Editing Decibel (section)

== Definition ==
{| class="wikitable floatright" style="width:0; font-size:85%; margin-left:1em"
|-
! scope="col" style="text-align:right;" |  dB
! scope="col" colspan="2" | Power ratio
! scope="col" colspan="2" | Amplitude ratio
|-
| style="text-align:right; border:none;" | 100
| style="text-align:right; border:none; padding-right:0" | {{gaps|10|000|000|000}} || style="border:none;" |
| style="text-align:right; border:none; padding-right:0" |        {{gaps|100|000}} || style="border:none;" |
|-
| style="text-align:right; border:none;" |  90
| style="text-align:right; border:none; padding-right:0" |  {{gaps|1|000|000|000}} || style="border:none;" |
| style="text-align:right; border:none; padding-right:0" |         {{gaps|31|623}} || style="border:none;" |
|-
| style="text-align:right; border:none;" |  80
| style="text-align:right; border:none; padding-right:0" |    {{gaps|100|000|000}} || style="border:none;" |
| style="text-align:right; border:none; padding-right:0" |         {{gaps|10|000}} || style="border:none;" |
|-
| style="text-align:right; border:none;" |  70
| style="text-align:right; border:none; padding-right:0" |     {{gaps|10|000|000}} || style="border:none;" |
| style="text-align:right; border:none; padding-right:0" |          {{gaps|3|162}} || style="border:none;" |
|-
| style="text-align:right; border:none;" |  60
| style="text-align:right; border:none; padding-right:0" |      {{gaps|1|000|000}} || style="border:none;" |
| style="text-align:right; border:none; padding-right:0" |          {{gaps|1|000}} || style="border:none;" |
|-
| style="text-align:right; border:none;" |  50
| style="text-align:right; border:none; padding-right:0" |        {{gaps|100|000}} || style="border:none;" |
| style="text-align:right; border:none; padding-right:0" |                     316 || style="border:none; padding-left:0;" | .2
|-
| style="text-align:right; border:none;" |  40
| style="text-align:right; border:none; padding-right:0" |         {{gaps|10|000}} || style="border:none;" |
| style="text-align:right; border:none; padding-right:0" |                     100 || style="border:none;" |
|-
| style="text-align:right; border:none;" |  30
| style="text-align:right; border:none; padding-right:0" |          {{gaps|1|000}} || style="border:none;" |
| style="text-align:right; border:none; padding-right:0" |                      31 || style="border:none; padding-left:0;" | .62
|-
| style="text-align:right; border:none;" |  20
| style="text-align:right; border:none; padding-right:0" |                     100 || style="border:none;" |
| style="text-align:right; border:none; padding-right:0" |                      10 || style="border:none;" |
|-
| style="text-align:right; border:none;" |  10
| style="text-align:right; border:none; padding-right:0" |                      10 || style="border:none;" |
| style="text-align:right; border:none; padding-right:0" |                       3 || style="border:none; padding-left:0;" | .162
|-
| style="text-align:right; border:none;" |   6
| style="text-align:right; border:none; padding-right:0" |                       3 || style="border:none; padding-left:0;" | .981 ≈ 4
| style="text-align:right; border:none; padding-right:0" |                       1 || style="border:none; padding-left:0;" | .995 ≈ 2
|-
| style="text-align:right; border:none;" |   3
| style="text-align:right; border:none; padding-right:0" |                       1 || style="border:none; padding-left:0;" | .995 ≈ 2
| style="text-align:right; border:none; padding-right:0" |                       1 || style="border:none; padding-left:0;" | .413 ≈ {{sqrt|2}}
|-
| style="text-align:right; border:none;" |   1
| style="text-align:right; border:none; padding-right:0" |                       1 || style="border:none; padding-left:0;" | .259
| style="text-align:right; border:none; padding-right:0" |                       1 || style="border:none; padding-left:0;" | .122
|-
| style="text-align:right; border:none;" |   0
| style="text-align:right; border:none; padding-right:0" |                       1 || style="border:none;" |
| style="text-align:right; border:none; padding-right:0" |                       1 || style="border:none;" |
|-
| style="text-align:right; border:none;" |  −1
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | .794
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | .891
|-
| style="text-align:right; border:none;" |  −3
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | .501 ≈ {{sfrac|2}}
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | .708 ≈ {{sqrt|{{sfrac|2}}}}
|-
| style="text-align:right; border:none;" |  −6
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | .251 ≈ {{sfrac|4}}
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | .501 ≈ {{sfrac|2}}
|-
| style="text-align:right; border:none;" | −10
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | .1
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | {{gaps|.316|2}}
|-
| style="text-align:right; border:none;" | −20
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | .01
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | .1
|-
| style="text-align:right; border:none;" | −30
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | .001
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | {{gaps|.031|62}}
|-
| style="text-align:right; border:none;" | −40
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | {{gaps|.000|1}}
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | .01
|-
| style="text-align:right; border:none;" | −50
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | {{gaps|.000|01}}
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | {{gaps|.003|162}}
|-
| style="text-align:right; border:none;" | −60
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | {{gaps|.000|001}}
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | .001
|-
| style="text-align:right; border:none;" | −70
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | {{gaps|.000|000|1}}
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | {{gaps|.000|316|2}}
|-
| style="text-align:right; border:none;" | −80
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | {{gaps|.000|000|01}}
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | {{gaps|.000|1}}
|-
| style="text-align:right; border:none;" | −90
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | {{gaps|.000|000|001}}
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | {{gaps|.000|031|62}}
|-
| style="text-align:right; border:none;" | −100
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | {{gaps|.000|000|000|1}}
| style="text-align:right; border:none; padding-right:0" |                       0 || style="border:none; padding-left:0;" | {{gaps|.000|01}}
|-
| colspan="5" style="text-align:left; background:#f8f8ff;" | An example scale showing power ratios ''x'', amplitude ratios {{sqrt|''x''}}, and dB equivalents 10&nbsp;log<sub>10</sub>&nbsp;''x''
|}

The IEC Standard [[IEC 60027|60027-3:2002]] defines the following quantities. The decibel (dB) is one-tenth of a bel: 1&nbsp;dB&nbsp;= 0.1&nbsp;B. The bel (B) is {{1/2}}&nbsp;ln(10) [[neper]]s: 1&nbsp;B&nbsp;= {{1/2}}&nbsp;ln(10)&nbsp;Np. The neper is the change in the [[level (logarithmic quantity)|level]] of a [[root-power quantity]] when the root-power quantity changes by a factor of [[e (mathematical constant)|''e'']], that is {{nowrap|1=1 Np = ln(e) = 1}}, thereby relating all of the units as nondimensional [[Natural logarithm|natural ''log'']] of root-power-quantity ratios, {{val|1|u=dB}}&nbsp;= {{val|0.11513|end=...|u=Np}}&nbsp;= {{val|0.11513|end=...}}.  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 root-power quantities of {{radic|10}}:1.<ref>{{cite book |title=International Standard CEI-IEC 27-3 |chapter=Letter symbols to be used in electrical technology |at=Part 3: Logarithmic quantities and units |publisher=International Electrotechnical Commission}}</ref>

Two signals whose levels differ by one decibel have a power ratio of 10<sup>1/10</sup>, which is approximately {{val|1.25893}}, and an amplitude (root-power quantity) ratio of 10<sup>1/20</sup> ({{val|1.12202}}).<ref name="auto"/><ref name="auto1"/>

The bel is rarely used either without a prefix or with [[metric prefix|SI unit prefixes]] other than ''[[deci-|deci]]''; it is customary, for example, to use ''hundredths of a decibel'' rather than ''millibels''.  Thus, five one-thousandths of a bel would normally be written 0.05&nbsp;dB, and not 5&nbsp;mB.<ref>Fedor Mitschke, ''Fiber Optics: Physics and Technology'', Springer, 2010 {{ISBN|3642037038}}.</ref>

The method of expressing a ratio as a level in decibels depends on whether the measured property is a ''power quantity'' or a ''root-power quantity''; see ''[[Power, root-power, and field quantities]]'' for details.

=== Power quantities ===
When referring to measurements of ''[[Power (physics)|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 ''P''<sub>0</sub> (reference power) is represented by ''L''<sub>''P''</sub>, that ratio expressed in decibels,<ref>{{Cite book |title=Microwave Engineering |author-first=David M. |author-last=Pozar |edition=3rd |publisher=Wiley |date=2005 |author-link=David M. Pozar |isbn=978-0-471-44878-5 |page=63}}</ref> which is calculated using the formula:<ref>IEC 60027-3:2002</ref>
: <math>
L_P = \frac{1}{2} \ln\!\left(\frac{P}{P_0}\right)\,\text{Np} = 10 \log_{10}\!\left(\frac{P}{P_0}\right)\,\text{dB}
</math>

The base-10 logarithm of the ratio of the two power quantities 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 ''P''<sub>0</sub> must measure the same type of quantity, and have the same units before calculating the ratio. If {{nowrap|1=''P'' =}} ''P''<sub>0</sub> in the above equation, then ''L''<sub>''P''</sub>&nbsp;= 0. If ''P'' is greater than ''P''<sub>0</sub> then ''L''<sub>''P''</sub> is positive; if ''P'' is less than ''P''<sub>0</sub> then ''L''<sub>''P''</sub> is negative.

Rearranging the above equation gives the following formula for ''P'' in terms of ''P''<sub>0</sub> and ''L''<sub>''P''</sub> :
: <math>
P = 10^\frac{L_P}{10\,\text{dB}} P_0
</math>

=== Root-power (field) quantities ===
{{main|Power, root-power, and field quantities}}
When referring to measurements of root-power quantities, it is usual to consider the ratio of the squares of ''F'' (measured) and ''F''<sub>0</sub> (reference). This is because the definitions were originally formulated to give the same value for relative ratios for both power and root-power quantities. Thus, the following definition is used:
: <math>
L_F = \ln\!\left(\frac{F}{F_0}\right)\,\text{Np} = 10 \log_{10}\!\left(\frac{F^2}{F_0^2}\right)\,\text{dB} = 20 \log_{10} \left(\frac{F}{F_0}\right)\,\text{dB}
</math>

The formula may be rearranged to give
: <math>
F = 10^\frac{L_F}{20\,\text{dB}} F_0
</math>

Similarly, in [[Electronic circuit|electrical circuits]], dissipated power is typically proportional to the square of voltage or current when the [[Electrical impedance|impedance]] is constant. Taking voltage as an example, this leads to the equation for power gain level ''L''<sub>''G''</sub>:
: <math>
L_G = 20 \log_{10}\!\left (\frac{V_\text{out}}{V_\text{in}}\right)\,\text{dB}
</math>
where ''V''<sub>out</sub> is the [[root-mean-square]] (rms) output voltage, ''V''<sub>in</sub> is the rms input voltage. A similar formula holds for current.

The term ''root-power quantity'' is introduced by ISO Standard [[ISO/IEC 80000|80000-1:2009]] as a substitute of ''field quantity''.  The term ''field quantity'' is deprecated by that standard and ''root-power'' is used throughout this article.

=== Relationship between power and root-power levels ===
Although power and root-power quantities are different quantities, their respective levels are historically measured in the same units, typically decibels.  A factor of 2 is introduced to make ''changes'' in the respective levels match under restricted conditions such as when the medium is linear and the ''same'' waveform is under consideration with changes in amplitude, or the medium impedance is linear and independent of both frequency and time.  This relies on the relationship
:<math> \frac{P(t)}{P_0} = \left(\frac{F(t)}{F_0}\right)^2 </math>
holding.<ref>{{citation |author=I M Mills |author2=B N Taylor |author3=A J Thor |title=Definitions of the units radian, neper, bel and decibel |year=2001 |journal=Metrologia |volume=38 |page=353 |number=4 |doi=10.1088/0026-1394/38/4/8|bibcode=2001Metro..38..353M |s2cid=250827251 }}</ref>  In a nonlinear system, this relationship does not hold by the definition of linearity.  However, even in a [[linear system]] in which the power quantity is the product of two linearly related quantities (e.g. voltage and current), if the impedance is frequency- or time-dependent, this relationship does not hold in general, for example if the energy spectrum of the waveform changes.

For differences in level, the required relationship is relaxed from that above to one of proportionality (i.e., the reference quantities ''P''{{sub|0}} and ''F''{{sub|0}} need not be related), or equivalently,
: <math> \frac{P_2}{P_1} = \left(\frac{F_2}{F_1}\right)^2 </math>
must hold to allow the power level difference to be equal to the root-power level difference from power ''P''{{sub|1}} and ''F''{{sub|1}} to ''P''{{sub|2}} and ''F''{{sub|2}}.  An example might be an [[amplifier]] with unity voltage gain independent of load and frequency driving a load with a frequency-dependent impedance: the relative voltage gain of the amplifier is always 0&nbsp;dB, but the power gain depends on the changing spectral composition of the waveform being amplified.  Frequency-dependent impedances may be analyzed by considering the quantities [[power spectral density]] and the associated root-power quantities via the [[Fourier transform]], which allows elimination of the frequency dependence in the analysis by analyzing the system at each frequency independently.

=== Conversions ===
Since logarithm differences measured in these units often represent power ratios and root-power ratios, values for both are shown below. The bel is traditionally used as a unit of logarithmic power ratio, while the neper is used for logarithmic root-power (amplitude) ratio.

{| class="wikitable"
|+ Conversion between units of level and a list of corresponding ratios
!Unit !! In decibels !! In bels !! In [[neper]]s !! Power ratio !! Root-power ratio
|-
| 1&nbsp;dB || 1&nbsp;dB || 0.1 B || {{val|0.11513}}&nbsp;Np || 10<sup>1/10</sup> ≈ {{val|1.25893}} || 10<sup>1/20</sup> ≈ {{val|1.12202}}
|-
| 1 Np || {{val|8.68589}}&nbsp;dB || {{val|0.868589}}&nbsp;B || 1 Np || e<sup>2</sup> ≈ {{val|7.38906}} || [[e (mathematical constant)|e]] ≈ {{val|2.71828}}
|-
| 1 B || 10&nbsp;dB || 1 B || 1.151&nbsp;3 Np || 10 || 10<sup>1/2</sup> ≈ 3.162&nbsp;28
|}

=== Examples ===
The unit dBW is often used to denote a ratio for which the reference is 1&nbsp;W, and similarly [[dBm]] for a {{nowrap|1 mW}} reference point.
* Calculating the ratio in decibels of {{nowrap|1 kW}} (one kilowatt, or {{val|1000}} watts) to {{nowrap|1 W}} yields: <math display="block">
L_G = 10 \log_{10} \left(\frac{1\,000\,\text{W}}{1\,\text{W}}\right)\,\text{dB} = 30\,\text{dB}
</math>
* The ratio in decibels of {{nowrap|1={{radic|1000}} V ≈ 31.62 V}} to {{nowrap|1 V}} is: <math display="block">
L_G = 20 \log_{10} \left(\frac{31.62\,\text{V}}{1\,\text{V}}\right)\,\text{dB} = 30\,\text{dB}
</math>
{{nowrap|1=(31.62 V / 1 V)<sup>2</sup> ≈ 1 kW / 1 W}}, illustrating the consequence from the definitions above that ''L''<sub>''G''</sub> has the same value, 30&nbsp;dB, 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.
* The ratio in decibels of {{nowrap|10 W}} to {{nowrap|1 mW}} (one milliwatt) is obtained with the formula: <math display="block">
L_G = 10 \log_{10} \left(\frac{10\text{W}}{0.001\text{W}}\right)\,\text{dB} = 40\,\text{dB}
</math>
* The power ratio corresponding to a {{nowrap|3 dB}} change in level is given by: <math display="block">
G = 10^\frac{3}{10} \times 1 = 1.995\,26\ldots \approx 2
</math>

A change in power ratio by a factor of 10 corresponds to a change in level of {{nowrap|10 dB}}. A change in power ratio by a factor of 2 or {{sfrac|2}} is approximately a [[Half-power point|change of 3&nbsp;dB]]. More precisely, the change is ±{{val|3.0103}}&nbsp;dB, but this is almost universally rounded to 3&nbsp;dB in technical writing.{{Citation needed|date=January 2025}}  This implies an increase in voltage by a factor of {{nowrap|{{sqrt|2}} ≈}} {{val|1.4142}}.  Likewise, a doubling or halving of the voltage, corresponding to a quadrupling or quartering of the power, is commonly described as 6&nbsp;dB rather than ±{{val|6.0206}}&nbsp;dB.

Should it be necessary to make the distinction, the number of decibels is written with additional [[significant figures]].  3.000&nbsp;dB corresponds to a power ratio of 10<sup>3/10</sup>, or {{val|1.9953}}, about 0.24% different from exactly 2, and a voltage ratio of {{val|1.4125}}, about 0.12% different from exactly {{sqrt|2}}.  Similarly, an increase of 6.000&nbsp;dB corresponds to a power ratio of {{nowrap|10<sup>6/10</sup> ≈}} {{val|3.9811}}, about 0.5% different from 4.