Uninformed-to-Informed Estimation: A Ping-Pong Positioning Method for Multi-user Wideband mmWave Systems

Lin Guo; Mugen Peng; Tiejun Lv; Yashuai Cao

REVIEW 5 major objections 5 minor 45 references

Uninformed-to-Informed Estimation: A Ping-Pong Positioning Method for Multi-user Wideband mmWave Systems

T0 review · 5 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper presents a ping-pong positioning framework for wideband mmWave MIMO-OFDM systems, claiming that a multi-subcarrier error lower bound can be minimized through alternating hybrid beamforming to improve positioning accuracy by at lea

desk verdict A useful ping-pong beamforming idea with a definite but local FIM-scaling error in the MSCPEB derivation; worth serious review after a major revision. read the letter →

arxiv 2509.00727 v1 pith:7KUZEQBZ submitted 2025-08-31 eess.SP

Lin Guo , Tiejun Lv , Yashuai Cao , Mugen Peng This is my paper · ORCID

classification eess.SP

keywords mmWavepositioningCramér-Raolowerboundhybridbeamformingping-pongmulti-subcarrierwidebandMIMO-OFDMmultipathclosed-form

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

The paper tries to show that positioning accuracy in wideband millimeter-wave MIMO-OFDM systems can be improved by treating the signal configuration—not just the estimation algorithm—as a resource to be optimized. It introduces the multi-subcarrier collaborative positioning error lower bound (MSCPEB), a CRLB-type bound derived from all subcarriers jointly, and proves it is no larger than the average of the per-subcarrier bounds. On this foundation, it designs an alternating-optimization procedure that minimizes the MSCPEB through hybrid beamforming at both the base station and the user, plus a multipath collaborative positioning step that weights paths by reliability. The reported outcome is a positioning method that is more accurate and uses far fewer pilot symbols than conventional schemes, without needing prior channel state information or position knowledge.

What carries the argument

The key object is the multi-subcarrier collaborative positioning error lower bound (MSCPEB), a Cramér-Rao-type bound computed from the Fisher information matrix of the received signals concatenated across all OFDM subcarriers. The MSCPEB is built from a subcarrier correlation factor vector that captures frequency-dependent phase responses, allowing a unified expression for the mean and covariance of the concatenated signal and decoupling the beamformers per subcarrier. The optimization machinery is an alternating optimization algorithm that uses semidefinite relaxation, Schur complements, and matrix decomposition to convexify the non-convex MSCPEB minimization and produce closed-form hybrid

What would settle it

Compute the exact Fisher information matrix for a simple two-subcarrier, single-path system by differentiating the joint log-likelihood of the concatenated received signal with respect to position, and compare it with the paper's J = (1/Nc) Σ J_n. If the matrices differ beyond a constant factor, Lemma 2 and the beamformer design would need revision. An experimental alternative: run a maximum-likelihood position estimator at high SNR in a wideband multi-subcarrier system and check whether its RMSE approaches the MSCPEB, or whether a different bound—for example, the sum of per-subcarrier FIMs—is

Watch

Extended reading notes

Core claim

The central claim is that the positioning error lower bound for a multi-subcarrier wideband system can be collaboratively minimized across subcarriers, and that this bound is provably no greater than the arithmetic mean of the per-subcarrier PELBs. The paper derives this MSCPEB from the Fisher information of the concatenated received signals, then formulates and convexifies a hybrid beamforming design problem to minimize it. It further develops a multipath position estimation method that avoids explicit path resolution and triangulation, using weighted least squares with path-reliability weights. Simulations validate that the resulting ping-pong framework improves estimation accuracy by at l

Load-bearing premise

The load-bearing premise is that the Fisher information matrix of the concatenated multi-subcarrier signal is the arithmetic mean of the per-subcarrier Fisher information matrices; for independent subcarrier observations the standard FIM is the sum, so if this scaling is wrong the MSCPEB and the beamformers optimized against it would not correspond to the true positioning error bound.

Editorial extensions

If this is right

If the MSCPEB bound is correct, beamformers scheduled to minimize it will yield positioning accuracy at the bound, so the derived closed-form beamformers provide a practical lower-cost alternative to brute-force optimization.
The ping-pong procedure implies that positioning and beam tracking can bootstrap from uninformed to informed estimates without requiring CSI or position priors, simplifying deployment in dynamic urban environments.
The use of only about four OFDM symbols (a quarter of a 5G slot) suggests that high-accuracy positioning can be achieved with low pilot overhead, freeing resources for communication.
The multipath collaborative positioning method, by avoiding explicit path resolution and triangulation, may offer robustness in dense multipath and NLoS conditions.

Reading between the lines

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

The scalability of the MSCPEB approach suggests that similar collaborative bounds could be constructed across time, frequency, and space jointly, potentially unifying sensing and communication resource allocation—an extension the paper does not explore.
The proof of Lemma 2 relies on the FIM being the arithmetic mean across subcarriers; if the standard CRLB scaling (the sum, not the average) is used, the bound and the beamformer design would change by a factor equal to the number of subcarriers—a testable check for the theory.
The reported 16% improvement is for the smallest antenna array (Nt=8); for larger arrays the gains are 29–36%, so the true benefit may grow with array size, which could guide deployment choices.
The method's reliance on far-field array responses means it cannot capture near-field phase curvature; extending the bound to near-field models could unlock additional accuracy at short range.

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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

Quick take: the ping-pong uninformed-to-informed framework is genuinely new, and the multipath collaborative position estimator is clever. The paper deserves a serious referee, but the MSCPEB derivation has a real error that must be fixed before the theoretical claims can stand.

What is actually new: bidirectional beam training where BS and UE alternate PELB-driven beams; the multi-subcarrier collaborative bound as a design criterion; and the line/polyline-based multipath least-squares positioning that avoids explicit path resolution. The AO/SDR approach is standard but competently executed. Simulations are self-consistent and show the expected gains (16-36%, quarter slot usage) for large arrays.

The soft spots are in the theory. Appendix C, eq. (63), says the multi-subcarrier FIM is "the sum" and then writes J = (1/Nc)ΣJn. For independent subcarriers the FIM is additive; the 1/Nc makes the "MSCPEB" Nc times the true CRLB. The proof of Lemma 2 then proves an inequality for that inflated object. The good news: since the scaling is constant, the argmin of the beamformer optimization is unchanged, and if the formula is corrected to J = ΣJn, Jensen still gives the claimed inequality (in fact stronger). So the fix is local, but the bound values reported in Fig. 12 and the absolute comparisons are not valid CRLBs as written.

Also, eq. (22)(b) leaves the Bernoulli indicator δk inside the expectation, which is not the expectation; it should be pLoS times the LoS bound plus (1-pLoS) times the NLoS bound. And eq. (23) has dimension problems with the all-ones matrix in the Kronecker product. These should be cleaned up. The subcarrier correlation factor in eq. (10) simplifies to e^{-j2πfnτ}; as written it adds no new information, though that doesn't break the framework.

One more thing: no code or data are provided, and the simulation section omits some setup details (number of runs, exact channel generation, Lk values). That limits reproducibility but is not fatal.

Bottom line: this is a plausible and useful method paper, not a breakthrough. The core algorithm may survive the fixes. Send it to review, but require the FIM scaling and expectation errors to be corrected and the absolute bound comparisons to be redone. I would not cite it in its current form.

Referee Report

5 major / 5 minor

Summary. The manuscript proposes a PELB-driven ping-pong positioning framework for multi-user wideband mmWave MIMO-OFDM systems. The BS and UE alternately transmit/receive adaptive beamformed pilots, using locally evaluated position-error bounds to schedule beams. The theoretical centerpiece is a multi-subcarrier collaborative positioning error bound (MSCPEB), together with Lemma 2 claiming that the MSCPEB does not exceed the arithmetic mean of the per-subcarrier PELBs. An alternating-optimization algorithm with closed-form hybrid precoder/combiner updates is then developed, followed by an uninformed-to-informed channel and position estimator and a multipath collaborative positioning method. Simulations report at least 16% RMSE improvement over non-optimized baselines and about one-quarter slot-resource usage.

Significance. The general direction is timely and potentially useful: adaptively configuring beams according to a positioning-centric lower bound, exploiting frequency diversity, and avoiding the need for initial CSI are all worthwhile goals for mmWave positioning. The paper contains substantial algorithmic material, including a full signal model, a detailed optimization framework, complexity/convergence discussion, and extensive benchmarks. However, the theoretical foundation currently has several load-bearing errors that affect the definition and interpretation of the MSCPEB, the Gaussian likelihood, and the optimization relaxation. If these are repaired, the proposed framework could be a solid contribution; in its present form the central claims are not established.

major comments (5)

[Appendix C, Eq. (63)] The text before Eq. (63) says the multi-subcarrier FIM is the sum of the per-subcarrier FIMs, but the displayed formula is J = (1/Nc) Σ_n J_n. For independent subcarrier observations the FIM is additive: J_true = Σ_n J_n. With the displayed averaging, e(p) = tr(J^{-1}) equals Nc · tr(J_true^{-1}), i.e., the reported MSCPEB is Nc times the actual CRLB of the concatenated signal. This invalidates the numerical MSCPEB values and the comparison in Fig. 12. The constant scaling does not change the argmin in P1–P5, so the beamformer optimization may survive, but the bound interpretation must be corrected. If Eq. (63) is replaced by J_true, Jensen's inequality still gives tr(J_true^{-1}) ≤ (1/Nc^2)Σ tr(J_n^{-1}) ≤ (1/Nc)Σ tr(J_n^{-1}), so Lemma 2 remains true; the fix is local but essential.
[Section III, Eqs. (18)–(19)] The mean and covariance of the concatenated signal Y'_k[Nc] are not correctly specified. With C_k[n] ∈ C^{Ns×Ns} and ρ ∈ C^{1×Nc}, the expression C'_k[Nc] = diag(C_k[n] ⊗ ρ) is not a well-formed covariance for a stacked vector; independent subcarrier observations would give a block-diagonal covariance with blocks C_k[1], ..., C_k[Nc]. Similarly, µ'_k[Nc] = µ_k[n] ⊗ ρ^T assumes the per-subcarrier mean is independent of n, which is not true since H_k[n], F_BB[n], and s[n] vary with the subcarrier. These equations underpin the Gaussian PDF (17) and the FIM formula (25), so they must be re-derived before Lemma 1 and the subsequent FIM computation are valid.
[Section III, Eq. (20)] The expression E_{pr,k}{E_{H_k}{tr(J^{-1}(pr,k|H_k))}} is not a lower bound on the marginal MSE E[∥ˆp - p∥²] when pr,k and H_k are random. The CRLB is conditional on the true parameters; averaging the trace of the inverse FIM over a prior does not give the Bayesian CRLB (which involves E[J] plus prior information) and is not guaranteed to bound the averaged MSE. The MSCPEB should be defined as a conditional CRLB, or derived properly as a Bayesian bound, before it is claimed to be a lower bound on positioning error.
[Section IV, P3/P4, Eqs. (29)–(31)] The auxiliary variable is introduced as Ω ∈ R^{3Nc×3Nc} with E a 3Nc×3Nc identity matrix, but the FIM for the user position pr,k ∈ R^3 is 3×3 after the transformation in Eq. (23). The Schur complement constraint [Ω, E; E, J(V[n])] ⪰ 0 is therefore dimensionally inconsistent as written. If E is meant to be a selection matrix embedding the 3×3 position FIM into a 3Nc×3Nc space, this must be stated explicitly; otherwise trace{Ω} rescales the objective by Nc and the relaxation is not equivalent to P1.
[Section III, Eq. (22)] The step labeled (b) does not correctly marginalize the Bernoulli LoS indicator δ_k. The expression retains δ_k in the exponents pLoS(τc)^{δ_k} and (1−pLoS)^{1−δ_k}, so the right-hand side is still a random quantity, not the expectation E_{δ_k}. Marginalizing gives pLoS · tr(J^{-1}(·|tilde H_k)) + (1−pLoS) · tr(J^{-1}(·|H_0k)); the notation should be corrected. The mixture of '≈' and '=' in (a)–(b) also obscures whether this is an approximation or an identity.

minor comments (5)

[Appendix C, Eq. (63) and notation] The phrase 'the sum of the FIM across all subcarriers' contradicts the displayed average; also 1_{Nc×NcNs} is called a 'vector' although it has two dimensions.
[Section II, Eq. (10)] In Eq. (10b), the product e^{-j2π fn/fc} · (e^{-j2π fn/fc})^H equals 1, so the 'array phase difference' cancels and ρ_n reduces to e^{-j2π fn τ}. Please clarify whether the intended ρ_n includes additional coupling or whether the simplification is intentional.
[Section V-A] Typo: 'scretizing' should be 'discretizing'. Also in Eq. (42), the bracket notation [vec(ˆY'_1[n]), ..., vec(ˆY'_{ℓ_k}[n])] is confusing; the index should be over paths, not over ℓ_k itself.
[Section VI-C] The convergence argument is heuristic: it states that the SDP subproblems ensure monotonicity, but no formal proof is given that each subproblem is solved globally or that the objective is bounded away from −∞. The statement 'constrained by system parameters and SNR' is not a proof. This should be tightened, or the claim softened.
[Figs. 2, 11 and references] Several caption/reference inconsistencies: Fig. 2 lists 'Rank-1 EFIM [42]' and later 'Rank-1 EFIM [39]' in different panels; Fig. 11(a) labels both vehicle traces as 'Vehicle 1'; reference numbering for [42]–[44] is inconsistent in the text. Please proofread.

Circularity Check

1 steps flagged · score 6.0 of 10

The central Lemma 2 bound is self-definitional: Eq. (63) defines the multi-subcarrier FIM as the arithmetic mean of per-subcarrier FIMs, so Jensen's inequality makes the claimed bound a consequence of the definition; the 1/Nc factor also makes the MSCPEB not the true CRLB.

self definitional [Appendix C, eqs. (63)-(65); used in Lemma 2 (Section III) and P1 (Section IV)]
"For a multi-subcarrier system, the FIM of the received signal matrix Y'_k[n] is the sum of the FIM across all subcarriers: J(pr,k|Hk) = 1/Nc \sum_{n=1}^{Nc} Jn(pr,k|Hk). ... tr(((1/Nc)\sum_n Jn(pr,k|Hk))^{-1}) ≤ (1/Nc)\sum_n tr(Jn^{-1}(pr,k|Hk))."

The proof defines the multi-subcarrier FIM as the arithmetic mean of the per-subcarrier FIMs. Since the MSCPEB is defined in (20d) as tr(J^{-1}), Jensen's inequality immediately yields the claim in Lemma 2; it is a restatement of the definition plus convexity, not a property derived from the FIM of the concatenated observation. The sentence even says 'sum' but writes an average; for independent subcarrier observations the true FIM is \sum_n J_n, so the MSCPEB as defined is Nc times the true CRLB. Thus Lemma 2 and the numerical MSCPEB comparisons are forced by Eq. (63) by construction.

full rationale

The load-bearing theoretical step is Appendix C. Eq. (63) says that the multi-subcarrier FIM is J = (1/Nc)\sum_n J_n after describing it as the sum, and Eq. (20d) defines MSCPEB as e(p)=tr(J^{-1}). With that definition, f(X)=tr(X^{-1}) being convex gives tr(J^{-1}) ≤ (1/Nc)\sum tr(J_n^{-1}) directly, which is exactly Lemma 2. Therefore Lemma 2 is not an independent first-principles result about the concatenated multi-subcarrier observation; it is the definition of J plus Jensen. Moreover, for independent subcarrier observations the FIM is additive with no 1/Nc factor, so the quantity labeled MSCPEB is Nc times the actual CRLB and is not a valid lower bound on the MSE of an unbiased estimator. The word 'sum' in the same sentence is internally inconsistent with the formula. The beamformer optimization P1 minimizes tr(J^{-1}) = Nc tr((\sum_n J_n)^{-1}); the scalar does not change the argmin, so the AO algorithm may still be salvageable, but the reported MSCPEB values and Fig. 12 are not genuine CRLBs. No load-bearing self-citation chain was found: the only clear self-reference [40] is used as a comparison benchmark, and the other cited support is external. The multipath weighting rho=1/(tau c)^2 is a design choice validated by simulation, not a fitted parameter renamed as prediction. The circularity is localized to the MSCPEB definition/Lemma 2, but that is the paper's foundational theoretical claim, so the score is 6 rather than lower.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central theoretical result rests on an ad hoc FIM normalization (1/Nc) that is not derived, and the practical method depends on modeling assumptions such as quasi-static channels, known path counts, and discrete search grids. The path weighting exponent is chosen by simulation comparison. No new physical entities are introduced.

free parameters (4)

Multi-subcarrier FIM averaging factor 1/Nc = 1/Nc
Introduced without derivation in eq. (63) to express the multi-subcarrier FIM as the arithmetic mean of per-subcarrier FIMs; the Lemma 2 inequality depends on this scaling.
Path weight exponent = 2 (rho = 1/(tau*c)^2)
Chosen after comparing exponents 0.5, 1, and 2 in Fig. 9; the quadratic inverse-length weighting is selected for best RMSE.
Newton update step size gamma = not specified
In eq. (44), the step size gamma for informed estimation is not given a value, yet convergence of the Newton refinement depends on it.
LoS probability pLoS(tau*c) = not specified
Invoked in eqs. (21)-(22) as a Bernoulli probability depending on path distance, but no functional form is provided and the algorithm does not use it in the actual optimization.

assumptions (5)

domain assumption Far-field planar wavefront and channel reciprocity
Section II-A/B: AoA and AoD are identical across antennas and H_k[n] is reciprocal between BS and UE.
domain assumption Quasi-static channel during a ping-pong round
Section II-C and VII: the framework is designed for slow mobility; mobility is simulated with Jakes' model, but the MSCPEB derivation assumes fixed positions during optimization.
ad hoc to paper Subcarrier observations are independent and equally informative
Appendix C eq. (63) asserts the multi-subcarrier FIM is the average of per-subcarrier FIMs, implying independence and equal weighting; this is not derived from the covariance in eq. (19).
domain assumption Per-path parameters can be estimated on discrete grids and the number of paths Lk is known
Section V-A: uninformed estimation searches grids of N_tau x N_theta x N_phi; the method assumes Lk and path association are known or reliably estimated.
domain assumption LoS probability pLoS depends only on path distance
Eq. (21) models LoS presence as Bernoulli with probability pLoS(tau*c), but no model for pLoS is given.

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Cite this review

Pith. "Pith review of Uninformed-to-Informed Estimation: A Ping-Pong Positioning Method for Multi-user Wideband mmWave Systems." pith.science (2026). https://pith.science/paper/7KUZEQBZ

@misc{pith2026250900727,
  author       = {Pith},
  title        = {Pith review of: Uninformed-to-Informed Estimation: A Ping-Pong Positioning Method for Multi-user Wideband mmWave Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7KUZEQBZ}},
  note         = {Machine review of arXiv:2509.00727}
}

read the original abstract

To enhance the positioning and tracking performance of dynamic user equipment (UE) in wideband millimeter-wave (mmWave) systems, we propose a novel positioning error lower bound (PELB)-driven ping-pong positioning framework, where the base station (BS) and UE alternately transmit and receive adaptive beamforming signals for positioning. All beam-formers are scheduled based on the locally evaluated PELB. In this framework, we exploit multi-dimensional information fusion to assist in positioning. Firstly, a multi-subcarrier collaborative positioning error lower bound (MSCPEB) is proposed to evaluate the positioning error limits of wideband mmWave systems, which quantifies the contribution of all subcarriers to positioning accuracy. Moreover, we prove that the MSCPEB does not exceed the arithmetic mean of the PELBs of the individual subcarriers. Subsequently, we develop an alternating optimization (AO) algorithm to optimize the hybrid beamformers targeted for MSCPEB minimization. By convexifying this problem, closed-form solutions of beamformers are derived. Finally, we develop a multipath collaborative positioning method that quantifies the impact of path reliability on positioning accuracy, with a closed-form solution for user position derived. The proposed method does not rely on path resolution and traditional triangular relationships. Numerical results validate that the proposed method improves estimation accuracy by at least 16% compared to potential schemes without optimized beam configurations, while requiring only approximately one-quarter of the slot resources.

Figures

Figures reproduced from arXiv: 2509.00727 by the authors.

**Figure 2.** Comparison for different positioning methods: (a) R [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 4.** Positioning RMSE vs. number of subcarriers, with [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

Figures from the paper (7 more)

**Figure 3.** Figure 3: The impact of optimization for MSCPEB on RMSE under di [PITH_FULL_IMAGE:figures/full_fig_p011_3.png]

**Figure 6.** Figure 6: Positioning RMSE vs. number of OFDM symbols, with [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: Positioning RMSE vs. number of total multipath compo [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 10.** Figure 10: Convergence speed of MSCPEB, with SNR = 15 dB: detail [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 11.** Figure 11: The impact of dynamic user mobility on the robustnes [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

**Figure 12.** Figure 12: MSCPEB vs. single-subcarrier PELB, with SNR = 15 dB. [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

**Figure 13.** Figure 13: The cumulative distribution of positioning RMSE, w [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]

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