Decibel

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For other uses, see Decibel (disambiguation).

The decibel (dB) method of calculation , that uses a logarithm to allow very large or very small relations to be represented with a conveniently small number (similar to scientific notation). It is a dimensionless unit of ratio which is used compare two usually-variable quantities. When used to compare a variable quantity to a known reference quantity the measurement is qualified with a suffix such as with dBm where the reference quantity is one milliwatt. Decibels are useful for a wide variety of measurements in acoustics, physics, electronics and other disciplines.

The decibel is not 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 is the SI prefix deci-, and the B is capitalized, as it is an abbreviation of a name-derived unit, the bel, named for Alexander Graham Bell. Written out it becomes decibel. This is standard English capitalization.

[edit] History

The bel (symbol B) is mostly used in telecommunication, electronics, and acoustics. Invented by engineers of the Bell Telephone Laboratory to quantify the reduction in audio level over a 1 mile (1.6 km) length of standard telephone cable, it was originally called the transmission unit or TU, but was renamed in 1923 or 1924 in honor of the laboratory's founder and telecommunications pioneer Alexander Graham Bell.

The bel was too large for everyday use, so the decibel (dB), equal to 0.1 bel (B), became more commonly used. The bel is still used to represent noise power levels in hard drive specifications, for instance. The Richter scale uses numbers expressed in bels as well, though they are not labeled with a unit. 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 stars logarithmically, since just as the ear responds logarithmically to acoustic power, the eye responds logarithmically to brightness.

[edit] Definition

A decibel is defined in two equivalent ways.

When referring to measurements of power or intensity it is:

$X_\mathrm{dB} = 10 \log_{10} \bigg(\frac{X}{X_0}\bigg) \$

But when referring to measurements of amplitude it is:

$X_\mathrm{dB} = 20 \log_{10} \bigg(\frac{X}{X_0}\bigg) \$

where X₀ is a specified reference with the same units as X. Which reference is used depends on convention and context. When the impedance is held constant, the power is proportional to the square of the amplitude of either voltage or current, and so the above two definitions become consistent. ( See neper for a similar unit using natural logarithms to base e.)

In electrical circuits, the dissipated power is typically proportional to the square of the voltage V. Substituting a measured voltage and a reference voltage and rearranging terms leads to the following equations using a multiplier of 20 for voltage:

$V_\mathrm{dB} =20 \log_{10} \left (\frac{V_1}{V_0} \right ) \quad \mathrm \quad$

where V₀ is a specified reference voltage. This means a 20 dB increase for every factor 10 increase in the voltage ratio, or approximately 6 dB increase for every factor 2.

The ratio between a measured and reference current can be expressed in the same way.

[edit] Examples

As examples, if P_dB is 10 dB greater than P_dB0, then P is ten times P₀. If P_dB is 3 dB greater, the power ratio is very close to a factor of two $(10^{3 \over 10} = 1.99526)$ .

For sound intensity, I₀ is typically chosen to be 10⁻¹² W/m², which is roughly the threshold of human hearing in air. When this choice is made, the units are said to be "dB SIL". For sound power, P₀ is typically chosen to be 10⁻¹² W, and the units are then "dB SWL".

[edit] Merits

The use of decibels has a number of merits:

It is more convenient to add the decibel values of, for instance, two consecutive amplifiers rather than to multiply their amplification 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. (See Bode Plot and half logarithm graph.)
In acoustics, the decibel scale was adopted for measuring sound intensity, which approximates the perception of loudness.

[edit] Uses

[edit] Acoustics

Main article: Acoustics

The decibel unit is commonly used in acoustics to quantify sound levels relative to some 0 dB reference. See Sound#Examples of sound pressure and sound pressure levels. 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 — middle A and its higher harmonics (between 2 and 4 kHz) — are factored more heavily into sound descriptions using a process called frequency weighting.

[edit] 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 ear.

In radio electronics and telecommunications, the decibel is used to describe the ratio between two measurements of electrical power. It can also be combined with a suffix to create an absolute unit of electrical 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 V_RMS. 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 circuit.

Since there may be many different bases for a measurement expressed in decibels, a dB value is considered an absolute measurement only if the reference value (equivalent to 0 dB) is clearly stated. For example, the gain of an antenna system can only be given with respect to a reference antenna (often a theoretical a perfect isotropic antenna), in which case the measurement is expressed as dBi. If the reference is not stated, as in simply "dB", the value is a relative measurement, such as the gain of an amplifier.

[edit] 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.

[edit] Seismology

Earthquakes were formerly measured on the Richter scale, which is expressed in bels. (The units in this case are always assumed, rather than explicit.) The more modern moment magnitude scale is designed to produce values comparable to those of the Richter scale.

[edit] Typical abbreviations

[edit] Absolute measurements

The term "measurement relative to" means so many dB greater than or less than the quantity specified. Some examples:

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 2 mW.
−6 dBm means 6 dB less than 1 mW. -6 dBm is 250 μW (0.25 mW).

[edit] Electric power

dBm or dBmW

dB(1 mW) — power measurement relative to 1 milliwatt.

dBW

dB(1 W) — similar to dBm, except the reference level is 1 watt. 0 dBW = +30 dBm; -30 dBW = 0 dBm.

[edit] Electric voltage

A schematic showing the relationship between dBu (the voltage source) and dBm (the power dissipated as heat by the 600 Ω resistor)

dBu or dBv

dB(0.775 V_RMS) — voltage relative to 0.775 volts.^[1] Originally dBv, it was changed to dBu to avoid confusion with dBV.^{[citation needed]} 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).

dBV

dB(1 V_RMS) — voltage relative to 1 volt, regardless of impedance.^[2]

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 -48.75 dBm.

[edit] Acoustics

dB(SPL)

dB(Sound Pressure Level) — for sound in air, relative to 20 micropascals (μPa) = 2×10⁻⁵ Pa, the quietest sound a human can hear.^[3] 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.

[edit] Radio power or energy

dBJ

dB(J) - 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 dBJ/K.

dBm

dB(mW) — power relative to 1 milliwatt.

dBμ or dBu

dB(μV/m) — electric field strength relative to 1 microvolt per metre.

dBf

dB(fW) — power relative to 1 femtowatt.

dBW

dB(W) — power relative to 1 watt.

dBk

dB(kW) — power relative to 1 kilowatt.

[edit] Relative measurements

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 dB_A or dBA. According to ANSI standards, the preferred usage is to write L_A = x dB, as dBA implies a reference to an "A" unit, not an A-weighting. They are still used commonly as a shorthand for A-weighted measurements, however.

dBd

dB(dipole) — the forward gain of an antenna compared to a half-wave dipole antenna.

dBFS or dBfs

dB(full scale) — the amplitude of a signal (usually audio) compared to the maximum which a device can handle before clipping occurs. In digital systems, 0 dBFS would equal the highest level (number) the processor is capable of representing. This is an instantaneous (sample) value as compared to the dBm/dBu/dBv which are typically RMS.^{[dubious — see talk page]} Measured values are usually negative, since they should be less than the maximum.

dB-Hz

dB(Hertz) - Bandwidth relative to 1 Hz. E.g., 20 dB-Hz is equal to 100 Hz. Commonly used in link budget calculations.

dBi

dB(isotropic) — the forward gain of an antenna compared to an idealized isotropic antenna.

dBov or dBO

dB(overload) — 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) — 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 levels of noise or sideband peak power, compared to the carrier power.

[edit] 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.

[edit] Round numbers

The values of coins and banknotes are round numbers. The rules are:

One is a round number
Twice a round number is a round number: 2, 4, 8, 16, 32, 64
Ten times a round number is a round number: 10, 100
Half a round number is a round number: 50, 25, 12.5, 6.25
The tenth of a round number is a round number: 5, 2.5, 1.25, 1.6, 3.2, 6.4

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  1    1.25 1.6  2    2.5  3.2  4    5    6.3  8   10
dB     0    1    2    3    4    5    6    7    8    9   10

This useful approximate table of logarithms is easily reconstructed or memorized.

[edit] The 4 → 6 energy rule

To one decimal place of precision, 4.x is 6.x in dB (energy).

Examples:

4.0 → 6.0 dB
4.3 → 6.3 dB
4.7 → 6.7 dB

[edit] The "789" rule

To one decimal place of precision, x → (½ x + 5.0 dB) for 7.0 ≤ x ≤ 10.

Examples:

7.0 → ½ 7.0 + 5.0 dB = 3.5 + 5.0 dB = 8.5 dB
7.5 → ½ 7.5 + 5.0 dB = 3.75 + 5.0 dB = 8.75 dB
8.2 → ½ 8.2 + 5.0 dB = 4.1 + 5.0 dB = 9.1 dB
9.9 → ½ 9.9 + 5.0 dB = 4.95 + 5.0 dB = 9.95 dB
10.0 → ½ 10.0 + 5.0 dB = 5.0 + 5.0 dB = 10 dB

[edit] −3 dB ≈ ½ power

A level difference of ±3 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 is not 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 10^3/10. 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 10¹² 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 × 10¹², giving a 10% error.

[edit] 6 dB per bit

In digital audio linear pulse-code modulation, 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.

[edit] 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.

[edit] Commonly used dB values

dB level	power ratio	dB level	voltage ratio
−30 dB	1/1000 = 0.001	−30 dB	$\sqrt{1/1000}$ = 0.03162
−20 dB	1/100 = 0.01	−20 dB	$\sqrt{1/100}$ = 0.1
−10 dB	1/10 = 0.1	−10 dB	$\sqrt{1/10}$ = 0.3162
−3 dB	1/2 = 0.5 (approx.)	−3 dB	$\sqrt{1/2}$ = 0.7071
3 dB	2 (approx.)	3 dB	$\sqrt{2}$ = 1.414
10 dB	10	10 dB	$\sqrt{10}$ = 3.162
20 dB	100	20 dB	$\sqrt{100}$ = 10
30 dB	1000	30 dB	$\sqrt{1000}$ = 31.62