The ARQ-DMT Theorem

Fix an arbitrary $L$-round ARQ code operating at effective long-term rate $\bar R(\text{SNR}) = r \log_2 \text{SNR}$. Each round uses $N$ channel uses and targets per-round rate $R = \bar R / 1$ or $\bar R / L$ depending on the protocol flavour — but the information-theoretic bottleneck is the cumulative mutual information across all $L$ rounds, namely $$I_L \;=\; \sum_{\ell=1}^L \log_2 \det\!\left(\mathbf{I}_{n_t} + \tfrac{\text{SNR}}{n_t}\mathbf{H}_\ell^H \mathbf{H}_\ell\right).$$ The message $W$ carries $L \cdot R$ bits (it is decodable only after $L$ rounds if the first $L - 1$ fail), so decoding succeeds with vanishing error probability only if $$I_L \;\ge\; L R \;=\; L r \log_2 \text{SNR}.$$ The $L$-round outage event is $\mathcal{O}_L = \{I_L < L r \log_2 \text{SNR}\}$ and $P_e \ge P_\mathrm{out}(L r \log_2 \text{SNR}, L) = \Pr(\mathcal{O}_L)$.

Repeat the Ch. 12 large-deviations computation, now with $L$ independent eigenvalue exponent vectors $\boldsymbol{\alpha}^{(\ell)} \in \mathbb{R}_+^{\min(n_t, n_r)}$, one per round. Each round contributes (by independence) an additive term to the log-density, $$p(\boldsymbol{\alpha}^{(1)}, \ldots, \boldsymbol{\alpha}^{(L)}) \;\doteq\; \prod_{\ell=1}^L \text{SNR}^{-\sum_i (2i - 1 + n_r - n_t)\alpha_i^{(\ell)}}.$$ The outage event (in high-SNR limit) becomes $$\sum_{\ell=1}^L \sum_i (1 - \alpha_i^{(\ell)})^+ \;<\; L r,$$ and the outage exponent is the optimum of the linear program $$d_\mathrm{ARQ, out}(r, L) \;=\; \inf_{\{\boldsymbol{\alpha}^{(\ell)}\}} \sum_{\ell=1}^L \sum_i (2i - 1 + n_r - n_t)\alpha_i^{(\ell)}$$ subject to ordering and the outage constraint above.

The LP is separable: for a fixed allocation $r_\ell \ge 0$ with $\sum_\ell r_\ell = L r$, the problem decomposes into $L$ independent Zheng-Tse LPs, each evaluated at per-round multiplexing $r_\ell$. The inner optimum is $d^{*}(r_\ell)$, i.e., $$d_\mathrm{ARQ, out}(r, L) \;=\; \min_{\{r_\ell \ge 0,\,\sum_\ell r_\ell = L r\}} \sum_{\ell=1}^L d^{*}(r_\ell).$$ Since $d^{*}(\cdot)$ is convex (piecewise-linear, decreasing, convex on each linear piece), the sum $\sum_\ell d^{*}(r_\ell)$ under the sum constraint $\sum_\ell r_\ell = L r$ is minimised by equal allocation $r_\ell = r$ for all $\ell$ (Jensen in reverse for concave minimisation, which for convex functions becomes equal allocation at the inf). Wait — let's be careful. For a convex $f$, $\sum f(x_\ell)$ subject to $\sum x_\ell$ fixed is minimised at the equal allocation (by Jensen). $d^{*}$ is convex (it is piecewise linear with decreasing slopes, i.e., convex). So the minimum is at $r_\ell = r$ for all $\ell$, giving $$d_\mathrm{ARQ, out}(r, L) \;=\; L \cdot d^{*}(r).$$

The above gives $L \cdot d^{*}(r)$, not $L \cdot d^{*}(r/L)$. The discrepancy comes from how we parametrise the rate. The outage bound is $I_L < L R$, and we wrote $R = r \log_2 \text{SNR}$ — treating $r$ as the per-round multiplexing gain. With that convention, the exponent is $L \cdot d^{*}(r)$.

The ARQ-DMT formula $L \cdot d^{*}(r/L)$ uses $r$ as the long-term effective multiplexing, i.e., $r = \lim_{\text{SNR}\to\infty} \bar R / \log_2 \text{SNR}$ where $\bar R$ is the effective rate averaged over all rounds. For IR-HARQ with a mother code of rate $R$ per round, the $L$-round rate envelope is $LR$ bits transmitted over $LN$ channel uses, i.e., long-term rate $R$ per channel use — but expressed in the per-round-multiplexing variable this is $r = R / \log_2 \text{SNR}$ per round.

Re-parametrising so that $r$ is the long-term effective multiplexing gain, the constraint $I_L \ge L R$ becomes $I_L \ge r \log_2 \text{SNR}$ (since the mother code transmits $r \log_2 \text{SNR}$ information bits per round over $N$ channel uses, and the decoder needs $I_L \ge r \log_2 \text{SNR}$ for success). The per-round "effective rate" seen by the LP is now $r / L$ (spread across $L$ rounds), giving $$d_\mathrm{ARQ, out}(r, L) \;=\; L \cdot d^{*}(r/L).$$ This is the claimed formula. The two parametrisations — per-round and long-term — differ by a factor of $L$; the ARQ-DMT convention is to use the long-term rate for $r$, as this is what a system designer cares about.

For achievability, use a Gaussian random codebook of rate $r \log_2 \text{SNR}$ (long-term effective), with i.i.d. $\mathcal{CN} (0, \text{SNR}/n_t)$ entries per position. Partition the codeword into $L$ fragments $\mathbf{X}_1, \ldots, \mathbf{X}_L$ of $N$ channel uses each. Round $\ell$ transmits $\mathbf{X}_\ell$. The joint ML decoder after round $\ell$ operates on $(\mathbf{Y}_1, \ldots, \mathbf{Y}_\ell)$ against the virtual $(\ell n_r) \times n_t$ block-diagonal channel $\mathrm{diag}(\mathbf{H}_{1}, \ldots, \mathbf{H}_\ell)$.

By a Gallager random-coding argument (lifted from Ch. 12 with the modification that the decoder now sees a growing virtual channel across rounds), the ensemble-average PEP at any $\ell$ is bounded by the outage at rate $r \log_2 \text{SNR}$ plus an exponentially-small Gallager correction. The $L$-round error event is $\{I_L < r \log_2 \text{SNR}\}$, and the corresponding exponent matches the converse: $L \cdot d^{*}(r/L)$.

The block-length condition $N \ge n_t + n_r - 1$ (inherited from Ch. 12) ensures that each per-round error-matrix product $\boldsymbol{\Delta}_\ell \boldsymbol{\Delta}_\ell^H$ is full-rank with probability 1, so the Gallager correction does not dominate the outage exponent. $\blacksquare$