# Decibel: Map

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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 quantities with the same unit, it is a dimensionless unit. A decibel is one tenth of a bel, a seldom-used unit.

The term decibel is most widely known as a measure of sound pressure level, but is also useful for a wide variety of measurements in science and engineering (particularly acoustics, electronics, and control theory) 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 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, 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 RMS.

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 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 or TU was devised by engineers of the Bell Telephone Laboratories in the 1920s to replace the MSC. 1 TU was defined as the ten times the base-10 logarithm of the ratio of measured power to reference power. The definitions were conveniently chosen such that 1 TU approximately equalled 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, which was named the "bel" in honor of the Bell System's founder and telecommunications pioneer Alexander Graham Bell. The bel was a factor of ten larger than the TU, such that 1 TU equalled 1 decibel. In many situations, 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. 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.

## 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: however, the choice of scale for this common unit is irrelevant, as it changes both quantities by the same factor, and thus cancels in the ratio (the ratio of two quantities is scale-invariant). 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) areL_\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, 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 factor of 20 is explained as: 10 is because it is in decibels (10ths of bels), and 2 is because it is a ratio of powers (squares of amplitudes): the product is 20. Thus provided that power ratios equal amplitude ratios squared, the two definitions -- 10-log and 20-log rules -- yield the same result in decibels (L_\mathrm{dB}); bear in mind though, that in some usages the equivalence condition is not fulfilled and the equivalence does not hold as in, e.g., dBu and dBV. Note also that no constant factor is needed for the power (one can take power ratio to be the square of the amplitude ratio, whatever the units), since any constant cancels in the ratio.

The formula may be rearranged to giveA_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 \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) = 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) = -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 \,

• An example illustrating the subtleties of the 20-log vs. 10-log rules is given by the so-called polarization ellipticity, the minor-to-major-axis ratio of the polarization ellipse. It is an amplitude ratio, thus when reported in decibels, it follows the 20-log rule, \alpha_\text{dB} = 20 \log_{10}{\alpha} = 10 \log_{10}{\alpha^2}. A reader unfamiliar with the applicability of each 20- and 10-log rules might find \alpha_\text{dB} inconsistent, arguing that it redefines the original \alpha as a power ratio rather than an amplitude ratio. The crux is that the subscript dB might imply either of two different mathematical operations, thus it can only be interpreted unambiguously given additional information about the nature of the quantity being reported (power ratio or amplitude ratio).

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 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 \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 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. As with other decibel figures, normally the ratio expressed is a power ratio (rather than a pressure ratio).

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 during short exposure to the lower 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 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 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.

### 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.

## 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. 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.

dBW
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

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

dBV
dB(1 VRMS) — voltage relative to 1 volt, regardless of impedance. Analog Devices : Virtual Design Center : Interactive Design Tools : Utilities : VRMS / dBm / dBu / dBV calculator

dBu or dBv
dB(0.775 VRMS) — 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

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 −78.75 dBW (-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

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 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 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.

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

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.

dBsm
dBsm - decibel (referenced to one) square meter, measure of reflected energy from a target compared to the RCS of a smooth perfectly conducting sphere at least several wavelengths in size with a cross-sectional area of 1 square meter. "Stealth" aircraft and insects have negative values of 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 meter.

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. 0dBd = 2.15dBi

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. 0dBq = -0.85dBi

### 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

## Footnotes

1. Sound system engineering, p. 35, Carolyn Davis, 1997
2. "Transmission Circuits for Telephonic Communication", Bell Labs, 1925
3. 100 Years of Telephone Switching, p. 276, Robert J. Chapuis, Amos E. Joel, 2003
4. Consultative Committee for Units, Meeting minutes, Section 3
5. "Letter symbols to be used in electrical technology - Part 3: Logarithmic and related quantities, and their units", IEC 60027-3 Ed. 3.0, International Electrotechnical Commission, 19th July 2002.
6. A. Thompson and B. N. Taylor, "Comments on Some Quantities and Their Units", The NIST Guide for the use of the International System of Units, National Institute of Standards and Technology, May 1996.
7. Taylor 1995, SP811
8. 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
9. Morfey, C. L. (2001). Dictionary of Acoustics. Academic Press, San Diego.