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Impact of Channel Aging on Cell-Free Massive MIMO Over Spatially Correlated Channels

Jiakang Zheng, Jiayi Zhang, Emil Björnson, Bo Ai

arXiv:2104.11500v1cs.IT

TL;DR

The paper addresses how channel aging affects CF massive MIMO when spatial correlation, pilot contamination, and imperfect channel estimation are present. It derives closed-form uplink and downlink SE expressions and evaluates receivers, transmission modes, power control, energy efficiency, and resource-block design. CF systems are more robust to channel aging than SC systems, with coherent transmission outperforming non-coherent transmission and SCCPC benefits weakening as aging increases.

  • Problem

    The paper investigates the limited understanding of channel aging in CF massive MIMO, where channels evolve continuously within resource blocks and spatial correlation affects performance.

  • Method

    The paper derives exact closed-form uplink and downlink SE expressions incorporating imperfect channel estimation, spatial correlation, pilot contamination, and channel aging.

  • Results

    CF massive MIMO is less affected by channel aging than SC systems, while coherent transmission outperforms non-coherent transmission and SCCPC gains weaken as aging strengthens.

  • Takeaways & Limitations

    Multiple antennas and enough pilots can mitigate channel aging, whereas weak spatial correlation yields larger SE but is more easily affected by aging.

Abstract

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In this paper, we investigate the impact of channel aging on the performance of cell-free (CF) massive multiple-input multiple-output (MIMO) systems with both spatial correlation and pilot contamination. We derive novel closed-form uplink and downlink spectral efficiency (SE) expressions that take imperfect channel estimation into account. More specifically, we consider large-scale fading decoding and matched-filter receiver cooperation in the uplink. The uplink performance of a small-cell (SC) system is derived for comparison. The CF massive MIMO system achieves higher 95\%-likely uplink SE than the SC system. In the downlink, the coherent transmission has four times higher 95\%-likely per-user SE than the non-coherent transmission. Statistical channel cooperation power control (SCCPC) is used to mitigate the inter-user interference. SCCPC performs better than full power transmission, but the benefits are gradually weakened as the channel aging becomes stronger. Furthermore, strong spatial correlation reduces the SE but degrades the effect of channel aging. Increasing the number of antennas can improve the SE while decreasing the energy efficiency. Finally, we use the maximum normalized Doppler shift to design the SE-improved length of the resource block. Simulation results are presented to validate the accuracy of our expressions and prove that the CF massive MIMO system is more robust to channel aging than the SC system.

I. INTRODUCTION

This paper studies channel aging in CF massive MIMO with spatial correlation and pilot contamination, comparing receiver, transmission, power-control, energy-efficiency, and resource-block designs. It derives closed-form SE expressions and finds CF systems outperform SC systems in uplink performance while coherent downlink transmission is superior.

  • Motivation: Pilot contamination from non-orthogonal pilots significantly reduces CF massive MIMO performance, while spatial correlation also affects system performance.Non-orthogonal pilots are used because of short training phases and many UEs.
  • Motivation: Channel aging is analyzed for CF massive MIMO with spatial correlation, pilot contamination, and continuously evolving channels rather than block-fading channels.The paper identifies this as the first analysis of channel aging for CF massive MIMO.
  • Approach: The paper derives closed-form uplink and downlink SE expressions under channel aging, including imperfect channel estimation, LSFD and MF uplink receivers, and SC comparisons.It also compares coherent and non-coherent downlink transmission and applies SCCPC.
  • Results: CF massive MIMO performs better than SC systems in uplink scenarios, while coherent downlink transmission performs better than non-coherent transmission.These comparisons are reported for both static and mobile scenarios in the uplink.
  • Results: SCCPC improves performance over full-power transmission, but its benefits gradually weaken as channel aging becomes stronger.More antennas, more pilots, and reduced spatial correlation can compensate for channel-aging performance loss.
  • Energy efficiency: Increasing channel aging significantly reduces energy efficiency, while the paper proposes a method to design an SE-improved resource-block length.The study uses a realistic power-consumption model and investigates the preferred number of APs for the EE operating point.

II. SYSTEM MODEL

The system model contains geographically distributed multi-antenna APs serving mobile single-antenna UEs through resource blocks with separate uplink and downlink data phases. Channels evolve continuously within each resource block and remain correlated across time instants.

  • Network layout: The CF system consists of L APs and K mobile single-antenna UEs, with each AP equipped with N antennas.All APs are connected to a CPU through fronthaul links and simultaneously serve all UEs on the same time-frequency resource.
  • Resource blocks: Communication is divided into resource blocks of τc time instants, with separate uplink-data and downlink-data blocks under TDD.Each uplink block includes uplink training and uplink data, while each downlink block includes uplink training and downlink data.
  • Channel model: The model assumes a flat-fading narrowband system, either single-carrier or one narrowband subcarrier in a multicarrier system.The fraction of resource blocks allocated to uplink and downlink data can be dynamically changed.
  • Channel model: The Rayleigh channel h_kl[n] follows a circularly symmetric complex Gaussian distribution with spatial correlation matrix R_kl.The large-scale fading coefficient is β_kl = tr(R_kl)/N, and channels are independent across UE–AP pairs.
  • Channel aging: Channel aging makes channel realizations different but correlated across time instants within a resource block.The model therefore does not treat the channel as identical throughout the block.
  • Channel aging: Uplink and downlink data use different resource blocks so transmission can exploit portions with more accurate channel information and reduce guard intervals.The estimated channel information becomes increasingly outdated over time because of channel aging.

A. Channel Aging

The paper models channel aging as time-varying propagation within each resource block, using a Jakes-matched temporal correlation model and MMSE channel estimation. Pilot contamination and increasing time distance from pilot transmission reduce estimation quality, while zero aging recovers conventional block fading.

  • Channel model: Channel aging arises from relative UE–AP movement and causes channel coefficients to vary within a resource block.The channel is represented through an initial state and an innovation component.
  • Channel model: The temporal correlation coefficient follows a Jakes-matched Bessel-function model, ρ_k[n] = J_0(2πf_D,kT_sn).The Doppler shift depends on UE velocity, carrier frequency, and the speed of light.
  • Model scope: The model is realistic for small-scale movements but accurate only over a limited time period because wide-sense stationarity is not guaranteed for larger movements.The paper also does not exploit possible correlation between resource blocks.
  • Channel estimation: Estimates formed after pilot transmission are propagated to later instants, but their quality decreases as the time distance from the pilot increases.The effective channel is expressed relative to the channel at λ = τ_p + 1 using a temporal correlation term and an innovation component.
  • Channel estimation: Pilot sharing among UEs produces pilot contamination in the MMSE channel estimate.The network assumes K > τ_p, so multiple UEs may use the same pilot time instant.
  • Special case: When channel aging vanishes, the resource block becomes a conventional block-fading model and the estimation expressions reduce to prior results.The reduction is independent of the different time instants.

III. UPLINK DATA TRANSMISSION

The uplink analysis derives achievable spectral-efficiency expressions for cell-free massive MIMO under channel aging, spatial correlation, and imperfect estimation. LSFD maximizes the effective SINR, while channel aging degrades desired-signal and beamforming gains and increasing antennas improves SINR despite linearly increasing pilot-contaminated interference.

  • System model and objectives: Novel uplink spectral-efficiency expressions are derived for cell-free and small-cell systems, with SCCPC used to improve performance.The cell-free analysis includes channel aging and spatial correlation within each uplink resource block.
  • Receiver processing: Each AP locally combines the received signal with its channel estimate and forwards the resulting data estimate to the CPU for joint detection.The CPU applies LSFD weights to form the detected symbol.
  • Signal decomposition: The uplink signal decomposition separates desired signal, beamforming uncertainty, channel aging, inter-user interference, and receiver noise.These terms are used to construct the effective SINR and achievable spectral efficiency.
  • Channel-aging impact: Channel aging degrades both the desired signal and beamforming gain, causing SINR to decrease as the aging effect becomes stronger.The approximation SINR_k[n] ≈ 1/(aκ+b) makes the dependence on κ = 1/ρ_k^2 explicit.
  • Receiver cooperation: The effective SINR is maximized by optimized CPU weights, yielding the LSFD-based maximum spectral efficiency.Equal weights provide a lower-complexity matched-filter receiver-cooperation alternative.
  • Antenna scaling: Increasing the number of antennas raises desired-signal power and pilot-contaminated interference linearly, while the SINR remains monotonically increasing with antenna count.The interference and noise determine one component of the SINR, whereas pilot contamination determines another.

B. Small-Cell Systems

The small-cell uplink is formulated as a comparison system in which each AP estimates channels and detects its served UE using the local channel estimate. Its achievable spectral efficiency is obtained in closed form, with the small-cell system appearing as a special case of cell-free massive MIMO under suitable LSFD weights.

  • Small-cell receiver: In the small-cell uplink, each AP estimates UE channels and uses the estimate to detect the desired signal.The combined signal includes receiver noise and interference from estimation errors, channel aging, and other UEs.
  • Relation to cell-free MIMO: The small-cell system is a special case of cell-free massive MIMO when LSFD weights select only the SE-maximizing AP for each user.Maximum-ratio combining then provides a lower bound on the UE capacity.
  • Achievable spectral efficiency: The small-cell achievable spectral efficiency is expressed in closed form for arbitrary antenna counts per AP.The N = 1 case yields a further closed-form expression involving the exponential integral.

C. Uplink Statistical Channel Cooperation Power Control

The paper extends statistical channel cooperation power control to the considered systems and derives downlink spectral-efficiency expressions for coherent and non-coherent transmission. Coherent transmission uses a common data symbol across APs, while power-control coefficients regulate AP transmission under the downlink power constraint.

  • Power control: SCCPC is extended to mitigate near-far effects and account for the large-scale fading coefficients linking a UE with all APs.The resulting power-control coefficient reflects the UE’s effective connection to the complete AP set.
  • Downlink analysis: The downlink analysis derives spectral-efficiency expressions for both coherent and non-coherent cell-free transmissions under channel aging and spatial correlation.SCCPC is used to further improve downlink performance.
  • Power constraint: The coherent downlink signal uses power-control coefficients constrained between 0 and 1 so that each AP satisfies its maximum transmission-power constraint.The downlink resource block reserves τ_c − τ_p instants for data transmission.
  • Coherent transmission: In coherent transmission, every AP sends the same data symbol to a UE using maximum-ratio precoding.The symbol is common across APs, unlike the non-coherent case where APs transmit different symbols.
  • Coherent downlink SE: Theorem 2 provides a lower bound on the coherent downlink capacity through SINR_coh_k[n].The expression is derived in closed form and reduces to a prior result when the relevant aging and antenna conditions are imposed.
  • Special case: For N = 1 and vanishing channel aging, the downlink expression reduces to the cited conventional result.This provides a consistency check for the derived formula.

B. Non-coherent Downlink Transmission

Non-coherent downlink transmission assigns different data symbols to each AP and detects them successively, treating remaining AP signals as interference. The section derives its downlink SE and introduces statistical channel cooperation power control within a realistic energy-efficiency model.

  • Transmission model: Each AP transmits a different data symbol, avoiding the phase-synchronization requirements of coherent transmission.The transmitted symbols differ across APs, with maximum-ratio precoding and downlink power-control coefficients.
  • Detection: Successive interference cancellation detects AP signals one by one, treating undetected AP transmissions as interference.The decoding order does not affect SE, although individual signals must be encoded for a chosen order.
  • Spectral efficiency: Theorem 3 gives the non-coherent downlink SE of each UE under successive interference cancellation.The theorem provides the SE expression, with closed-form SINR evaluation developed subsequently.
  • Power control: Statistical channel cooperation power control extends an existing policy with a predetermined function based on global statistical channel information.The resulting SCCPC coefficients are used for power control across users and APs.
  • Energy-efficiency model: The energy-efficiency analysis uses a realistic model combining radiated transmit power, circuit consumption, fronthaul power, and sum spectral efficiency.The total energy efficiency is evaluated using uplink LSFD and downlink coherent transmission.

B. Total Energy Efficiency

The paper defines total energy efficiency as network throughput divided by total power consumption and evaluates it under channel aging using a practical simulation setup. It also examines how resource-block length affects spectral efficiency as Doppler shift changes.

  • Energy-efficiency definition: Total energy efficiency is defined as the sum throughput in bit/s divided by total network power consumption in Watt.The model includes uplink and downlink power consumption and circuit costs.
  • Channel aging: The average uplink and downlink SE is tracked over time-indexed channel aging within resource blocks.The analysis considers the first 500 time instants of an infinitely long resource block.
  • Resource-block design: Increasing normalized Doppler shift moves the first zero of average SE earlier, motivating a shorter resource block.The resource-block length is chosen not to exceed the first-zero position based on the maximum normalized Doppler shift.
  • Resource-block design: A practical resource-block-length rule makes sum SE more stable across the considered Doppler-frequency range.The evaluated setting uses 0 ≤ f_D T_s ≤ 0.002 and τ_c = 200.

B. Spectral Efficiency and Total Energy Efficiency Analysis

The numerical analysis compares CF and SC systems, transmission modes, spatial correlations, antenna counts, and AP densities under channel aging. CF generally retains stronger uplink performance, coherent transmission outperforms non-coherent transmission, and energy efficiency has an interior AP-count optimum.

  • Downlink transmission: 42% and 49% median SE losses occur for coherent and non-coherent downlink transmission, respectively, as f_D T_s increases from 0 to 0.002.Coherent transmission nevertheless provides substantially higher SE than non-coherent transmission, whose sequential detection is more sensitive to channel aging.
  • Uplink SE comparison: CF with LSFD achieves larger 95%-likely uplink SE than SC across both low- and high-mobility conditions.The paper therefore identifies CF massive MIMO as more suitable for mobility scenarios than SC systems.
  • Power control: SCCPC becomes less influential as channel aging strengthens because self-interference dominates the inter-user interference it mitigates.Its 95%-likely SE approaches full-power performance as f_D T_s varies from 0 to 0.002, especially for SC systems.
  • Downlink transmission: Coherent transmission has at least four times the 95%-likely per-user SE of non-coherent transmission under both low- and high-mobility conditions.Increasing τ_p provides 25% and 10% gains at the 95%-likely point for coherent and non-coherent transmission, respectively, without Doppler shift.
  • Antenna scaling: Uplink SE grows with antennas per AP, and in CF systems the SINR increases nearly proportionally to the antenna count even with pilot contamination.Pilot-contaminated SE grows almost as if pilot contamination were absent in the evaluated regime.
  • Energy efficiency: Total energy efficiency first increases and then decreases with AP count, yielding an optimal number of APs.Power consumption grows linearly with APs while SE grows logarithmically; larger Doppler shifts lower total EE and favor more APs at the optimum.
  • Energy efficiency: Increasing AP count reduces channel-aging impact on total energy efficiency, with 32% loss at L = 100 versus 37% at L = 10 when f_D T_s rises from 0.001 to 0.002.For K = 20, the larger network is less sensitive to the specified Doppler-shift increase.

VII. CONCLUSIONS

The paper derives exact uplink and downlink spectral-efficiency expressions for cell-free massive MIMO under channel aging, spatial correlation, pilot contamination, and imperfect channel estimation. It finds that cell-free systems are less affected by aging than small-cell systems, while coherent transmission and practical power control improve performance.

  • Contributions: The analysis accounts for channel aging, spatial correlation, pilot contamination, and imperfect channel estimation in uplink and downlink cell-free massive MIMO.The uplink considers LSFD and matched-filter receiver cooperation with small-cell comparison; the downlink considers coherent and non-coherent transmission.
  • Performance comparison: Cell-free massive MIMO is less affected by channel aging than small-cell systems, while coherent transmission outperforms non-coherent transmission.The conclusions identify these as the main comparative performance findings.
  • Power control: SCCPC improves spectral efficiency, but its gain gradually decreases as channel aging becomes stronger.The scheme is proposed as a practical method for improving spectral efficiency under the considered conditions.
  • Spatial correlation and aging: Weak spatial correlation achieves larger spectral efficiency but is more easily affected by channel aging.The conclusions distinguish the spectral-efficiency benefit of weak correlation from its greater sensitivity to aging.
  • Energy efficiency: Increasing channel aging significantly reduces energy efficiency and favors more access points at the optimal energy-efficiency operating point.The paper separately evaluates energy efficiency and spectral efficiency under aging.
  • Resource-block design: The paper provides a method to design the spectral-efficiency-improved resource-block length for more uniform performance under channel aging.The design targets resource-block operation under varying aging conditions.
  • Scope and future work: The flat-fading treatment can apply methods independently per subcarrier, while future work may exploit frequency-domain correlation between subcarriers.The stated scope includes narrowband single-carrier operation or one narrowband subcarrier in a multicarrier system.

APPENDIX A PROOF OF THEOREM 1

The proof of Theorem 1 computes the uplink spectral efficiency by evaluating the terms of the SINR under channel estimation and channel aging. It uses estimation properties, independence relations, variance identities, and the capacity bound to obtain the stated expression.

  • Capacity bound: The resulting achievable spectral efficiency is obtained by applying the use-and-then-forget capacity bound at each time instant and averaging over the block.The proof concludes after substituting the evaluated terms into the target expression.
  • SINR-to-SE derivation: The proof computes every term of the uplink SINR and uses those terms to obtain the spectral-efficiency expression.The derivation proceeds by evaluating expectations and substituting them into the SINR.
  • Channel-estimation relations: MMSE channel estimates and estimation errors are independent, while pilot-sharing users produce correlated channel estimates.These relations determine which expectation terms differ for users inside and outside the pilot-sharing set.
  • Expectation evaluation: The proof evaluates expectations using Gaussian-moment results, trace expressions, and the variance sum rule for independent random variables.The cited steps include trace terms such as tr(Q_kl Q_il) and separate variance calculations.
  • Channel aging: Channel aging contributes through the temporal correlation model, with the innovation component uncorrelated with the earlier channel estimate.The proof substitutes the resulting aging-related expressions into the variance calculation.

APPENDIX B PROOF OF THEOREM 2

The proof of Theorem 2 derives the downlink spectral-efficiency expression by modeling sequential signal detection at the user equipment and evaluating the resulting desired-signal and interference terms. The derivation treats residual terms as uncorrelated noise and applies a capacity lower bound.

  • Signal decomposition: For a single-antenna access point, the received uplink signal is decomposed into an estimated-channel term, estimation error, aging innovation, and noise.The decomposition introduces the channel-aging components used in the subsequent SINR calculation.
  • Signal and channel model: The transmit signal includes user power and an independent innovation component with variance determined by the large-scale fading coefficient.The proof also distinguishes pilot-sharing and non-sharing users through their estimate dependence.
  • Expectation evaluation: The proof computes the expectations needed for the achievable spectral-efficiency bound from the decomposed received signal.These calculations provide the terms used in the final expression.
  • Sequential detection: Successive interference cancellation detects access-point signals sequentially, subtracting previously detected signals before detecting the next one.The desired signal is treated as a deterministic-channel term, while remaining terms are treated as uncorrelated noise.
  • Downlink SE: The downlink SINR and total spectral efficiency are formed from the desired-signal power and the variance of the remaining uncorrelated terms.The proof then uses the resulting SINR in the spectral-efficiency expression.
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