Hierarchical Distribution Matcher Design for Probabilistic Constellation Shaping Based on a Novel Semi-Analytical Optimization Approach

A novel design procedure for practical hierarchical distribution matchers (HiDMs) in probabilistically shaped constellation systems is presented. The proposed approach enables the determination of optimal parameters for any target distribution matcher rate. Specifically, lower bounds on energy loss, rate loss, and memory requirements are analytically estimated for HiDM architectures approximating the Maxwell Boltzmann (MB) distribution. A semi analytical optimization framework is employed to jointly optimize rate and energy loss, allowing the selection of the number of hierarchical layers, memory size, and block length required to optimize channel capacity. The accuracy of the proposed model is validated through probabilistic amplitude shaping of 16QAM (PAS 16QAM), showing good agreement between analytical predictions and simulated results. The proposed analytical tool facilitates the design of HiDM structures that are compatible with practical hardware and implementation constraints, such as those imposed by state-of-the-art application-specific integrated circuits (ASICs) and field-programmable gate arrays (FPGAs). Furthermore, the performance of the optimized HiDM structure, incorporating layer selection based on lower-bound energy loss, is evaluated over the AWGN channel in terms of normalized generalized mutual information (NGMI) as a function of the optical signal-to-noise ratio (OSNR). At a net data rate of 200 Gbps with 25% forward error correction (FEC) overhead, the proposed scheme achieves a shaping gain improvement of 2.8% compared to previously reported solutions.