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Spreading over OFDM for Integrated Sensing and Communications (ISAC) Ranging: Multi-user Interference Mitigation

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

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Pith's one-line read The paper claims that in a two-user OFDMA ISAC system, inter-band aperiodic cross-correlation leakage dominates ranging sidelobes at high interference and is suppressible by a P-DPSS spreading layer.

desk verdict Solid closed-form analysis of OFDMA ISAC inter-band leakage, but the P-DPSS spreading claim does not follow from Theorem 1 and needs either a real proof or a more modest abstract. read the letter →

arxiv 2505.02160 v1 pith:TRQF2LNI submitted 2025-05-04 eess.SP

Karim A. Said , Ibraheem Alturki , Husheng Li , Elizabeth Bentley , Lingjia Liu This is my paper · ORCID
classification eess.SP
keywords OFDMAISACintegratedsensingandcommunicationsaperiodiccorrelationinter-bandcross-correlationinterferencesidelobelevelP-DPSSspreadingOFDMrangingmulti-usermitigation
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

Ranging in OFDMA ISAC assumes each user's correlation is clean because users occupy disjoint subcarriers, but that is only true for circular correlation within the cyclic prefix. For target delays beyond the CP, the receiver performs aperiodic correlation, and the zero-padding that makes this operation spectrum-friendly up-samples the bands so that disjoint subcarrier allocations leak into each other. The paper derives the inter-band leakage energy E_IB, shows it dominates integrated sidelobe energy when the interfering user is much stronger than the backscatter, and proves an upper bound on it, then constructs a P-DPSS spreading layer that minimizes that bound. The central claim is that spreading keeps EISL nearly flat with interference amplitude, at a price in spectral efficiency: another concrete instance of the sensing-versus-communication trade-off.

What carries the argument

The central objects are the inter-band cross-correlation energy $E_\mathrm{IB}$ and the upper bound on it, together with P-DPSS spreading: the eigenvectors of the discrete Dirichlet kernel that concentrate signal energy within a user's assigned band. The zero-padding used to express aperiodic correlation as a $2N$-point frequency-domain product up-samples each user's spectrum by a factor of two, so frequency-disjoint bands acquire overlapping tails; $E_\mathrm{IB}$ captures the resulting leakage, and Theorem 1 bounds it by the cross-band Frobenius norms of $W^{(1)}$, $W^{(2)}$. The P-DPSS eigenvectors are the global leakage-minimizing signaling directions for the Dirichlet kernel, and retaining only the most concentrated fraction $\eta$ of them forms the orthogonal spreading layer. The spectral-utilization factor $\eta$ is the dial that trades sidelobe suppression against communication capacity.

What would settle it

In a two-user OFDMA ISAC experiment or simulation, pass user-2's signal through a multipath channel with two or more taps at different delays instead of a single scaled copy, then measure EISL versus alpha with and without P-DPSS spreading. If spreading no longer keeps EISL flat as alpha grows, or if the measured leakage energy no longer tracks Eq. (17), the paper's central claim is contradicted.

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Extended reading notes

Core claim

In a two-user OFDMA ISAC setting, user-1's aperiodic correlation sidelobes split into in-band auto-correlation terms plus an inter-band leakage term $E_\mathrm{IB}=\alpha^2(2M-1)\operatorname{tr}(W^{(1)}W^{(1)H}\circ W^{(2)}W^{(2)H})$, where $W^{(1)}$, $W^{(2)}$ are the zero-padded, up-sampled basis/spreading matrices of the two users and $\alpha$ is the amplitude of user-2's signal at user-1. As $\alpha$ grows, this term dominates the integrated sidelobe energy, so the ranging sidelobes rise roughly quadratically with the interferer's amplitude. The paper proves an upper bound on $E_\mathrm{IB}$ and shows that the bound is controlled by the Frobenius norms of the cross-band blocks of the spreading matrices. Choosing orthonormal spreading columns from the primary eigenvectors of the discrete Dirichlet kernels, i.e. periodic DPSS sequences, minimizes the leakage for a given spectral utilization. Simulations show that with both users spread, EISL remains nearly constant as $\alpha$ grows to 20 dB and higher, whereas plain OFDM's EISL rises sharply.

Load-bearing premise

The interference path is modeled as a single delay-free, frequency-flat, noiseless scaled copy alpha times the other user's signal added to the sensing user's received signal; real multipath, delay, or frequency-selective interference would change the correlation-leakage structure and is not covered by the analysis.

Editorial extensions

If this is right

  • With both users spread, EISL stays nearly constant as the interferer amplitude rises to at least 20 dB, while plain OFDM's EISL climbs; the crossover benefit appears near 12 dB for L=16, 15 dB for L=32, and 19 dB for L=64 subcarriers.
  • Increasing the number of OFDM symbols M lowers sidelobes in both cases, and spreading outperforms plain OFDM at high interference for M > 1; at M=10 and alpha=20 dB the gap narrows to about 1 dB.
  • The interference suppression is not free: lowering the spectral utilization eta below 0.9 reduces E_IB but raises autocorrelation sidelobes, so eta near 0.9 is the practical sweet spot in the simulated settings.
  • Higher-order modulation such as 16QAM and 64QAM worsens EISL under plain OFDM but has almost no effect under P-DPSS spreading at high interference, making the spread waveform insensitive to modulation choice in strong-interference regimes.

Reading between the lines

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

  • The pairwise trace form of E_IB suggests that a K-user OFDMA extension would simply sum pairwise leakage terms, one per interfering user with its own alpha; the paper does not derive this, but the P-DPSS mechanism would likely carry over if each user spreads.
  • P-DPSS spreading acts effectively as a soft guard band: instead of blanking edge subcarriers, it shapes each user's energy into a spectrally concentrated subspace, so the same leakage reduction can be traded with a continuous parameter eta rather than an integer number of unused subcarriers.
  • Because the inter-band term vanishes in the M-to-infinity limit, long coherent frames already suppress the leakage; spreading therefore matters most for short frames or fast-moving targets where M cannot be made large.
  • A frequency-selective interference channel would weight the terms inside the trace by per-subcarrier gains, so a fixed P-DPSS basis may not be optimal under multipath; adapting the spreading subspace to the measured interference channel is a natural testable extension.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies a two-user OFDMA ISAC system in which user 1 performs ranging by aperiodic correlation between its transmitted OFDM signal and a received signal that is modeled as the user's own backscatter plus a scaled version of user 2's transmission. The authors derive closed-form expressions for the expected squared aperiodic autocorrelation function (Proposition 1), the expected integrated sidelobe energy EISL (Corollary 1), and an inter-band interference energy term EIB (Eq. 17). They propose an orthogonal spreading layer based on periodic discrete prolate spheroidal sequences (P-DPSS) with a spectral utilization back-off parameter η<1, and claim that this spreading minimizes EIB and keeps EISL nearly constant as the interferer strength α grows. Simulation results compare OFDM with and without spreading for varying α, M, L, and modulation order, and show the claimed robustness at η=0.9.

Significance. If the central design claim holds, the paper would provide a useful analytical characterization of inter-band correlation leakage in OFDMA-based ISAC and a concrete waveform modification that trades a small amount of spectral efficiency for sidelobe robustness. The derivations of Proposition 1 and Corollary 1 appear to be validated by the simulations, which sweep α, M, L, and modulation order, and the agreement between the theoretical A-ACF curves and simulated curves in Fig. 4 is a clear strength. However, the paper's main analytical claim about the optimality of the P-DPSS spreading is not established by the presented theorem, and the received-signal model is highly idealized. The empirical comparison at η=0.9 is suggestive but does not by itself separate the effect of the spreading transform from the effect of reducing the active signaling dimensions.

major comments (3)
  1. [Section IV, Theorem 1, Eqs. (17)-(21)] Theorem 1 cannot support the claim that P-DPSS spreading minimizes EIB. For square orthonormal P, W^(i)W^(i)H = D B^(i) U_L D P^(i)P^(i)H D^H U_L^H B^(i)H D^H = D B^(i) U_L D D^H U_L^H B^(i)H D^H, so EIB in Eq. (17) and the upper bound in Eq. (18) are independent of the spreading matrix. The proposed benefit appears only for rectangular P_η with η<1, but Theorem 1 is stated and proved for square P, and no analogous bound is derived for the truncated case. Moreover, no argument is given that the leading eigenvectors of \bar B^(1) minimize the Hadamard-product trace in Eq. (17). Section IV-A explicitly concedes that "how the auto-correlation energy changes with η is not known, and thus the overall impact of η on EISL is not fully known." The simulations may support an empirical claim at η=0.9, but the analytical claim that the proposed spreading itself minimizes IB energy is not established.
  2. [Section III-A, Eq. (11)] The received signal model y_t = x_t^(1) + α x_t^(2) omits delay, multipath, frequency selectivity, and noise on the interference path. Equations (13)-(17) and the spreading design in Eqs. (20)-(21) are therefore tied to a single-tap, frequency-flat interferer with a single scalar amplitude α. For a frequency-selective or delayed interfering path, the leakage distribution over the rows of W changes, and the P-DPSS eigenvectors of the in-band Dirichlet kernel need not be the minimizer of EIB. This is a scope restriction that should be stated prominently, and the claimed robustness of spreading should be tested at least with a one-tap delayed interferer or a simple two-tap channel.
  3. [Section V, first paragraph and Figs. 4-8] The fairness of the η=0.9 comparison is not fully established at the modeling level. For OFDM, η is implemented by zeroing edge subcarriers as guard bands, while for P-DPSS it is implemented by truncating the spreading basis to the most concentrated eigenvectors. However, the analytical W^(1) and W^(2) in Eqs. (8)-(9) are derived for all L (respectively N-L) subcarriers being active. It is not specified whether the theoretical curves for the OFDM η=0.9 case use modified W matrices that account for the guard bands or use the full-band formulas. This matters because Fig. 8 shows that reducing η alone degrades EISL at small α, so the reader cannot tell how much of the apparent improvement of P-DPSS is due to the spreading transform itself and how much is due to the different way the active dimensions are reduced.
minor comments (5)
  1. [Eqs. (20)-(21)] The definitions of P_η^(1) and P_η^(2) contain an undefined variable k in the diagonal exponent (z^{kL(N-1)/(2N)} and z^{k(N-L)(N-1)/(2N)}). This should be corrected to a fixed notation consistent with Eq. (8).
  2. [Section IV-A, Eq. (22)] The statement "the supported capacity is MtηLu" is dimensionally incorrect: with modulation order M and ⌈ηL⌉ dimensions, the number of bits per OFDM symbol is ⌈ηL⌉ log2 M, not M⌈ηL⌉. This should be corrected since the paper uses η as a spectral-efficiency back-off.
  3. [Abstract and Section V] The abstract's claim that spreading results in ISL levels "significantly lower" than OFDM without spreading is an overstatement without the qualifiers that the advantage appears at sufficiently large α and sufficiently large M; at low α and small M, the simulations in Figs. 4-7 show OFDM without spreading is better. The abstract should state the condition under which the benefit holds.
  4. [Fig. 4 caption] The caption contains an incomplete sentence: "for example α = 15 dB, sie the amplitude difference ... is 20 dB." This should be rewritten.
  5. [Theorem 1 statement] The theorem statement begins with a stray "E" before "For an in-band user signal spreading matrix..." which should be removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EISL/EIB derivation is self-contained, and the P-DPSS design is an externally motivated ansatz rather than a fitted or self-referential input.

full rationale

The paper's derivation chain is not circular. The EISL expression in Corollary 1 and the EIB term in Eq. (17) are derived from the assumed received-signal model in Eq. (11) and the zero-padded frequency-domain representation of aperiodic correlation; Proposition 1 and Corollary 1 are then checked against Monte Carlo simulations across independent sweeps of alpha, M, L, and modulation order. No parameter is fitted to the target EISL curve and then renamed a prediction; the spreading-matrix choice in Eqs. (20)-(21) is imported from the known spectral-concentration property of periodic DPSS eigenvectors (external reference [23]), not obtained by minimizing the paper's own EIB expression. The paper even concedes the relevant limitation: "how the auto-correlation energy changes with eta is not known, and thus the overall impact of eta on EISL is not fully known" (Section IV-A), and the upper bound in Theorem 1 is admitted to be unaffected by square orthogonal spreading ("their specific choices will have no impact on the upper bound (18)"). Those are gaps in the theoretical support for P-DPSS, but they do not make the claimed EISL prediction equivalent to its inputs by construction. The choice of eta = 0.9 is selected from the EISL-versus-eta sweep in Fig. 8, but it is thereafter used as a fixed design parameter when the paper evaluates behavior versus alpha, M, L, and modulation, so it is not a fitted input called a prediction. No load-bearing self-citation was found: the only same-author citations [17], [21], [24] support ancillary definitions or a continuous-time analogue, while the load-bearing DPSS optimality claim is attributed to [23].

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the idealized two-user signal model, the moment factorization for random symbols, and the design choice η=0.9. The P-DPSS spreading itself is not an invented entity; it is a known basis cited from [23] and [24]. No new physical entities are postulated.

free parameters (1)
  • η (spectral utilization back-off) = 0.9
    Selected from the EISL-versus-η simulation curve (Fig. 8) to maximize the gain of P-DPSS spreading at α = 20 dB; this is a hand-chosen design parameter rather than a derived constant.
assumptions (4)
  • standard math DFT and zero-padding properties: aperiodic correlation equals periodic correlation of zero-padded signals (Eqs. 6-7).
    This is a standard signal processing identity used to represent aperiodic correlation in the frequency domain.
  • domain assumption Received signal at the sensing user is y_t = x_t^(1) + α x_t^(2), a noiseless sum of the user's own backscatter and the other user's LOS signal with a single scalar amplitude α (Section III-A).
    This model is used to compute the correlation in Eq. (11) and underlies all A-ACF and EISL expressions. It ignores noise, delay, and multipath on the interference path.
  • domain assumption Symbols are independent across users and across symbols, with zero mean and known kurtosis μ4, enabling the moment decomposition S = I + S1 + S2 in Eq. (34) cited from [16].
    This probabilistic model is needed to take the expectation in Proposition 1 and to express the dependence on modulation format only through μ4.
  • domain assumption The target delay is longer than the cyclic prefix but shorter than one OFDM symbol, so only same-index and adjacent-index symbol correlations contribute (Eq. 12).
    This assumption defines the aperiodic correlation regime and the structure of the M-symbol correlation sum.

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Pith. "Pith review of Spreading over OFDM for Integrated Sensing and Communications (ISAC) Ranging: Multi-user Interference Mitigation." pith.science (2026). https://pith.science/paper/TRQF2LNI

@misc{pith2026250502160,
  author       = {Pith},
  title        = {Pith review of: Spreading over OFDM for Integrated Sensing and Communications (ISAC) Ranging: Multi-user Interference Mitigation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TRQF2LNI}},
  note         = {Machine review of arXiv:2505.02160}
}
read the original abstract

In the context of communication-centric integrated sensing and communication (ISAC), the orthogonal frequency division multiplexing (OFDM) waveform was proven to be optimal in minimizing ranging sidelobes when random signaling is used. A typical assumption in OFDM-based ranging is that the max target delay is less than the cyclic prefix (CP) length, which is equivalent to performing a \textit{periodic} correlation between the signal reflected from the target and the transmitted signal. In the multi-user case, such as in Orthogonal Frequency Division Multiple Access (OFDMA), users are assigned disjoint subsets of subcarriers which eliminates mutual interference between the communication channels of the different users. However, ranging involves an aperiodic correlation operation for target ranges with delays greater than the CP length. Aperiodic correlation between signals from disjoint frequency bands will not be zero, resulting in mutual interference between different user bands. We refer to this as \textit{inter-band} (IB) cross-correlation interference. In this work, we analytically characterize IB interference and quantify its impact on the integrated sidelobe levels (ISL). We introduce an orthogonal spreading layer on top of OFDM that can reduce IB interference resulting in ISL levels significantly lower than for OFDM without spreading in the multi-user setup. We validate our claims through simulations, and using an upper bound on IB energy which we show that it can be minimized using our proposed spreading. However, for orthogonal spreading to be effective, a price must be paid in terms of spectral utilization, which is yet another manifestation of the trade-off between sensing accuracy and data communication capacity

Figures

Figures reproduced from arXiv: 2505.02160 by the authors.

Figure 1
Figure 1. (Left) Periodic/circular convolution. (Right) Aperiodic/linear convo [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Two-user OFDMA-ISAC where J˜ k “ F H N Dpz k N N qFN (4) where zN “ rz 0 , .., zN´1 s, z “ e jπ, and Dpaq is a diagonal matrix with a on the diagonal. We distinguish between ape￾riodic correlation cxy, and periodic correlation c˜xy using˜for the latter. Using (4), it is possible to express the periodic correlation in the frequency domain as a point-wise multiplicative operation in (5) c˜xy “ F H N px ˚ f d yf q (5) … view at source ↗
Figure 3
Figure 3. Correlation of a block of M symbols The first term in (11) corresponds to the auto-correlation of user 1’s transmitted signal, and the second term corresponds to adjacent frequency correlation interference from user 2 to user 1. We note that the main correlation lobe energy is unaffected by the adjacent frequency interference and remains the same as in the single user case cxyr0s “ ||s||4 2 . B. Correlation using Fr… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Comparison of OFDM and DPSS performance for varying [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 8
Figure 8. Figure 8: EISL vs spectral efficiency η for different values of relative out-of￾band user power α using one symbol per frame. 20 dB, respectively. For no spreading, η represents the fraction of on subcarriers, where p1´ηq{2 edge subcarriers are turned off from both sides. For α …
Figure 6
Figure 6. Figure 6: EISL vs number of symbols in frame, M, for different values of relative out-of-band user power α [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: EISL vs relative out-of-band user power α for different number of symbols in a frame M [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 9
Figure 9. Figure 9: EISL vs spectral efficiency η for different values of relative out-of￾band user power α using 10 symbol per frame [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: EISL vs relative out-of-band user power α for different modulation schemes. for ranging accuracy in the presence of multiple targets. To address this, we proposed a spreading scheme that transforms the OFDM basis to a DPSS basis, thereby minimizing OBCL and achieving …

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