The decibel (symbol: dB) is a relative unit of measurement corresponding to one tenth of a bel (B). It is used to express the ratio of one value of a power or root-power quantity to another, on a logarithmic scale. A logarithmic quantity in decibels is called a level. 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^{.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1⁄20} (approximately 1.12).^{[1]}^{[2]}
The unit is used to express a change in value (e.g., +1 dB or −1 dB) or an absolute value. In the latter case, the numeric value expresses 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 in 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 in acoustics, electronics, and control theory. In electronics, the gains of amplifiers, attenuation of signals, and signal-to-noise ratios are often expressed in decibels.
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 log_{10} x. |
History
The decibel originates from methods used to quantify signal loss in telegraph and telephone circuits. The unit for loss was originally Miles of Standard Cable (MSC). 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 matched closely the smallest attenuation detectable to the average listener. The standard telephone cable implied 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 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
ISO 80000-3 describes definitions for quantities and units of space and time.
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 = 0.115 13… Np = 0.115 13…. 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).^{[17]}^{[18]}
The bel is rarely used either without a prefix or with SI unit prefixes other than deci; it is preferred, 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.^{[19]}
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 P_{0} (reference power) is represented by L_{P}, that ratio expressed in decibels,^{[20]} which is calculated using the formula:^{[21]}
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 P_{0} must measure the same type of quantity, and have the same units before calculating the ratio. If P = P_{0} in the above equation, then L_{P} = 0. If P is greater than P_{0} then L_{P} is positive; if P is less than P_{0} then L_{P} is negative.
Rearranging the above equation gives the following formula for P in terms of P_{0} and L_{P}:
Root-power (field) quantities
When referring to measurements of root-power quantities, it is usual to consider the ratio of the squares of F (measured) and F_{0} (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:
The formula may be rearranged to give
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}:
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
holding.^{[22]} 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 P_{0} and F_{0} need not be related), or equivalently,
must hold to allow the power level difference to be equal to the root-power level difference from power P_{1} and V_{1} to P_{2} and V_{2}. 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.
Unit | In decibels | In bels | In nepers | Power ratio | Root-power ratio |
---|---|---|---|---|---|
1 dB | 1 dB | 0.1 B | 0.11513 Np | 10^{1⁄10} ≈ 1.25893 | 10^{1⁄20} ≈ 1.12202 |
1 Np | 8.68589 dB | 0.868589 B | 1 Np | e^{2} ≈ 7.38906 | e ≈ 2.71828 |
1 B | 10 dB | 1 B | 1.151 3 Np | 10 | 10^{1⁄2} ≈ 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:
- The ratio in decibels of √1 000 V ≈ 31.62 V to 1 V is
(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
- The power ratio corresponding to a 3 dB change in level is given by
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, and 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 multiplied effects such as attenuation from multiple sources along a signal chain. Its application in systems with additive effects is less intuitive.
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.
- 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:
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.^{[23]}^{[24]} Quantities in decibels are not necessarily additive,^{[25]}^{[26]} thus being "of unacceptable form for use in dimensional analysis".^{[27]} Thus, units require special care in decibel operations. Take, for example, carrier-to-noise-density ratio C/N_{0} (in hertz), involving carrier power C (in watts) and noise power spectral density N_{0} (in W/Hz). Expressed in decibels, this ratio would be a subtraction (C/N_{0})_{dB} = C_{dB} - N_{0dB}. 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
According to Mitschke,^{[28]} "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:^{[29]}
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:
Note that the logarithmic mean is obtained from the logarithmic sum by subtracting , 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.^{[30]}^{[31]}^{[32]}^{[33]}^{[34]}^{[35]}
Acoustics
The decibel is commonly used in acoustics as a unit of 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:
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.^{[36]}
Use of the decibel in underwater acoustics leads to confusion, in part because of this difference in reference value.^{[37]}
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 (10^{12}).^{[38]} Such large measurement ranges are conveniently expressed in logarithmic scale: the base-10 logarithm of 10^{12} is 12, which is expressed as 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.^{[39]}
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.^{[40]}
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.^{[41]}
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.^{[42]} 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.^{[43]} 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.^{[44]}
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 suffices to dB is widespread in practice, albeit being against the rules promulgated by standards bodies (ISO and IEC):^{[15]} A change in level of 1 dB always corresponds to a power ratio of approximately 1 : 1.259. The suffix merely indicates the method of measurement, not a different scaling or type of unit. 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).
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.
- dBV
- 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.^{[45]}
- dBu or dBv
- RMS voltage relative to (i.e. the voltage that would dissipate 1 mW into a 600 Ω load). An RMS voltage of 1 V therefore corresponds to ^{[3]} Originally dBv, it was changed to dBu to avoid confusion with dBV.^{[46]} The "v" comes from "volt", while "u" comes from the volume unit used in the VU meter.^{[47]}dBu can be used as a measure of voltage, regardless of impedance, but is derived from a 600 Ω load dissipating 0 dBm (1 mW). The reference voltage comes from the computation where is the resistance and 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 dBV.^{[48]} 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 dBu and −10 dBV is common in professional equipment.
- dBm0s
- Defined by Recommendation ITU-R V.574.; dBmV: dB(mV_{RMS}) – voltage relative to 1 millivolt across 75 Ω.^{[49]} 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 approximately 13 nW.
- dBμV or dBuV
- dB(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 level is dB SPL, sound pressure level referenced to the nominal threshold of human hearing:^{[50]} 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.^{[51]} 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/m^{2}, which is roughly the threshold of human hearing in air.
- dB SWL
- dB sound power level – relative to 10^{−12} W.
- dBA, dBB, and dBC
- 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)^{[52]} and the American National Standards Institute, ANSI S1.4,^{[53]} 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
- 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]}
- dBpp
- relative to the peak to peak sound pressure.^{[54]}
- dBG
- G‑weighted spectrum^{[55]}
Audio electronics
See also dBV and dBu above.
- dBm
- dB(mW) – power relative to 1 milliwatt. In audio and telephony, dBm is typically referenced relative to a 600 Ω impedance,^{[56]} which corresponds to a voltage level of 0.775 volts or 775 millivolts.
- dBm0
- Power in dBm (described above) measured at a zero transmission level point.
- dBFS
- 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 dBFS(fullscale square wave).
- dBVU
- dB volume unit^{[57]}
- dBTP
- dB(true peak) – peak amplitude of a signal compared with the maximum which a device can handle before clipping occurs.^{[58]} In digital systems, 0 dBTP 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
- dBZ
- dB(Z) – decibel relative to Z = 1 mm^{6}⋅m^{−3}:^{[59]} energy of reflectivity (weather radar), related to the amount of transmitted power returned to the radar receiver. Values above 20 dBZ usually indicate falling precipitation.^{[60]}
- dBsm
- dB(m^{2}) – 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 dBsm, large flat plates or non-stealthy aircraft have positive values.^{[61]}
Radio power, energy, and field strength
- dBc
- relative to carrier – in telecommunications, this indicates the relative levels of noise or sideband power, compared with the carrier power. Compare dBC, used in acoustics.
- dBpp
- relative to the maximum value of the peak power.
- dBJ
- energy relative to 1 joule. 1 joule = 1 watt second = 1 watt per hertz, so power spectral density can be expressed in dBJ.
- dBm
- dB(mW) – power relative to 1 milliwatt. In the radio field, dBm is usually referenced to a 50 Ω load, with the resultant voltage being 0.224 volts.^{[62]}
- dBμV/m, dBuV/m, or dBμ
- ^{[63]} 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 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.
- dBe
- dB electrical.
- dBo
- dB optical. A change of 1 dBo in optical power can result in a change of up to 2 dBe in electrical signal power in a system that is thermal noise limited.^{[64]}
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
- dBsm
- dB(m^{2}) – decibel relative to one square meter: measure of the antenna effective area.^{[65]}
- dBm^{−1}
- dB(m^{−1}) – decibel relative to reciprocal of meter: measure of the antenna factor.
Other measurements
- 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).
- 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. According to ITU-T Rec. G.100.1 the level in dBov of a digital system is defined as:
- ,
- with the maximum signal power , for a rectangular signal with the maximum amplitude . The level of a tone with a digital amplitude (peak value) of is therefore .^{[66]}
- 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
- dBrnC
- dBrnC represents an audio level measurement, typically in a telephone circuit, relative to a -90 dBm 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. See Psophometric weighting to see a comparison of frequency response curves for the C-message weighting and Psophometric weighting filters.^{[67]}
- dBK
- dB(K) – decibels relative to 1 K; used to express noise temperature.^{[68]}
- dB/K
- dB(K^{−1}) – decibels relative to 1 K^{−1}.^{[69]} — not decibels per kelvin: Used for the G/T factor, a figure of merit utilized in satellite communications, relating the antenna gain G to the receiver system noise equivalent temperature T.^{[70]}^{[71]}
List of suffixes in alphabetical order
Unpunctuated suffixes
- dBA
- see dB(A).
- dBa
- see dBrn adjusted.
- dBB
- see dB(B).
- dBc
- relative to carrier – in telecommunications, this indicates the relative levels of noise or sideband power, compared with the carrier power.
- dBC
- see dB(C).
- dBD
- see dB(D).
- dBd
- dB(dipole) – the forward gain of an antenna compared with a half-wave dipole antenna. 0 dBd = 2.15 dBi
- dBe
- dB electrical.
- dBf
- dB(fW) – power relative to 1 femtowatt.
- dBFS
- 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 dBFS(fullscale square wave).
- dBG
- G-weighted spectrum
- 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.
- 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.
- dBJ
- energy relative to 1 joule. 1 joule = 1 watt second = 1 watt per hertz, so power spectral density can be expressed in dBJ.
- dBk
- dB(kW) – power relative to 1 kilowatt.
- dBK
- dB(K) – decibels relative to kelvin: Used to express noise temperature.
- dBm
- dB(mW) – power relative to 1 milliwatt.
- dBm0
- Power in dBm measured at a zero transmission level point.
- dBm0s
- Defined by Recommendation ITU-R V.574.
- dBmV
- dB(mV_{RMS}) – voltage relative to 1 millivolt across 75 Ω.
- dBo
- dB optical. A change of 1 dBo in optical power can result in a change of up to 2 dBe in electrical signal power in system that is thermal noise limited.
- dBO
- see dBov
- dBov or dBO
- dB(overload) – the amplitude of a signal (usually audio) compared with the maximum which a device can handle before clipping occurs.
- dBpp
- relative to the peak to peak sound pressure.
- dBpp
- relative to the maximum value of the peak power.
- 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
- 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
- dBrnC
- dBrnC 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.
- dBsm
- dB(m^{2}) – decibel relative to one square meter
- dBTP
- dB(true peak) – peak amplitude of a signal compared with the maximum which a device can handle before clipping occurs.
- dBu or dBv
- RMS voltage relative to .
- dBu0s
- Defined by Recommendation ITU-R V.574.
- dBuV
- see dBμV
- dBuV/m
- see dBμV/m
- dBv
- see dBu
- dBV
- dB(V_{RMS}) – voltage relative to 1 volt, regardless of impedance.
- dBVU
- dB volume unit
- dBW
- dB(W) – power relative to 1 watt.
- dBZ
- dB(Z) – decibel relative to Z = 1 mm^{6}⋅m^{−3}
- dBμ
- see dBμV/m
- dBμV or dBuV
- dB(μV_{RMS}) – voltage relative to 1 microvolt.
- dBμV/m, dBuV/m, or dBμ
- 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/m^{2}
- 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
- dB(Hz) – bandwidth relative to one hertz.
- dB/K
- dB(K^{−1}) – decibels relative to reciprocal of kelvin
- dBm^{−1}
- dB(m^{−1}) – decibel relative to reciprocal of meter: measure of the antenna factor.
Related units
- mBm
- mB(mW) – power relative to 1 milliwatt, in millibels (one hundredth of a decibel). 100 mBm = 1 dBm. 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)
- dB drag racing
- Decade (log scale)
- Loudness
- One-third octave § Base 10
- pH
- Phon
- Richter magnitude scale
- Sone
References
- ^ Mark, James E. (2007). Physical Properties of Polymers Handbook. Springer. p. 1025. Bibcode:2007ppph.book.....M.
[…] the decibel represents a reduction in power of 1.258 times […]
- ^ Yost, William (1985). Fundamentals of Hearing: An Introduction (Second ed.). Holt, Rinehart and Winston. p. 206. ISBN 978-0-12-772690-8.
[…] a pressure ratio of 1.122 equals + 1.0 dB […]
- ^ ^{a} ^{b} ^{c} Utilities : V_{RMS} / dBm / dBu / dBV calculator, Analog Devices, retrieved 16 September 2016
- ^ Thompson and Taylor 2008, Guide for the Use of the International System of Units (SI), NIST Special Publication SP811 Archived 2016-06-03 at the Wayback Machine.
- ^ IEEE Standard 100: a dictionary of IEEE standards and terms (7th ed.). New York: The Institute of Electrical and Electronics Engineering. 2000. p. 288. ISBN 978-0-7381-2601-2.
- ^ Johnson, Kenneth Simonds (1944). Transmission Circuits for Telephonic Communication: Methods of analysis and design. New York: D. Van Nostrand Co. p. 10.
- ^ Davis, Don; Davis, Carolyn (1997). Sound system engineering (2nd ed.). Focal Press. p. 35. ISBN 978-0-240-80305-0.
- ^ Hartley, R. V. L. (December 1928). "'TU' becomes 'Decibel'". Bell Laboratories Record. AT&T. 7 (4): 137–139.
- ^ Martin, W. H. (January 1929). "DeciBel—The New Name for the Transmission Unit". Bell System Technical Journal. 8 (1).
- ^ 100 Years of Telephone Switching, p. 276, at Google Books, Robert J. Chapuis, Amos E. Joel, 2003
- ^ Harrison, William H. (1931). "Standards for Transmission of Speech". Standards Yearbook. National Bureau of Standards, U. S. Govt. Printing Office. 119.
- ^ Horton, J. W. (1954). "The bewildering decibel". Electrical Engineering. 73 (6): 550–555. doi:10.1109/EE.1954.6438830. S2CID 51654766.
- ^ "Meeting minutes" (PDF). Consultative Committee for Units. Section 3.
- ^ ^{a} ^{b} "Letter symbols to be used in electrical technology". International Electrotechnical Commission. 19 July 2002. Part 3: Logarithmic and related quantities, and their units. IEC 60027-3, Ed. 3.0.
- ^ ^{a} ^{b} Thompson, A. and Taylor, B. N. sec 8.7, "Logarithmic quantities and units: level, neper, bel", Guide for the Use of the International System of Units (SI) 2008 Edition, NIST Special Publication 811, 2nd printing (November 2008), SP811 PDF
- ^ "Letter symbols to be used in electrical technology". International Standard CEI-IEC 27-3. International Electrotechnical Commission. Part 3: Logarithmic quantities and units.
- ^ Mark, James E. (2007). Physical Properties of Polymers Handbook. Springer. p. 1025. Bibcode:2007ppph.book.....M.
[…] the decibel represents a reduction in power of 1.258 times […]
- ^ Yost, William (1985). Fundamentals of Hearing: An Introduction (Second ed.). Holt, Rinehart and Winston. p. 206. ISBN 978-0-12-772690-8.
[…] a pressure ratio of 1.122 equals + 1.0 dB […]
- ^ Fedor Mitschke, Fiber Optics: Physics and Technology, Springer, 2010 ISBN 3642037038.
- ^ Pozar, David M. (2005). Microwave Engineering (3rd ed.). Wiley. p. 63. ISBN 978-0-471-44878-5.
- ^ IEC 60027-3:2002
- ^ I M Mills; B N Taylor; A J Thor (2001), "Definitions of the units radian, neper, bel and decibel", Metrologia, 38 (4): 353, Bibcode:2001Metro..38..353M, doi:10.1088/0026-1394/38/4/8
- ^ R. Hickling (1999), Noise Control and SI Units, J Acoust Soc Am 106, 3048
- ^ Hickling, R. (2006). Decibels and octaves, who needs them?. Journal of sound and vibration, 291(3-5), 1202-1207.
- ^ Nicholas P. Cheremisinoff (1996) Noise Control in Industry: A Practical Guide, Elsevier, 203 pp, p. 7
- ^ Andrew Clennel Palmer (2008), Dimensional Analysis and Intelligent Experimentation, World Scientific, 154 pp, p.13
- ^ J. C. Gibbings, Dimensional Analysis, p.37, Springer, 2011 ISBN 1849963177.
- ^ Fiber Optics. Springer. 2010.
- ^ R. J. Peters, Acoustics and Noise Control, Routledge, 12 November 2013, 400 pages, p. 13
- ^ Sensation and Perception, p. 268, at Google Books
- ^ Introduction to Understandable Physics, Volume 2, p. SA19-PA9, at Google Books
- ^ Visual Perception: Physiology, Psychology, and Ecology, p. 356, at Google Books
- ^ Exercise Psychology, p. 407, at Google Books
- ^ Foundations of Perception, p. 83, at Google Books
- ^ Fitting The Task To The Human, p. 304, at Google Books
- ^ ISO 1683:2015
- ^ C. S. Clay (1999), Underwater sound transmission and SI units, J Acoust Soc Am 106, 3047
- ^ "Loud Noise Can Cause Hearing Loss". cdc.gov. Centers for Disease Control and Prevention. Retrieved 30 July 2020.
- ^ Richard L. St. Pierre, Jr. and Daniel J. Maguire (July 2004), The Impact of A-weighting Sound Pressure Level Measurements during the Evaluation of Noise Exposure (PDF), retrieved 13 September 2011
- ^ Reeve, William D. (1992). Subscriber Loop Signaling and Transmission Handbook – Analog (1st ed.). IEEE Press. ISBN 0-87942-274-2.
- ^ Chomycz, Bob (2000). Fiber optic installer's field manual. McGraw-Hill Professional. pp. 123–126. ISBN 978-0-07-135604-6.
- ^ Stephen J. Sangwine and Robin E. N. Horne (1998). The Colour Image Processing Handbook. Springer. pp. 127–130. ISBN 978-0-412-80620-9.
- ^ Francis T. S. Yu and Xiangyang Yang (1997). Introduction to optical engineering. Cambridge University Press. pp. 102–103. ISBN 978-0-521-57493-8.
- ^ Junichi Nakamura (2006). "Basics of Image Sensors". In Junichi Nakamura (ed.). Image sensors and signal processing for digital still cameras. CRC Press. pp. 79–83. ISBN 978-0-8493-3545-7.
- ^ Winer, Ethan (2013). The Audio Expert: Everything You Need to Know About Audio. Focal Press. p. 107. ISBN 978-0-240-82100-9.
- ^ stason.org, Stas Bekman: stas (at). "3.3 – What is the difference between dBv, dBu, dBV, dBm, dB SPL, and plain old dB? Why not just use regular voltage and power measurements?". stason.org.
- ^ Rupert Neve, Creation of the dBu standard level reference
- ^ deltamedia.com. "DB or Not DB". Deltamedia.com. Retrieved 16 September 2013.
- ^ The IEEE Standard Dictionary of Electrical and Electronics terms (6th ed.). IEEE. 1996 [1941]. ISBN 978-1-55937-833-8.
- ^ Jay Rose (2002). Audio postproduction for digital video. Focal Press. p. 25. ISBN 978-1-57820-116-7.
- ^ Morfey, C. L. (2001). Dictionary of Acoustics. Academic Press, San Diego.
- ^ IEC 61672-1:2013 Electroacoustics - Sound Level meters - Part 1: Specifications. Geneva: International Electrotechnical Committee. 2013.
- ^ ANSI S1.4-19823 Specification for Sound Level Meters, 2.3 Sound Level, p. 2–3.
- ^ Zimmer, Walter MX, Mark P. Johnson, Peter T. Madsen, and Peter L. Tyack. "Echolocation clicks of free-ranging Cuvier’s beaked whales (Ziphius cavirostris)." The Journal of the Acoustical Society of America 117, no. 6 (2005): 3919–3927.
- ^ http://oto2.wustl.edu/cochlea/wt4.html
- ^ Bigelow, Stephen (2001). Understanding Telephone Electronics. Newnes. p. 16. ISBN 978-0750671750.
- ^ Tharr, D. (1998). Case Studies: Transient Sounds Through Communication Headsets. Applied Occupational and Environmental Hygiene, 13(10), 691–697.
- ^ ITU-R BS.1770
- ^ "Glossary: D's". National Weather Service. Archived from the original on 8 August 2019. Retrieved 25 April 2013.
- ^ "RIDGE Radar Frequently Asked Questions". Archived from the original on 31 March 2019. Retrieved 8 August 2019.
- ^ "Definition at Everything2". Archived from the original on 10 June 2019. Retrieved 8 August 2019.
- ^ Carr, Joseph (2002). RF Components and Circuits. Newnes. pp. 45–46. ISBN 978-0750648448.
- ^ "The dBµ vs. dBu Mystery: Signal Strength vs. Field Strength?". radio-timetraveller.blogspot.com. 24 February 2015. Retrieved 13 October 2016.
- ^ Chand, N., Magill, P. D., Swaminathan, S. V., & Daugherty, T. H. (1999). Delivery of digital video and other multimedia services (> 1 Gb/s bandwidth) in passband above the 155 Mb/s baseband services on a FTTx full service access network. Journal of lightwave technology, 17(12), 2449–2460.
- ^ David Adamy. EW 102: A Second Course in Electronic Warfare. Retrieved 16 September 2013.
- ^ ITU-T Rec. G.100.1 The use of the decibel and of relative levels in speechband telecommunications https://www.itu.int/rec/dologin_pub.asp?lang=e&id=T-REC-G.100.1-201506-I!!PDF-E&type=items
- ^ dBrnC is defined on page 230 in "Engineering and Operations in the Bell System," (2ed), R.F. Rey (technical editor), copyright 1983, AT&T Bell Laboratories, Murray Hill, NJ, ISBN 0-932764-04-5
- ^ K. N. Raja Rao (31 January 2013). Satellite Communication: Concepts And Applications. Retrieved 16 September 2013.
- ^ Ali Akbar Arabi. Comprehensive Glossary of Telecom Abbreviations and Acronyms. Retrieved 16 September 2013.
- ^ Mark E. Long. The Digital Satellite TV Handbook. Retrieved 16 September 2013.
- ^ Mac E. Van Valkenburg (19 October 2001). Reference Data for Engineers: Radio, Electronics, Computers and Communications. Retrieved 16 September 2013.
- ^ "en:users:documentation:iw [Linux Wireless]". wireless.kernel.org.
- ^ "Is your WiFi AP Missing Channels 12 & 13?". wordpress.com. 16 May 2013.
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.