A Frequency Domain Channel Estimation Based on Atomic Norm Minimization for Frequency Selective MmWave MIMO Systems

Hamidreza Bakhshi; Mahdi Eskandari

REVIEW 3 major objections 5 minor 19 references

A Frequency Domain Channel Estimation Based on Atomic Norm Minimization for Frequency Selective MmWave MIMO Systems

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Atomic norm minimization outperforms grid-based estimator for wideband mmWave channels

desk verdict A plausible but rough per-subcarrier ANM extension to wideband mmWave channel estimation; the idea is sound but the paper is not reproducible as written. read the letter →

arxiv 1908.04278 v1 pith:B6G4B5L3 submitted 2019-08-12 eess.SP

Mahdi Eskandari , Hamidreza Bakhshi This is my paper · ORCID

classification eess.SP

keywords channelestimationmillimeterwaveMIMOatomicnormminimizationreweightedfrequency-selectiveoff-gridOFDMcompressedsensing

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 estimator for wideband millimeter-wave MIMO-OFDM systems that treats frequency-selective channel estimation as an off-grid sparse recovery problem rather than compressed sensing on a discrete angle grid. The authors claim this is the first formulation of frequency-selective mmWave channel estimation as an atomic norm minimization problem, and they adapt the reweighted atomic norm minimization technique to push the solution closer to the atomic $\ell_0$ norm. Working with the vectorized frequency-domain channel at each subcarrier, the estimator solves a semidefinite program whose atoms represent continuous angles of arrival and departure. Monte Carlo simulations under hybrid-architecture constraints, including quantized phase shifters and few RF chains, show that both ANM and RAM achieve lower normalized mean squared error than the on-grid OMP-based estimator of [6], with RAM the more accurate. If the claim is right, wideband mmWave channels can be estimated without discretizing the angle domain, removing the grid mismatch that limits dictionary-based methods.

What carries the argument

The central object is the two-dimensional atomic norm of the vectorized channel, defined by the atom set $\{\mathbf{a}_T(\theta) \otimes \mathbf{a}_R(\phi)\}$ for continuous angles $(\theta,\phi)$. The machinery is an equivalent semidefinite program: minimize a data-fit term plus a regularization term built from the trace of the Toeplitz matrix $\mathcal{S}(\mathbf{U})$ and the trace of the Hermitian variable, subject to a block positive-semidefinite constraint, with the noise-aware regularization parameter $\zeta$ from (18). The reweighted variant replaces $\mathcal{S}(\mathbf{U})$ with a reweighted version and iterates, which sharpens sparsity. This machinery moves angle estimation off the grid because the Toeplitz matrix encodes all possible continuous angles without enumerating them.

What would settle it

Rerun the Figure 2 Monte Carlo setup with five paths instead of three, or with two paths whose angles differ by less than the array's nominal resolution; the central claim fails if the ANM and RAM NMSE advantage over OMP disappears or reverses in either regime.

Watch

Extended reading notes

Core claim

The central claim is that the vectorized frequency-selective channel $\mathbf{h}_v[k]$ has a useful two-dimensional atomic norm representation, with atom $\mathbf{g}(\theta,\phi) = \mathbf{a}_T(\theta) \otimes \mathbf{a}_R(\phi)$, and that minimizing this norm against the linear observation model (15) estimates the channel at continuous, off-grid angles. The paper derives this as the semidefinite program (17), whose blocks include a multilevel Toeplitz matrix $\mathcal{S}(\mathbf{U})$ acting as the atomic-norm dual variable, and solves it once per subcarrier. The RAM version (19) iteratively reweights the Toeplitz penalty, using $\boldsymbol{\Theta}_j = \xi(\mathbf{U}^{(j-1)} + \xi \mathbf{I})^{-1}$, to make the convex program behave more like the nonconvex atomic $\ell_0$ norm. Simulation results show that both estimators outperform the on-grid OMP estimator of [6] across SNR and training-frame counts, which is the paper's evidence that the formulation works.

Load-bearing premise

The results assume that the semidefinite program recovers the true multipath angles from the small number of training frames used in the simulations, with no proven recovery condition behind that assumption.

Editorial extensions

If this is right

Frequency-selective mmWave channels can be estimated in the continuous angular domain, avoiding the grid-mismatch error that on-grid compressed-sensing estimators suffer by construction.
The RAM variant improves on plain ANM by promoting sparsity more aggressively, giving lower NMSE at the same SNR and training overhead.
The method works under practical hybrid-architecture constraints such as 7-bit quantized phase shifters, few RF chains, and zero-padded OFDM training frames.
Increasing the number of training frames improves NMSE for ANM and RAM and widens the gap over OMP, so the estimator can trade training overhead against accuracy.
Because the per-subcarrier SDP is separable, the formulation extends directly to any OFDM-based mmWave system with $K$ subcarriers.

Reading between the lines

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

A natural extension the paper does not pursue is to couple the SDPs across subcarriers, sharing a single set of delay and angle atoms; this would likely improve accuracy at the cost of a much larger optimization.
The simulations use small arrays and only three paths, so whether the NMSE advantage survives at massive-array sizes or with many closely spaced paths is untested, since no recovery guarantee is proven.
The per-subcarrier SDP is computationally heavy, so a practical implementation would likely need a fast solver or an explicit factorization of the Toeplitz structure, which the paper leaves implicit.
Because the angle atoms are continuous, the estimated angles could be fed directly into beamforming design, avoiding a separate grid-search stage.

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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: this is a straightforward per-subcarrier application of 2D atomic norm minimization to frequency-selective mmWave MIMO-OFDM channel estimation. The application is new, but the machinery is imported wholesale from prior ANM and RAM work. The simulations suggest the approach beats an on-grid OMP baseline, and the RAM variant helps as expected. The core idea is plausible; the paper just isn't reproducible as written.

What it does well: the wideband channel model with delay taps and the hybrid ZP-OFDM setup are sensible. There are no fitted coefficients, so the performance comparison is not circular. The Monte Carlo curves in Figures 2 and 3 are consistent with what you would expect from off-grid versus on-grid estimation.

Soft spots, in order of severity. First, Eq. (18) is garbled. It defines κ as a covariance matrix, then puts κ inside a scalar regularization parameter ζ. At the simulated sizes that matrix is 120×120, so the printed formula cannot be what was implemented. The authors need to state the actual scalar regularization rule. Second, no recovery guarantee is given for the quantized-beam observation operator. That is a fair criticism, though it matters less for an empirical paper; the NMSE curves do show the SDP working in the tested regime. Third, the baseline is a single OMP from [6], and the grid size G is never reported. That makes the claimed gains look thinner than they should. They should add at least one more frequency-selective estimator, report G, and give error bars or a Monte Carlo count. Fourth, the computational cost of solving K SDPs plus RAM iterations is not discussed, which is a minor omission.

The stress-test note asks whether the recovery condition might fail in this regime. I don't think that invalidates the paper; the empirical results are what they are. But the lack of any condition means the results do not generalize beyond the exact simulation settings.

Who is this for? Someone wanting a quick example of ANM applied to wideband mmWave channel estimation. It is not a theory paper, and it does not break new mathematical ground.

My recommendation: send it to peer review, but with major revision. The topic is appropriate and the idea is sound; the revision must fix Eq. (18), add comparison details, and make the simulations reproducible. Without those, the paper should not be accepted.

Referee Report

3 major / 5 minor

Summary. The paper proposes channel estimators for frequency-selective mmWave MIMO-OFDM systems based on atomic norm minimization (ANM) and reweighted atomic norm minimization (RAM). The frequency-domain channel vector at each subcarrier is expressed as a sparse linear combination of Kronecker products of array response vectors, and estimation is posed as a convex SDP (Eqs. (16)-(17)), with a RAM variant in Eq. (19). The authors claim this is the first ANM formulation for frequency-selective mmWave channel estimation, and they compare the proposed estimators against an on-grid OMP baseline from [6] using NMSE curves as a function of SNR and number of training frames (Figs. 2-3). The paper contains no theoretical recovery guarantee and no error bars or stated Monte Carlo counts.

Significance. If the performance claims hold, the paper makes a modest but useful contribution: it transfers the off-grid ANM framework, previously applied to flat channels, to the frequency-selective mmWave setting, and it demonstrates numerically that off-grid ANM/RAM can beat a grid-based OMP estimator. The formulation is standard and imported from cited theory, the RAM implementation follows [13] with explicit parameters (J=5, xi=1), and the comparison target is an external OMP baseline rather than a self-tuned method. However, the significance is limited by the absence of any recovery guarantee or incoherence condition for the random quantized measurement operator, by the unverifiable regularization formula in Eq. (18), and by missing experimental details (grid size, number of realizations, error bars), so the central claim is currently supported only by single-curve simulations.

major comments (3)

[Section 3, Eq. (18)] The printed regularization formula is garbled and the definition of kappa is inconsistent with its use. Eq. (18) reads zeta = (1 + ... ) sqrt(N' log(N)) log(4 log(N)) / log(N) with kappa = E[q[k] q[k]^H], but kappa is a matrix (at the stated parameters it would be MN_r x MN_r), while zeta appears as a scalar in Eqs. (17) and (19). As written, the implemented estimator cannot be reproduced, and this directly affects the central performance claim because the NMSE curves depend on the chosen regularization. The authors must correct Eq. (18), clarify what kappa is (scalar noise power vs. covariance matrix), and state the exact SDP objective used in CVX.
[Section 3, Eqs. (15)-(17)] No condition is stated or verified under which the random quantized measurement operator Phi[k] = [(x_1[k]^T tensor W_1^H); ...; (x_M[k]^T tensor W_M^H)] supports atomic-norm recovery. The cited ANM theory requires incoherence or a dual certificate for the observation operator; for M=60, N_t=N_r=16, N_RF=2, and 7-bit quantized phase shifters this is not guaranteed. Without such a condition, the reported advantage over OMP could reflect an insufficiently fine grid or a regularization-bias artifact rather than accurate off-grid recovery. At minimum, please add a noiseless recovery test and report the atomic-norm dual residue or a sample-complexity discussion; ideally, state the recovery condition satisfied by Phi[k].
[Section 4, Figs. 2-3] The OMP baseline's grid size G is never reported, although the caption of Fig. 2 says both AoAs and AoDs are quantized with G points. Without G, the reader cannot judge whether the comparison is fair or whether the ANM/RAM gain is simply an artifact of a coarse grid. Additionally, the figures show single NMSE curves with no error bars, and the text only says results are 'averaged over many independent realizations' without stating the number of trials. Please report G, the number of Monte Carlo runs, and error bars or confidence intervals.

minor comments (5)

[Section 1] The phrase 'simulation results verify the accuracy' should be phrased as 'validate' or 'demonstrate', since simulations do not verify a claim in the mathematical sense.
[Section 2, Eq. (2) and Section 4] The array response definitions in Section 4 contain swapped subscripts (the transmitter array response is written with an 'R' superscript and the receiver with 'T'), which conflicts with the notation in Eq. (2). Please correct the subscripts.
[Throughout] The spelling 'reweighed' is inconsistent with the standard term 'reweighted' used in the title and references; please unify the spelling.
[Section 2.1] Eq. (9) defines the noise vector z_m[k] but the subscript for the noise covariance is not explicitly stated; please clarify the noise vector dimension and its independence across frames and subcarriers, since this affects the SNR definition.
[Fig. 1 and Section 2] Figure 1 is referenced before it is introduced in the text; the figure caption is also terse. Please renumber or move the reference so it first appears after the figure is mentioned.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the ANM/RAM estimators are validated against an external OMP baseline with no fitted parameters or self-citation chain.

full rationale

The paper's derivation chain does not reduce to its own inputs. The ANM formulation is taken from external theory: the atomic-norm relaxation and SDP equivalence are cited to Bhaskar et al. [9], Chandrasekaran et al. [14], Yang and Xie [15], and Yang et al. [16], none of which are authored by Eskandari and Bakhshi. The reweighted atomic norm procedure is imported from Yang and Xie [13], also external. The regularization parameter in Eq. (17) is stated via Eq. (18) with a reference to [9], and the RAM parameters are explicit simulation settings (J=5 iterations, ζ=1); nothing is calibrated to the NMSE curves. The performance claim is tested against the on-grid OMP estimator of Venugopal et al. [6], an external baseline, using a standard NMSE metric computed on simulated channels generated from the stated model. No coefficient, constant, or prior distribution is fitted to the data being predicted, and no load-bearing conclusion relies on a self-citation. The absence of a recovery-condition analysis, the unreported OMP grid size, and the apparent mismatch between the matrix-valued κ in Eq. (18) and the scalar regularization ζ are reproducibility and correctness concerns, not circularity. Therefore no circular steps are identified.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on imported SDP theory and a synthetic channel model; the paper itself contributes the application and simulation, not new recovery theory. Two practical tuning parameters (xi, J) are chosen by hand, and the ANM regularization value is effectively unspecified due to garbled Eq. (18).

free parameters (3)

RAM regularization parameter xi = 1 (used in Fig. 2 and Fig. 3)
Chosen by hand, not derived; it controls the l0/l1 trade-off in the reweighted objective in Eq. (19), so RAM performance depends on it.
RAM iteration count J = 5
Chosen for simulations without convergence or sensitivity analysis in Section 4.
ANM regularization parameter zeta = not specified in readable form
The formula in Eq. (18) is garbled in the manuscript; the actual value used in simulations is not stated, making the effective tuning a hidden free parameter.

assumptions (4)

standard math The 2D atomic norm has the SDP characterization in Eq. (12), proven in [15] and [16].
The paper invokes it without proof; the reliability of the convex relaxation is prior work.
domain assumption The wideband mmWave channel follows the L-path, N_c-tap model of Eq. (2) from [6],[7] with known pulse shape and statistically known AoAs/AoDs.
All claims and simulations assume this generative model.
ad hoc to paper The measurement operator Phi[k] built from random quantized hybrid precoders and combiners is sufficient for atomic norm recovery.
No recovery condition or sample-complexity bound is provided; the paper relies on simulation outcomes.
domain assumption Noise after combining q[k] is circularly symmetric complex Gaussian with covariance kappa.
Standard assumption, but colored noise from W_m is not analyzed; the regularization parameter depends on kappa.

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Cite this review

Pith. "Pith review of A Frequency Domain Channel Estimation Based on Atomic Norm Minimization for Frequency Selective MmWave MIMO Systems." pith.science (2026). https://pith.science/paper/B6G4B5L3

@misc{pith2026190804278,
  author       = {Pith},
  title        = {Pith review of: A Frequency Domain Channel Estimation Based on Atomic Norm Minimization for Frequency Selective MmWave MIMO Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B6G4B5L3}},
  note         = {Machine review of arXiv:1908.04278}
}

read the original abstract

In this study, a channel estimator for millimeter wave (mmWave) systems is proposed. By considering the sparse nature of channels in millimeter wave band, the channel estimation problem formulated as an atomic norm minimization problem. Previous studies on mmWave channel estimation have focused on frequency flat channels. However, in this study the work is based on a frequency selective channel and adopted atomic norm minimization and reweighed atomic norm minimization technique for channel estimation in the frequency domain. Simulation results verify the accuracy of the atomic and reweighted atomic norm minimization techniques.

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Reference graph

Works this paper leans on

19 extracted references · 18 canonical work pages

[6]

Venugopal, A

K. Venugopal, A. Alkhateeb, N. Gonzalez -Prelcic, and R. W. Heath Jr.,” Channel estimation for hybrid architecture based wideband millimeter wave systems,” IEEE J. Sel. Areas Commun., vol. 35, no. 9, Sept. 2017

work page 2017
[13]

Yang and L

Z. Yang and L. Xie ,” Enhancing Sparsity and Resolution via Reweighted Atomic Norm Minimization,” IEEE Trans. Signal Process., vol. 64, no. 4, Feb. 2016. 6

work page 2016
[1]

T. S. Rappaport, S. Sun, R Mayzus, H. Zhao, Y Azar, K. Wang, G.N. Figure 3: Evolution of NMSE vs. the number of training steps ( M ) for the OMP, ANM and RAM algorithms with SNR 10dB= . Wong, J.K. Schulz, M. Samimi, and F Gutierrez,” Millimeter wave mobile communications for 5G cellular: It will work!,” IEEE Access, vol. 1, pp. 335-349, 2013

work page 2013
[2]

R. W. Heath, N. Gonzalez -Prelcic, S. Rangan, W. Roh, and A. M. Sayeed,” An overview of signal processing techniques for millimeter wave MIMO systems,” IEEE J. Sel. Topics Signal Process., vol. 10, pp. 436–453, April 2016

work page 2016
[3]

Alkhateeb, O

A. Alkhateeb, O. E. Ayach, G. Leus, and R. W. Heath Jr.,” Channel estimation and hybrid precoding for millimeter wave cellular systems,”IEEE J. Sel. Topics Signal Process. , vol. 8, no. 5, pp. 831 – 846, Oct. 2014

work page 2014
[4]

J. Lee, G. T. Gil, and Y. H. Lee,” Channel estimation via orthogonal matching pursui t for hybrid MIMO systems in millimeter wave communications,” IEEE Trans. Commun. , vol. 64, no. 6, pp 2370 - 2386, 2016

work page 2016
[5]

Z. Gao, C. Hu, L. Dai, and Z. Wang,” Channel estimation for millimeter wave massive MIMO with hybrid precoding over frequency-selective fading channels,” IEEE Commun. Lett, vol. 20, no. 6, pp. 1259 1262, June 2016

work page 2016
[7]

J. R. Fern´andez, N. G. Prelcic, K. Venugopal and R. W. Heath Jr., ” Frequency-domain Compressive Channel Estimation for Frequency selective Hybrid mmWave MIMO Systems,” IEEE Trans. Wireless Commun., vol., no., pp., 2018

work page 2018

Show all 19 references

[8]

G. Tang, B. Bhaskar, P. Shah, and B. Recht,” Compressed sensing off the grid,” IEEE Trans. Inf. Theory , vol. 59, no. 11, pp. 7465 -7490, Nov 2013

2013
[9]

B. N. Bhaskar, G. Tang and B. Recht,” Atomic Norm Denoising With Applications to Line Spectral Estim ation,” IEEE Trans. Signal Process., vol. 61, no. 23, Dec. 2013

2013
[10]

J. Deng, O. Tirkkonen and Ch. Studer,” MmWave Channel Estimation via Atomic Norm Minimization for Multi -User Hybrid Precoding,”. Available: https://arxiv.org/abs/1801.08601

arXiv
[11]

H. Chu, L. Zheng, X. Wang,” Super-Resolution mmWave Channel Estimation using Atomic Norm Minimization, ”. Available: https://arxiv.org/abs/1801.07400

arXiv
[12]

Y. Wang, P. Xu, and Z. Tian,” Efficient channel estimation for massive MIMO systems via truncated two -dimensional atomic norm minimization,” in Proc. IEEE Conf. Commun. , Paris, France, May 2017, pp. 1–6

2017
[14]

Chandrasekaran, B

V. Chandrasekaran, B. Recht, P. A. Parrilo, and A. S. Willsky ,” The convex geometry of linear inverse problems,” Foundations of Computational mathematics, vol. 12, no. 6, pp. 805-849, 2012

2012
[15]

Yang and L

Z. Yang and L. Xie ,” Exact Joint Sparse F requency Recovery via Optimization Methods,”. Available: http://arxiv.org/abs/ 1405.6585

arXiv
[16]

Z. Yang, L. Xie, and P. Stoica ,” Vandermonde decomposition of multilevel Toeplitz matrices with application to multidimensional superresolution,” IEEE Trans. Inf. Theory, vol. 62, no. 6, pp. 3685 - 3701, 2016

2016
[17]

He received his Bachelor degree in electrical engineering from university of Zanjan, Iran in

Mahdi Eskandari Mahdi Eskandari was born in Zanjan, Iran on January 21, 1994. He received his Bachelor degree in electrical engineering from university of Zanjan, Iran in

1994
[19]

He received the B.Sc

Hamidreza Bakhshi Hamidreza Bakhshi was born in Tehran, Iran on April 25, 1971. He received the B.Sc. degree in electrical engineering from Tehran University, Iran in 1992, and his M.Sc. and Ph.D. degree in Electrical Engineering from Tarbiat Modarres University, Iran in 1995 ...

1971
[2016]

student in department of electrical engineering, Shahed University, Tehran, Iran

Currently, he is a Msc. student in department of electrical engineering, Shahed University, Tehran, Iran. His research interest s include signal processing, array signal processing , MIMO systems, and mmWave communication. E-mail: m-eskandari@shahed.ac.ir

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [6]

Venugopal, A

K. Venugopal, A. Alkhateeb, N. Gonzalez -Prelcic, and R. W. Heath Jr.,” Channel estimation for hybrid architecture based wideband millimeter wave systems,” IEEE J. Sel. Areas Commun., vol. 35, no. 9, Sept. 2017

work page 2017

[2] [13]

Yang and L

Z. Yang and L. Xie ,” Enhancing Sparsity and Resolution via Reweighted Atomic Norm Minimization,” IEEE Trans. Signal Process., vol. 64, no. 4, Feb. 2016. 6

work page 2016

[3] [1]

T. S. Rappaport, S. Sun, R Mayzus, H. Zhao, Y Azar, K. Wang, G.N. Figure 3: Evolution of NMSE vs. the number of training steps ( M ) for the OMP, ANM and RAM algorithms with SNR 10dB= . Wong, J.K. Schulz, M. Samimi, and F Gutierrez,” Millimeter wave mobile communications for 5G cellular: It will work!,” IEEE Access, vol. 1, pp. 335-349, 2013

work page 2013

[4] [2]

R. W. Heath, N. Gonzalez -Prelcic, S. Rangan, W. Roh, and A. M. Sayeed,” An overview of signal processing techniques for millimeter wave MIMO systems,” IEEE J. Sel. Topics Signal Process., vol. 10, pp. 436–453, April 2016

work page 2016

[5] [3]

Alkhateeb, O

A. Alkhateeb, O. E. Ayach, G. Leus, and R. W. Heath Jr.,” Channel estimation and hybrid precoding for millimeter wave cellular systems,”IEEE J. Sel. Topics Signal Process. , vol. 8, no. 5, pp. 831 – 846, Oct. 2014

work page 2014

[6] [4]

J. Lee, G. T. Gil, and Y. H. Lee,” Channel estimation via orthogonal matching pursui t for hybrid MIMO systems in millimeter wave communications,” IEEE Trans. Commun. , vol. 64, no. 6, pp 2370 - 2386, 2016

work page 2016

[7] [5]

Z. Gao, C. Hu, L. Dai, and Z. Wang,” Channel estimation for millimeter wave massive MIMO with hybrid precoding over frequency-selective fading channels,” IEEE Commun. Lett, vol. 20, no. 6, pp. 1259 1262, June 2016

work page 2016

[8] [7]

J. R. Fern´andez, N. G. Prelcic, K. Venugopal and R. W. Heath Jr., ” Frequency-domain Compressive Channel Estimation for Frequency selective Hybrid mmWave MIMO Systems,” IEEE Trans. Wireless Commun., vol., no., pp., 2018

work page 2018