Neper

Like the decibel, the neper is a unit in a logarithmic scale. While the bel uses the decadic (base-10) logarithm to compute ratios, the neper uses the natural logarithm, based on Euler's number ({{nowrap|e ≈ 2.71828}}). The level of a ratio of two signal amplitudes or root-power quantities, with the unit neper, is given by {{cite tech report |title=Letter symbols to be used in electrical technology – Part 3: Logarithmic and related quantities, and their units |number=IEC 60027-3:2002|type=International standard |institution= International Electrotechnical Commission |date=2002-07-19}}

:$$

L = \ln\frac{x_1}{x_2}\mathrm{~Np},

where $x\_1$ and $x\_2$ are the signal amplitudes, and {{math|ln}} is the natural logarithm. The level of a ratio of two power quantities, with the unit neper, is given by

:$$

L = \frac{1}{2} \ln\frac{p_1}{p_2}\mathrm{~Np},

where $p\_1$ and $p\_2$ are the signal powers.

In the International System of Quantities, the neper is defined as {{nowrap|1=1 Np = 1}}.{{Cite journal |last=Thor |first=A J |date=1994-01-01 |title=New International Standards for Quantities and Units |url=https://iopscience.iop.org/article/10.1088/0026-1394/30/5/010 |journal=Metrologia |volume=30 |issue=5 |pages=517–522 |doi=10.1088/0026-1394/30/5/010 |issn=0026-1394|url-access=subscription }}

{{main|Level (logarithmic quantity)}}

The neper is defined in terms of ratios of field quantities — also called root-power quantities — (for example, voltage or current amplitudes in electrical circuits, or pressure in acoustics), whereas the decibel was originally defined in terms of power ratios. A power ratio 10 log r dB is equivalent to a field-quantity ratio 20 log r dB, since power in a linear system is proportional to the square (Joule's laws) of the amplitude. Hence the decibel and the neper have a fixed ratio to each other:{{cite journal |date=January 2022 |orig-date=2021-11-09 |title=A terminology standard for underwater acoustics and the benefits of international standardization. |url=https://ieeexplore.ieee.org/document/9607022 |journal=IEEE Journal of Oceanic Engineering |publisher=IEEE |volume=47 |issue=1 |pages=179-200 [Appendix B Decibel: Past, Present, and Future – Section D] |doi=10.1109/JOE.2021.3085947 |issn=0364-9059 |eissn=1558-1691 |s2cid=243948953 |access-date=2022-12-20 |doi-access=free |author-last1=Ainslie |author-first1=Michael A. |author-last2=Halvorsen |author-first2=Michele B. |author-last3=Robinson |author-first3=Stephen P.}} [https://web.archive.org/web/20221220112042/https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9607022] (22 pages)

:$1\backslash \; \backslash text\{Np\}\; =\; 20\; \backslash log\_\{10\}\; e\backslash \; \backslash text\{dB\}\; \backslash approx\; \backslash text\{8.685889638\; dB\}$

and

:$$

1\ \mathrm{dB} = \frac{1}{20} \ln(10)\ \mathrm{Np} \approx \text{0.115129255 Np} .

The (voltage) level ratio is

:$$

\begin{align}

L & = 10 \log_{10} \frac{x_1^2}{x_2^2} & \text{dB} \\

& = 10 \log_{10} {\left(\frac{x_1}{x_2}\right)}^2 & \text{dB} \\

& = 20 \log_{10} \frac{x_1}{x_2} & \text{dB} \\

& = \ln \frac{x_1}{x_2} & \text{Np}. \\

\end{align}

Like the decibel, the neper is a dimensionless unit. The International Telecommunication Union (ITU) recognizes both units. Only the neper is coherent with the SI.ISO 80000-3:2007 §0.5

The neper is a natural linear unit of relative difference, meaning in nepers (logarithmic units) relative differences add rather than multiply. This property is shared with logarithmic units in other bases, such as the bel.

The derived units decineper (1 dNp = 0.1 neper) and centineper (1 cNp = 0.01 neper) are also used.{{cite book |title=Glossary of Telecommunication Terms |date=1980 |publisher=General Services Administration, Federal Supply Service |page=73 |url=https://books.google.com/books?id=3cQ65jxbaTAC&pg=PA73 |language=en}} The centineper for root-power quantities corresponds to a log point or log percentage, see {{section link|Relative change and difference#Logarithmic scale}}.{{cite journal |last1=Karjus |first1=Andres |last2=Blythe |first2=Richard A. |last3=Kirby |first3=Simon |last4=Smith |first4=Kenny |title=Quantifying the dynamics of topical fluctuations in language |journal=Language Dynamics and Change |date=10 February 2020 |volume=10 |issue=1 |pages=86–125 |doi=10.1163/22105832-01001200 |s2cid=46928080 |url=https://brill.com/view/journals/ldc/10/1/article-p86_5.xml|doi-access=free |arxiv=1806.00699 }}

• Nat (unit)
• Nepers per metre

{{Reflist}}

File:Napier, John – Mirifici logarithmorum canonis constructio, 1825 – BEIC 6555650.jpg

• {{Cite book|title=Mirifici logarithmorum canonis constructio|volume=|publisher=Librairie scientifique Hermann et C.ie|location=Paris|year=1825|language=fr|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=6555650}}

• {{cite journal |author-last=Tuffentsammer |author-first=Karl |title=Das Dezilog, eine Brücke zwischen Logarithmen, Dezibel, Neper und Normzahlen |language=de |trans-title=The decilog, a bridge between logarithms, decibel, neper and preferred numbers |journal=VDI-Zeitschrift |volume=98 |date=1956 |pages=267–274}}
• {{cite book |title=Logarithmen, Normzahlen, Dezibel, Neper, Phon - natürlich verwandt! |language=de |trans-title=Logarithms, preferred numbers, decibel, neper, phon - naturally related! |author-first=Eugen |author-last=Paulin |date=2007-09-01 |url=http://www.rechenschieber.org/Normzahlen.pdf |access-date=2016-12-18 |url-status=live |archive-url=https://web.archive.org/web/20161218223050/http://www.rechenschieber.org/Normzahlen.pdf |archive-date=2016-12-18}}

• [https://web.archive.org/web/20140716201523/http://www.microwaves101.com/ENCYCLOPEDIA/neper.cfm What's a neper?]
• [http://www.sengpielaudio.com/calculator-neper.htm Conversion of level gain and loss: neper, decibel, and bel]
• [https://web.archive.org/web/20140429141423/http://www.microwaves101.com/encyclopedia/transmission_loss.cfm Calculating transmission line losses]

{{SI units}}

Category:Units of level