Impact of Insufficient CP on Sensing Performance in OFDM-ISAC Systems

Ming Li; Peishi Li; Qian Liu; Rang Liu

REVIEW 3 major objections 5 minor 14 references

Impact of Insufficient CP on Sensing Performance in OFDM-ISAC Systems

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper derives closed-form formulas for how a cyclic prefix shorter than target delays raises the sidelobe floor of OFDM range-Doppler maps under both matched and reciprocal filtering.

desk verdict Useful incremental theory paper that quantifies insufficient-CP ISI/ICI in OFDM-ISAC, but Eq. (20) is wrong as printed for multi-target scenes and needs a fix before publication. read the letter →

arxiv 2505.01125 v1 pith:KJ26I5GP submitted 2025-05-02 eess.SP

Peishi Li , Rang Liu , Qian Liu , Ming Li This is my paper · ORCID

classification eess.SP

keywords OFDM-ISACcyclicprefixsensingperformancepeaksidelobelevelratiointegratedmatchedfilteringreciprocalinter-carrierinterference

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks what happens to an OFDM integrated sensing and communication system when the cyclic prefix is too short to cover target delays, and it answers with explicit statistical formulas. The authors derive closed-form expressions for the average power of every cell in the range-Doppler map under two standard filters: matched filtering and reciprocal filtering. From these, they obtain peak-sidelobe and integrated-sidelobe ratios that depend on CP length, constellation shape, and the number and strength of targets. If the formulas are right, engineers can predict when a too-short CP will hide weak targets and can choose between reciprocal filtering's noise amplification and matched filtering's inter-target leakage. The paper also shows that with constant-modulus modulation the two filters are equivalent, while with QAM their sensing performance diverges in a quantified way.

What carries the argument

The load-bearing object is the second-order moment of the range-Doppler map, $E\{|\chi(l,\nu)|^2\}$, computed separately for reciprocal filtering (dividing by the transmitted symbol before correlating) and matched filtering (multiplying by its conjugate). The argument is carried by decomposing each echo into a useful term, ISI, ICI, and noise, treating the interference as circularly symmetric complex Gaussian with variances $P_{\mathrm{ISI}}=\sum \rho_q|\alpha_q|^2$ and $P_{\mathrm{ICI}}=\sum \rho_q(1-\rho_q)|\alpha_q|^2$, and then tracking how the random data symbols $s_{n,m}$ survive the filter: RF leaves a constellation-dependent factor $\xi_s=E\{1/|s_{n,m}|^2\}$, while MF leaves a leakage floor proportional to $\mu_4-1$. The PSLR step then invokes order statistics: for $MN-Q$ independent exponential off-peak bins, the expected maximum is the harmonic number $H_{MN-Q}$ times the per-bin mean.

What would settle it

Run a noiseless matched-filter simulation with two QAM targets beyond the CP and many random symbol draws, then compare the empirical mean of the largest off-peak bin to $H_{MN-Q}\big((\mu_4-1)\sum|\tilde\alpha_q|^2\big)$; if the ratio deviates from $H_{MN-Q}$ as $MN$ or $\mu_4$ changes, the exponential assumption behind Eq. (26) is false.

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Extended reading notes

Core claim

The central result is Eq. (20): after CP removal and FFT, the mean power of a matched-filter range-Doppler bin is $(\mu_4-1)|\tilde\alpha_q|^2+\sigma_{\mathrm{IN}}^2$ away from a target and $(MN+\mu_4-1)|\tilde\alpha_q|^2+\sigma_{\mathrm{IN}}^2$ at the target bin, where $\mu_4=E\{|s_{n,m}|^4\}$ is the constellation fourth moment, $\tilde\alpha_q$ is the target amplitude attenuated by $(1-\rho_q)$ when its delay exceeds the CP, and $\sigma_{\mathrm{IN}}^2$ is the summed ISI, ICI, and noise power. For reciprocal filtering, the corresponding values are $\xi_s\sigma_{\mathrm{IN}}^2$ off-peak and $MN|\tilde\alpha_q|^2+\xi_s\sigma_{\mathrm{IN}}^2$ at the peak, with $\xi_s=E\{1/|s_{n,m}|^2\}$ amplifying the interference-plus-noise. Using an exponential approximation for off-peak bins, these lead to the PSLR formulas in Eqs. (24) and (26). The paper's claim is that these expressions capture, for the first time, how an insufficient CP raises sidelobes and how RF and MF trade noise amplification against inter-target interference.

Load-bearing premise

The PSLR formulas treat all off-peak range-Doppler bins as independent, exponentially distributed samples with the computed mean; if real sidelobes are correlated or heavier-tailed, the harmonic-number peak formula is only an approximation.

Editorial extensions

If this is right

When the modulation is constant-envelope, such as PSK, both $\xi_s$ and $\mu_4$ equal one, so RF and MF produce identical PSLR and ISLR and the filter choice makes no sensing difference.
With QAM, RF is the better filter at short range or high echo SNR because it removes inter-target leakage, while MF is better at long range or low SNR because it does not amplify the noise floor; the crossover is computable from the derived formulas.
An insufficient CP does two quantifiable things: it shrinks the effective target amplitude by $1-\rho_q$ and injects ISI/ICI power $\rho_q|\alpha_q|^2+\rho_q(1-\rho_q)|\alpha_q|^2$, so PSLR and ISLR rise sharply once the target range exceeds the CP-limited unambiguous range.
The framework extends earlier interference-power analyses, which were limited to reciprocal filtering, to matched filtering in multi-target scenarios.
For a fixed CP length and modulation order, the formulas give a closed-form prediction of the range beyond which sensing performance crosses a required PSLR, which is useful for system dimensioning.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the exponential off-peak assumption fails for matched filtering, the PSLR formula in Eq. (26) will misstate the true peak sidelobe; a high-SNR simulation measuring the distribution of off-peak bins would show whether the discrepancy grows with $\mu_4$.
The same moment-based accounting could be rerun for windowed range-Doppler processing or for other multicarrier waveforms, since the paper's decomposition into useful signal, ISI, ICI, and leakage terms is generic.
The matched-filter sidelobe floor $(\mu_4-1)\sum|\tilde\alpha_q|^2$ suggests a testable design rule: constellations with lower fourth moment should produce cleaner matched-filter range-Doppler maps even when the cyclic prefix is short.
Because reciprocal filtering amplifies noise by $\xi_s$, the formulas imply that high-order QAM pays a sensing penalty under RF that grows with the reciprocal-power moment; ISAC systems may need to budget for this penalty when choosing constellations for communication.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: a competent incremental theory paper that puts the insufficient-CP ISI/ICI floor into closed-form PSLR/ISLR formulas for both matched and reciprocal filtering. It is not a breakthrough, and Eq. (20) has a genuine multi-target omission that should have been caught.

What is new: the paper takes the existing Gaussian approximation for CP-induced interference from [8], [9], extends the second-order moment calculation for random OFDM symbols from RF to MF, and turns those moments into PSLR/ISLR formulas. That combination is new and practically useful for 6G sensing design. The simulation support is honest: 5000 realizations, realistic NR parameters, and the curves match the formulas. The qualitative conclusion—RF suppresses inter-target interference but amplifies noise, MF keeps the noise floor but leaks symbol energy—is correct and worth having stated cleanly.

Soft spots. First, Eq. (20) as printed is not the second-order moment of the MF range-Doppler map in a multi-target scene. From (12b) there is a sum over Q targets, and each target contributes (μ4−1)|α~_q'|² away from its own bin. The correct off-peak mean is (μ4−1)Σ_{q'}|α~_q'|² + σ_IN², and the target-bin mean includes the same sum, not just the q-th target term. The later PSLR/ISLR formulas in (25)–(26) use the summed version, so they are internally consistent with the simulations; the issue is that Eq. (20) is written as if only one target exists. Fixable, but wrong as published. Second, the conversion of second-order moments into PSLR assumes the off-peak MF samples are exponentially distributed. That is not proven in the text. For the large M and N used here the CLT argument is plausible, and the simulations bear it out, so I would call it a minor caveat rather than a flaw. There is also a typo in Eq. (16a) where reciprocal symbol expectations are misprinted; the final RF result is correct.

The citation pattern is fine. The ISI/ICI variance is taken from independent work, no parameter fitting, no circularity. The novelty claim is appropriately limited.

This paper is for readers working on CP-aware OFDM-ISAC analysis or 6G sensing waveform design. It deserves a serious referee: the central moment derivation is useful and likely correct, but the manuscript needs the Eq. (20) correction and a clearer flag on the exponential assumption. I would accept with revisions, not desk-reject.

Referee Report

3 major / 5 minor

Summary. The manuscript analyzes OFDM-ISAC sensing when target round-trip delays exceed the cyclic prefix (CP) duration, so that inter-symbol interference (ISI) and inter-carrier interference (ICI) contaminate the received echo. Assuming the interference-plus-noise (IN) term is circularly symmetric complex Gaussian (CSCG) with variances taken from prior work, the authors derive closed-form expressions for the second-order moments of the range-Doppler map (RDM) under both matched filtering (MF) and reciprocal filtering (RF). They then convert these moments into peak sidelobe level ratio (PSLR) and integrated sidelobe level ratio (ISLR) formulas, leading to the conclusion that RF suppresses inter-target interference but amplifies noise (via a constellation-dependent factor xi_s) while MF has a lower noise floor but suffers from symbol-induced leakage. The theoretical predictions are validated by Monte Carlo simulations with 5000 realizations for 1024-QAM.

Significance. If the technical issues raised below are resolved, this is a useful contribution to OFDM-ISAC sensing. It extends the ISI/ICI modeling of prior work on reciprocal filtering to a unified treatment of both RF and MF, and it offers closed-form PSLR/ISLR expressions that quantify the CP-length/constellation trade-off. The paper provides reproducible simulation validation for the main formulas and explicitly identifies a design trade-off (noise amplification in RF versus inter-target interference in MF). However, the central moment derivation contains an inconsistency in Eq. (20) and the PSLR step relies on an unproven distributional assumption for MF; these need to be fixed before the claims of exact closed-form characterization are fully supported.

major comments (3)

[Sec. III-A, Eq. (20)]
[Sec. III-B, Eqs. (22)-(25)]
[Sec. III-A, Eq. (16a)]

minor comments (5)

[Sec. III-A, Eq. (14)]
[Sec. III-B, Eq. (22)]
[Sec. II and Sec. III]
[Abstract and Conclusion]
[Sec. II-B, Eq. (10)]

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning identified; the derivation is self-contained and uses only external, independently published results for ISI/ICI variances and standard order statistics.

full rationale

The central derivation chain is not circular. The paper takes the ISI/ICI variance expressions in Eq. (11) directly from the externally authored works [8], [9] ('According to the findings in [8], [9]'), uses those variances as fixed inputs, and then obtains the RDM second-order moments in Eqs. (16)-(20) by straightforward expectations over the random data symbols, with no fitted parameters. The PSLR/ISLR step in Eqs. (22)-(26) relies on the standard expectation of the maximum of iid exponential random variables from [12], [13]; this is an imported mathematical fact, and the exponential assumption itself is an approximation, not a quantity fitted to the target PSLR/ISLR. There are no self-citations that carry a load-bearing premise: none of the cited sources [1]-[13] is authored by Li, Liu, Liu, or Li. The Monte Carlo validation is genuinely external. Two non-circular technical concerns should be noted for correctness: (i) Eq. (20) omits the multi-target inter-target-interference sum (µ4-1)Σ_{q'≠q}|α_q'|² that should be present at both the mainlobe and off-peak bins, although Eqs. (25)-(26) silently include it; and (ii) the exponential-distribution assumption for MF off-peak samples is asserted without proof. Neither concern makes a prediction equivalent to an input by construction, so the circularity score remains 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on several modeling assumptions: the Gaussian approximation of CP-induced interference (from prior work), the independence of interference across subcarriers and symbols, and the exponential distribution of off-peak RDM samples for PSLR. No parameters are fitted to data; the constellation-dependent constants xi_s and mu_4 are computed from the modulation, not estimated. No new entities are introduced.

assumptions (5)

domain assumption ISI and ICI from targets with delay exceeding CP are modeled as independent circularly symmetric complex Gaussian with variances P_ISI and P_ICI (Eq. 11), per prior work [8], [9].
This is the foundation of the interference-plus-noise model used in all subsequent second-order moment derivations; it holds only approximately for large N.
domain assumption The interference-plus-noise terms are independent across subcarriers and symbols, and independent of the transmitted data symbols.
Required to factorize expectations in Eqs. (16)-(20); the paper asserts i.i.d. properties without proof.
ad hoc to paper Off-peak range-Doppler map samples are i.i.d. exponential with mean equal to the derived second-order moment, allowing E[max] = H_Q * mean (Eqs. 22-25).
Invoked without proof to derive PSLR; especially questionable for MF, where off-peak values include random-symbol leakage, so the PSLR formulas are approximate.
domain assumption Target delays and Doppler shifts are integer multiples of the resolution (lq = tau_q B, nu_q = f_d,q T_obs).
On-grid parameters make the deterministic target response zero at off-grid bins; the authors note windowing can mitigate fractional leakage, so this is an idealization.
standard math Standard OFDM signal model with rectangular pulses, point targets, and AWGN.
Standard assumptions in OFDM radar analysis; not controversial.

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Cite this review

Pith. "Pith review of Impact of Insufficient CP on Sensing Performance in OFDM-ISAC Systems." pith.science (2026). https://pith.science/paper/KJ26I5GP

@misc{pith2026250501125,
  author       = {Pith},
  title        = {Pith review of: Impact of Insufficient CP on Sensing Performance in OFDM-ISAC Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KJ26I5GP}},
  note         = {Machine review of arXiv:2505.01125}
}

read the original abstract

Orthogonal frequency-division multiplexing (OFDM) is widely considered a leading waveform candidate for integrated sensing and communication (ISAC) in 6G networks. However, the cyclic prefix (CP) used to mitigate multipath effects in communication systems also limits the maximum sensing range. Target echoes arriving beyond the CP length cause inter-symbol interference (ISI) and inter-carrier interference (ICI), which degrade the mainlobe level and raise sidelobe levels in the range-Doppler map (RDM). This paper presents a unified analytical framework to characterize the ISI and ICI caused by an insufficient CP length in multi-target scenarios. For the first time, we derive closed-form expressions for the second-order moments of the RDM under both matched filtering (MF) and reciprocal filtering (RF) processing with insufficient CP length. These expressions quantify the effects of CP length, symbol constellation, and inter-target interference (ITI) on the mainlobe and sidelobe levels. Based on these results, we further derive explicit formulas for the peak sidelobe level ratio (PSLR) and integrated sidelobe level ratio (ISLR) of the RDM, revealing a fundamental trade-off between noise amplification in RF and ITI in MF. Numerical results validate our theoretical derivations and illustrate the critical impact of insufficient CP length on sensing performance in OFDM-ISAC systems.

Figures

Figures reproduced from arXiv: 2505.01125 by the authors.

**Figure 2.** The range profile under RF and MF processing. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** RMSE for range and velocity estimation versus the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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