AbstractFarid-Hranilovic (FH),in an interesting way, found a capacity-achieving discrete input distribution for free space optical (FSO) channel by numerically maximizing the input-parameter (β) dependent mutual information between channel input and the scaled output. In this paper, first, by using a simple mathematical inequality, we find an upper bound for FH input-scaled output mutual information and then maximize the obtained upper bound to reach to a third order equation for the optimum β as β*. Our equation (i) determines β* exactly in contrary to the FH work where β* is found numerically through an exhaustive search and also, (ii) is consistent with the estimated equation for β* in the FH work. Our upper bound is shown to be tighter than the proposed upper bound in the FH work that is found through sphere packing argument at very high SNRs. Using numerical illustrations at different SNRs, we compare our β*s, mass point spacing as *, and upper boun

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