Neper

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:

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

and

${\displaystyle 1\mathrm{d}\mathrm{B}=\frac{1}{20}\ln (10)\mathrm{N}\mathrm{p}\approx \text{0.115129255 Np}.}$

The (voltage) level ratio is

${\displaystyle \begin{array}{ccc}L& =10{\log }_{10}\frac{x_1^2}{x_2^2}& \text{dB}\\ & =10{\log }_{10}{\left(\frac{x_1}{x_2}\right)}^2& \text{dB}\\ & =20{\log }_{10}\frac{x_1}{x_2}& \text{dB}\\ & =\ln \frac{x_1}{x_2}& \text{Np}.\end{array}}$

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.

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