Chapter 16: Lattice Codes for the AWGN Channel

Learning Objectives

• Define a nested lattice code $\mathcal{C} = \Lambda_c \cap \mathcal{V}(\Lambda_s)$ by intersecting a fine coding lattice $\Lambda_c$ with a Voronoi shaping region of a coarse lattice $\Lambda_s \subset \Lambda_c$, and compute its rate $R = \tfrac{1}{n} \log_2 |\Lambda_c / \Lambda_s|$
• State and prove the Erez–Zamir theorem: nested lattice codes with MMSE scaling $\alpha = \text{SNR}/(1 + \text{SNR})$ plus uniform dither achieve the AWGN capacity $\tfrac12 \log_2(1 + \text{SNR})$
• Decompose the capacity gap of any lattice-coded system into an (additive) coding-gain shortfall of $\Lambda_c$ and a shaping-gain shortfall of $\Lambda_s$, with asymptotic limits $\pi e / 6 \approx 1.53$ dB (shaping) and the Poltyrev capacity gap (coding)
• Explain the Costa dirty-paper coding theorem and its lattice implementation: a modulo-lattice precoder cancels any non-causally known interference $\mathbf{s}$ without a rate penalty, realising the same capacity as on the clean channel
• Analyse the closest-lattice-point (CLP) problem, state the Pohst / Schnorr–Euchner sphere decoder, and use Viterbo–Boutros's result that average CLP complexity is polynomial in $n$ at high SNR despite worst-case NP-hardness
• Trace the operational impact of these results: MMSE-DFE in DSL/V.34, Tomlinson–Harashima precoding, MU-MIMO vector perturbation, and the forward-link to DMT-optimal LAST codes in Ch. 17