[Paper Review] Achievable Rates for Probabilistic Shaping

Georg Böcherer|arXiv (Cornell University)|Jul 4, 2017

Probabilistic and Robust Engineering Design7 references56 citations

TL;DR

This paper derives achievable rates for layered probabilistic shaping (PS), including PAS, using Gallager’s error exponent framework, and analyzes decoding metrics and their impact on rate.

ABSTRACT

For a layered probabilistic shaping (PS) scheme with a general decoding metric, an achievable rate is derived using Gallager's error exponent approach and the concept of achievable code rates is introduced. Several instances for specific decoding metrics are discussed, including bit-metric decoding, interleaved coded modulation, and hard-decision decoding. It is shown that important previously known achievable rates can also be achieved by layered PS. A practical instance of layered PS is the recently proposed probabilistic amplitude shaping (PAS).

Motivation & Objective

• Motivate probabilistic shaping as a way to match non-uniform capacity-achieving inputs for channels.
• Introduce layered PS and achievable code rates to distinguish between encoding and transmission rates.
• Derive general achievable rate expressions using Gallager’s error exponent approach for a generic decoding metric.
• Instantiate the framework for specific decoding metrics such as bit-metric decoding, interleaved coded modulation, and hard-decision decoding.

Proposed method

• Define layered probabilistic shaping with a shaping set and a random codebook.
• Use Gallager’s error exponent method to derive conditions for successful encoding and decoding in the layered PS setting.
• Introduce achievable code rates and derive the transmission-rate expression from encoding and divergence terms.
• Provide a generic uncertainty-based rate expression and specialize it for several decoding metrics.
• Present practical instances, including PAS as a concrete layered PS scheme, and relate results to existing transceiver concepts.

Experimental results

Research questions

• RQ1 What achievable rate can be guaranteed for layered PS under a general decoding metric?
• RQ2 How do encoding and decoding constraints interact with the input distribution P_X and the uniform codebook to determine achievable rates?
• RQ3 What are the specific achievable rates for common metrics such as bit-metric decoding, interleaved coded modulation, and hard-decision decoding?
• RQ4 How does the choice of decoding metric q influence rate via uncertainty, divergence, and output perspectives?
• RQ5 How do PAS and similar layered PS schemes fit within the derived framework and compare to classical transceivers?

Key findings

• A closed-form condition for successful layered PS encoding is given by R_tx < [R_c − D(P_X || P_U)]^+.
• The decoding-rate bound is R_c < T_c(x^n, y^n, q) with an explicit expression for T_c involving log and a normalized q-term.
• For memoryless channels, a single-letter achievable rate T_c is derived, depending on the expected uncertainty term involving q and P_XY.
• Bit-metric decoding yields a practical achievable rate that matches the ABC (Achievable Binary Code) rate formulation.
• Interleaved coded modulation is analyzed with a unified rate expression that accounts for interleaving and metric design.
• Hard-decision decoding yields a rate expression involving binary/|X|-ary entropy terms, illustrating how quantization affects PS rates.
• The framework recovers the classical LM-Rate as a special case and shows how non-uniform inputs can be incorporated into layered PS to achieve comparable rates.
• Generalized mutual information (GMI) is discussed as a tool for analyzing rate with metric q under uniform input, and a metric transformation shows equivalence with PS rates in certain settings.
• The work clarifies how PAS can be interpreted within a rigorous information-theoretic layer, connecting practical shaping to achievable rates.