The
decibel (
dB) is a
logarithmic unit of measurement that
expresses the magnitude of a physical quantity (usually
power or
intensity) relative to a specified or
implied
reference level. Since it expresses a ratio of two
quantities with the same unit, it is a
dimensionless unit. A decibel is one
tenth of a
bel, a seldom-used unit.
The term decibel is most widely known as a measure of
sound pressure level, but is also useful for a wide variety of
measurements in science and
engineering
(particularly
acoustics,
electronics, and
control theory) and other disciplines. It
confers a number of advantages, such as the ability to conveniently
represent very large or small numbers, a logarithmic scaling that
roughly corresponds to the human perception of sound and light, and
the ability to carry out multiplication of ratios by simple
addition and subtraction.
The decibel symbol is often qualified with a suffix, which
indicates which reference quantity or
frequency weighting function has been used. For
example, "dBm" indicates that the reference quantity is one
milliwatt, while "dBu" is referenced to
0.775
volts RMS.
The definitions of the decibel and bel use base-10 logarithms. For
a similar unit using natural logarithms to base
e, see
neper.
History
The decibel originates from methods used to quantify reductions in
audio levels in telephone circuits. These losses were originally
measured in units of Miles of Standard Cable (MSC), where 1 MSC
corresponded to the loss of power over a 1
mile
(approximately 1.6 km) length of standard
telephone cable at a frequency of 5000
radians per second (795.8 Hz) and roughly matched
the smallest attenuation detectable to an average listener.
Standard telephone cable was defined as "a cable having uniformly
distributed resistances of 88 ohms per loop mile and uniformly
distributed
shunt capacitance of .054 microfarad per mile"
(approximately 19 gauge).
The
transmission unit or TU was devised by engineers
of the Bell Telephone
Laboratories in the 1920s to replace the MSC. 1 TU was
defined as the ten times the base-10 logarithm of the ratio of
measured power to reference power. The definitions were
conveniently chosen such that 1 TU approximately equalled 1 MSC
(specifically, 1.056 TU = 1 MSC). Eventually, international
standards bodies adopted the base-10 logarithm of the power ratio
as a standard unit, which was named the "bel" in honor of the
Bell System's founder and
telecommunications pioneer
Alexander Graham Bell. The bel was a
factor of ten larger than the TU, such that 1 TU equalled 1
decibel. In many situations, the bel proved inconveniently large,
so the decibel has become more common.
In April 2003, the
International
Committee for Weights and Measures (CIPM) considered a
recommendation for the decibel's inclusion in the SI system, but
decided not to adopt the decibel as an SI unit. However, the
decibel is recognized by other international bodies such as the
International
Electrotechnical Commission (IEC). The IEC permits the use of
the decibel with field quantities as well as power and this
recommendation is followed by many national standards bodies, such
as
NIST, which justifies the use of the decibel
for voltage ratios.
Definitions
Power
When referring to measurements of
power or
intensity, a ratio can be expressed in
decibels by evaluating ten times the base-10
logarithm of the ratio of the measured quantity to
the reference level. Thus, if
L represents the ratio of a
power value
P_{1} to another power value
P_{0}, then
L_{dB} represents
that ratio expressed in decibels and is calculated using the
formula:
L_\mathrm{dB} = 10 \log_{10} \bigg(\frac{P_1}{P_0}\bigg) \,
Naturally,
P_{1} and
P_{0} must
have the same
dimension (that is, must
measure the same type of quantity), and must as necessary, be
converted to the same units before calculating the ratio of their
numerical values: however, the choice of scale for this common unit
is irrelevant, as it changes both quantities by the same factor,
and thus cancels in the ratio (the ratio of two quantities is
scale-invariant). Note that if
P_{1} =
P_{0} in the above equation, then
L_{dB} = 0. If
P_{1} is greater
than
P_{0} then
L_{dB} is
positive; if
P_{1} is less than
P_{0} then
L_{dB} is
negative.
Rearranging the above equation gives the following formula for
P_{1} in terms of
P_{0} and
L_{dB}:P_1 = 10^\frac{L_\mathrm{dB}}{10} P_0
\,.
Since a bel is equal to ten decibels, the corresponding formulae
for measurement in bels (
L_{B}) areL_\mathrm{B} =
\log_{10} \bigg(\frac{P_1}{P_0}\bigg) \,
P_1 = 10^{L_\mathrm{B}} P_0 \,.
Amplitude, voltage and current
When referring to measurements of
amplitude it is usual to consider the ratio
of the squares of
A_{1} (measured amplitude) and
A_{0} (reference amplitude). This is because in
most applications power is proportional to the square of amplitude,
and it is desirable for the two decibel formulations to give the
same result in such typical cases. Thus the following definition is
used:L_\mathrm{dB} = 10 \log_{10} \bigg(\frac{A_1^2}{A_0^2}\bigg) =
20 \log_{10} \bigg(\frac{A_1}{A_0}\bigg). \,This formula is
sometimes called the
20 log rule, and similarly
the formula for ratios of powers is the
10 log
rule, and similarly for other factors. The factor of 20 is
explained as: 10 is because it is in decibels (10ths of bels), and
2 is because it is a ratio of powers (squares of amplitudes): the
product is 20. Thus provided that power ratios equal amplitude
ratios squared, the two definitions -- 10-log and 20-log rules --
yield the same result in decibels (L_\mathrm{dB}); bear in mind
though, that in some usages the equivalence condition is not
fulfilled and the equivalence does not hold as in, e.g., dBu and
dBV. Note also that no constant factor is needed for the power (one
can take power ratio to be the square of the amplitude ratio,
whatever the units), since any constant cancels in the ratio.
The formula may be rearranged to giveA_1 =
10^\frac{L_\mathrm{dB}}{20} A_0 \,
Similarly, in
electrical
circuits, dissipated power is typically proportional to the
square of
voltage or
current when the
impedance is held constant. Taking
voltage as an example, this leads to the equation:G_\mathrm{dB} =20
\log_{10} \left (\frac{V_1}{V_0} \right ) \quad \mathrm \quad
where
V_{1} is the voltage being measured,
V_{0} is a specified reference voltage, and
G_{dB} is the power gain expressed in decibels. A
similar formula holds for current.
Examples
Note that all of these examples yield dimensionless answers in dB
because they are relative ratios expressed in decibels.
- To calculate the ratio of 1 kW (one kilowatt, or 1000 watts) to
1 W in decibels, use the formula
G_\mathrm{dB} = 10 \log_{10} \bigg(\frac{1000 \mathrm{W}}{1
\mathrm{W}}\bigg) = 30 \mathrm{dB} \,
- To calculate the ratio of \sqrt(1000) \mathrm{V} \approx 31.62
\mathrm{V} to 1 \mathrm{V} in decibels, use the formula
G_\mathrm{dB} = 20 \log_{10} \bigg(\frac{31.62 \mathrm{V}}{1
\mathrm{V}}\bigg) = 30 \mathrm{dB} \,Notice that
({31.62\mathrm{V}}/{1\mathrm{V}})^2 \approx
{1\mathrm{kW}}/{1\mathrm{W}}, illustrating the consequence from the
definitions above that G_\mathrm{dB} has the same value, (30
\mathrm{dB}), regardless of whether it is obtained with the 10-log
or 20-log rules; provided that in the specific system being
considered power ratios are equal to amplitude ratios
squared.
- To calculate the ratio of 1 mW (one milliwatt) to 10 W in
decibels, use the formula
G_\mathrm{dB} = 10 \log_{10} \bigg(\frac{0.001 \mathrm{W}}{10
\mathrm{W}}\bigg) = -40 \mathrm{dB} \,
- To find the power ratio corresponding to a 3 dB change in
level, use the formula
G = 10^\frac{3}{10} \times 1\ = 1.99526... \approx 2 \,
- An example illustrating the subtleties of the 20-log vs. 10-log
rules is given by the so-called polarization ellipticity, the
minor-to-major-axis ratio of the polarization ellipse. It is an
amplitude ratio, thus when reported in decibels, it follows the
20-log rule, \alpha_\text{dB} = 20 \log_{10}{\alpha} = 10
\log_{10}{\alpha^2}. A reader unfamiliar with the applicability of
each 20- and 10-log rules might find \alpha_\text{dB} inconsistent,
arguing that it redefines the original \alpha as a power ratio
rather than an amplitude ratio. The crux is that the subscript dB
might imply either of two different mathematical operations, thus
it can only be interpreted unambiguously given additional
information about the nature of the quantity being reported (power
ratio or amplitude ratio).
A change in power ratio by a factor of 10 is a 10 dB change.
A change in power ratio by a factor of two is approximately a 3 dB
change.
(More precisely, the factor is 10
^{3/10}, or 1.9953, about
0.24% different from exactly 2.) Similarly, an increase of 3 dB
implies an increase in voltage by a factor of approximately
\scriptstyle\sqrt{2}, or about 1.41, an increase of 6 dB
corresponds to approximately four times the power and twice the
voltage, and so on.
(In exact terms the power ratio is 10
^{6/10}, or about
3.9811, a relative error of about 0.5%.)
Merits
The use of the decibel has a number of merits:
- The decibel's logarithmic nature means
that a very large range of ratios can be represented by a
convenient number, in a similar manner to scientific notation. This allows one to
clearly visualize huge changes of some quantity. (See Bode Plot and half logarithm graph.)
- The mathematical properties of logarithms mean that the overall
decibel gain of a multi-component system (such as consecutive
amplifiers) can be calculated simply by
summing the decibel gains of the individual components, rather than
needing to multiply amplification factors. Essentially this is
because log(A × B × C × ...) = log(A)
+ log(B) + log(C) + ...
- The human perception of, for example, sound or light, is,
roughly speaking, such that a doubling of actual intensity causes
perceived intensity to always increase by the same amount,
irrespective of the original level. The decibel's logarithmic scale, in which a doubling of
power or intensity always causes an increase of approximately 3 dB,
corresponds to this perception.
Uses
Acoustics
The decibel is commonly used in
acoustics
to quantify
sound levels relative to some 0 dB
reference. The reference level is typically set at the threshold of
perception of an average human and there are
common comparisons used to illustrate different levels of sound
pressure. As with other decibel figures, normally the ratio
expressed is a power ratio (rather than a pressure ratio).
A reason for using the decibel is that the ear is capable of
detecting a very large range of
sound
pressures. The ratio of the sound
pressure that causes
permanent damage during short exposure to the lower limit that
(undamaged) ears can hear is above a
million. Because the
power in a sound wave
is proportional to the
square of the pressure, the ratio
of the maximum power to the minimum power is above one (
short scale)
trillion. To deal with such a range,
logarithmic units are useful: the log of a trillion is 12, so this
ratio represents a difference of 120 dB. Since the human ear is not
equally sensitive to all the frequencies of sound within the entire
spectrum, noise levels at maximum human sensitivity — for example,
the higher
harmonics of middle
A (between 2 and 4
kHz) — are factored more heavily into sound
descriptions using a process called
frequency weighting.
Electronics
In electronics, the decibel is often used to express power or
amplitude ratios (
gains), in preference to
arithmetic ratios or
percentages. One advantage is that the total decibel
gain of a series of components (such as
amplifiers and
attenuators) can be calculated simply by summing
the decibel gains of the individual components. Similarly, in
telecommunications, decibels are used to account for the gains and
losses of a signal from a transmitter to a receiver through some
medium (
free space,
wave guides,
coax,
fiber optics, etc.) using a
link budget.
The decibel unit can also be combined with a suffix to create an
absolute unit of electric power. For example, it can be combined
with "m" for "milliwatt" to produce the "
dBm".
Zero dBm is the power level corresponding to a power of one
milliwatt, and 1 dBm is one decibel greater (about 1.259 mW).
In professional audio, a popular unit is the dBu (see below for all
the units). The "u" stands for "unloaded", and was probably chosen
to be similar to lowercase "v", as dBv was the older name for the
same thing. It was changed to avoid confusion with dBV. This unit
(dBu) is an
RMS measurement of
voltage which uses as its reference 0.775 V
_{RMS}. Chosen
for historical reasons, it is the voltage level which delivers
1 mW of power in a 600 ohm resistor, which used to be the
standard reference impedance in telephone audio circuits.
The bel is used to represent noise power levels in
hard drive specifications. It shares the same
symbol (
B) as the
byte.
Optics
In an
optical link, if a known amount
of
optical power, in
dBm
(referenced to 1 mW), is launched into a
fiber, and the losses, in dB (decibels), of
each
electronic component
(e.g., connectors, splices, and lengths of fiber) are known, the
overall link loss may be quickly calculated by addition and
subtraction of decibel quantities.
In spectrometry and optics, the
blocking
unit used to measure
optical
density is equivalent to −1 B. In astronomy, the
apparent magnitude measures the
brightness of a star logarithmically, since, just as the ear
responds logarithmically to acoustic power, the eye responds
logarithmically to brightness; however astronomical magnitudes
reverse the sign with respect to the bel, so that the brightest
stars have the
lowest magnitudes, and the magnitude
increases for
fainter stars.
Video and digital imaging
In connection with digital and video
image
sensors, decibels generally represent ratios of video voltages
or digitized light levels, using 20 log of the ratio, even when the
represented optical power is directly proportional to the voltage
or level, not to its square. Thus, a camera
signal-to-noise ratio of 60 dB
represents a power ratio of 1000:1 between signal power and noise
power, not 1,000,000:1.
Common reference levels and corresponding units
"Absolute" and "relative" decibel measurements
Although decibel measurements are always relative to a reference
level, if the numerical value of that reference is explicitly and
exactly stated, then the decibel measurement is called an
"absolute" measurement, in the sense that the exact value of the
measured quantity can be recovered using the formula given earlier.
For example, since dBm indicates power measurement relative to 1
milliwatt,
- 0 dBm means no change from 1 mW. Thus, 0 dBm is the power level
corresponding to a power of exactly 1 mW.
- 3 dBm means 3 dB greater than 0 dBm. Thus, 3 dBm is the power
level corresponding to 10^{3/10} × 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.
X_{dBm} = X_{dBW} + 30.
dBW
- dB(1 W) — similar to dBm, except the reference level is 1
watt. 0 dBW = +30 dBm; −30 dBW = 0 dBm;
X_{dBW} = X_{dBm} − 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 V_{RMS}) —
voltage relative to 1 volt, regardless of
impedance. Analog Devices : Virtual Design Center :
Interactive Design Tools : Utilities : V_{RMS} / dBm / dBu
/ dBV calculator
dBu or
dBv
- dB(0.775 V_{RMS}) —
voltage relative to 0.775 volts. Originally
dBv, it was changed to dBu to avoid confusion with dBV. The "v"
comes from "volt", while "u" comes from "unloaded". dBu can be used
regardless of impedance, but is derived from a 600 Ω load
dissipating 0 dBm (1 mW). Reference voltage V = \sqrt{600
\, \Omega \cdot 0.001\,\mathrm W}\, \approx 0.7746\,\mathrm V
dBmV
- dB(1 mV_{RMS}) —
voltage relative to 1 millivolt, regardless of
impedance. Widely used in cable
television networks, where the nominal strength of a single TV
signal at the receiver terminals is about 0 dBmV. Cable TV uses 75
Ω coaxial cable, so 0 dBmV corresponds to −78.75 dBW (-48.75 dBm)
or ~13 nW.
dBμV or
dBuV
- dB(1 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
loudness is dB SPL, referenced to the nominal threshold of human
hearing:
dB(SPL)
- dB (sound pressure level) —
for sound in air and other gases, relative to 20 micropascals (μPa)
= 2×10^{−5} Pa, the quietest sound a human can hear. This
is roughly the sound of a mosquito flying 3 metres away. This is
often abbreviated to just "dB", which gives some the erroneous
notion that "dB" is an absolute unit by itself. For sound in water and other liquids, a
reference pressure of 1 μPa is used.
dB SIL
- dB sound intensity level —
relative to 10^{−12} W/m^{2}, 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 dB_{A} or dBA. According to ANSI standards, 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 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 — relative to carrier — in telecommunications, this indicates the
relative levels of noise or sideband peak power, compared with the
carrier power. Compare dBC, used in acoustics.
dBJ
- dB(J) — energy relative to 1 joule. 1
joule = 1 watt per hertz, so power spectral density can be
expressed in dBJ.
dBm
- dB(mW) — power relative to 1 milliwatt. When used in audio work the milliwatt
is referenced to a 600 ohm load, with the resultant voltage being
0.775 volts. When used in the 2-way
radio field, the dB is referenced to a 50 ohm load, with the
resultant voltage being 0.224 volts. There are times when spec
sheets may show the voltage & power level e.g. -120 dBm = 0.224
microvolts.
dBμV/m or
dBuV/m
- dB(μV/m) — electric field
strength relative to 1 microvolt per
meter.
dBf
- dB(fW) — power relative to 1 femtowatt.
dBW
- dB(W) — power relative to 1 watt.
dBk
- dB(kW) — power relative to 1 kilowatt.
Antenna measurements
dBi
- dB(isotropic) — the forward gain of an antenna compared with
the hypothetical isotropic
antenna, which uniformly distributes energy in all directions.
Linear polarization of the EM
field is assumed unless noted otherwise.
dBd
- dB(dipole) — the forward gain of an antenna compared with a half-wave
dipole antenna. 0dBd = 2.15dBi
dBiC
- dB(isotropic circular) — the forward gain of an antenna
compared to a circularly
polarized isotropic antenna. There is no fixed conversion rule
between dBiC and dBi, as it depends on the receiving antenna and
the field polarization.
dBq
- dB(quarterwave) — the forward gain of an antenna compared to a
quarter wavelength whip. Rarely used, except in some marketing
material. 0dBq = -0.85dBi
Other measurements
dBFS or
dBfs
- dB(full scale) — the amplitude of a signal (usually audio) compared
with the maximum which a device can handle before clipping occurs. In digital
systems, 0 dBFS (peak) would equal the highest level (number) the
processor is capable of representing. Measured values are always
negative or zero, since they are less than the maximum or
full-scale. Full-scale is typically defined as the power level of a
full-scale sinusoid, though some systems will have extra headroom
for peaks above the nominal full scale.
dB-Hz
- dB(hertz) — bandwidth relative to 1 Hz.
E.g., 20 dB-Hz corresponds to a bandwidth of 100 Hz. Commonly used
in link budget calculations. Also used
in carrier-to-noise-density
ratio (not to be confused with carrier-to-noise ratio, in dB).
dBov or
dBO
- dB(overload) — the amplitude of a signal (usually audio) compared
with the maximum which a device can handle before clipping occurs. Similar to
dBFS, but also applicable to analog systems.
dBr
- dB(relative) — simply a relative difference from something
else, which is made apparent in context. The difference of a
filter's response to nominal levels, for instance.
dBrn
- dB above reference noise. See
also dBrnC.
See also
Footnotes
- Sound system engineering, p. 35,
Carolyn Davis, 1997
- "Transmission Circuits for Telephonic Communication", Bell
Labs, 1925
- 100 Years of Telephone Switching, p.
276, Robert J. Chapuis, Amos E. Joel, 2003
- Consultative Committee for Units, Meeting
minutes, Section 3
- "Letter symbols to be used in electrical technology
- Part 3: Logarithmic and related quantities, and their units",
IEC 60027-3 Ed. 3.0, International Electrotechnical
Commission, 19th July 2002.
- A. Thompson and B. N. Taylor, "Comments on Some Quantities and Their Units", The
NIST Guide for the use of the International System of Units,
National Institute of Standards and Technology, May 1996.
- Taylor 1995, SP811
- What is the difference between dBv, dBu, dBV, dBm,
dB SPL, and plain old dB? Why not just use regular voltage and
power measurements? - rec.audio.pro Audio Professional
FAQ
- Morfey, C. L. (2001). Dictionary of Acoustics. Academic Press,
San Diego.
References
External links
- What is a decibel? With sound
files and animations
- Conversion of dBu to volts, dBV to volts, and volts
to dBu, and dBV
- Working with decibels - a tutorial
- Conversion of sound level units: dBSPL or dBA to sound
pressure p and sound intensity J
- Conversion of voltage V to dB, dBu, dBV, and
dBm
- OSHA Regulations on Occupational Noise
Exposure
- V_{peak}, V_{RMS}, Power, dBm, dBu, dBV
online converter at Analog Devices
- Use of the decibel with respect to aerials and aerial
systems