Chapter 22: Open Problems

Learning Objectives

• State the Marzetta-Hochwald non-coherent MIMO result: in a block-fading model with coherence time $T$, the pre-log of the non-coherent capacity is $n_t^* (1 - n_t^*/T)$ with $n_t^* = \min(n_t, n_r, \lfloor T/2 \rfloor)$, and identify the Grassmannian geometry of the capacity-achieving input
• Reproduce the Polyanskiy-Poor-Verdú normal approximation $R(n, \epsilon) \approx C - \sqrt{V/n}\, Q^{-1}(\epsilon) + O(\log n / n)$ for the finite-blocklength AWGN channel, compute the dispersion $V = \rho(\rho+2)/(2(\rho+1)^2 \log_2^2 e)$, and quantify the finite-blocklength gap for URLLC operating points at BLER $10^{-5}$ and blocklength $n = 100$
• Explain the autoencoder approach to end-to-end physical-layer learning (O'Shea-Hoydis 2017): encoder and decoder as jointly-trained neural networks with a channel layer in between, contrast learned constellations on AWGN versus on nonlinear HPA / phase-noise channels, and identify the open theoretical question of generalisation guarantees outside the training channel
• State the nonlinear Shannon peak for optical fibre coded modulation (Essiambre-Kramer-Winzer-Foschini-Goebel 2010, GN model): the achievable rate rises then falls with launch power, with a peak at a fibre-length-dependent optimum, and identify why the fibre channel has no Shannon-type capacity theorem
• Recognise the arc of coded modulation from Shannon 1948 through Ungerboeck 1982, Caire-Taricco-Biglieri 1998, Zheng-Tse 2003, El Gamal-Caire-Damen 2004, and Elia-Kumar-Caire 2006 to today's PAS-based optical systems, deep-learned physical-layer codes, and near-field 6G MIMO open problems
• Identify at least four major open research directions — DMT-optimal non-coherent STC beyond $2 \times 2$, finite-blocklength MIMO fading bounds, generalisation theory for learned codes, and beyond-GN models for optical fibre — and place each on a maturity axis from conjecture to deployed technology