Decibel

This article is about the logarithmic unit. For use of this unit in sound measurements, see Sound pressure level. For other uses, see Decibel (disambiguation).

The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a power ratio of 10^1/10 (approximately 1.26) or root-power ratio of 10^1/20 (approximately 1.12).^[1][2]

decibel
Unit system	Non-SI accepted unit
Symbol	dB
Named after	Alexander Graham Bell
Conversions
1 dB in ...	... is equal to ...
bel	⁠1/10⁠ bel

The unit fundamentally expresses a relative change but may also be used to express an absolute value as the ratio of a value to a fixed reference value; when used in this way, the unit symbol is often suffixed with letter codes that indicate the reference value. For example, for the reference value of 1 volt, a common suffix is "V" (e.g., "20 dBV").^[3][4]

Two principal types of scaling of the decibel are in common use. When expressing a power ratio, it is defined as ten times the logarithm with base 10.^[5] That is, a change in power by a factor of 10 corresponds to a 10 dB change in level. When expressing root-power quantities, a change in amplitude by a factor of 10 corresponds to a 20 dB change in level. The decibel scales differ by a factor of two, so that the related power and root-power levels change by the same value in linear systems, where power is proportional to the square of amplitude.

The definition of the decibel originated in the measurement of transmission loss and power in telephony of the early 20th century in the Bell System in the United States. The bel was named in honor of Alexander Graham Bell, but the bel is seldom used. Instead, the decibel is used for a wide variety of measurements in science and engineering, most prominently for sound power in acoustics, in electronics and control theory. In electronics, the gains of amplifiers, attenuation of signals, and signal-to-noise ratios are often expressed in decibels.

History

The decibel originates from methods used to quantify signal loss in telegraph and telephone circuits. Until the mid-1920s, the unit for loss was miles of standard cable (MSC). 1 MSC corresponded to the loss of power over one mile (approximately 1.6 km) of standard telephone cable at a frequency of 5000 radians per second (795.8 Hz), and matched closely the smallest attenuation detectable to a listener. A standard telephone cable was "a cable having uniformly distributed resistance of 88 ohms per loop-mile and uniformly distributed shunt capacitance of 0.054 microfarads per mile" (approximately corresponding to 19 gauge wire).^[6]

In 1924, Bell Telephone Laboratories received a favorable response to a new unit definition among members of the International Advisory Committee on Long Distance Telephony in Europe and replaced the MSC with the Transmission Unit (TU). 1 TU was defined such that the number of TUs was ten times the base-10 logarithm of the ratio of measured power to a reference power.^[7] The definition was conveniently chosen such that 1 TU approximated 1 MSC; specifically, 1 MSC was 1.056 TU. In 1928, the Bell system renamed the TU into the decibel,^[8] being one tenth of a newly defined unit for the base-10 logarithm of the power ratio. It was named the bel, in honor of the telecommunications pioneer Alexander Graham Bell.^[9] The bel is seldom used, as the decibel was the proposed working unit.^[10]

The naming and early definition of the decibel is described in the NBS Standard's Yearbook of 1931:^[11]

Since the earliest days of the telephone, the need for a unit in which to measure the transmission efficiency of telephone facilities has been recognized. The introduction of cable in 1896 afforded a stable basis for a convenient unit and the "mile of standard" cable came into general use shortly thereafter. This unit was employed up to 1923 when a new unit was adopted as being more suitable for modern telephone work. The new transmission unit is widely used among the foreign telephone organizations and recently it was termed the "decibel" at the suggestion of the International Advisory Committee on Long Distance Telephony.

The decibel may be defined by the statement that two amounts of power differ by 1 decibel when they are in the ratio of 10^0.1 and any two amounts of power differ by N decibels when they are in the ratio of 10^N(0.1). The number of transmission units expressing the ratio of any two powers is therefore ten times the common logarithm of that ratio. This method of designating the gain or loss of power in telephone circuits permits direct addition or subtraction of the units expressing the efficiency of different parts of the circuit ...

In 1954, J. W. Horton argued that the use of the decibel as a unit for quantities other than transmission loss led to confusion, and suggested the name logit for "standard magnitudes which combine by multiplication", to contrast with the name unit for "standard magnitudes which combine by addition".^{[12][clarification needed]}

In April 2003, the International Committee for Weights and Measures (CIPM) considered a recommendation for the inclusion of the decibel in the International System of Units (SI), but decided against the proposal.^[13] However, the decibel is recognized by other international bodies such as the International Electrotechnical Commission (IEC) and International Organization for Standardization (ISO).^[14] The IEC permits the use of the decibel with root-power 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.^[15] In spite of their widespread use, suffixes (such as in dBA or dBV) are not recognized by the IEC or ISO.

Definition

dB	Power ratio		Amplitude ratio
100	10000000000		100000
90	1000000000		31623
80	100000000		10000
70	10000000		3162
60	1000000		1000
50	100000		316	.2
40	10000		100
30	1000		31	.62
20	100		10
10	10		3	.162
6	3	.981 ≈ 4	1	.995 ≈ 2
3	1	.995 ≈ 2	1	.413 ≈ √2
1	1	.259	1	.122
0	1		1
−1	0	.794	0	.891
−3	0	.501 ≈ ⁠1/2⁠	0	.708 ≈ √⁠1/2⁠
−6	0	.251 ≈ ⁠1/4⁠	0	.501 ≈ ⁠1/2⁠
−10	0	.1	0	.3162
−20	0	.01	0	.1
−30	0	.001	0	.03162
−40	0	.0001	0	.01
−50	0	.00001	0	.003162
−60	0	.000001	0	.001
−70	0	.0000001	0	.0003162
−80	0	.00000001	0	.0001
−90	0	.000000001	0	.00003162
−100	0	.0000000001	0	.00001
An example scale showing power ratios x, amplitude ratios √x, and dB equivalents 10 log10 x

The IEC Standard 60027-3:2002 defines the following quantities. The decibel (dB) is one-tenth of a bel: 1 dB = 0.1 B. The bel (B) is 1⁄2 ln(10) nepers: 1 B = 1⁄2 ln(10) Np. The neper is the change in the level of a root-power quantity when the root-power quantity changes by a factor of e, that is 1 Np = ln(e) = 1, thereby relating all of the units as nondimensional natural log of root-power-quantity ratios, 1 dB = 0.11513... Np = 0.11513.... Finally, the level of a quantity is the logarithm of the ratio of the value of that quantity to a reference value of the same kind of quantity.

Therefore, the bel represents the logarithm of a ratio between two power quantities of 10:1, or the logarithm of a ratio between two root-power quantities of √10:1.^[16]

Two signals whose levels differ by one decibel have a power ratio of 10^1/10, which is approximately 1.25893, and an amplitude (root-power quantity) ratio of 10^1/20 (1.12202).^[1][2]

The bel is rarely used either without a prefix or with SI unit prefixes other than deci; it is customary, for example, to use hundredths of a decibel rather than millibels. Thus, five one-thousandths of a bel would normally be written 0.05 dB, and not 5 mB.^[17]

The method of expressing a ratio as a level in decibels depends on whether the measured property is a power quantity or a root-power quantity; see Power, root-power, and field quantities for details.

Power quantities

When referring to measurements of power quantities, a ratio can be expressed as a level in decibels by evaluating ten times the base-10 logarithm of the ratio of the measured quantity to reference value. Thus, the ratio of P (measured power) to P₀ (reference power) is represented by L_P, that ratio expressed in decibels,^[18] which is calculated using the formula:^[19]

${\displaystyle L_{P}={\frac {1}{2}}\ln \!\left({\frac {P}{P_{0}}}\right)\,{\text{Np}}=10\log _{10}\!\left({\frac {P}{P_{0}}}\right)\,{\text{dB}}}$

The base-10 logarithm of the ratio of the two power quantities is the number of bels. The number of decibels is ten times the number of bels (equivalently, a decibel is one-tenth of a bel). P and P₀ must measure the same type of quantity, and have the same units before calculating the ratio. If P = P₀ in the above equation, then L_P = 0. If P is greater than P₀ then L_P is positive; if P is less than P₀ then L_P is negative.

Rearranging the above equation gives the following formula for P in terms of P₀ and L_P :

${\displaystyle P=10^{\frac {L_{P}}{10\,{\text{dB}}}}P_{0}}$

Root-power (field) quantities

Main article: Power, root-power, and field quantities

When referring to measurements of root-power quantities, it is usual to consider the ratio of the squares of F (measured) and F₀ (reference). This is because the definitions were originally formulated to give the same value for relative ratios for both power and root-power quantities. Thus, the following definition is used:

${\displaystyle L_{F}=\ln \!\left({\frac {F}{F_{0}}}\right)\,{\text{Np}}=10\log _{10}\!\left({\frac {F^{2}}{F_{0}^{2}}}\right)\,{\text{dB}}=20\log _{10}\left({\frac {F}{F_{0}}}\right)\,{\text{dB}}}$

The formula may be rearranged to give

${\displaystyle F=10^{\frac {L_{F}}{20\,{\text{dB}}}}F_{0}}$

Similarly, in electrical circuits, dissipated power is typically proportional to the square of voltage or current when the impedance is constant. Taking voltage as an example, this leads to the equation for power gain level L_G:

${\displaystyle L_{G}=20\log _{10}\!\left({\frac {V_{\text{out}}}{V_{\text{in}}}}\right)\,{\text{dB}}}$

where V_out is the root-mean-square (rms) output voltage, V_in is the rms input voltage. A similar formula holds for current.

The term root-power quantity is introduced by ISO Standard 80000-1:2009 as a substitute of field quantity. The term field quantity is deprecated by that standard and root-power is used throughout this article.

Relationship between power and root-power levels

Although power and root-power quantities are different quantities, their respective levels are historically measured in the same units, typically decibels. A factor of 2 is introduced to make changes in the respective levels match under restricted conditions such as when the medium is linear and the same waveform is under consideration with changes in amplitude, or the medium impedance is linear and independent of both frequency and time. This relies on the relationship

${\displaystyle {\frac {P(t)}{P_{0}}}=\left({\frac {F(t)}{F_{0}}}\right)^{2}}$

holding.^[20] In a nonlinear system, this relationship does not hold by the definition of linearity. However, even in a linear system in which the power quantity is the product of two linearly related quantities (e.g. voltage and current), if the impedance is frequency- or time-dependent, this relationship does not hold in general, for example if the energy spectrum of the waveform changes.

For differences in level, the required relationship is relaxed from that above to one of proportionality (i.e., the reference quantities P₀ and F₀ need not be related), or equivalently,

${\displaystyle {\frac {P_{2}}{P_{1}}}=\left({\frac {F_{2}}{F_{1}}}\right)^{2}}$

must hold to allow the power level difference to be equal to the root-power level difference from power P₁ and F₁ to P₂ and F₂. An example might be an amplifier with unity voltage gain independent of load and frequency driving a load with a frequency-dependent impedance: the relative voltage gain of the amplifier is always 0 dB, but the power gain depends on the changing spectral composition of the waveform being amplified. Frequency-dependent impedances may be analyzed by considering the quantities power spectral density and the associated root-power quantities via the Fourier transform, which allows elimination of the frequency dependence in the analysis by analyzing the system at each frequency independently.

Conversions

Since logarithm differences measured in these units often represent power ratios and root-power ratios, values for both are shown below. The bel is traditionally used as a unit of logarithmic power ratio, while the neper is used for logarithmic root-power (amplitude) ratio.

Conversion between units of level and a list of corresponding ratios

Unit	In decibels	In bels	In nepers	Power ratio	Root-power ratio
1 dB	1 dB	0.1 B	0.11513 Np	101/10 ≈ 1.25893	101/20 ≈ 1.12202
1 Np	8.68589 dB	0.868589 B	1 Np	e2 ≈ 7.38906	e ≈ 2.71828
1 B	10 dB	1 B	1.151 3 Np	10	101/2 ≈ 3.162 28

Examples

The unit dBW is often used to denote a ratio for which the reference is 1 W, and similarly dBm for a 1 mW reference point.

• Calculating the ratio in decibels of 1 kW (one kilowatt, or 1000 watts) to 1 W yields: $${\displaystyle L_{G}=10\log _{10}\left({\frac {1\,000\,{\text{W}}}{1\,{\text{W}}}}\right)\,{\text{dB}}=30\,{\text{dB}}}$$
• The ratio in decibels of √1000 V ≈ 31.62 V to 1 V is: $${\displaystyle L_{G}=20\log _{10}\left({\frac {31.62\,{\text{V}}}{1\,{\text{V}}}}\right)\,{\text{dB}}=30\,{\text{dB}}}$$

(31.62 V / 1 V)² ≈ 1 kW / 1 W, illustrating the consequence from the definitions above that L_G has the same value, 30 dB, regardless of whether it is obtained from powers or from amplitudes, provided that in the specific system being considered power ratios are equal to amplitude ratios squared.

• The ratio in decibels of 10 W to 1 mW (one milliwatt) is obtained with the formula: $${\displaystyle L_{G}=10\log _{10}\left({\frac {10{\text{W}}}{0.001{\text{W}}}}\right)\,{\text{dB}}=40\,{\text{dB}}}$$
• The power ratio corresponding to a 3 dB change in level is given by: $${\displaystyle G=10^{\frac {3}{10}}\times 1=1.995\,26\ldots \approx 2}$$

A change in power ratio by a factor of 10 corresponds to a change in level of 10 dB. A change in power ratio by a factor of 2 or ⁠1/2⁠ is approximately a change of 3 dB. More precisely, the change is ±3.0103 dB, but this is almost universally rounded to 3 dB in technical writing. This implies an increase in voltage by a factor of √2 ≈ 1.4142. Likewise, a doubling or halving of the voltage, corresponding to a quadrupling or quartering of the power, is commonly described as 6 dB rather than ±6.0206 dB.

Should it be necessary to make the distinction, the number of decibels is written with additional significant figures. 3.000 dB corresponds to a power ratio of 10^3/10, or 1.9953, about 0.24% different from exactly 2, and a voltage ratio of 1.4125, 0.12% different from exactly √2. Similarly, an increase of 6.000 dB corresponds to the power ratio is 10^6/10 ≈ 3.9811, about 0.5% different from 4.

Properties

The decibel is useful for representing large ratios and for simplifying representation of multiplicative effects, such as attenuation from multiple sources along a signal chain. Its application in systems with additive effects is less intuitive, such as in the combined sound pressure level of two machines operating together. Care is also necessary with decibels directly in fractions and with the units of multiplicative operations.

Reporting large ratios

The logarithmic scale nature of the decibel means that a very large range of ratios can be represented by a convenient number, in a manner similar to scientific notation. This allows one to clearly visualize huge changes of some quantity. See Bode plot and Semi-log plot. For example, 120 dB SPL may be clearer than "a trillion times more intense than the threshold of hearing".^{[citation needed]}

Representation of multiplication operations

Level values in decibels can be added instead of multiplying the underlying power values, which means that the overall gain of a multi-component system, such as a series of amplifier stages, can be calculated by summing the gains in decibels of the individual components, rather than multiply the amplification factors; that is, log(A × B × C) = log(A) + log(B) + log(C). Practically, this means that, armed only with the knowledge that 1 dB is a power gain of approximately 26%, 3 dB is approximately 2× power gain, and 10 dB is 10× power gain, it is possible to determine the power ratio of a system from the gain in dB with only simple addition and multiplication. For example:

• A system consists of 3 amplifiers in series, with gains (ratio of power out to in) of 10 dB, 8 dB, and 7 dB respectively, for a total gain of 25 dB. Broken into combinations of 10, 3, and 1 dB, this is:
  25 dB = 10 dB + 10 dB + 3 dB + 1 dB + 1 dB
  With an input of 1 watt, the output is approximately
  1 W × 10 × 10 × 2 × 1.26 × 1.26 ≈ 317.5 W
  Calculated precisely, the output is 1 W × 10^25/10 ≈ 316.2 W. The approximate value has an error of only +0.4% with respect to the actual value, which is negligible given the precision of the values supplied and the accuracy of most measurement instrumentation.

However, according to its critics, the decibel creates confusion, obscures reasoning, is more related to the era of slide rules than to modern digital processing, and is cumbersome and difficult to interpret.^[21][22] Quantities in decibels are not necessarily additive,^[23][24] thus being "of unacceptable form for use in dimensional analysis".^[25] Thus, units require special care in decibel operations. Take, for example, carrier-to-noise-density ratio C/N₀ (in hertz), involving carrier power C (in watts) and noise power spectral density N₀ (in W/Hz). Expressed in decibels, this ratio would be a subtraction (C/N₀)_dB = C_dB − N_0 dB. However, the linear-scale units still simplify in the implied fraction, so that the results would be expressed in dB-Hz.

Representation of addition operations

Further information: Logarithmic addition

According to Mitschke,^[26] "The advantage of using a logarithmic measure is that in a transmission chain, there are many elements concatenated, and each has its own gain or attenuation. To obtain the total, addition of decibel values is much more convenient than multiplication of the individual factors." However, for the same reason that humans excel at additive operation over multiplication, decibels are awkward in inherently additive operations:^[27]

if two machines each individually produce a sound pressure level of, say, 90 dB at a certain point, then when both are operating together we should expect the combined sound pressure level to increase to 93 dB, but certainly not to 180 dB!; suppose that the noise from a machine is measured (including the contribution of background noise) and found to be 87 dBA but when the machine is switched off the background noise alone is measured as 83 dBA. [...] the machine noise [level (alone)] may be obtained by 'subtracting' the 83 dBA background noise from the combined level of 87 dBA; i.e., 84.8 dBA.; in order to find a representative value of the sound level in a room a number of measurements are taken at different positions within the room, and an average value is calculated. [...] Compare the logarithmic and arithmetic averages of [...] 70 dB and 90 dB: logarithmic average = 87 dB; arithmetic average = 80 dB.

Addition on a logarithmic scale is called logarithmic addition, and can be defined by taking exponentials to convert to a linear scale, adding there, and then taking logarithms to return. For example, where operations on decibels are logarithmic addition/subtraction and logarithmic multiplication/division, while operations on the linear scale are the usual operations:

${\displaystyle 87\,{\text{dBA}}\ominus 83\,{\text{dBA}}=10\cdot \log _{10}{\bigl (}10^{87/10}-10^{83/10}{\bigr )}\,{\text{dBA}}\approx 84.8\,{\text{dBA}}}$

${\displaystyle {\begin{aligned}M_{\text{lm}}(70,90)&=\left(70\,{\text{dBA}}+90\,{\text{dBA}}\right)/2\\&=10\cdot \log _{10}\left({\bigl (}10^{70/10}+10^{90/10}{\bigr )}/2\right)\,{\text{dBA}}\\&=10\cdot \left(\log _{10}{\bigl (}10^{70/10}+10^{90/10}{\bigr )}-\log _{10}2\right)\,{\text{dBA}}\approx 87\,{\text{dBA}}\end{aligned}}}$

The logarithmic mean is obtained from the logarithmic sum by subtracting ${\displaystyle 10\log _{10}2}$, since logarithmic division is linear subtraction.

Fractions

Attenuation constants, in topics such as optical fiber communication and radio propagation path loss, are often expressed as a fraction or ratio to distance of transmission. In this case, dB/m represents decibel per meter, dB/mi represents decibel per mile, for example. These quantities are to be manipulated obeying the rules of dimensional analysis, e.g., a 100-meter run with a 3.5 dB/km fiber yields a loss of 0.35 dB = 3.5 dB/km × 0.1 km.

Uses

Perception

The human perception of the intensity of sound and light more nearly approximates the logarithm of intensity rather than a linear relationship (see Weber–Fechner law), making the dB scale a useful measure.^{[28][29][30][31][32][33]}

Acoustics

The decibel is commonly used in acoustics as a unit of sound power level or sound pressure level. The reference pressure for sound in air 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 sound pressure is a root-power quantity, the appropriate version of the unit definition is used:

${\displaystyle L_{p}=20\log _{10}\!\left({\frac {p_{\text{rms}}}{p_{\text{ref}}}}\right)\,{\text{dB}},}$

where p_rms is the root mean square of the measured sound pressure and p_ref is the standard reference sound pressure of 20 micropascals in air or 1 micropascal in water.^[34]

Use of the decibel in underwater acoustics leads to confusion, in part because of this difference in reference value.^[35][36]

Sound intensity is proportional to the square of sound pressure. Therefore, the sound intensity level can also be defined as:

${\displaystyle L_{p}=10\log _{10}\!\left({\frac {I}{I_{\text{ref}}}}\right)\,{\text{dB}},}$

The human ear has a large dynamic range in sound reception. The ratio of the sound intensity that causes permanent damage during short exposure to that of the quietest sound that the ear can hear is equal to or greater than 1 trillion (10¹²).^[37] Such large measurement ranges are conveniently expressed in logarithmic scale: the base-10 logarithm of 10¹² is 12, which is expressed as a sound intensity level of 120 dB re 1 pW/m². The reference values of I and p in air have been chosen such that this corresponds approximately to a sound pressure level of 120 dB re 20 μPa.

Since the human ear is not equally sensitive to all sound frequencies, the acoustic power spectrum is modified by frequency weighting (A-weighting being the most common standard) to get the weighted acoustic power before converting to a sound level or noise level in decibels.^[38]

Further information: Sound pressure § Examples of sound pressure

Telephony

The decibel is used in telephony and audio. Similarly to the use in acoustics, a frequency weighted power is often used. For audio noise measurements in electrical circuits, the weightings are called psophometric weightings.^[39]

Electronics

In electronics, the decibel is often used to express power or amplitude ratios (as for 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 denote signal gain or loss from a transmitter to a receiver through some medium (free space, waveguide, coaxial cable, fiber optics, etc.) using a link budget.

The decibel unit can also be combined with a reference level, often indicated via a suffix, to create an absolute unit of electric power. For example, it can be combined with "m" for "milliwatt" to produce the "dBm". A power level of 0 dBm corresponds to one milliwatt, and 1 dBm is one decibel greater (about 1.259 mW).

In professional audio specifications, a popular unit is the dBu. This is relative to the root mean square voltage which delivers 1 mW (0 dBm) into a 600-ohm resistor, or √1 mW × 600 Ω ≈ 0.775 V_RMS. When used in a 600-ohm circuit (historically, the standard reference impedance in telephone circuits), dBu and dBm are identical.

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 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.^[40]

In spectrometry and optics, the blocking unit used to measure optical density is equivalent to −1 B.

Video and digital imaging

In connection with video and digital image sensors, decibels generally represent ratios of video voltages or digitized light intensities, using 20 log of the ratio, even when the represented intensity (optical power) is directly proportional to the voltage generated by the sensor, not to its square, as in a CCD imager where response voltage is linear in intensity.^[41] Thus, a camera signal-to-noise ratio or dynamic range quoted as 40 dB represents a ratio of 100:1 between optical signal intensity and optical-equivalent dark-noise intensity, not a 10,000:1 intensity (power) ratio as 40 dB might suggest.^[42] Sometimes the 20 log ratio definition is applied to electron counts or photon counts directly, which are proportional to sensor signal amplitude without the need to consider whether the voltage response to intensity is linear.^[43]

However, as mentioned above, the 10 log intensity convention prevails more generally in physical optics, including fiber 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 dB, 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 typically use an alternative base-2 log unit, the stop, to describe light intensity ratios or dynamic range.

Suffixes and reference values

Suffixes are commonly attached to the basic dB unit in order to indicate the reference value by which the ratio is calculated. For example, dBm indicates power measurement relative to 1 milliwatt.

In cases where the unit value of the reference is stated, the decibel value is known as "absolute". If the unit value of the reference is not explicitly stated, as in the dB gain of an amplifier, then the decibel value is considered relative.

This form of attaching suffixes to dB is widespread in practice, albeit being against the rules promulgated by standards bodies (ISO and IEC),^[15] given the "unacceptability of attaching information to units"^[a] and the "unacceptability of mixing information with units".^[b] The IEC 60027-3 standard recommends the following format:^[14] L_x (re x_ref) or as L_x/xref, where x is the quantity symbol and x_ref is the value of the reference quantity, e.g., L_E (re 1 μV/m) = 20 dB or L_{E/(1 μV/m)} = 20 dB for the electric field strength E relative to 1 μV/m reference value. If the measurement result 20 dB is presented separately, it can be specified using the information in parentheses, which is then part of the surrounding text and not a part of the unit: 20 dB (re: 1 μV/m) or 20 dB (1 μV/m).

Outside of documents adhering to SI units, the practice is very common as illustrated by the following examples. There is no general rule, with various discipline-specific practices. Sometimes the suffix is a unit symbol ("W","K","m"), sometimes it is a transliteration of a unit symbol ("uV" instead of μV for microvolt), sometimes it is an acronym for the unit's name ("sm" for square meter, "m" for milliwatt), other times it is a mnemonic for the type of quantity being calculated ("i" for antenna gain with respect to an isotropic antenna, "λ" for anything normalized by the EM wavelength), or otherwise a general attribute or identifier about the nature of the quantity ("A" for A-weighted sound pressure level). The suffix is often connected with a hyphen, as in "dB‑Hz", or with a space, as in "dB HL", or enclosed in parentheses, as in "dB(sm)", or with no intervening character, as in "dBm" (which is non-compliant with international standards).

List of suffixes

Voltage

Since the decibel is defined with respect to power, not amplitude, conversions of voltage ratios to decibels must square the amplitude, or use the factor of 20 instead of 10, as discussed above.

A schematic showing the relationship between dB_u (the voltage source) and dB_m (the power dissipated as heat by the 600 Ω resistor)

dB_V

dB(V_RMS) – voltage relative to 1 volt, regardless of impedance.^[3] This is used to measure microphone sensitivity, and also to specify the consumer line-level of −10 dBV, in order to reduce manufacturing costs relative to equipment using a +4 dBu line-level signal.^[44]

dB_u or dB_v

RMS voltage relative to ${\displaystyle V={\sqrt {600\ \Omega \ \cdot \ 0.001\ {\mathsf {W}}\;}}\approx 0.7746\ {\mathsf {V}}\ }$ (i.e. the voltage that would dissipate 1 mW into a 600 Ω load). An RMS voltage of 1 V therefore corresponds to ${\displaystyle \ 20\cdot \log _{10}\left({\frac {\ 1\ V_{\mathsf {RMS}}\ }{{\sqrt {0.6\ }}\ V}}\right)=2.218\ {\mathsf {dB_{u}}}~.}$^[3] Originally dB_v , it was changed to dB_u to avoid confusion with dB_V.^[45] The v comes from volt, while u comes from the volume unit displayed on a VU meter.^[46]

dB_u can be used as a measure of voltage, regardless of impedance, but is derived from a 600 Ω load dissipating 0 dB_m (1 mW). The reference voltage comes from the computation ${\displaystyle \ 7{\mathsf {V}}={\sqrt {R\cdot P\ }}\ }$ where ${\displaystyle \ R\ }$ is the resistance and ${\displaystyle \ P\ }$ is the power.

In professional audio, equipment may be calibrated to indicate a "0" on the VU meters some finite time after a signal has been applied at an amplitude of +4 dBu. Consumer equipment typically uses a lower "nominal" signal level of −10 dB_V .^[47] Therefore, many devices offer dual voltage operation (with different gain or "trim" settings) for interoperability reasons. A switch or adjustment that covers at least the range between +4 dB_u and −10 dB_V is common in professional equipment.

dB_m0s

Defined by Recommendation ITU-R V.574 ; dB_mV: dB(mV_RMS) – root mean square voltage relative to 1 millivolt across 75 Ω.^[48] Widely used in cable television networks, where the nominal strength of a single TV signal at the receiver terminals is about 0 dB_mV. Cable TV uses 75 Ω coaxial cable, so 0 dB_mV corresponds to −78.75 dB_W ( −48.75 dB_m ) or approximately 13 nW.

dB_μV or dB_uV

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

Acoustics

Probably the most common usage of "decibels" in reference to sound level is dB_SPL, sound pressure level referenced to the nominal threshold of human hearing:^[49] The measures of pressure (a root-power quantity) use the factor of 20, and the measures of power (e.g. dB_SIL and dB_SWL) use the factor of 10.

dB_SPL

dB_SPL (sound pressure level) – for sound in air and other gases, relative to 20 micropascals (μPa), or 2×10⁻⁵ Pa, approximately the quietest sound a human can hear. For sound in water and other liquids, a reference pressure of 1 μPa is used.^[50]

An RMS sound pressure of one pascal corresponds to a level of 94 dB SPL.

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 its effects on humans and other animals, and they are widely used in industry while discussing noise control issues, regulations and environmental standards. Other variations that may be seen are dB_A or dB(A). According to standards from the International Electro-technical Committee (IEC 61672-2013)^[51] and the American National Standards Institute, ANSI S1.4,^[52] the preferred usage is to write L_A = x dB . Nevertheless, the units dB_A and dB(A) are still commonly used as a shorthand for A‑weighted measurements. Compare dB_c, used in telecommunications.

dB_HL

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

sometimes used to denote weighted noise level, commonly using the ITU-R 468 noise weighting^{[citation needed]}

dB_pp

relative to the peak to peak sound pressure.^[53]

dB_G

G‑weighted spectrum^[54]

Audio electronics

See also dB_V and dB_u above.

dB_m

dB(mW) – power relative to 1 milliwatt. In audio and telephony, dB_m is typically referenced relative to a 600 Ω impedance,^[55] which corresponds to a voltage level of 0.775 volts or 775 millivolts.

dB_m0

Power in dB_m (described above) measured at a zero transmission level point.

dB_FS

dB(full scale) – the amplitude of a signal compared with the maximum which a device can handle before clipping occurs. Full-scale may be defined as the power level of a full-scale sinusoid or alternatively a full-scale square wave. A signal measured with reference to a full-scale sine-wave appears 3 dB weaker when referenced to a full-scale square wave, thus: 0 dBFS(fullscale sine wave) = −3 dB_FS (fullscale square wave).

dB_VU

dB volume unit^[56]

dB_TP

dB(true peak) – peak amplitude of a signal compared with the maximum which a device can handle before clipping occurs.^[57] In digital systems, 0 dB_TP would equal the highest level (number) the processor is capable of representing. Measured values are always negative or zero, since they are less than or equal to full-scale.

Radar

dB_Z

dB(Z) – decibel relative to Z = 1 mm⁶⋅m⁻³:^[58] energy of reflectivity (weather radar), related to the amount of transmitted power returned to the radar receiver. Values above 20 dB_Z usually indicate falling precipitation.^[59]

dB_sm

dB(m²) – decibel relative to one square meter: measure of the radar cross section (RCS) of a target. The power reflected by the target is proportional to its RCS. "Stealth" aircraft and insects have negative RCS measured in dB_sm, large flat plates or non-stealthy aircraft have positive values.^[60]

Radio power, energy, and field strength

dB_c

relative to carrier – in telecommunications, this indicates the relative levels of noise or sideband power, compared with the carrier power. Compare dB_C, used in acoustics.

dB_pp

relative to the maximum value of the peak power.

dB_J

energy relative to 1 joule. 1 joule = 1 watt second = 1 watt per hertz, so power spectral density can be expressed in dB_J.

dB_m

dB(mW) – power relative to 1 milliwatt. In the radio field, dB_m is usually referenced to a 50 Ω load, with the resultant voltage being 0.224 volts.^[61]

dB_μV/m , dB_uV/m , or dB_μ 

^[62] dB(μV/m) – electric field strength relative to 1 microvolt per meter. The unit is often used to specify the signal strength of a television broadcast at a receiving site (the signal measured at the antenna output is reported in dBμ_V).

dB_f

dB(fW) – power relative to 1 femtowatt.

dB_W

dB(W) – power relative to 1 watt.

dB_k

dB(kW) – power relative to 1 kilowatt.

dB_e

dB electrical.

dB_o

dB optical. A change of 1 dB_o in optical power can result in a change of up to 2 dB_e in electrical signal power in a system that is thermal noise limited.^[63]

Antenna measurements

dB_i

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

dB_d

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

dB_iC

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

dB_q

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

dB_sm

dB_m², dB(m²) – decibels relative to one square meter: A measure of the effective area for capturing signals of the antenna.^[64]

dB_m⁻¹

dB(m⁻¹) – decibels relative to reciprocal of meter: measure of the antenna factor.

Other measurements

dB_Hz or dB‑Hz

dB(Hz) – bandwidth relative to one hertz. 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).

dB_ov or dB_O

dB(overload) – the amplitude of a signal (usually audio) compared with the maximum which a device can handle before clipping occurs. Similar to dB FS, but also applicable to analog systems. According to ITU-T Rec. G.100.1 the level in dB ov of a digital system is defined as: $${\displaystyle \ L_{\mathsf {ov}}=10\log _{10}\left({\frac {P}{\ P_{\mathsf {max}}\ }}\right)\ [{\mathsf {dB_{ov}}}]\ ,}$$ with the maximum signal power ${\displaystyle \ P_{\mathsf {max}}=1.0\ ,}$ for a rectangular signal with the maximum amplitude ${\displaystyle \ x_{\mathsf {over}}~.}$ The level of a tone with a digital amplitude (peak value) of ${\displaystyle \ x_{\mathsf {over}}\ }$ is therefore ${\displaystyle \ L_{\mathsf {ov}}=-3.01\ {\mathsf {dB_{ov}}}~.}$^[65]

dB_r

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.

dB_rn

dB above reference noise. See also dB_rnC

dB_rnC

dB(rnC) represents an audio level measurement, typically in a telephone circuit, relative to a −90 dB_m reference level, with the measurement of this level frequency-weighted by a standard C-message weighting filter. The C-message weighting filter was chiefly used in North America. The psophometric filter is used for this purpose on international circuits.^[c][66]

dB_K

dB(K) – decibels relative to 1 K; used to express noise temperature.^[67]

dB_K⁻¹ or dB_/K

dB(K⁻¹) – decibels relative to 1 K⁻¹.^[68] — not decibels per Kelvin: Used for the ⁠ G / T ⁠ (G/T) factor, a figure of merit used in satellite communications, relating the antenna gain G to the receiver system noise equivalent temperature T.^[69][70]

List of suffixes in alphabetical order

Unpunctuated suffixes

dB_A

see dB(A).

dB_a

see dB_rn adjusted.

dB_B

see dB(B).

dB_c

relative to carrier – in telecommunications, this indicates the relative levels of noise or sideband power, compared with the carrier power.

dB_C

see dB(C).

dB_D

see dB(D).

dB_d

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

dB_e

dB electrical.

dB_f

dB(fW) – power relative to 1 femtowatt.

dB_FS

dB(full scale) – the amplitude of a signal compared with the maximum which a device can handle before clipping occurs. Full-scale may be defined as the power level of a full-scale sinusoid or alternatively a full-scale square wave. A signal measured with reference to a full-scale sine-wave appears 3 dB weaker when referenced to a full-scale square wave, thus: 0 dB_FS (fullscale sine wave) = −3 dB_FS (full-scale square wave).

dB_G

G-weighted spectrum

dB_i

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.

dB_iC

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

dB_J

energy relative to 1 joule: 1 joule = 1 watt-second = 1 watt per hertz, so power spectral density can be expressed in dB_J .

dB_k

dB(kW) – power relative to 1 kilowatt.

dB_K

dB(K) – decibels relative to kelvin: Used to express noise temperature.

dB_m

dB(mW) – power relative to 1 milliwatt.

dB_m² or dB_sm

dB(m²) – decibel relative to one square meter

dB_m0

Power in dB_m measured at a zero transmission level point.

dB_m0s

Defined by Recommendation ITU-R V.574.

dB_mV

dB(mV_RMS) – voltage relative to 1 millivolt across 75 Ω.

dB_o

dB optical. A change of 1 dB_o in optical power can result in a change of up to 2 dB_e in electrical signal power in system that is thermal noise limited.

dB_O

see dB_ov

dB_ov or dB_O

dB(overload) – the amplitude of a signal (usually audio) compared with the maximum which a device can handle before clipping occurs.

dB_pp

relative to the peak to peak sound pressure.

dB_pp

relative to the maximum value of the peak electrical power.

dB_q

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 dB_i

dB_r

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.

dB_rn

dB above reference noise. See also dB_rnC

dB_rnC

dB_rnC represents an audio level measurement, typically in a telephone circuit, relative to the circuit noise level, with the measurement of this level frequency-weighted by a standard C-message weighting filter. The C-message weighting filter was chiefly used in North America.

dB_sm

see dB_m²

dB_TP

dB(true peak) – peak amplitude of a signal compared with the maximum which a device can handle before clipping occurs.

dB_u or dB_v

RMS voltage relative to ${\displaystyle \ {\sqrt {0.6\;}}\ {\mathsf {V}}\ \approx 0.7746\ {\mathsf {V}}\ \approx -2.218\ {\mathsf {dB_{V}}}~.}$

dB_u0s

Defined by Recommendation ITU-R V.574.

dB_uV

see dB_μV

dB_uV/m

see dB_μV/m

dB_v

see dB_u

dB_V

dB(V_RMS) – voltage relative to 1 volt, regardless of impedance.

dB_VU

dB(VU) dB volume unit

dB_W

dB(W) – power relative to 1 watt.

dB_{W·m⁻²·Hz⁻¹}

spectral density relative to 1 W·m⁻²·Hz⁻¹^[71]

dB_Z

dB(Z) – decibel relative to Z = 1 mm⁶⋅m⁻³

dB_μ

see dB_μV/m

dB_μV or dB_uV

dB(μV_RMS) – voltage relative to 1 root mean square microvolt.

dB_μV/m , dB_uV/m , or dB_μ 

dB(μV/m) – electric field strength relative to 1 microvolt per meter.

Suffixes preceded by a space

dB HL

dB hearing level is used in audiograms as a measure of hearing loss.

dB Q

sometimes used to denote weighted noise level

dB SIL

dB sound intensity level – relative to 10⁻¹² W/m²

dB SPL

dB SPL (sound pressure level) – for sound in air and other gases, relative to 20 μPa in air or 1 μPa in water

dB SWL

dB sound power level – relative to 10⁻¹² W.

Suffixes within parentheses

dB(A), dB(B), dB(C), dB(D), dB(G), and dB(Z)

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 its effects on humans and other animals, and they are widely used in industry while discussing noise control issues, regulations and environmental standards. Other variations that may be seen are dB_A or dBA.

Other suffixes

dB_Hz or dB-Hz

dB(Hz) – bandwidth relative to one Hertz

dB_K⁻¹ or dB_/K

dB(K⁻¹) – decibels relative to reciprocal of kelvin

dB_m⁻¹

dB(m⁻¹) – decibel relative to reciprocal of meter: measure of the antenna factor

mB_m

mB(mW) – power relative to 1 milliwatt, in millibels (one hundredth of a decibel). 100 mB_m = 1 dB_m . This unit is in the Wi-Fi drivers of the Linux kernel^[72] and the regulatory domain sections.^[73]

See also

• Apparent magnitude
• Cent (music)
• Day–evening–night noise level (L_den) and day-night average sound level (Ldl), European and American standards for expressing noise level over an entire day
• dB drag racing
• Decade (log scale)
• Loudness
• Neper
• One-third octave § Base 10
• pH
• Phon
• Richter magnitude scale
• Sone

Notes

1. "When one gives the value of a quantity, it is incorrect to attach letters or other symbols to the unit in order to provide information about the quantity or its conditions of measurement. Instead, the letters or other symbols should be attached to the quantity."^[15]: 16
2. "When one gives the value of a quantity, any information concerning the quantity or its conditions of measurement must be presented in such a way as not to be associated with the unit. This means that quantities must be defined so that they can be expressed solely in acceptable units..."^[15]: 17
3. See psophometric weighting to see a comparison of frequency response curves for the C-message weighting and psophometric weighting filters.

References

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Further reading

• Tuffentsammer, Karl (1956). "Das Dezilog, eine Brücke zwischen Logarithmen, Dezibel, Neper und Normzahlen" [The decilog, a bridge between logarithms, decibel, neper and preferred numbers]. VDI-Zeitschrift (in German). 98: 267–274.
• Paulin, Eugen (1 September 2007). Logarithmen, Normzahlen, Dezibel, Neper, Phon - natürlich verwandt! [Logarithms, preferred numbers, decibel, neper, phon - naturally related!] (PDF) (in German). Archived (PDF) from the original on 18 December 2016. Retrieved 18 December 2016.

External links

• What is a decibel? With sound files and animations
• Conversion of sound level units: dBSPL or dBA to sound pressure p and sound intensity J
• OSHA Regulations on Occupational Noise Exposure
• Working with Decibels (RF signal and field strengths)