Editing Decibel (section)

== Properties ==
The decibel is useful for representing large ratios and for simplifying representation of multiplicative effects, such as attenuation from multiple sources along a signal chain. Its application in systems with additive effects is less intuitive, such as in the combined [[sound pressure level]] of two machines operating together. Care is also necessary with decibels directly in fractions and with the units of multiplicative operations.

=== Reporting large ratios ===
The [[logarithmic scale]] nature of the decibel means that a very large range of ratios can be represented by a convenient number, in a manner similar to [[scientific notation]]. This allows one to clearly visualize huge changes of some quantity. See ''[[Bode plot]]'' and ''[[Semi-log plot]]''.  For example, 120&nbsp;dB SPL may be clearer than "a trillion times more intense than the threshold of hearing".{{citation needed|date=February 2021}}

=== Representation of multiplication operations ===
Level values in decibels can be added instead of multiplying the underlying power values, which means that the overall gain of a multi-component system, such as a series of amplifier stages, can be calculated by summing the gains in decibels of the individual components, rather than multiply the amplification factors; that is, {{nowrap|log(''A'' × ''B'' × ''C'') }}= log(''A'') + log(''B'') + log(''C''). Practically, this means that, armed only with the knowledge that 1&nbsp;dB is a power gain of approximately 26%, 3&nbsp;dB is approximately 2× power gain, and 10&nbsp;dB is 10× power gain, it is possible to determine the power ratio of a system from the gain in dB with only simple addition and multiplication. For example:
*A system consists of 3 amplifiers in series, with gains (ratio of power out to in) of 10&nbsp;dB, 8&nbsp;dB, and 7&nbsp;dB respectively, for a total gain of 25&nbsp;dB. Broken into combinations of 10, 3, and 1&nbsp;dB, this is: {{block indent | em = 1.5 | text =
 25&nbsp;dB&nbsp;= 10&nbsp;dB&nbsp;+ 10&nbsp;dB&nbsp;+ 3&nbsp;dB&nbsp;+ 1&nbsp;dB&nbsp;+ 1&nbsp;dB
}} With an input of 1 watt, the output is approximately {{block indent | em = 1.5 | text =
 1&nbsp;W&nbsp;× 10&nbsp;× 10&nbsp;× 2&nbsp;× 1.26&nbsp;× 1.26&nbsp;≈ 317.5&nbsp;W
}} Calculated precisely, the output is 1&nbsp;W&nbsp;× 10<sup>25/10</sup>&nbsp;≈ 316.2&nbsp;W. The approximate value has an error of only +0.4% with respect to the actual value, which is negligible given the precision of the values supplied and the accuracy of most measurement instrumentation.

However, according to its critics, the decibel creates confusion, obscures reasoning, is more related to the era of [[slide rule]]s than to modern digital processing, and is cumbersome and difficult to interpret.<ref name="Hickling">R. Hickling (1999), Noise Control and SI Units, J Acoust Soc Am 106, 3048</ref><ref>Hickling, R. (2006). Decibels and octaves, who needs them?. Journal of sound and vibration, 291(3-5), 1202-1207.</ref>
Quantities in decibels are not necessarily [[Dimensional homogeneity|additive]],<ref>Nicholas P. Cheremisinoff (1996) Noise Control in Industry: A Practical Guide, Elsevier, 203 pp, p. [{{Google books |plainurl=yes |id=rrpEuUOkT3UC |page=7}} 7]</ref><ref>Andrew Clennel Palmer (2008), Dimensional Analysis and Intelligent Experimentation, World Scientific, 154 pp, p.13</ref>  thus being "of unacceptable form for use in [[dimensional analysis]]".<ref>J. C. Gibbings, ''Dimensional Analysis'', [{{Google books |plainurl=yes |id=Q6iflrgVaWcC |page=37}} p.37], Springer, 2011 {{ISBN|1849963177}}.</ref>
Thus, units require special care in decibel operations. Take, for example, [[carrier-to-noise-density ratio]] ''C''/''N''<sub>0</sub> (in hertz), involving carrier power ''C'' (in watts) and noise power spectral density ''N''<sub>0</sub> (in W/Hz). Expressed in decibels, this ratio would be a subtraction (''C''/''N''<sub>0</sub>)<sub>dB</sub> = ''C''<sub>dB</sub> − ''N''<sub>0&nbsp;dB</sub>. However, the linear-scale units still simplify in the implied fraction, so that the results would be expressed in dB-Hz.

=== Representation of addition operations <span class="anchor" id="Addition"></span> ===
{{Further|Logarithmic addition}}
According to Mitschke,<ref>{{cite book |title=Fiber Optics |publisher=Springer |date=2010}}</ref> "The advantage of using a logarithmic measure is that in a transmission chain, there are many elements concatenated, and each has its own gain or attenuation. To obtain the total, addition of decibel values is much more convenient than multiplication of the individual factors." However, for the same reason that humans excel at additive operation over multiplication, decibels are awkward in inherently additive operations:<ref>R. J. Peters, ''Acoustics and Noise Control'', Routledge, 12 November 2013, 400 pages, p.&nbsp;13</ref><blockquote>if two machines each individually produce a sound pressure level of, say, 90&nbsp;dB at a certain point, then when both are operating together we should expect the combined sound pressure level to increase to 93&nbsp;dB, but certainly not to 180&nbsp;dB!; suppose that the noise from a machine is measured (including the contribution of background noise) and found to be 87&nbsp;dBA but when the machine is switched off the background noise alone is measured as 83&nbsp;dBA. [...] the machine noise [level (alone)] may be obtained by 'subtracting' the 83&nbsp;dBA background noise from the combined level of 87&nbsp;dBA; i.e., 84.8&nbsp;dBA.; in order to find a representative value of the sound level in a room a number of measurements are taken at different positions within the room, and an average value is calculated. [...] Compare the logarithmic and arithmetic averages of [...] 70&nbsp;dB and 90&nbsp;dB: [[logarithmic average]] = 87&nbsp;dB; [[arithmetic average]] = 80&nbsp;dB.</blockquote>

Addition on a logarithmic scale is called [[logarithmic addition]], and can be defined by taking exponentials to convert to a linear scale, adding there, and then taking logarithms to return. For example, where operations on decibels are logarithmic addition/subtraction and logarithmic multiplication/division, while operations on the linear scale are the usual operations:
:<math>87\,\text{dBA} \ominus 83\,\text{dBA} = 10 \cdot \log_{10}\bigl(10^{87/10} - 10^{83/10}\bigr)\,\text{dBA} \approx 84.8\,\text{dBA}</math>
:<math>
\begin{align}
M_\text{lm}(70, 90) &= \left(70\,\text{dBA} + 90\,\text{dBA}\right)/2 \\
&= 10 \cdot \log_{10}\left(\bigl(10^{70/10} + 10^{90/10}\bigr)/2\right)\,\text{dBA} \\
&= 10 \cdot \left(\log_{10}\bigl(10^{70/10} + 10^{90/10}\bigr) - \log_{10} 2\right)\,\text{dBA}
\approx 87\,\text{dBA}
\end{align}
</math>
The [[logarithmic mean]] is obtained from the logarithmic sum by subtracting <math>10\log_{10} 2</math>, since logarithmic division is linear subtraction.

=== Fractions ===
[[Attenuation]] constants, in topics such as [[optical fiber]] communication and [[radio propagation]] [[path loss]], are often expressed as a [[Fraction (mathematics)|fraction]] or ratio to distance of transmission. In this case, {{nowrap|dB/m}} represents decibel per meter, {{nowrap|dB/mi}} represents decibel per mile, for example. These quantities are to be manipulated obeying the rules of dimensional analysis, e.g., a 100-meter run with a {{nowrap|3.5 dB/km}} fiber yields a loss of {{nowrap|0.35 dB =}} {{nowrap|3.5 dB/km ×}} 0.1&nbsp;km.