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ΙΝΝΝΝΝΝΝΝΝΝΝΝΝ»
Ί dB Notation Ί
ΘΝΝΝΝΝΝΝΝΝΝΝΝΝΌ
The decibel (dB) is a logarithmic unit of measurement that expresses
the magnitude of a physical quantity (usually power) relative to a specified
or implied reference level. Its logarithmic nature allows very large or very
small ratios to be represented by a convenient number, in a similar manner to
scientific notation. Being essentially a ratio, it is a dimensionless unit.
Decibels are useful for a wide variety of measurements in acoustics, physics,
electronics and other disciplines because it linearizes a physical value
-- e.g. light intensity or level of noise -- whose variation is exponential,
but perceived as linear (in fact, as a logarithm of the original) by humans.
The decibel isn't an SI unit, although the International Committee for
Weights and Measures (CIPM) has recommended its inclusion in the SI system.
Following the SI convention, the d is lowercase, as it represents the SI
prefix deci, and the B is capitalized, as it is an abbreviation of a name
derived unit, the bel (see below). The full name decibel follows the usual
English capitalization rules for a common noun. The decibel symbol is often
qualified with a suffix, which indicates which reference quantity has been
assumed. For example, "dBm" indicates that the reference quantity is one
milliwatt. The practice of attaching a suffix in this way, though not
permitted by SI, is widely followed.
A decibel is one tenth of a bel (B). Devised by engineers of the Bell
Telephone Laboratory quantify the reduction in audio level over a 1 mile
(approximately 1.6 km) length of standard telephone cable, the bel was
originally called the transmission unit or TU, but was renamed by 1924 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.
The definitions of the decibel and bel use base-10 logarithms. For a
similar unit using natural logarithms to base e, see neper. An increase of 3
dB corresponds to an approximate doubling of power. (In exact terms, the
factor is 103/10, or 1.9953, about 0.24% different from exactly 2.) Since in
many electrical applications power is proportional to the square of voltage,
an increase of 3 dB implies an increase in voltage by a factor of
approximately ϋ2, or about 1.41. Similarly, an increase of 6 dB corresponds to
approximately four times the power and twice the voltage, and so on. (In
exact terms the power factor is 106/10, or about 3.9811, a relative error of
about 0.5%.) See the formulas below for further details.
Contents:
ΝΝΝΝΝΝΝΝΝ
1) Definitions:
1.1) Power.
1.2) Amplitude, voltage and current.
1.3) Examples
2) Merits
3) Uses
3.1) Acoustics
3.2) Electronics
3.3) Optics
4) Common reference levels and corresponding units
4.1) "Absolute" and "relative" decibel measurements
4.2) Absolute measurements
4.2.1) Electric power
4.2.2) Voltage
4.2.3) Acoustics
4.2.4) Radio power or energy
4.3) Relative measurements
5) Reckoning
5.1) Round numbers
5.2) The 4 6 energy rule
5.3) The "789" rule
5.4) -3 dB ~ « power
5.5) 6 dB per bit
5.6) dB chart
5.6.1) Commonly used dB values
1) Definitions
ΔΔΔΔΔΔΔΔΔΔΔΔΔΔ
1.1) 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, X[dB] is calculated
using the formula:
X[dB]= 10 log (X / Xo) [1]
where X is the actual value of the quantity being measured, Xo is a specified
or implied reference level, and then X[dB] is the quantity expressed in units
of decibels, relative to Xo. Which reference is used depends on convention and
context (see later in this article). X and Xo must have the same dimensions
(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. The reference level itself is always at 0 dB, as shown by setting
X = Xo in the above equation. If X is greater than Xo then XdB is positive;
if X is less than Xo then XdB is negative.
Rearranging the above equation gives the following formula for X in
terms of Xo and X[dB]:
X = Xo * { 10 Exp (x[dB] / 10 )} [2]
1.2) Amplitude, voltage and current.
When referring to measurements of amplitude it is usual to consider the
ratio of the squares of X (measured amplitude) and Xo (reference amplitude).
This is because in most applications power is proportional to the square of
amplitude. Thus the following definition is used:
X[dB]= 10 log (Xύ / Xoύ)
X[dB]= 20 log (X / Xo) [3]
Similarly, in electrical circuits, dissipated power is typically
proportional to the square of voltage or current:
P[W] = E[V]ύ / R[κ] = I[A]ύ * R[κ] [4]
when the impedance or resitance is held constant. Taking voltage as an
example, from [3] and [4] this leads to the equation:
P[dB] = 10 log (P / Po)
= 10 log (Eύ/R / Eoύ / R)
V[dB] = 10 log (Eύ / Eoύ)
= 20 log (E / Eo) [5]
where V is the voltage being measured, Vo is a specified reference voltage,
and V[dB] is the voltage gain expressed in decibels. A similar formula holds
for current.
1.3) Examples
These examples assume that X in the formulas above measures power
relative to 1 W (one watt); i.e. Xo = 1 W.
a) To convert 1 kW (one kilowatt, or 1000 W) to decibels, use the formula [1]:
X[dB]= 10 log (X / Xo)
X[dB]= 10 log (1000 / 1)
X[dB]= 10 log (1000)
X[dB]= 10 * 3
X[dB]= +30 dB
b) To convert 1 mW (one milliwatt, or .001 W) to decibels, use the same
formula:
X[dB]= 10 log (X / Xo)
X[dB]= 10 log (.001 / 1)
X[dB]= 10 log (.001)
X[dB]= 10 * (-3)
X[dB]= -30 dB
It will be seen that there is a 10 dB increase (decrease) for each
factor 10 increase (decrease) in the ratio of X to Xo, and approximately a 3
dB increase (decrease) for every factor 2 increase (decrease).
2) Merits
The use of decibels has a number of merits:
* The mathematical laws of exponents mean that the overall decibel gain of a
multi-component system (such as consecutive amplifiers and/or attenuators)
can be calculated simply by summing the decibel gains or loss of the
individual components, rather than needing to multiply amplification or
attenuation factors.
* A very large range of ratios can be expressed with decibel values in a range
of moderate size, allowing one to clearly visualize huge changes of some
quantity. (For example: the Bode plot and half logarithm graph.)
In acoustics, the decibel scale approximates the human perception of loudness
(which is itself roughly logarithmic).
3) Uses
3.1) 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.
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 trillion (1,000,000,000,000). 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 - middle A and its higher harmonics
(between 2 and 4 kHz) - are factored more heavily into sound descriptions
using a process called frequency weighting.
3.2) Electronics
The decibel is used rather than arithmetic ratios or percentages
because when certain types of circuits, such as amplifiers and attenuators,
are connected in series, expressions of power level in decibels may be
arithmetically added and subtracted. It is also common in disciplines such as
audio, in which the properties of the signal are best expressed in logarithms
due to the response of the human ear.
In radio electronics and telecommunications, the decibel is used to
describe the ratio between two measurements of electric power. It 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 one milliwatt, and 1 dBm is one decibel greater than 0 dBm, or
about 1.259 mW.
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.
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 at which you get 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.
3.3) Optics
In an optical link, if a known amount of optical power, in dBm
(referenced to 1 mW), is launched into a fibre, and the losses, in dB
(decibels), of each electronic component (e.g., connectors, splices, and
lengths of fibre) are known, the overall link loss may be quickly calculated
by simple 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.
4) Common reference levels and corresponding units
4.1) "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 formulas given earlier. For example, since dBm indicates power measurement
relative to 1 milliwatt,
0 dBm means no change from 1 mW, in other words 0 dBm is 1 mW.
3 dBm means 3 dB greater than 1 mW. 3 dBm is (10E(3/10)) * 1 mW, or
approximately 2 mW.
-6 dBm means 6 dB less than 1 mW. -6 dBm is (10E(-6/10)) * 1 mW, or
approximately 250 ζW (0.25 mW).
If the numerical value of the reference isn't 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, isn't permitted by SI. However, the practice is
very common, as illustrated by the following examples.
4.2) Absolute measurements
4.1.1) Electric power
dBm or dBmW
dB(1 mW) is power measurement relative to 1 milliwatt. XdBm = XdBW + 30.
dBW
dB(1 W) is similar to dBm, except the reference level is 1 watt.
0 dBW = +30 dBm; -30 dBW = 0 dBm; XdBW = XdBm - 30.
4.1.2) Voltage
Note that the decibel has a different definition when applied to voltage (as
contrasted with power). See the "Definitions" section above.
dBu or dBv:
dB(0.775 VRMS is the 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 κ load dissipating 0 dBm (1 mW).
dBV
dB(1 VRMS is the voltage relative to 1 volt, regardless of impedance.
dBmV
dB(1 mVRMS is the 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 Ohms coaxial cable, so 0 dBmV corresponds to -48.75 dBm or ~13 nW.
4.1.3) Acoustics
dB(SPL)
dB (Sound Pressure Level) is for sound in air and other gases, relative to 20
micropascals (ζPa) = 2 * 10E-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 ζPa is used.
dB SIL
dB Sound Intensity Level is relative to 10E-12 W/m2, which is roughly the
threshold of humal is relative to 10E-12 W.
dB(A), dB(B), and dB(C)
These symbols are often used to denote the use of different frequency
weightings, used to approximate the human ear's response to sound, although
the measurement is still in dB (SPL). Other variations that may be seen are
dBA or dBA. According to ANSI standards, the preferred usage is to write
LA = x dB. Nevetheless, the units dBA and dB(A) are still commonly used as a
shorthand for A-weighted measurements. Compare dBc, used in
telecommunications.
4.2.4) Radio power or energy
dBJ
dB(J) is the energy relative to 1 joule. 1 joule = 1 watt-second, so noise
spectral density can be expressed in dBJ, where 0 dBJ = 0 dBW/Hz. Boltzmann's
constant is -228.6 dBKJ or dBW/KHz.
dBm
dB(mW) is the power relative to 1 milliwatt.
dBζ or dBu
dB(ζV/m) is the electric field strength relative to 1 microvolt per metre.
dBf
dB(fW) is the power relative to 1 femtowatt.
dBW
dB(W) is the power relative to 1 watt.
dBk
dB(kW) is the power relative to 1 kilowatt.
4.3) Relative measurements
dBd
dB(dipole) is the the forward gain of an antenna compared to a half-wave
dipole antenna.
dBFS or dBfs
dB(full scale) is the the amplitude of a signal (usually audio) compared to
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 usually negative, since they
should be less than the maximum.
dB-Hz
dB(Hertz) The Bandwidth relative to 1 Hz. E.g., 20 dB-Hz is equal to 100 Hz.
Commonly used in link budget calculations.
dBi
dB(isotropic) is the the forward gain of an antenna compared to a fictitious
isotropic antenna, which uniformly distributes energy in all directions.
dBiC
dB(isometric circular) is the power measurement relative to a circularly
polarized isometric antenna.
dBov or dBO
dB(overload) is the the amplitude of a signal (usually audio) compared to the
maximum which a device can handle before clipping occurs. Similar to dBFS,
but also applicable to analog systems.
dBr
dB(relative) is the simply a relative difference to 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
evels of noise or sideband peak power, compared to the carrier power. Compare
dBC, used in acoustics.
5) Reckoning
Decibels are handy for mental calculation, because adding them is
easier than multiplying ratios. First, however, one has to be able to convert
easily between ratios and decibels. The most obvious way is to memorize the
logs of small primes, but there are a few other tricks that can help.
The values of coins and banknotes are roud number: 10, 100, etc.
- Half a round number is a round number: 50, 25, 12.5, 6.25, etc.
- The tenth of a round number is a round number: 5, 2.5, 1.25, 1.6, 3.2, 6.4,
etc.
- Now 6.25 and 6.4 are approximately equal to 6.3, so we don't care. Thus the
round numbers between 1 and 10 are these:
Ratio ³ 1 ³ 1.25 ³ 1.6 ³ 2 ³ 2.5 ³ 3.2 ³ 4 ³ 5 ³ 6.3 ³ 8 ³ 10
ΔΔΔΔΔΔΕΔΔΔΕΔΔΔΔΔΔΕΔΔΔΔΔΕΔΔΔΔΕΔΔΔΔΔΕΔΔΔΔΔΕΔΔΔΕΔΔΔΕΔΔΔΔΔΕΔΔΔΕΔΔΔΔΔ
dB ³ 0 ³ 1 ³ 2 ³ 3 ³ 4 ³ 5 ³ 6 ³ 7 ³ 8 ³ 9 ³ 10
This useful approximate table of logarithms is easily reconstructed or
memorized.
5.2) The 4 6 energy rule
To one decimal place of precision, 4.x is 6.x in dB (energy).
Examples:
10 log(4.0) 6.0 dB
10 log(4.3) 6.3 dB
10 log(4.7) 6.7 dB
10 log(4.9) 6.9 dB
5.3) The "789" rule
To one decimal place of precision, x ? (« x + 5.0 dB) for 7.0 = x = 10.
Examples:
10 log10(7.0) ? « 7.0 + 5.0 dB = 3.5 + 5.0 dB = 8.5 dB
10 log10(7.5) ? « 7.5 + 5.0 dB = 3.75 + 5.0 dB = 8.75 dB
10 log10(8.2) ? « 8.2 + 5.0 dB = 4.1 + 5.0 dB = 9.1 dB
10 log10(9.9) ? « 9.9 + 5.0 dB = 4.95 + 5.0 dB = 9.95 dB
10 log10(10.0) ? « 10.0 + 5.0 dB = 5.0 + 5.0 dB = 10 dB
5.4) -3 dB ~ « power
A level difference of ρ3 dB is roughly double/half power (equal to a
ratio of 1.995). That is why it is commonly used as a marking on sound
equipment and the like.
Another common sequence is 1, 2, 5, 10, 20, 50 ... . These preferred
numbers are very close to being equally spaced in terms of their logarithms.
The actual values would be 1, 2.15, 4.64, 10 ... .
The conversion for decibels is often simplified to: "+3 dB means two
times the power and 1.414 times the voltage", and "+6 dB means four times the
power and two times the voltage ".
While this is accurate for many situations, it isn't exact. As stated
above, decibels are defined so that +10 dB means "ten times the power". From
this, we calculate that +3 dB actually multiplies the power by 10E(3/10).
This is a power ratio of 1.9953 or about 0.25% different from the "times 2"
power ratio that is sometimes assumed. A level difference of +6 dB is 3.9811,
about 0.5% different from 4.
To contrive a more serious example, consider converting a large decibel
figure into its linear ratio, for example 120 dB. The power ratio is correctly
calculated as a ratio of 10E12 or one trillion. But if we use the assumption
that 3 dB means "times 2", we would calculate a power ratio of 2(120/3)
= 240 = 1.0995 * 10E12, giving a 10% error. (NOTE: read 240 as 2 to 40th
power)
5.5) 6 dB per bit
In digital audio linear pulse-code modulation (PCM), the first bit
(least significant bit, or LSB) produces residual quantization noise (bearing
little resemblance to the source signal) and each subsequent bit offered by
the system doubles the (voltage) resolution, corresponding to a 6 dB (power)
ratio. So for instance, a 16-bit (linear) audio format offers 15 bits beyond
the first, for a dynamic range (between quantization noise and clipping) of
(15 * 6) = 90 dB, meaning that the maximum signal (see 0 dBFS, above) is 90 dB
above the theoretical peak(s) of quantization noise. The negative impacts of
quantization noise can be reduced by implementing "dither".
5.4) dB chart
As is clear from the above description, the dB level is a logarithmic
way of expressing not only power ratios, but also voltage ratios. The
following tables are cheat-sheets that provide values for various dB power
ratios and also "voltage" ratios.
5.4.1) Commonly used dB values
dB level power ratio dB level voltage ratio
-30 dB 1/1000 = .001 -30 dB .03162
-20 dB 1/100 = .01 -20 dB .1
-10 dB 1/10 = .1 -10 dB .3162
-3 dB 1/2 = .5 -3 dB .7071
3 dB 2 3 dB 1.414
10 dB 10 10 dB 3.162
20 dB 100 20 dB 10
30 dB 1000 30 dB 31.62
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Ί Compilled from Wikipedia.com . Translatted to ASCII by LW1DSE Osvaldo Ί
Ί F. Zappitor (edit.com) in my AMD's 80486. Ί
Ί November 08, 2007 Ί
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