Deep Learning based Channel Estimation for Massive MIMO with Mixed-Resolution ADCs

Geoffrey Ye Li; Peihao Dong; Shen Gao; Zhiwen Pan

REVIEW 3 major objections 4 minor 18 references

Deep Learning based Channel Estimation for Massive MIMO with Mixed-Resolution ADCs

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that a deep network using only the high-resolution ADC antennas' pilot signals can predict the low-resolution antennas' channels and outperform LMMSE and EM-GM-GAMP in mixed-ADC massive MIMO.

desk verdict The SIP-DNN idea is a real contribution, but the paper's headline superiority claims are compromised by comparing curves computed with two different NMSE definitions. read the letter →

arxiv 1908.06245 v1 pith:NZUMGZPA submitted 2019-08-17 cs.IT eess.SPmath.IT

Shen Gao , Peihao Dong , Zhiwen Pan , Geoffrey Ye Li This is my paper · ORCID

classification cs.ITeess.SPmath.IT

keywords massiveMIMOmixed-ADCchannelestimationdeeplearningneuralnetworkone-bitADCuplinkspatialcorrelation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether deep learning can estimate uplink channels in a massive MIMO base station whose antenna array mixes high-resolution ADCs with cheap one-to-three-bit ADCs. It proposes two fully connected networks: DI-DNN, which feeds the pilot observations of all antennas into the network, and SIP-DNN, which feeds only the high-resolution antennas' observations and then predicts the channels of the low-resolution antennas. The central claim is that SIP-DNN achieves lower normalized MSE than the LMMSE and EM-GM-GAMP baselines across the tested SNR range, with no significant error floor at high SNR, and that it beats DI-DNN at medium and high SNR, especially when the low-resolution side uses one-bit ADCs. This matters because low-resolution ADCs cut hardware cost and power consumption, and if a trained network can predict the badly quantized antennas' channels from their undistorted neighbors, accurate estimation no longer requires every antenna to carry an expensive high-resolution converter.

What carries the argument

The central object is the selective-input prediction mapping from the channels of the high-resolution ADC antennas to the channels of the low-resolution ADC antennas. It is realized by two parallel fully connected networks: the R-DNN, which takes the least-squares estimates $\bar{r}_A$ from the high-resolution antennas and outputs refined channel estimates $\hat{h}_A$, and the MP-DNN, which maps $\bar{r}_A$ directly to predictions $\hat{h}_B$ for the low-resolution antennas. The paper replaces a serial prediction network with this parallel structure so the two sub-networks can be trained offline without backpropagating through each other. This machinery carries the argument because it removes the severely quantized signals from the estimation path entirely: instead of trying to invert one-bit quantization distortion, the network only has to exploit spatial correlation among antennas, which the paper models as a consequence of a limited number of scattering clusters and finite antenna spacing.

What would settle it

Train and test the same SIP-DNN with more propagation paths than high-resolution antennas, or with test AoAs drawn from a dense continuous range absent from training; if the trained network's normalized MSE then matches or exceeds that of LMMSE at high SNR, the central claim of learnable prediction from high-resolution antennas would be refuted.

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Extended reading notes

Core claim

The paper asserts that the channels seen by low-resolution ADC antennas are learnable functions of the channels seen by high-resolution ADC antennas in a mixed-ADC massive MIMO uplink, and that a fully connected network can learn this mapping from data. In the simulated setting—64 antennas, 8 propagation paths, a uniform linear array, and AoAs drawn from a 20-point grid—the SIP-DNN, built from an R-DNN that refines the high-resolution antenna estimates and an MP-DNN that predicts the remaining channels, attains a lower normalized MSE than LMMSE and EM-GM-GAMP at every SNR tested. Unlike methods that ingest the coarsely quantized one-bit pilots, the SIP-DNN does not exhibit a significant high-SNR error floor, because the quantized observations are excluded from its input. The paper also claims the approach is effective across different ADC resolution patterns, including random antenna placement, and that a convolutional variant offers no accuracy gain over the fully connected version.

Load-bearing premise

The standing assumption is that the low-resolution antennas' channels are a learnable function of the high-resolution antennas' channels; the paper only tests this under 8 paths and a fixed 20-angle AoA grid, so a richer scattering environment or different array geometry could make the mapping unlearnable.

Editorial extensions

If this is right

In a mixed-ADC massive MIMO uplink, the channels of one-bit ADC antennas can be estimated from the high-resolution antennas' pilots without any error floor at high SNR, so system designers can tolerate a large fraction of cheap ADCs.
A plain fully connected network is sufficient for this prediction task, since the CNN variant performs about the same; no specialized architecture is required.
Because random and block ADC resolution patterns give nearly identical NMSE, the proposed estimators do not depend on a particular antenna layout.
The preferred network depends on operating conditions: DI-DNN suits low SNR or a very small fraction of high-resolution antennas, while SIP-DNN wins at medium-to-high SNR and larger fractions.

Reading between the lines

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

Implicit in the paper but not tested: the prediction gain should shrink as the angular degrees of freedom of the channel grow, because the information carried by the high-resolution antennas becomes insufficient once the number of significant paths exceeds the number of those antennas.
A natural but unrun extension is cross-condition generalization—training on one ADC resolution pattern or SNR and testing on another—which would reveal whether the network has learned the physical antenna correlation rather than the training distribution.
In the paper's own terms, the same selective-input idea could transfer to other quantized sensor arrays: use a few fine sensors to predict many coarse ones whenever the underlying field is spatially correlated, provided the fine sensors are informative enough.

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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 genuinely new thing here is the SIP-DNN: instead of feeding all quantized observations into a network, it predicts the low-resolution ADC channels from the high-resolution ADC channels alone. That is a sensible way to avoid the distortion from one-bit quantizers, and the paper deserves credit for framing it clearly. The system model is standard, the baselines (LMMSE and EM-GM-GAMP) are appropriate, and the plots showing robustness to block versus random ADC patterns are useful. The SIP-CNN comparison is also a nice touch, showing the fully-connected architecture is sufficient.

That said, the central quantitative claims are not currently supported. The paper defines NMSE one way for DI-DNN and another way for LMMSE, EM-GM-GAMP, and SIP-DNN. Those are different quantities, and nothing in the text shows they are interchangeable for the tested channel statistics. So the claims that DI-DNN "always outperforms" LMMSE/EM-GM-GAMP and that the DI/SIP crossover picks the better architecture compare curves on different scales. The stress-test note is correct, and this is not a nitpick — it affects the paper's main conclusion.

Two other soft spots, in order of importance. First, the claimed generalization in footnote 2 is weak: the test AoAs are drawn from the same 20-point grid as the training AoAs, so the network may have memorized the grid rather than learned continuous spatial structure. Testing on a different grid, or ideally a different array geometry, would make the claim real. Second, the serial-to-parallel transformation is justified only by "simulation trails" with no detail. That might be true, but an unexplained empirical claim carries little weight. Also, no code or data are provided, and there are no error bars, which matters for a paper whose evidence is entirely experimental.

The citation pattern is fine. The LMMSE Bussgang coefficients come from [5], which shares an author, but the baseline algorithm itself is independent and standard; I do not see a circularity problem.

Who is this for? Physical-layer ML researchers working on mixed-ADC or low-resolution MIMO. They will find the selective-input idea worth taking seriously, but not the numerical superiority claims in their current form.

Recommendation: send to peer review, but insist on a revision that uses one common NMSE definition for all methods, tests on held-out channel statistics, documents the parallel-training equivalence, and ideally releases code and data. With that fixed, the paper would make a solid incremental contribution.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes two deep-neural-network-based channel estimators for massive MIMO uplink with mixed-resolution ADCs: a direct-input DNN (DI-DNN) that uses the quantized received signals from all antennas, and a selective-input prediction DNN (SIP-DNN) that uses only the high-resolution-ADC antennas' LS estimates to predict the channels of both antenna groups. The system model is a single-user, 64-antenna uniform linear array with L=8 propagation paths, and the paper compares the proposed methods against LMMSE and EM-GM-GAMP baselines under block and random ADC resolution patterns. Numerical results in Section IV claim that DI-DNN outperforms the baselines at all SNRs and that SIP-DNN outperforms DI-DNN at medium/high SNR, with crossover points in SNR and in the high-resolution antenna ratio η used as design guidance. The paper also reports robustness to ADC resolution pattern and, in a footnote, claims generalization to different channel statistics.

Significance. If the performance claims were established on a common metric, the paper would be a useful contribution: it demonstrates a data-driven alternative to model-based estimation in mixed-ADC massive MIMO, with clearly described architectures, explicit simulation settings, and comparisons to two recognized baselines. The proposed SIP-DNN idea of predicting low-resolution-ADC channels from high-resolution-ADC observations is interesting and practically motivated, and the paper gives enough detail (architectures, training hyperparameters, dataset sizes) for the experiments to be reproduced. However, the central numerical comparison is weakened by the use of two different NMSE definitions for the methods being compared, and a key architectural-equivalence claim in Section III.B is unverified. These issues are load-bearing for the paper's headline claims, so the current evidence does not support the stated conclusions.

major comments (3)

[Section IV, NMSE definitions and Fig. 3] The paper compares DI-DNN against LMMSE, EM-GM-GAMP, and SIP-DNN using two different normalized mean-squared-error statistics. For DI-DNN the NMSE is E{||h−ĥ||²/||h||²}, while for LMMSE, EM-GM-GAMP, and SIP-DNN it is η·E{||hA−ĥA||²/||hA||²} + (1−η)·E{||hB−ĥB||²/||hB||²}. These are not the same quantity, and no argument is given that they are interchangeable for the simulated channel statistics. Consequently, the claims in Fig. 3(a) that DI-DNN always outperforms LMMSE and EM-GM-GAMP, and the crossover analysis in Fig. 3(b) that selects between DI-DNN and SIP-DNN, compare curves measured on different scales. The crossover in particular could be an artifact of the different normalizations rather than a genuine performance ordering. Please report results with one common NMSE definition, for example E{||h−ĥ||²/||h||²} for every method, or instead present per-antenna-set errors separately for all methods.
[Section III.B, paragraph after Eq. (9)] The manuscript states that replacing the serial R-DNN plus P-DNN structure with the parallel R-DNN plus MP-DNN structure 'will not cause performance loss but facilitate the offline training significantly,' yet no simulation result, figure, or analytical argument in the paper supports this claim. Since all SIP-DNN numerical results are obtained with the parallel MP-DNN configuration, the claimed performance of SIP-DNN depends on this unverified equivalence. Please supply supporting evidence, or explicitly weaken the claim to state that the parallel structure is an approximation whose performance is evaluated empirically.
[Footnote 2 and Section IV first paragraph] The generalization statement in footnote 2 is not supported by the experimental design. The AoAs in both the training and testing sets are drawn from the same discrete set of 20 angles, namely 2π/20 × {0,1,...,19}; with 90,000 training samples and only 20 possible angles per path, the training set almost certainly contains all grid values many times. Thus the testing procedure does not demonstrate that the networks 'can learn the inherent channel structure and are suitable for different channel statistics.' It only shows generalization to different random combinations of the same discrete angle grid. Please test on angles outside the training grid, or on different values of L, different array geometries, or different spatial correlation models, before making the broader generalization claim.

minor comments (4)

[Section III.B, text near Eq. (7)] There is a typo: 'simulation trails' should be 'simulation trials.'
[Section IV, first paragraph] The LMMSE baseline description says 'liner minimum mean-squared error'; this should be 'linear minimum mean-squared error.'
[Section IV and Fig. 3] The SIP-CNN architecture is mentioned but not specified in the same detail as the DNN architectures; a sentence describing its layers, kernel sizes, and number of parameters would make the comparison reproducible.
[Section IV, Fig. 4 caption] The caption says 'mixed 1, 2, 3 bits' but the main text explains the patterns; it would be clearer to state in the caption that the figure combines results for mixed 1-bit, 2-bit, and 3-bit ADC cases.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the DNN training and evaluation are self-contained; the shared-author citation supplies only a baseline coefficient.

full rationale

The paper's channel estimation methods are built by standard supervised learning: DI-DNN minimizes the MSE between its output and the true channel, and SIP-DNN trains R-DNN and MP-DNN on labeled pairs (r_A, h_A) and (r_A, h_B), respectively. No parameter fitted to a subset of data is renamed as a prediction, and no defining equation reduces a claimed output to its input. The only shared-author citation, [5], provides Bussgang linearization coefficients used in the LMMSE baseline; this is an external published result and does not determine the DNN outputs or the central claim. The held-out test set uses different random realizations from the same discrete AoA grid, which limits generalization claims but is not circular, since the DNNs are not evaluated on the training samples. The paper's separate NMSE definitions for DI-DNN versus LMMSE/EM-GM-GAMP/SIP-DNN are a comparison-validity concern rather than a circular derivation: the figures may compare curves on different scales, but this does not make any result equivalent to its inputs by construction. Accordingly, no circular step is identified.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The channel-estimation problem is entirely supervised, so the main free parameters are design choices: the output scaling constant c=3 and the hand-chosen hidden layer sizes per eta. None are fitted to an external benchmark. The axioms are the synthetic channel model, the ideal high-resolution ADC assumption, and the spatial-correlation premise that makes predicting the low-resolution ADC channels from the high-resolution ADC inputs learnable. No new physical entities are introduced; the DNNs are algorithms, not entities with independent falsifiable handles.

free parameters (2)

Output scaling constant c = 3
Section III.A: 'c > 0 is a scaling constant to make the range of all target data match the tangent activation function of the output layer.' Chosen by hand; the paper gives no criterion for its value.
Hidden layer neuron counts for R-DNN and MP-DNN per eta = Table II values, e.g., 50/100/50 for R-DNN at eta=0.2
Footnote 1: 'We set different numbers of neurons in hidden layers for R-DNN and MP-DNN in different values of eta to match the corresponding input and output dimensions.' These are hand-chosen architecture sizes, not derived from an optimization.

assumptions (4)

domain assumption Multipath channel model with L=8 paths, AoAs drawn uniformly from a fixed 20-direction grid, and gains alpha_l ~ CN(0, sigma^2_alpha) with unit variance.
Invoked through Eq. (1) and the simulation setup in Section IV. All training, validation, and test samples are generated from this model, so performance under other channel statistics is not measured.
domain assumption High-resolution ADCs are ideal, i.e., their quantization error is ignored.
Section II, Eq. (4): 'we have ignored the quantization error of the high-resolution ADCs in antenna set A.' This makes the high-resolution inputs clean by construction.
domain assumption The channel has low enough spatial dimensionality that the low-resolution ADC channels can be predicted from high-resolution ADC observations.
Section III.A: 'the limited number of scattering clusters in the propagation environment and finite physical space between antennas at the BS introduce the correlation among the received signal of each antenna.' The whole SIP-DNN approach depends on this learnable mapping.
domain assumption Bussgang decomposition coefficients alpha for the LMMSE baseline are taken from [5, Table I].
Section IV, LMMSE formula. [5] shares an author with this paper; the coefficients are presented without derivation here, but they are standard quantizer correction constants in the low-resolution ADC literature.

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Cite this review

Pith. "Pith review of Deep Learning based Channel Estimation for Massive MIMO with Mixed-Resolution ADCs." pith.science (2026). https://pith.science/paper/NZUMGZPA

@misc{pith2026190806245,
  author       = {Pith},
  title        = {Pith review of: Deep Learning based Channel Estimation for Massive MIMO with Mixed-Resolution ADCs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NZUMGZPA}},
  note         = {Machine review of arXiv:1908.06245}
}

read the original abstract

In this article, deep learning is applied to estimate the uplink channels for mixed analog-to-digital converters (ADCs) massive multiple-input multiple-output (MIMO) systems, where a portion of antennas are equipped with high-resolution ADCs while others employ low-resolution ones at the base station. A direct-input deep neural network (DI-DNN) is first proposed to estimate channels by using the received signals of all antennas. To eliminate the adverse impact of the coarsely quantized signals, a selective-input prediction DNN (SIP-DNN) is developed, where only the signals received by the high-resolution ADC antennas are exploited to predict the channels of other antennas as well as to estimate their own channels. Numerical results show the superiority of the proposed DNN based approaches over the existing methods, especially with mixed one-bit ADCs, and the effectiveness of the proposed approaches on different ADC resolution patterns.

Figures

Figures reproduced from arXiv: 1908.06245 by the authors.

**Figure 2.** DNN based channel estimation. Without loss of generality, we set x = 1 since it is known to the BS. Then the received pilots at the BS is written as y = √ Ph + n, (3) where n denotes the additive white Gaussian noise vector at the BS with independent and identically distributed CN (0, σ2 0 ) elements. Then the elements of y corresponding to A and B are quantized by the high- and low-resolution uniform quantizers, re… view at source ↗

**Figure 3.** NMSE for LMMSE, EM-GM-GAMP, and the proposed NN based [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: NMSE for the proposed DNN based approaches with block [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

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

[1] [5]

Efﬁcient low-resolu tion ADC relaying for multiuser massive MIMO system,

P . Dong, H. Zhang, W. Xu, and X. Y ou, “Efﬁcient low-resolu tion ADC relaying for multiuser massive MIMO system,” IEEE Trans. V eh. Technol., vol. 66, no. 12, pp. 11039 − 11056, Dec. 2017

work page 2017

[2] [1]

An overview of massive MIMO: beneﬁts and challenges,

L. Lu, G. Y . Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zh ang, “An overview of massive MIMO: beneﬁts and challenges,” IEEE J. Sel. Topics in Signal Process. , vol. 8, no. 5, pp. 742 − 758, Oct. 2014

work page 2014

[3] [2]

Low-complexity hybrid prec oding in massive multiuser MIMO systems,

L. Liang, W. Xu, and X. Dong, “Low-complexity hybrid prec oding in massive multiuser MIMO systems,” IEEE Wireless Commun. Lett. , vol. 3, no. 6, pp. 653 − 656, Dec. 2014

work page 2014

[4] [3]

Hybrid precoding-b ased millimeter-wave massive MIMO-NOMA with simultaneous wire less information and power transfer,

L. Dai, B. Wang, M. Peng, and S. Chen, “Hybrid precoding-b ased millimeter-wave massive MIMO-NOMA with simultaneous wire less information and power transfer,” IEEE J. Sel. Areas Commun. , vol. 37, no. 1, pp. 131 − 141, Jan. 2019

work page 2019

[5] [4]

Uplink achievabl e rate for massive MIMO systems with low-resolution ADC,

L. Fan, S. Jin, C.-K. Wen, and H. Zhang, “Uplink achievabl e rate for massive MIMO systems with low-resolution ADC,” IEEE Commun. Lett., vol. 19, no. 12, pp. 2186 − 2189, Dec. 2015

work page 2015

[6] [6]

Mixed-ADC massive MIMO,

N. Liang and W. Zhang, “Mixed-ADC massive MIMO,” IEEE J. Sel. Areas Commun. , vol. 34, no. 4, pp. 983 − 997, Apr. 2016

work page 2016

[7] [7]

On low-resol ution ADCs in practical 5G millimeter-wave massive MIMO systems,

J. Zhang, L. Dai, X. Li, Y . Liu, and L. Hanzo, “On low-resol ution ADCs in practical 5G millimeter-wave massive MIMO systems, ” IEEE Commun. Mag. , vol. 56, no. 7, pp. 205 − 211, Jul. 2018

work page 2018

[8] [8]

Performance ana lysis of mixed-ADC massive MIMO systems over Rician fading channels ,

J. Zhang, L. Dai, Z. He, S. Jin, and X. Li, “Performance ana lysis of mixed-ADC massive MIMO systems over Rician fading channels ,” IEEE J. Sel. Areas Commun. , vol. 35, no. 6, pp. 1327 − 1338, Jun. 2017

work page 2017