Neper

Last updated January 31, 2024

The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As is the case for the decibel and bel, the neper is a unit defined in the international standard ISO 80000. It is not part of the International System of Units (SI), but is accepted for use alongside the SI.^[1]

Definition

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 ( e ≈ 2.71828). The level of a ratio of two signal amplitudes or root-power quantities, with the unit neper, is given by^[2]

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

where $x_{1}$ and $x_{2}$ are the signal amplitudes, and $ln$ is the natural logarithm. The level of a ratio of two power quantities, with the unit neper, is given by^[2]

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 1 Np = 1.^[3]

Units

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:^[4]

1\ {\text{Np}}=20\log _{10}e\ {\text{dB}}\approx {\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{aligned}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{aligned}}

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.^[5]

Applications

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.^[6] The centineper for root-power quantities corresponds to a log point or log percentage, see Relative change and difference § Logarithmic scale .^[7]

Related Research Articles

The decibel is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a power ratio of 10^1/10 or root-power ratio of 10^1⁄20.

In mathematics, the logarithm is the inverse function to exponentiation. That means that the logarithm of a number $x$ to the base $b$ is the exponent to which $b$ must be raised to produce $x$ . For example, since $1000 = 10 3$ , the logarithm base 10 of $1000$ is $3$ , or $log 10 (1000) = 3$ . The logarithm of $x$ to base $b$ is denoted as $log b (x)$ , or without parentheses, $log b x$ , or even without the explicit base, $log x$ , when no confusion is possible, or when the base does not matter such as in big O notation.

The natural logarithm of a number is its logarithm to the base of the mathematical constant $e$ , which is an irrational and transcendental number approximately equal to $2.718 281828459$ . The natural logarithm of $x$ is generally written as $ln x$ , $log e x$ , or sometimes, if the base $e$ is implicit, simply $log x$ . Parentheses are sometimes added for clarity, giving $ln(x)$ , $log e (x)$ , or $log(x)$ . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

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The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the dimensionless change in magnitude or phase per unit length. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

Signal-to-noise ratio is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to noise power, often expressed in decibels. A ratio higher than 1:1 indicates more signal than noise.

<span class="mw-page-title-main">Gain (electronics)</span> Ability of a circuit to increase the power or amplitude of a signal

In electronics, gain is a measure of the ability of a two-port circuit to increase the power or amplitude of a signal from the input to the output port by adding energy converted from some power supply to the signal. It is usually defined as the mean ratio of the signal amplitude or power at the output port to the amplitude or power at the input port. It is often expressed using the logarithmic decibel (dB) units. A gain greater than one, that is, amplification, is the defining property of an active component or circuit, while a passive circuit will have a gain of less than one.

In electromagnetics, an antenna's gain is a key performance parameter which combines the antenna's directivity and radiation efficiency. The term power gain has been deprecated by IEEE. In a transmitting antenna, the gain describes how well the antenna converts input power into radio waves headed in a specified direction. In a receiving antenna, the gain describes how well the antenna converts radio waves arriving from a specified direction into electrical power. When no direction is specified, gain is understood to refer to the peak value of the gain, the gain in the direction of the antenna's main lobe. A plot of the gain as a function of direction is called the antenna pattern or radiation pattern. It is not to be confused with directivity, which does not take an antenna's radiation efficiency into account.

In information theory, the Shannon–Hartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise. It is an application of the noisy-channel coding theorem to the archetypal case of a continuous-time analog communications channel subject to Gaussian noise. The theorem establishes Shannon's channel capacity for such a communication link, a bound on the maximum amount of error-free information per time unit that can be transmitted with a specified bandwidth in the presence of the noise interference, assuming that the signal power is bounded, and that the Gaussian noise process is characterized by a known power or power spectral density. The law is named after Claude Shannon and Ralph Hartley.

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References

↑ The International System of Units (SI) (9 ed.). International Bureau of Weights and Measures. 2019. pp. 145–146. Archived from the original on 2022-10-09.
1 2 Letter symbols to be used in electrical technology – Part 3: Logarithmic and related quantities, and their units (International standard). International Electrotechnical Commission. 2002-07-19. IEC 60027-3:2002.
↑ Thor, A J (1994-01-01). "New International Standards for Quantities and Units". Metrologia. 30 (5): 517–522. doi:10.1088/0026-1394/30/5/010. ISSN 0026-1394.
↑ Ainslie, Michael A.; Halvorsen, Michele B.; Robinson, Stephen P. (January 2022) [2021-11-09]. "A terminology standard for underwater acoustics and the benefits of international standardization". IEEE Journal of Oceanic Engineering . IEEE. 47 (1): 179-200 [Appendix B Decibel: Past, Present, and Future – Section D]. doi: 10.1109/JOE.2021.3085947 . eISSN 1558-1691. ISSN 0364-9059. S2CID 243948953 . Retrieved 2022-12-20. (22 pages)
↑ ISO 80000-3:2007 §0.5
↑ Glossary of Telecommunication Terms. General Services Administration, Federal Supply Service. 1980. p. 73.
↑ Karjus, Andres; Blythe, Richard A.; Kirby, Simon; Smith, Kenny (10 February 2020). "Quantifying the dynamics of topical fluctuations in language". Language Dynamics and Change. 10 (1): 86–125. arXiv: 1806.00699 . doi: 10.1163/22105832-01001200 . S2CID 46928080.

Works

Mirifici logarithmorum canonis constructio (in French). Paris: Librairie scientifique Hermann et C.ie. 1825.

External links

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[BIPM-1] The International System of Units (SI) (9 ed.). International Bureau of Weights and Measures. 2019. pp. 145–146. Archived from the original on 2022-10-09.

[IEC-2] 1 2 Letter symbols to be used in electrical technology – Part 3: Logarithmic and related quantities, and their units (International standard). International Electrotechnical Commission. 2002-07-19. IEC 60027-3:2002.

[3] Thor, A J (1994-01-01). "New International Standards for Quantities and Units". Metrologia. 30 (5): 517–522. doi:10.1088/0026-1394/30/5/010. ISSN 0026-1394.

[4] Ainslie, Michael A.; Halvorsen, Michele B.; Robinson, Stephen P. (January 2022) [2021-11-09]. "A terminology standard for underwater acoustics and the benefits of international standardization". IEEE Journal of Oceanic Engineering . IEEE. 47 (1): 179-200 [Appendix B Decibel: Past, Present, and Future – Section D]. doi: 10.1109/JOE.2021.3085947 . eISSN 1558-1691. ISSN 0364-9059. S2CID 243948953 . Retrieved 2022-12-20. (22 pages)

[5] ISO 80000-3:2007 §0.5

[6] Glossary of Telecommunication Terms. General Services Administration, Federal Supply Service. 1980. p. 73.

[Blythe2020-7] Karjus, Andres; Blythe, Richard A.; Kirby, Simon; Smith, Kenny (10 February 2020). "Quantifying the dynamics of topical fluctuations in language". Language Dynamics and Change. 10 (1): 86–125. arXiv: 1806.00699 . doi: 10.1163/22105832-01001200 . S2CID 46928080.

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v t e SI units
Base units	ampere candela kelvin kilogram metre mole second
Derived units with special names	becquerel coulomb degree Celsius farad gray henry hertz joule katal lumen lux newton Nm ohm pascal radian siemens sievert steradian tesla volt watt weber
Other accepted units	astronomical unit dalton day decibel degree of arc electronvolt hectare hour litre minute minute and second of arc neper tonne
See also	Conversion of units Metric prefixes Historical definitions of the SI base units 2019 redefinition System of units of measurement
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