Interference-Aware Multiuser Hybrid Precoding for Coexistence with LEO Satellite Communication

Murat Bayraktar; Nima Razavi; Nuria Gonzalez Prelcic; Robert W. Heath Jr

REVIEW 4 major objections 6 minor 19 references

Interference-Aware Multiuser Hybrid Precoding for Coexistence with LEO Satellite Communication

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Hybrid precoding nulls LEO satellite interference while keeping sum-rate close to baselines.

desk verdict Relevant coexistence problem and a plausible BCD/PGD approach, but the printed gradient expressions have the wrong sign for the sum-rate term, so the reported results cannot be reproduced from the manuscript. read the letter →

arxiv 2506.03438 v1 pith:M7KIVQVU submitted 2025-06-03 eess.SP

Nima Razavi , Murat Bayraktar , Nuria Gonzalez Prelcic , Robert W. Heath Jr This is my paper

classification eess.SP

keywords hybridprecodingLEOsatellitecoexistencespectrumsharinginterferencenullingprojectedgradientdescentblockcoordinateuppermid-bandFR3multiuserMIMO

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 proposes a beamforming algorithm for hybrid MIMO base stations that suppresses interference to LEO satellites in shared spectrum while preserving downlink sum-rate. The algorithm alternates between analog and digital precoder updates using block coordinate descent with projected gradient steps, guided by a cost function that penalizes satellite interference. Simulations show it reduces mean satellite interference-to-noise ratio by 22.4 dB versus the best hybrid baseline and meets the -20 dB protection threshold in 99.95% of trials, with only a 2.69% rate loss. The significance is a practical path to spectrum coexistence in the upper mid-band without exclusion zones or full digital arrays.

What carries the argument

The central mechanism is a block coordinate descent (BCD) algorithm with projected gradient descent (PGD) applied alternately to the analog precoder $\mathbf{F}_{\mathrm{RF}}$ and the digital precoder $\mathbf{F}_{\mathrm{BB}}$. The cost function is $C(\mathbf{F}_{\mathrm{RF}}, \mathbf{F}_{\mathrm{BB}}) = -\sum_u \log(1+\mathrm{SINR}_u) + \lambda_{\mathrm{Sat}} \operatorname{tr}(\mathbf{F}_{\mathrm{BB}}^* \mathbf{F}_{\mathrm{RF}}^* \mathbf{H}_{\mathrm{Sat}}^* \mathbf{H}_{\mathrm{Sat}} \mathbf{F}_{\mathrm{RF}} \mathbf{F}_{\mathrm{BB}})$, whose closed-form gradients with respect to both precoders are derived. Each update projects the analog entries onto the unit-modulus circle and normalizes the digital precoder to the power constraint, allowing the algorithm to null satellite directions while maximizing user rates.

What would settle it

Run the same algorithm in a scenario where the satellite channel includes a strong multipath component or a small angle error, and measure the fraction of trials in which the satellite interference-to-noise ratio exceeds -20 dB; if that fraction rises well above 0.05%, the nulling claim fails.

Watch

Extended reading notes

Core claim

The paper establishes that a hybrid MIMO precoder can simultaneously maintain near-baseline multiuser sum-rate and dramatically suppress interference to LEO satellites by optimizing a penalty-based objective. The key result is a block coordinate descent algorithm with projected gradient descent that alternates between the analog and digital precoders, using closed-form gradients of a cost function that includes both negative sum-rate and a satellite interference penalty weighted by a tunable parameter. Across simulated urban scenarios with two satellites and two UEs, the proposed method achieves satellite interference-to-noise ratios far below the -20 dB protection threshold in 99.95% of trials, outperforming both DFT-codebook and hybrid factorization baselines while staying within 2.69% of the best hybrid sum-rate.

Load-bearing premise

The paper assumes the base station-to-satellite channel is a single line-of-sight steering vector determined only by the satellite's known position, ignoring multipath, satellite antenna pattern, and ephemeris errors.

Editorial extensions

If this is right

Base stations in the upper mid-band can share spectrum with LEO satellites without relying on exclusion zones or coordination overhead.
Hybrid MIMO architectures, which are practical at FR3 frequencies, can provide satellite protection comparable to fully digital nulling at a fraction of the hardware cost.
The penalty parameter $\lambda_{\mathrm{Sat}}$ gives operators a tunable trade-off between satellite interference and terrestrial sum-rate, enabling dynamic coexistence policies.
The algorithm's fixed-step projected gradient design keeps computational complexity low enough for real-time precoder updates as satellites move.
The approach can be extended to arbitrary numbers of satellites and users, as demonstrated with two satellites and two UEs, while maintaining the protection threshold.

Reading between the lines

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

If the satellite channel departs from the single-line-of-sight steering vector assumption, the nulling may degrade; tracking the satellite's angle and adding robustness to channel uncertainty would likely preserve the protection guarantee.
The same penalty-based BCD framework could be adapted to protect other incumbent receivers, such as fixed satellite service earth stations or radio astronomy sites, by replacing the satellite channel matrix with the corresponding interference channel.
A testable extension is to combine the proposed precoder with dynamic $\lambda_{\mathrm{Sat}}$ scheduling based on satellite ephemeris updates, potentially improving sum-rate during satellite passes while still satisfying the protection criterion.

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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

Colleague,

The paper tackles a real problem: letting terrestrial base stations in FR3 coexist with LEO satellites by designing hybrid precoders that null interference to the satellite. That is worth reading. The system model is standard, the simulation setup uses Wireless InSite urban channels, and the authors compare against four baselines including fully digital and hybrid nulling schemes. On the surface, the reported INR improvement (22.4 dB mean reduction, 0.05% threshold violation vs 7.7% for DFT+BD) is striking, and the sum-rate loss is small.

But there is a load-bearing problem in the algorithm description. Equations (11)-(12) claim to be gradients of the cost function C in (9), but they have the wrong sign for the sum-rate term. For U=1 and λ=0, the printed update FBB ← FBB - α∇FBB C moves the digital precoder in the direction that reduces |w*H F b|^2, i.e., it decreases the UE rate rather than increasing it. The auxiliary variable N_{u,j} in (10e) also uses e_u instead of e_j, so it does not correspond to any term in the cost. The authors omit the derivation "because of a lack of space" and provide no code. So as printed, Algorithm 1 cannot produce the claimed rate/INR results. Either the equations are typesetting errors and the implementation used correct gradients, or the implementation has the same sign error; we have no way to verify.

This is a major issue, not a minor one, because the central empirical claim rests on an unverifiable and internally inconsistent description. The omission of the derivation and code would be problematic even if the signs were right, since the step size and penalty parameter are hand-tuned.

What is genuinely new: the specific problem formulation (penalty-based hybrid precoding for satellite nulling) and the BCD/PGD approach seem not to appear in prior work, and the comparison against multiple hybrid baselines is fair.

If the gradient equations can be corrected and verified (ideally with code or a complete derivation), this could be a useful contribution. As it stands, I would not trust the results. The paper deserves a serious referee, but only with the expectation of heavy revision—the authors need to fix the math, include a derivation, and share the implementation.

For a reading group, it is a good example of why printed gradient expressions need checking. For citing, I'd hold off until a corrected version appears.

Recommendation: send it to peer review, but the reviewers should be instructed to verify the gradients and require code or a full derivation.

Referee Report

4 major / 6 minor

Summary. This paper addresses spectrum coexistence between a terrestrial base station (BS) with a hybrid MIMO architecture and LEO satellite uplinks in the upper mid-band. The authors propose a block coordinate descent (BCD) algorithm with projected gradient descent that alternately updates the analog and digital precoders to maximize UE sum-rate while penalizing the beamforming gain toward known satellite directions. The cost function in (9) combines the negative sum-rate and a satellite interference penalty weighted by λ_sat. Simulations using Wireless InSite channels claim that the proposed method reduces the mean satellite INR by 22.4 dB compared to the DFT-Codebook + BD baseline and stays within 2.69% of the HF baseline sum-rate, while violating the -20 dB INR protection threshold in only 0.05% of trials.

Significance. The problem of enabling terrestrial-satellite coexistence in FR3 is timely and relevant, and the proposed approach is a natural extension of existing hybrid precoding techniques with a satellite interference penalty. If the algorithm performs as claimed, the paper would offer a useful engineering contribution. However, the central technical content currently rests on the closed-form gradients in (11) and (12), which are stated without derivation and appear to be incorrect. Because the derivation is omitted and no code is provided, the reported numerical results are not reproducible from the manuscript as written. The simulation methodology itself is standard (Wireless InSite channels, CDF comparisons, multiple baselines), and the authors are explicit about the hand-tuned parameters, which is a strength in transparency but also a limitation in generality.

major comments (4)

[Sec. IV-B, Eqs. (10)-(12) and Algorithm 1] The closed-form gradients (11) and (12) are not gradients of the cost function C in (9), as printed. For the rate term, the gradient of -log(1+SINR_u) has a negative coefficient on the desired-signal derivative and a positive coefficient on the inter-user interference derivative; equations (11) and (12) have the opposite signs. In addition, the auxiliary variable N_{u,j} in (10e) is defined with e_u instead of e_j, so the interference-part term in (11) is not the derivative of any component of (9). Concretely, for U=1 and λ_sat=0, Algorithm 1 updates F_BB in the direction that reduces the received signal power, lowering the UE rate rather than maximizing it. Therefore the results in Figs. 2-5 cannot be reproduced from the manuscript as written. The authors must correct the gradients, supply the full derivation, and ideally provide the code or a reproducible description of the implemented update.
[Sec. IV-B, after Eq. (9)] The statement "Because of a lack of space, we omit the derivation here" is not acceptable for the paper's central algorithmic contribution. The correctness of (11) and (12) is load-bearing: Algorithm 1 explicitly relies on these expressions, and the simulation results depend on their validity. The derivation should be included in the manuscript or in a supplementary document, with the chain rule for complex Wirtinger derivatives spelled out.
[Sec. V-C and Figs. 4-5] The performance improvement is reported for specific hand-tuned values of the penalty parameter λ_sat (5, 7, 9) and a fixed step size α=10^-4, with no sensitivity analysis. Since the INR reduction is a direct consequence of minimizing the penalty term in (9), the reported 22.4 dB improvement is not an independent prediction but a property of the chosen objective. The authors should report how the results vary with λ_sat and α, and justify the selection mechanism, or the claims should be tempered to reflect the dependence on these free parameters.
[Sec. II-A, Eq. (2)] The BS-to-satellite channel is modeled as a single line-of-sight steering vector determined solely by the satellite position, ignoring multipath, satellite antenna pattern/orientation, atmospheric effects beyond a pathloss factor, and ephemeris errors. The authors should discuss the robustness of the nulling performance to deviations from this idealized model, since a mismatch between the assumed steering vector and the actual channel could substantially weaken the interference suppression. At minimum, a sensitivity analysis with angle errors or a discussion of the LOS-only assumption's scope would substantiate the practical claims.

minor comments (6)

[Abstract and Sec. I] The abstract and introduction state that the algorithm "optimizes the precoding within the set of precoders that null the interference to the satellite," but the actual formulation in (9) is a penalty-based method that does not guarantee exact nulling. The wording should be aligned with the mathematical formulation.
[Sec. II-A, Eq. (2)] The steering vector notation a_T(θ,φ) is used for the URA but the angle convention (azimuth and elevation) is only loosely described. Please define the array response explicitly, including the element spacing and the meaning of θ and φ.
[Sec. V-B] The text states that the LEO satellite orbits at an altitude of 100 km, which is unrealistically low for a LEO satellite (typical altitudes are 400-1200 km). This is likely a typo and should be corrected, as it materially affects the channel model and pathloss.
[Fig. 2 and Fig. 3] The curves for "Proposed w/o nulling" and "HF w/o nulling" are described but are difficult to distinguish in the figures due to similar line styles. Please use more distinct markers or colors, and add a legend entry for each baseline in every subplot.
[Algorithm 1, line 12] The update F_RF ← exp(j ∠(F_RF - α∇_{F_RF} C)) is a standard unit-modulus projection, but for complex matrices the notation should be clarified: the exponential and angle are applied entry-wise. Please state this explicitly.
[Sec. V-C] The claim that the proposed method "contributes to reducing harmful interference" is based on a single ITU-style protection threshold of -20 dB INR. The manuscript should clarify whether this threshold is frequency-dependent and whether the conclusions are sensitive to the chosen threshold.

Circularity Check

2 steps flagged · score 6.0 of 10

The 22.4 dB INR improvement is the minimized penalty of the paper's own cost function with a hand-tuned lambdaSat, and the rate-maintenance half of the central claim is not reproducible because Eqs. (11)-(12) as printed have the wrong sign relative to the objective in Eq. (9).

fitted input called prediction [Sec. IV-B, Eq. (9); Sec. V-C (INR definition); Sec. III (penalty parameter lambdaSat)]
"C(FRF,FBB) = - Sigma_u log(1 + SINRu) + lambdaSat tr(F*_BB F*_RF H*_Sat HSat FRF FBB) (9). ... INRi = 10 log10(Pt |h*_i FRF FBB|^2 / (Li sigma_i^2)). ... 'our proposed nulling algorithm achieves within 2.69% of the average HF rate ... while reducing the mean satellite INR by 22.4 dB compared to the DFT-Codebook + BD baseline.'"

The penalty in (9), tr(F*_BB F*_RF H*_Sat HSat FRF FBB) = Sigma_i |h*_i FRF FBB|^2, is, apart from the fixed constants Pt, Li and sigma_i^2, exactly the numerator of the reported INRi metric. The headline 22.4 dB mean-INR improvement is therefore the value of the cost-function term that the algorithm is constructed to minimize, scaled by a hand-tuned lambdaSat (values 5, 7, 9, 10 in Figs. 2-5). Any successful minimization of (9) must lower the reported INR; the reduction is a consequence of the objective definition, not an independent prediction. The only non-constructed element of the central claim, maintaining sum-rate within 2.69%, is the trade-off term, and it is not reproducible from the printed equations (next step).
other [Sec. IV-B, Eqs. (10)-(12), and Algorithm 1 (digital update step)]
"Because of a lack of space, we omit the derivation here. ... nabla_FBB C = 2lambdaSat F*_RF H*_Sat HSat FRF FBB + Sigma_u [ 2/(denu+numu) F*_RF Mu FRF FBB eu eu^T - 2numu/(denu(denu+numu)) Sigma_{j!=u} F*_RF Mu FRF FBB ej ej^T ] (12). ... FBB <- FBB - alpha nabla_FBB C."

Algorithm 1 performs FBB <- FBB - alpha nabla_FBB C. For U=1 and lambdaSat=0 the objective is C = -log(1+SINR1), whose correct conjugate gradient is -F*_RF H*_u wu wu* Hu FRF FBB e1 e1^T/(den1+num1); the printed (12) has the opposite sign, so the documented update ascends C and drives the UE rate down, contradicting Figs. 3 and 5. The auxiliary Nu,j in (10e) uses eu instead of ej, so the cross-user term is not a derivative of any term in (9). Because the derivation is explicitly omitted, the reported sum-rate-maintenance result cannot be produced by the algorithm as documented; what remains derivable from the printed update is only reduction of the satellite penalty, i.e., the INR, which is the objective itself.

full rationale

The manuscript's central claim is a two-part result: a 22.4 dB mean-INR reduction relative to the best hybrid baseline and a sum-rate within 2.69% of the highest hybrid rate. The first part is by construction: the INR metric and the penalty term of the cost function (9) are the same quadratic form, tr(F*_BB F*_RF H*_Sat HSat FRF FBB) = Sigma_i |h*_i FRF FBB|^2, with the reported INRi differing only by fixed pathloss and noise constants, and the penalty weight lambdaSat is hand-tuned (5, 7, 9, 10). Minimizing (9) must reduce that quantity, so reporting the reduction as the headline result recapitulates the objective. The second part, the rate trade-off, is the only element that would make the claim non-circular, but it is not reproducible from the manuscript: Eq. (12) has the wrong sign for the sum-rate derivative (for U=1 it is the negative of the true gradient of C), Eq. (10e) uses eu where ej belongs, and the derivation is omitted ('Because of a lack of space, we omit the derivation here'). As printed, Algorithm 1 would lower UE rate while lowering INR, contradicting the reported results. No load-bearing self-citation is present: Ref. [13] (with two present co-authors) is cited only for a standard hybrid-precoding feasibility constraint and is not an argument in the derivation chain, and Ref. [7] (external) is used only for the LOS satellite channel convention, a stated modeling assumption rather than a smuggled ansatz. No uniqueness theorem is imported. If the gradient signs were corrected and the derivation supplied, the paper would be an ordinary multi-objective optimization study whose INR-vs-baseline comparison is legitimate algorithmic content, warranting a score of 0-2 here; as documented, the central claim reduces to the optimized penalty, with the informative half left unverifiable, giving a partial circularity score of 6.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The algorithm relies on several modeling assumptions (LOS satellite channel, perfect CSI, narrowband) and on hand-tuned hyperparameters (lambda_sat, step size, iterations). No new physical entities are introduced. The most fragile element is the LOS-only satellite channel model, since nulling performance is only as good as the channel estimate.

free parameters (3)

lambda_sat = 5, 7, 9 (in Figs. 4-5); also reported with lambda_sat=10 in Figs. 2-3
Penalty weight for satellite interference in cost function (7a)/(9). Hand-tuned; no principled selection or sensitivity threshold. Performance results depend on its value.
step size alpha = 1e-4
Fixed PGD step size, stated as hand-tuned common practice in Sec. IV-B. No line search or convergence guarantee.
iteration counts (iterBB, iterRF, T) = 5, 5, 20
Inner and outer BCD iterations chosen in Sec. V; no convergence analysis or stopping criterion.

assumptions (7)

domain assumption BS has perfect knowledge of all UE downlink channels
Stated in Sec. II-A; no estimation error or feedback overhead considered.
domain assumption BS-to-satellite channel is a single LOS steering vector h_i = a_T(theta_tilde_i, phi_tilde_i) from ephemeris data
Sec. II-A, Eq. (2). Ignores multipath, satellite antenna pattern, orientation, and ephemeris errors. This is the load-bearing premise for nulling.
domain assumption Narrowband channel model for both UE and satellite links
Sec. II-A; no frequency selectivity or time variation within a block.
domain assumption No coordination between BS and LEO satellite, and satellite antenna geometry is unknown
Sec. II-A; used to justify steering-vector-only satellite channel.
domain assumption Interference from satellite ground stations is neglected in UE SINR
Sec. II-D; justified by high directionality of satellite ground station antennas.
ad hoc to paper Closed-form gradients in (11) and (12) are correct
Sec. IV-B omits the derivation due to lack of space; the algorithm and reported results depend on these gradients.
domain assumption Projected gradient descent with fixed step size converges to a useful solution
Sec. IV-B/IV-C; no convergence proof provided, step size hand-tuned.

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Cite this review

Pith. "Pith review of Interference-Aware Multiuser Hybrid Precoding for Coexistence with LEO Satellite Communication." pith.science (2026). https://pith.science/paper/M7KIVQVU

@misc{pith2026250603438,
  author       = {Pith},
  title        = {Pith review of: Interference-Aware Multiuser Hybrid Precoding for Coexistence with LEO Satellite Communication},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7KIVQVU}},
  note         = {Machine review of arXiv:2506.03438}
}

read the original abstract

Interference from terrestrial networks can reduce the communication rate for low Earth orbit (LEO) satellites in the upper mid-band. To coexist in frequency, MIMO precoding can be used to reduce the signal that impinges on the LEO satellite. We present a beamforming algorithm designed for the hybrid architecture that incorporates a satellite interference penalty while optimizing the analog and digital precoders. Our algorithm optimizes the precoding at the base station (BS) within the set of precoders that null the interference to the satellite. Simulations demonstrate that our algorithm reduces the interference at the satellites and lowers the probability of violating prescribed LEO satellite protection thresholds, outperforming prior hybrid nulling algorithms. Results indicate that the algorithm maintains sum-rate within 3\% of the existing hybrid solutions, while effectively improving interference to noise power by 22.4 dB.

Figures

Figures reproduced from arXiv: 2506.03438 by the authors.

**Figure 2.** CDF of satellite INR for 2 UE, 8x8 BS array with 8 RF chai [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** CDF of UE Sum-Rate for 2 UEs, 8x8 BS array with 8 RF chain [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: BS array transmit power and the resulting satellite I [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: BS Array transmit power and the resulting UE sum-rate [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: Example Wireless Insite channel simulation from 1 LE [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

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Reference graph

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F. Ortiz, E. Lagunas, A. Saifaldawla, M. Jalali, L. Emili ani, and S. Chatzinotas, “Emerging NGSO constellations: Spectral c oexistence with gso satellite communication systems,” 2024. [Online] . Available: https://arxiv.org/abs/2404.12651

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