Prerequisites & Notation

Before You Begin

This chapter generalises the Zheng-Tse diversity-multiplexing tradeoff of Chapter 12 to MIMO channels equipped with an ARQ feedback loop: the receiver can request up to $L$ retransmissions, and the system accumulates evidence across rounds until it decodes successfully or runs out of budget. The point of view is that of El Gamal, Caire, and Damen (2006): an ARQ round is a new channel realisation, each round incrementally lowers the effective rate, and both together reshape the DMT into a three-dimensional object $d_\mathrm{ARQ}(r, L)$ . To follow the arguments the reader needs the static-DMT machinery of Ch. 12 (outage exponent, exponential equality, Gaussian random-coding achievability), the block-fading MIMO channel model of Ch. 10, and the pairwise-error / rank-determinant analysis of space-time codes from Ch. 11. On the practical side, the reader should know what a hybrid ARQ (HARQ) protocol looks like in LTE/NR — an LDPC-coded transport block, a circular-buffer rate matcher, and a small number of redundancy versions transmitted in sequence.

Zheng-Tse DMT for block-fading MIMO(Review ch12)
Self-check: Can you write $d^{*}(r) = (n_t - r)(n_r - r)$ at integer corners $r \in \{0, 1, \ldots, \min(n_t, n_r)\}$ for an $n_t \times n_r$ i.i.d. Rayleigh channel with $L \ge n_t + n_r - 1$ , and explain why the curve is piecewise-linear between corners?
Outage probability of MIMO block-fading channels(Review ch10)
Self-check: Can you write $P_\mathrm{out}(R, \text{SNR}) = \Pr[\log_2 \det(\mathbf{I} + \tfrac{\text{SNR}}{n_t}\mathbf{H}\mathbf{H}^{H}) < R]$ and identify its high-SNR exponent with the DMT curve value?
Rank and determinant criteria for space-time codes(Review ch11)
Self-check: Can you state the rank criterion $\mathrm{rank}(\boldsymbol{\Delta}) \ge r \Rightarrow d \ge r n_r$ for the codeword-difference matrix $\boldsymbol{\Delta} = \mathbf{X} - \hat{\mathbf{X}}$ ?
Exponential equality $\doteq$ and large-deviations asymptotics(Review ch12)
Self-check: Can you apply the definition $f \doteq \text{SNR}^{a} \iff \lim \log f / \log \text{SNR} = a$ and manipulate sums, products, and integrals of the form $\int \text{SNR}^{-d(\alpha)}\,d\alpha \doteq \text{SNR}^{-\inf_\alpha d(\alpha)}$ ?
Basic ARQ / HARQ protocols (CC-HARQ, IR-HARQ)(Review ch12)
Self-check: Can you explain what ACK/NACK feedback does, what "stop-and-wait" means, and how Chase combining differs operationally from incremental redundancy?
Forward-error correction via LDPC and rate matching(Review ch08)
Self-check: Can you describe how a mother LDPC code is punctured into several redundancy versions via a circular buffer, and why RV_0 typically contains the systematic bits?

Notation for This Chapter

Symbols specific to the ARQ-DMT analysis. The Chapters 10–12 MIMO notation (channel matrix $\mathbf{H}$ , codeword matrix $\mathbf{X}$ , SNR $\text{SNR}$ , noise $\mathbf{w}$ , capacity $C$ , mutual information $I$ ) continues to apply and is not repeated here.

Symbol	Meaning	Introduced
$L$	Maximum number of ARQ rounds (delay budget). In Chapter 12 $L$ denoted the block length; here it denotes the ARQ round budget.	s01
$\ell$	Round index, $\ell = 1, 2, \ldots, L$	s01
$\mathbf{X}_\ell$	Codeword matrix transmitted in round $\ell$ ; for IR, each round sends fresh parity	s01
$\mathbf{H}_\ell$	MIMO channel realisation in round $\ell$ ; assumed i.i.d. across $\ell$	s01
$d_\mathrm{ARQ}(r, L)$	ARQ-DMT curve: optimal diversity at effective multiplexing $r$ with at most $L$ rounds	s02
$r$	Effective (long-term) multiplexing gain, $r = \lim_{\text{SNR}\to\infty} \bar R(\text{SNR}) / \log_2 \text{SNR}$	s01
$P_\mathrm{out}(R, L)$	$L$ -round outage probability — the probability that the accumulated $L$ -round mutual information is less than $R$	s02
$\eta_\mathrm{eff}$	Effective throughput (bits per channel use) accounting for ACK/NACK and retransmissions	s04
$T_\mathrm{rtt}$	HARQ round-trip time — the delay between transmission and the arrival of feedback	s05
$CC / IR$	Chase combining / incremental redundancy — the two canonical HARQ flavours	s01
$RV$	Redundancy version — the LTE/NR identifier (RV_0, RV_1, RV_2, RV_3) specifying which portion of the circular buffer is transmitted	s05
$BLER$	Block error rate — the probability of decoding failure after all available HARQ rounds	s04

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