10^6/10, 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 the intensity of, for example, sound or light, is more nearly proportional to the logarithm of intensity than to the intensity itself, per the Weber–Fechner law, so the dB scale can be useful to describe perceptual levels or level differences.

Uses

Acoustics

Main article: Sound pressure

The decibel is commonly used in acoustics to quantify sound levels relative to a 0 dB reference which has been defined as a sound pressure level of .0002 microbar. The reference level is set at the typical threshold of perception of an average human and there are common comparisons used to illustrate different levels of sound pressure. As with other decibel figures, normally the ratio expressed is a power ratio (rather than a pressure ratio).

The human ear has a large dynamic range in audio perception. The ratio of the sound intensity that causes permanent damage during short exposure to the quietest sound that the ear can hear is above a trillion. Such large measurement ranges are conveniently expressed in logarithmic units: the base-10 logarithm of one trillion (10¹²) is 12, which is expressed as an audio level of 120 dB. Since the human ear is not equally sensitive to all sound frequencies, noise levels at maximum human sensitivity—somewhere between 2 and 4 kHz—are factored more heavily into some measurements using frequency weighting.

Further information: [[:Examples of sound pressure and sound pressure levels]]

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, fibre 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 V_RMS. 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 telephone 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 fibre, 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.

Video and digital imaging

In connection with digital and video image sensors, decibels generally represent ratios of video voltages or digitized light levels, using 20 log of the ratio, even when the represented optical power is directly proportional to the voltage or level, not to its square. Thus, a camera signal-to-noise ratio of 60 dB represents a power ratio of 1000:1 between signal power and noise power, not 1,000,000:1. However, as mentioned above, the 10 log convention prevails more generally in physical optics, so the terminology can become murky between the conventions of digital photographic technology and physics. Most commonly, quantities called "dynamic range" or "signal-to-noise" (of the camera) would be specified in 20 log dBs, but in related contexts (e.g. attenuation, gain, intensifier SNR, or rejection ratio) the term should be interpreted cautiously, as confusion of the two units can result in very large misunderstandings of the value. Photographers also often use an alternative base-2 log unit, the f-stop, and in software contexts these image level ratios, particularly dynamic range, are often loosely referred to by the number of bits needed to represent the quantity, such that 60 dB (digital photographic) is roughly equal to 10 f-stops or 10 bits.

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 10^3/10 × 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. 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_dBm = X_dBW + 30.

dBW

dB(1 W) – similar to dBm, except the reference level is 1 watt. 0 dBW = +30 dBm; −30 dBW = 0 dBm; X_dBW = X_dBm − 30.

Voltage

Since the decibel is defined with respect to power, not amplitude, conversions of voltage ratios to decibels must square the amplitude, as discussed above.

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_RMS) – voltage relative to 1 volt, regardless of impedance.

dBu or dBv

dB(0.775 V_RMS) – voltage relative to 0.775 volts. Originally dBv, it was changed to dBu to avoid confusion with dBV. 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). Reference voltage $V={\sqrt {600\,\Omega \cdot 0.001\,\mathrm {W} }}\,\approx 0.7746\,\mathrm {V}$

In pro audio most equipment for processing signals will operate at +4 dBu. Consumer equipment will typically operate at -10 dBu. Therefore, many devices offer dual voltage operation for interoperability reasons, typically with a switch for +4/-10 dBu.

dBmV

dB(1 mV_RMS) – voltage relative to 1 millivolt across 75 Ω. 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 −78.75 dBW (−48.75 dBm) or ~13 nW.

dBμV or dBuV

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

Acoustics

Probably the most common usage of "decibels" in reference to sound loudness is dB SPL, referenced to the nominal threshold of human hearing:

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.

dB(PA)

dB – relative to 1 Pa, often used in telecommunications.

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). These measurements usually refer to noise and noisome effects on humans and animals, and are in widespread use in the industry with regard to noise control issues, regulations and environmental standards. Other variations that may be seen are dB_A or dBA. According to ANSI standards, the preferred usage is to write L_A = 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.

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

Radar

dBZ

dB(Z) – energy of reflectivity (weather radar), related to the amount of transmitted power returned to the radar receiver; the reference level for Z is 1 mm⁶ m⁻³. Values above 15–20 dBZ usually indicate falling precipitation.

dBsm

dBsm – decibel measure of the radar cross section (RCS) of a target relative one square metre. The power reflected by the target is proportional to its RCS. "Stealth" aircraft and insects have negative RCS measured in dBsm, large flat plates or non-stealthy aircraft have positive values.

Radio power, energy, and field strength

dBc

dBc – relative to carrier—in telecommunications, this indicates the relative levels of noise or sideband peak power, compared with the carrier power. Compare dBC, 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. When used in audio work the milliwatt is referenced to a 600 ohm load, with the resultant voltage being 0.775 volts. When used in the 2-way radio field, the dB is referenced to a 50 ohm load, with the resultant voltage being 0.224 volts. There are times when spec sheets may show the voltage & power level e.g. −120 dBm = 0.224 microvolts.

dBμV/m or dBuV/m

dB(μV/m) – electric field strength relative to 1 microvolt per metre. Often used to specify the signal strength from a television broadcast at a receiving site (the signal measured at the antenna output will be in dBμV).

dBf

dB(fW) – power relative to 1 femtowatt.

dBW

dB(W) – power relative to 1 watt.

dBk

dB(kW) – power relative to 1 kilowatt.

Antenna measurements

dBi

dB(isotropic) – the forward gain of an antenna compared with the hypothetical isotropic antenna, which uniformly distributes energy in all directions. Linear polarization of the EM field is assumed unless noted otherwise.

dBd

dB(dipole) – the forward gain of an antenna compared with a half-wave dipole antenna. 0 dBd = 2.15 dBi

dBiC

dB(isotropic circular) – the forward gain of an antenna compared to a circularly polarized isotropic antenna. There is no fixed conversion rule between dBiC and dBi, as it depends on the receiving antenna and the field polarization.

dBq

dB(quarterwave) – the forward gain of an antenna compared to a quarter wavelength whip. Rarely used, except in some marketing material. 0 dBq = −0.85 dBi

Other measurements

dBFS or dBfs

dB(full scale) – the amplitude of a signal (usually audio) compared with 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 always negative or zero, since they are less than the maximum or full-scale. Full-scale is typically defined as the power level of a full-scale sinusoid, though some systems will have extra headroom for peaks above the nominal full scale.

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. Also used in carrier-to-noise-density ratio (not to be confused with carrier-to-noise ratio, in dB).

dBov or dBO

dB(overload) – the amplitude of a signal (usually audio) compared with 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 from 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.

See also

• apparent magnitude
• Cent in music
• dB drag racing
• Equal-loudness contour
• Noise (environmental)
• Richter magnitude scale
• Signal noise
• Weighting filter—discussion of dBA

Further reading

• 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. doi:10.1037/h0046162. PMID 13441853.

External links

• What is a decibel? With sound files and animations
• Conversion of dBu to volts, dBV to volts, and volts to dBu, and dBV
• Working with decibels – a tutorial
• Conversion of sound level units: dBSPL or dBA to sound pressure p and sound intensity J
• Conversion of voltage V to dB, dBu, dBV, and dBm
• OSHA Regulations on Occupational Noise Exposure
• V_peak, V_RMS, Power, dBm, dBu, dBV online converter at Analog Devices
• Use of the decibel with respect to aerials and aerial systems