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From IR-LAST to LDPC-HARQ: What Real Systems Actually Do

Section 3 gave us an information-theoretic reference construction — IR-LAST — that achieves the ARQ-DMT at every $(r, L)$ . But IR-LAST is not what LTE or 5G NR actually deploys. Real systems use LDPC codes (NR) or Turbo codes (LTE) with a circular-buffer rate matcher that generates a small number of predefined redundancy versions. The question of this section is: how close does the practical scheme get to the ARQ-DMT, and where does it fall short?

The short answer is that a well-designed LDPC-based IR-HARQ scheme achieves the ARQ-DMT asymptotically — the NVD-type property of a capacity-achieving LDPC code, combined with puncturing-compatible rate matching, gives the same $L \cdot d^{*}(r/L)$ exponent at high SNR. The gap from IR-LAST is a matter of coding gain and finite-length effects, not asymptotic exponent.

This section develops the LTE/NR HARQ mechanism in three pieces. First, we define the redundancy versions and the circular-buffer rate matcher. Second, we state the LDPC-IR achieves-the-ARQ-DMT theorem. Third, we examine the practical tradeoffs between CC-HARQ and IR-HARQ in terms of BLER and latency.

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Definition:
Redundancy Version (RV)

A redundancy version (RV) is an integer index $\mathrm{RV} \in \{0, 1, 2, 3\}$ that specifies which consecutive fragment of the mother code's circular buffer is transmitted in a given HARQ round.

Formally: the mother LDPC encoder produces a long codeword $\mathbf{c} \in \{0, 1\}^{N_c}$ of rate $R_m = K / N_c$ where $K$ is the number of information bits. The circular-buffer rate matcher lays out $\mathbf{c}$ along a ring of $N_c$ positions and defines four starting offsets $k_0^{(\mathrm{RV})}$ for $\mathrm{RV} \in \{0, 1, 2, 3\}$ : $k_0^{(0)} = 0,\qquad k_0^{(1)} = \lceil N_c / 4 \rceil,\qquad k_0^{(2)} = \lceil N_c / 2 \rceil,\qquad k_0^{(3)} = \lceil 3 N_c / 4\rceil.$ Round $\ell$ transmits a fragment of length $E = K / R_\ell$ starting from position $k_0^{(\mathrm{RV}_\ell)}$ and wrapping around the ring. $R_\ell$ is the effective per-round code rate (chosen by the MCS — modulation and coding scheme — and the resource allocation).

The RV_0 fragment is chosen so that the systematic bits (the original information bits before encoding) are always at the start; hence RV_0 contains mainly systematic bits and RV_2 contains mainly parity bits from the far side of the ring. The typical RV sequence across retransmissions is 0, 2, 3, 1 — chosen to maximise the coverage of the circular buffer across four rounds.

The circular-buffer + RV mechanism elegantly realises incremental redundancy: each RV transmits a different portion of the mother code's output, and after all four RVs have been transmitted the receiver has seen (most of) the full mother code of rate $R_m$ . The ARQ-DMT effective rate drops as $R_m \to 0$ in the limit of many retransmissions.

Not every HARQ round bumps the RV: if the transport block (TB) is retransmitted verbatim (Chase combining mode), the RV stays at 0 and the scheme degenerates to CC-HARQ. In NR, the scheduler chooses the RV per retransmission based on the estimated channel quality — aggressive IR at low SNR (bigger coverage gain from new parity) vs CC-style at high SNR (simpler LLR combining).

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Theorem: LDPC-IR Asymptotically Achieves the ARQ-DMT

Consider an IR-HARQ protocol on an $n_t \times n_r$ i.i.d. Rayleigh block-fading MIMO channel with BICM signalling (LDPC mother code + circular-buffer rate matching + higher-order modulation). As the LDPC block length $K \to \infty$ with effective rate $R_m \to r / L$ , the $L$ -round BLER satisfies $P_e^\mathrm{LDPC\text{-}IR}(\text{SNR}, L) \;\doteq\; \text{SNR}^{-L \cdot d^{*}(r/L)} \;=\; \text{SNR}^{-d_\mathrm{ARQ}(r, L)}.$ In words: LDPC-based IR-HARQ achieves the ARQ-DMT in the limit of long block length.

The proof uses the BICM capacity lifting of Ch. 5–7 together with the ARQ-DMT of §2: BICM is DMT-optimal under uniform input and Gray labelling, and the LDPC+IR system realises BICM plus the IR mechanism.

A capacity-achieving LDPC code at rate $R_m$ achieves outage- limited performance on any memoryless channel with BICM capacity $\ge R_m$ . On the $L$ -round ARQ channel, the combined observation after $\ell$ rounds is equivalent to a BICM channel with capacity $\sum_{k \le \ell} I_{\rm BICM}(\mathbf{H}_{k})$ ; outage happens iff this sum is less than $L R_m = r \log_2 \text{SNR}$ . The outage exponent of this event is (by the same argument as §2) $L \cdot d^{*}(r/L)$ . A long enough LDPC code gets arbitrarily close to this outage bound — hence the ARQ-DMT is asymptotically achieved.

Show Hint

Recall from Ch. 7 that LDPC + BICM achieves the BICM capacity on any memoryless channel.

The $L$ -round BICM capacity is additive over rounds: $I_{\rm BICM}^{(L)} = \sum_{\ell} I_{\rm BICM}(\mathbf{H}_\ell)$ .

Apply the ARQ-DMT large-deviations argument of Thm. TARQ-DMT (El Gamal-Caire-Damen 2006) with $I_{\rm BICM}$ replacing $\log\det$ .

Proof

BICM per-round mutual information

Under BICM signalling (Gray-QAM + uniform-input bit channels), the per-round mutual information is $I_{\rm BICM}(\mathbf{H}_\ell) \;=\; \sum_{i,j} I(Y_{ij}; B_{ij}\mid \mathbf{H}_\ell),$ where the sum is over channel uses and bit positions of the label. At high SNR, $I_{\rm BICM}$ differs from the Gaussian-input capacity $\log\det(\mathbf{I} + \tfrac{\text{SNR}} {n_t}\mathbf{H}_\ell\mathbf{H}_\ell^H)$ by a constant shaping gap — which is invisible to the exponential-equality $\doteq$ notation. Hence the BICM outage exponent equals the Gaussian outage exponent: $d^{*}_{\rm BICM}(r) = d^{*}(r)$ .

$L$-round outage exponent under BICM

Repeating the argument of Thm. TARQ-DMT (El Gamal-Caire-Damen 2006) with $I_{\rm BICM}$ replacing $\log\det$ , the $L$ -round outage exponent is $L \cdot d^{*}(r/L)$ .

LDPC achieves outage at long block length

Density-evolution analysis (Richardson-Urbanke) shows that a rate- $R_m$ LDPC code with protograph thresholds meeting the BICM capacity achieves zero-error decoding whenever the empirical channel's BICM capacity exceeds $R_m + \epsilon$ for any $\epsilon > 0$ , in the $K \to \infty$ limit. Hence the BLER at round $\ell$ is $\le P_{\rm out}(R_m, \ell) + o(1)$ as $K \to \infty$ .

Combine — LDPC-IR achieves ARQ-DMT

After $L$ rounds, the BLER is bounded by the $L$ -round outage $P_\mathrm{out}^{\rm BICM}(L r \log_2 \text{SNR}, L) \doteq \text{SNR}^{-L \cdot d^{*}(r/L)}$ . The long-block-length limit closes the gap, giving the claimed $P_e \doteq \text{SNR}^{- d_\mathrm{ARQ}(r, L)}$ . $\blacksquare$

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BLER vs Number of HARQ Rounds: CC vs IR

Block error rate after $\ell \in \{1, 2, 3, 4\}$ HARQ rounds for CC-HARQ and IR-HARQ at fixed SNR. The IR curve decays with a slope proportional to $d^{*}(r/L)$ per round, while CC decays with the (smaller) slope of $d^{*}(r)$ per round. At moderate SNR the gap between the two is $\sim 2$ – $4$ dB per round; at high SNR the gap widens as the ARQ-DMT asymptote takes over.

Parameters

SNR [dB]5

Example: IR-HARQ Effective Throughput: $r = 1$ bit/ch.use on $2\times 2$

For a $2 \times 2$ MIMO channel with IR-HARQ, long-term rate $r = 1$ bit/channel use, and delay budget $L = 4$ , compute the effective throughput $\eta_\mathrm{eff}$ as a function of SNR. What is the SNR needed for $\eta_\mathrm{eff} \ge 0.95$ ?

Solution

Throughput formula

$\eta_\mathrm{eff} = r \cdot (1 - P_{\rm out}^{\rm IR}(L r, L))$ , where $P_{\rm out}^{\rm IR}$ is the $L$ -round IR outage. At high SNR, $P_{\rm out}^{\rm IR} \doteq \text{SNR}^{-L \cdot d^{*}(r/L)} = \text{SNR}^{-13}$ (using $d^{*}(1/4) = 3.25$ on $2 \times 2$ from §3).

SNR for 95% throughput

Require $1 - P_{\rm out} \ge 0.95$ , i.e., $P_{\rm out} \le 0.05$ . Numerical outage at $r = 1$ , $L = 4$ , $2 \times 2$ reaches $0.05$ at $\text{SNR} \approx 3$ dB, and $0.01$ at $\approx 4.5$ dB (from Monte Carlo on the outage integral). The IR-HARQ scheme is solidly in the useful-throughput regime by $\sim 4$ dB — a $\sim 8$ dB advantage over one-shot transmission at the same rate.

Comparison to CC

CC-HARQ at the same parameters has $P_{\rm out}^{\rm CC} \doteq \text{SNR}^{-4}$ (exponent $L \cdot d^{*}(1) = 4 \cdot 1 = 4$ ). To reach $P_{\rm out} = 0.05$ , CC needs $\text{SNR} \approx 7$ dB — a $\sim 4$ dB penalty vs IR.

Takeaway

At $r = 1$ on $2 \times 2$ with $L = 4$ , IR-HARQ delivers 95% throughput at $\sim 3$ dB, vs $\sim 7$ dB for CC-HARQ and $\sim 11$ dB for no-HARQ. The ARQ-DMT payoff is $\sim 4$ dB of SNR per HARQ flavour upgrade at this operating point.

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Common Mistake: Budget $L$ by End-to-End Latency, Not by DMT Gain

Mistake:

Choosing the ARQ round budget $L$ by maximising the DMT gain $L \cdot d^{*}(r/L)$ , which grows unboundedly in $L$ . This leads to $L = 8$ or higher — far beyond any practical latency budget.

Correction:

The ARQ round budget is fundamentally latency-bound, not diversity-bound. The end-to-end latency of a transmission that uses $L$ HARQ rounds is approximately $L \cdot T_\mathrm{rtt}$ , and this must fit inside the application-level latency budget.

eMBB (mobile broadband, ~100 ms budget): $L \le 4$ is typical. The marginal DMT gain past $L = 4$ is theoretically huge (each round adds $\sim d^{*}(0) = n_t n_r$ to the exponent), but the scheme spends most of its time in the cold-start latency regime before the gain materialises.
URLLC (ultra-reliable low latency, ~1 ms budget): $L \le 1$ – $2$ . Sometimes the preferred choice is no HARQ with aggressive one-shot transmission at very low rate — this simply cannot accommodate even one retransmission within the budget.
Satellite / NTN (non-terrestrial, RTT = 20–600 ms): $L \le 2$ is typical despite the long latency budget, because ACK/NACK takes hundreds of milliseconds round-trip.

The ARQ-DMT formula tells you the reliability per $L$ ; the latency budget tells you the feasible $L$ . A system design picks the smallest $L$ that meets a target reliability at the given SNR — not the largest.

🔧Engineering Note

HARQ Soft Buffer Sizing

A receiver performing IR-HARQ must store the soft LLRs of previously-received rounds so that they can be combined with fresh rounds. The required storage is roughly proportional to the number of coded bits transmitted across all rounds — a $K$ -bit transport block, rate-matched to a circular buffer of $N_c \approx K / R_m$ positions, needs up to $N_c$ LLR slots of typically 6–8 bits each.

For a typical 5G NR transport block of $K = 8448$ information bits and mother-code rate $R_m = 1/3$ , the soft buffer is $\approx 25{,}000$ LLRs. A UE supporting $N_{\rm HARQ} = 16$ parallel HARQ processes (see §5) thus needs $\approx 400{,}000$ LLRs $\times$ 8 bits $= 3.2$ Mbit of RAM just for HARQ.

NR's limited-buffer rate matching (LBRM) caps the soft-buffer size for low-tier UEs by limiting the effective mother-code rate they can use. A Category 4 UE (the mid-range class) is allowed up to 25% of the full mother code; a Category 20 UE (flagship) gets the full mother code. This is a real-world example where the ARQ-DMT prediction of "arbitrarily many IR rounds" is capped by silicon, not by information theory.

Practical Constraints

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IR-HARQ soft buffer $\propto N_c \times \text{LLR bit-width}$ per transport block per process.
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5G NR LBRM limits the fraction of the mother code that can be stored.
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Buffer size directly caps the achievable $L$ -round effective rate.

📋 Ref: 3GPP TS 38.214 §5.4 (limited-buffer rate matching)

Quick Check

In 5G NR HARQ, RV_0 is designed to contain

The systematic bits of the LDPC codeword

Only parity bits from the far side of the circular buffer

A random subset of the codeword bits

The same bits as RV_1, shifted by half the buffer length

Correction:

The systematic bits of the LDPC codeword

RV_0 starts at offset 0 of the circular buffer, which by design includes the systematic bits. If a single RV is transmitted (the common one-shot case), RV_0 alone is often decodable on its own at high SNR.

Quick Check

In the high-SNR limit, the BLER of an IR-HARQ scheme with $L = 3$ rounds on $2 \times 2$ MIMO at long-term rate $r = 1$ decays as

$\text{SNR}^{-9}$ — a slope of $9$ on a $\log$ - $\log$ BLER plot

$\text{SNR}^{-3}$ — the number of rounds $L$

$\text{SNR}^{-1}$ — the static DMT

$\text{SNR}^{-4}$ — the maximum single-round diversity

Correction:

\text{SNR}^{-9}

— a slope of

9

on a

\log

-

\log

BLER plot

$d_\mathrm{ARQ}(r, L) = L \cdot d^{*}(r/L) = 3 \cdot d^{*}(1/3) = 3 \cdot 3 = 9$ (using $d^{*}(1/3) = 4 - 3 \cdot 1/3 = 3$ on $2 \times 2$ ).

Why This Matters: Forward Link: BICM-OFDM in Wireless Standards

The LDPC-IR-HARQ architecture of this section is one piece of a larger wireless-standards ecosystem. In Chapter 21 we will see how it composes with OFDM (for frequency-selective channelisation) and space-time coding (Alamouti / V-BLAST) to form the full physical-layer pipeline of LTE and 5G NR. The flow of operations is: information bits $\to$ LDPC encoder $\to$ rate matcher (RV selection) $\to$ BICM interleaver $\to$ QAM mapper $\to$ layer mapping (spatial streams) $\to$ OFDM modulator $\to$ antenna ports. The ARQ-DMT of this chapter characterises the information-theoretic ceiling of the whole pipeline; Chapter 21 examines where real standards fall short of the ceiling and why.

Practical HARQ: Chase Combining vs Incremental Redundancy

From IR-LAST to LDPC-HARQ: What Real Systems Actually Do

Definition: Redundancy Version (RV)

Theorem: LDPC-IR Asymptotically Achieves the ARQ-DMT

BICM per-round mutual information

$L$-round outage exponent under BICM

LDPC achieves outage at long block length

Combine — LDPC-IR achieves ARQ-DMT

BLER vs Number of HARQ Rounds: CC vs IR

Parameters

Example: IR-HARQ Effective Throughput: r=1r = 1r=1 bit/ch.use on 2×22\times 22×2

Throughput formula

SNR for 95% throughput

Comparison to CC

Takeaway

Common Mistake: Budget LLL by End-to-End Latency, Not by DMT Gain

HARQ Soft Buffer Sizing

Quick Check

Quick Check

Why This Matters: Forward Link: BICM-OFDM in Wireless Standards

Definition:
Redundancy Version (RV)

Example: IR-HARQ Effective Throughput: $r = 1$ bit/ch.use on $2\times 2$

Common Mistake: Budget $L$ by End-to-End Latency, Not by DMT Gain