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Shaping gain is attained in schemes where a shaped subcode is chosen from alarger codebook by a codeword selection process. This includes the popularmethod of Trellis Shaping (TS), originally proposed by Forney for average powerreduction. The decoding process of such schemes is mismatched, since it isaware of only the large codebook. This study models such schemes by a randomcode construction and derives achievable bounds on the transmission rate undermatched and mismatched decoding. For matched decoding the bound is obtainedusing a modified asymptotic equipartition property (AEP) theorem derived tosuit this particular code construction. For mismatched decoding, relying on thelarge codebook performance is generally wrong, since the performance of thenon-typical codewords within the large codebook may differ substantially fromthe typical ones. Hence, we present two novel lower bounds on the capacityunder mismatched decoding. The first is based upon Gallager's random exponent,whereas the second on a modified version of the joint-typicality decoder.

Theoretic Shaping Bounds for Single Letter Constraints and Mismatched Decoding