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Exploiting Both Pilots and Data Payloads for Integrated Sensing and Communications

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Random data payloads can be reused for sensing, and a new asymptotic expression makes the joint pilot-plus-data precoding design tractable, with up to 5.6 dB sensing-error gain over pilot-only operation.

desk verdict Solid RMT-based asymptotic result for ISAC precoding with pilots plus data, but the high-SNR closed form hides a full-rank assumption that breaks the convex reformulation when the rate constraint is tight. read the letter →

arxiv 2506.15998 v1 pith:LWKTG4NR submitted 2025-06-19 eess.SP

Chen Xu , Xianghao Yu , Fan Liu , Shi Jin This is my paper
classification eess.SP
keywords integratedsensingandcommunicationsrandomdatapayloadergodicLMMSEmatrixtheoryprecodingoptimizationsuccessiveconvexapproximationhigh-SNRanalysis
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

Integrated sensing and communications frames dedicate only a small slice of time-frequency resources to deterministic pilots, so most of the frame—random data symbols—is idle from a sensing point of view. This paper tries to establish that those data symbols can be reused for sensing, and that the resulting average sensing error can be computed and optimized without tracking the random symbol realizations. The authors derive a semi-closed-form asymptotic expression for the ergodic linear minimum mean square error (ELMMSE), which turns the joint precoding design into an optimization problem solvable by successive convex approximation, and a closed-form high-SNR version that is convex. If this is right, the practical payoff is up to 5.6 dB lower sensing error than pilot-only sensing while keeping communication rate above a chosen threshold.

What carries the argument

The load-bearing tool is a random-matrix-theory deterministic equivalent for matrices of the form $G = A + B S S^H B^H$ with i.i.d. zero-mean entries: through the Stieltjes transform, the trace of $G^{-1}$ is replaced by a deterministic expression involving a fixed-point equation for $e$. In this paper that expression yields the asymptotic ELMMSE $J_{\mathrm{ae}}$ and the scalar $\alpha = \frac{L_d}{L_d + N_t e}$, which quantifies how much the randomness of the data symbols degrades sensing relative to deterministic signals. At high SNR, Proposition 4 collapses the fixed point to $e = \frac{L_d}{L_d - N_t}$, giving $\alpha = 1 - \frac{N_t}{L_d}$ and making the precoding optimization convex.

What would settle it

Take $N_t = 2$, $L_d = 4$, and a rank-1 precoder $W$, and evaluate the high-SNR limit of the fixed-point equation (30) numerically; if the resulting $\alpha$ approaches $3/4$ rather than the Proposition 4 value $1 - N_t/L_d = 1/2$, then the high-SNR closed form and the convex problem built on it fail for rank-deficient precoders.

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Extended reading notes

Core claim

The paper's central claim is that random data payloads are not a nuisance for monostatic ISAC sensing but a usable resource, provided the precoder is optimized against the right performance metric. Replacing the random-symbol-dependent ELMMSE by a deterministic asymptotic equivalent $J_{\mathrm{ae}} = \operatorname{Tr}\big((R^{-1} + \frac{1}{N_r \sigma_s^2}(\frac{P_p}{N_t} I_{N_t} + \alpha W W^H))^{-1}\big)$, with $\alpha = \frac{L_d}{L_d + N_t e}$ defined through a fixed-point equation, removes the expectation over data symbols and makes precoding design tractable. At high SNR the paper derives the explicit degradation factor $\alpha = 1 - \frac{N_t}{L_d}$, so the only penalty for using random symbols instead of deterministic ones is governed by the ratio of transmit antennas to data-symbol length, and the precoding problem becomes convex with a globally optimal solution. Numerical results are claimed to confirm that the asymptotic formulas match Monte-Carlo ELMMSE even for moderate $N_t$ and $L_d$, and that the resulting precoders outperform pilot-only sensing and the stochastic-gradient benchmark.

Load-bearing premise

The high-SNR closed form assumes the data precoder continues to use all $N_t$ transmit dimensions in the limit; when the rate constraint is tight and the user has fewer antennas than the base station, the optimal precoder can become rank-deficient, in which case the factor $\alpha = 1 - N_t/L_d$ is not guaranteed to hold.

Editorial extensions

If this is right

  • Reusing random data symbols for sensing can cut the sensing error by up to 5.6 dB compared with pilot-only sensing when the communication rate requirement is relaxed to 85 percent of its maximum.
  • The asymptotic ELMMSE expression is accurate enough at moderate sizes, for example 32 transmit antennas and 64 data symbols, to replace Monte-Carlo averaging in precoding design.
  • At high SNR, the sensing penalty caused by random data is fixed by the ratio $N_t/L_d$: increasing transmit power cannot remove it, only lengthening the data block can.
  • In the high-SNR regime the precoding problem is convex and globally solvable, with complexity $O(N_t^{3.25})$, versus the iterative fixed-point and SCA steps needed at general SNR.
  • The proposed SCA-based design satisfies the communication rate constraint at every iteration and runs about 27 percent faster than the stochastic-gradient alternating-optimization baseline.

Reading between the lines

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

  • Beyond the paper: the same deterministic-equivalent technique should extend to other stochastic sensing metrics such as Bayesian Cramér-Rao bounds or sensing mutual information, giving similar fixed-point factors that depend only on antenna counts and data-block length.
  • Beyond the paper: the result suggests a frame-design rule—keep $L_d$ comfortably larger than $N_t$ so that the random-data penalty $1 - N_t/L_d$ stays close to one—and this can be tested against constellation-constrained data symbols, not just Gaussian ones.
  • Beyond the paper: if the hidden full-rank assumption fails under a tight rate constraint, adding a small full-rank regularization or explicitly optimizing the rank of the precoder would likely restore the validity of the high-SNR approximation; this is a numerical experiment, not a claim of the paper.
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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 / 5 minor

Summary. This paper considers a mono-static MIMO ISAC system in which both deterministic pilot symbols and random data symbols are used for sensing. The authors derive an asymptotic expression for the ergodic LMMSE using random matrix theory (Proposition 2, Eq. (28)), which is independent of the random data symbols, and validate it by Monte Carlo simulation. Based on this expression, they formulate an ISAC precoding problem under a communication rate constraint and a power budget, and propose an SCA algorithm (Algorithm 1). They then derive a high-SNR closed form (Proposition 4, Eq. (50)) in which the data-induced degradation factor is alpha = 1 - Nt/Ld, leading to a convex reformulation (51) claimed to be globally optimal. Simulations report sensing gains up to 5.6 dB over pilot-only sensing and show agreement with the asymptotic expressions.

Significance. Subject to the correctness caveats below, the RMT-based asymptotic expression is a useful and nontrivial contribution: it replaces a difficult stochastic optimization problem with a deterministic expression and is supported by Monte Carlo validation. The comparison with the SGD-based baseline [21] is appropriate and shows clear computational advantages. The high-SNR closed form, if valid, would provide clean engineering insight. However, the high-SNR reformulation rests on a hidden full-rank assumption, and the SCA convergence claims are not proven; these issues affect the advertised third contribution and therefore temper the significance. The paper does not provide code, but the numerical validation is reproducible in principle.

major comments (2)
  1. [Section V, Proposition 4 (Eqs. (43)-(50))] The derivation of ebar = Ld/(Ld - Nt) assumes that Phi = WW^H/Tr(WW^H) is invertible, since Eq. (45) uses Phi^{-1}. This is not guaranteed by the optimization problem (35)/(51): when the rate constraint is tight (e.g., R0 = Rmax with Nc < Nt), an optimal M = WW^H can have rank r < Nt. For rank-r M, the correct high-SNR limit of the fixed-point equation (30) is e = rLd/(Nt(Ld - r)), hence alpha = 1 - r/Ld. For the concrete example Nt = 2, Ld = 4, r = 1, this gives alpha = 3/4 instead of 1/2. Consequently, the closed form (50) and the convex reformulation (51) do not describe the true high-SNR ELMMSE for rank-deficient precoders, and the global-optimality claim for (51) fails. The same limit in (45) also holds the pilot term fixed while letting the data SNR diverge; if both Pp and Pd scale with SNR, the high-SNR limit changes even for full-rank W. The authors should either prove that the optimal M in the regime of interest is full rank, restrict the claims to the full-rank case, or derive the rank-dependent version.
  2. [Section IV-B, Eq. (37)-(40) and Section VI-D] The SCA algorithm minimizes a first-order Taylor approximation Jhat(M|M0) of a non-convex function. A first-order Taylor expansion is not a global upper bound of a non-convex function, and no line search, regularization, or trust region is introduced; therefore the claimed monotonic decrease and convergence to a locally optimal solution are not established. The statement in Section VI-D that the algorithm 'guarantees a strict monotonic decrease' is unsupported. Please provide a convergence proof (e.g., showing that the surrogate satisfies the SCA conditions in [31]) or soften the claims to empirical convergence.
minor comments (5)
  1. [Abstract] The abstract defines 'successive convex approximation (SAC)', but the acronym used throughout the rest of the paper is SCA; please make this consistent.
  2. [Section VI-D and Fig. 9] The baseline algorithm is referred to inconsistently as 'SGD-AO' and 'SGP-AO'; please use a single name consistently in the text and figure legends.
  3. [Section VI-A, Fig. 5] The paper says that SNR refers to the transmitter-side SNR but does not state how Pp and Pd scale with SNR; this matters for interpreting the high-SNR approximation and should be clarified.
  4. [Eq. (38b)] The notation Tr(MTMT) is ambiguous; please write Tr(M T M^T) or clarify whether a Hermitian transpose is intended.
  5. [Fig. 9] One legend entry is truncated as 'R0 = 0.95Rm'; the full label 'Rmax' should be restored.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the ELMMSE asymptotics come from an external RMT theorem, and the high-SNR closed form is an algebraic limit of the derived fixed-point equation rather than a fitted or self-referential input.

full rationale

The central derivation chain is externally anchored and does not reduce to its own inputs. Proposition 2 (Eqs. (27)-(30)) is a direct specialization of Theorem 1 from [30] (Couillet, Debbah, and Silverstein, IEEE Trans. Inf. Theory 2011), an external RMT deterministic-equivalent result; the identifications in (67a)-(67e) map the ELMMSE in (15) onto the general model (20), and the asymptotic expression (28) with the fixed point (30) follows by taking z=0 in (23)-(24). No quantity in this chain is defined in terms of the ELMMSE it purports to predict, and e is evaluated by fixed-point iteration, not fitted to simulation data. Proposition 4 obtains e = Ld/(Ld - Nt) by taking gamma -> infinity in the rewritten fixed-point equation (45); Eq. (49) alpha = 1 - Nt/Ld then follows algebraically from (29). This is an asymptotic evaluation of the derived fixed point, not an ansatz or a renamed known result. The only author-overlapping citation is [21], used for the ELMMSE metric definition and as the SGD-AO benchmark; that citation is not load-bearing for the asymptotic derivation. The reviewer's objection that (45) requires the normalized precoder matrix to be invertible and fails for rank-deficient optimal precoders is a correctness gap in the high-SNR approximation, not a circularity: the claim does not reduce to its own input by definition or by self-citation. Accordingly, no circular step is identified; the paper is a good-faith derivation from an external theorem with simulation validation.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central derivations rest on established RMT theorems, standard Gaussian signaling assumptions, and two unproved technical conditions: convergence of the fixed-point iteration and full-rankness of the precoder in the high-SNR limit. There are no free parameters fitted to data and no invented physical entities.

assumptions (7)
  • standard math Theorem 1 (deterministic equivalent for G = A + B S S^H B^H) from Couillet et al. [30] is valid and applicable at z = 0.
    Invoked in Section III and Appendix B to derive the asymptotic ELMMSE; the proof does not re-derive the theorem, and the z = 0 evaluation is taken without discussing the support boundary.
  • domain assumption Data symbols are i.i.d. complex Gaussian with variance 1/Ld.
    Equation (5) in Section II-A; used to compute E[Sd Sd^H] = I and to apply RMT.
  • domain assumption Pilot matrix is row-orthogonal, Sp Sp^H = I_Nt.
    Equation (4) in Section II-A; used to reduce the pilot contribution to a scaled identity.
  • domain assumption Perfect communication CSI and known target correlation R at the BS.
    Assumed in Section II-B and footnote 1; used to formulate the LMMSE estimator and the rate constraint.
  • standard math Nt and Ld grow large with finite ratio Nt/Ld for the asymptotic expressions.
    Stated in Proposition 2; the finite-size accuracy is only checked by simulation.
  • ad hoc to paper The fixed-point equation (30) has a unique solution and the iteration (31)-(33) converges.
    Algorithm 1 repeatedly solves (30) by fixed-point iteration without a convergence proof.
  • domain assumption In Proposition 4, Phi = W W^H / Tr(W W^H) is invertible (W has full rank).
    Equation (44)-(45); without it the limit argument fails, and rank-deficient precoders (e.g., R0 = Rmax) give a different alpha.

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Cite this review

Pith. "Pith review of Exploiting Both Pilots and Data Payloads for Integrated Sensing and Communications." pith.science (2026). https://pith.science/paper/LWKTG4NR

@misc{pith2026250615998,
  author       = {Pith},
  title        = {Pith review of: Exploiting Both Pilots and Data Payloads for Integrated Sensing and Communications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LWKTG4NR}},
  note         = {Machine review of arXiv:2506.15998}
}
read the original abstract

Integrated sensing and communications (ISAC) is one of the key enabling technologies in future sixth-generation (6G) networks. Current ISAC systems predominantly rely on deterministic pilot signals within the signal frame to accomplish sensing tasks. However, these pilot signals typically occupy only a small portion, e.g., 0.15% to 25%, of the time-frequency resources. To enhance the system utility, a promising solution is to repurpose the extensive random data payload signals for sensing tasks. In this paper, we analyze the ISAC performance of a multi-antenna system where both deterministic pilot and random data symbols are employed for sensing tasks. By capitalizing on random matrix theory (RMT), we first derive a semi-closed-form asymptotic expression of the ergodic linear minimum mean square error (ELMMSE). Then, we formulate an ISAC precoding optimization problem to minimize the ELMMSE, which is solved via a specifically tailored successive convex approximation (SAC) algorithm. To provide system insights, we further derive a closed-form expression for the asymptotic ELMMSE at high signal-to-noise ratios (SNRs). Our analysis reveals that, compared with conventional sensing implemented by deterministic signals, the sensing performance degradation induced by random signals is critically determined by the ratio of the transmit antenna size to the data symbol length. Based on this result, the ISAC precoding optimization problem at high SNRs is transformed into a convex optimization problem that can be efficiently solved. Simulation results validate the accuracy of the derived asymptotic expressions of ELMMSE and the performance of the proposed precoding schemes. Particularly, by leveraging data payload signals for sensing tasks, the sensing error is reduced by up to 5.6 dB compared to conventional pilot-based sensing.

Figures

Figures reproduced from arXiv: 2506.15998 by the authors.

Figure 3
Figure 3. An example of the permutation matrix J of a frame structure with Lp = 2 and Ld = 4. where J is an orthogonal matrix that has exactly one entry of 1 in both each row and each column with all other entries being 0 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 2
Figure 2. Frame structure, where “Pilot” and “Comm. Data” represent the pilot [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 6
Figure 6. ELMMSE comparison among different settings versus 160 180 200 220 240 260 16 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: ELMMSE, its lower bound Jlb, and two approximate expressions Jae and J˜ae, versus the transmit SNR. while the legend “Simulation” represents the numerical results of ELMMSE, as defined in (14), obtained through Monte￾Carlo simulations over 5,000 realizations. It can be…
Figure 8
Figure 8. Figure 8: Sensing and communication performance tradeoff under different rate [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: ELMMSE comparison among different precoding schemes versus [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: ELMMSE comparison between the SCA-based and high-SNR [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

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