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Variational Bayesian Data Detection for Multiuser MIMO Systems Corrupted by Phase Noises

Toan-Van Nguyen, Duy H. N. Nguyen

arXiv:2609.08252v1cs.ITmath.IT

TL;DR

Independent oscillators and phase-noise distortion complicate uplink multiuser MIMO detection, while exact maximum-likelihood detection becomes computationally difficult at scale. The paper applies variational Bayes to jointly infer symbols and phase noise, with iterative updates for phase estimates. Its improved detector has lower symbol error rates than competing methods in the reported channel and modulation settings.

  • Problem

    Independent oscillators create a high-dimensional phase-noise process, and exact maximum-likelihood detection requires an intractable phase-noise integral and a symbol search that grows with user count.

    Source: §I · p. 1, A. System Description · p. 3

  • Method

    The VB updates use a residual formed from current estimates and update a phase posterior whose mean is steered by the likelihood and whose concentration increases with observed evidence.

    Source: A. MF-VB Algorithm · p. 4, §IV · p. 8

  • Results

    The improved MF-VB achieves the lowest SER across the reported i.i.d. and correlated Rayleigh fading and mmWave settings, with a greater advantage at high SNR, high-order modulation, and stronger phase noise.

    Source: §VI · p. 13

  • Takeaways & Limitations

    The improved MF-VB’s advantage over SIW is more pronounced under 64-QAM, underscoring the value of an accurate transmit-signal posterior for high-order constellations.

    Source: §V · p. 11

Abstract

from arXiv · show

Phase noise (PN), arising from imperfect local oscillators, introduces multiplicative distortions that degrade the performance of communication systems. In uplink multiuser multiple-input multiple-output (MIMO) systems, this impairment is further compounded by the presence of independent oscillators at each transmit and receive antenna, each contributing an uncorrelated noise component. Existing PN compensation algorithms at the receiver either rely on linearization approximations that lose accuracy under severe PN conditions, or incur computational complexity that scales prohibitively with the number of antennas. To address these limitations, we propose a variational Bayes (VB) framework for joint PN estimation and data detection in uplink MIMO systems. We develop VB-based detectors that treat noise statistics as latent variables, and reformulate the inference problem by absorbing the transmitter PN into the transmitted signal, treating the resulting composite variable as the inference target. Under von Mises priors, this reformulation yields exact closed-form conjugate posterior updates, from which we derive an improved detector achieving superior performance at low complexity. Simulation results demonstrate that the proposed VB algorithm achieves lower symbol error rates than the Self-Interference Whitening (SIW) algorithm and conventional phase-noise-unaware detectors across a wide range of channel conditions, modulation orders, and PN severities, while remaining computationally scalable to large MIMO deployments.

I. INTRODUCTION

Independent transmitter and receiver oscillators make phase-noise compensation difficult in multiuser MIMO, while existing methods trade accuracy under strong phase noise against computational tractability. The paper develops closed-form variational Bayes detectors, including a reformulation that absorbs transmitter phase noise into the inferred signal.

I. INTRODUCTION

  • In massive MIMO, independent oscillators create a high-dimensional phase-noise process coupled with interference and discrete symbols, making exact maximum-likelihood detection computationally intractable.The challenge is especially pronounced for high-order constellations and at mmWave and terahertz frequencies, and in geostationary satellite links.Source: §I · p. 1

A. Motivations and Contributions

  • Linearization-based methods lose accuracy as phase-noise variance increases, while factor-graph methods can impose high complexity and approximate methods can suffer posterior mismatch.The section contrasts these accuracy–tractability trade-offs across prior approaches.Source: §I · pp. 1–2, A. Motivations and Contributions · p. 2
  • The proposed VB framework absorbs transmitter phase noise into the transmitted signal, avoiding the posterior mismatch of treating phase noise and data symbols as independent inference targets.Under a von Mises prior, the composite variable has an exact closed-form conjugate posterior as a finite mixture of von Mises distributions.Source: A. Motivations and Contributions · p. 2
  • MF-VB uses von Mises phase priors and latent noise precision, while LMMSE-VB models the full noise covariance to capture residual inter-user interference.The contribution list also states that MF-VB and improved MF-VB scale linearly with users, receive antennas, constellation size, and iterations.Source: A. Motivations and Contributions · p. 2

A. System Description

The system model considers an uplink with K single-antenna users and an M-antenna base station, with phase noise affecting transmitted and received signals. Detection involves high-dimensional phase-noise integration and a constellation search that grows with the number of users.

A. System Description

  • The uplink comprises K single-antenna users transmitting symbols from a discrete constellation to a base station with M antennas.The symbols have zero mean and normalized power.Source: A. System Description · p. 2
  • User symbols are perturbed by transmitter phase noise, and the received signal also includes base-station phase noise and Gaussian background noise.The model assumes a known channel matrix and describes channel vectors using covariance matrices that can represent spatial correlation and large-scale fading.Source: A. System Description · p. 3, Figure 1 · p. 3
  • Exact maximum-likelihood detection requires a high-dimensional phase-noise integral and a search over |A|^K symbol combinations, making it difficult for massive MIMO.If all base-station antennas share one oscillator, the model has fewer phase-noise parameters to estimate, and the proposed VB frameworks can be adopted for that scenario.Source: A. System Description · p. 3

B. Preliminaries

The preliminaries establish notation and probability identities used in the variational derivations, including parameters for von Mises variables and expectations of quadratic forms.

B. Preliminaries

  • The section introduces preliminary results from real analysis and probability for use in the subsequent derivations.It also states a large-concentration approximation involving the modified Bessel function of the first kind.Source: B. Preliminaries · p. 3
  • For a von Mises random variable, the mean direction is the angle of its natural parameter, and its concentration is the parameter magnitude, analogous to inverse variance.The expectation and variance of e^jnx are also introduced, with expressions involving a modified Bessel function.Source: B. Preliminaries · p. 3
  • The preliminaries give identities for expectations involving independent random variables and their variational means and covariances.One stated identity concerns the expected quadratic form (y − Ax)^H B(y − Ax) for Hermitian B.Source: B. Preliminaries · p. 3

III. VARIATIONAL BAYES FOR MIMO DETECTION WITH PHASE NOISES

This section develops variational Bayes methods to infer transmitted symbols and phase noises from the received observation when the channel is known. It considers estimating the input signal under postulated scalar-noise and full-covariance models.

III. VARIATIONAL BAYES FOR MIMO DETECTION WITH PHASE NOISES

  • The VB methods jointly infer transmitted symbols, transmitter phase noises, and receiver phase noises from the observation, assuming a known channel.Source: §III · p. 3
  • The section estimates the input signal under two postulated noise models: a noise variance and a noise covariance matrix.The covariance-model description continues in the next passage.Source: §III · p. 3
  • The methods account for noise statistics that are not known a priori by estimating them as part of the inference.Source: §III · p. 3

A. MF-VB Algorithm

The MF-VB algorithm uses coordinate-ascent variational updates with von Mises phase priors and a Gamma noise-precision prior. A residual vector carries current estimates into the updates while limiting computation to the contribution of the most recently updated variable.

A. MF-VB Algorithm

  • MF-VB obtains variational factors by minimizing KL divergence to the true posterior, updating each variable using expectations over the others.The framework uses a mean-field factorization and a Gamma prior for noise precision.Source: A. MF-VB Algorithm · p. 4
  • Von Mises priors for transmitter and receiver phase noise yield von Mises variational factors for their respective phase updates.The phase means and concentrations are updated from the prior and likelihood parameters.Source: A. MF-VB Algorithm · p. 4
  • The residual r uses current variational estimates, and updating only the latest variable’s contribution avoids recomputing the whole vector.The algorithm description identifies full residual-vector updates as a source of high complexity.Source: A. MF-VB Algorithm · pp. 4–5

B. LMMSE-VB Algorithm

The LMMSE-VB algorithm treats the noise precision matrix as random and approximates the posterior with mean-field variational factors. It updates phase, symbol, and noise-covariance variables, with a special update described for a single base-station phase-noise source.

B. LMMSE-VB Algorithm

  • LMMSE-VB treats the noise precision matrix Γ = (Cpost)^−1 as a random matrix.The algorithm is presented with a postulated noise covariance and von Mises prior.Source: B. LMMSE-VB Algorithm · pp. 5, 7
  • Its mean-field updates cover receiver phase, transmitter phase, symbols, and the noise-precision matrix.The variational distributions are updated for φm, θi, si, and Γ.Source: B. LMMSE-VB Algorithm · pp. 5–6
  • For one phase-noise source at the base station, the algorithm modifies both the phase update and the precision-matrix update.The special-case update refers to the previous iteration’s phase estimate.Source: B. LMMSE-VB Algorithm · p. 6

IV. VARIATIONAL BAYES FOR INFERRING x

This section absorbs transmitter phase noise into the transmitted signal, defining a composite variable x as the inference target instead of estimating s and θ separately. The reformulation yields tractable closed-form posteriors, including a finite mixture of von Mises distributions, and a posterior mean for x.

IV. VARIATIONAL BAYES FOR INFERRING x

  • Inferring the composite variable x instead of s and θ separately yields tractable closed-form posteriors and is presented as reducing posterior mismatch and improving detection.The transmitter phase noise is absorbed into the transmitted signal.Source: §IV · p. 7
  • The posterior signal distribution is a finite mixture of von Mises distributions on concentric circles with radii {|sk|}.The theorem also gives the prior of x and the symbol posterior p(sk | z).Source: §IV · p. 7
  • As γ increases, the likelihood pulls the phase posterior mean toward ∠z−∠sk, and its concentration is always at least the prior concentration.The posterior natural parameter combines the baseline phase uncertainty with a matched-filter term.Source: §IV · p. 8
  • For each symbol component, the posterior mean of x shrinks its amplitude by dk and rotates it to phase ∠sk + ∠ηk.The component’s mean resultant length dk lies in [0, 1).Source: §IV · p. 8

A. Improved MF-VB Algorithm

The improved MF-VB algorithm infers the composite variable x so its posterior encodes phase-noise information without separate tracking variables. This also simplifies its residual update relative to Algorithm 1.

A. Improved MF-VB Algorithm

  • The improved algorithm updates xi without separate phase-noise tracking variables dt,i and θi because they are encoded in xi’s composite posterior.The passage identifies this as a structural advantage over Algorithms 1 and 2.Source: A. Improved MF-VB Algorithm · p. 9
  • Its residual update simplifies to r ← r + hi(…), reducing per-iteration overhead compared with Algorithm 1.Final symbol and phase decisions are recovered from parts (II) and (III) of Theorem 1.Source: A. Improved MF-VB Algorithm · p. 9
  • The algorithm recovers symbol and phase decisions from the corresponding posteriors in parts (II) and (III) of Theorem 1.The update for xi uses its variational distribution, with the variational mean and variance obtained from Corollary 1.Source: A. Improved MF-VB Algorithm · pp. 8–9

B. Computational Complexity Analysis

The improved MF-VB detector has dominant complexity O(KT(M+|S|)), while LMMSE-VB has a cubic-in-M term. The improved detector has the same dominant order as MF-VB but runs longer in simulations because of a higher constant factor.

B. Computational Complexity Analysis

  • Naive ML and SIW incur exhaustive-search costs, with SIW also requiring a cubic-in-M Cholesky decomposition.Naive ML searches |S|^K candidate vectors; SIW’s stated complexity is O(M^3 + M|S|^K).Source: B. Computational Complexity Analysis · p. 9
  • MF-VB’s per-user projections and residual updates cost O(M), while scalar von Mises updates cost O(1) and constellation lookups span |S| points.Across K users and T iterations, these operations yield dominant complexity O(KT(M+|S|)).Source: B. Computational Complexity Analysis · p. 9
  • LMMSE-VB has total complexity O(M^3T + KT(M^2+|S|)), dominated by inverting an M × M effective noise covariance matrix.Its inner loop costs O(KM^2 + K|S|) per iteration.Source: B. Computational Complexity Analysis · p. 9
  • Improved MF-VB has dominant complexity O(KT(M+|S|)), matching MF-VB’s order while taking longer in simulations because of a higher constant factor.It absorbs transmitter phase noise into the composite variable and removes two von Mises update steps per user per iteration.Source: B. Computational Complexity Analysis · p. 9

V. SIMULATION RESULTS

Across the evaluated channel and modulation settings, improved MF-VB generally achieves the lowest SER, with its advantage especially visible at high SNR and under stronger phase noise. The simulations also assess convergence, antenna scaling, and phase-noise estimation.

Convergence and simulation setup

  • The three VB algorithms converge reliably within fewer than 20 iterations in the reported setup.The convergence simulation uses i.i.d. Rayleigh fading, M = 16, K = 8, 16-QAM, and receive-PN standard deviation 5°; the plots average 1,000 trials.Source: §V · pp. 9–10, Figure 2 · p. 10

i.i.d. Rayleigh fading, 16-QAM

  • SERs on the order of 10^-3 at high SNR accompany improved MF-VB’s best performance in the i.i.d. Rayleigh 16-QAM comparison; its gap over SIW widens above 20 dB.Naive ML and naive LMMSE-VB reach error floors around 10^-2 at high SNR.Source: §V · p. 10, Figure 3 · p. 10

i.i.d. Rayleigh fading, 64-QAM

  • Under i.i.d. Rayleigh 64-QAM, improved MF-VB has the lowest high-SNR SER, followed by SIW, LMMSE-VB, and MF-VB.Its advantage over SIW is more pronounced than under 16-QAM.Source: Figure 4 · p. 10, §V · p. 11

Correlated Rayleigh fading

  • With correlated Rayleigh fading, QPSK results are nearly identical across algorithms, while improved MF-VB and SIW lead for 16-QAM and improved MF-VB is superior at high SNR.For correlated 64-QAM, SIW degrades and improved MF-VB retains a substantial performance advantage.Source: §V · p. 11, Figure 5 · p. 11, Figure 6 · p. 11

Few-path mmWave channels

  • In few-path mmWave channels, improved MF-VB maintains the lowest SER across the SNR range for both 16-QAM and 64-QAM.LMMSE-VB narrows its gap with improved MF-VB relative to i.i.d. Rayleigh channels, while SIW degrades more, especially under 64-QAM.Source: §V · pp. 11–12, Figure 7 · p. 11

Phase-noise severity

  • As PN standard deviation increases from 4° to 10°, improved MF-VB maintains the lowest SER, while SIW’s gap widens as PN severity increases.The proposed method’s von Mises posterior concentration adapts to the observed evidence.Source: §V · p. 12, Figure 8 · p. 12

Receive-antenna scaling

  • As receive antennas increase from 12 to 60, improved MF-VB retains the lowest SER and its advantage over SIW and LMMSE-VB.All algorithms benefit from enhanced spatial diversity and MIMO degrees of freedom.Source: §V · p. 12, Figure 9 · p. 12

Phase-noise estimation

  • Improved MF-VB achieves the lowest MSE for transmitter and receiver PN across the SNR range, with error continuing to decrease at high SNR.MF-VB saturates at low MSE; SIW does not produce a PN estimate.Source: §V · pp. 12–13, Figure 10 · p. 12

VI. CONCLUSIONS

The paper presents a variational Bayes framework for joint phase-noise estimation and data detection in uplink multiuser MIMO. Its improved MF-VB detector achieves the lowest SER across the evaluated channel types, with advantages growing under challenging conditions.

VI. CONCLUSIONS

  • The improved MF-VB achieves the lowest SER across i.i.d. Rayleigh, correlated Rayleigh, and mmWave channels, especially at high SNR, with higher-order modulation, and under stronger phase noise.The performance advantage becomes more pronounced in those conditions.Source: §VI · p. 13
  • Absorbing the phase-noise-corrupted transmit signal into a composite variable yields exact conjugate closed-form posteriors under von Mises priors.The framework jointly estimates phase noise and detects data in uplink multiuser MIMO systems.Source: §VI · p. 13
  • The paper develops MF-VB, LMMSE-VB, and improved MF-VB detectors, whose detection performance increases across that sequence.Source: §VI · p. 13
  • All proposed algorithms converge reliably within 20 iterations and scale well with the number of antennas.Future work will explore joint channel estimation, Wiener phase-noise temporal modeling, and hardware-efficient implementations for massive MIMO receivers.Source: §VI · p. 13
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