[Paper Review] Comparison of Geometric and Probabilistic Shaping with Application to ATSC 3.0

Fabian Steiner, Georg Böcherer|arXiv (Cornell University)|Aug 1, 2016

Telecommunications and Broadcasting Technologies36 citations

TL;DR

This paper compares geometric shaping (GS) and probabilistic shaping (PS) for the AWGN channel using symbol-metric (SMD) and bit-metric decoding (BMD), proposing a differential evolution-based optimization for GS. It shows that PS with BMD achieves near-capacity performance, while GS incurs a 0.4 dB gap to capacity under BMD; simulations confirm PS outperforms ATSC 3.0 GS modcods by over 0.5 dB at spectral efficiencies >3.2 bpcu.

ABSTRACT

In this work, geometric shaping (GS) and probabilistic shaping (PS) for the AWGN channel is reviewed. Both approaches are investigated in terms of symbol-metric decoding (SMD) and bit-metric decoding (BMD). For GS, an optimization algorithm based on differential evolution is formulated. Achievable rate analysis reveals that GS suffers from a 0.4 dB performance degradation compared to PS when BMD is used. Forward-error correction simulations of the ATSC 3.0 modulation and coding formats (modcods) confirm the theoretical findings. In particular, PS enables seamless rate adaptation with one single modcod and it outperforms ATSC 3.0 GS modcods by more than 0.5 dB for spectral efficiencies larger than 3.2 bits per channel use.

Motivation & Objective

• To compare the achievable rate performance of geometric shaping (GS) and probabilistic shaping (PS) on the AWGN channel under both symbol-metric decoding (SMD) and bit-metric decoding (BMD).
• To develop a differential evolution (DE)-based optimization algorithm for designing optimal non-uniform, non-equidistant constellations in GS.
• To evaluate the practical coded performance of ATSC 3.0’s GS modcods against a single PS modcod using the PAS (Pulse Amplitude Shaping) framework.
• To assess whether theoretical information-theoretic gains from shaping translate into real-coded performance improvements in practical systems.
• To demonstrate that PS enables seamless rate adaptation with a single modcod, unlike ATSC 3.0’s multiple GS modcods.

Proposed method

• Formulates an optimization problem for GS using differential evolution (DE) to find non-uniform, non-equidistant constellation points that maximize mutual information under BMD.
• Derives the achievable rate under BMD as $ R_{\text{BMD}}(P_X, \text{SNR}) = \left[ \operatorname{H}(\bm{B}) - \sum_{i=1}^{m} \operatorname{H}(B_i|Y_i) \right]^+ $, enabling performance comparison across shaping schemes.
• Uses the PAS (Pulse Amplitude Shaping) framework to implement PS, where shaping is applied after FEC encoding and reversed at the receiver, enabling low-complexity receiver design.
• Employs 5/6-rate IRA LDPC codes of blocklength 64,800 for both ATSC 3.0 GS modcods and the PAS-based PS system, with identical interleaving and bit-mapping for fair comparison.
• Calculates required SNR for target spectral efficiencies using inverse of the BMD achievable rate function, $ \text{SNR}_{\text{req}} = \mathsf{R}_{\text{BMD}}^{-1}(P_X, R) $, to compute gap-to-capacity.
• Performs iterative BP decoding with 50 iterations and sum-product update rules to evaluate coded performance across different spectral efficiencies (2.13, 3.2, 5.33 bpcu).

Experimental results

Research questions

• RQ1 How do geometric shaping (GS) and probabilistic shaping (PS) compare in terms of achievable rate under symbol-metric decoding (SMD) and bit-metric decoding (BMD)?
• RQ2 What is the performance gap of GS relative to capacity when using bit-metric decoding (BMD), and how does it compare to PS?
• RQ3 Can the theoretical information-theoretic gains of PS over GS be realized in practical coded systems with finite blocklengths?
• RQ4 To what extent does PS with a single modcod outperform ATSC 3.0’s multiple GS modcods in terms of coded SNR gain at high spectral efficiencies?
• RQ5 Does the use of PAS enable seamless rate adaptation while maintaining performance close to the Shannon limit?

Key findings

• Geometric shaping (GS) incurs a 0.4 dB gap to capacity when using bit-metric decoding (BMD), while probabilistic shaping (PS) with BMD achieves near-capacity performance.
• Theoretical analysis shows that PS closes both the shaping and BMD gaps to capacity, whereas GS fails to close the BMD gap due to suboptimal bit-level metrics.
• Forward-error correction (FEC) simulations confirm that PS with a single PAS modcod outperforms ATSC 3.0’s GS modcods by more than 0.5 dB for spectral efficiencies above 3.2 bpcu.
• At 3.2 bpcu, the coded SNR gain of PS over ATSC 3.0 GS is 0.43 dB, exceeding the predicted 0.39 dB from information-theoretic analysis.
• For 5.33 bpcu, the coded gain is 0.44 dB, slightly exceeding the predicted 0.40 dB, indicating that practical gains can surpass theoretical expectations.
• The PAS-based PS system achieves a gap to capacity of only 0.06 dB across the entire spectral efficiency range of 1.0 to 5.33 bpcu, demonstrating its suitability for rate-adaptive, near-capacity-approaching systems.