The Performance of Coded Modulation with Gallager Mapping in the Finite Length Regime
Chong Zhang, Xijin Mu, Jinhong Yuan, Yiqing Zhou, Jinglin Shi · 2019
This paper investigates the capacity of coded modulation (CM) schemes with Gallager mapping based constellations in the finite length regime. We first present the channel capacity for M-ary modulations with a finite block-length. Then, we discuss the performance of Gallager mapping based constellation with probabilistic shaping. We show that at an asymptotic high signal-to-noise-ratio (SNR), the channel capacity of equiprobable constellations with a finite block-length can reach the capacity with infinite block-length while the channel capacity of nonequiprobable constellations can not approach the capacity in the infinite length regime. Finally, we discuss the channel capacity loss of non-equiprobable constellations in the finite length regime. At an asymptotic high SNR, the channel dispersion of nonequiprobable constellations equals to a positive constant, which causes the channel capacity loss for non-equiprobable constellations. We also derive an upper bound and a tight approximation of the asymptotic capacity loss (ACL) of the constellations in the finite length regime.