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Decibel
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The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity) relative to a specified or implied reference level. Since it expresses a ratio of two quantities with the same unit, it is a dimensionless unit. A decibel is one tenth of a bel (B).
The decibel is useful for a wide variety of measurements in science and engineering (e.g., acoustics and electronics) and other disciplines.
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Encyclopedia
The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity) relative to a specified or implied reference level. Since it expresses a ratio of two quantities with the same unit, it is a dimensionless unit. A decibel is one tenth of a bel (B).
The decibel is useful for a wide variety of measurements in science and engineering (e.g., acoustics and electronics) and other disciplines. It confers a number of advantages, such as the ability to conveniently represent very large or small numbers, a logarithmic scaling that roughly corresponds to the human perception of, for example, sound and light, and the ability to carry out multiplication of ratios by simple addition and subtraction.
The decibel symbol is often qualified with a suffix, which indicates which reference quantity or frequency weighting function has been used. For example, "dBm" indicates that the reference quantity is one milliwatt, while "dBu" is referenced to 0.7746 volts RMS.
The definitions of the decibel and bel use base-10 logarithms. For a similar unit using natural logarithms to base e, see neper.
History
The transmission unit or TU was devised by engineers of the Bell Telephone Laboratories to quantify the reduction in audio level over a 1 mile (approximately 1.6 km) length of standard telephone cable. A mile of standard cable was defined as a certain reduction of power at a frequency of 5000 radians per second (796.4 Hz). Standard telephone cable was defined as "a cable having uniformly distributed resistances of 88 ohms per loop mile and uniformly distributed shunt capacitance of .054 microfarad per mile" (approximately 19 gauge). Note that an actual mile is 1.056 TU.. Eventually, the approximate attenuation of 10 miles (10 TU) was renamed the 'bel' in honor of the Bell System's founder and telecommunications pioneer Alexander Graham Bell. In many situations, however, the bel proved inconveniently large, so the decibel has become more common.
In April 2003, the International Committee for Weights and Measures (CIPM) considered a recommendation for the decibel's inclusion in the SI system, but decided not to adopt the decibel as an SI unit.
Definitions
Power
When referring to measurements of power or intensity, a ratio can be expressed in decibels by evaluating ten times the base-10 logarithm of the ratio of the measured quantity to the reference level. Thus, if L represents the ratio of a power value P1 to another power value P0, then LdB represents that ratio expressed in decibels and is calculated using the formula:
Naturally, P1 and P0 must have the same dimension (that is, must measure the same type of quantity), and must as necessary be converted to the same units before calculating the ratio of their numerical values. Note that if P1 = P0 in the above equation, then LdB = 0. If P1 is greater than P0 then LdB is positive; if P1 is less than P0 then LdB is negative.
Rearranging the above equation gives the following formula for P1 in terms of P0 and LdB:
.
Since a bel is equal to ten decibels, the corresponding formulae for measurement in bels (LB) are
.
Amplitude, voltage and current
When referring to measurements of amplitude it is usual to consider the ratio of the squares of A1 (measured amplitude) and A0 (reference amplitude). This is because in most applications power is proportional to the square of amplitude. 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 V1 is the voltage being measured, V0 is a specified reference voltage, and GdB is the power gain expressed in decibels. A similar formula holds for current.
Examples
Note that all of these examples yield dimensionless answers in dB because they are relative ratios expressed in decibels.
- To calculate the ratio of 1 kW (one kilowatt, or 1000 watts) to 1 W in decibels, use the formula
- To calculate the ratio of 1 mW (one milliwatt) to 10 W in decibels, use the formula
- To find the power ratio corresponding to a 3 dB change in level, use the formula
A change in power ratio by a factor of 10 is a 10 dB change. A change in power ratio by a factor of two is approximately a 3 dB change. (More precisely, the factor is 103/10, or 1.9953, about 0.24% different from exactly 2.) Similarly, an increase of 3 dB implies an increase in voltage by a factor of approximately v2, or about 1.41, an increase of 6 dB corresponds to approximately four times the power and twice the voltage, and so on. (In exact terms the power ratio is 106/10, or about 3.9811, a relative error of about 0.5%.)
Merits
The use of the decibel has a number of merits:
- The decibel's logarithmic nature means that a very large range of ratios can be represented by a convenient number, in a similar manner to scientific notation. This allows one to clearly visualize huge changes of some quantity. (See Bode Plot and half logarithm graph.)
- The mathematical properties of logarithms mean that the overall decibel gain of a multi-component system (such as consecutive amplifiers) can be calculated simply by summing the decibel gains of the individual components, rather than needing to multiply amplification factors. Essentially this is because log(A × B × C × ...) = log(A) + log(B) + log(C) + ...
- The human perception of, for example, sound or light, is, roughly speaking, such that a doubling of actual intensity causes perceived intensity to always increase by the same amount, irrespective of the original level. The decibel's logarithmic scale, in which a doubling of power or intensity always causes an increase of approximately 3 dB, corresponds to this perception.
Uses
Acoustics
The decibel is commonly used in acoustics to quantify sound levels relative to some 0 dB reference. The reference level is typically set at the threshold of perception of an average human and there are common comparisons used to illustrate different levels of sound pressure. As with other decibel figures, normally the ratio expressed is a power ratio (rather than a pressure ratio).
A reason for using the decibel is that the ear is capable of detecting a very large range of sound pressures. The ratio of the sound pressure that causes permanent damage from short exposure to the limit that (undamaged) ears can hear is above a million. Because the power in a sound wave is proportional to the square of the pressure, the ratio of the maximum power to the minimum power is above one (short scale) trillion. To deal with such a range, logarithmic units are useful: the log of a trillion is 12, so this ratio represents a difference of 120 dB. Since the human ear is not equally sensitive to all the frequencies of sound within the entire spectrum, noise levels at maximum human sensitivity — for example, the higher harmonics of middle A (between 2 and 4 kHz) — are factored more heavily into sound descriptions using a process called frequency weighting.
Electronics
In electronics, the decibel is often used to express power or amplitude ratios (gains), in preference to arithmetic ratios or percentages. One advantage is that the total decibel gain of a series of components (such as amplifiers and attenuators) can be calculated simply by summing the decibel gains of the individual components. Similarly, in telecommunications, decibels are used to account for the gains and losses of a signal from a transmitter to a receiver through some medium (free space, wave guides, coax, fiber optics, etc.) using a link budget.
The decibel unit can also be combined with a suffix to create an absolute unit of electric power. For example, it can be combined with "m" for "milliwatt" to produce the "dBm". Zero dBm is the power level corresponding to a power of one milliwatt, and 1 dBm is one decibel greater (about 1.259 mW).
In professional audio, a popular unit is the dBu (see below for all the units). The "u" stands for "unloaded", and was probably chosen to be similar to lowercase "v", as dBv was the older name for the same thing. It was changed to avoid confusion with dBV. This unit (dBu) is an RMS measurement of voltage which uses as its reference 0.775 VRMS. Chosen for historical reasons, it is the voltage level which delivers 1 mW of power in a 600 ohm resistor, which used to be the standard reference impedance in almost all professional low-impedance audio circuits.
The bel is used to represent noise power levels in hard drive specifications. It shares the same symbol (B) as the byte.
Optics
In an optical link, if a known amount of optical power, in dBm (referenced to 1 mW), is launched into a fiber, and the losses, in dB (decibels), of each electronic component (e.g., connectors, splices, and lengths of fiber) are known, the overall link loss may be quickly calculated by addition and subtraction of decibel quantities.
In spectrometry and optics, the blocking unit used to measure optical density is equivalent to -1 B. In astronomy, the apparent magnitude measures the brightness of a star logarithmically, since, just as the ear responds logarithmically to acoustic power, the eye responds logarithmically to brightness; however astronomical magnitudes reverse the sign with respect to the bel, so that the brightest stars have the lowest magnitudes, and the magnitude increases for fainter stars.
Common reference levels and corresponding units
"Absolute" and "relative" decibel measurements
Although decibel measurements are always relative to a reference level, if the numerical value of that reference is explicitly and exactly stated, then the decibel measurement is called an "absolute" measurement, in the sense that the exact value of the measured quantity can be recovered using the formula given earlier. For example, since dBm indicates power measurement relative to 1 milliwatt,
- 0 dBm means no change from 1 mW. Thus, 0 dBm is the power level corresponding to a power of exactly 1 mW.
- 3 dBm means 3 dB greater than 0 dBm. Thus, 3 dBm is the power level corresponding to 103/10 × 1 mW, or approximately 2 mW.
- -6 dBm means 6 dB less than 0 dBm. Thus, −6 dBm is the power level corresponding to 10−6/10 × 1 mW, or approximately 250 µW (0.25 mW).
If the numerical value of the reference is not explicitly stated, as in the dB gain of an amplifier, then the decibel measurement is purely relative. The practice of attaching a suffix to the basic dB unit, forming compound units such as dBm, dBu, dBA, etc, is not permitted by SI. However, outside of documents adhering to SI units, the practice is very common as illustrated by the following examples.
Absolute measurements
Electric power
dBm or dBmW
- dB(1 mW) — power measurement relative to 1 milliwatt. XdBm = XdBW + 30.
dBW
- dB(1 W) — similar to dBm, except the reference level is 1 watt. 0 dBW = +30 dBm; −30 dBW = 0 dBm; XdBW = XdBm − 30.
Voltage
Since the decibel is defined with respect to power, not amplitude, conversions of voltage ratios to decibels must square the amplitude, as discussed above.
dBV
- dB(1 VRMS) — voltage relative to 1 volt, regardless of impedance.
dBu or dBv
- dB(0.775 VRMS) — voltage relative to 0.775 volts. Originally dBv, it was changed to dBu to avoid confusion with dBV. The "v" comes from "volt", while "u" comes from "unloaded". dBu can be used regardless of impedance, but is derived from a 600 Ω load dissipating 0 dBm (1 mW). Compare ambiguous use of dBu in radio engineering.
dBmV
- dB(1 mVRMS) — voltage relative to 1 millivolt, regardless of impedance. Widely used in cable television networks, where the nominal strength of a single TV signal at the receiver terminals is about 0 dBmV. Cable TV uses 75 O coaxial cable, so 0 dBmV corresponds to −78.75 dBW (-48.75 dBm) or ~13 nW.
dBµV or dBuV
- dB(1 µVRMS) — voltage relative to 1 microvolt. Widely used in television and aerial amplifier specifications. 60 dBµV = 0 dBmV.
Acoustics
This usage is probably the most common in colloquial speech and a layperson referring to "decibels" in general is most likely to be referring to sound pressure level as referenced to human hearing:
dB(SPL)
- dB (Sound Pressure Level) — for sound in air and other gases, relative to 20 micropascals (µPa) = 2×10-5 Pa, the quietest sound a human can hear. This is roughly the sound of a mosquito flying 3 metres away. This is often abbreviated to just "dB", which gives some the erroneous notion that "dB" is an absolute unit by itself. For sound in water and other liquids, a reference pressure of 1 µPa is used.
dB SIL
- dB Sound Intensity Level — relative to 10-12 W/m2, which is roughly the threshold of human hearing in air.
dB SWL
- dB Sound Power Level — relative to 10-12 W.
dB(A), dB(B), and dB(C)
- These symbols are often used to denote the use of different weighting filters, used to approximate the human ear's response to sound, although the measurement is still in dB (SPL). These measurements usually refer to noise and noisome effects on humans and animals, and are in widespread use in the industry with regard to noise control issues, regulations and environmental standards. Other variations that may be seen are dBA or dBA. According to ANSI standards, the preferred usage is to write LA = x dB. Nevertheless, the units dBA and dB(A) are still commonly used as a shorthand for A-weighted measurements. Compare dBc, used in telecommunications.
dB HL or dB hearing level is used in audiograms as a measure of hearing loss. The reference level varies with frequency according to a Minimum audibility curve as defined in ANSI and other standards, such that the resulting audiogram shows deviation from what is regarded as 'normal' hearing.
dB Q is sometimes used to denote weighted noise level, commonly using the ITU-R 468 noise weighting
Radar
dBZ
- dB(Z) - energy of reflectivity (weather radar), or the amount of transmitted power returned to the radar receiver. Values above 15-20 dBZ usually indicate falling precipitation.
dBsm
- dBsm - decibel (referenced to one)square meter, measure of reflected energy from a target compared to the RCS of a smooth perfectly conducting sphere at least several wavelengths in size with a cross-sectional area of 1 square meter. "Stealth" aircraft and insects have negative values of dBsm, large flat plates or non-stealthy aircraft have positive values.
Radio power, energy, and field strength
dBc
- dBc — power relative to the power of the main carrier frequency; typically used to describe spurs, noise, channel crosstalk, and intermodal signals which may interfere with the carrier. Compare dB(C), used in acoustics.
dBJ
- dB(J) — energy relative to 1 joule. 1 joule = 1 watt per hertz, so power spectral density can be expressed in dBJ.
dBm
- dB(mW) — power relative to 1 milliwatt. When used in audio work the milliwatt is referenced to a 600 ohm load, with the resultant voltage being 0.775 volts. When used in the 2-way_radio field, the dB is referenced to a 50 ohm load, with the resultant voltage being 0.224 volts. There are times when spec sheets may show the voltage & power level e.g. -120 dBm = 0.224 microvolts.
dBµV/m or dBuV/m
- dB(µV/m) — electric field strength relative to 1 microvolt per meter. Compare the ambiguous use of dBu as a unit of voltage level.
dBf
- dB(fW) — power relative to 1 femtowatt.
dBW
- dB(W) — power relative to 1 watt.
dBk
- dB(kW) — power relative to 1 kilowatt.
Antenna measurements
dBi
- dB(isotropic) — the forward gain of an antenna compared with the hypothetical isotropic antenna, which uniformly distributes energy in all directions. Linear polarization of the EM field is assumed unless noted otherwise.
dBd
- dB(dipole) — the forward gain of an antenna compared with a half-wave dipole antenna. 0dBd = 2.15dBi
dBiC
- dB(isometric circular) — the forward gain of an antenna compared to a circularly polarized isotropic antenna. There is no fixed conversion rule between dBiC and dBi, as it depends on the receiving antenna and the field polarization.
dBq
- dB(quarterwave) — the forward gain of an antenna compared to a quarter wavelength whip. Rarely used, except in some marketing material. 0dBq = -0.85dBi
Other measurements
dBFS or dBfs
- dB(full scale) — the amplitude of a signal (usually audio) compared with the maximum which a device can handle before clipping occurs. In digital systems, 0 dBFS (peak) would equal the highest level (number) the processor is capable of representing. Measured peak values are always negative or zero, since they are less than the maximum. Measured rms values are ambiguous, and depend on whether you are using a sine or square wave's rms value as the reference.
dB-Hz
- dB(hertz) — bandwidth relative to 1 Hz. E.g., 20 dB-Hz corresponds to a bandwidth of 100 Hz. Commonly used in link budget calculations.
dBov or dBO
- dB(overload) — the amplitude of a signal (usually audio) compared with the maximum which a device can handle before clipping occurs. Similar to dBFS, but also applicable to analog systems.
dBr
- dB(relative) — simply a relative difference from something else, which is made apparent in context. The difference of a filter's response to nominal levels, for instance.
dBrn
- dB above reference noise. See also dBrnC.
dBc
- dB relative to carrier — in telecommunications, this indicates the relative levels of noise or sideband peak power, compared with the carrier power. Compare dBC, used in acoustics.
See also
Footnotes
External links
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