# Neper

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Neper

The neper is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. It has the unit symbol Np. 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 not a unit in the International System of Units (SI), but it 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 value of a ratio in nepers is given by

$L_{Np} = \ln\frac{x_1}{x_2} = \ln x_1 - \ln x_2. \,$

where x1 and x2 are the values of interest, and ln is the natural logarithm.

The neper is defined in terms of ratios of field 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 10log(ratio) dB is equvalent to a field-quantity ratio 20log(ratio) dB, since power is proportional to the square (Joule's laws) of the amplitude. Hence the neper and dB are related via:

$1\ \mbox{Np} = 20 \log_{10} e\ \mbox{dB} \approx 8{.}685889638 \ \mbox{dB} \,$

and

$1\ \mbox{dB} = \frac{1}{20 \log_{10} e}\ \mbox{Np} \approx 0{.}115129254 \ \mbox{Np}. \,$

The decibel and the neper have a fixed ratio to each other. The (voltage) level ratio is

\begin{align} L & = 10 \log_{10} \frac{x_1^2}{x_2^2} & \mathrm{dB} \\ & = 10 \log_{10} {\left(\frac{x_1}{x_2}\right)}^2 & \mathrm{dB} \\ & = 20 \log_{10} \frac{x_1}{x_2} & \mathrm{dB} \\ & = \ln \frac{x_1}{x_2} & \mathrm{Np}. \\ \end{align}

Like the decibel, the neper is a dimensionless unit. The International Telecommunication Union (ITU) recognizes both units.

## References

1. ^ Bureau International des Poids et Mesures (2006), The International System of Units (SI) Brochure, 8th edition, pp 127-128.

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