Chapter Summary

Chapter Summary

Key Points

• 1.

  Two currencies: bandwidth and power. Every AWGN operating point is a pair $(\eta, E_b/N_0)$, and the Shannon limit $(E_b/N_0)_{\min}(\eta) = (2^\eta - 1)/\eta$ carves the plane into achievable and unachievable regions. Power-limited ($\eta \ll 1$) and bandwidth-limited ($\eta \gg 1$) regimes require qualitatively different code families, and the boundary at $\eta \approx 1$ bit/2D marks the transition from "use bandwidth" to "use coded modulation."

• 2.

  The ultimate energy cost: $-1.59$ dB. No scheme of any spectral efficiency can operate below $E_b/N_0 = \ln 2$, i.e., $-1.59$ dB. In the bandwidth-limited regime, every extra bit/2D costs roughly $3$ dB of power. Uncoded QAM sits $7$-$10$ dB above capacity in the bandwidth-limited regime — a gap that coding, shaping, and sharp engineering close to within about 1 dB in modern systems.

• 3.

  The gap decomposes cleanly. $\gamma_{\rm total} = \gamma_{\rm coding} + \gamma_{\rm shaping} + \gamma_{\rm finite-block}$. Coding gain (5-8 dB) is won by placing the transmitted vectors to maximize Euclidean distance; shaping gain (up to $\pi e / 6 \approx 1.53$ dB) is won by making the input distribution Gaussian-like; the finite-blocklength residual (0.3-1 dB) is an unavoidable implementation cost. The three are nearly independent design knobs.

• 4.

  On AWGN, minimum Euclidean distance is the design criterion. The pairwise error probability depends only on $\|\boldsymbol{\Delta}\|$: $P(\mathbf{x} \to \hat{\mathbf{x}}) = Q(\|\boldsymbol{\Delta}\|/(2\sigma))$. At high SNR the union bound gives $P_e \approx K_{\min} Q(d_{\rm E, \min}/(2\sigma))$, so the normalized ratio $d_{\rm E, \min}^2 / E_s$ controls the asymptotic error exponent and defines the asymptotic coding gain.

• 5.

  Hamming $\ne$ Euclidean. Classical binary coding theory targets Hamming distance; coded modulation targets Euclidean distance of the transmitted signal. These are not the same, and a binary code designed for $d_H$ concatenated with an arbitrary QAM mapper is strictly suboptimal — the SC+M bound $d_{\rm E, \min}^2 \le d_H \cdot d_{\rm E, \min}^2(\mathcal{X}, \mu)_{1\text{-bit}}$ quantifies the loss. The code must live in signal space, as Ungerboeck argued in 1982.

• 6.

  Gray labeling saves BICM. If the separated coding-plus-modulation scheme uses Gray labeling, the one-bit-neighbor distance equals $d_{\rm E, \min}^2(\mathcal{X})$ and the binary channel seen by the code is a close approximation to the true signal-space channel. The Caire-Taricco-Biglieri BICM analysis (1998) showed that BICM capacity is within 0.3-0.5 dB of CM capacity under Gray labeling at all practical rates — a result that justifies the widespread use of BICM in LTE, 5G NR, Wi-Fi, and DVB.

• 7.

  Shaping: $1.53$ dB and never more. A uniform distribution over a cubic QAM boundary is Gaussian-mismatched; a Gaussian-like distribution (or a ball-shaped boundary) recovers up to $\pi e / 6$ in energy efficiency, but never more. Probabilistic amplitude shaping, adopted in DVB-S2X and under study for 6G, is the modern practical realization. The shaping-coding decomposition makes PAS a clean add-on rather than a joint redesign of the code.

• 8.

  The roadmap ahead. Chapter 2 (TCM) and Chapter 3 (MLC/MSD) develop the first CM schemes that close the coding gain in the bandwidth-limited regime. Chapter 4 introduces coset codes and Voronoi shaping. Chapters 5-9 build the BICM framework. Chapters 10-14 extend the distance criterion to fading MIMO via rank and determinant, yielding the diversity-multiplexing tradeoff. Chapters 15-18 develop lattice codes and compute-and-forward. Chapters 19-22 cover modern extensions (probabilistic shaping, 1-bit MIMO, 5G/6G). The Chapter 1 signal-space view is the thread that runs through all of them.