How to Compute GMI

Assuming that
i) the code bits are independent and uniformly distributed, and
ii) log-likelihood ratios (LLRs) for the code bits were correctly calculated, the generalized mutual information (GMI) can be approximated as

where m is the number of bit positions in the multilevel modulation, n is the number of  transmitted symbols, and ck,l and λk,l are the code bits and LLRs, respectively.

When the LLRs are “mismatched”, i.e., when they were incorrectly calculated or when approximations were used (like for example the max-log approximation), the GMI should be estimated as

For more details, see:

[R1] L. Szczecinski, A. Alvarado, “Bit-Interleaved Coded Modulation: Fundamentals, Analysis and Design,” John Wiley & Sons, ISBN: 978-0-470-68617-1, January 2015.
[R2] A. Alvarado, and E. Agrell, “Four-Dimensional Coded Modulation with Bit-wise Decoders for Future Optical Communications“, J. Lightw. Technol., vol. 33, no. 10, pp. 1993–2003, May 2015.
[R3] A. Alvarado, E. Agrell, D. Lavery, R. Maher, and P. Bayvel, “Replacing the Soft FEC Limit Paradigm in the Design of Optical Communication Systems“, J. Lightw. Technol., 2015.

Please note that the expressions in Sec. III-D of [R3] are valid only if the constellation is normalized to unit energy. We erroneously claimed in Sec. III-D of [R3] that those expressions are valid for any average symbol energy.

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