Ferkans — Interactive Telecom Tutor

Every Real Wireless System Retransmits

Chapter 12 gave us the DMT for a single block of a block-fading MIMO channel. A single-shot transmission is a mathematical fiction: it is the right abstraction for analysing capacity, but no operational wireless system on Earth actually works that way. LTE, 5G NR, Wi-Fi, DVB-S2X, and WiMAX all retransmit on decoding failure. The receiver sends back an ACK or a NACK, and the transmitter reacts — re-sending the same codeword, or sending fresh parity bits, until either the packet is decoded or a deadline is hit.

The point is that each retransmission happens on a new channel realisation — the fading coefficients in $\mathbf{H}_{2}$ are (to a very good approximation) independent of those in $\mathbf{H}_{1}$ , provided the gap between rounds exceeds the coherence time. That independence is operationally a free resource. Zheng and Tse showed what you can extract from one Wishart matrix; the question of this chapter is what you can extract from $L$ of them.

The answer, due to El Gamal, Caire, and Damen (2006), is elegant and surprising: with optimal incremental redundancy, the diversity exponent scales linearly in $L$ at fixed effective rate. The tradeoff generalises from the two-dimensional curve $d^{*}(r)$ of Ch. 12 to a three-dimensional surface $d_\mathrm{ARQ}(r, L)$ , with the beautiful product structure $d_\mathrm{ARQ}(r, L) \;=\; L \cdot d^{*}(r/L).$ Each additional round multiplies the diversity exponent by $1/L$ of the static curve evaluated at the same rate — each round earns its keep, and the earnings compound.

This chapter develops the protocol (§1), proves the ARQ-DMT theorem (§2), constructs the incremental-redundancy lattice codes that achieve it (§3), and maps the whole story onto the HARQ mechanism used in 3GPP LTE and 5G NR (§§4–5).

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Definition:
ARQ Protocol over a Block-Fading MIMO Channel

An $L$ -round ARQ protocol over an $n_t \times n_r$ block-fading MIMO channel is the following sequence of operations, parametrised by a target rate $R$ and a maximum number of rounds $L \ge 1$ :

The transmitter has a single information message $W \in \{1, 2, \ldots, 2^{LNR}\}$ of rate $R$ per round (so total rate $LR$ over the $L$ -round block).
In round $\ell = 1$ , the transmitter sends a codeword matrix $\mathbf{X}_1 \in \mathbb{C}^{n_t \times N}$ through channel $\mathbf{H}_{1}$ ; the receiver gets $\mathbf{Y}_1 \;=\; \sqrt{\tfrac{\text{SNR}}{n_t}}\,\mathbf{H}_{1} \mathbf{X}_1 + \mathbf{w}_{1}, \qquad \mathbf{H}_{1} \sim \mathcal{CN}(0, \mathbf{I}).$
The receiver attempts to decode using all observations so far $(\mathbf{Y}_1, \ldots, \mathbf{Y}_\ell)$ . If decoding succeeds, it sends an ACK and the protocol halts. Otherwise, it sends a NACK on an (assumed error-free, zero-delay) feedback link.
On NACK, the transmitter sends $\mathbf{X}_{\ell+1}$ — a new codeword matrix, possibly a function of the message $W$ alone or of $W$ together with $(\mathbf{X}_1, \ldots, \mathbf{X}_\ell)$ . The channel $\mathbf{H}_{\ell+1}$ is drawn independently.
The protocol halts at round $L$ whether or not decoding has succeeded; if it has not, the block is declared in error.

The protocol is characterised by the round codebooks $\mathbf{X}_1, \mathbf{X}_2, \ldots, \mathbf{X}_L$ and the decoding rule. Chase combining (CC) uses $\mathbf{X}_\ell = \mathbf{X}_1$ for all $\ell$ — the same codeword retransmitted. Incremental redundancy (IR) uses $\mathbf{X}_\ell$ containing fresh parity symbols of a common mother code. The mother-code rate is $R$ per round, and after $\ell$ rounds the effective rate seen by the decoder is $R_\mathrm{eff}(\ell) = LR/\ell$ (the same information bits carried by $\ell N n_t$ instead of $L N n_t$ channel uses).

Three subtleties. First, the channel matrices $\mathbf{H}_\ell$ are assumed i.i.d. across rounds — this is the independence assumption that makes ARQ buy diversity. If the coherence time is longer than the HARQ round-trip time, consecutive rounds see correlated fading and the diversity claim weakens (see ⚠ARQ Diversity Requires Independent Retransmissions). Second, the feedback link is idealised: error-free, zero-delay, single-bit ACK/NACK. In practice, NACK-to-ACK misinterpretation rates are $\sim 10^{-3}$ and ACK/NACK travel over PUCCH/PDCCH with their own reliability budget. Third, the decoder is a joint ML decoder across all received rounds — it does not decode round-by-round, except as a complexity-limited approximation.

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Definition:
Chase Combining (CC-HARQ)

A Chase-combining HARQ protocol transmits $\mathbf{X}_\ell = \mathbf{X}_1$ at every round $\ell \in \{1, \ldots, L\}$ — i.e., it repeats the same codeword. The receiver combines the received signals at the log-likelihood level. For a binary-input AWGN round model $y_\ell = h_\ell x + w_\ell$ , the combined LLR after $\ell$ rounds is $\mathrm{LLR}_\ell(x) \;=\; \sum_{k=1}^\ell \mathrm{LLR}_k(x) \;=\; \sum_{k=1}^\ell \frac{2|h_k|^2}{{\sigma^2}^{2}}\,\mathrm{Re}(h_k^* y_k).$ The combined LLR is equivalent to coherent combining of $\ell$ independent copies of the same symbol, so the effective SNR grows linearly in $\ell$ : $\text{SNR}_\mathrm{eff}(\ell) = \ell \cdot \text{SNR}$ . In the MIMO setting, round $\ell$ sees $\mathbf{H}_\ell \mathbf{X}_1 + \mathbf{w}_\ell$ , and the combined decoder sees a virtual $\ell n_r \times n_t$ channel stacking the individual $\mathbf{H}_{k}$ .

Chase combining is simple, legacy, and still the workhorse of low-complexity HARQ. Its name honours David Chase's 1985 IEEE Trans. Comm. paper which proved that summing LLRs across repetitions is asymptotically the ML decoder for the concatenated-packet ensemble. The rate after $\ell$ rounds is $R/\ell$ — the decoder is trying to decode the same $R$ bits spread across $\ell$ times as many channel uses.

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Definition:
Incremental Redundancy (IR-HARQ)

An incremental-redundancy HARQ protocol transmits a different codeword $\mathbf{X}_\ell$ at each round, typically obtained by puncturing a common mother code of low rate $R_m = R/L$ into $L$ consecutive parity fragments. Round $\ell$ sends the $\ell$ -th fragment. After $\ell$ rounds, the decoder has observed an effective code of rate $R_\ell = R (L / \ell)$ — the systematic bits plus $\ell - 1$ additional parity fragments.

In the block-fading MIMO setting, round $\ell$ uses codeword $\mathbf{X}_\ell \in \mathbb{C}^{n_t \times N}$ and the receiver sees $\mathbf{Y}_\ell \;=\; \sqrt{\tfrac{\text{SNR}}{n_t}}\,\mathbf{H}_\ell \mathbf{X}_\ell + \mathbf{w}_\ell.$ A joint ML decoder treats the concatenation $(\mathbf{X}_1, \ldots, \mathbf{X}_\ell)$ as a single long codeword passed through the block-diagonal channel $\mathrm{diag}(\mathbf{H}_{1}, \ldots, \mathbf{H}_\ell)$ . The mutual information accumulated over $\ell$ rounds is $I_\ell \;=\; \sum_{k=1}^\ell \log_2 \det\!\left(\mathbf{I} + \tfrac{\text{SNR}}{n_t} \mathbf{H}_{k} \mathbf{H}_{k}^{H}\right),$ independently over the $\ell$ rounds. Decoding succeeds when $I_\ell \ge LR$ , i.e., when the cumulative mutual information exceeds the total number of information bits $LR$ .

The IR flavour earns its name from the fact that round $\ell > 1$ carries no new information bits (the message $W$ is already fully determined by $\mathbf{X}_1$ in principle) but does carry new redundancy — fresh parity on the same underlying bits. The decoder strictly benefits from each additional fragment.

Two equivalent pictures of IR-HARQ are worth holding in mind. First, puncturing: round $\ell$ unpunctures additional positions of a fixed rate- $R_m$ mother code, lowering the effective rate. Second, rate matching via a circular buffer: the encoded bits of the mother code are laid out along a circular buffer, and each round reads out a fragment of length $N n_t$ starting at a round-dependent offset. LTE and 5G NR adopt the circular-buffer formulation — see §5.

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Chase Combining vs Incremental Redundancy vs No HARQ

Property	No HARQ (one-shot)	Chase combining (CC)	Incremental redundancy (IR)
Rounds allowed	$L = 1$	$L \ge 1$	$L \ge 1$
Round $\ell$ payload	Codeword $\mathbf{X}_1$	Repeat $\mathbf{X}_1$	Fresh parity fragment $\mathbf{X}_\ell$
Effective rate after $\ell$ rounds	$R$	$R/\ell$	$R(L/\ell)$ of a rate- $R/L$ mother code
Combining at receiver	None	Sum LLRs (coherent)	Joint ML across $(\mathbf{X}_1, \ldots, \mathbf{X}_\ell)$
Diversity per round at fixed rate	$d^{*}(r)$	$\le L \cdot d^{*}(r)$ (strictly)	$= L \cdot d^{*}(r/L)$ (achieves ARQ-DMT)
Buffer at transmitter	None	None (resend)	Mother-code circular buffer
Use case	URLLC one-shot	Legacy / low-complexity	LTE, 5G NR eMBB, Wi-Fi
Computational complexity	Baseline	Low (LLR sum)	Higher (full decoder on each round)

Theorem: Chase Combining Achieves at Most $L \cdot d^{*}(r)$ Diversity

Consider an $L$ -round CC-HARQ protocol on an $n_t \times n_r$ i.i.d. Rayleigh MIMO channel with fixed per-round rate $R(\text{SNR}) = r \log_2 \text{SNR}$ for all rounds. The $L$ -round error probability satisfies $P_e^{\mathrm{CC}}(\text{SNR}, L) \;\doteq\; \text{SNR}^{-d^\mathrm{CC}(r, L)}, \qquad d^\mathrm{CC}(r, L) \;\le\; L \cdot d^{*}(r).$ In words: CC-HARQ can achieve at most $L$ times the static DMT evaluated at the original rate $r$ . The bound is attained only when the decoder is ML and the $L$ round channels are independent.

Comparison to IR. For the same $L$ and the same effective long-term rate $\bar r = r$ , IR achieves diversity $L \cdot d^{*}(\bar r / L) = L \cdot d^{*}(r/L)$ , which is strictly greater than $L \cdot d^{*}(r)$ for all $r > 0$ because $d^{*}(\cdot)$ is strictly decreasing on $[0, \min(n_t, n_r)]$ .

CC resends the same codeword $L$ times. Each round adds an independent fading realisation, so the effective SNR after $L$ rounds is $\text{SNR}_\mathrm{eff} \doteq L \cdot \text{SNR}$ — i.e., the combined $L$ -round channel is like a single round at SNR $L \cdot \text{SNR}$ . But the rate has stayed at $R$ , not $LR$ : CC trades the per-round rate for SNR, it does not use the extra channel uses to lower the rate slope. In DMT language, CC operates the original curve at a shifted SNR — which gives diversity $L \cdot d^{*}(r)$ , not $L \cdot d^{*}(r/L)$ . Since $d^{*}(r/L) \ge d^{*}(r)$ with equality only at $r = 0$ , IR is strictly better.

Show Hint

Stack the $L$ rounds: the CC-HARQ decoder sees a virtual $L n_r \times n_t$ channel with block-diagonal structure $\mathrm{diag}(\mathbf{H}_{1}, \ldots, \mathbf{H}_{L})$ all multiplying the same codeword $\mathbf{X}_1$ .

The outage event is $\sum_k \log_2 \det(\mathbf{I} + \tfrac{\text{SNR}}{n_t}\mathbf{H}_{k}\mathbf{H}_{k}^{H}) < R$ . Note the rate budget is $R$ , not $LR$ : each round targets the same $R$ as a single-shot code.

The outage exponent is a sum of $L$ i.i.d. static-DMT exponents evaluated at $r$ : $d^\mathrm{CC}(r, L) \le L \cdot d^{*}(r)$ .

Proof

Stack the rounds

In CC-HARQ, round $\ell$ transmits $\mathbf{X}_\ell = \mathbf{X}_1$ over channel $\mathbf{H}_\ell$ . Write the stacked observation $\bar{\mathbf{Y}} = (\mathbf{Y}_1^T, \ldots, \mathbf{Y}_L^T)^T$ as $\bar{\mathbf{Y}} \;=\; \sqrt{\tfrac{\text{SNR}}{n_t}}\, \underbrace{\begin{pmatrix}\mathbf{H}_{1} \\ \vdots \\ \mathbf{H}_{L}\end{pmatrix}}_{\bar{\mathbf{H}}} \mathbf{X}_1 + \bar{\mathbf{w}},$ with $\bar{\mathbf{H}}$ of size $L n_r \times n_t$ .

Outage event

The mutual information of the stacked channel is $I_\mathrm{stack}(\bar{\mathbf{H}}) \;=\; \log_2 \det\!\left( \mathbf{I}_{n_t} + \tfrac{\text{SNR}}{n_t} \bar{\mathbf{H}}^H \bar{\mathbf{H}}\right).$ Decoding at target rate $R$ per round succeeds iff $I_\mathrm{stack} \ge R$ (the code occupies $N$ channel uses per round, so the normalising denominator is $N$ , not $LN$ — the rate is per-round, not per-block).

High-SNR exponent

For independent rounds, $\bar{\mathbf{H}}^H \bar{\mathbf{H}} = \sum_{\ell=1}^L \mathbf{H}_\ell^H \mathbf{H}_\ell$ . By the same Wishart large-deviations argument as Ch. 12, the outage exponent of this sum is a convolution of $L$ i.i.d. copies of the single-round outage exponent. At rate $r \log_2 \text{SNR}$ , the cumulative outage exponent satisfies $-\log P_\mathrm{out}^\mathrm{CC}(r \log_2 \text{SNR}, L) / \log \text{SNR} \;\xrightarrow{\text{SNR}\to\infty}\; L \cdot d^{*}(r).$ This is because each round independently pays $d^{*}(r)$ in exponent, and the outage event requires all $L$ rounds to jointly fail — which gives an exponent of $L \cdot d^{*}(r)$ .

Converse inequality

Any code operating on the stacked channel must satisfy $P_e \ge P_\mathrm{out}^\mathrm{CC}(R)$ , so $d^\mathrm{CC}(r, L) \le L \cdot d^{*}(r)$ holds for every CC-HARQ scheme. Equality is attained by a Gaussian random code of rate $r \log_2 \text{SNR}$ and ML decoding across the $L$ rounds.

Comparison to IR

Fix a long-term effective rate $\bar r$ ; CC sets per-round rate $r = \bar r$ and achieves exponent $L \cdot d^{*}(\bar r)$ . IR sets per-round rate $r = L \bar r$ (mother-code rate $\bar r$ stretched across $L$ rounds) and achieves exponent $L \cdot d^{*}(\bar r)$ — wait, this looks the same, but the rate per round is different. The proper comparison: at long-term effective rate $\bar r$ , IR operates the static DMT at $\bar r / L$ per round (because the mother code rate is $\bar r$ and the $L$ rounds carry $L$ fragments), giving exponent $L \cdot d^{*}(\bar r / L) > L \cdot d^{*}(\bar r)$ . This is the ARQ-DMT advantage of §2. $\blacksquare$

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ARQ Diversity Contribution vs Number of Rounds $L$

Sweep the number of ARQ rounds $L \in \{1, 2, \ldots, 6\}$ and plot the ARQ-DMT diversity $d_\mathrm{ARQ}(r, L) = L \cdot d^{*}(r/L)$ at several effective rates $r \in \{0.5, 1, 1.5, 2\}$ for an $n_t \times n_r$ channel. The curves are strictly increasing in $L$ for $r > 0$ — each round earns positive diversity — and the marginal gain per round is largest at high $r$ (where $d^{*}$ is steep) and smallest at low $r$ (where $d^{*}$ is flat). This is the operational illustration of the ARQ-DMT product rule.

Parameters

n_t

2

n_r

2

Example: $2\times 2$ MIMO with $L = 3$ : CC vs IR at $r = 1$

Compare the diversity exponents of CC-HARQ and IR-HARQ on a $2 \times 2$ i.i.d. Rayleigh channel with $L = 3$ ARQ rounds at long-term effective rate $r = 1$ . Use the $2 \times 2$ static DMT $d^{*}(r) = (2 - r)^2$ at integer corners with piecewise-linear interpolation between.

Solution

Static DMT values

On $2 \times 2$ : $d^{*}(0) = 4$ , $d^{*}(1) = 1$ , $d^{*}(2) = 0$ ; interpolation linear between. We will need $d^{*}(1) = 1$ and $d^{*}(1/3) = 4 - 3 \cdot (1/3) = 3$ (the linear segment from $(0, 4)$ to $(1, 1)$ has slope $-3$ ).

CC-HARQ exponent at $r = 1$, $L = 3$

CC operates the static DMT at per-round rate $r = 1$ and sums exponents across $L = 3$ independent rounds: $d^\mathrm{CC}(1, 3) \;=\; 3 \cdot d^{*}(1) \;=\; 3 \cdot 1 \;=\; 3.$

IR-HARQ exponent at $r = 1$, $L = 3$

IR uses a mother-code rate $r / L = 1/3$ per round (a lower per-round rate than CC) and pays diversity $d^{*}(1/3) = 3$ per round: $d_\mathrm{ARQ}(1, 3) \;=\; L \cdot d^{*}(r/L) \;=\; 3 \cdot d^{*}(1/3) \;=\; 3 \cdot 3 \;=\; 9.$

Interpretation

IR triples the diversity exponent of CC ( $9$ vs $3$ ) at the same long-term rate. Operationally, at high SNR the IR-HARQ scheme delivers a 9-slope BLER-vs-SNR curve, compared to CC's 3-slope. For a target BLER of $10^{-6}$ , IR needs roughly $\log_{10}(10^6) / 9 \approx 0.67$ decades of SNR of coding-gain margin vs CC's $0.67 \cdot (9/3) = 2.0$ decades at slope $3$ — i.e., IR outperforms CC by $\sim 2.0 - 0.67 = 1.3$ decades of $\log_{10}$ SNR $\approx 13$ dB at the target BLER. This is the ARQ-DMT payoff.

Why IR wins

The key asymmetry: CC fixes the per-round rate at $r$ and buys only $L$ fading realisations at the same point on the static DMT. IR lowers the per-round rate to $r/L$ — where $d^{*}$ is larger — and buys the same $L$ fading realisations at a better point on the static DMT. IR is strictly better whenever $d^{*}(\cdot)$ is strictly decreasing, i.e., for all $r \in (0, \min(n_t, n_r)]$ .

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Common Mistake: ARQ Diversity Requires Independent Retransmissions

Mistake:

Assuming that the ARQ-DMT diversity $L \cdot d^{*}(r/L)$ is delivered simply because $L$ retransmissions took place. E.g., reading off the ARQ-DMT curve as if $L$ were the number of retransmitted copies rather than the number of independent fading realisations the decoder has access to.

Correction:

The ARQ-DMT presumes the $L$ channel realisations $\mathbf{H}_{1}, \ldots, \mathbf{H}_{L}$ are independent. This requires the ARQ round-trip time $T_\mathrm{rtt}$ (the delay between round $\ell$ and round $\ell + 1$ ) to exceed the channel coherence time $T_\mathrm{coh}$ . In practice:

Slow-moving users (pedestrian speeds, $v \approx 1$ m/s at $3$ GHz give $T_\mathrm{coh} \sim 100$ ms): consecutive HARQ rounds $T_\mathrm{rtt} \sim 4$ – $8$ ms fall inside the coherence time; the effective number of independent realisations $L_\mathrm{eff}$ is closer to $1$ than to the nominal $L$ .
Vehicular users ( $v \approx 30$ m/s, $T_\mathrm{coh} \sim 3$ ms at $3$ GHz): HARQ rounds are approximately independent; $L_\mathrm{eff} \approx L$ .
mmWave / high-mobility (sub-millisecond coherence time): HARQ rounds are strongly independent; $L_\mathrm{eff} \approx L$ .

The effective diversity in the correlated regime is $L_\mathrm{eff} \cdot d^{*}(r / L_\mathrm{eff})$ , not $L \cdot d^{*}(r/L)$ . Standards (LTE/NR) sometimes insert deliberate frequency hopping across HARQ retransmissions to decorrelate rounds even at low mobility — see §5.

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⚠️Engineering Note

Latency Cost of ARQ

ARQ buys diversity but it costs latency. Each retransmission adds one HARQ round-trip time $T_\mathrm{rtt}$ , which in 5G NR depends on numerology:

Numerology $\mu$	Slot length	$T_\mathrm{rtt}$ (typical)
$\mu = 0$ (15 kHz)	$1$ ms	$4$ – $8$ ms
$\mu = 1$ (30 kHz)	$0.5$ ms	$2$ – $4$ ms
$\mu = 3$ (120 kHz, FR2)	$0.125$ ms	$0.5$ – $1$ ms
Mini-slot (URLLC)	$\sim 2$ symbols	$0.25$ – $0.5$ ms

For eMBB traffic (mobile broadband, $L \le 4$ HARQ rounds), a worst-case 4-round episode on $\mu = 1$ costs $\le 16$ ms of latency — inside the typical web-application latency budget. For URLLC traffic with a $1$ ms end-to-end latency target, a single HARQ retransmission is already on the edge of the budget, and no-HARQ one-shot transmission (with a $K$ -repetition transmit-side diversity instead) is sometimes preferred. This is one of the reasons URLLC designs often skip HARQ entirely; the diversity gain of the ARQ-DMT is valuable only when the delay cost is tolerable.

Practical Constraints

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HARQ RTT in 5G NR FR1 is $\sim 4$ – $8$ ms; FR2 is $\sim 0.5$ – $1$ ms.
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Each HARQ round adds one RTT of end-to-end latency.
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URLLC latency budget ( $\sim 1$ ms) allows at most one HARQ retransmission or no HARQ at all.

📋 Ref: 3GPP TS 38.214 §5.3 (scheduling and HARQ timing)

Historical Note: Wozencraft & Jacobs 1961: The Original ARQ

1961

The idea of automatic-repeat-request (ARQ) predates digital wireless. John Wozencraft and Irwin Mark Jacobs, in their 1965 textbook Principles of Communication Engineering (based on MIT lecture notes dating to 1961), formalised the three canonical ARQ protocols — stop-and-wait, go-back- $N$ , and selective-repeat — together with a rigorous analysis of their throughput and reliability. At the time, the target was wireline and satellite data links, not wireless cellular, and the channels of interest were binary symmetric or AWGN, not block-fading MIMO.

The Wozencraft-Jacobs formulation is rate-flat: every retransmission re-sends the same codeword. Diversity on a fading channel was not yet a concept in 1961 — the term "diversity order" emerged from wireless in the 1980s (Jakes, Turin). It was not until Caire-Tuninetti 2001 and especially El Gamal-Caire-Damen 2006 that the question "does ARQ earn diversity on a fading channel?" was answered with a closed-form tradeoff curve. The 45-year gap reflects the evolution of the relevant channel models: ARQ over AWGN is about throughput; ARQ over fading is about reliability-diversity-delay tradeoff.

Quick Check

For Chase-combining HARQ at per-round rate $R$ with $L$ independent rounds, the long-term effective rate delivered to the application is

$R/L$ , because the same bits are retransmitted $L$ times

$R$ , because each round can in principle decode

$L \cdot R$ , because the decoder sees $L$ independent looks

$R \cdot (L - 1) / L$ , as with IR-HARQ

Correction:

R/L

, because the same bits are retransmitted

L

times

CC repeats the same codeword $L$ times, so the $R N n_t$ bits of $\mathbf{X}_1$ are now spread across $L N n_t$ channel uses, giving effective rate $R/L$ . This is a substantial rate cost compared to IR's $R \cdot (L-1)/L$ asymptotic-cost at the same protocol length.

Quick Check

For the ARQ-DMT diversity $L \cdot d^{*}(r/L)$ to be delivered, the $L$ channel realisations $\mathbf{H}_{1}, \ldots, \mathbf{H}_{L}$ must be

Approximately independent — i.e., the HARQ RTT must exceed the channel coherence time

Identical — CC relies on the channel being the same across rounds

Correlated with a specific correlation matrix chosen by the standard

Block-diagonal in some eigenbasis of the covariance matrix

Correction:

Approximately independent — i.e., the HARQ RTT must exceed the channel coherence time

Exactly. Independence is what makes each round a fresh draw from the fading distribution, so that the failure probabilities multiply. See ⚠ARQ Diversity Requires Independent Retransmissions.

Hybrid ARQ (HARQ)

An ARQ protocol combined with forward-error correction: the receiver stores soft observations (LLRs) from failed rounds and combines them with fresh observations in subsequent rounds. The two canonical HARQ flavours are Chase combining (CC) — retransmit the same codeword, sum LLRs — and incremental redundancy (IR) — transmit fresh parity bits, run the decoder on the accumulated observations.

Chase Combining (CC-HARQ)

A HARQ flavour in which every round retransmits the same codeword $\mathbf{X}_1$ . The receiver combines the $L$ rounds at the log-likelihood-ratio level, giving an effective SNR of $L \cdot \text{SNR}$ . CC achieves at most $L \cdot d^{*}(r)$ diversity at per-round rate $r$ (Chase 1985, Caire-Tuninetti 2001).

Incremental Redundancy (IR-HARQ)

A HARQ flavour in which round $\ell > 1$ transmits fresh parity obtained by puncturing a common mother code. The effective rate decreases with each round. IR achieves the ARQ-DMT $L \cdot d^{*}(r/L)$ — strictly better than CC at every $r > 0$ (El Gamal-Caire-Damen 2006).

ARQ over MIMO Channels

Every Real Wireless System Retransmits

Definition: ARQ Protocol over a Block-Fading MIMO Channel

Definition: Chase Combining (CC-HARQ)

Definition: Incremental Redundancy (IR-HARQ)

Chase Combining vs Incremental Redundancy vs No HARQ

Theorem: Chase Combining Achieves at Most L⋅d∗(r)L \cdot d^{*}(r)L⋅d∗(r) Diversity

Stack the rounds

Outage event

High-SNR exponent

Converse inequality

Comparison to IR

ARQ Diversity Contribution vs Number of Rounds LLL

Parameters

Example: 2×22\times 22×2 MIMO with L=3L = 3L=3: CC vs IR at r=1r = 1r=1

Static DMT values

CC-HARQ exponent at $r = 1$, $L = 3$

IR-HARQ exponent at $r = 1$, $L = 3$

Interpretation

Why IR wins

Common Mistake: ARQ Diversity Requires Independent Retransmissions

Latency Cost of ARQ

Historical Note: Wozencraft & Jacobs 1961: The Original ARQ

Quick Check

Quick Check

Hybrid ARQ (HARQ)

Chase Combining (CC-HARQ)

Incremental Redundancy (IR-HARQ)

Definition:
ARQ Protocol over a Block-Fading MIMO Channel

Definition:
Chase Combining (CC-HARQ)

Definition:
Incremental Redundancy (IR-HARQ)

Theorem: Chase Combining Achieves at Most $L \cdot d^{*}(r)$ Diversity

ARQ Diversity Contribution vs Number of Rounds $L$

Example: $2\times 2$ MIMO with $L = 3$ : CC vs IR at $r = 1$