|dB||Power ratio||Amplitude ratio|
|6||3||.981 ≈ 4||1||.995 ≈ 2|
|3||1||.995 ≈ 2||1||.413 ≈ √|
|−3||0||.501 ≈ 1⁄2||0||.708 ≈ √|
|−6||0||.251 ≈ 1⁄4||0||.501 ≈ 1⁄2|
|An example scale showing power ratios x, amplitude ratios √, and dB equivalents 10 log10 x.|
The decibel (symbol: dB) is a logarithmic unit used to express the ratio of one value of a physical property to another, and may used to express a change in value (e.g., +1 dB or -1 dB) or an absolute value. In the latter case, it expresses the ratio of a value to a reference value; when used in this way, the decibel symbol should be appended with a suffix that indicates the reference value or some other property. For example, if the reference value is 1 volt, then the suffix is "dBV" (i.e., "20 dBV"), and if the reference value is one milliwatt, then the suffix is "dBm" (i.e., "20 dBm").  However, sound pressure level is referenced to the "threshold of hearing" (generally given as 20 microPascals at 1 kHz), and the suffix is "dB-SPL" (i.e., "60 dB-SPL").
There are two different scales used when expressing a ratio in decibels depending on the nature of the quantities: field quantity ratio or power quantity ratio. (Field quantity ratio is also referred to as root-power ratio or amplitude ratio.) When expressing power quantities, the number of decibels is ten times the logarithm to base 10 of the ratio of two power quantities. That is, a change in power by a factor of 10 corresponds to a 10 dB change in level. When expressing field quantities, a change in amplitude by a factor of 10 corresponds to a 20 dB change in level. The extra factor of two is due to the logarithm of the quadratic relationship between power and amplitude. The decibel scales differ so that direct comparisons can be made between related power and field quantities when they are expressed in decibels.
The definition of the decibel is based on the measurement of power in telephony of the early 20th century in the Bell System in the United States. One decibel is one tenth (deci-) of one bel, named in honor of Alexander Graham Bell; however, the bel is seldom used. Today, 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.
In the International System of Quantities, the decibel is defined as a unit of measurement for quantities of type level or level difference, which are defined as the logarithm of the ratio of power- or field-type quantities.
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 microfarad per mile" (approximately 19 gauge).
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 level. 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, 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. The bel is seldom used, as the decibel was the proposed working unit.
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 100.1 and any two amounts of power differ by N decibels when they are in the ratio of 10N(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 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. However, the decibel is recognized by other international bodies such as the International Electrotechnical Commission (IEC) and International Organization for Standardization (ISO). The IEC permits the use of the decibel with field quantities as well as power and this recommendation is followed by many national standards bodies, such as NIST, which justifies the use of the decibel for voltage ratios. The term field quantity is deprecated by ISO 80000-1, which favors root-power. In spite of their widespread use, suffixes (such as in dBA or dBV) are not recognized by the IEC or ISO.
ISO 80000-3 describes definitions for quantities and units of space and time. The decibel for use in acoustics is defined in ISO 80000-8. The major difference from the article below is that for acoustics the decibel has no absolute value.
The ISO Standard 80000-3:2006 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 field quantity when the field 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 field-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 field quantities of √10:1.
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'.
The method of expressing a ratio as a level in decibels depends on whether the measured property is a power quantity or a field quantity; see Field, power, and root-power quantities for details.
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 LP, that ratio expressed in decibels, which is calculated using the formula:
The base-10 logarithm of the ratio of the two power levels 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 LP = 0. If P is greater than P0 then LP is positive; if P is less than P0 then LP is negative.
Rearranging the above equation gives the following formula for P in terms of P0 and LP:
When referring to measurements of field quantities, it is usual to consider the ratio of the squares of F (measured field) and F0 (reference field). This is because in most applications power is proportional to the square of field, and it is desirable for the two decibel formulations to give the same result in such typical cases. 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 held constant. Taking voltage as an example, this leads to the equation:
where V is the voltage being measured, V0 is a specified reference voltage, and GdB is the power gain expressed in decibels. A similar formula holds for current.
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.
Since logarithm differences measured in these units are used to represent power ratios and field ratios, the values of the ratios represented by each unit are also included in the table.
|Unit||In decibels||In bels||In nepers||Power ratio||Field 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||e2 ≈ 7.38906||e ≈ 2.71828|
|1 B||10 dB||1 B||1.1513 Np||10||10 1⁄2 ≈ 3.16228|
All of these examples yield dimensionless answers in dB because they are relative ratios expressed in decibels. 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.
(31.62 V/1 V)2 ≈ 1 kW/1 W, illustrating the consequence from the definitions above that GdB has the same value, 30, 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.
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 √ ≈ 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. 3.00 dB is 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 √. Similarly, an increase of 6.00 dB is the power ratio is 10 6⁄10 ≈ 3.9811, about 0.5% different from 4.
The decibel has the following properties:
According to Mitschke, "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."
The human perception of the intensity of sound and light approximates the logarithm of intensity rather than a linear relationship (Weber–Fechner law), making the dB scale a useful measure.
Decibels are still the commonly used units to express ratios in a number of fields, even when the original meaning of the term is obscured. Decibels are the traditional way of expressing gain or margin in such diverse disciplines as control theory, antenna and radio frequency transmission theory, and even assessment of nuclear hardness.[dubious ]
Various published articles have criticized the unit decibel as having shortcomings that hinder its understanding and use: 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.
Representing the equivalent of zero watts is not possible, causing problems in conversions. Hickling concludes "Decibels are a useless affectation, which is impeding the development of noise control as an engineering discipline".
A common source of confusion in using the decibel occurs when deciding about the use of 10 × log or 20 × log. In the original definition, it was a power measurement, and as employed in that context, the formulation 10 × log should be used, as deci means one tenth. The user must be clear whether the quantity expressed is power or amplitude. It is useful to consider how power or energy is expressed, e.g., current × current × resistance, 1⁄2 × velocity × velocity × mass. Where the power is a square function of a field variable (such as voltage, current, or pressure), then 10 × log is the correct expression for the square, or 20 × log for the field variable itself.
For the same reason that humans excel at additive operation over multiplication, decibels are awkward in inherently additive operations: "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."
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. Sound pressure is a field quantity, therefore the field version of the unit definition is used:
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 greater than or equal to 1 trillion (1012). Such large measurement ranges are conveniently expressed in logarithmic scale: the base-10 logarithm of 1012 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, noise levels at maximum human sensitivity, somewhere between 2 and 4 kHz, are factored more heavily into some measurements using frequency weighting. (See also Stevens' power law.)
In electronics, the decibel is often used to express power or amplitude ratios (gains), in preference to arithmetic ratios or percentages. One advantage is that the total decibel gain of a series of components (such as amplifiers and attenuators) can be calculated simply by summing the decibel gains of the individual components. Similarly, in telecommunications, decibels 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 suffix to create an absolute unit of electric power. For example, it can be combined with "m" for "milliwatt" to produce the "dBm". Zero dBm is the level corresponding 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 √ ≈ 0.775 VRMS. When used in a 600-ohm circuit (historically, the standard reference impedance in telephone circuits), dBu and dBm are identical.
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.
In connection with video and digital image sensors, decibels generally represent ratios of video voltages or digitized light levels, using 20 log of the ratio, even when the represented optical power is directly proportional to the voltage or level, not to its square, as in a CCD imager where response voltage is linear in intensity. Thus, a camera signal-to-noise ratio or dynamic range of 40 dB represents a power ratio of 100:1 between signal power and noise power, not 10,000:1. Sometimes the 20 log ratio definition is applied to electron counts or photon counts directly, which are proportional to intensity without the need to consider whether the voltage response is linear.
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 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.
The SI does not permit attaching qualifiers to units, whether as suffix or prefix, other than standard SI prefixes. Therefore, even though the decibel is accepted for use alongside SI units, the practice of attaching a suffix to the basic dB unit, forming compound units such as dBm, dBu, dBA, etc., is not. The proper way, according to the IEC 60027-3, is either as Lx (re xref) or as Lx/xref, where x is the quantity symbol and xref is the value of the reference quantity, e.g., LE (re 1 μV/m) = LE/(1 μV/m) for the electric field strength E relative to 1 μV/m reference value.
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 dash (dB-Hz), with a space (dB HL), with no intervening character (dBm), or enclosed in parentheses, dB(sm).
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.
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: The measures of pressure (a field quantity) use the factor of 20, and the measures of power (e.g. dB SIL and dB SWL) use the factor of 10.
See also dBV and dBu above.
Np or cNp
Attenuation constants, in fields such as optical fiber communication and radio propagation path loss, are often expressed as a fraction or ratio to distance of transmission. dB/m means decibels per meter, dB/mi is decibels 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.
… the decibel represents a reduction in power of 1.258 times.
… a pressure ratio of 1.122 equals + 1.0 dB
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