Pith. sign in

REVIEW 3 major objections 6 minor 46 references

Massive MIMO-OFDM Channel Acquisition with Time-Frequency Phase-Shifted Pilots

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Time-frequency phase-shifted pilots reduce massive MIMO channel estimation error by over 8 dB while serving more users with the same pilot overhead.

desk verdict Interesting pilot design and plausible empirical gains, but Theorem 1's proof is wrong for the oversampled DFT dictionaries used in the simulations. read the letter →

arxiv 2505.04933 v1 pith:YRS7BI25 submitted 2025-05-08 cs.IT eess.SPmath.IT

Jinke Tang , Xiqi Gao , Li You , Ding Shi , Jiyuan Yang , Xiang-Gen Xia , Xinwei Zhao , Peigang Jiang This is my paper · ORCID
classification cs.ITeess.SPmath.IT
keywords massiveMIMO-OFDMchannelestimationtriple-beamtensortime-frequencyphase-shiftedpilotspilotschedulinginformationgeometryinter-userinterference
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 channel acquisition method for massive MIMO-OFDM that lets many user terminals share the same pilot resource without a proportional loss in accuracy. The key move is to shift pilots in both frequency and time, not just frequency, and to model the channel as a sparse tensor over spatial, delay, and Doppler beams. The authors show that when each user's equivalent triple-beam power distribution does not overlap any other's, inter-user pilot interference disappears and the MMSE lower bound is reached. Simulations with 300 users report normalized mean-square error more than 8 dB lower than the adjustable phase-shifted pilot baseline, along with a lower-complexity estimator.

What carries the argument

The triple-beam channel tensor model: the space-frequency-time channel is written as $\mathbf{H}^{\mathrm{SFT}}_{u,T} = \mathcal{V}_T *_3 \mathcal{H}^{\mathrm{TB}}_u$, a product of DFT-structured beam matrices with a sparse tensor whose axes are spatial, delay, and Doppler beams. Phase-shifted pilots act as tensor cyclic shifts on $\mathcal{H}^{\mathrm{TB}}_u$, so scheduling pilots becomes the combinatorial task of choosing shifts that make the equivalent power tensors $\mathcal{W}_{u,L_{\varphi_u},L_{\phi_u}}$ pairwise non-overlapping. The information geometry estimator projects the posterior onto a tractable Gaussian manifold, and the DFT structure of the beam matrices lets each projection step run through fast Fourier transforms.

What would settle it

Give each UT a channel with off-grid delay and Doppler values that do not sit on the discretized triple-beam grid, estimate $\mathcal{W}_u$ from finite samples instead of assuming it known, then run the TFPSP scheduler and tensor IGA at 20 dB SNR with 300 UTs. If the pairwise overlap measure $\eta(\cdot,\cdot)$ of the scheduled equivalent power tensors is strictly positive for most pairs and the NMSE advantage over APSP-IGA drops below 8 dB, the central claim as stated fails in that regime.

Watch

Extended reading notes

Core claim

The central claim is that time-frequency phase-shifted pilots (TFPSPs) convert pilot separation into a tensor cyclic shift in the triple-beam domain, so two users stop interfering exactly when their shifted power tensors $\mathcal{W}_{u,L_{\varphi_u},L_{\phi_u}}$ have disjoint supports. Theorem 1 states that as $M, K, N_p \to \infty$, this non-overlap condition achieves the MMSE lower bound for channel estimation without increasing pilot overhead. The paper also claims that the tensor-based information geometry estimator reaches near-MMSE accuracy with much lower complexity than direct tensor inversion, and that in high-user-count simulations the TFPSP pipeline outperforms the prior APSP-IGA approach by more than 8 dB in NMSE at 20 dB SNR.

Load-bearing premise

The load-bearing premise is that the base station knows every user's triple-beam power tensor $\mathcal{W}_u$ exactly and that each channel is exactly sparse on the discretized angle-delay-Doppler grid, so the pairwise non-overlap condition can be certified; if the covariance is estimated with error or energy leaks off the grid, the interference suppression and the 8 dB gain can shrink or vanish.

Editorial extensions

If this is right

  • A base station can serve many more UTs with the same pilot overhead, because each UT uses two phase-shift indices instead of one and only needs its sparse triple-beam support to avoid others.
  • The TFPSP scheduling criterion, pairwise zero Hadamard product of shifted power tensors, gives a direct and checkable design rule for pilot assignment.
  • The tensor-based IGA converges to the MMSE estimate at its fixed point, and its FFT-based implementation lowers complexity so a 300-UT system can be estimated in roughly a quarter of the runtime of GAMP or EPV.
  • The estimated triple-beam tensor directly supports channel prediction over the data segment of the slot, with growing benefit as UT speed increases from 30 to 60 km/h.

Reading between the lines

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

  • If $\mathcal{W}_u$ is estimated rather than known exactly, the non-overlap condition cannot be certified; a natural robustness test is to feed noisy covariance estimates into the scheduler and observe when the NMSE gain falls below 8 dB.
  • The same two-dimensional phase-shift idea may transfer to delay-Doppler grids such as OTFS, where fractional off-grid Doppler leakage would stress the grid-sparsity assumption.
  • The threshold $\gamma$ in the DSatur grouping trades estimation performance for scheduling complexity, so an adaptive threshold based on the current UT population could improve pilot reuse without rerunning the full graph coloring.
  • Theorem 1 is asymptotic; finite arrays will retain residual interference from sinc-like leakage, so the practical reach of the claim depends on how fast the DFT factors $\alpha_M(\cdot)\alpha_K(\cdot)\alpha_{N_p}(\cdot)$ decay at finite $M$, $K$, and $N_p$.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper proposes a complete uplink channel acquisition pipeline for massive MIMO-OFDM. Section II-C introduces a triple-beam (TB) tensor channel model that represents the space-frequency-time channel as beam matrices multiplying a sparse TB-domain channel tensor. Section III proposes time-frequency phase-shifted pilots (TFPSPs), derives an 'optimal' interference-free condition (Theorem 1, condition (40)) under which the MMSE lower bound (39) is claimed to be achieved, and gives a DSatur-based pilot scheduling algorithm (Algorithm 1). Section IV extends the authors' earlier information geometry approach (IGA) to tensors for estimating the TB channel, with an FFT-accelerated low-complexity implementation and channel prediction. Section V reports QuaDRiGa simulations showing large NMSE gains over APSP-based channel acquisition (more than 8 dB at 300 UTs and SNR = 20 dB), faster convergence than GAMP and EPV, and lower complexity.

Significance. The high-level idea is strong and timely: exploiting joint angular, delay, and Doppler sparsity through a two-dimensional pilot phase-shift schedule to serve many UTs with fixed pilot overhead is well motivated, and the reported gains are substantial. The paper is clearly written, the tensor machinery is developed carefully, and the simulation study is reasonably broad (48 and 300 UTs, two speeds, three estimation baselines, complexity and convergence comparisons). The FFT-based simplification of the tensor IGA is a genuine algorithmic contribution, and the use of a physically motivated QuaDRiGa channel model rather than synthetic sparse channels strengthens the empirical claims. The estimator is inherited from the authors' prior work [9], [25], [26] rather than fitted to the data in this paper, which removes any circularity concern. However, the theoretical anchor of the pilot design, Theorem 1, is not correct for the oversampled DFT dictionaries used throughout the simulations (see Major Comment 1), so the 'optimal' terminology overstates what is proven; the headline practical claims rest on the simulations alone.

major comments (3)
  1. [Appendix A, Eqs. (63)-(65); Section III-B (Theorem 1)] The proof of sufficiency of condition (40) is incorrect for the oversampled beam dictionaries used in all the simulations, and the counterexample below shows that the claimed sufficiency fails in that regime. The proof states that α_A(x) → δ(x) as A → ∞ and concludes that whenever any of p1 ≠ q1, p2 ≠ q2, p3 ≠ q3 holds, at least one of α_M(·), α_K(·), α_Np(·) in (65) vanishes in the limit. With Nϑ = FϑM, Nτ = FτK, Nν = FνNp (Section II-C) and Fϑ, Fτ, Fν ≥ 1, this is false: α_A(x) = Σ_{a=0}^{A−1} exp(−j2πax/A) is an unnormalized Dirichlet kernel, and for fixed nonzero x one has |α_A(x)| = O(A), not o(1); only the normalized kernel α_A(x)/A converges to a periodic delta train in distribution. The orthogonality the proof requires holds only in the critically sampled case Fϑ = Fτ = Fν = 1. Explicit counterexample in the paper's F = 2 setting: let Fϑ = Fτ = Fν = 2, take two users with single non-zero TB entries Wu = δ_{(0,0,0)} and Wu′ = δ_{(1,1,1)} and identical (zero) phase shifts. Condition (40) holds because the supports are disjoint, yet the cross-covariance in (36)-(32) is the rank-one tensor with entries [V^p_T]_{a,(1,1,1)}[V^p_T]^∗_{b,(1,1,1)}, of unit magnitude and independent of M, K, Np; equivalently, the kernel product in (65) equals α_M(1/2)α_K(1/2)α_Np(1/2), whose magnitude is O(MKNp), not zero. Hence the assertion lim_{M,K,Np→∞} Q_{u,u′} = 0 in Appendix A fails, C_{u,all} ≠ C_u, and the claimed achievability of εMSE,min is not established. Because Algorithm 1 schedules against condition (40) and the abstract advertises an 'optimal TFPSP design', this is a load-bearing error. Since Figs. 4-9 use F = 2 and F = 4, and Fig. 4 shows that the F = 2 gain over F = 1 is the main source of the reported performance, the theorem cannot be rescued by falling back to F = 1 without giving up the paper's central performance claims. Please either restrict Theorem 1 to critically sampled dictionaries (where the claim holds exactly for all finite M, K, Np), or replace the sufficiency proof with a quantitative analysis of the residual interference for redundant beam dictionaries.
  2. [Section II-C (last paragraph); Sections III-C and IV-A] The 'optimal' claim is conditional on ideal statistics and exact grid matching, but the paper never analyzes the gap between the ideal setting and the simulation setting. The analysis assumes that Wu of all UTs is available to the BS (end of Section II-C) and that every physical path is exactly on the discretized TB grid (the approximation in (12)); in the simulations, by contrast, Wu is estimated from data via the method of [46] and the QuaDRiGa channels have continuous angles, delays, and Doppler shifts, so measured power leaks off-grid and the empirical Wu is not sparse in the exact sense used to define Snz and condition (40). Under mismatch, condition (40) can be neither certified nor verified, and Theorem 1 provides no finite-system or mismatch-robust guarantee. The threshold γ in Algorithm 1 is also a free parameter with no recommended value or sensitivity study. I recommend adding a robustness study (imperfect Wu, off-grid channels) and softening the 'optimal' terminology in the abstract and Section III-B unless it is explicitly restricted to the idealized case.
  3. [Section IV-B and Fig. 10] The low-complexity claim is not fully substantiated. The O(C̄1) figure is stated for a single application of A∗3B or AH_3∗3C ((57)-(58)), but the total cost of Algorithm 2 depends on the number of iterations, which is governed by the unspecified damping factor α and is only reported empirically (300 iterations); no bound or convergence-rate analysis is given for the tensor recursion (56). The convergence results cited from [9], [25], [26] are for the (simplified) IGA of those papers, not for the coupled D/F update in (56), so the paper should either prove convergence of (56) or state explicitly that convergence is inherited only at the level of the prior analyses. In addition, the complexity comparison of Fig. 10 does not include the cost of Algorithm 1 or the acquisition of Wu, although both are part of the proposed acquisition pipeline, and the runtime figures are reported without platform details. Please provide a formal per-iteration and total complexity statement and declare the settings of α and γ used in the simulations.
minor comments (6)
  1. [Appendix A, Eq. (64)] The first Dirichlet factor is written as α_M((p1−q1)/Fν), but consistency with Eq. (65) and with the exponent e^{−j2πr1(p1−q1)/Nϑ} requires Fϑ.
  2. [Eq. (24) and Algorithm 1] The symbol γ is used both for the integer coprime to Np in the ZC sequence (24) and for the scheduling overlap threshold in Algorithm 1; please rename one of them.
  3. [Algorithm 1, step 15] The assignment 'φUi′ = φ' presumably should read 'φUi = φ'.
  4. [Section V] The text 'as shown in Fig. V' should refer to Fig. 9, and the legend of Fig. 6 lists 'VEP' where 'EPV' is meant.
  5. [Throughout] Typographical issues: 'implys' after Theorem 1; 'has an affect on' in Section III-B; 'acquition' in the Fig. 4 caption; 'normalized factor' and 'iteraction terms' in Section IV-A; 'seperation' and 'accomodated' in Section III.
  6. [Figs. 5 and 8] The paper would benefit from including a genie-aided reference curve corresponding to εMSE,min in (39), so that the reader can calibrate how close the heuristic scheduling (with its free threshold γ) comes to the claimed bound.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: optimal pilot condition and simulations are self-contained; Appendix A delta-limit gap is a correctness issue, not a circular reduction.

full rationale

After walking the derivation chain, I find no circular reduction. The optimal TFPSP condition (40) is derived from the MMSE expression (38) and is stated in terms of the power tensors Wu, which are assumed inputs, not parameters fitted to the NMSE target. The scheduling algorithm (Algorithm 1) uses the same Wu, and the simulations obtain Wu with an external GAMP/SBL method [46] on QuaDRiGa channels, so the reported >8 dB gain is an external-benchmark comparison, not the paper's own estimator generating its target. The tensor-based IGA does import convergence and fixed-point optimality from the authors' prior work ([9, Thm. 2], [25, Thms. 1-2], [26]), but those are published external results used as algorithmic support, and the core TFPSP/interference-suppression claim is independently evaluated against APSP-IGA, GAMP, and EPV. The one substantive flaw is in Appendix A: the proof of Theorem 1 asserts alpha_A(x)->delta(x) as A->infinity for the unnormalized Dirichlet kernel, which is false for the oversampled (F>1) beam dictionaries used in Figs. 4-9; e.g., with F=2 and index difference 1 in all three dimensions, the product in Eq. (65) is O(MKNp), not 0. Consequently Eq. (65) does not force Qu,u' to vanish under condition (40). This is a mathematical correctness risk in the asymptotic optimality claim, not a circularity: no fitted parameter is renamed as a prediction and no conclusion is identical to an input by construction. Score 1 reflects the absence of circularity, with the proof gap left to a correctness review.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on strong statistical priors (known Wu, exact grid sparsity) and on asymptotic arguments. The paper introduces no new physical entity; the triple-beam tensor and TFPSPs are modeling and signaling constructs, not newly postulated physical objects.

free parameters (3)
  • Fine factors Fϑ, Fτ, Fν = F=2 (main results)
    Grid oversampling factors chosen in Section V; F=2 is used for Figs. 5-10 because it gives large gains over F=1 while F=4 gives only marginal improvement. The reported performance depends on this choice.
  • Scheduling threshold γ = not stated
    Algorithm 1 requires a threshold γ for deciding when UTs can reuse a pilot; no value is given in the simulation setup, so results may depend on unreported tuning.
  • Damping factor α = not stated
    The tensor IGA update in Algorithm 2 uses a damping factor α; the simulation section does not report its value, and convergence and performance depend on it.
assumptions (5)
  • domain assumption Channel is sparse on the discretized triple-beam grid; true angle, delay, and Doppler parameters are approximated by nearest grid points without significant error.
    Equations (8a)-(8c) and (12)-(14) quantize continuous parameters to the grid; the sparsity support is then used for pilot scheduling and as the IGA prior.
  • domain assumption The BS has perfect knowledge of the statistical power tensor Wu for every UT.
    Stated at the end of Section II-C: 'we assume that Wu of all the UTs are available at the BS in the rest of this paper.' Both Algorithm 1 and the IGA estimator require Wu.
  • standard math M, K, and Np are asymptotically large so that sums of complex exponentials behave as delta functions in Theorem 1.
    The optimality proof in Appendix A uses αA(x)→δ(x) as A→∞; finite-system performance relies on approximate separation rather than exact orthogonality.
  • domain assumption The tensor-based IGA converges to a fixed point whose expectation equals the MMSE estimate.
    The paper relies on [9, Theorem 2] and [26] for convergence and optimality; the proof is not reproduced here, so the central estimator inherits this previously established result.
  • domain assumption Channel coefficients in different beams are independent circular-symmetric complex Gaussian with known variances.
    Assumed in Section II-C via (19) and the surrounding text, citing [33], [35]; this underlies the Gaussian prior in the IGA and the MMSE formulas.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Massive MIMO-OFDM Channel Acquisition with Time-Frequency Phase-Shifted Pilots." pith.science (2026). https://pith.science/paper/YRS7BI25

@misc{pith2026250504933,
  author       = {Pith},
  title        = {Pith review of: Massive MIMO-OFDM Channel Acquisition with Time-Frequency Phase-Shifted Pilots},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YRS7BI25}},
  note         = {Machine review of arXiv:2505.04933}
}
read the original abstract

In this paper, we propose a channel acquisition approach with time-frequency phase-shifted pilots (TFPSPs) for massive multi-input multi-output orthogonal frequency division multiplexing (MIMO-OFDM) systems. We first present a triple-beam (TB) based channel tensor model, allowing for the representation of the space-frequency-time (SFT) domain channel as the product of beam matrices and the TB domain channel tensor. By leveraging the specific characteristics of TB domain channels, we develop TFPSPs, where distinct pilot signals are simultaneously transmitted in the frequency and time domains. Then, we present the optimal TFPSP design and provide the corresponding pilot scheduling algorithm. Further, we propose a tensor-based information geometry approach (IGA) to estimate the TB domain channel tensors. Leveraging the specific structure of beam matrices and the properties of TFPSPs, we propose a low-complexity implementation of the tensor-based IGA. We validate the efficiency of our proposed channel acquisition approach through extensive simulations. Simulation results demonstrate the superior performance of our approach. The proposed approach can effectively suppress inter-UT interference with low complexity and limited pilot overhead, thereby enhancing channel estimation performance. Particularly in scenarios with a large number of UTs, the channel acquisition method outperforms existing approaches by reducing the normalized mean square error (NMSE) by more than 8 dB.

Figures

Figures reproduced from arXiv: 2505.04933 by the authors.

Figure 1
Figure 1. Frame structure for massive MIMO-OFDM transmission [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Pilots transmitted in the current frame at the time sl [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Flowchart of the channel estimation framework [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: NMSE of the proposed channel acquition approach unde [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: NMSE of different channel acquisition approaches ve [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Convergence performance versus the number of iterat [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 8
Figure 8. Figure 8: NMSE of different channel acquisition approaches ve [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: NMSE of different channel acquisition approaches ve [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [9]

    Channel estimation for massive MIMO: An information geome try approach,

    J. Y ang, A.-A. Lu, Y . Chen, X. Q. Gao, X.-G. Xia, and D. T. M. Slock, “Channel estimation for massive MIMO: An information geome try approach,” IEEE Trans. Signal Process. , vol. 70, pp. 4820–4834, 2022. 15

    work page 2022

  2. [25]

    Efficient Information Geometry Approach for Massive MIMO-OFDM Channel Estimation

    J. Y ang, Y . Chen, A.-A. Lu, W. Zhong, X. Q. Gao, X. H. Y ou, X .- G. Xia, and D. Slock, “Simplified information geometry appro ach for massive MIMO-OFDM channel estimation–Part I: Algorithm an d fixed point analysis,” arXiv preprint arXiv:2401.02035 , 2024

    work page Pith review arXiv 2024

  3. [26]

    Simplified Information Geometry Approach for Massive MIMO-OFDM Channel Estimation -- Part II: Convergence Analysis

    J. Y ang, Y . Chen, M. Fan, X. Q. Gao, X.-G. Xia, and D. Slock , “Simplified information geometry approach for massive MIMO -OFDM channel estimation–Part II: Convergence analysis,” arXiv preprint arXiv:2401.02037, 2024

    work page Pith review arXiv 2024

  4. [46]

    A GAMP-bas ed low complexity sparse Bayesian learning algorithm,

    M. Al-Shoukairi, P . Schniter, and B. D. Rao, “A GAMP-bas ed low complexity sparse Bayesian learning algorithm,” IEEE Trans. Signal Process., vol. 66, no. 2, pp. 294–308, Jan. 2018

    work page 2018

  5. [1]

    Time-frequency phase-shifted pilots for mas sive MIMO- OFDM channel estimation,

    J. Tang, X. Q. Gao, L. Y ou, D. Shi, J. Y ang, X.-G. Xia, X. Zha o, and P . Jiang, “Time-frequency phase-shifted pilots for mas sive MIMO- OFDM channel estimation,” in Proc. IEEE GLOBECOM , Cape Town, South Africa, Dec. 2024, pp. 974–979

    work page 2024

  6. [2]

    On the road to 6G: Visions, requirements, key technologies, and testbeds,

    C.-X. Wang, X. H. Y ou, X. Q. Gao et al. , “On the road to 6G: Visions, requirements, key technologies, and testbeds,” IEEE Commun. Surveys Tuts., vol. 25, no. 2, pp. 905–974, 2nd Quart 2023

    work page 2023

  7. [3]

    Towards 6G wireless com- munication networks: Vision, enabling technologies, and n ew paradigm shifts,

    X. H. Y ou, C.-X. Wang, J. Huang, X. Q. Gao, Z. Zhang, M. Wang , Y . Huang, C. Zhang, Y . Jiang, J. Wanget al., “Towards 6G wireless com- munication networks: Vision, enabling technologies, and n ew paradigm shifts,” Sci. China Inf. Sci. , vol. 64, pp. 1–74, Jan. 2021

    work page 2021

  8. [4]

    6G wireless channel measurements and models: Trends and chall enges,

    C.-X. Wang, J. Huang, H. Wang, X. Q. Gao, X. H. Y ou, and Y . Ha o, “6G wireless channel measurements and models: Trends and chall enges,” IEEE V eh. Technol. Mag., vol. 15, no. 4, pp. 22–32, Dec. 2020

    work page 2020

Show all 46 references
  1. [5]

    5G-advanced toward 6G: Past, present, and future,

    W. Chen, X. Lin, J. Lee, A. Toskala, S. Sun, C. F. Chiasseri ni, and L. Liu, “5G-advanced toward 6G: Past, present, and future,” IEEE J. Sel. Areas Commun. , vol. 41, no. 6, pp. 1592–1619, Jun. 2023

    2023

  2. [6]

    Tra ining signal design and tradeoffs for spectrally-efficient multi-user M IMO-OFDM systems,

    Y . Chi, A. Gomaa, N. Al-Dhahir, and A. R. Calderbank, “Tra ining signal design and tradeoffs for spectrally-efficient multi-user M IMO-OFDM systems,” IEEE Trans. Wireless Commun. , vol. 10, no. 7, pp. 2234– 2245, Jul. 2011

    2011

  3. [7]

    Spars e channel estimation via hierarchical hybrid message passing for mas sive MIMO- OFDM systems,

    X. Liu, W. Wang, X. Song, X. Q. Gao, and G. Fettweis, “Spars e channel estimation via hierarchical hybrid message passing for mas sive MIMO- OFDM systems,” IEEE Trans. Wireless Commun. , vol. 20, no. 11, pp. 7118–7134, Nov 2021

    2021

  4. [8]

    St ructured hybrid message passing based channel estimation for massiv e MIMO- OFDM systems,

    X. Liu, W. Wang, X. Gong, X. Fu, X. Q. Gao, and X.-G. Xia, “St ructured hybrid message passing based channel estimation for massiv e MIMO- OFDM systems,” IEEE Trans. V eh. Technol., vol. 72, no. 6, pp. 7491– 7507, June 2023

    2023

  5. [10]

    Low- overhead hierarchically-sparse channel estimation for mu ltiuser wide- band massive MIMO,

    G. Wunder, S. Stefanatos, A. Flinth, I. Roth, and G. Cair e, “Low- overhead hierarchically-sparse channel estimation for mu ltiuser wide- band massive MIMO,” IEEE Trans. Wireless Commun. , vol. 18, no. 4, pp. 2186–2199, Apr. 2019

    2019

  6. [11]

    Pilot reu se for massive MIMO transmission over spatially correlated rayle igh fading channels,

    L. Y ou, X. Q. Gao, X.-G. Xia, N. Ma, and Y . Peng, “Pilot reu se for massive MIMO transmission over spatially correlated rayle igh fading channels,” IEEE Trans. Wireless Commun. , vol. 14, no. 6, pp. 3352– 3366, Jun. 2015

    2015

  7. [12]

    Noncooperative cellular wireless wit h unlimited num- bers of base station antennas,

    T. L. Marzetta, “Noncooperative cellular wireless wit h unlimited num- bers of base station antennas,” IEEE Trans. Wireless Commun. , vol. 9, no. 11, pp. 3590–3600, Nov. 2010

    2010

  8. [13]

    Chan nel ac- quisition for massive MIMO-OFDM with adjustable phase shif t pilots,

    L. Y ou, X. Q. Gao, A. L. Swindlehurst, and W. Zhong, “Chan nel ac- quisition for massive MIMO-OFDM with adjustable phase shif t pilots,” IEEE Trans. Signal Process. , vol. 64, no. 6, pp. 1461–1476, Mar. 2016

    2016

  9. [14]

    Chann el estimation and robust detection for IQ imbalanced uplink ma ssive MIMO-OFDM with adjustable phase shift pilots,

    Y . Chen, L. Y ou, A.-A. Lu, X. Q. Gao, and X.-G. Xia, “Chann el estimation and robust detection for IQ imbalanced uplink ma ssive MIMO-OFDM with adjustable phase shift pilots,” IEEE Access , vol. 9, pp. 35 864–35 878, 2021

    2021

  10. [15]

    Efficient c oordinated recovery of sparse channels in massive MIMO,

    M. Masood, L. H. Afify, and T. Y . Al-Naffouri, “Efficient c oordinated recovery of sparse channels in massive MIMO,” IEEE Trans. Signal Process., vol. 63, no. 1, pp. 104–118, Jan. 2015

    2015

  11. [16]

    Pilot reuse for vehicle-to-vehicle underlay massive MIMO transmis- sion,

    L. Y ou, M. Xiao, X. Song, Y . Liu, W. Wang, X. Q. Gao, and G. F ettweis, “Pilot reuse for vehicle-to-vehicle underlay massive MIMO transmis- sion,” IEEE Trans. V eh. Technol., vol. 69, no. 5, pp. 5693–5697, May. 2020

    2020

  12. [17]

    P . J. Schreier and L. L. Scharf, Statistical Signal Processing of Complex- valued Data: The Theory of Improper and Noncircular Signals . Cam- bridge Univ. press, Cambridge, UK, 2010

    2010

  13. [18]

    A coordinat ed approach to channel estimation in large-scale multiple-antenna sys tems,

    H. Yin, D. Gesbert, M. Filippou, and Y . Liu, “A coordinat ed approach to channel estimation in large-scale multiple-antenna sys tems,” IEEE J. Sel. Areas Commun. , vol. 31, no. 2, pp. 264–273, Feb. 2013

    2013

  14. [19]

    Channel acquisition for HF skywave massive MIMO-OFDM comm uni- cations,

    D. Shi, L. Song, W. Zhou, X. Q. Gao, C.-X. Wang, and G. Y e Li , “Channel acquisition for HF skywave massive MIMO-OFDM comm uni- cations,” IEEE Trans. Wireless Commun., vol. 22, no. 6, pp. 4074–4089, Jun. 2023

    2023

  15. [20]

    Inform ation geometry approach for ultra-massive MIMO signal detection ,

    Y . Chen, J. Y ang, X. Gao, D. Slock, and X.-G. Xia, “Inform ation geometry approach for ultra-massive MIMO signal detection ,” in Proc. Int. Conf. Wireless Commun. Signal Process. (WCSP) , Nov 2023, pp. 1–6

    2023

  16. [21]

    Constructing fre e-energy approx- imations and generalized belief propagation algorithms,

    J. Y edidia, W. Freeman, and Y . Weiss, “Constructing fre e-energy approx- imations and generalized belief propagation algorithms,” IEEE Trans. Inf. Theory , vol. 51, no. 7, pp. 2282–2312, Jul. 2005

    2005

  17. [22]

    Message-pas sing algo- rithms for compressed sensing,

    D. L. Donoho, A. Maleki, and A. Montanari, “Message-pas sing algo- rithms for compressed sensing,” Proc. Nat. Acad. Sci. USA , vol. 106, no. 45, pp. 18 914–18 919, Oct. 2009

    2009

  18. [23]

    Generalized approximate message passing f or estimation with random linear mixing,

    S. Rangan, “Generalized approximate message passing f or estimation with random linear mixing,” in Proc. IEEE ISIT , St. Petersburg, Russia„ Jul. 2011, pp. 2168–2172

    2011

  19. [24]

    Un ifying message passing algorithms under the framework of constrai ned bethe free energy minimization,

    D. Zhang, X. Song, W. Wang, G. Fettweis, and X. Q. Gao, “Un ifying message passing algorithms under the framework of constrai ned bethe free energy minimization,” IEEE Trans. Wireless Commun. , vol. 20, no. 7, pp. 4144–4158, Jul. 2021

    2021

  20. [27]

    Amari, Information Geometry and Its Applications

    S.-I. Amari, Information Geometry and Its Applications. Springer, 2016, vol. 194

    2016

  21. [28]

    Channel es timation for orthogonal time frequency space (OTFS) massive MIMO,

    W. Shen, L. Dai, J. An, P . Fan, and R. W. Heath, “Channel es timation for orthogonal time frequency space (OTFS) massive MIMO,” IEEE Trans. Signal Process. , vol. 67, no. 16, pp. 4204–4217, Aug. 2019

    2019

  22. [29]

    Deterministic pilot design and channel estimation for dow nlink massive MIMO-OTFS systems in presence of the fractional doppler,

    D. Shi, W. Wang, L. Y ou, X. Song, Y . Hong, X. Q. Gao, and G. F ettweis, “Deterministic pilot design and channel estimation for dow nlink massive MIMO-OTFS systems in presence of the fractional doppler,” IEEE Trans. Wireless Commun. , vol. 20, no. 11, pp. 7151–7165, Nov. 2021

    2021

  23. [30]

    A tensor framework for multi-lin ear complex MMSE estimation,

    D. Pandey and H. Leib, “A tensor framework for multi-lin ear complex MMSE estimation,” IEEE Open J. Signal Process. , vol. 2, pp. 336–358, 2021

    2021

  24. [31]

    Solving mul tilinear systems via tensor inversion,

    M. Brazell, N. Li, C. Navasca, and C. Tamon, “Solving mul tilinear systems via tensor inversion,” SIAM J. Matrix Anal. Appl. , vol. 34, no. 2, pp. 542–570, 2013

    2013

  25. [32]

    Tensor inversio n and its application to the tensor equations with Einstein product,

    M.-L. Liang, B. Zheng, and R.-J. Zhao, “Tensor inversio n and its application to the tensor equations with Einstein product, ” Linear and Multilinear Algebra , vol. 67, no. 4, pp. 843–870, 2019

    2019

  26. [33]

    First- and second-order characterization of direction disper- sion and space selectivity in the radio channel,

    B. Fleury, “First- and second-order characterization of direction disper- sion and space selectivity in the radio channel,” IEEE Trans. Inf. Theory, vol. 46, no. 6, pp. 2027–2044, Sep. 2000

    2027

  27. [34]

    Channel estimation f or LEO satellite massive MIMO-OFDM communications,

    K.-X. Li, X. Q. Gao, and X.-G. Xia, “Channel estimation f or LEO satellite massive MIMO-OFDM communications,” IEEE Trans. Wireless Commun., vol. 22, no. 11, pp. 7537–7550, Nov. 2023

    2023

  28. [35]

    3D MIMO-OFDM channel estimation,

    G. Auer, “3D MIMO-OFDM channel estimation,” IEEE Trans. Com- mun., vol. 60, no. 4, pp. 972–985, Apr. 2012

    2012

  29. [36]

    Polyphase codes with good periodic correlatio n properties (corresp.),

    D. Chu, “Polyphase codes with good periodic correlatio n properties (corresp.),” IEEE Trans. Inf. Theory , vol. 18, no. 4, pp. 531–532, Jul. 1972

    1972

  30. [37]

    Dahlman, S

    E. Dahlman, S. Parkvall, and J. Skold, 5G NR: The Next Generation Wireless Access Technology . Academic Press, New Y ork, NY , USA, 2020

    2020

  31. [38]

    Optimal training de sign for MIMO OFDM systems in mobile wireless channels,

    I. Barhumi, G. Leus, and M. Moonen, “Optimal training de sign for MIMO OFDM systems in mobile wireless channels,” IEEE Trans. Signal Process., vol. 51, no. 6, pp. 1615–1624, Jun. 2003

    2003

  32. [39]

    Low complexity modem structure for OFDM-based orthog- onal time frequency space modulation,

    A. Farhang, A. RezazadehReyhani, L. E. Doyle, and B. Far hang- Boroujeny, “Low complexity modem structure for OFDM-based orthog- onal time frequency space modulation,” IEEE Trans. Wireless Com- mun.L, vol. 7, no. 3, pp. 344–347, Jun. 2018

    2018

  33. [40]

    Multi-domain com munication systems and networks: A tensor-based approach,

    D. Pandey, A. V enugopal, and H. Leib, “Multi-domain com munication systems and networks: A tensor-based approach,” Network, vol. 1, no. 2, pp. 50–74, 2021

    2021

  34. [41]

    Weighted-Graph-Co loring- Based pilot decontamination for multicell massive MIMO sys tems,

    X. Zhu, L. Dai, Z. Wang, and X. Wang, “Weighted-Graph-Co loring- Based pilot decontamination for multicell massive MIMO sys tems,” IEEE Trans. V eh. Technol., vol. 66, no. 3, pp. 2829–2834, Mar. 2017

    2017

  35. [42]

    Tensor decomposition for signal pr ocessing and machine learning,

    N. D. Sidiropoulos, L. De Lathauwer, X. Fu, K. Huang, E. E . Papalex- akis, and C. Faloutsos, “Tensor decomposition for signal pr ocessing and machine learning,” IEEE Trans. Signal Process. , vol. 65, no. 13, pp. 3551–3582, Jul. 2017

    2017

  36. [43]

    Amari and H

    S.-I. Amari and H. Nagaoka, Methods of Information Geometry . Amer- ican Mathematical Soc., 2000, vol. 191

    2000

  37. [44]

    Information geometrical framework to analy ze the belief propagation algorithm,

    S. Ikeda, “Information geometrical framework to analy ze the belief propagation algorithm,” in Proc. W orkshop Math. Stat. Inference , 2004

    2004

  38. [45]

    Q uaDRiGa: A 3- D multi-cell channel model with time evolution for enabling virtual field trials,

    S. Jaeckel, L. Raschkowski, K. Börner, and L. Thiele, “Q uaDRiGa: A 3- D multi-cell channel model with time evolution for enabling virtual field trials,” IEEE Trans. Antennas Propag. , vol. 62, no. 6, pp. 3242–3256, Jun. 2014

    2014

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.