Beyond Single-Band: Analysis and Resource Allocation for Multi-band ISAC Systems

Haotian Liu; Qixun Zhang; Xingwang Li; Yunxin Geng; Zhiqing Wei; Zhiyong Feng

REVIEW 3 major objections 4 minor 17 references

Beyond Single-Band: Analysis and Resource Allocation for Multi-band ISAC Systems

T0 review · 3 major / 4 minor · reviewed 2026-07-10 · glm-5.2

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Pith's one-line read 18 dB detection gain from multi-band radar-communication signals

desk verdict Closed-form multi-band OFDM detection expressions are a genuine new result; the 18 dB gain claim is inflated by a misleading single-band baseline. read the letter →

arxiv 2607.08068 v1 pith:5KBEG5CL submitted 2026-07-09 eess.SP

Haotian Liu , Zhiqing Wei , Xingwang Li , Yunxin Geng , Qixun Zhang , Zhiyong Feng This is my paper

classification eess.SP

keywords multi-bandisacdetectionresourcesensingallocationperformancesingle-band

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 argues that splitting a radar-sensing signal across multiple non-contiguous frequency bands, rather than concentrating it in one band, yields large detection gains for weak targets. The central mechanism is frequency diversity: when the frequency gap between bands exceeds the target's coherence bandwidth, the radar cross-section fluctuations at each band become statistically independent. Under this independence assumption, the paper derives exact closed-form expressions for the detection and false-alarm probabilities of multi-band OFDM signals by factoring the characteristic function of the sum of independent-but-not-identically-distributed exponential random variables. It then uses these expressions as an optimization objective, jointly allocating transmit power and time-frequency resources across bands via an ADMM-based algorithm to maximize detection probability. The paper reports an 18 dB detection gain over single-band baselines at 90% detection probability.

What carries the argument

The characteristic function (CF) method applied to a sum of independent-but-not-identically-distributed (i.n.i.d.) exponential random variables. Each band contributes a weighted exponential detection statistic whose rate parameter depends on the per-band SNR. The CF of the joint statistic factorizes into a product of per-band CFs (Eq. 11, Eq. 16), which is then inverted via partial fraction expansion to yield closed-form PDFs and integration to CDFs. The ADMM algorithm splits the non-convex resource allocation problem into three subproblems (auxiliary variable, time-frequency resource, power) solved iteratively with SQP and CVX.

What would settle it

If the inter-band scattering correlation is non-negligible for realistic targets or frequency configurations, the characteristic function factorization in Eq. 11 fails, the closed-form detection probability in Eq. 18 no longer holds, and the optimization objective becomes invalid.

Watch

Extended reading notes

Core claim

The detection and false-alarm probabilities of a multi-band OFDM ISAC system with a Swerling II target can be expressed in exact closed form as sums of weighted exponentials (Eq. 14 and Eq. 18), provided the inter-band scattering coefficients are statistically independent. This independence holds when the frequency gap between non-contiguous bands far exceeds the target's coherence bandwidth. The optimal combining weight for each band is the per-band average SNR, which maximizes the deflection coefficient. Using these closed forms as an optimization objective, a joint power and time-frequency resource allocation scheme solved by ADMM achieves a reported 18 dB detection gain over single-band,

Load-bearing premise

The inter-band statistical independence assumption — that radar cross-section fluctuations at different non-contiguous bands are uncorrelated — is the load-bearing premise. The entire characteristic function factorization and resulting closed-form expressions depend on it. The paper justifies this via a coherence bandwidth argument, but its validity depends on target geometry, bistatic angle, and specific frequency gaps.

Editorial extensions

If this is right

Multi-band ISAC systems using carrier aggregation can achieve substantially higher weak-target detection sensitivity than single-band systems with equivalent total resources, supporting the case for fragmented spectrum as an asset rather than a liability.
The closed-form expressions enable rapid resource optimization without Monte Carlo simulation, making real-time adaptive sensing frame design tractable.
The 18 dB gain claim, if validated, suggests that dynamic resource allocation across heterogeneous bands (e.g., sub-6 GHz and mmWave) can compensate for frequency-selective fading that would otherwise degrade single-band detection.

Reading between the lines

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

If inter-band correlation is non-negligible (e.g., for targets with large projected lengths or small bistatic angles), the CF factorization breaks down and the closed-form expressions would require modification to account for correlated exponential components, potentially reducing the diversity gain.
The framework could be extended to multi-target or MIMO scenarios, but the independence assumption and the GLRT grid search would need re-examination, as multi-target interference and spatial correlation would introduce additional coupling between bands.

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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

The real contribution here is the closed-form detection and false alarm probabilities for multi-band OFDM ISAC (Eqs. 14, 18). The characteristic function method for sums of i.n.i.d. exponentials is correctly applied, the partial fraction expansion is clean, and the deflection-coefficient optimality proof for the weights (Appendix A) is tight. This is a genuine analytical result that I haven't seen in the prior multi-band ISAC literature, which has focused on CRB analysis and signal processing rather than detection probability expressions. The GLRT framework and the inter-band independence assumption are well-motivated — for bands separated by GHz-level gaps, the coherence bandwidth argument is physically sound and not the weak point. The reader flagged independence as load-bearing, but I think that concern is overstated; the approximation is standard and defensible for non-contiguous CA bands. The Monte Carlo validation in Fig. 1 matches the closed forms almost perfectly, which is good evidence the math is right. The ADMM-based resource allocation is reasonable but has a real gap: Step 1 (Eq. 23) is non-convex, solved via SQP with no global optimality guarantee, and convergence is shown only empirically for three noise levels. That's a soft spot but not fatal — the algorithm clearly works in practice, and the 6 dB gain of optimized over uniform multi-band (Fig. 4) is the meaningful number that isolates the optimization benefit. The 18 dB gain over single-band is where I part ways with the paper. The stress-test note is correct: the single-band baseline randomly selects among {2.6, 3.5, 26, 28} GHz, and since path loss goes as 1/f², the SNR spread between 2.6 and 28 GHz is roughly 20 dB. The paper's own variance numbers (30 dB for single-band in Fig. 2, 26 dB in Fig. 4) confirm this. So the 18 dB gain largely reflects the multi-band system's ability to access the favorable sub-6 GHz band while the single-band baseline is stuck with whatever it randomly draws — not frequency diversity or optimization per se. The paper should have used a best-single-band baseline (all resources on 2.6 GHz) to make a fair comparison. No code or data is shipped, which limits reproducibility of the optimization results specifically. The closed-form expressions themselves are parameter-free and independently verifiable, which is the stronger part of the work. This paper is for ISAC researchers who need analytical detection performance expressions for multi-band systems. The analytical core deserves a serious referee; the optimization and simulation claims need tightening before publication.

Referee Report

3 major / 4 minor

Summary. This paper derives closed-form detection and false alarm probabilities for multi-band OFDM ISAC signals via the generalized likelihood ratio test (GLRT) and the characteristic function (CF) method for sums of independent non-identically distributed (i.n.i.d.) exponential random variables. Building on these expressions, the authors formulate a joint power and time-frequency resource allocation problem to maximize detection probability and solve it using an ADMM-based algorithm. Simulations validate the theoretical derivations and demonstrate detection gains for multi-band signals over single-band baselines.

Significance. The derivation of closed-form detection performance for multi-band OFDM ISAC is a solid contribution. The mathematical treatment of the GLRT statistic as a sum of i.n.i.d. exponential variables and the subsequent use of partial fraction expansion to obtain tractable closed-form expressions (Eqs. 11-18) is rigorous and provides a useful theoretical foundation. The deflection-coefficient optimality proof for the weights (Appendix A) is clean. The proposed ADMM-based resource allocation scheme is a practical approach to exploiting the non-linear characteristics of the derived detection probability.

major comments (3)

Section V-C, Fig. 4: The claim of an '18 dB detection gain over traditional single-band baselines' is misleading due to the construction of the single-band baseline. As detailed in Appendix B, the single-band baseline selects a random band $s$ from the set $B$ and concentrates all resources there. Because path loss $L_b$ scales with $1/f_b^2$ (Eq. 1), the SNR varies drastically across the chosen bands (2.6, 3.5, 26, 28 GHz). A random draw that lands on 26 or 28 GHz will suffer enormous path loss compared to 2.6 GHz. The paper's own variance results (26-30 dB for single-band) confirm that the single-band mean is heavily dragged down by these high-frequency draws. Consequently, the 18 dB gain largely reflects the multi-band system's ability to access the favorable sub-6 GHz bands while the baseline cannot, rather than isolating the benefits of frequency diversity or the proposed resource优化
Section V-C, Fig. 4: The comparison between optimized multi-band and uniform multi-band (6 dB gain) is the more meaningful metric for evaluating the proposed ADMM optimization scheme, as it holds the available frequency bands constant. The authors should reframe the primary contribution and abstract to emphasize this 6 dB optimization gain, rather than the 18 dB gain which is heavily conflated with path loss disparities.
Section IV-B, Eq. (20)-(23): The ADMM update for $E_b$ in Step 1 (Eq. 23) is noted to be a non-convex problem solved via sequential quadratic programming (SQP). Because the overall problem (P2) is non-convex, the standard convergence guarantees of ADMM do not apply. The manuscript should briefly discuss the theoretical or empirical convergence behavior of alternating between SQP for the non-convex subproblem and CVX for the convex subproblems, as this impacts the reliability of the optimization results.

minor comments (4)

Section II-B, Eq. (2): The correlation coefficient $ho_{i,j}$ involves Bessel functions $J_0$ and $J_2$. It would help the reader to briefly state the physical intuition behind the transition from this continuous correlation model to the binary intra-band/independent inter-band assumptions.
Section IV-A, Eq. (19): The notation $S_b = M_b N_b$ is introduced, but it is not explicitly defined that $M_b$ corresponds to $M^b_{sym}$ and $N_b$ to $N^b_c$ from Section II. Please clarify.
Section V: The simulation parameters mention 'average target RCS $[-20, 0]$ dBsm'. It is unclear if this means a uniform distribution across this range or a specific mean value within it. Please specify the distribution used for the Swerling II model.
Figures 2 and 4: The axis labels and legends are somewhat sparse. For instance, Fig. 2 lacks explicit axis titles in the provided text. Ensure all figures have clearly labeled axes, units, and legends for standalone readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: closed-form expressions are genuinely derived from the system model via standard probability theory, and the optimization targets the derived expression rather than a fitted input.

full rationale

The paper's derivation chain is self-contained and non-circular. The closed-form detection probability (Eq. 18) and false alarm probability (Eq. 14) follow from standard probability theory: the GLRT statistic (Eq. 9) is a weighted sum of per-band normalized energy statistics U_b, each exponentially distributed (Eq. 10) with parameters determined by the physical SNR γ̄_b (Eq. 5), which is defined independently in terms of transmit power, path loss, RCS, and noise. The characteristic function factorization (Eq. 11) relies on the inter-band independence assumption (Eq. 3), which is justified physically via the coherence bandwidth argument citing [13] (Zhou and Liu, 2010 — different authors). The partial fraction expansion yielding the closed-form is the standard hypoexponential distribution derivation. The optimal weight w_b = γ̄_b is proven via the deflection coefficient criterion using Cauchy-Schwarz (Appendix A) — a genuine mathematical argument, not a definition. The resource allocation (Section IV) maximizes the derived P_D expression over physical parameters (power, subcarriers, symbols), which is a legitimate optimization of an independently derived objective. Self-citations [8], [9], [16] by the present authors are used for system model context and tool references, not as load-bearing mathematical results. The Monte Carlo validation (Fig. 1) provides external confirmation of the closed-form expressions. The skeptic's concern about the single-band baseline comparison fairness is a validity issue, not a circularity issue — the baseline is mathematically consistent and the comparison, while potentially misleading, does not involve any derivation that reduces to its own inputs.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

No new physical entities, forces, or mathematical objects are introduced. The framework uses standard radar detection theory constructs.

free parameters (3)

ρ (ADMM penalty parameter) = 0.5
Chosen empirically for the ADMM algorithm; affects convergence speed but not the analytical results.
εpri (convergence tolerance) = 0.001
Algorithmic parameter for stopping criterion; does not affect analytical derivations.
ϑmin, ϑmax (per-band resource scaling factors) = 0.05, 0.85
Boundary constraints on per-band power and time-frequency allocation; chosen for simulation, not derived.

assumptions (6)

domain assumption Inter-band statistical independence: E[α_i α_j*] = 0 for i≠j (Eq. 3)
Section II-B. Justified by coherence bandwidth argument (Δf >> Bc for non-contiguous bands), citing [13]. This is the load-bearing assumption for the CF factorization in Eq. 11 and all subsequent closed-form results.
domain assumption Intra-band correlation: ρ→1 within a single band
Section II-B. Justified by Δf << Bc within a band. Allows treating each band's coherent accumulation as producing a single exponential statistic.
domain assumption Swerling II model for target RCS: α_b ~ CN(0, σ²_rcs,b)
Section II-B. Standard radar assumption; makes per-band detection statistic exponentially distributed, enabling the CF method.
standard math GLRT framework for unknown target parameters (τ, f_d)
Section III-A. Standard detection theory; the grid search over candidate parameters is a known GLRT approach [14].
domain assumption Known noise variance σ²_b,n,m and prior average RCS σ²_rcs,b
Section IV-A. The resource allocation scheme assumes these are acquired before optimization. Limits applicability in scenarios with unknown or rapidly varying noise/RCS.
domain assumption Low-SNR approximation: w_glrt_b = γ̄b/(γ̄b+1) ≈ γ̄b
Section III-A, between Eqs. 8-9. Justified for weak-target scenarios. The deflection coefficient proof (Appendix A) shows γ̄b is optimal under a different criterion, providing criterion compatibility.

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Cite this review

Pith. "Pith review of Beyond Single-Band: Analysis and Resource Allocation for Multi-band ISAC Systems." pith.science (2026). https://pith.science/paper/5KBEG5CL

@misc{pith2026260708068,
  author       = {Pith},
  title        = {Pith review of: Beyond Single-Band: Analysis and Resource Allocation for Multi-band ISAC Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5KBEG5CL}},
  note         = {Machine review of arXiv:2607.08068}
}

read the original abstract

Integrated sensing and communication (ISAC) has emerged as a pivotal technology for sixth-generation wireless networks to empower high-precision sensing. The demand for superior sensing resolution and the reality of spectrum fragmentation have driven the research of multi-band ISAC. Multi-band ISAC provides frequency diversity through independent observations across disparate bands, mitigating sensing performance fluctuations caused by frequency-selective radar cross-section compared to single-band counterparts. In this paper, we propose a framework for analytical performance characterization and resource optimization in multi-band ISAC systems. Specifically, analytical closed-form detection and false alarm probabilities for multi-band OFDM signals are derived, providing a theoretical foundation for subsequent resource allocation. Then, a joint power and time-frequency resource allocation scheme is developed and solved via a proposed ADMM-based algorithm to maximize detection performance. Numerical results validate the accuracy of the closed-form derivations and demonstrate the superior robustness of multi-band signals. Notably, the proposed optimization scheme achieves an 18 dB detection gain over traditional single-band baselines at a 90\% detection probability.

Figures

Figures reproduced from arXiv: 2607.08068 by the authors.

**Figure 3.** The mean detection probability and the mean normal [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 2.** illustrates the mean and variance of the detection probability at a false alarm probability of 10−8 , where variance characterizes the robustness and mean characterizes the average performance. The multi-band signal outperforms its single-band counterpart when the noise power is below -83 dBm. More importantly, the variance results indicate that the performance fluctuation of the multi-band signal is approximately… view at source ↗

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Pith tools

Reviewed July 10, 2026 · model on record in the stance chip above.

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Integrated sensing and communications: Toward dual-func tional wire- less networks for 6G and beyond,

F. Liu, Y . Cui, C. Masouros, J. Xu, T. X. Han, Y . C. Eldar, an d S. Buzzi, “Integrated sensing and communications: Toward dual-func tional wire- less networks for 6G and beyond,” IEEE J. Sel. Areas Commun. , vol. 40, no. 6, pp. 1728–1767, 2022

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work page 2024

[3] [3]

Deep cooperation in ISAC system: Resource, node and infras tructure perspectives,

Z. Wei, H. Liu, Z. Feng, H. Wu, F. Liu, Q. Zhang, and Y . Du, “Deep cooperation in ISAC system: Resource, node and infras tructure perspectives,” IEEE Internet Things Mag. , vol. 7, no. 6, pp. 118–125, Nov. 2024

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Multi-node multi-band cooperative integrated s ensing and communications: State-of-the-art, challenges and opport unities,

H. Li, K. Qu, C. Sun, S. Wang, X. Wang, Y . Zhang, B. Priyanto , and H. Zhang, “Multi-node multi-band cooperative integrated s ensing and communications: State-of-the-art, challenges and opport unities,” IEEE Wirel. Commun., vol. 32, no. 4, pp. 180–188, Aug. 2025

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H. Luo, F. Gao, F. Liu, and S. Jin, “6D Radar Technology in M onostatic Integrated Sensing and Communications System,” in 2024 IEEE/CIC Int. Conf. Commun. China (ICCC) , Hangzhou, China, Sep. 2024, pp. 1104–1109

work page 2024

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T. Kou, Z. Zhou, H. Liu, and Y . Y ang, “Multi-band composit e detection and recognition of aerial infrared point targets,” Infrared Phys. Technol., vol. 94, pp. 102–109, Sep. 2018

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Detection performance using frequency diversity with distributed se nsors,

B. W. Jung, R. S. Adve, J. Chun, and M. C. Wicks, “Detection performance using frequency diversity with distributed se nsors,” IEEE Trans. Aerosp. Electron. Syst. , vol. 47, no. 3, pp. 1800–1813, Jul. 2011

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Carrier a ggregation enabled MIMO-OFDM integrated sensing and communication,

H. Liu, Z. Wei, J. Piao, H. Wu, X. Li, and Z. Feng, “Carrier a ggregation enabled MIMO-OFDM integrated sensing and communication,” IEEE Trans. Wireless Commun. , vol. 24, no. 6, pp. 4532–4548, Jun. 2025

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