Packet-Level Encoded L-HARQ Scheme

• Packet-level Encoded L-HARQ is an advanced retransmission scheme that dynamically injects new data mixed with redundancy to boost throughput in block-fading channels.
• It employs layered and joint encoding strategies along with adaptive mixing and backtrack decoding to recover both current and previous transmissions.
• The protocol integrates AMC-based rate selection and off-the-shelf codes like turbo codes, achieving significant gains in high-SNR regimes compared to traditional HARQ.

Packet-level encoded Layer-coded Hybrid Automatic Repeat reQuest (L-HARQ) schemes are advanced retransmission protocols designed to address the inherent throughput limitations of conventional HARQ under block-fading channels. Unlike standard Incremental Redundancy HARQ (IR-HARQ), which seeks to recover a fixed set of information bits via repeated transmissions of additional coded redundancy, packet-level L-HARQ protocols enable the dynamic injection of new, jointly encoded information across multiple HARQ rounds. These designs exploit mutual information accumulation and adaptive mixing of past and new packets, attaining substantial throughput improvements, especially in the high-throughput, high-SNR regime where classic HARQ protocols offer diminishing returns. L-HARQ utilizes joint or layered coding, backtrack decoding, and empirical packet error rate-based adaptation, often leveraging off-the-shelf codes such as turbo codes (Jabi et al., 2018, Jabi et al., 2016, Jabi et al., 2016).

1. System Model and Channel Assumptions

Packet-level L-HARQ operates on block-fading channels, where each HARQ round $k$ transmits a block of $n$ or $N$ channel symbols $x_k$ over a fading coefficient or SNR, denoted by $h_k$ or $\Gamma_k$, respectively:

$$y_k = \sqrt{\text{snr}_k}\ x_k + z_k \quad \text{or} \quad y[n] = \sqrt{\Gamma[n]}\ x[n] + z[n]$$

with $z_k, z[n]$ modeled as AWGN and snr or SNR blockwise i.i.d. Codewords are independently channel-coded in each round. Perfect receiver-side CSI for each block is assumed, while transmitter CSI is limited to statistical knowledge. Feedback for each round consists of a single ACK/NACK bit per block and potentially additional rate or mixing information (Jabi et al., 2018, Jabi et al., 2016, Jabi et al., 2016).

The packet/block duration is $n$ or $N$ symbols, and adaptive modulation and coding (AMC) is supported via SNR-dependent selection of rates $R(\Gamma)$ from a finite set $\mathcal{R} = \{ R_1, \dots, R_L \}$.

2. Protocol Architecture and Encoding Structure

Packet-level L-HARQ, also known as cross-packet HARQ, deviates from IR-HARQ by allowing the coded payload in round $k$ to be a composite of bits from prior (unsuccessful) packets and new information bits. There are two principal realization methodologies:

• Layered L-HARQ (Editor’s term): Each round’s payload concatenates a redundancy layer—typically a punctured fragment ($\rho_{k-1}$ bits) from the previous failed packet—and $R_k - \rho_{k-1}$ new bits for a fresh packet. Encoding is performed using a standard or off-the-shelf code, e.g., turbo codes, at the new total rate (Jabi et al., 2018, Jabi et al., 2016).
• Joint/Accumulating L-HARQ: Each round $k$ introduces $R_k$ new bits $m_k$ and forms a compound message $m_{[k]} = (m_1, \dots, m_k)$ of joint length $n (R_1 + \dots + R_k )$. This is encoded via a potentially distinct codebook for that round, allowing aggregate mutual information exploitation (Jabi et al., 2016).

The key encoding operations can be summarized as:

HARQ Round	Encoded Bits per Block	Channel Coding
1	$m_1$ of size $n R_1$	Standard (e.g., turbo)
$k > 1$	$[ \text{punctured}(m_{[k-1]});$ $m_k ]$	Rate-$R_k$ channel code

This approach enables consecutive HARQ rounds to share channel resources, embedding “old” information within new transmissions.

3. Decoding Strategy and Backtrack Mechanism

Decoding in L-HARQ interleaves forward and backtrack procedures:

• Direct Decode (Forward): In each round $k$, attempt to decode the composite payload $m_{[k]}$ from $y_k$. Success yields the new bits and, through the redundancy layer, a part of previous packets (Jabi et al., 2018).
• Backtrack Decode: Upon a successful forward decode, the known redundancy (e.g., $m'_{[k-1]}$) enables backtrack decoding of $m_{[k-1]}$ using both the prior channel observation $y_{k-1}$ and the recovered side information. This process may be recursively applied for earlier rounds (Jabi et al., 2018, Jabi et al., 2016).
• Decoding Conditions: Joint decoding asserts that cumulative mutual information must satisfy $$I^{(k)} := \sum_{\ell=1}^k I_\ell \geq \sum_{\ell=1}^k R_\ell$$, and $I_k \geq R_k$ for current-layer recovery (Jabi et al., 2016).
• Backtrack Error Probability: The backtrack failure probability is represented as $\mathrm{PER}^{b}(\gamma; R, \rho)$, empirically determined via Monte-Carlo simulations or BCJR-based decoding (Jabi et al., 2018, Jabi et al., 2016). Threshold decoding may be used: if $I(\tilde{\Gamma}) \geq R - \rho$, backtrack succeeds.
• Successful Recovery: Correct recovery requires not only direct decode success in a round but also backtrack decode success for all prior rounds in the chain.

4. Throughput Analysis and Optimization

Packet-level L-HARQ schemes optimize throughput by exploiting mutual information aggregation and adaptive rate/mixing selection. Throughput ($\eta$) expressions depend on the specific architecture:

• AMC-only (baseline):

$$\eta^{\text{amc}} = \mathbb{E}_{\Gamma} [ R(\Gamma) \cdot (1 - \mathrm{PER}(\Gamma; R(\Gamma))) ]$$

• Layered L-HARQ:

$$\eta^{L-\mathrm{HARQ}} = \mathbb{E} [ (R_k + J_{k-1}) (1 - \mathrm{PER}(\Gamma_k; R_k)) ]$$

where $J_{k-1}$ is the cumulative reward from successful backtracks (Jabi et al., 2018).

• Accumulating L-HARQ:

$$\eta_K^{\mathrm{xp}} = \left[ \sum_{k=1}^K \bar{R}_k \cdot (f_{k-1} - f_k) \right] / \left[ 1 + \sum_{k=1}^{K-1} f_k \right]$$

where $\bar{R}_k = \sum_{\ell=1}^k R_\ell$, and $f_k$ is the probability of decode failure after $k$ rounds (Jabi et al., 2016).

Rate and mixing parameters are optimized via dynamic programming or heuristics based on empirical $\mathrm{PER}$ and $\mathrm{PER}^{b}$ curves. A pragmatic strategy is to select the smallest $\rho$ so that $\mathrm{PER}^{b} (\gamma; R, \rho) \leq \epsilon$, with $\epsilon \approx 0.1$ shown to be effective (Jabi et al., 2018, Jabi et al., 2016).

5. Practical Implementations with Off-the-Shelf Codes

Packet-level L-HARQ is amenable to implementation with pragmatic codes, notably turbo codes:

• Turbo Code Structure: Standard 3GPP turbo codes with generator polynomials [13, 15]$_8$ and pseudo-random interleaving are used. The mother code rate is $1/3$, and actual rates $R$ are achieved via puncturing systematic and/or parity bits (Jabi et al., 2016).
• Compressor/Multiplexer: For layering, a systematic compressor selects the first $\rho_k$ systematic bits for the redundancy layer. The remaining payload is filled with fresh bits (Jabi et al., 2016).
• Decoder Strategy: Uses a single BCJR turbo decoder per NACK for backtrack steps; for joint/accumulated schemes, a serial or parallel turbo-graph combining LLRs across rounds may be constructed (Jabi et al., 2016).
• Feedback: In addition to ACK/NACK and AMC index, a few bits (e.g., 2–4 bits) describe $\rho$ for mixing adaptation per round (Jabi et al., 2018, Jabi et al., 2016).
• Empirical PER Curves: Required for all rates of interest, these are measured offline using Monte-Carlo simulation for both direct and backtrack conditions.

6. Performance Gains and Comparative Results

Comprehensive numerical examples demonstrate that L-HARQ yields distinct throughput advantages:

• Rayleigh fading with 16-QAM and practical turbo codes: L-HARQ with $K=2$ achieves approximately $1$–$1.5$ dB gain at $\eta \approx 3$ bits/symbol over IR-HARQ; with $K=4$, gains grow to $2$–$2.5$ dB (Jabi et al., 2018, Jabi et al., 2016, Jabi et al., 2016).
• Idealized threshold decoding: With increasing $K$ (HARQ round limit), L-HARQ approaches ergodic capacity, reducing the gap by $3$–$4$ dB in the high-throughput region (Jabi et al., 2018).
• Practical regime: Gains are most pronounced when conventional IR-HARQ saturates (i.e., at high SNR where single-round PER is small), as L-HARQ reactivates throughput improvement by exploiting mutual information accumulation (Jabi et al., 2016).
• Complexity: L-HARQ requires only one additional BCJR decode per NACK (for layering) and maintains feedback/encoding complexity that is comparable to IR-HARQ plus AMC (Jabi et al., 2018).

7. Relationship to Other HARQ Schemes and Design Insights

L-HARQ generalizes IR-HARQ, which transmits punctured portions of a fixed mother code. It also differs from Chase Combining, which simply repeats the original codeword and saturates early. In L-HARQ, each HARQ round brings in new information, and the design matches the aggregate code rate to the realized mutual information, improving outage robustness (Jabi et al., 2016).

Key design features include:

• Separation of Rate Control and Mixing: AMC rate selection and HARQ mixing operate independently, allowing leveraging of existing optimized codes (Jabi et al., 2018).
• Layered vs. Joint Coding: Layered approaches offer simplicity by using conventional encoders with puncturing, while joint accumulation may offer higher information-theoretic throughput but at increased decoding complexity (Jabi et al., 2016, Jabi et al., 2016).
• Adaptation Heuristics: Simple fixed-outage rules for setting mixing/compression rates yield performance near optimal dynamic programming solutions (Jabi et al., 2016).

A plausible implication is that L-HARQ enables the reuse of off-the-shelf coding/decoding infrastructure with modest protocol extensions, rendering it a practical and effective solution for modern block-fading wireless systems.