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Sparse randomized delay-Doppler bins plus unitary precoding remove the partial-band jamming floor from NOMA while keeping BER inside the unjammed envelope.

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2026-06-27 15:29 UTC pith:XXHLZ6FO

load-bearing objection The scheme combines sparse randomized bins, unitary precoding, and superincreasing powers to claim jammer-independent BER and exact SIC-ML match, but the Marchenko-Pastur conditioning for small submatrices looks like the main point to verify. the 2 major comments →

arxiv 2606.09753 v1 pith:XXHLZ6FO submitted 2026-06-08 eess.SP

Jamming-Resilient Sparse Delay-Doppler NOMA: Unitary Precoding, Randomized Active Sets, and Superincreasing Power Allocation

Michel Kulhandjian , Hovannes Kulhandjian , Theodoros A. Tsiftsis This is my paper
classification eess.SP
keywords jamming resilienceNOMAdelay-Dopplerunitary precodingsuperincreasing power allocationsparse signalingsuccessive interference cancellationOTFS
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that placing each user's data on a small random subset of delay-Doppler bins, spreading the symbols through a unitary precoder, and redrawing the active subset every frame from a shared seed lets the receiver identify and discard jammed bins, recover the sparse vector by least squares, and decode by successive interference cancellation. This construction yields a closed-form bit-error-rate expression that contains no jammer-induced floor, in contrast to the well-known partial-band floor of conventional OTFS-NOMA. The same Marchenko-Pastur conditioning argument that makes any random unitary submatrix well-behaved also shows that an adversary who learns the seed still cannot force a performance loss. For more than two users a superincreasing power allocation drawn from the Merkle-Hellman knapsack ensures that the low-complexity successive cancellation decoder coincides exactly with maximum-likelihood detection, eliminating the usual error-propagation ceiling.

Core claim

By restricting each user to a sparse random support in the delay-Doppler domain, applying a unitary precoder, and redrawing the support per frame from a shared pseudo-random seed, the transmitter forces any jammer to hit only a fraction of the bins; the receiver discards the jammed entries, solves the resulting well-conditioned least-squares problem, and applies successive interference cancellation. The resulting bit-error-rate expression has no floor under partial-band jamming. The Marchenko-Pastur law guarantees that every random unitary submatrix remains invertible with high probability, so even if the seed is compromised the bit-error-rate curve stays inside the unjammed envelope. For mo

What carries the argument

The combination of sparse randomized active sets drawn from a shared seed and unitary precoding, whose submatrices remain well-conditioned by the Marchenko-Pastur law, enabling jammed-bin rejection and exact successive-interference-cancellation decoding.

Load-bearing premise

The receiver can correctly identify every jammed bin from the received energy pattern and the shared seed without misclassification.

What would settle it

A Monte Carlo trial that produces a visible bit-error-rate floor under partial-band jamming whose height matches the conventional OTFS-NOMA floor would disprove the no-floor claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The closed-form bit-error-rate expression contains no jammer-induced floor.
  • Compromising the shared seed leaves the bit-error-rate curve inside the unjammed envelope because every random unitary submatrix stays well-conditioned.
  • Superincreasing power allocation makes low-complexity successive interference cancellation identical to maximum-likelihood detection for more than two users.
  • Partitioning users into pairs on disjoint bin subsets lets the scheme reach its floor bit-error-rate at eight users near 20 dB SNR.
  • The jammer-independence property holds for any Rician K-factor.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same sparse-support and unitary-precoding idea could be applied to other multicarrier waveforms that admit a delay-Doppler representation.
  • The exact equivalence between successive cancellation and maximum-likelihood under superincreasing powers may reduce receiver complexity in power-limited jammed links.
  • Hardware validation would need to confirm that least-squares recovery remains accurate when channel estimation errors and synchronization offsets are present.
  • The 40 dB reported improvement against pattern-aware jammers suggests the scheme could be attractive for low-probability-of-intercept links.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper proposes a sparse delay-Doppler NOMA scheme for jamming resilience: user data is placed on a small random subset of delay-Doppler bins drawn from a shared pseudo-random seed, spread via a unitary precoder (Hadamard/DFT/Haar), recovered by least-squares after discarding jammed bins, and decoded by SIC. It asserts a closed-form BER with no jammer-induced floor (unlike conventional OTFS-NOMA), that Marchenko-Pastur conditioning ensures any random unitary submatrix remains well-conditioned even under seed compromise, exact equivalence of low-complexity SIC to ML detection for >2 users via superincreasing (Merkle-Hellman) power allocation, an OMA-friendly partitioning rule for >4 users, and extension to Rician fading. Monte Carlo results are stated to track the analysis within 3 dB with large gains versus pattern-aware jammers.

Significance. If the central claims hold, the work would offer a concrete, parameter-free approach to eliminating the partial-band jammer floor in OTFS-NOMA while preserving low-complexity decoding, with the exact SIC-ML equivalence and seed-compromise resilience as notable technical strengths. The Rician extension and multi-user partitioning broaden applicability, though the magnitude of the reported 40 dB BER-ratio improvement would need to be weighed against the sparsity overhead.

major comments (2)
  1. [Abstract (precoder performance and seed compromise paragraph)] Abstract (Marchenko-Pastur conditioning argument): the claim that this law controls the conditioning of any random unitary submatrix (including small sparse ones when the shared seed is compromised) is load-bearing for both the no-jammer-floor BER and the seed-compromise resilience statements. Marchenko-Pastur governs bulk eigenvalue distributions in the large-dimension limit; for finite row dimension equal to the sparsity level the extreme singular values are not guaranteed to remain bounded away from zero and infinity with high probability, so an adversary knowing the seed could in principle select a jammed pattern that produces an ill-conditioned effective channel and induces a BER floor. A concrete finite-dimensional bound or counter-example check is required.
  2. [Abstract (multi-user extension paragraph)] Abstract (partitioning rule for >4 users): the OMA-friendly partitioning into pairs on disjoint bin subsets is presented as reaching floor BER at eight users by ~20 dB, but the rule appears post-hoc and its effect on the overall cross-user BER and the claimed jammer-independence must be derived explicitly; without this the multi-user scaling claim is not fully supported.
minor comments (2)
  1. [Abstract (simulation paragraph)] The abstract states that simulations track analytical predictions within 3 dB but provides no error bars, number of Monte Carlo trials, or data-exclusion criteria; these details should be added to the simulation section for reproducibility.
  2. Notation for the active-set selection and the precise definition of the superincreasing sequence (Merkle-Hellman construction) should be introduced with an equation number in the main text rather than left implicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract (precoder performance and seed compromise paragraph)] Abstract (Marchenko-Pastur conditioning argument): the claim that this law controls the conditioning of any random unitary submatrix (including small sparse ones when the shared seed is compromised) is load-bearing for both the no-jammer-floor BER and the seed-compromise resilience statements. Marchenko-Pastur governs bulk eigenvalue distributions in the large-dimension limit; for finite row dimension equal to the sparsity level the extreme singular values are not guaranteed to remain bounded away from zero and infinity with high probability, so an adversary knowing the seed could in principle select a jammed pattern that produces an ill-conditioned effective channel and induces a BER floor. A concrete finite-dimensional bound or counter-example check is required.

    Authors: We acknowledge that the Marchenko-Pastur law is an asymptotic result and that a finite-dimensional guarantee on the extreme singular values of random unitary submatrices would strengthen the argument. Our current justification relies on the fact that the three specific unitary precoders (Hadamard, DFT, Haar) produce submatrices whose empirical condition numbers remain bounded in all simulated regimes, including small sparsity levels, with no observed BER floor even under seed-compromise scenarios. In the revision we will add either a finite-dimensional bound (leveraging known results on the singular-value distribution of submatrices of these structured unitaries) or an expanded set of Monte-Carlo checks that explicitly test adversarial bin selection when the seed is known. This addresses the referee's request for a concrete check. revision: yes

  2. Referee: [Abstract (multi-user extension paragraph)] Abstract (partitioning rule for >4 users): the OMA-friendly partitioning into pairs on disjoint bin subsets is presented as reaching floor BER at eight users by ~20 dB, but the rule appears post-hoc and its effect on the overall cross-user BER and the claimed jammer-independence must be derived explicitly; without this the multi-user scaling claim is not fully supported.

    Authors: The partitioning assigns users to disjoint delay-Doppler bin subsets so that each pair operates exactly as the two-user case with its own superincreasing power allocation; because the supports are disjoint, cross-pair interference is zero and the jammer acts independently on each pair. Consequently the no-floor property and the SIC-ML equivalence carry over directly. We agree that an explicit derivation of the aggregate cross-user BER under this rule is needed to support the scaling claim. In the revision we will insert a short derivation showing that the overall BER is the average of the per-pair BERs and remains independent of the jammer pattern. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on external Marchenko-Pastur law and standard Merkle-Hellman construction

full rationale

The derivation invokes the established Marchenko-Pastur law from random matrix theory to bound conditioning of random unitary submatrices (including under seed compromise) and uses the standard Merkle-Hellman superincreasing sequence to prove exact SIC-ML equivalence. These are external references, not self-definitions or fitted inputs renamed as predictions. No load-bearing self-citations, ansatz smuggling, or renaming of known results appear in the provided text. The closed-form BER and Monte Carlo agreement are presented as independent verification against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard results from random matrix theory and cryptographic constructions for power allocation, with no new free parameters or invented entities introduced in the abstract.

axioms (1)
  • standard math Marchenko-Pastur law governs the singular value distribution and conditioning of random unitary submatrices
    Invoked to argue that Hadamard, DFT, and Haar-random precoders all yield essentially the same BER and that seed compromise does not degrade performance.

pith-pipeline@v0.9.1-grok · 5864 in / 1450 out tokens · 37175 ms · 2026-06-27T15:29:30.201662+00:00 · methodology

0 comments
read the original abstract

We propose a sparse delay-Doppler NOMA scheme resilient to intentional jamming. The transmitter places user data on a small random subset of delay-Doppler bins, spreads the result through a unitary precoder, and re-draws the active subset per frame from a pseudo-random seed shared with the receiver. The receiver detects and discards jammed bins, recovers the sparse signal by least squares, and decodes per bin via SIC. Hadamard, DFT, and Haar-random precoders all yield essentially the same BER, because a Marchenko-Pastur conditioning argument controls any random unitary submatrix. The closed-form BER has no jammer-induced floor, unlike the well-known partial-band floor of conventional OTFS-NOMA. The same argument shows that compromising the shared seed does not break the system: random unitary submatrices remain well-conditioned, so BER stays within the unjammed envelope. For more than two users we use a superincreasing power allocation (Merkle-Hellman knapsack) and prove the resulting low-complexity SIC matches maximum-likelihood detection exactly, removing the usual SIC propagation ceiling. For more than four users we partition them into pairs assigned to disjoint bin subsets; this OMA-friendly NOMA rule reaches floor BER at eight users by SNR around 20 dB. We extend the framework to Rician fading and show the jammer-independence property holds for arbitrary Rician K-factor. Monte Carlo simulations track the analytical predictions within 3 dB and show at least a 40 dB BER-ratio improvement against pattern-aware jammers, with about 24 dB of cumulative gain over conventional OTFS-NOMA under oracle jamming.

Figures

Figures reproduced from arXiv: 2606.09753 by Hovannes Kulhandjian, Michel Kulhandjian, Theodoros A. Tsiftsis.

Figure 1
Figure 1. Figure 1: End-to-end system architecture of the proposed spar [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Composite-constellation geometry for K = 2 at unit Ps. (a) Equal-power PA places the cross-terms (−, +) and (+, −) at the same location, eliminating SIC’s stage-1 discriminability. (b) Superincreasing PA at ε = 1 separates the constellation into two halves of two points each, with the half-separation 2δ1 √ Ps > 0 ensuring sign-based decoding of b1 in the noise-free case (Thm. 1). Bob’s minimum distance 2 √… view at source ↗
Figure 3
Figure 3. Figure 3: Bob BER versus jamming fraction ρJ for four sparsity values s at SNR= 20 dB, K = 2. M-P-predicted curves overlaid as dash-dotted. The cliff at ρJ ≈ 1 − s confirms the sparsity-loading design rule. 2) For s = 0.5, BER catastrophically saturates beyond ρJ ≈ 0.1, consistent with the conditioning threshold ρ cond J (s = 0.5, X = 10dB) ≈ 0.0 (operating point is already outside the conditioning-safe region). 3) … view at source ↗
Figure 5
Figure 5. Figure 5: Bob BER versus SNR for the four unitary precoders unde [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Bob BER versus SNR for geometric versus superincreas [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: shows the Bob BER versus SNR waterfall at K = 4 for four representative margins ε ∈ {0.1, 0.3, 0.5, 1.0} in AWGN, 50 000 frames per operating point. Three observa￾tions follow. First, at the optimum ε ⋆ = 0.5 Bob BER decays from 3.9 × 10−1 at SNR = 15 dB to 2 × 10−5 at SNR = 35 dB, a clean waterfall with no error floor—the operational signature of Proposition 1. Second, the small-ε curve (ε = 0.1) saturate… view at source ↗
Figure 9
Figure 9. Figure 9: Active-pattern study at K = 4 under (left) partial-band random jammer and (right) oracle jammer. C5 (per-frame random) is the only pattern safe under both. Uniform-spaced is oracle-vulnerable; clustered and bit-reversed are partial-band￾vulnerable (Sylvester replication). -5 0 5 10 15 20 JSR (dB) per jammed bin 10-4 10-3 10-2 10-1 100 Bob BER Intelligent-jammer test: N=64, na =16 (na /N=0.25), SNR=20 dB, |… view at source ↗
Figure 11
Figure 11. Figure 11: C5 protocol validation: BER versus JSR at SNR [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Cluster design taxonomy validation under HT+C5 + Ra [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Bob BER versus CSI-error variance σ 2 ǫ at SNR= 25 dB, Ktot = 4, Γ = 10 dB, oracle jammer. The HT+C5 architecture beats the same-PA T-NOMA baseline across the entire imperfect-CSI range; the multi-cluster Kg = 2 recipe (green ▽) is the most CSI-robust, retaining a large gap over T-NOMA even at σ 2 ǫ = 0.10. the LS keep set and inflating σ 2 e ; at high TJ the miss-detect rate climbs to ∼ 15%, leaving jamm… view at source ↗
Figure 16
Figure 16. Figure 16: Bob BER versus fractional-Doppler leakage [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: reports Monte Carlo simulation at K = 2, s = 0.25, ρJ = 0.25 (oracle jammer), Γ = 10 dB, 100,000 frames per operating point, sweeping Kr ∈ {Rayleigh, 0, 3, 6, 10} dB. Three observations confirm the analytical framework: (i) Waterfall ordering matches Rician fading averaging. In the waterfall region (SNR ∈ [10, 20] dB), empirical BER decreases monotonically with Kr, from 1.5×10−2 at Rayleigh to 5.5 × 10−3 … view at source ↗

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