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Passive Detection in Multi-Static ISAC Systems: Performance Analysis and Joint Beamforming Optimization

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Pith's one-line read In multi-static ISAC systems, passive detection with random unknown communication signals is feasible and analytically tractable: the paper derives a GLRT detector whose asymptotic detection probability is controlled by a closed-form…

desk verdict Useful passive-detection analysis for multi-static ISAC, but the claimed asymptotic DOF ν=2MC is invalid for C>M, and the paper never simulates that regime. read the letter →

arxiv 2506.07019 v1 pith:DHVILDIN submitted 2025-06-08 cs.IT eess.SPmath.IT

Renjie He , Yiqiu Wang , Meixia Tao , Shu Sun This is my paper
classification cs.ITeess.SPmath.IT
keywords integratedsensingandcommunicationpassiveradardetectiongeneralizedlikelihoodratiotestmulti-staticISACjointbeamformingdesignrandomunknownsignalsasymptoticprobabilitynon-centralityparameter
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 whether a set of sensing receivers can detect a target using ordinary downlink communication data that the receivers never see in advance, and it answers with a quantitative yes. The authors derive a generalized likelihood ratio test for the two-channel passive radar setup and prove that, for long data blocks, its detection probability is a monotone function of a single non-centrality parameter built from the target-path and direct-path signal-to-noise ratios. The formula shows a clear hierarchy: target-path SNR dominates the achievable detection performance, while direct-path SNR acts as a gate that must be strong enough for the unknown-signal reference to be usable. On this basis the paper formulates and solves two transmit beamforming problems that jointly serve communication users and improve detection probability, and validates the analysis with simulations including OFDM signals. The payoff would be ISAC networks that sense passively without dedicating pilot or sensing waveforms.

What carries the argument

The central object carrying the argument is the non-centrality parameter $\kappa$ of the asymptotic GLRT statistic. It condenses the whole passive-detection geometry into one scalar that is monotonically equivalent to the detection probability: $\kappa = \frac{2L}{\sigma_r^2} \mathrm{tr}\left[ \tilde{H}_t^\mathrm{H} \tilde{H}_d^\mathrm{H} (\sigma_r^2 I_M + \tilde{H}_d \tilde{H}_d^\mathrm{H})^{-1} \tilde{H}_d \tilde{H}_t^\mathrm{H} \right]$. The argument chain uses a two-channel reception model with isolated surveillance and reference arrays, the spectral-decomposition-based MLE of the covariance to form the GLRT, Wilk's theorem for the asymptotic chi-square and non-central chi-square distributions, a Woodbury expansion that separates the target-path energies $\{\delta_n\}$ from the direct-path spectral factors $\{\bar{\sigma}_n/(1+\bar{\sigma}_n)\}$, and a quadratic-transform/SDR reformulation that turns $\kappa$-maximization into an alternating convex optimization. The same $\kappa$ serves as the objective for the beamforming problems, which is why the two designs target either $\kappa$ directly or its two ingredients.

What would settle it

In a testbed with a two-channel sensing receiver, deliberately inject a controlled amount of direct-path leakage into the surveillance array (or a controlled mutual coupling between the two arrays), then compare the empirical distribution of the GLRT statistic under the target-present hypothesis with the predicted $\frac{1}{2}\chi'^2(2MC,\kappa)$ using $\kappa$ from Eq. (23); a systematic deviation that grows with the leakage level would falsify the perfect-decoupling premise on which the analysis rests.

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Extended reading notes

Core claim

The paper's central claim is that in a multi-static ISAC system with one multi-antenna base station, $M$ two-channel sensing receivers, and $C$ single-antenna users, passive detection using random unknown Gaussian communication signals is feasible and its asymptotic performance admits a closed form. The GLRT statistic $\Lambda(Y)$ from Proposition 1, built from the sample covariances of the aggregated surveillance and reference channels, converges in distribution to $\frac{1}{2}\chi^2(2MC)$ under the null hypothesis and to $\frac{1}{2}\chi'^2(2MC, \kappa)$ under the alternative, where the non-centrality parameter is $\kappa = \frac{2L}{\sigma_r^2} \mathrm{tr}\left[ \tilde{H}_t^\mathrm{H} \tilde{H}_d^\mathrm{H} (\sigma_r^2 I_M + \tilde{H}_d \tilde{H}_d^\mathrm{H})^{-1} \tilde{H}_d \tilde{H}_t^\mathrm{H} \right]$. For a single user this reduces to $\kappa = 2LM^2 \mathrm{SNR}_t \mathrm{SNR}_d / (1 + M \mathrm{SNR}_d)$, showing that detection probability rises monotonically with both SNRs but is dominated by the target-path SNR; the direct-path SNR enters as a saturating factor, so that when it is large the passive detector approaches the active-detection upper bound $\kappa_{\mathrm{act}} = 2LM \mathrm{SNR}_t$. In the multi-user case $\kappa$ decomposes as $\frac{2L}{\sigma_r^2} \sum_{n=1}^{C} \frac{\bar{\sigma}_n}{1+\bar{\sigma}_n} \delta_n$, with $\bar{\sigma}_n$ the eigenvalues of the direct-path Gram matrix and $\delta_n$ the target energy along the corresponding eigenvectors, and approaches the upper bound when the smallest $\bar{\sigma}_n$ is large. These formulas motivate two beamforming designs: one maximizes the asymptotic detection probability under per-user SINR and total power constraints via an alternating quadratic transform and semidefinite relaxation, and the other maximizes target energy subject to a direct-path SNR threshold, with lower complexity.

Load-bearing premise

The entire detection model depends on the assumption that, after receive beamforming, the two arrays at each sensing receiver are perfectly decoupled: the target echo appears only in the surveillance array, the direct-path signal only in the reference array, and the noises in the two arrays are statistically independent.

Editorial extensions

If this is right

  • In the single-user case, detection probability depends only on the number of SRs $M$, the block length $L$, and the two SNRs, so a network operator can predict and guarantee passive detection performance without knowing the transmitted symbols.
  • When the smallest eigenvalue of the direct-path Gram matrix is much larger than 1, passive detection approaches the active-detection upper bound $\kappa \approx 2LM\,\mathrm{SNR}_t$, meaning the random communication signal becomes an essentially perfect reference.
  • Increasing the number of SRs $M$ improves detection probability and narrows the gap to active detection, while increasing the number of users $C$ widens that gap for both proposed designs.
  • The 'max $\tilde{P}_d$' beamforming design outperforms the heuristic SNR-threshold design and all benchmarks in the simulated regimes, and the asymptotic approximation becomes accurate as $L$ grows and the direct-path SNR is high.

Reading between the lines

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

  • Since $\kappa$ saturates in the direct-path SNR, a practical operating rule suggested by the analysis is to set the direct-path SNR threshold in design P2 just above the knee of the saturation region; beyond that, extra direct-path energy is wasted, and the remaining gains must come from target-path SNR.
  • The OFDM 16-QAM simulation suggests the asymptotic distribution may be robust beyond Gaussian symbols; one testable extension is to derive the characteristic function of the GLRT statistic for finite-alphabet signals and check whether $\kappa$ remains the correct non-centrality parameter.
  • The dependence of $\kappa$ on the direct-path eigenvectors implies that the identity of the served users matters: choosing which $C$ user streams are used for sensing could be a future scheduling lever that the paper does not optimize.
  • If the perfect-decoupling assumption fails in practice, the covariance in Eq. (12) would acquire off-diagonal blocks; a natural extension is to add a residual-leakage matrix and study how much direct-path suppression is needed for the derived formulas to remain within an acceptable error margin.
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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

2 major / 4 minor

Summary. The manuscript studies passive target detection in a multi-static ISAC downlink, where a base station transmits Gaussian (and later OFDM) data symbols to C single-antenna users and M two-channel sensing receivers jointly detect a target using random, unknown communication signals. The paper derives a GLRT detector in Proposition 1, then uses Wilks-type asymptotics in Proposition 2 to claim that the test statistic is asymptotically half a central chi-square with 2MC degrees of freedom under H0 and half a noncentral chi-square with noncentrality parameter kappa in Eq. (23) under H1. Closed-form insight is given for the single-CU case (Eq. (26)) and for the high-direct-path-SNR case (Proposition 3). Based on this analysis, two joint transmit beamforming designs are proposed: a max-Pd design using quadratic transform, SDR, and alternating optimization, and a lower-complexity heuristic that maximizes target energy subject to a direct-path SNR threshold. Numerical simulations with Gaussian and OFDM signals validate the asymptotic approximations in the tested regimes and show performance gains over the benchmarks.

Significance. If the asymptotic model is valid, the paper offers a genuinely useful, analytically tractable treatment of passive detection with unknown multi-user communication signals, together with a concrete performance metric kappa that can be optimized. The single-CU formula (26), the eigen-decomposition in Proposition 3, the two beamforming algorithms, and the OFDM validation are positive strengths. The central qualitative conclusion that target-path SNR dominates while direct-path SNR acts as a constraint is well supported in the C <= M regime. However, as detailed below, a load-bearing generality claim about the degrees of freedom is not correct for C > M, so the contribution as stated is overstated until that scope is fixed.

major comments (2)
  1. [Section III-A, Proposition 2 and Appendix B (Eqs. (21)-(23), (24)-(25))] The asserted DOF nu = 2MC is not valid for C > M. The likelihood in Eq. (13) depends on Ht and Hd only through the covariance Ry, which is a function of HH^H with H = [Ht; Hd]; the model is invariant under (Ht, Hd) -> (Ht U, Hd U) for any unitary U. Consequently, the Fisher information J(xi) in Eqs. (19)-(20) is singular along gauge directions when C exceeds the rank of Hd. At Ht = 0, the first-order effect of Ht enters only through Ht Hd^H, so the tangent dimension of the alternative relative to the null is 2M * rank(Hd) real; for full-rank Hd this is 2M^2, not 2MC, when C > M. The proof in Appendix B uses J_{xid xid}^{-1} in Eq. (22) without verifying invertibility, and the Wilks limit is therefore not a single chi-square with 2MC degrees of freedom in this regime. The threshold (24) and detection probability (25) inherit the error. All simulations in Section V use C < M (C = 2, M = 4; Figs. 5 and 8 use C <= 3, M = 4; the OFDM setup has C = 1, M = 2), so the overcount is never exposed. The paper should either restrict all statements to C <= M with rank(Hd) = C, or re-derive the correct boundary distribution for C > M.
  2. [Section III-B, Proposition 3 (Eqs. (29), (66))] Proposition 3 states that when the minimum eigenvalue sigma_bar_C of Hd^H Hd / sigma_r^2 is large, the noncentrality parameter satisfies kappa approx 2L M SNR_t. For C > M, however, the C x C matrix Hd^H Hd has rank at most M, so its minimum eigenvalue is exactly zero; the condition sigma_bar_C >> 1 is never satisfied in that regime. Thus the claimed approach to the active-detection upper bound via increasing direct-path SNR holds only when C <= M (and Hd has full column rank). This is the same overcounting as in Proposition 2 and should be corrected in the same revision, ideally by explicitly stating the rank condition or by deriving the C > M counterpart of Eq. (29).
minor comments (4)
  1. [Eq. (68)] The denominator in the second inequality appears to be 1 + M SNR_t; it should be 1 + M SNR_d to be consistent with the preceding inequality sigma_bar_1/(1+sigma_bar_1) <= M SNR_d/(1+M SNR_d) and with the single-CU formula (26).
  2. [Appendix C (Eq. (62))] The statistic Lambda_a defined in Eq. (62) is gamma(MC, 1) under H0; it is 2*Lambda_a that follows a central chi-square distribution with 2MC degrees of freedom. The stated noncentrality parameter kappa_act likewise applies to 2*Lambda_a. Since the thresholds in Fig. 4 are Monte-Carlo calibrated, the numerical conclusions are unaffected, but the distributional statement should be corrected.
  3. [Section II-B, Eq. (8)] The assumption that receive beamforming perfectly separates the target echo from the direct path is not explicitly acknowledged as an idealization. Residual direct-path leakage or correlated noise between the surveillance and reference arrays would introduce cross terms in the covariance (12) and would invalidate the exact GLRT formula (15). The paper flags the clutter-free assumption as an upper bound; a similar caveat for the direct-path separation would be helpful.
  4. [Algorithm 1 and Eq. (38)] The monotonicity proof (38) is written as if a beamforming matrix W exists at every iteration, but the SDR subproblem (P1.2) returns covariance matrices R_n that are not guaranteed to be rank-one, and the Gaussian randomization step is only applied after the loop. Please state explicitly that the convergence guarantee applies to the SDR-relaxed problem, and that the final rank-1 extraction is a heuristic post-processing step. Also specify the number of Gaussian randomizations used in the simulations.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the GLRT, asymptotic chi-square approximation, and beamforming objectives are derived from the stated Gaussian model, with the sole self-citation being a non-load-bearing provenance note.

full rationale

The paper's derivation chain is self-contained and does not reduce to its inputs by construction. The GLRT statistic in Proposition 1 (Eq. 15) follows from the Gaussian likelihood in Eq. (13) and standard covariance MLE results; the asymptotic distribution in Proposition 2 (Eqs. 21-23) is obtained from Wilks' theorem and a Fisher-information computation under the stated model, with no parameter fitted to detection data. The detection probability expressions in Eqs. (24)-(25) are standard chi-square and Marcum Q evaluations using the derived degrees of freedom and non-centrality parameter. Beamforming design P1 maximizes the derived kappa (Eq. 31) rather than a fitted surrogate, and P2 is explicitly a heuristic relaxation (Eq. 39) whose relationship to P1 is analyzed in Appendix E; this is model-based optimization, not circular prediction. The only self-citation is reference [1], a workshop presentation of part of this work, cited in a footnote to acknowledge prior dissemination. It is not used to justify the GLRT, the asymptotic analysis, or the beamforming algorithms, so it is not load-bearing. Numerical comparisons use independent empirical Monte-Carlo simulations against the derived asymptotics, providing an external check rather than a circular fit. Therefore no circular step is identified.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper relies on standard Gaussian signal modeling, idealized channel separation after receive beamforming, exact delay-Doppler matching, and Wilks-type regularity conditions. The only manually tuned quantity in the optimization is the direct-path SNR threshold Gamma_d. No new physical entities are postulated.

free parameters (1)
  • Gamma_d (direct-path SNR threshold in P2) = not fixed analytically; selected by search over a finite interval in simulations
    The second beamforming design depends on this manually chosen threshold, and the paper reports performance variations with it; no principled closed-form choice is given.
assumptions (6)
  • domain assumption Communication symbols s[l] are i.i.d. circularly symmetric complex Gaussian with unit power; the likelihood in (13) and the FIM in (19) are Gaussian.
    Used in Section II-A and throughout; the GLRT statistic and its chi-squared asymptotics are derived under this distribution. OFDM 16-QAM simulations are a robustness check, not part of the derivation.
  • domain assumption The environment is clutter-free and, after receive beamforming, the surveillance and reference channels are perfectly separated with independent AWGN (Eq. (8)).
    Section II-B states that clutter suppression techniques exist and that the results serve as an upper bound; any residual direct-path leakage would change the covariance in (12) and the GLRT.
  • domain assumption The delay-Doppler compensation assumes exact matching of the true target and direct path cells by the discretized tuple grid (Section II-B, Step 2).
    The paper assumes D(hat tau, hat f) = D_t,i and D = D_d,i; grid mismatch is cited to [26] and not analyzed.
  • standard math Standard Wilks-type regularity conditions hold for the MLEs of the rank-constrained Gaussian covariance model.
    Section III-A invokes [27] and states that the MLEs attain the asymptotic PDF; the conditions such as identifiability of Htilde_t at H0 and interior parameter space are not checked explicitly.
  • domain assumption Communication channels h_m and noise variances sigma_c^2 and sigma_r^2 are perfectly known at the BS and receivers.
    Used in the SINR constraints (4) and the detection model (7); no channel estimation error is modeled.
  • standard math Theorem 9.4.1 of [30] gives the MLE of a covariance with rank constraint used to derive the GLRT statistic.
    Used in Appendix A, Eq. (44); treated as an unproved standard result.

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Pith. "Pith review of Passive Detection in Multi-Static ISAC Systems: Performance Analysis and Joint Beamforming Optimization." pith.science (2026). https://pith.science/paper/DHVILDIN

@misc{pith2026250607019,
  author       = {Pith},
  title        = {Pith review of: Passive Detection in Multi-Static ISAC Systems: Performance Analysis and Joint Beamforming Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DHVILDIN}},
  note         = {Machine review of arXiv:2506.07019}
}
read the original abstract

This paper investigates the passive detection problem in multi-static integrated sensing and communication (ISAC) systems, where multiple sensing receivers (SRs) jointly detect a target using random unknown communication signals transmitted by a collaborative base station. Unlike traditional active detection, the considered passive detection does not require complete prior knowledge of the transmitted communication signals at each SR. First, we derive a generalized likelihood ratio test detector and conduct an asymptotic analysis of the detection statistic under the large-sample regime. We examine how the signal-to-noise ratios (SNRs) of the target paths and direct paths influence the detection performance. Then, we propose two joint transmit beamforming designs based on the analyses. In the first design, the asymptotic detection probability is maximized while satisfying the signal-to-interference-plus-noise ratio requirement for each communication user under the total transmit power constraint. Given the non-convex nature of the problem, we develop an alternating optimization algorithm based on the quadratic transform and semi-definite relaxation. The second design adopts a heuristic approach that aims to maximize the target energy, subject to a minimum SNR threshold on the direct path, and offers lower computational complexity. Numerical results validate the asymptotic analysis and demonstrate the superiority of the proposed beamforming designs in balancing passive detection performance and communication quality. This work highlights the promise of target detection using unknown communication data signals in multi-static ISAC systems.

Figures

Figures reproduced from arXiv: 2506.07019 by the authors.

Figure 1
Figure 1. System illustrations. (a): Mono-static ISAC system; (b): Bi [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Geometry illustration. path, while the reference array collects direct path signals. Each SR sends locally processed signals to a central processor via dedicated links for joint detection. Suppose that each CU has a single antenna and the BS is equipped with a uniform linear array with Nt antenna elements, where Nt ≥ C. In the following, we first establish the downlink communication model, then introduce the multi-s… view at source ↗
Figure 4
Figure 4. Pd = 0.9 contour for the single-CU case where C = 1, M = 4, Pfa = 10−3 and L = [125, 250, 500]. probability, denoted as P˜ d, of the GLRT detector at L → ∞. Specifically, for a decision threshold ρ, we have: P˜ fa = P(Λ(Y) > ρ; H0) = Fχ2(ν)(2ρ), (24) P˜ d = P(Λ(Y) > ρ; H1) = Qν 2 ( √ κ, p 2ρ), (25) where Qx(a, b) is the generalized Marcum Q-function with order x, and Fχ2(ν)(·) is the right-tail probability of the di… view at source ↗
Figures from the paper (5 more)
Figure 6
Figure 6. Figure 6: The receiver operating characteristic with [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 8
Figure 8. Figure 8: Trade-off curves with varying numbers of CUs. Here [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 11
Figure 11. Figure 11: Detection probability versus the transmit power. Here [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 10
Figure 10. Figure 10: Detection probability versus the target RCS. Here [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 13
Figure 13. Figure 13: Detection heatmap. APPENDIX A. Proof of Proposition 1 Under hypothesis H1, the log-likelihood is given by: l1 = −Lln det Ry,1 − Ltr(R−1 y,1X), (42) where Ry,1 = H˜ H˜ H + σ 2 r I2M, H˜ = [H˜ T t , H˜ T d ] T , and the constant term is omitted for brevity. We first cal…

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