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Why Fine-Grained Rate Adaptation Matters

The point is that real wireless and optical links have VARYING SNR. Discrete MCS tables (5G NR has ~30 MCS levels; Wi-Fi has ~12) force the scheduler to jump between rates in 0.5-1 bit/symbol steps — a coarse quantisation that wastes spectral efficiency between MCS boundaries. Probabilistic shaping delivers CONTINUOUS rate adaptation at fixed constellation and code.

Theorem: PAS Achieves Continuous Rate Adaptation

Given a fixed LDPC code of rate $R_c$ and a fixed constellation of size $M$ , the PAS rate is $R_{\rm PAS}(\lambda) = H(P_A(\lambda)) + R_c - 1$ where $P_A(\lambda)$ is the Maxwell-Boltzmann distribution with parameter $\lambda$ , and $H(P_A(\lambda))$ ranges continuously in $[0, \log_2 \sqrt{M}]$ as $\lambda$ varies in $[0, \infty)$ . Hence $R_{\rm PAS}$ takes every value in $[R_c - 1, (\log_2 M) R_c + R_c - 1]$ .

Proof

Entropy continuity

$H(P_A(\lambda))$ is a continuous monotonically-decreasing function of $\lambda$ : $\lambda = 0$ gives uniform ( $H = \log_2 \sqrt{M}$ ); $\lambda \to \infty$ concentrates all mass on the smallest amplitude ( $H \to 0$ ).

PAS rate formula

$R_{\rm PAS} = R_c \log_2 M + R_c - 1$ for uniform input. For shaped input, replace $\log_2 M$ with $H(P_A) + \log_2 \sqrt{M}$ (the amplitude bits are shaped; sign bits remain uniform via the LDPC). Expanding: $R_{\rm PAS} = R_c (H(P_A) + \log_2 \sqrt{M}) + R_c - 1$ . At $\lambda = 0$ : $R_{\rm PAS} = R_c \log_2 M + R_c - 1$ (max). At $\lambda = \infty$ : $R_{\rm PAS} = R_c \log_2 \sqrt{M} + R_c - 1$ (min).

Intermediate values

As $\lambda$ varies continuously, $R_{\rm PAS}$ sweeps its full range. There is NO discrete jump. $\blacksquare$

PAS Continuous Rate vs MCS Staircase

Throughput vs SNR for two rate-adaptation strategies: (1) switching between 8 discrete MCS levels (staircase), (2) continuous PAS with a single fixed 64-QAM constellation. PAS closes the gap to Shannon more smoothly.

Parameters

Fixed constellation size

M

Example: PAS Rate Adaptation in 400ZR Optical

A 400ZR link operates at 64 GBaud with dual-polarisation. The fibre link varies SNR from 14 to 22 dB depending on span length. The system uses 16-QAM (4 bits/symbol per pol, 8 bits/symbol total) and LDPC rate 0.75. Estimate the PAS rate range achievable by tuning $\lambda$ .

Solution

Rate range

Max rate (uniform): $R_c \cdot 4 + R_c - 1 = 3 + 0.75 - 1 = 2.75$ bits/symbol/pol. Dual-pol: 5.5 bits/symbol. Min rate (shaping $\to 0$ ): $R_c \cdot 2 + R_c - 1 = 1.5 + 0.75 - 1 = 1.25$ bits/symbol/pol. Dual-pol: 2.5 bits/symbol.

At 64 GBaud

Max line rate $\approx 5.5 \cdot 64 = 352$ Gb/s; min line rate $\approx 160$ Gb/s. Continuous tuning of $\lambda$ spans this entire range WITHOUT switching constellation or code.

Link adaptation

As SNR varies from 14 → 22 dB, $\lambda$ is tuned from high (shaping aggressive, low rate) to zero (uniform, max rate). The transition is seamless — no re-configuration of the FEC encoder/decoder.

Online PAS Rate Adaptation

Complexity: O(

\log

iterations) per adaptation step.

Input: Measured SNR

\gamma

(dB); target BLER

\epsilon_0

.

Output: MB shaping parameter

\lambda^*

to apply at the CCDM.

1. Compute target mutual information

I^* =

PAS inverse lookup at

\gamma

,

\epsilon_0

.

2. Solve

H(P_A(\lambda^*)) + \log_2 \sqrt{M} = I^* / R_c

for

\lambda^*

.

(Binary search on

\lambda \in [0, \lambda_{\max}]

,

\lambda_{\max} \sim 1

.)

3. Configure the CCDM with shaping parameter

\lambda^*

.

4. The FEC encoder and QAM mapper remain UNCHANGED.

5. On next channel state update, goto step 1.

The inversion at step 2 uses a pre-computed table indexed by $\lambda$ . Real systems (e.g., 400ZR) update every 10-100 ms.

🔧Engineering Note

400ZR Rate Adaptation via PAS

The 400ZR specification standardises PAS-based rate adaptation for coherent optical links:

Fixed constellation: dual-polarisation 16-QAM or shaped 16-QAM.
Fixed LDPC: rate 15/16 (Concatenated LDPC + staircase code).
Shaping parameter: 32 discrete $\lambda$ levels, selected by the transponder's DSP per 100 ms.
Monitoring: per-block BER triggers $\lambda$ increment/decrement.
Line rate: 300-400 Gb/s depending on $\lambda$ . This is the FIRST mass-market deployment of PAS. Later 800ZR (2022) and 1.6T (2025) extend the framework to larger constellations.

📋 Ref: OIF-400ZR-01.0 Section 7

Common Mistake: Adding MCS Levels Is Not the Same as PAS

Mistake:

"We don't need PAS — just add more MCS levels to the table for finer rate granularity."

Correction:

Each new MCS level requires: (a) a new (M, R) pair; (b) SNR threshold calibration; (c) receiver-side verification. Adding many MCS levels balloons the standard's complexity. PAS provides continuous adaptation with a SINGLE (M, R) pair and ONE tunable parameter — massively simpler for the standard and for the transceiver hardware.

Why This Matters: Future Shaping Methods

Ch 22 discusses emerging alternatives: autoencoder-learned constellations (equivalent to learned geometric shaping) and adversarial-trained DMs (adaptive to unknown channels). The principle of tunable shaping extends naturally to AI-driven link adaptation.

Key Takeaway

PAS delivers continuous rate adaptation with a single (M, R) pair — a significant simplification over discrete MCS tables. The tunable parameter is the MB shaping coefficient $\lambda$ , translating directly into shaped-input entropy and hence achievable rate. 400ZR, DVB-S2X, and ATSC 3.0 all use this principle; 5G NR Rel-18+ is studying PAS for future standards.

Rate-Adaptive Transmission via Shaping