Geometric Shaping

An Alternative to Probabilistic Shaping

The point is that probabilistic shaping (Chs 19.1–3) keeps the constellation uniform and changes the INPUT distribution. An alternative is to keep the input distribution uniform but change the CONSTELLATION LOCATIONS — make the outer points closer together than the inner points. This is "geometric shaping". At high SNR the two approaches are asymptotically equivalent, but they differ in implementation cost and short-block behaviour.

Definition:
Geometric Shaping

A geometric shaped constellation $\mathcal{X}_{\rm gs}$ is a set of $M$ points in $\mathbb{C}$ chosen to minimise BER (or maximise mutual information) over a target channel, with UNIFORM input probabilities $p(x) = 1/M$ for all $x \in \mathcal{X}_{\rm gs}$ . Typical construction: non-uniform QAM where outer points are pulled inward, OR non-uniform QAM where the amplitude grid is replaced by a non-linearly-spaced grid.

Theorem: Geometric and Probabilistic Shaping Are Asymptotically Equivalent

At high SNR and large constellation size $M \to \infty$ , the shaping gain achievable by geometric shaping and by probabilistic shaping converges to the same asymptotic value $\gamma_s \to \pi e / 6 \approx 1.53$ dB. The two methods are DUAL in the sense that either a non-uniform INPUT on a uniform constellation, or a uniform INPUT on a non-uniform constellation, realises the same capacity- approaching signalling.

Proof

Mutual information equivalence

Both approaches target the same capacity-achieving input density: a Gaussian-like distribution in signal space. PS does this by reweighting uniform QAM points; GS does this by moving points to locations that are more densely packed near the centre.

Duality at high SNR

At high SNR, the MI-maximising distribution is matched (in distribution) to a continuous Gaussian. Any discrete approximation (PS or GS) converges to the continuous optimum as $M \to \infty$ .

Gap to ceiling

Both approaches converge at rate $O(1/\log M)$ to the ceiling $\pi e / 6$ . $\blacksquare$

Geometric vs Probabilistic Shaping on 16-QAM

Compare baseline 16-QAM (grey squares) with geometric shaping (red circles, non-uniform point locations) and probabilistic shaping (blue triangles, point sizes encoding probability). The two approaches are visually different but achieve the same asymptotic shaping gain.

Parameters

QAM size

M

Example: Shaping Gain at 16-QAM, SNR 15 dB

At SNR 15 dB, compute the shaping gain (in dB) of: (a) uniform 16-QAM (baseline), (b) geometric-shaped 16-QAM with points pulled radially inward by a factor $(1 - 0.06 |x|^2)$ , (c) probabilistic-shaped 16-QAM with MB parameter $\lambda = 0.5$ .

Solution

Uniform baseline

Rate $\approx 3.85$ bits/symbol at 15 dB (uniform 16-QAM MI via Gauss-Hermite).

Geometric shaping

Rate $\approx 3.91$ bits/symbol — a +0.06 bit gain, equivalent to $\sim 0.15$ dB of shaping gain.

Probabilistic shaping

Rate $\approx 3.92$ bits/symbol — very similar to GS. PS has a slight edge at matched complexity due to finer-grained control over the distribution.

Conclusion

At 16-QAM both approaches deliver ~0.1-0.15 dB shaping gain. At 256-QAM the gains approach 1 dB; at 1024-QAM, 1.3 dB.

Why PAS/PS Dominates in Standards

Despite the theoretical equivalence (Thm. 1), PS has won in modern standards (400ZR, DVB-S2X, ATSC 3.0) for PRACTICAL reasons:

The DEMAPPER is UNCHANGED: same uniform QAM detector works on the shaped signal.
The FEC code is UNCHANGED: PS composes with existing LDPC codes via the PAS architecture (§2).
Rate adaptation is CONTINUOUS: tuning $\lambda$ changes rate without switching MCS. Geometric shaping, in contrast, requires a new demapper and receiver calibration for each constellation — a migration cost that standards bodies have declined to pay.

🔧Engineering Note

Non-Uniform Constellations in ATSC 3.0

ATSC 3.0 (US digital TV broadcast, 2017) IS an exception: it uses Non-Uniform Constellations (NUC) — a form of geometric shaping. The NUCs are optimised per (modulation order, code rate) via off-line numerical search. This is the first mass-market deployment of geometric shaping. Over a broadcast channel (single transmitter, no adaptive feedback), NUCs simplify operation vs PAS. Over cellular with adaptive MCS, PAS is preferred.

Probabilistic Shaping vs Geometric Shaping

Property	Probabilistic (PS)	Geometric (GS)
Constellation	Uniform (QAM)	Non-uniform
Input distribution	MB (non-uniform)	Uniform
Encoder	Bits → CCDM → QAM	Bits → NUC mapper
Decoder	Standard QAM + FEC	Custom NUC demapper + FEC
Rate adaptation	Continuous via $\lambda$	Discrete (per MCS)
FEC integration	PAS (clean)	Separate design per NUC
Short-block loss	$O(\log n/n)$ (CCDM)	None at code level
Asymptotic gain	$\pi e/6 \approx 1.53$ dB	Same
Deployed in	400ZR, DVB-S2X, 5G research	ATSC 3.0

Key Takeaway

Geometric shaping changes the CONSTELLATION; probabilistic shaping changes the INPUT DISTRIBUTION. Both approach the 1.53 dB shaping ceiling asymptotically. PS has won in cellular/optical standards for its compatibility with existing QAM hardware and its continuous rate adaptation. GS appears in ATSC 3.0 digital broadcast where per-link adaptation is unnecessary.

CCDM and Distribution Matching Rate-Adaptive Transmission via Shaping