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Optimal Power Allocation for OFDM-based Ranging Using Random Communication Signals

T0 review · 4 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Uniform power allocation is the unique optimum for random-OFDM ranging sidelobes, the paper proves.

desk verdict The P-ACF half of the paper is solid and useful; the A-ACF global-optimality claim is true but not proved, because the Appendix D lower-bound argument minimizes the wrong quantity. read the letter →

arxiv 2504.18016 v1 pith:T4LU63FC submitted 2025-04-25 eess.SP

Ying Zhang , Fan Liu , Tao Liu , Shi Jin This is my paper · ORCID
classification eess.SP
keywords integratedsensingandcommunicationOFDMpowerallocationrandomsignalingrangingsidelobeperiodicautocorrelationfunctionaperiodicfrequencyzero-padding
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 subcarrier power allocation can improve the ranging sidelobe behavior of OFDM signals whose subcarriers carry random communication data, a setup central to 6G integrated sensing and communication (ISAC). It establishes that, under the standard circular-symmetric constellation assumption (unit power, zero mean, zero pseudo-variance, which excludes BPSK and 8-QAM), uniform power allocation is the only scheme that minimizes the normalized expected integrated sidelobe level for both cyclic-prefix (periodic) and non-cyclic (aperiodic) autocorrelations of the OFDM signal. For the periodic case, uniform power also minimizes the average sidelobe at every delay lag. The practical consequence is that no power-allocation optimization is needed for the basic OFDM ranging problem, saving computation and preserving communication performance. The exception is frequency-domain zero-padding for finer range resolution, where uniform power is suboptimal and optimized allocations trade a wider mainlobe for lower sidelobes.

What carries the argument

The carrying object is the exact expression for the average squared periodic autocorrelation, $\mathbb{E}(|\tilde{r}_k|^2) = (\mu_4 - 1)\sum_{i=1}^N P_i^2 + |\sum_{n=1}^N P_n e^{j2\pi k n/N}|^2$, where $\mu_4 = \mathbb{E}(|s_n|^4)$ is the kurtosis of the unit-power, zero-pseudo-variance constellation. Summing this over nonzero lags and normalizing by the expected mainlobe makes the normalized EISL a strictly monotone function of $\sum_i P_i^2$, and the Cauchy-Schwarz inequality forces that sum to be minimized only at $P_1 = \cdots = P_N = 1$. The aperiodic case is reduced to the same structure by embedding the $N$-sample signal in a $2N$-periodic shift, after which a second application of Cauchy-Schwarz on the odd and even column blocks delivers the same uniqueness conclusion. The kurtosis parameter $\mu_4$ is what carries constellation dependence: all PSK constellations have $\mu_4 = 1$ and all QAM constellations have $1 \le \mu_4 \le 2$, so the theorems cover the whole allowed family at once.

What would settle it

For a small system, say $N=8$ subcarriers with 16-QAM symbols (satisfying Assumption 1), numerically minimize the normalized EISL in equation (16) over the simplex $\sum_i P_i = N$, $P_i \ge 0$ by exhaustive grid search or random sampling; the theorem says the global minimum occurs only at $P_1=\cdots=P_8=1$, with value $N\mu_4/((\mu_4-1)+N)-1$. Finding any admissible power vector with a strictly smaller normalized EISL, or with a smaller normalized $\mathbb{E}(|\tilde{r}_k|^2)$ at some lag $k\neq 0$ in the periodic case, would refute the uniqueness claim.

Watch

Extended reading notes

Core claim

The paper proves that uniform power allocation, $P_i = 1$ on every subcarrier, is the unique minimizer of the normalized expected integrated sidelobe level (EISL) for both the periodic and aperiodic autocorrelation functions of a random OFDM ISAC signal, whenever the constellation satisfies the unit-power, zero-mean, zero-pseudo-variance condition. It further proves that for the periodic case the same uniform scheme minimizes the average squared sidelobe at every delay lag, not just the integrated total. The proof works uniformly across all such constellations because the normalized EISL depends on the power vector only through $\sum_i P_i^2$, which is minimized exactly when all subcarrier powers are equal. Frequency-domain zero-padding breaks this conclusion: the paper shows that with zero-padding, uniform power is no longer optimal, and it gives a projected-gradient-descent algorithm that lowers sidelobes at the cost of a wider mainlobe, plus a constrained variant that makes the tradeoff tunable.

Load-bearing premise

The proofs assume the random data symbols come from a constellation with zero mean, unit power, and no correlation between a symbol and its complex conjugate (zero pseudo-variance), which excludes BPSK and 8-QAM; if that assumption fails, the closed-form sidelobe expressions gain extra terms and uniform power may stop being optimal.

Editorial extensions

If this is right

  • For the basic zero-Doppler ranging setup with cyclic-prefix or non-cyclic OFDM, the power-allocation subproblem has a closed-form answer: set every subcarrier power to 1, so no iterative PA optimization is required.
  • Uniform power gives an average sidelobe floor of $(\mu_4-1)N$ per lag for the periodic ACF; with PSK constellations, where $\mu_4=1$, the expected periodic ACF is impulse-like with zero sidelobes.
  • Any non-uniform power vector strictly raises the normalized EISL for both P-ACF and A-ACF, so communication-oriented power loading carries a ranging sidelobe penalty in this metric.
  • Frequency zero-padding changes the optimal design: the PGD allocation reduces sidelobes relative to uniform power, and the SCA variant lets an operator choose how much mainlobe widening to accept for a given sidelobe reduction.

Reading between the lines

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

  • Because the theorems depend on vanishing pseudo-variance, allowing improper constellations such as BPSK or 8-QAM is the natural stress test; extra conjugate-correlation terms may make non-uniform power strictly optimal even without zero-padding, which would extend rather than contradict the paper's domain.
  • The optimality applies to the expectation over random symbols; a sensing system concerned with per-realization peak sidelobes or worst-case ambiguity could still profit from non-uniform power, a different objective than EISL.
  • The zero-padding case effectively turns the power vector into a spectral window on the interpolated ranging response; the mainlobe-versus-sidelobe tradeoff found here has the same shape as classical window design, suggesting the PGD and SCA allocations could be compared against standard window families in future work.
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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

4 major / 3 minor

Summary. The paper studies power allocation (PA) for OFDM-based monostatic ISAC ranging under random communication symbols. It derives closed-form expressions for the expected squared periodic and aperiodic autocorrelation functions (P-ACF and A-ACF), formulates the normalized expected integrated sidelobe level (EISL), and claims that uniform PA is globally optimal for both the P-ACF and A-ACF and for every P-ACF delay index. It then extends the analysis to frequency-domain zero-padding, where uniform PA is no longer optimal, and proposes projected gradient descent and successive convex approximation algorithms to optimize PA under a mainlobe-width constraint. Simulations validate the closed-form expressions and illustrate the sidelobe-versus-mainlobe tradeoff.

Significance. If the main claims are correct, the paper establishes a strong and useful negative result for basic OFDM ranging with random communication signals: no PA optimization is needed, since uniform PA minimizes the average sidelobe level of both P-ACF and A-ACF under the stated constellation assumptions. The closed-form characterizations in Propositions 1-3 are valuable and carefully derived. The zero-padding extension, with its explicit tradeoff between sidelobe level and mainlobe width, is also a worthwhile contribution. The paper is self-contained except for one decomposition from the authors' companion work [26], and the simulation results support the derived expressions. However, the proof of the A-ACF global-optimality theorem contains a substantial gap, and the theorem statements overclaim the constellation domain relative to Assumption 1.

major comments (4)
  1. [Appendix D, Eq. (87)] The proof of Theorem 3 does not establish global optimality of uniform PA. The displayed Cauchy-Schwarz inequality is a lower bound on the even-column contribution, not an expression for it. Equality at uniform PA only makes the lower bound tight at that point; it does not show that the true even-column term at any nonuniform P is at least its value at uniform P. Moreover, for a fixed value of sum_n A_n, the Cauchy-Schwarz right-hand side is maximized, not minimized, when all A_n are equal, so the direction of the asserted minimization is incorrect. The theorem may be true, but it needs a direct proof, for example by writing the even-column term as a monotone function of S = sum P_n^2 once the column norms of W2 are taken into account.
  2. [Section IV-A, Theorem 2 proof, Eq. (33)] The claim that uniform PA is the only power vector for which |sum P_n exp(j2pi k n / N)|^2 is minimized is false. For N=4 and k=2, p=(1.5, 0.5, 0.5, 1.5) satisfies sum P_n exp(j pi n)=0 while being nonuniform. The theorem is recoverable because the denominator grows with sum P_n^2, but the 'only when' conclusion requires an additional argument, and the current proof is incomplete as written.
  3. [Section II-A and Theorems 1-3] Assumption 1 restricts the analysis to zero-mean, zero-pseudo-variance constellations and explicitly excludes BPSK and 8-QAM, yet the theorem statements and the abstract claim optimality 'for all constellations.' The A-ACF derivations in Appendix C rely on the decomposition S = I + S1 + S2 in Eq. (62), which uses the vanishing pseudo-variance. The authors should either prove the extension to improper constellations or restate the theorems with the explicit restriction.
  4. [Section V, Algorithm 1, Eq. (52) and line 5] The update rule in Algorithm 1 contains a leading minus sign before ProjB, which would drive every iterate negative and violate the constraint Pi >= 0. In addition, ProjB in Eq. (52) is a radial scaling that forces the sum to N but is not the standard Euclidean projection onto the simplex. The pseudocode should be corrected and the projection actually used in the simulations should be specified precisely.
minor comments (3)
  1. [Section II-B, Eq. (16)] The same symbol for the normalized objective is reused for the P-ACF, the A-ACF, and the zero-padding cases; introducing separate notation for E(|r0|^2) in each case would improve readability.
  2. [Section V, Algorithm 2, Eq. (54)] The first-order Taylor surrogate in Eq. (54) is not necessarily a majorant of the nonconvex objective, and no convergence proof is given; the paper should state whether the reported convergence is empirical or provide a supporting argument.
  3. [Notations and Eq. (39)] The notation 'a_m.n' should be 'a_{m,n}', and the matrix norm in Eq. (39) should be written as ||FN ~F2N^H||_4^4 with parentheses to avoid ambiguity.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central optimality claims follow from derived closed-form statistics, with only a minor non-load-bearing self-citation.

full rationale

The main claims are derived from first principles rather than assumed. Proposition 1 derives the average squared P-ACF directly from the constellation moment assumptions, Corollary 1 sums it via Parseval, and Theorem 1 minimizes the resulting normalized EISL by Cauchy-Schwarz over the fixed-total-power simplex; no fitted parameter is renamed as a prediction. Theorem 2 uses the same derived per-lag formula. The A-ACF analysis in Appendix C uses the fourth-order moment decomposition S = I + S1 + S2, cited from the authors' prior work [26] in Eq. (62). This is a self-citation, but the decomposition is a standard consequence of Assumption 1's unit-power, zero-mean, and zero-pseudo-variance conditions, and it does not assume uniform power allocation or the target optimality result, so it is not a circular reduction. The proof of Theorem 3 in Appendix D contains a Cauchy-Schwarz lower-bound argument around Eq. (87) that does not directly compare the actual even-column term at nonuniform power allocations with its value under uniform allocation; this is a proof gap and a correctness concern, but it is not circularity. Likewise, Assumption 1's exclusion of BPSK and 8-QAM is a domain restriction rather than a circular input. The zero-padding results are algorithmic and are validated against simulations rather than fitted to the conclusions. Overall, the derivation chain is self-contained apart from a minor, non-load-bearing self-citation.

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

No free parameters fitted to data; µ4 is a known constellation moment, not an adjustable constant. The load-bearing assumptions are the proper-constellation condition and i.i.d. symbols. The zero-padding sidelobe definition is a modeling choice that influences the optimization objective.

assumptions (5)
  • domain assumption Constellations satisfy Assumption 1: unit power, zero mean, zero pseudo-variance, E(|s_n|^2)=1, E(s_n)=0, E(s_n^2)=0.
    Central to all closed-form EISL derivations; excludes BPSK and 8-QAM. Load-bearing for Theorems 1-3.
  • domain assumption Subcarrier symbols are independent and identically distributed.
    The expectation factorizations E(|s_n|^2|s_m|^2)=1 for n≠m require independence across subcarriers.
  • domain assumption The sensing receiver has full knowledge of the transmitted random symbols.
    Monostatic ISAC setup; standard but load-bearing for matched filtering and the ACF metric.
  • ad hoc to paper In the zero-padded P-ACF, the sidelobe region is defined as k from L to NL/2 - 1.
    This choice sets the mainlobe width to 2L-1 and directly shapes the EISL_ZP objective and the claimed tradeoff.
  • standard math Standard mathematical tools: Cauchy-Schwarz, Parseval's theorem, properties of DFT matrices.
    Used in the proofs of Theorem 1-3 and the closed-form propositions.

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Cite this review

Pith. "Pith review of Optimal Power Allocation for OFDM-based Ranging Using Random Communication Signals." pith.science (2026). https://pith.science/paper/T4LU63FC

@misc{pith2026250418016,
  author       = {Pith},
  title        = {Pith review of: Optimal Power Allocation for OFDM-based Ranging Using Random Communication Signals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T4LU63FC}},
  note         = {Machine review of arXiv:2504.18016}
}
read the original abstract

High-precision ranging plays a crucial role in future 6G Integrated Sensing and Communication (ISAC) systems. To improve the ranging performance while maximizing the resource utilization efficiency, future 6G ISAC networks have to reuse data payload signals for both communication and sensing, whose inherent randomness may deteriorate the ranging performance. To address this issue, this paper investigates the power allocation (PA) design for an OFDM-based ISAC system under random signaling, aiming to reduce the ranging sidelobe level of both periodic and aperiodic auto-correlation functions (P-ACF and A-ACF) of the ISAC signal. Towards that end, we first derive the closed-form expressions of the average squared P-ACF and A-ACF, and then propose to minimize the expectation of the integrated sidelobe level (EISL) under arbitrary constellation mapping. We then rigorously prove that the uniform PA scheme achieves the global minimum of the EISL for both P-ACF and A-ACF. As a step further, we show that this scheme also minimizes the P-ACF sidelobe level at every lag. Moreover, we extend our analysis to the P-ACF case with frequency-domain zero-padding, which is a typical approach to improve the ranging resolution. We reveal that there exists a tradeoff between sidelobe level and mainlobe width, and propose a project gradient descent algorithm to seek a locally optimal PA scheme that reduces the EISL. Finally, we validate our theoretical findings through extensive simulation results, confirming the effectiveness of the proposed PA methods in reducing the ranging sidelobe level for random OFDM signals.

Figures

Figures reproduced from arXiv: 2504.18016 by the authors.

Figure 3
Figure 3. The resultant normalized EISL of A-ACF for 16PSK/16Q [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 2
Figure 2. The P-ACF of 64-QAM under OFDM signaling and uniform p [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 7
Figure 7. The convergence results of the SCA algorithm for solv [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: The P-ACF of 16-QAM under OFDM signaling and differen [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 8
Figure 8. Figure 8: The P-ACF of 16-QAM under OFDM signaling and differen [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 6
Figure 6. Figure 6: The P-ACF of 16-PSK under OFDM signaling and differen [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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