Dear 802.3dh members,
There was a question following my presentation in yesterday’s 802.3dh ad hoc
In a number of presentations in 802.3 over the years, including cz and dh,
I use the approximation BW[GHz.km] = 0.2/sigma [nsec/km] to give a rough approximation of the expected bandwidth BW of a fiber given the RMS sigma in the time domain.
The question was where does the “0.2” come from because others use 0.35 etc.
The simplest, shortest answer is that 0.187/sigma is the correct relation if sigma is the standard deviation (RMS sigma).
In some cases it may make sense to use 2*sigma which will look more like the ‘width’ of the pulse (sigma is sometimes called RMS width which is confusing).
In that case one would have (2*0.187)/(2*sigma) which is consistent with the 0.35 number Watanabe-san mentioned.
===most readers can stop there ===
I have also seen 0.44/T where “T” is the full-width half max. This is not 2*.187 because the full-width half max will not be exactly 2-sigma. This has the advantage of not requiring any analysis of the pulse but just locating the leading and trailing 50% points.
The details of the 0.187 number are complicated; the “exact” number will depend on whether one literally means 3dB BW is the 3dB point (which gives 0.187) or one means transfer function has come down to |H(f)|=0.5. Because pulses are seldom perfectly Gaussian the 0.2 number works fine.
For this presentation since we are only interested in the exponent g in BW [GHz] = (L[m])^-g, it doesn’t really matter.
I should also mention that sigma and T above refer to the impulse response and assume the width of the input pulse has been subtracted off or is negligible etc. This can become a challenge with measurements at very short lengths.
References:
I’m sorry I didn’t find good open access discussion
For the 0.187/sigma number this is mentioned in various textbooks I would mention the 1992 JLT paper by Gair Brown which is part of the basis for the “IEEE link model”: (sigma is equation 2)
Gair Brown, “Bandwidth and Rise Time Calculations for Multimode Fiber-Optic Data Links”, JLT Volume 10 No. 5 May 1992 p.672.
For the 0.44/T number this is mentioned in the chapter by C. Bunge in the book Polymer Optical Fibres eqn. 3.92:
Equation 3.92 in C.A. Bunge et al., “Chapter 3-Basic Principles of optical fibres”, in Polymer Optical Fibres, ed. Bunge, Gries, Beckers. New York: Elsevier, 2017
More explanation
A nice feature about the 0.187/sigma number is that it can be translated into analysis of glass optical fibers like OM3 and OM4 which can be understand as supporting 18 mode groups each with a mode delay tau_g and a model power P_g
If the mode delays are normalized relative to the mean delay (sum of Pg*tau_g), then the RMS variance sigma^2 is (sum of Pg*tau_g^2) and sigma is the square root of this. Obviously the pulse itself is not Gaussian but what is important to the link is the low frequency content.
Dear 802.3dh participants,
The minutes of November 16th 802.3dh plenary meeting has been uploaded.
If need correction, please let me know.
Best regards,
Yuji Watanabe, AGC
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