Neper

Definition

Like the

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

where ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ are the signal amplitudes, and ln is the natural logarithm. The level of a ratio of two

${\displaystyle L={\frac {1}{2}}\ln {\frac {p_{1}}{p_{2}}}\mathrm {~Np} ,}$

where ${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$ are the signal powers.

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

and

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

The (voltage) level ratio is

${\displaystyle {\begin{aligned}L&=10\log _{10}{\frac {x_{1}^{2}}{x_{2}^{2}}}&{\text{dB}}\\&=10\log _{10}{\left({\frac {x_{1}}{x_{2}}}\right)}^{2}&{\text{dB}}\\&=20\log _{10}{\frac {x_{1}}{x_{2}}}&{\text{dB}}\\&=\ln {\frac {x_{1}}{x_{2}}}&{\text{Np}}.\\\end{aligned}}}$

See also