GRAND for Gaussian Intersymbol Interference Channels

Abstract: Channel decoding is a challenging task in communication channels exhibiting memory effects. In this work, we apply the recently proposed decoding paradigm of guessing random additive noise decoding (GRAND) to channels with memory, focusing on linear Gaussian intersymbol interference (ISI) channels. For describing error patterns (EPs), we introduce the concept of error burst to account for the memory effect, and define sequence reliability to characterize the likelihood of EP. Based on sequence reliability, we obtain the optimal GRAND algorithm as a generalization of soft GRAND (SGRAND) for linear Gaussian ISI channels, termed SGRAND-ISI, which is equivalent to the maximum-likelihood (ML) decoding algorithm. We then develop order-reliability-bit (ORB) GRAND algorithms based on SGRAND-ISI, to facilitate implementation. In numerical experiments, our proposed algorithms achieve multiple-dB improvements compared to GRAND algorithms which ignore channel memory, and can often attain performance within 0.1--0.2dB of the ML lower bound. We also compare our proposed algorithms with the recently proposed ORBGRAND-Approximate Independence algorithm for handling channel memory, and observe a performance gain of at least 0.5dB at block error rate of $10<sup>{-3}$, meanwhile incurring a substantially lower computational complexity.

• The paper extends GRAND to Gaussian ISI channels by ordering error patterns through error bursts and sequence reliability, with SGRAND-ISI proven equivalent to maximum-likelihood decoding.
• Rank-based ORBGRAND-ISI variants achieve 0.5–1 dB gains over ORBGRAND-AI at BLER 10⁻³, remain within 0.1–0.2 dB of ML, and reduce reliability computations by about fivefold.
• For higher-order ISI, the method truncates partially decomposable bursts—typically at size 3—providing strong simulated performance but leaving approximation guarantees, achievable rates, and robustness to channel-estimation errors open.

Overview

This paper extends guessing random additive noise decoding (GRAND) to linear Gaussian intersymbol interference (ISI) channels, addressing a gap in the GRAND literature, which has largely focused on memoryless channels (2603.08325). The central contribution is a principled way to order error patterns (EPs) when channel memory invalidates the per-symbol reliability ordering used in memoryless GRAND. The authors introduce two constructs: error bursts, contiguous (or $L$-separated) groups of flipped bit positions that reflect how ISI couples adjacent symbols, and sequence reliability, defined as the log-likelihood drop incurred by flipping a set of positions in the hard detection sequence. Building on these, they derive SGRAND-ISI, prove it equivalent to maximum-likelihood (ML) decoding, and develop hardware-oriented variants ORBGRAND-ISI and CDF-ORBGRAND-ISI that use only ranks of sequence reliability.

System model and background

The channel is BPSK over an $L$-th order FIR ISI channel with known coefficients $h_0,\ldots,h_L$, normalized so SNR is $1/\sigma^2$. The weight function $\Lambda(\underline{x},\underline{y})$ is the sum of log conditional probabilities $P_{\mathsf{Y}_i|\underline{\mathsf{X}}_{i-L:i}}$, and the hard detection sequence $\underline{x}^*$ maximizes $\Lambda$ via the Viterbi algorithm. GRAND then queries EPs $\underline{e}$ in some order, testing whether $\underline{x}^*\oplus\underline{e}$ satisfies the parity-check matrix, terminating on success or at a query cap.

The key difficulty relative to memoryless channels is that EP likelihood cannot be decomposed into per-symbol terms: flipping one bit perturbs the likelihoods of its $L$0-neighborhood. The paper's unified GRAND-ISI framework assigns each error burst $L$1 a non-negative cost $L$2 and orders candidate EPs by the sum of burst costs; choices of $L$3 recover SGRAND-ISI (exact sequence reliability), ORBGRAND-ISI (rank), and CDF-ORBGRAND-ISI (inverse-CDF-companded rank).

SGRAND-ISI and ML equivalence

The main theoretical result is that SGRAND-ISI, with $L$4 as in Definition 2, is exactly ML decoding when the query limit is unbounded (Theorem 1). The proof shows that the sum of sequence reliabilities over the bursts of an EP equals $L$5, so minimizing the burst-cost sum is identical to maximizing codeword likelihood. This is a clean generalization of the known SGRAND/ML equivalence for memoryless channels, and it holds for arbitrary ISI order.

Implementation is made tractable by three lemmas characterizing sequence reliability. For first-order ISI, decomposable sets (whose parts are separated by more than one index) have additive reliability; for non-decomposable (contiguous) sets, the excess over the sum of singleton reliabilities telescopes into pairwise correction terms $L$6, which equal $L$7 — a constant depending only on the hard detection sequence, not on $L$8 beyond $L$9. Consequently all $h_0,\ldots,h_L$0 burst reliabilities can be computed from the $h_0,\ldots,h_L$1 singleton reliabilities plus these corrections. Candidate EP assembly must additionally expurgate patterns whose bursts overlap or are not properly separated.

Higher-order extensions and approximation

For $h_0,\ldots,h_L$2, bursts are defined with gaps up to $h_0,\ldots,h_L$3 within a burst and at least $h_0,\ldots,h_L$4 separation between bursts. Reliability remains additive across fully separated components, but a new class of partially-decomposable bursts arises (gaps between 2 and $h_0,\ldots,h_L$5), whose reliabilities include cross-terms that do not telescope simply. The number of such bursts grows prohibitively with $h_0,\ldots,h_L$6, so the paper adopts an approximation: only partially-decomposable bursts of size at most $h_0,\ldots,h_L$7 are enumerated, yielding algorithms labeled "-$h_0,\ldots,h_L$8". This is explicitly a complexity-driven truncation; the simulations indicate the loss is small when $h_0,\ldots,h_L$9, but no general bound on the approximation error is provided.

Numerical results

Experiments use CA-Polar(128,114+6) with CRC-6 and BCH(127,113) over first-order ($1/\sigma^2$0; and stronger, $1/\sigma^2$1) and second-order ($1/\sigma^2$2) ISI channels, against baselines of memoryless ORBGRAND, ORBGRAND-AI (block-partitioned approximate-independence decoding), and a genie-aided ML lower bound. Key findings:

Comparison	Result
CDF-ORBGRAND-ISI vs. memoryless ORBGRAND	≥ 2 dB gain even at BLER $1/\sigma^2$3
CDF-ORBGRAND-ISI vs. ORBGRAND-AI12, first-order ISI	≥ 0.5–0.6 dB gain at BLER $1/\sigma^2$4; gap to ML ≤ 0.1 dB
CDF-ORBGRAND-ISI vs. ORBGRAND-AI12, strong first-order ISI	1 dB gain at BLER $1/\sigma^2$5; gap to ML ≤ 0.15 dB
CDF-ORBGRAND-ISI-3 vs. ORBGRAND-AI12, second-order ISI	0.8 dB gain; gap to ML ≤ 0.2 dB
SGRAND-ISI vs. ML bound	negligible deviation (truncation-only)

The failure of memoryless ORBGRAND is explained quantitatively via the average logistic weight of target EPs: it stays below ~30 for the memoryless case but exceeds 800 for $1/\sigma^2$6, far beyond what $1/\sigma^2$7 queries can reach — a strong, concrete diagnosis rather than a qualitative claim. Complexity comparisons show CDF-ORBGRAND-ISI computes roughly five times fewer sequence-reliability values than ORBGRAND-AI12 at comparable performance, with correspondingly fewer queries (e.g., 1.8 vs. 10.7 valid-EP queries at 7 dB in the second-order channel). SGRAND-ISI requires the fewest queries but the highest real-time computation due to exact reliability evaluation.

Limitations and open questions

The paper concedes several points. First, the higher-order treatment relies on truncating partially-decomposable bursts to size $1/\sigma^2$8; while simulations suggest minimal loss at $1/\sigma^2$9, neither a performance guarantee nor a principled rule for choosing $\Lambda(\underline{x},\underline{y})$0 is given. Second, the total number of required reliability computations grows exponentially in $\Lambda(\underline{x},\underline{y})$1 (in closed form via Fibonacci numbers for $\Lambda(\underline{x},\underline{y})$2), making the approximation strategy necessary rather than optional for long blocks or large $\Lambda(\underline{x},\underline{y})$3. Third, although CDF-ORBGRAND is exactly capacity-achieving for memoryless binary-input channels, whether CDF-ORBGRAND-ISI retains this property for channels with memory is unaddressed, as are achievable-rate characterizations of the proposed decoders generally. Finally, all results assume perfect knowledge of the impulse response coefficients; robustness to channel estimation error is not examined.

Conclusion

The paper provides an optimal, ML-equivalent instantiation of GRAND for Gaussian ISI channels by restructuring EP ordering around error bursts and sequence reliability, together with rank-based variants suited to implementation. Reported gains — multiple dB over memoryless GRAND, 0.5–1 dB over ORBGRAND-AI at substantially lower complexity, and near-ML performance — are consistent across code families and channel strengths. The remaining questions concern achievable rates under ISI, approximation-error control for high-order channels, and behavior under imperfect channel state information.