Chapter Summary

Chapter Summary

Key Points

• 1.

  ARQ is the third axis of the MIMO tradeoff surface. Zheng-Tse (Ch. 12) gave us a two-dimensional diversity-multiplexing tradeoff $d^{*}(r)$ for a single block. El Gamal-Caire-Damen (2006) added the delay dimension $L$ — the number of allowed ARQ rounds — and produced the three-dimensional ARQ-DMT $d_\mathrm{ARQ}(r, L)$. Every real wireless system (LTE, 5G NR, Wi-Fi, DVB-S2X) retransmits; ARQ is not an afterthought but a fundamental resource reshaping the tradeoff curve.

• 2.

  The ARQ-DMT has a beautiful product structure. On an $n_t \times n_r$ i.i.d. Rayleigh block-fading channel with per-round block length $N \ge n_t + n_r - 1$ and at most $L$ ARQ rounds, the optimal diversity-multiplexing-delay tradeoff is (Thm. TARQ-DMT (El Gamal-Caire-Damen 2006)) $$d_\mathrm{ARQ}(r, L) \;=\; L \cdot d^{*}(r/L),$$ where $d^{*}(\cdot)$ is the static Zheng-Tse DMT. Each additional round multiplies the diversity exponent by the factor $d^{*}(r/L) / d^{*}(r/(L-1)) \ge 1$; the gain compounds across rounds.

• 3.

  Incremental redundancy strictly beats Chase combining. CC-HARQ retransmits the same codeword and achieves at most $L \cdot d^{*}(r)$ diversity (Thm. $L \cdot d^{*}(r)$ Diversity" data-ref-type="theorem">TChase Combining Achieves at Most $L \cdot d^{*}(r)$ Diversity) — inferior to IR-HARQ's $L \cdot d^{*}(r/L)$ whenever $d^{*}$ is strictly decreasing, i.e., for all $r > 0$. IR earns more diversity by lowering the per-round rate to $r/L$ where the static DMT is steeper, while still collecting $L$ independent fading realisations.

• 4.

  IR-LAST codes achieve the ARQ-DMT explicitly. The 2006 El Gamal-Caire-Damen paper constructs nested lattice codes over a CDA-structured alphabet with common random dithering — the IR-LAST family — and proves they attain $d_\mathrm{ARQ}(r, L)$ at every $(r, L)$ with MMSE-GDFE lattice decoding (Thm. TIR-LAST Codes Achieve the ARQ-DMT). This was the first explicit ARQ-DMT-optimal code family; it remains the canonical information-theoretic reference construction.

• 5.

  LDPC-based IR-HARQ asymptotically achieves the ARQ-DMT. Real systems use LDPC mother codes + circular-buffer rate matching + redundancy versions (RVs) instead of IR-LAST. Under BICM signalling with Gray labelling, the LDPC-IR scheme achieves the ARQ-DMT in the long-block-length limit (Thm. TLDPC-IR Asymptotically Achieves the ARQ-DMT). The gap from IR-LAST is a matter of coding gain at moderate SNR, not of asymptotic exponent.

• 6.

  The round budget $L$ is latency-bound, not DMT-bound. The ARQ-DMT grows unboundedly in $L$, but the feasible $L$ is set by the end-to-end latency budget: $L \cdot T_\mathrm{rtt} \le T_\mathrm{budget}$. For 5G NR eMBB with $T_\mathrm{rtt} \sim 4$ ms, $L \le 4$ is typical; for URLLC with $T_\mathrm{budget} = 1$ ms, $L \le 1$–$2$ or no HARQ at all. The system design picks the smallest $L$ meeting a target reliability — not the largest.

• 7.

  Independence of retransmissions is a real assumption. The ARQ-DMT formula presumes $\mathbf{H}_{1}, \ldots, \mathbf{H}_{L}$ are independent — which requires the HARQ round-trip time $T_\mathrm{rtt}$ to exceed the channel coherence time $T_\mathrm{coh}$. At pedestrian speeds with legacy numerology, consecutive HARQ rounds are strongly correlated and the effective diversity gain is $L_\mathrm{eff} \cdot d^{*}(r / L_\mathrm{eff})$ with $L_\mathrm{eff} \ll L$. 5G NR addresses this via per-round frequency hopping, which decorrelates rounds even at low mobility.

• 8.

  5G NR HARQ is a direct realisation of the ARQ-DMT. NR supports 16 parallel HARQ processes per UE per direction, each running stop-and-wait with up to 4 redundancy versions drawn from a circular buffer of the LDPC mother code. Per-round MCS adaptation + RV selection + frequency hopping together implement the IR-HARQ design that the ARQ-DMT theorem analyses. The empirical BLER-vs-SNR slope of NR HARQ closely tracks the ARQ-DMT prediction in the 20–30 dB regime typical of cellular operation.

• 9.

  The $L \to \infty$ limit is ergodic capacity. As $L \to \infty$ with fixed long-term rate $r$, the ARQ-DMT diverges: $d_\mathrm{ARQ}(r, L) \to \infty$. Operationally this means the channel averages to its ergodic capacity via the infinite-repetition law of large numbers — so ARQ interpolates smoothly between the single-shot block-fading regime (L = 1, static DMT) and the ergodic-capacity regime ($L \to \infty$, Telatar). The ARQ-DMT is a genuine unification of these two canonical channel models.