PaperlayerView exact revision on arXiv

Source-linked AI summary

Uplink Performance of Time-Reversal MRC in Massive MIMO Systems Subject to Phase Noise

Antonios Pitarokoilis, Saif Khan Mohammed, Erik G. Larsson

arXiv:1306.4495v2cs.IT

TL;DR

Phase noise can undermine coherent reception in Massive MIMO uplinks with imperfect CSI. The paper models synchronous and non-synchronous oscillator operation, derives TR-MRC achievable sum-rates, and reports O(sqrt(M)) array gain alongside trade-offs involving data duration and scheduled users.

  • Problem

    The paper examines how transmitter and receiver oscillator phase noise affects frequency-selective Massive MIMO uplinks with imperfect CSI.

  • Method

    It compares synchronous and non-synchronous BS phase-noise processes using low-complexity TR-MRC reception and derives achievable sum-rates for both.

  • Results

    O(sqrt(M)) array gain is achievable with TR-MRC and estimated CSI in the phase-noise-impaired uplink.

  • Takeaways & Limitations

    The achievable rates guide the choice of data-interval length and scheduled-user count because phase-noise drift progressively makes channel estimates outdated.

  • Takeaways & Limitations

    The reported rates are lower bounds whose interpretation relies on a Gaussian approximation for the effective noise.

Abstract

from arXiv · show

Multi-user multiple-input multiple-output (MU-MIMO) cellular systems with an excess of base station (BS) antennas (Massive MIMO) offer unprecedented multiplexing gains and radiated energy efficiency. Oscillator phase noise is introduced in the transmitter and receiver radio frequency chains and severely degrades the performance of communication systems. We study the effect of oscillator phase noise in frequency-selective Massive MIMO systems with imperfect channel state information (CSI). In particular, we consider two distinct operation modes, namely when the phase noise processes at the $M$ BS antennas are identical (synchronous operation) and when they are independent (non-synchronous operation). We analyze a linear and low-complexity time-reversal maximum-ratio combining (TR-MRC) reception strategy. For both operation modes we derive a lower bound on the sum-capacity and we compare their performance. Based on the derived achievable sum-rates, we show that with the proposed receive processing an $O(\sqrt{M})$ array gain is achievable. Due to the phase noise drift the estimated effective channel becomes progressively outdated. Therefore, phase noise effectively limits the length of the interval used for data transmission and the number of scheduled users. The derived achievable rates provide insights into the optimum choice of the data interval length and the number of scheduled users.

I. INTRODUCTION

The paper analyzes how Wiener phase noise affects frequency-selective Massive MIMO uplinks with imperfect CSI, comparing synchronous and non-synchronous BS oscillators using TR-MRC. It derives achievable sum-rates and identifies array-gain, transmission-interval, and user-scheduling implications.

  • Motivation: Phase noise causes partial loss of coherency between estimated and true channel gains, potentially degrading system performance.The impairment arises from random carrier-phase drift in transmitter and receiver local oscillators.
  • Scope and approach: The paper studies a single-cell frequency-selective MU-MIMO uplink with imperfect CSI, uplink training, and phase noise at users and BS antennas.The BS uses TR-MRC reception and achievable sum-rates are derived for both oscillator operation modes.
  • Operation modes: The two operation modes are synchronous BS phase noise, with identical antenna processes, and non-synchronous phase noise, with independent antenna processes.They correspond to common and independent phase references, respectively.
  • Results: 2 is the reported transmit-power reduction factor when doubling BS antennas at fixed per-user information rate, matching the no-phase-noise scaling law.The result is stated from the derived achievable sum-rates.
  • Results: Independent phase-noise sources can yield higher sum-rate performance than fully synchronous sources, according to the derived rates and a supporting toy example.The paper also identifies a trade-off between data-interval length and sum-rate and provides insight into the number of scheduled users.
  • Model assumptions: The channel model assumes i.i.d. channel gains whose captured energy increases linearly with M, a deficiency relevant only for exorbitantly large antenna counts.The authors restrict the practical regime of interest accordingly.

B. Received Signal

The system uses block-based uplink transmission with pilot training followed by data transmission, while phase-noise-impaired channel estimates support TR-MRC detection. The estimated effective channel is distorted by both additive noise and oscillator phase noise.

  • Transmission block: Each coherence block contains KL channel uses for uplink training, followed by preamble, data, and postamble intervals.The preamble and postamble account for intersymbol-interference edge effects.
  • Channel estimation: Users transmit training signals sequentially, with one active user at each training time and the others silent.The training sequence uses an impulse for each user, simplifying channel estimation and achievable-rate derivations.
  • Operation modes: The synchronous-mode expressions follow from the non-synchronous model by setting all BS phase processes equal.The paper gives both expressions when their differences are not obvious.
  • Channel estimation: The BS forms maximum-likelihood estimates of the effective channel from the received pilot signals.The effective channel includes the propagation channel and transmitter–receiver oscillator phase difference.
  • Estimation impairment: The channel estimate is distorted by additive white Gaussian noise and by phase noise at both user and BS local oscillators.This distortion affects the effective channel used for subsequent detection.

B. Time-Reversal Maximum Ratio Combining (TR-MRC)

The paper models TR-MRC detection as an effective SISO channel with known desired-signal gain and uncorrelated additive noise, then derives achievable rates for synchronous and non-synchronous phase-noise operation. The resulting bounds show better performance for non-synchronous operation and explain how phase drift reduces coherent combining over time.

  • TR-MRC convolves received symbols with the complex conjugate of the time-reversed estimated channel impulse response to detect each symbol.
  • The detected signal is decomposed into desired signal, intersymbol interference, multi-user interference, and aggregate noise terms.
  • The equivalent channel uses a deterministic mean desired-signal coefficient known at the BS and zero-mean uncorrelated effective noise with computable variance.
  • Phase drift creates partial non-coherency because data detection occurs after channel estimation, reducing the effective amplitude gain exponentially with the estimation-to-detection delay.
  • The achievable-rate proposition gives separate expressions for synchronous and non-synchronous operation using the corresponding effective-noise variance.
  • The proposed TR-MRC receiver has better performance under non-synchronous phase noise, although the comparison is between achievable lower bounds whose tightness relies on a Gaussian effective-noise approximation.

1) Achievable Sum-Rate:

Phase noise lowers the achievable sum-rate relative to the no-phase-noise case in both operation modes. The derived rate expressions nevertheless provide a direct comparison of the two modes and show that phase noise degrades performance through the effective rate bound.

  • The achievable sum-rate is obtained by summing the per-user, per-channel-code achievable rates.
  • Phase noise degrades sum-rate performance under both synchronous and non-synchronous operation compared with the no-phase-noise case.
  • The rate comparison follows from the derived bounds, with the no-phase-noise expression recovered by setting the phase-noise variance to zero.

A. Exact Analysis of Synchronous versus Non-Synchronous Operation for a Toy Channel Model

A toy channel with binary inputs and phase-noise rotations is analyzed exactly under synchronous and independent phase-noise processes. In this example, the non-synchronous channel has strictly larger capacity, while the authors caution that this is not universal.

  • The toy channel uses binary input X ∈ {±1}, with phase noise rotating the input into two observed outputs.
  • Synchronous operation sets the two phase-noise processes equal, producing an output alphabet with four possible symbols.
  • Independent phase-noise processes produce a larger output alphabet whose probabilities depend on the binary-input distribution.
  • The exact calculation gives a synchronous-case capacity of 5/9 bits.
  • The example establishes Cs < Cns, so non-synchronous operation has strictly higher capacity for this channel.
  • The example does not show that independent phase noise always increases capacity, only that it can be beneficial in some cases.

V. ASYMPTOTIC RESULTS

The asymptotic analysis shows that TR-MRC can preserve an O(sqrt(M)) array gain, while phase noise affects achievable rates through SNR-dependent saturation and transmission-design trade-offs.

  • Low-SNR behavior: At low SNR, phase noise causes only a small performance loss relative to the no-phase-noise scenario.The result is especially relevant to the low-SNR-per-degree-of-freedom operating point emphasized for Massive MIMO.
  • High-SNR behavior: In the high-SNR regime, phase noise causes the effective information rate to saturate in both synchronous and non-synchronous operation.MRC also exhibits high-SNR saturation without phase noise because intersymbol and multi-user interference dominate the effective noise term.
  • Array gain: O(sqrt(M)) array gain is achievable for phase-noise-impaired single-carrier Massive MIMO uplinks using TR-MRC and estimated CSI.This is the same scaling law as for flat-fading channels without phase noise.
  • Array-gain scaling: The achievable-rate expressions expose a trade-off between transmission-power scaling and maintaining non-zero rates as the antenna count grows.The asymptotic argument evaluates the effective SINR under per-user power scaling proportional to 1/M^η.
  • Array-gain scaling: A fixed non-zero rate remains achievable when per-user transmit power scales as 1/M^η only up to η = 1/2.For η > 1/2, the achievable rates approach zero as M increases.

VI. IMPACT OF PHASE NOISE SEPARATELY AT THE BS AND AT THE USER TERMINALS

The analysis separates phase noise at the user terminals from phase noise at the base station, preparing two special-case studies of their distinct effects on achievable rates.

  • Special cases: The paper examines sum-rate performance when phase noise occurs only at the user terminals or only at the base station.These cases are contrasted with the general model containing phase noise at both transmitter and receiver chains.
  • Special cases: The two special cases isolate whether transmitter-side or receiver-side oscillator phase noise drives the observed rate behavior.The subsequent sections analyze these configurations separately.

A. Special Case 1: Phase Noise Only at the UTs, σ2φ = 0

With phase noise only at the user terminals, synchronous and non-synchronous base-station operation become indistinguishable, while high-SNR rates saturate.

  • Operation modes: When the base-station oscillators are ideal, there is no distinction between synchronous and non-synchronous operation.The achievable-rate lower bound then reduces to a common expression for both modes.
  • High-SNR behavior: In the high-SNR limit, the achievable rate saturates under user-terminal-only phase noise.The saturation reflects the limiting value of the corresponding rate expression.
  • Interference mechanism: User-terminal phase-noise distortion adds across antennas after TR-MRC and creates an additional interference term with standard deviation O(M).This behavior is stated to hold regardless of whether phase noise is present at the base station.

B. Special Case 2: Phase Noise Only at the BS, (σ2

With phase noise only at the base station, synchronous and non-synchronous operation differ fundamentally: independent base-station oscillators can become asymptotically benign under TR-MRC.

  • Operation modes: For base-station-only phase noise, the achievable-rate expressions are provided separately for synchronous and non-synchronous operation.The section compares their high-SNR limits and large-array behavior.
  • Synchronous operation: The synchronous and user-terminal-only cases have qualitatively similar rate behavior.The paper attributes this similarity to analogous distortion mechanisms.
  • Non-synchronous operation: In non-synchronous operation, increasing the number of base-station antennas can increase the high-SNR saturation value arbitrarily.The large-array expression also grows through appropriate selection of E_u.
  • Non-synchronous operation: Independent base-station oscillator distortions asymptotically vanish when TR-MRC reception is used.The paper connects this behavior to the dominance of user-equipment hardware impairments over independent base-station impairments.

VII. NUMERICAL EXAMPLES

The numerical examples show how phase noise affects required transmit power, sum-rate, data-interval length, and user scheduling. Independent BS phase-noise processes can outperform synchronous operation, while phase-noise drift creates an optimum data interval and limits scheduled users.

  • Sum-rate versus transmit power: Low-SNR sum-rate loss is insignificant, while BS-dominated phase noise makes non-synchronous operation significantly better than synchronous operation.When terminal phase noise is dominant, the two operation modes have similar performance.
  • Array power gain: Doubling the number of BS antennas reduces the required per-user transmit power for r = 2 bpcu by approximately 1.5 dB for sufficiently large M.This extends the massive-MIMO power-scaling behavior to the phase-noise-impaired setting.
  • Simulation setup: 0.22°, 0.49°, and 1.1° phase-noise innovation standard deviations correspond to oscillator constants 9.4 × 10−19, 4.7 × 10−18, and 2.35 × 10−17 (rad Hz)−1.The study uses σφ = σθ = 0.49° as a reference value in its numerical examples.
  • Required transmit power: The performance gap from phase noise is minimal for small phase-noise drift and increases as the accumulated drift standard deviation increases.The gap is evaluated over ND + L − 1 channel uses, from the end of training to the end of data transmission.
  • Data-interval length: A longer data interval initially improves sum-rate by amortizing training overhead, but excessive ND causes phase-noise drift and loss of coherency.The no-phase-noise optimum is ND = infinity, whereas phase-noise-impaired operation has a finite trade-off.
  • Number of scheduled users: Phase-noise-impaired sum-rate is unimodal in the number of scheduled users, so phase noise imposes an upper bound beyond the coherence-interval training constraint.Without phase noise, sum-rate increases monotonically with user count in the considered arbitrarily long-coherence-interval case.

VIII. CONCLUSIONS

The paper studies phase noise in single-carrier Massive MU-MIMO uplinks with synchronous and non-synchronous BS oscillators using TR-MRC and imperfect CSI. It shows that array power gains remain available, while phase-noise drift creates a trade-off between data-interval length and sum-rate.

  • Scope and approach: The study considers synchronous and non-synchronous BS phase-noise processes in a single-carrier Massive MU-MIMO uplink with CSI acquired through uplink training.The receiver uses time-reversal maximum-ratio combining.
  • Main findings: The results establish an O(√M) array power gain under phase noise, extending earlier phase-noise-free results.The conclusion attributes the gain to the studied Massive MIMO reception strategy.
  • Main findings: Progressive oscillator phase-noise drift creates a fundamental trade-off between data-transmission interval length and sum-rate performance.This trade-off is a central consequence of the effective channel becoming progressively outdated during transmission.

APPENDIX

The appendix derives variance terms used in the achievable-rate analysis for the two phase-noise operation modes. It decomposes effective noise and interference contributions using independence assumptions and treats the operation-mode-specific term separately.

  • Variance decomposition: The derivation decomposes the effective noise variance into inter-symbol interference, multi-user interference, additive noise, and an operation-mode-dependent interference term.The appendix computes the common terms first and then evaluates the phase-noise-dependent term separately.
  • Non-synchronous case: The non-synchronous variance calculation begins with the inter-symbol-interference, multi-user-interference, and additive-noise terms shared by both operation modes.The phase-noise-dependent variance term is calculated afterward.
  • Assumptions: The derivations use mutual independence of channel coefficients, phase-noise processes, and data symbols, together with normalization of the power-delay profile.These facts are reused throughout the appendix calculations.
  • Operation modes: The phase-noise-dependent variance term is evaluated separately for synchronous and non-synchronous operation.The appendix explicitly starts the mode-specific calculation with synchronous operation and also states the non-synchronous variance.
Loading 1306.4495v2…