Ferkans — Interactive Telecom Tutor

Why Shaping Is Orthogonal to Coding

Coding and shaping contribute independently to the gap to capacity. The CM capacity decomposition is

$\underbrace{\tfrac{1}{2} \log_2(1 + \text{SNR})}_{\text{Shannon}} \;=\; \underbrace{R_{\rm code}}_{\text{coding side}} \;+\; \underbrace{\gamma_c}_{\text{coding gain}} \;+\; \underbrace{\gamma_s}_{\text{shaping gain}} \;+\; \underbrace{\Delta_{\rm gap}}_{\text{residual}},$

where the three gains combine additively in dB. Coding gain $\gamma_c$ arises from the structure of the codewords — it depends on the code and the lattice, not on the constellation boundary. Shaping gain $\gamma_s$ arises from the shape of the constellation boundary — it depends on the marginal distribution of the transmitted symbols, not on the code. The two improvements are additive in dB because they act on orthogonal degrees of freedom, and either one can be improved without touching the other.

The practical impact: a $6$ -dB coding gain + a $1$ -dB shaping gain gives a $7$ -dB total gain over uncoded QAM, and we can chase each independently. Shaping is, in a very concrete sense, the "free" dB we leave on the table when we use a square QAM alphabet with uniform probability.

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Definition:
Voronoi Constellation

Let $\Lambda_c$ be a coding lattice and let $\Lambda_s \subset \Lambda_c$ be a shaping lattice — a sublattice whose Voronoi region $\mathcal{V}(\Lambda_s)$ will play the role of the constellation boundary. The Voronoi constellation with coding lattice $\Lambda_c$ and shaping lattice $\Lambda_s$ is

$\mathcal{X}(\Lambda_c, \Lambda_s) \;=\; \Lambda_c \cap \bigl(\mathcal{V}(\Lambda_s) + \boldsymbol{\delta}\bigr),$

where $\boldsymbol{\delta}$ is an arbitrary offset (usually chosen for convenience — e.g., to make the constellation zero-mean). The constellation has $|\Lambda_c / \Lambda_s|$ points.

The average energy of the constellation equals the per-dimension second moment of a uniform distribution over $\mathcal{V}(\Lambda_s)$ , which depends only on $\Lambda_s$ .

The idea is to shape the "boundary" of the constellation to look like a Voronoi region of some good lattice — something close to a sphere — rather than the square boundary of ordinary QAM. The sphere has minimum energy for a given number of points, so a sphere-like boundary shapes the marginal distribution of transmitted symbols toward the Gaussian ideal (high-entropy, low-energy) that Shannon capacity demands.

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Definition:
Normalised Second Moment $G(\Lambda)$

The second moment of a lattice $\Lambda$ is the per-dimension expected squared norm of a uniformly distributed point in its Voronoi region:

$\sigma^2(\Lambda) \;=\; \frac{1}{n \, V(\Lambda)} \int_{\mathcal{V}(\Lambda)} \|\mathbf{x}\|^2 \, d\mathbf{x}.$

The normalised second moment is the dimensionless ratio

$G(\Lambda) \;=\; \frac{\sigma^2(\Lambda)}{V(\Lambda)^{2/n}}.$

By construction $G(\Lambda)$ is scale-invariant — rescaling $\Lambda \to \alpha \Lambda$ does not change it. Hence $G(\Lambda)$ measures the "shape" of the Voronoi region, not its size.

Low $G(\Lambda)$ means the Voronoi region is close to a sphere (small second moment per unit volume). The cube $\mathbb{Z}^n$ has $G(\mathbb{Z}^n) = 1/12$ ; the hexagonal lattice $A_2$ has $G(A_2) = 5/(18\sqrt{3}) \approx 0.0802$ ; $E_8$ has $G(E_8) \approx 0.0717$ ; the best known lattice at $n = 24$ achieves $G(\Lambda_{24}) \approx 0.0658$ . The infimum as $n \to \infty$ is $G_\infty = \lim_{n \to \infty} G(n\text{-ball}) = 1/(2 \pi e) \approx 0.0585$ .

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Definition:
Shaping Gain

The shaping gain of a lattice $\Lambda_s$ (or equivalently of a Voronoi constellation shaped by $\Lambda_s$ ) is the ratio of the second moment of $\mathbb{Z}^n$ (the cubic reference) to the second moment of $\Lambda_s$ at the same volume:

$\gamma_s(\Lambda_s) \;=\; \frac{1/12}{G(\Lambda_s)} \;=\; \frac{G(\mathbb{Z}^n)}{G(\Lambda_s)}, \qquad \gamma_s[\mathrm{dB}] = 10 \log_{10} \gamma_s.$

Because $G$ is scale-invariant, $\gamma_s$ measures exclusively how much energy we save by using $\Lambda_s$ 's Voronoi region instead of a cube of the same volume.

Equivalently, $\gamma_s = 1/(12 G(\Lambda_s))$ . A sphere-like Voronoi region (small $G$ ) gives a large $\gamma_s$ . A cube-like region ( $G = 1/12$ ) gives $\gamma_s = 1$ , i.e.\ $0$ dB — no shaping gain, as expected for axis-aligned QAM.

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Voronoi Constellation: Square vs Circle vs Hex Shaping

A 2D Voronoi constellation formed by intersecting $\mathbb{Z}^2$ with a shaping region: square ( $\mathbb{Z}^2$ -Voronoi, giving standard QAM), circle (sphere-shaped, giving the $\pi e / 6$ -approaching ideal), or hexagonal ( $A_2$ -Voronoi, giving the optimal 2D shaping). The "radius" of the shaping region is chosen so each option contains the same number of constellation points. Notice how the average energy $\bar{E}_s$ reported in the title drops as the boundary becomes more sphere-like — that is the shaping gain made visible.

Parameters

Shaping region

Constellation size64

Theorem: Shaping Gain $=$ Reciprocal of $12 G(\Lambda_s)$

Let $\mathcal{X}$ be a Voronoi constellation shaped by a lattice $\Lambda_s$ in $n$ dimensions, with constellation size $|\Lambda_c / \Lambda_s| = M$ . As $M \to \infty$ (continuous-approximation limit), the average transmit energy per dimension of $\mathcal{X}$ satisfies

$\bar{E}_s(\mathcal{X}) \;=\; \sigma^2(\Lambda_s) \;=\; G(\Lambda_s) \cdot V(\Lambda_s)^{2/n}.$

Consequently, at a fixed rate (equivalently, fixed $V(\Lambda_s)^{2/n}$ per point), the energy saving over a $\mathbb{Z}^n$ -shaped constellation of the same size is exactly

$\gamma_s \;=\; \frac{1/12}{G(\Lambda_s)}.$

In the continuous approximation, the discrete points of $\mathcal{X}$ are asymptotically uniformly distributed over $\mathcal{V}(\Lambda_s)$ , so the average energy is the uniform-distribution second moment $\sigma^2(\Lambda_s) = G(\Lambda_s) V(\Lambda_s)^{2/n}$ . Fixing the point density ( $V(\Lambda_s)$ ) makes the energy a pure function of $G(\Lambda_s)$ , and taking the ratio to the cubic case gives the claim.

Show Hint

Count the constellation points and relate to $V(\Lambda_s)$ .

Write the average energy as an integral over the Voronoi region.

Normalise by point density to compare fairly to the cubic reference.

Proof

Point density

The constellation $\mathcal{X} = \Lambda_c \cap \mathcal{V}(\Lambda_s)$ has exactly $|\Lambda_c/\Lambda_s| = V(\Lambda_s)/V(\Lambda_c)$ points (by TIndex Equals Volume Ratio). Equivalently, the spatial density of constellation points is $1/V(\Lambda_c)$ — independent of $\Lambda_s$ .

Average energy as an integral

In the continuous approximation (large $M$ ), summing over $\mathcal{X}$ is well-approximated by integrating over $\mathcal{V}(\Lambda_s)$ and dividing by the volume: $\bar{E}_{\rm tot} = \frac{1}{|\mathcal{X}|} \sum_{\mathbf{x} \in \mathcal{X}} \|\mathbf{x}\|^2 \;\approx\; \frac{1}{V(\Lambda_s)} \int_{\mathcal{V}(\Lambda_s)} \|\mathbf{x}\|^2 \, d\mathbf{x}.$ Dividing by $n$ to obtain per-dimension energy and invoking the definition of $\sigma^2(\Lambda_s)$ gives $\bar{E}_s = \sigma^2(\Lambda_s)$ .

Ratio to the cubic reference

For the cube $\Lambda_s = \alpha \mathbb{Z}^n$ at the same volume $\alpha^n = V(\Lambda_s)$ , we have $\sigma^2(\alpha \mathbb{Z}^n) = \alpha^2 / 12 = V(\Lambda_s)^{2/n} / 12$ . The ratio of energies is $\frac{\sigma^2(\alpha \mathbb{Z}^n)}{\sigma^2(\Lambda_s)} \;=\; \frac{V(\Lambda_s)^{2/n} / 12}{G(\Lambda_s) V(\Lambda_s)^{2/n}} \;=\; \frac{1}{12 G(\Lambda_s)},$ which is exactly $\gamma_s$ . $\blacksquare$

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Example: Shaping Gain of the Hexagonal Lattice $A_2$

Compute the shaping gain $\gamma_s$ of the hexagonal lattice $A_2$ in $\mathbb{R}^2$ . Compare to the ultimate $\pi e / 6$ ceiling.

Solution

Normalised second moment of $A_2$

$A_2$ is the densest 2D lattice. Its Voronoi region is a regular hexagon. A direct integration over this hexagon (see Conway–Sloane, Ch. 2) gives $G(A_2) = \frac{5}{18\sqrt{3}} \approx 0.080188.$

Shaping gain

$\gamma_s(A_2) = \frac{1/12}{5/(18\sqrt{3})} = \frac{18 \sqrt 3}{60} = \frac{3 \sqrt 3}{10} \approx 1.0392$ , i.e.\ $\gamma_s[\mathrm{dB}] = 10 \log_{10}(1.0392) \approx 0.167$ dB.

Ceiling comparison

The ultimate shaping gain as $n \to \infty$ is $\pi e / 6 \approx 1.4233$ , i.e.\ $\approx 1.533$ dB. So $A_2$ gets about $11\%$ of the way to the ceiling — modest but non-zero. The lesson is that shaping gain grows slowly with dimension: we need to go to $n \sim 16$ to recover a full dB, and we can never exceed $1.53$ dB even in infinite dimensions. Higher-dimensional lattices close the remaining gap but pay in complexity.

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Shaping Gain of Canonical Lattices

Lattice $\Lambda_s$	$n$	$G(\Lambda_s)$	$\gamma_s$ [dB]	% of $\pi e / 6$
$\mathbb{Z}^n$ (cube)	any	$0.0833$	$0.00$	$0\%$
$A_2$ (hexagonal)	$2$	$0.0802$	$0.17$	$11\%$
$D_4$	$4$	$0.0766$	$0.37$	$24\%$
$E_8$	$8$	$0.0717$	$0.65$	$42\%$
$\Lambda_{16}$ (Barnes–Wall)	$16$	$0.0683$	$0.86$	$56\%$
$\Lambda_{24}$ (Leech)	$24$	$0.0658$	$1.03$	$67\%$
$n$ -sphere (limit)	$\infty$	$1/(2\pi e) \approx 0.0585$	$1.53$	$100\%$

Why This Matters: Shaping Gain in Lattice Space–Time Codes (Ch. 17)

The shaping gain we derive here is not an academic quantity — it is a key ingredient in the LAST (Lattice Space–Time) codes of Part IV. The construction of Erez–Zamir for the AWGN channel and its MIMO extension by El Gamal, Caire, and Damen use nested lattices where the outer lattice plays the role of the coding lattice ( $\Lambda_c$ ) and the inner lattice plays the role of the shaping lattice ( $\Lambda_s$ ). The MMSE-scaled modulo-lattice operation on the receiver side separates the two: the inner lattice determines the shaping gain (at most $1.53$ dB) while the outer lattice determines the coding gain. Chapter 17 will revisit the same decomposition for the fading channel, where shaping plays a crucial role in achieving the optimal DMT.

🔧Engineering Note

Shaping Gain in Modern Wireless Standards

In modern digital-communications standards, shaping is almost always implemented as probabilistic shaping (Ch. 19) rather than lattice-shell mapping. The rationale: probabilistic shaping allows a capacity-approaching binary code (LDPC or polar) and a simple QAM alphabet to be used unchanged, with shaping achieved by changing only the input distribution. This buys about $0.8$ – $1.0$ dB of shaping gain over uniform QAM at rates $\ge 4$ bits/symbol — well short of the ultimate $1.53$ dB but bought at minimal system-level cost. Voronoi/shell-mapping constellations of the kind discussed in this chapter are used in practice almost exclusively in legacy dial-up modems (V.34) and satellite systems (DVB-S2X shaping extension).

Practical Constraints

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Legacy: V.34 modem ( $16$ D shell mapping, $\sim 0.8$ dB shaping gain)
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Modern: 5G NR uses uniform QAM — no shaping in the standard
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Emerging: probabilistic amplitude shaping in DVB-S2X and optical coherent links ( $\sim 1$ dB shaping gain)

📋 Ref: DVB-S2X optional annex M

Voronoi constellation

A constellation formed by intersecting a coding lattice $\Lambda_c$ with the Voronoi region of a shaping lattice $\Lambda_s \subset \Lambda_c$ . Its average energy is determined by the shaping lattice alone.

Shaping gain

The dB energy advantage of a non-cubic constellation boundary over a square (QAM) boundary, at the same point density. Equal to $1/(12 G(\Lambda_s))$ . Bounded above by $\pi e / 6 \approx 1.53$ dB.

Normalised second moment

The scale-invariant quantity $G(\Lambda) = \sigma^2(\Lambda) / V(\Lambda)^{2/n}$ that measures how sphere-like the Voronoi region of a lattice is. Bounded below by the ball value $1/(2\pi e)$ as $n \to \infty$ .

Quick Check

What is the shaping gain $\gamma_s$ of the cubic lattice $\mathbb{Z}^n$ ?

$0$ dB

$1.53$ dB

Depends on $n$

$\pi e / 6$ dB

Correction:

0

dB

By definition, $\gamma_s(\mathbb{Z}^n) = (1/12) / G(\mathbb{Z}^n) = (1/12) / (1/12) = 1$ , i.e.\ $0$ dB. The cube is our baseline — ordinary QAM has no shaping gain.

Key Takeaway

Shaping gain depends only on the shape of the constellation boundary. Build the boundary from the Voronoi region of a shaping lattice $\Lambda_s$ ; the resulting shaping gain is $\gamma_s = 1 / (12 G(\Lambda_s))$ , where $G(\Lambda_s)$ is the dimensionless normalised second moment. The sphere is optimal, giving the ceiling $\pi e / 6$ ; all finite-dimensional lattices fall short. The next section proves this ceiling rigorously and shows where it comes from.

Voronoi Constellations and Shaping

Why Shaping Is Orthogonal to Coding

Definition: Voronoi Constellation

Definition: Normalised Second Moment G(Λ)G(\Lambda)G(Λ)

Definition: Shaping Gain

Voronoi Constellation: Square vs Circle vs Hex Shaping

Parameters

Theorem: Shaping Gain === Reciprocal of 12G(Λs)12 G(\Lambda_s)12G(Λs​)

Point density

Average energy as an integral

Ratio to the cubic reference

Example: Shaping Gain of the Hexagonal Lattice A2A_2A2​

Normalised second moment of $A_2$

Shaping gain

Ceiling comparison

Shaping Gain of Canonical Lattices

Why This Matters: Shaping Gain in Lattice Space–Time Codes (Ch. 17)

Shaping Gain in Modern Wireless Standards

Voronoi constellation

Shaping gain

Normalised second moment

Quick Check

Key Takeaway

Definition:
Voronoi Constellation

Definition:
Normalised Second Moment $G(\Lambda)$

Definition:
Shaping Gain

Theorem: Shaping Gain $=$ Reciprocal of $12 G(\Lambda_s)$

Example: Shaping Gain of the Hexagonal Lattice $A_2$