Robust Distributed Phase Retrieval for Multi-View Compressive Networked Sensing With Outliers

Ming-Hsun Yang

REVIEW 3 major objections 4 minor 17 references

Robust Distributed Phase Retrieval for Multi-View Compressive Networked Sensing With Outliers

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proposes a two-stage distributed algorithm that exactly recovers each sensor's observable sparse signal from intensity-only measurements corrupted by sparse outliers, by first recovering global signal amplitudes through a…

desk verdict Problem setup is new and Theorem 1 is sound, but Theorem 2's majority-rule guarantee is false—masked devices and colluding outliers make the wrong value win. read the letter →

arxiv 2505.24651 v1 pith:PWJSNTUB submitted 2025-05-30 cs.IT eess.SPmath.IT

Ming-Hsun Yang This is my paper

classification cs.ITeess.SPmath.IT MSC 94A12

keywords distributedphaseretrievalmulti-viewsensingsparseoutliersdisjunctmatricesmajorityrulecompressivewirelesssensornetworkssignalrecovery

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

Each sensor in this paper sees only intensity measurements of a masked, sparse global signal, and some measurements are corrupted by arbitrary outliers. The paper proposes a two-stage distributed algorithm: local devices send only counts of zero measurements to a central server, which identifies the support of the global signal, then send amplitude ratios from which the server recovers the magnitude of every nonzero entry by majority vote. Once the amplitudes are known, each device recovers its own visible signal by solving a binary optimization over signs, without sharing raw data. The paper proves both stages exact under conditions on outlier sparsity, group count, and per-component visibility; its guarantees are for locally observed signals, and the author explicitly leaves global reconstruction as future work. If the guarantees hold, phase-less sensor networks with occlusion can reconstruct their local data through low-bandwidth, partially privacy-preserving collaboration.

What carries the argument

The load-bearing object is a $K$-disjunct sensing matrix, partitioned into $B$ disjoint groups, each of size large enough that every column has $t+1$ measurements whose supports avoid the union of the supports of any other $K$ columns. This combinatorial isolation lets the group-wise counting rule in (4) identify the signal support and lets the amplitude ratios in (5) be collected for the majority vote in (6). A sparse Bernoulli random design is used to approximate disjunctness, and the threshold $\eta$ in (4) separates the counts produced by true signal entries from those produced by empty coordinates under $K_o < t/2$ outliers.

What would settle it

Run the proposed scheme with a masked device whose column support for a nonzero coordinate $n$ produces measurements equal to zero; because $\sqrt{0}/|\varphi| = 0$ deterministically, the union of ratios will contain repeated zeros. If enough masked devices contribute zero ratios in a group that otherwise sees $s_n$, the majority rule in (6) returns $0$ instead of $|s_n|$, so exact amplitude recovery fails even when $K_o < t/2$ and $\alpha \le B-1$ hold.

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Extended reading notes

Core claim

The central claim is Theorem 2: under Assumption 1, if the group-wise sensing matrices are $K$-disjunct with parameter $t$, the number of outliers in any group satisfies $K_o < t/2$, and every signal component is blocked from view in at most $B-1$ of the $B$ groups, then the majority rule in (6) exactly recovers the amplitude $|s_n|$ of every nonzero entry of the global sparse signal. The argument uses ratios $\sqrt{y^{(i)}_m}/|\varphi^{(i)}_{m,n}|$ built from measurements that isolate column $n$; in a fully observable, outlier-free group all such ratios equal $|s_n|$, and sparse outliers contaminate at most $K_o$ of them. The proof requires that any incorrect ratio value appears essentially uniquely, so the correct value's $t+1-K_o$ votes outnumber every wrong value. Theorem 3 then shows that, given these exact amplitudes and a connected sensing graph, each local device recovers its observable signal exactly, up to a global sign, by solving a $\ell_0$-minimization whose solution has the same absolute values as the true local signal.

Load-bearing premise

The majority rule for amplitudes relies on the assumption that every incorrect ratio value is almost surely unique across the collected votes, unless it comes from one of the at most $K_o$ outliers; masked devices break this because $\sqrt{0}/|\varphi|$ equals $0$ with probability one, and many identical zero ratios can outvote the true amplitude.

Editorial extensions

If this is right

Exact local recovery is achievable without sharing raw intensity data, only zero-counts and amplitude ratios, so the protocol is communication-efficient and partially privacy-preserving.
The conditions $K_o < t/2$ and $\alpha \le B-1$ give a quantitative trade-off: increasing the disjunctness parameter $t$ tolerates more outliers, while adding groups relaxes the per-component visibility requirement.
With exact amplitudes and a connected sensing graph, the remaining sign-recovery problem is a finite ternary search of size $3^K$, which the paper approximates by projecting a phase-retrieval solution onto ternary vectors.
Support identification succeeds whenever at least one group has full visibility of each active component and that group's outliers stay below $t/2$; network coverage, not total measurement count, is the main bottleneck.

Reading between the lines

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

An implicit requirement of the majority rule is that wrong ratios almost never repeat; excluding zero ratios from masked devices would restore the theorem because the true amplitude is nonzero.
The sign-recovery stage fixes only relative signs up to a global flip; applications needing absolute orientation would require a reference measurement or a known pilot entry.
A natural testable variant replaces the majority vote with a median or trimmed mean of the ratios; the same disjunctness machinery should yield a similar guarantee with weaker uniqueness demands.
The support-counting stage could be reused in dynamic networks: if the partition into $B$ groups changes over time, only the counts $u^{(i)}_n$, not raw measurements, need to be resent, so the protocol remains communication-light.

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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

The paper is worth a look for its problem statement, but the central theoretical guarantee does not hold. The multi-view, unknown-mask, phaseless distributed sensing model with sparse outliers is a genuine addition to the distributed sparse recovery literature, and the two-stage protocol is clearly laid out. Theorem 1, the support identification result, is sound in the stated conditional setting. Theorem 3 is conditional on perfect amplitude recovery and its proof is sketchy but plausible. The problem is Theorem 2.

The flaw sits in Appendix B, around inequality (14). The proof claims that any wrong amplitude ratio in the union of the F_b sets is almost surely distinct, so no wrong value can receive more than Ko+1 votes. That assertion fails. Measurements from masked devices with d_n=0 and w=0 give y_m=0, so the ratio sqrt(y_m)/|phi| is exactly 0, independent of the continuous phi distribution. Worse, outlier values are arbitrary, so an adversary can repeat the same wrong ratio across many groups; the 'almost surely' argument only covers clean measurements. Inequality (14) is false.

Concrete counterexample: set B=10, t=7, Ko=3, alpha=9, K=1, and choose group matrices of size 8 that satisfy 1 7-disjunctness. One group has eight clean observing rows. In each of the other nine groups, take three rows from a masked device, three rows with outliers engineered to produce a distinct wrong ratio, and two clean rows. Each bad group has u_n=3 < eta, so all groups are selected. The value 0 appears 27 times, the true |s_n| appears 26 times, so the argmax in (6) returns 0. This satisfies all assumptions of Theorem 2, so the theorem is false as stated. Since Theorem 3 assumes perfect global amplitude recovery, the end-to-end exact recovery claim collapses.

The simulations do not check the disjunctness condition, so the numerics demonstrate average-case usefulness, not the advertised adversarial guarantee. The paper could be repaired, for example by changing the aggregation rule or adding assumptions that prevent outlier values from colluding across groups, but as submitted the main advertised result is false. I'd still send it to a serious referee rather than desk-reject: the problem formulation and the support identification result have value, and the flaw is a well-defined proof error with a clear revision path. My recommendation is reject in current form.

Referee Report

3 major / 4 minor

Summary. The paper studies a distributed multi-view compressive phase retrieval problem in which each sensor observes squared-magnitude measurements of a masked version of a global K-sparse signal, corrupted by a sparse outlier vector. The proposed two-stage protocol first identifies the global support at a central server by counting zero measurements per group (Theorem 1), then recovers the amplitudes |s_n| by a majority vote over ratios sqrt(y_m)/|phi_mn| from selected groups (Theorem 2), and finally recovers each local signal by solving a ternary l0/l1 optimization (Theorem 3). The authors claim exact end-to-end recovery in the noise-free case under disjunct sensing, bounded outlier sparsity, and bounded masking.

Significance. The problem formulation is timely and the architecture is attractive: it preserves raw measurements on-device, exploits combinatorial disjunctness, and the support-identification step (Theorem 1) is clean and correctly proved. If the amplitude-recovery guarantee were valid, the paper would be a useful contribution to distributed phase retrieval, and the simulations do suggest practical promise. However, the central amplitude-recovery theorem is not established and, under the multiset reading required by its proof, is false; since Theorem 3 assumes perfect amplitude recovery, the paper's main end-to-end guarantee is unsupported.

major comments (3)

[Appendix B, Eq. (13)] The proof counts inner-sum contributions as if F_bn were a multiset of measurement ratios, but in (6) F_bn is defined as a set, so the inner sum over s' in F_bn of 1{s=s'} is at most 1 for each group. Consequently the lower bound of t+1-Ko on the objective value at s=|s_n| does not follow from the existence of t+1-Ko qualifying measurements in a single group. The statement of (6) and the counting in the proof are inconsistent; one of them must be changed.
[Appendix B, inequality (14) and Section III-B] The claim that any two wrong ratios are almost surely distinct is false. Since w(i) in (2) is an arbitrary sparse vector, an outlier row can force sqrt(y_m^(i))/|phi^(i)_{m,n}| to equal any prescribed value c by setting w_m^(i)=c^2|phi^(i)_{m,n}|^2 - |phi^(i)_{m,n} s_n|^2; the same c can be inserted in every selected group, so a wrong value can receive one vote per group (set reading) or one vote per outlier row (multiset reading). Additionally, when d_n^(i)=0 and w_m^(i)=0, (5) gives y_m^(i)=0 and the ratio equals 0 deterministically, independently of the continuous distribution of Phi. Under the multiset reading required by the proof, take K=1, t=7, Ko=3, B=10, alpha=9, eta=4; let one group have 8 clean rows observing the active entry, and in each of the other nine groups let 3 rows come from a single masked device (ratio 0), 3 rows be outliers with distinct engineered ratios c not equal to |s_n|, and 2 rows be clean. Each group has 8 rows with nonzero phi for the single active column, so K=1 t-disjunctness holds; every group has u_n<eta and is selected; 0 receives 27 votes and |s_n| receives 26 votes, so the argmax in (6) is 0. This instance satisfies Assumption 1 and the conditions of Theorem 1, so Theorem 2 is false as stated. Under the literal set definition, the same engineered-outlier construction gives a tie between |s_n| and c, so exact recovery is still not guaranteed.
[Theorem 3 and Appendix C] The inference from (16) to |xhat|=|h| is not justified in the text, because the sign in (16) may depend on m; a generic-position argument is needed to rule out ternary x not equal to +/- h satisfying |Phi x|=|Phi h|. This gap is repairable, but the more serious issue is that Theorem 3 is stated conditional on perfect global signal amplitude recovery, which is exactly the guarantee that Theorem 2 was supposed to provide. Since Theorem 2 is false under the proof's own counting, the end-to-end exact-recovery claim in the abstract and Section IV collapses.

minor comments (4)

[Section V] The simulations do not report the quantities Ko, alpha, t, and the disjunctness parameters for the randomly generated group matrices, so the reader cannot verify whether the operating points satisfy the conditions of Theorems 1 and 2.
[Section III-A, Eq. (4)] The threshold eta is a free parameter; the paper proves an admissible interval but never states how eta was chosen in the simulations.
[Figures 1 and 2] The captions are terse and should state the plotted quantities and the fixed parameters (K, B, M, q) used in each figure.
[General presentation] There are minor typographical issues, such as the spacing in the author byline and line-break artifacts in the introduction; the notation I_n with an overbar is introduced immediately before (6) but would benefit from a more standard symbol.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is conditional and the known flaw is a proof error, not a circular step.

full rationale

The paper's derivation chain is conditional throughout: Theorem 1 establishes support recovery under Assumption 1 and explicit numeric bounds; Theorem 2 assumes the support estimate is exact and proves amplitude recovery by a majority rule; Theorem 3 explicitly assumes perfect global amplitude recovery and a connected graph before proving local recovery. The decision threshold eta is not fitted to data but is required to lie in the analytic interval (Ko, t + 1 - Ko), and the recovered amplitudes are computed from the measurement ratios rather than being tuned to match a target output. There are no load-bearing self-citations: the cited disjunct-matrix constructions, group-testing bounds, and phase retrieval algorithms are external. The known weakness identified in the reader's take is that Appendix B claims wrong amplitude ratios are almost surely distinct, which fails for masked devices with w_m = 0 and d_n = 0, producing deterministic zero ratios that can outvote the true amplitude. That is a correctness gap in the proof of Theorem 2, not a circularity: the conclusion is not assumed as an input, and no fitted parameter is renamed as a prediction. The end-to-end guarantee may indeed be unsupported because Theorem 3 depends on Theorem 2, but that dependency is a standard conditional proof structure rather than a circular reduction. Therefore the circularity score is 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No new physical entities are invented. The masking matrices D(i) are part of the input model. The main load-bearing assumptions are the group-wise disjunctness property and the unstated uniqueness of wrong amplitude ratios; the latter is the source of the proof failure.

free parameters (1)

support decision threshold eta = not specified in simulations
Rules (4) and (6) require eta, with a theoretical range (Ko, t+1-Ko). The paper does not state how eta is chosen in the experiments, and the proof of Theorem 2 is sensitive to this choice.

assumptions (4)

domain assumption Assumption 1: after partitioning devices into B groups, each group-wise sensing matrix Phi_b is K t-disjunct with t satisfying Ko < t/2.
Invoked in Theorems 1 and 2 to guarantee at least t+1-Ko uncontaminated rows per column. Not verified in the simulations, and the simulation parameters make the required t unlikely.
domain assumption Assumption 2: the partition I_1,...,I_B is known at the central server.
Needed for the group-wise counting rule in (4) and for the definition of the groups used in amplitude recovery.
domain assumption Nonzero entries of s and Phi are independently drawn from continuous distributions, and outliers are sparse with arbitrary nonzero values.
Used for probability-one distinctions in Theorems 2 and 3. The continuous distribution of Phi does not remove the deterministic zero ratios from masked devices with w_m = 0.
domain assumption A connected graph Gi is sufficient for uniqueness in noiseless binary phase retrieval.
Theorem 3 asserts that connectedness plus continuous coefficients forces x = +/- h, but the proof only sketches the sign-propagation argument and does not fully handle zero entries in h.

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Cite this review

Pith. "Pith review of Robust Distributed Phase Retrieval for Multi-View Compressive Networked Sensing With Outliers." pith.science (2026). https://pith.science/paper/PWJSNTUB

@misc{pith2026250524651,
  author       = {Pith},
  title        = {Pith review of: Robust Distributed Phase Retrieval for Multi-View Compressive Networked Sensing With Outliers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PWJSNTUB}},
  note         = {Machine review of arXiv:2505.24651}
}

read the original abstract

This work examines the multi-view compressive phase retrieval problem in a distributed sensor network, where each sensor device, limited by storage and sensing capabilities, can access only intensity measurements from an unknown part of the global sparse vector. The goal is to enable each sensor to recover its observable sparse signal when measurements are corrupted by outliers. To achieve reliable local signal recovery with limited data access, we propose a distributed reconstruction algorithm that enables collaboration among sensor devices without the need to share individual raw data. The proposed scheme employs a two-stage approach that first recovers the amplitude of the global signal (at a central server) and subsequently estimates the observable nonzero signal entries (at each local device). Our analytic results show that perfect global signal amplitude recovery can be achieved under mild conditions on the support size of sparse outliers and the view blockage level. In addition, the exact reconstruction of locally observed signal components is shown to be attainable in the noise-free case by solving a binary optimization problem, subject to a mild requirement on the structure of the sensing matrix. Computer simulations are provided to illustrate the effectiveness of the proposed scheme.

Figures

Figures reproduced from arXiv: 2505.24651 by the authors.

**Figure 2.** Performance comparisons of all methods for differen [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

Heterogeneous networked data recovery from compressi ve measurements using a copula prior,

N. Deligiannis, J. F. C. Mota, E. Zimos, and M. R. D. Rodrig ues “Heterogeneous networked data recovery from compressi ve measurements using a copula prior,” IEEE Trans. Commun. , vol. 65, no. 12, pp. 5333–5347, Dec. 2017

work page 2017
[2]

Big data reduction for a smart city’s critical infrastructural health monitoring,

T. Wang, M. Z. A. Bhuiyan, G. Wang, M. A. Rahman, J. Wu, and J . Cao “Big data reduction for a smart city’s critical infrastructural health monitoring,” IEEE Commun. Mag. , vol. 56, no. 3, pp. 128–133, Mar. 2018

work page 2018
[3]

Y . C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications . Cambridge University Press, 2011

work page 2011
[4]

Unmann ed aerial vehicle-assisted sparse sensing in wireless sensor networks,

C. Lv, Y . Ren, X. Li, P . Wang, Z. Du, G. Ma, and H. Chi “Unmann ed aerial vehicle-assisted sparse sensing in wireless sensor networks,” IEEE Wireless Commun. Lett. , vol. 12, no. 6, pp. 977–981, Jun. 2023

work page 2023
[5]

Amplitude-Aided 1-bit compressi ve sensing over noisy wireless sensor networks,

C. H. Chen and J. Y . Wu, “Amplitude-Aided 1-bit compressi ve sensing over noisy wireless sensor networks,” IEEE Wireless Commun. Lett. , vol. 4, no. 5, pp. 473–476, Oct. 2015

work page 2015
[6]

Distributed multi-view sparse vector recovery,

Z. Tian, Z. Zhang, and L. Hanzo, “Distributed multi-view sparse vector recovery,” IEEE Trans. Signal Process. , vol. 71, pp. 1448–1463, 2023

work page 2023
[7]

Environment sensing considering the occlusion effect: A multi-view approach,

X. Tong, Z. Zhang, Y . Zhang, Z. Yang, C. Huang, K.-K. Wong, and M. Debbah, “Environment sensing considering the occlusion effect: A multi-view approach,” IEEE Trans. Signal Process. , vol. 70, pp. 3598–3615, 2022

work page 2022
[8]

Modulation-Free M2M communicati ons for mission-critical applications,

S. Cai and V . K. N. Lau, “Modulation-Free M2M communicati ons for mission-critical applications,” IEEE Trans. Signal and Information Processing over Networks , vol. 4, no. 2, pp. 248–263, Jun. 2018

work page 2018

Show all 17 references

[9]

Towards atomic MIMO receiv ers,

M. Cui, Q. Zeng, and K. Huang, “Towards atomic MIMO receiv ers,” IEEE J. Sel. Areas Commun. , vol. 43, no. 3, pp. 659–673, Mar. 2025

2025
[10]

Quantum-MUSIC: Multiple si gnal classiﬁcation for quantum wireless sensing,

H. Kim, H. Park, and S. Kim, “Quantum-MUSIC: Multiple si gnal classiﬁcation for quantum wireless sensing,” IEEE Wireless Commun. Lett. , early access, 2025

2025
[11]

Security of the internet of things: Vulnerabilities, attacks, and countermeasures,

I. Butun, P . ¨Osterberg, and H. Song, “Security of the internet of things: Vulnerabilities, attacks, and countermeasures,” IEEE Commun. Surveys Tuts. , vol. 22, no. 1, pp. 616–644, Firstquarter 2020

2020
[12]

Error-correcting nonadaptive group tes ting with de-disjunct matrices,

A. J. Macula, “Error-correcting nonadaptive group tes ting with de-disjunct matrices,” Discrete Applied Mathematics , vol. 80, no. 2, pp. 217–222, Dec. 1997

1997
[13]

Improved suppor t recovery in universal 1-bit compressed sensing,

N. Matsumoto, A. Mazumdar, and S. Pal, “Improved suppor t recovery in universal 1-bit compressed sensing,” IEEE Trans. Information Theory , vol. 70, no. 2, pp. 1453–1472, Feb. 2024. 14

2024
[14]

On the all-o r-nothing behavior of Bernoulli group testing,

L. V . Truong, M. Aldridge, and J. Scarlett, “On the all-o r-nothing behavior of Bernoulli group testing,” IEEE Journal on Selected Areas in Information Theory , vol. 1, no. 3, pp. 669–680, Nov. 2020

2020
[15]

Improved algorithms and bounds for list union-free families,

A. A. Rescigno and U. V accaro, “Improved algorithms and bounds for list union-free families,” IEEE Trans. Information Theory, vol. 70, no. 4, pp. 2456–2463, Apr. 2024

2024
[16]

Low-rank positive semideﬁnit e matrix recovery from corrupted rank-one measurements,

Y . Li, Y . Sun, and Y . Chi, “Low-rank positive semideﬁnit e matrix recovery from corrupted rank-one measurements,” IEEE Trans. Signal Process. , vol. 65, no. 2, pp. 397–408, Jan. 2017

2017
[17]

Sparse phase retrieval via truncated amplitude ﬂow,

G. Wang, L. Zhang, G. B. Giannakis, M. Akc ¸akaya, and J. C hen, “Sparse phase retrieval via truncated amplitude ﬂow,” IEEE Trans. Signal Process. , vol. 66, no. 2, pp. 479–491, Jan. 2018

2018

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Heterogeneous networked data recovery from compressi ve measurements using a copula prior,

N. Deligiannis, J. F. C. Mota, E. Zimos, and M. R. D. Rodrig ues “Heterogeneous networked data recovery from compressi ve measurements using a copula prior,” IEEE Trans. Commun. , vol. 65, no. 12, pp. 5333–5347, Dec. 2017

work page 2017

[2] [2]

Big data reduction for a smart city’s critical infrastructural health monitoring,

T. Wang, M. Z. A. Bhuiyan, G. Wang, M. A. Rahman, J. Wu, and J . Cao “Big data reduction for a smart city’s critical infrastructural health monitoring,” IEEE Commun. Mag. , vol. 56, no. 3, pp. 128–133, Mar. 2018

work page 2018

[3] [3]

Y . C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications . Cambridge University Press, 2011

work page 2011

[4] [4]

Unmann ed aerial vehicle-assisted sparse sensing in wireless sensor networks,

C. Lv, Y . Ren, X. Li, P . Wang, Z. Du, G. Ma, and H. Chi “Unmann ed aerial vehicle-assisted sparse sensing in wireless sensor networks,” IEEE Wireless Commun. Lett. , vol. 12, no. 6, pp. 977–981, Jun. 2023

work page 2023

[5] [5]

Amplitude-Aided 1-bit compressi ve sensing over noisy wireless sensor networks,

C. H. Chen and J. Y . Wu, “Amplitude-Aided 1-bit compressi ve sensing over noisy wireless sensor networks,” IEEE Wireless Commun. Lett. , vol. 4, no. 5, pp. 473–476, Oct. 2015

work page 2015

[6] [6]

Distributed multi-view sparse vector recovery,

Z. Tian, Z. Zhang, and L. Hanzo, “Distributed multi-view sparse vector recovery,” IEEE Trans. Signal Process. , vol. 71, pp. 1448–1463, 2023

work page 2023

[7] [7]

Environment sensing considering the occlusion effect: A multi-view approach,

X. Tong, Z. Zhang, Y . Zhang, Z. Yang, C. Huang, K.-K. Wong, and M. Debbah, “Environment sensing considering the occlusion effect: A multi-view approach,” IEEE Trans. Signal Process. , vol. 70, pp. 3598–3615, 2022

work page 2022

[8] [8]

Modulation-Free M2M communicati ons for mission-critical applications,

S. Cai and V . K. N. Lau, “Modulation-Free M2M communicati ons for mission-critical applications,” IEEE Trans. Signal and Information Processing over Networks , vol. 4, no. 2, pp. 248–263, Jun. 2018

work page 2018