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arxiv: 2605.14451 · v1 · submitted 2026-05-14 · 💻 cs.IT · eess.SP· math.IT

Recognition: 2 theorem links

· Lean Theorem

CP-OFDM Achieves Lower Ranging CRB Than Frequency-Spread Waveforms in the Large-Sample Regime

Fan Liu , Yifeng Xiong , Ya-Feng Liu , Jie Yang , Christos Masouros , Shi Jin

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:23 UTC · model grok-4.3

classification 💻 cs.IT eess.SPmath.IT
keywords CP-OFDMCramér-Rao BoundIntegrated Sensing and CommunicationsWaveform DesignRanging EstimationFisher Information MatrixAsymptotic AnalysisStochastic CRB
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The pith

CP-OFDM achieves a lower ranging CRB than frequency-spread waveforms when the number of symbols grows large.

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

The paper examines how the choice of modulation waveform influences ranging accuracy in systems that reuse random communication symbols for sensing. It identifies a factorization of the Fisher information matrix that splits the deterministic geometry of the target from the random power fluctuations caused by the data symbols. This factorization produces a universal lower bound on the CRB that CP-OFDM attains exactly for PSK constellations. For QAM and other sub-Gaussian constellations, an asymptotic analysis shows that CP-OFDM outperforms every other frequency-spread orthogonal waveform in the large-N limit, almost surely whenever the random matrix is invertible, and the advantage carries over to amplitude and joint delay-amplitude estimation.

Core claim

The central claim is that a structural factorization of the FIM for joint delay-amplitude estimation separates the deterministic Jacobian of target geometry from the random frequency-domain signal power induced by data symbols. This structure yields a Jensen-type universal lower bound on the CRB that is exactly attained by CP-OFDM under PSK constellations. For QAM and broader sub-Gaussian constellations, an asymptotic perturbation analysis of the inverse FIM proves that, as the number of transmitted symbols N grows large, CP-OFDM achieves a lower ranging CRB than any frequency-spread orthogonal waveform over the almost-sure event where the random FIM is invertible. The superiority extends to

What carries the argument

The structural factorization of the Fisher information matrix that isolates the random frequency-domain power fluctuations induced by the data symbols from the deterministic target geometry.

If this is right

  • CP-OFDM attains the universal lower bound on ranging CRB for PSK constellations.
  • In the large-sample regime the superiority of CP-OFDM holds for amplitude estimation and for joint delay-amplitude estimation.
  • CP-OFDM is a stationary point of the stochastic CRB minimization problem over the unitary group for any finite N.
  • The Riemannian Hessian of CP-OFDM is positive semidefinite for all sufficiently large N, establishing asymptotic local optimality.
  • Numerical comparisons confirm that CP-OFDM outperforms SC, OTFS, and AFDM.

Where Pith is reading between the lines

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

  • The result supplies a concrete criterion for selecting waveforms in dual-function systems that must support both data transmission and ranging.
  • If the same factorization structure appears for other sensing parameters such as Doppler or angle, the same asymptotic preference for CP-OFDM would follow.
  • For practical deployment it would be useful to quantify the finite-N gap between CP-OFDM and competing waveforms on typical channel realizations.

Load-bearing premise

The random Fisher information matrix remains invertible with probability approaching one as the number of symbols grows large.

What would settle it

A concrete counter-example in which, for sufficiently large N and with positive probability, some frequency-spread orthogonal waveform yields a strictly smaller ranging CRB than CP-OFDM on an invertible realization of the random FIM.

Figures

Figures reproduced from arXiv: 2605.14451 by Christos Masouros, Fan Liu, Jie Yang, Shi Jin, Ya-Feng Liu, Yifeng Xiong.

Figure 1
Figure 1. Figure 1: Ranging CRB for SC, AFDM, OTFS and OFDM under 16-QAM an [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
read the original abstract

The inherent randomness of communication symbols creates a fundamental tension in Integrated Sensing and Communications (ISAC). On the one hand, they enable data transmission while allowing sensing to fully reuse communication resources. On the other hand, their randomness induces waveform-dependent fluctuations that directly affect sensing accuracy. This paper investigates a foundational question arising from this tradeoff: \textit{How does the modulation waveform affect the ranging Cram\'er--Rao Bound (CRB) when sensing reuses random data symbols?} We address this question by revealing a structural factorization of the Fisher information matrix (FIM) for joint delay-amplitude estimation, which separates the deterministic Jacobian of the target geometry from the random frequency-domain signal power induced by the data symbols. This structure yields a Jensen-type universal lower bound on the CRB, which is exactly attained by CP-OFDM under PSK constellations. For QAM and broader sub-Gaussian constellations, we develop an asymptotic perturbation analysis of the inverse FIM and prove that, when the number of transmitted symbols $N$ grows large, CP-OFDM achieves a lower ranging CRB than any frequency-spread orthogonal waveform over the almost-sure event where the random FIM is invertible. This superiority is further extended to amplitude estimation and full joint delay-amplitude estimation. We also characterize the local geometry of the stochastic CRB minimization problem over the unitary group. The analysis reveals that CP-OFDM is a stationary point for finite $N$, and its Riemannian Hessian is positive semidefinite for sufficiently large $N$, establishing its asymptotic local optimality. Numerical results confirm that OFDM outperforms representative waveforms including SC, OTFS, and AFDM.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript claims that CP-OFDM achieves a strictly lower ranging CRB than any other frequency-spread orthogonal waveform in the large-N regime for joint delay-amplitude estimation when random communication symbols are reused for sensing. The central technical step is a factorization of the FIM into a deterministic geometric Jacobian and a random diagonal frequency-domain power matrix; Jensen's inequality then supplies a universal lower bound attained exactly by CP-OFDM under PSK. For sub-Gaussian constellations an asymptotic perturbation expansion of the inverse FIM shows that the leading correction term is positive whenever the power profile is non-constant, yielding almost-sure superiority on the event that the random FIM is invertible. The result is extended to amplitude-only and joint estimation, and the stochastic CRB minimization problem is shown to have CP-OFDM as a stationary point whose Riemannian Hessian is positive semidefinite for large N. Numerical comparisons with SC, OTFS and AFDM corroborate the ordering.

Significance. If the derivations hold, the work supplies a parameter-free, asymptotically rigorous ranking of waveforms for ISAC ranging that favors CP-OFDM without fitting constants to the target CRB. The clean separation of geometry and power fluctuations, the exact attainment of the Jensen bound, and the standard large-N perturbation technique together constitute a solid theoretical contribution. The local-optimality analysis over the unitary group and the numerical validation further increase the result's relevance for waveform design in integrated sensing and communications.

major comments (1)
  1. The almost-sure superiority is stated only on the event that the random FIM remains invertible. A quantitative bound on the probability of singularity for finite but large N (or at least a reference to existing random-matrix results that control this probability) would make the practical scope of the asymptotic claim clearer.
minor comments (3)
  1. The definition of 'frequency-spread orthogonal waveform' appears only implicitly through the frequency-domain power matrix; an explicit one-sentence characterization in the introduction, together with the precise unitary constraint, would help readers outside the immediate subfield.
  2. In the numerical section the Monte-Carlo setup (number of trials, SNR range, exact constellation parameters) should be stated explicitly so that the plotted CRB curves can be reproduced from the given theoretical expressions.
  3. A few instances of inconsistent bold-face notation for vectors versus matrices appear in the early sections; a quick consistency pass would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment and the recommendation of minor revision. We address the point below and will incorporate the suggested clarification.

read point-by-point responses

  1. Referee: The almost-sure superiority is stated only on the event that the random FIM remains invertible. A quantitative bound on the probability of singularity for finite but large N (or at least a reference to existing random-matrix results that control this probability) would make the practical scope of the asymptotic claim clearer.

    Authors: We agree that quantifying the probability of the singularity event strengthens the practical interpretation. In the revised manuscript we will add a remark (in the paragraph following the statement of the almost-sure result) noting that, under the sub-Gaussian assumption on the constellation, the diagonal entries of the random power matrix are i.i.d. sub-Gaussian random variables. Standard concentration and random-matrix results (e.g., Rudelson–Vershynin bounds on the smallest singular value) then imply that the probability of singularity decays exponentially in N: P(FIM singular) ≤ exp(−cN) for a positive constant c that depends only on the sub-Gaussian norm of the constellation. Consequently, the almost-sure superiority holds with overwhelmingly high probability already for moderate but finite N. We will cite the relevant random-matrix literature. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation starts from the signal model to obtain an exact structural factorization of the FIM separating the deterministic geometric Jacobian from the random diagonal frequency-domain power matrix induced by data symbols. Jensen's inequality is then applied directly to this factorization to produce a universal lower bound on the CRB that is attained exactly when the power matrix is deterministic (CP-OFDM with PSK). For sub-Gaussian constellations an asymptotic perturbation expansion of the inverse FIM is performed; the leading correction term is shown strictly positive for any non-constant power profile, yielding the almost-sure superiority for large N on the invertible-FIM event. These steps rely on external standard tools (Jensen, second-order perturbation, sub-Gaussian concentration) without fitting any parameter to the target CRB or invoking load-bearing self-citations. The local-geometry analysis on the unitary group independently establishes stationarity and positive-semidefiniteness of the Riemannian Hessian for large N. The entire chain is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivations rely on standard properties of the Fisher information matrix for Gaussian noise models, the definition of sub-Gaussian random variables, and the almost-sure invertibility of the random FIM for large N; no free parameters are fitted to data and no new physical entities are postulated.

axioms (2)
  • domain assumption The noise is circularly symmetric complex Gaussian and independent across observations.
    Standard model for communication and sensing receivers; invoked when writing the likelihood and FIM.
  • domain assumption The data symbols are i.i.d. from a constellation with finite moments (PSK or sub-Gaussian).
    Required for the random power term to have well-defined mean and for the law of large numbers to apply in the large-N limit.

pith-pipeline@v0.9.0 · 5622 in / 1616 out tokens · 37939 ms · 2026-05-15T01:23:57.212831+00:00 · methodology

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