I have trouble trying to interpret the following formula:

$$G_{SCI} = \mu N^{span} G^3 \mathrm{arcsinh}\,(\rho \Delta f^2)$$

$$G_{SCI} = \frac{P_{SCI}}{B},$$ where $P_{SCI}$ is the self-channel interference noise ($W$), $B$ is the bandwidth (Hz), so the unit of $G_{SCI}$ is $W \cdot Hz^{-1}$, $\Delta f$ is also in Hz

$$\mu = \frac{3 \gamma^2}{2\pi \alpha |\beta_2|},\qquad \rho = \frac{\pi^2 |\beta_2|}{2 \alpha}$$

where $\gamma = 1.3 W^{-1} \cdot km^{-1}$, $\beta_2 = -21.3\ ps^2 \cdot km^{-1}$, $\alpha = 0.22 \ dB \cdot km^{-1}$, $N^{span}$ is the number of spans, which is linear, $G$ is the power spectral density ($W/Hz$).

The unit of $\mu$ will be $W^{-2} \cdot Hz^2 \cdot dB^{-1}$, the unit of $\rho$ will be $Hz^{-2} \cdot dB^{-1}$, the overall unit will be :

$$W^{-2} \cdot Hz^2 \cdot dB^{-1} \cdot W^3 \cdot Hz^{-3} \mathrm{arcsinh}\,(dB^{-1}) = W \cdot Hz^{-1} \cdot dB^{-1} \mathrm{arcsinh}\,(dB^{-1})$$

So it seems to me that the $\mathrm{arcsinh}\,(dB^{-1})$ will produce a measure with unit in $dB$, which I don't understand why. Could anybody please explain that to me ?

  • $\begingroup$ I don't have too much experience with arcsinh, but I have found that dB units tend to behave very differently than others because they represent a logairthmic quantity rather than a linear one. This is actually a major challenge in metrology at this time. $\endgroup$
    – Cort Ammon
    Mar 7, 2020 at 2:02

1 Answer 1


Functions that can be written as infinite polynomials don't have dimensions. So arcsinh(x) is dimensionless and x has to be dimensionless.

  • $\begingroup$ But then if you take a look at the term inside arcsinh, the unit is $dB^{-1}$ $\endgroup$ Mar 7, 2020 at 3:36
  • 2
    $\begingroup$ @NguyenQuangAnh dB is dimensionless $\endgroup$ Mar 7, 2020 at 21:34
  • $\begingroup$ Then arcsinh($dB^{-1}$) has unit as $dB^{-1}$ ? If so then for the first formula, the unit at LHS is $W \cdot Hz^{-1}$ while the RHS unit is $W \cdot Hz^{-1} \cdot dB^{-1} \cdot dB^{-1}$, which is very strange $\endgroup$ Mar 8, 2020 at 4:55

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