On the performance of coded/shaped modulation

An upper bound on the error probability of a shaped modulation scheme is derived. The bound shows that nonequiprobable signaling affects the performance in two ways. First, it reduces the average signal energy which contributes positively to the shaping gain. Second, it reduces the "equivalent minimum distance" of the signal set relative to the noise level which contributes negatively to the shaping gain. The "equivalent minimum distance" becomes the true minimum distance in the signal set when the noise level is near zero (SNR/spl rarr//spl infin/). Thus, the conventional definition of shaping gain is in fact the asymptotic shaping gain.<
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