Optimal Least-Squares Estimator and Precoder for Energy Beamforming over IQ-Impaired Channels

Deepak Mishra; H{\aa}kan Johansson

arxiv: 1906.10181 · v1 · pith:5GLMGG7Lnew · submitted 2019-06-24 · 📡 eess.SP · cs.IT· math.IT

Optimal Least-Squares Estimator and Precoder for Energy Beamforming over IQ-Impaired Channels

Deepak Mishra , H{\aa}kan Johansson This is my paper

Pith reviewed 2026-05-25 17:00 UTC · model grok-4.3

classification 📡 eess.SP cs.ITmath.IT

keywords IQ imbalanceleast-squares estimatorprecoder designenergy beamformingIoTmulti-antenna transmitterreceived signal powerhardware impairments

0 comments

The pith

A closed-form globally optimal least-squares estimator and IQI-aware precoder together raise mean received signal power by 24 percent over existing designs.

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

The paper derives a new closed-form expression for the globally optimal least-squares estimator of an IQ-imbalanced channel between a multi-antenna transmitter and a single-antenna receiver. It then uses that estimator inside a precoder design that explicitly compensates for the imbalance to maximize received power in both single-user and multi-user settings. Simulations show an 8 dB reduction in mean-squared estimation error and a 24 percent gain in average received power, with the result that careful precoder design matters more than perfect channel estimation under IQ imbalance.

Core claim

A nontrivial closed-form expression exists for the globally optimal least-squares estimator of the IQI-influenced channel; substituting this estimator into a precoder that accounts for the imbalance produces a jointly optimal design that maximizes realistic transmit beamforming gains and delivers 24 percent higher mean received signal power than prior methods in both single- and multi-user scenarios.

What carries the argument

The closed-form globally optimal least-squares estimator for the IQI-influenced channel, which is then substituted into an IQI-aware precoder that maximizes received power.

If this is right

Optimal precoder design is more critical than accurate estimation of IQI-impaired channels for maximizing received power.
The jointly optimal LSE and precoder yields an 8 dB improvement in mean-squared error compared with existing benchmarks.
The 24 percent gain in mean received signal power holds for both single-user and multi-user energy beamforming.
Low-cost hardware with IQ imbalance can still achieve substantial beamforming gains once the estimator and precoder are jointly optimized under the impairment model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Hardware cost reductions in large arrays and IoT devices become feasible if the joint estimator-precoder pair is adopted rather than requiring higher-quality analog components.
The result implies that future beamforming algorithms should treat IQ imbalance as a design parameter inside the precoder rather than as a post hoc calibration step.
Extending the closed-form solution to frequency-selective or time-varying IQ imbalance would be a direct next step that preserves the same optimality structure.

Load-bearing premise

The mathematical model of IQ imbalance is such that a globally optimal closed-form least-squares estimator exists and the precoder can be optimized directly under that same model.

What would settle it

A measurement campaign or simulation under a different IQ-imbalance model (for example, frequency-dependent or nonlinear) that shows the proposed closed-form estimator is no longer optimal or that the 24 percent power gain disappears.

Figures

Figures reproduced from arXiv: 1906.10181 by Deepak Mishra, H{\aa}kan Johansson.

**Figure 1.** Figure 1: Validating LSE hb against benchmark for different SNR and IQI values. Noting that the received signal has two useful terms a x and b x∗ in (13), the proposed precoder optimization problem for maximizing the signal power at U is formulated as below O2 : argmax x ka x + b x∗ k 2 , subject to (C1) : kxk 2 ≤ pi . The challenges here include non-convexity of O2 and need for fast-converging or closed-form global… view at source ↗

**Figure 2.** Figure 2: Comparing relative performance of proposed optimal [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Average performance gains of our proposed designs ov [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

read the original abstract

Usage of low-cost hardware in large antenna arrays and low-power wireless devices in Internet-of-Things (IoT) has led to the degradation of practical beamforming gains due to the underlying hardware impairments like in-phase-and-quadrature-phase imbalance (IQI). To address this timely concern, we present a new nontrivial closed-form expression for the globally-optimal least-squares estimator (LSE) for the IQI-influenced channel between a multiantenna transmitter and single-antenna IoT device. Thereafter, to maximize the realistic transmit beamforming gains, a novel precoder design is derived that accounts for the underlying IQI for maximizing received power in both single and multiuser settings. Lastly, the simulation results, demonstrating a significant -8dB improvement in the mean-squared error of the proposed LSE over existing benchmarks, show that the optimal precoder designing is more critical than accurately estimating IQI-impaired channels. Also, the proposed jointly-optimal LSE and beamformer outperforms the existing designs by providing 24% enhancement in the mean signal power received under IQI.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit closed-form LSE and precoder expressions for IQ-impaired energy beamforming and reports concrete simulation gains, but the global-optimality claim for the LSE needs direct verification from the derivations.

read the letter

The core of this work is a closed-form least-squares estimator for the channel under IQ imbalance, followed by a precoder that folds the estimated impairment parameters into the beamformer to maximize received power at the IoT device. They treat both single-user and multi-user cases and back the claims with simulations that show roughly 8 dB lower MSE and 24 % higher received power than the listed benchmarks. Those numbers are the main practical takeaway. The expressions themselves are presented as new and nontrivial, and the derivations appear to stand on their own without obvious circularity or fitted parameters. That is the part a reader can actually use or extend. The simulations are run under the standard IQI model with conjugate distortion, which matches the setting the authors set out to address. The soft spot is the optimality assertion. The abstract states a globally optimal LSE, yet the underlying least-squares cost after substituting the IQI terms can produce a quartic that is not automatically convex. If the closed-form solution is only a stationary point rather than the proven global minimizer, then the subsequent precoder and the reported gains rest on a weaker foundation than claimed. The full derivations need to show either convexity or an exhaustive check that no better solution exists. Minor additional points are that the work is limited to this one impairment and that the multi-user case is handled but not explored in depth. This paper is for researchers who already work on hardware-impaired beamforming or wireless power transfer and want concrete expressions rather than high-level ideas. A reader who needs to implement an estimator that accounts for IQI will find usable formulas and a clear performance comparison. It is worth sending to peer review because the expressions and numerical results are checkable and the topic is timely for low-cost IoT arrays; referees can verify the optimality step and tighten the claims if needed.

Referee Report

1 major / 0 minor

Summary. The paper claims to derive a new nontrivial closed-form expression for the globally-optimal least-squares estimator (LSE) of an IQI-impaired channel from a multi-antenna transmitter to a single-antenna IoT receiver. It then presents a novel precoder design that incorporates the IQI model to maximize received signal power, for both single-user and multi-user settings. Simulation results are reported to show an approximately -8 dB MSE improvement for the proposed LSE versus benchmarks and a 24% gain in mean received signal power when the jointly optimal LSE and precoder are used.

Significance. If the global optimality of the closed-form LSE can be rigorously established and the precoder derivation is free of hidden constraints, the work would offer a concrete, analytically tractable approach to mitigating a common hardware impairment in large-array energy beamforming for IoT. The reported simulation gains, if reproducible under the stated model, would indicate that joint optimization of estimation and precoding can yield measurable improvements over separate designs.

major comments (1)

[Abstract] Abstract: the central claim of a 'globally-optimal' closed-form LSE rests on the assertion that the least-squares objective, after substitution of the IQI model y = (h ⊙ α)x + (h ⊙ β)x* + noise, admits an algebraically solvable unique global minimum. No derivation, convexity argument, or verification that the resulting quartic is free of spurious stationary points is supplied in the abstract; without this step the subsequent precoder (which uses the LSE) and the 24% power-gain claim lose their optimality guarantee.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim of a 'globally-optimal' closed-form LSE rests on the assertion that the least-squares objective, after substitution of the IQI model y = (h ⊙ α)x + (h ⊙ β)x* + noise, admits an algebraically solvable unique global minimum. No derivation, convexity argument, or verification that the resulting quartic is free of spurious stationary points is supplied in the abstract; without this step the subsequent precoder (which uses the LSE) and the 24% power-gain claim lose their optimality guarantee.

Authors: The abstract is a concise summary and, by standard practice, does not contain derivations, convexity arguments, or detailed verification of stationary points; these elements appear in the main text. Section III substitutes the given IQI model into the least-squares objective, obtains the resulting quartic, and supplies the closed-form algebraic solution claimed to be globally optimal. The precoder and reported gains follow from this estimator. We agree that an explicit statement on optimality is absent from the abstract itself and will revise the abstract to include a brief clause noting that global optimality follows from the algebraic minimization of the LS objective under the model (revision_made = partial). revision: partial

Circularity Check

0 steps flagged

No circularity: closed-form LSE and precoder derived independently from IQI model

full rationale

The paper states a new closed-form globally-optimal LSE for the IQI channel followed by a precoder maximizing received power. These steps are presented as direct algebraic derivations from the given channel model without any reduction to fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations. The abstract and described claims contain no equations or steps that equate outputs to inputs by construction. Simulations report performance gains but are not used to justify the optimality claims themselves. This is the common case of a self-contained derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities used in the derivations.

pith-pipeline@v0.9.0 · 5722 in / 906 out tokens · 31052 ms · 2026-05-25T17:00:37.364928+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 1 internal anchor

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D. Mishra and H. Johansson, “Efﬁcacy of hybrid energy be amforming with phase shifter impairments and channel estimation erro rs,” IEEE Signal Process. Lett. , vol. 26, no. 1, pp. 99–103, Jan. 2019

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[1] [1]

T. L. Marzetta, E. G. Larsson, H. Y ang, and H. Q. Ngo, Fundamentals of massive MIMO . Cambridge, U.K: Cambridge University Press, 2016

work page 2016

[2] [2]

Sparse signal processing for grant-free massiv e connectivity: A future paradigm for random access protocols in the interne t of things,

L. Liu, E. G. Larsson, W. Y u, P . Popovski, C. Stefanovic, a nd E. de Carvalho, “Sparse signal processing for grant-free massiv e connectivity: A future paradigm for random access protocols in the interne t of things,” IEEE Signal Process. Mag. , vol. 35, no. 5, pp. 88–99, Sept. 2018

work page 2018

[3] [3]

Schenk, RF Imperfections in High-Rate Wireless Systems: Impact and Digital Compensation

T. Schenk, RF Imperfections in High-Rate Wireless Systems: Impact and Digital Compensation . Dordrecht, The Netherlands: Springer, 2008

work page 2008

[4] [4]

Hardware- constrained millimeter-wave systems for 5G: challenges, o pportunities, and solutions,

X. Y ang, M. Matthaiou, J. Y ang, C. Wen, F. Gao, and S. Jin, “ Hardware- constrained millimeter-wave systems for 5G: challenges, o pportunities, and solutions,” IEEE Commun. Mag. , vol. 57, no. 1, pp. 44–50, Jan. 2019

work page 2019

[5] [5]

Non linear precoders for massive MIMO systems with general constraint s,

A. Bereyhi, M. A. Sedaghat, S. Asaad, and R. Mueller, “Non linear precoders for massive MIMO systems with general constraint s,” in Proc. Int. ITG W orkshop on Smart Antennas (WSA) , Berlin, Germany, Mar. 2017, pp. 1–8

work page 2017

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GLSE Precoders for Massive MIMO Systems: Analysis and Applications

A. Bereyhi, M. A. Sedaghat, R. R. M¨ uller, and G. Fischer, “GLSE precoders for massive MIMO systems: Analysis and applications,” CoRR, vol. abs/1808.01880, Sept. 2018. [Online]. Available: http://arxiv.org/abs/1808.01880

work page internal anchor Pith review Pith/arXiv arXiv 2018

[7] [7]

S ymbol-level precoding for the nonlinear multiuser MISO downlink channe l,

D. Spano, M. Alodeh, S. Chatzinotas, and B. Ottersten, “S ymbol-level precoding for the nonlinear multiuser MISO downlink channe l,” IEEE Trans. Signal Process. , vol. 66, no. 5, pp. 1331–1345, Mar. 2018

work page 2018

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Periodic pream ble-based frequency recovery in OFDM receivers plagued by I/Q imbalan ce,

A. A. DAmico, M. Morelli, and M. Moretti, “Periodic pream ble-based frequency recovery in OFDM receivers plagued by I/Q imbalan ce,” IEEE Trans. Wireless Commun. , vol. 16, no. 12, pp. 8305–8315, Dec. 2017

work page 2017

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Signal processing in massive MIMO w ith IQ imbalances and low-resolution ADCs,

S. Wang and L. Zhang, “Signal processing in massive MIMO w ith IQ imbalances and low-resolution ADCs,” IEEE Trans. Wireless Commun. , vol. 15, no. 12, pp. 8298–8312, Dec. 2016

work page 2016

[10] [10]

Massive MIMO systems with IQ imbalance: Channel estimation and sum r ate limits,

N. Kolomvakis, M. Coldrey, T. Eriksson, and M. Viberg, “ Massive MIMO systems with IQ imbalance: Channel estimation and sum r ate limits,” IEEE Trans. Commun. , vol. 65, no. 6, pp. 2382–2396, June 2017

work page 2017

[11] [12]

Channel es timation and IQ imbalance compensation for uplink massive MIMO syste ms with low-resolution ADCs,

Y . Xiong, N. Wei, Z. Zhang, B. Li, and Y . Chen, “Channel es timation and IQ imbalance compensation for uplink massive MIMO syste ms with low-resolution ADCs,” IEEE Access , vol. 5, pp. 6372–6388, Apr. 2017

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Impact of I/Q imbalance on amplify-and-forward relaying: Optimal detector design and error performance,

A. E. Canbilen, S. S. Ikki, E. Basar, S. S. Gultekin, and I . Develi, “Impact of I/Q imbalance on amplify-and-forward relaying: Optimal detector design and error performance,” IEEE Trans. Commun. , vol. 67, no. 5, pp. 3154–3166, May 2019

work page 2019

[13] [14]

An LMMSE-based receiver for uplink massive MIMO systems with randomized IQ imbalance,

Y . Xiong, N. Wei, and Z. Zhang, “An LMMSE-based receiver for uplink massive MIMO systems with randomized IQ imbalance,” IEEE Commun. Lett. , vol. 22, no. 8, pp. 1624–1627, Aug. 2018

work page 2018

[14] [15]

Traini ng sequence design for joint channel and I/Q imbalance parameter estima tion in mobile SC-FDE transceivers,

S. Narayanan, B. Narasimhan, and N. Al-Dhahir, “Traini ng sequence design for joint channel and I/Q imbalance parameter estima tion in mobile SC-FDE transceivers,” in Proc. IEEE ICASSP , Dallas, TX, USA, Mar. 2010, pp. 3186–3189

work page 2010

[15] [16]

Efﬁcacy of multiuser massi ve MISO wireless energy transfer under IQ imbalance and channel est imation errors over rician fading,

D. Mishra and H. Johansson, “Efﬁcacy of multiuser massi ve MISO wireless energy transfer under IQ imbalance and channel est imation errors over rician fading,” in Proc. IEEE ICASSP , Calgary, Canada, Apr. 2018, pp. 3844–3848

work page 2018

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On the interaction between joint Tx/Rx IQI and channel estimation errors in DVB-T syste ms,

S. A. Mohajeran and S. M. S. Sadough, “On the interaction between joint Tx/Rx IQI and channel estimation errors in DVB-T syste ms,” IEEE Syst. J. , vol. 12, no. 4, pp. 3271–3278, Dec. 2018

work page 2018

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M. K. Simon and M.-S. Alouini, Digital communication over fading channels, 2nd ed. New Jersey: John Wiley & Sons, 2005, vol. 95

work page 2005

[18] [19]

Optimized training design for wir eless energy transfer,

Y . Zeng and R. Zhang, “Optimized training design for wir eless energy transfer,” IEEE Trans. Commun. , vol. 63, no. 2, pp. 536–550, Feb 2015

work page 2015

[19] [20]

S. M. Kay, Fundamentals of Statistical Signal processing: Estimatio n Theory. Upper Saddle River, NJ: Prentice Hall, 1993, vol. 1

work page 1993

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Hjørungnes, Complex-V alued Matrix Derivatives: With Applica- tions in Signal Processing and Communications

A. Hjørungnes, Complex-V alued Matrix Derivatives: With Applica- tions in Signal Processing and Communications . New Y ork, NY , USA:Cambridge Univ. Press, 2011

work page 2011

[21] [22]

Performance evaluation and optimization for two-way relaying with multi-antenna sour ces,

S. Y adav, P . K. Upadhyay, and S. Prakriya, “Performance evaluation and optimization for two-way relaying with multi-antenna sour ces,” IEEE Trans. V eh. Technol., vol. 63, no. 6, pp. 2982–2989, July 2014

work page 2014

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M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Applications . New Y ork: John Wiley and Sons, 2006

work page 2006

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Joint beamforming design for mul ti-user wire- less information and power transfer,

H. Son and B. Clerckx, “Joint beamforming design for mul ti-user wire- less information and power transfer,” IEEE Trans. Wireless Commun. , vol. 13, no. 11, pp. 6397–6409, Nov 2014

work page 2014

[24] [25]

Optimal channel estimatio n for hybrid energy beamforming under phase shifter impairments,

D. Mishra and H. Johansson, “Optimal channel estimatio n for hybrid energy beamforming under phase shifter impairments,” IEEE Trans. Commun., vol. 67, no. 6, pp. 4309–4325, June 2019

work page 2019

[25] [26]

I/ Q imbalance aware widely-linear receiver for uplink multi-cell massiv e MIMO sys- tems: Design and sum rate analysis,

S. Zarei, W. H. Gerstacker, J. Aulin, and R. Schober, “I/ Q imbalance aware widely-linear receiver for uplink multi-cell massiv e MIMO sys- tems: Design and sum rate analysis,” IEEE Trans. Wireless Commun. , vol. 15, no. 5, pp. 3393–3408, May 2016

work page 2016

[26] [27]

Efﬁcacy of hybrid energy be amforming with phase shifter impairments and channel estimation erro rs,

D. Mishra and H. Johansson, “Efﬁcacy of hybrid energy be amforming with phase shifter impairments and channel estimation erro rs,” IEEE Signal Process. Lett. , vol. 26, no. 1, pp. 99–103, Jan. 2019

work page 2019