Chapter Summary

Chapter Summary

Key Points

• 1.

  Non-coherent space-time coding. The Marzetta-Hochwald (1999) pre-log result gives capacity $n_t^*(1 - n_t^*/T) \log \text{SNR}$ with $n_t^* = \min(n_t, n_r, \lfloor T/2 \rfloor)$. Capacity- achieving inputs live on Grassmannian manifolds. Explicit DMT- optimal codes for arbitrary $(n_t, n_r, T)$ remain OPEN.

• 2.

  URLLC and finite-blocklength theory. Shannon's capacity is asymptotic. At URLLC blocklengths ($n \sim 100$, BLER $\sim 10^{-5}$) the Polyanskiy-Poor-Verdú normal approximation $R^*(n, \epsilon) \approx C - \sqrt{V/n}\,Q^{-1}(\epsilon)$ is the correct design target. 5G NR URLLC MCS tables reflect this gap.

• 3.

  Autoencoder-based code design. Neural networks can learn end-to-end codes that outperform hand-design on NONLINEAR channels (HPA, phase noise). The open problem is GENERALISATION: learned codes are trained for a specific channel and may not retain their advantage out-of-distribution. Certification for safety-critical systems remains unresolved.

• 4.

  Optical fibre coded modulation. The Kerr nonlinearity produces a rate-vs-power PEAK (Essiambre et al. 2010 GN model). Probabilistic amplitude shaping (PAS, Ch 19) delivers 0.8-1 dB of the 1.53 dB shaping gap and is now deployed in 400ZR coherent optical systems. Beyond-GN capacity remains open.

• 5.

  Seven CommIT contributions. This book tagged seven Caire- affiliated landmark papers: BICM (Ch 5, 6), BICM error exponents (Ch 7), CDA codes (Ch 13), IR-LAST for ARQ (Ch 14), LAST (Ch 17), structured LAST (Ch 17), BICM-OFDM-STBC (Ch 21). Together they define 25+ years of coded-modulation research at CommIT.

• 6.

  The 70-year arc. Shannon 1948 → Ungerboeck 1982 → Caire- Taricco-Biglieri 1998 → Zheng-Tse 2003 → El Gamal-Caire-Damen 2004 → Elia-Kumar-Caire 2006 → Akay-Ayanoglu-Caire 2006 → PAS 2015 → today's autoencoder / quantum / near-field frontier. The book ends where current research begins.