Asynchronous Cell-Free Massive MIMO With Rate-Splitting

I. INTRODUCTION

• Rate-splitting divides each user’s message into common and private parts to mitigate interference under imperfect CSI; the paper analyzes its performance and robust common-message precoding.It derives an asynchronous-reception sum-SE expression for arbitrary common-message precoding and uses bisection to maximize the minimum individual common-message SE.Source: §I · pp. 3, 5
• Asynchronous reception degrades cell-free massive MIMO through impaired pilot orthogonality, disrupted coherent data transmission, and increased multi-user interference.The paper derives closed-form downlink spectral-efficiency expressions for coherent and non-coherent transmission under imperfect CSI caused by asynchronous phase and pilot contamination.Source: §I · pp. 3–4

II. SYSTEM MODEL

• The modeled system has L access points with N antennas each serving K single-antenna users over shared time-frequency resources using time-division duplexing.A coherence block has τc channel uses, with τp for uplink training and τc − τp for downlink transmission; channels are frequency-flat Rayleigh fading.Source: §II · p. 6

A. Asynchronous Reception

• Asynchronous reception is modeled through multiplicative random phases caused by propagation-delay differences and hardware oscillator errors.The analysis focuses on phase asynchronization; time and frequency synchronization are noted as additional requirements for coherent processing.Source: A. Asynchronous Reception · p. 6

1) Delay Phase:

• Different AP-to-user distances cause asynchronous signal arrivals and constant delay-phase shifts, with the first-arriving AP’s timing offset set to zero.The timing offset is specified for the signal intended for user k and transmitted by AP l; it depends on propagation delay and symbol duration.Source: 1) Delay Phase · p. 7

2) Oscillator Phase:

• Independent AP and user oscillators contribute a time-varying phase modeled by a discrete-time Wiener process.The overall phase combines the UE and AP oscillator phases; the analysis assumes mutually independent oscillator processes and identically distributed statistics across devices.Source: 2) Oscillator Phase · p. 7
• The channel at each time instant combines the propagation-delay phase, oscillator phase, and underlying channel.The delay-dependent factor is treated as constant across multiple coherent blocks, while the oscillator-combined channel varies over time.Source: 2) Oscillator Phase · pp. 7–8

B. Channel Estimation

• With K > τp, UEs may reuse one of τp mutually orthogonal time-multiplexed pilots, creating pilot contamination.Each pilot is transmitted at a single time instant, and Pk denotes the UEs sharing UE k’s pilot time.Source: B. Channel Estimation · p. 8
• The AP uses the received pilot to estimate the channel at time λ = τp + 1, with accuracy decreasing as the gap from pilot transmission grows.The derivation applies MMSE estimation to the channel realization at the selected time; the delay phase is assumed perfectly known, though later precoding considers using or ignoring it.Source: B. Channel Estimation · pp. 8–9
• MMSE estimation worsens as oscillator phase variance increases, while LS is generally less accurate and performs very poorly at large variances.The authors attribute the MMSE degradation to random oscillator phases destroying received-pilot orthogonality; in extreme asynchronous cases, non-orthogonal pilots can yield more accurate estimates.Source: Figure 2 · p. 10, B. Channel Estimation · p. 10
• LS estimation NMSE can exceed one because its estimate and estimation error are correlated, unlike the independent MMSE estimate and error.NMSE ranges from 0 for perfect estimation to 1 for extremely impaired estimation; under large oscillator phase variances, LS performs very poorly.Source: B. Channel Estimation · pp. 9–10

C. Rate-Splitting Strategy

• Rate-splitting divides each UE message into private and common parts, combines the common parts into one stream, and uses SIC to decode it before private streams.The analysis assumes error-free common-stream decoding because this one-layer scheme requires each UE to perform SIC only once.Source: C. Rate-Splitting Strategy · p. 11
• The common and private messages share each AP’s power through a common factor ρ, with pdc = ρpd and pdp = (1 − ρ)pd; ρ = 0 recovers conventional transmission.The factors are assumed identical across APs due to small channel differences; optimizing them per AP is left for future work.Source: C. Rate-Splitting Strategy · p. 12, §III · p. 12
• Rate-splitting can decode part of interference through the common message instead of treating all imperfect-CSI multi-user interference as noise, boosting sum spectral efficiency.The passage describes this as promising for multi-user transmission with imperfect CSI.Source: C. Rate-Splitting Strategy · p. 12

III. DOWNLINK ASYNCHRONOUS TRANSMISSION

• The asynchronous-transmission analysis sets ρ = 0 and derives closed-form SE expressions for coherent and non-coherent transmission with DU- and DF-MR private precoding.These conventional-system results serve as a building block for the later rate-splitting analysis.Source: §III · p. 12

A. Coherent Downlink Transmission

• Coherent joint transmission has each AP send the same data symbol to each UE, and low-complexity MR private precoding is used to obtain analytical results.Source: A. Coherent Downlink Transmission · p. 13
• In the large-antenna approximation, UE oscillator phase has a greater effect on SINR than AP oscillator phase, and adding APs can significantly reduce the AP-phase effect.Source: A. Coherent Downlink Transmission · p. 14
• With DU-MR precoding, propagation delay does not affect the SINR, whereas DF-MR introduces random desired-signal phases that reduce spectral efficiency.The analysis also derives a lower bound on the achievable rate for coherent transmission using DU-MR private precoding.Source: A. Coherent Downlink Transmission · pp. 13–14

B. Non-Coherent Downlink Transmission

• To relax AP phase-synchronization requirements, each AP sends a different data symbol to each UE than the other APs.Source: B. Non-Coherent Downlink Transmission · pp. 14–15
• Each UE uses successive interference cancellation to detect signals from the APs; the decoding order does not affect spectral efficiency but must be chosen before encoding.Source: B. Non-Coherent Downlink Transmission · p. 15
• With DF-MR private precoding, the non-coherent SINR has the same expression as with DU-MR private precoding.Source: B. Non-Coherent Downlink Transmission · p. 15
• Non-coherent transmission overcomes asynchronous reception and can eliminate delay-phase effects, but it comes at the cost of poor spectral efficiency.The paper suggests a hybrid of coherent and non-coherent transmission as a future trade-off between performance and robustness.Source: B. Non-Coherent Downlink Transmission · pp. 15–16

C. Power Control for Private Messages

• Channel-inversion rate α = -1 is chosen to enhance SE for poor UEs, with power-control coefficients constrained by downlink power.Source: C. Power Control for Private Messages · p. 16
• Power-control schemes have limited sum-SE gains in high-density CF massive MIMO because increasing AP numbers reduce global statistical channel differences among UEs.Source: C. Power Control for Private Messages · p. 16

IV. RATE-SPLITTING ASSISTED CF MASSIVE MIMO

• The RS analysis treats private messages as noise when deriving common-message SE, then decodes private messages after common-message decoding.Source: §IV · p. 16
• A binary-search method finds the optimal power-splitting factor for downlink sum SE, and robust common precoding is designed to mitigate asynchronization.Source: §IV · p. 16

A. Coherent Downlink Transmission

• In the high-antenna-density regime, the analysis assumes asymptotically orthogonal UE channels and uses a common precoder formed as a linear sum of private precoders.Source: A. Coherent Downlink Transmission · p. 17
• Oscillator phase weakens common-message decoding, especially at the UE, negatively affecting RS spectral efficiency.Source: A. Coherent Downlink Transmission · p. 17
• The common-message delay-phase effect depends on both APs and UEs, including AP physical positions and pilot sharing among some UEs.Source: A. Coherent Downlink Transmission · p. 18

B. Non-Coherent Downlink Transmission

• Each access point sends different common and private data symbols, removing inter-access-point cooperation and greatly reducing synchronization requirements.The received signal includes contributions from the access points and noise.Source: B. Non-Coherent Downlink Transmission · p. 18
• Under DU-MR private precoding and simple common precoding, Theorem 4 lower-bounds the achievable common-message rate.The proof treats private messages as interference.Source: B. Non-Coherent Downlink Transmission · pp. 18–19
• With unknown delay phase, private-message SE is unaffected, whereas common-message SE remains affected because all UEs decode the common message.Private messages are decoded only by their corresponding UE.Source: B. Non-Coherent Downlink Transmission · p. 19

C. Power Splitting Factor to Maximize the Downlink Sum SE

• After SIC removes the common messages, private messages are decoded from the remaining received signal to obtain downlink sum SE.Source: C. Power Splitting Factor to Maximize… · p. 19
• A simple binary search is proposed to find the optimal power split factor ρ that maximizes downlink sum SE.The section states that the downlink sum SE is a monotonic function and has an optimal ρ.Source: C. Power Splitting Factor to Maximize… · p. 20

D. Robust Precoding Design for Common Message

• The design chooses common-message precoding weights to maximize the minimum downlink common SE across UEs under a power constraint, with equal user SE at the optimum.The design focuses on common-message precoding because private-message precoders such as MR, zero-forcing, and MMSE are established in CF massive MIMO,.Source: D. Robust Precoding Design for Common… · p. 20
• The max-min problem is quasi-concave and solved by bisection over convex feasibility problems, halving the SINR search range each iteration for rapid convergence.Source: D. Robust Precoding Design for Common… · pp. 21–22
• The design accounts for residual interference in common-message precoding, unlike, whose design does not.Source: D. Robust Precoding Design for Common… · p. 21

• Under coherent transmission, DU-MMSE outperforms DU-MR through interference cancellation, oscillator phase degrades SE, and known delay phase preserves DU-MR performance; RS adds 2.2 bit/s/Hz with DU-MR.The simulations also confirm the derived closed-form SE expressions.Source: Figure 4 · p. 23, §V · p. 23
• At median SE, the UatF bound is 0.3 bit/s/Hz below achievable rate and the estimation bound is 0.8 bit/s/Hz below; dense AP deployment tightens UatF, and statistical cooperation helps poorly served UEs.High antenna density strengthens channel hardening, while statistical channel-cooperation power control improves SE for UEs with poor channel conditions.Source: Figure 5 · p. 23, §V · pp. 23–24
• As oscillator variance rises from −50 dB to −20 dB, DU-MR RS gain falls from 2.2 to 0.5 bit/s/Hz, while DF-MR RS gain remains below 0.1 bit/s/Hz.For non-coherent transmission, delay phase has no impact without RS, and non-coherent transmission outperforms coherent transmission when delay phase is unknown; RS still requires delay synchronization.Source: Figure 6 · p. 24, §V · p. 25
• Dynamic cooperation clustering has only a modest positive effect when delay phase is unknown and cannot eliminate asynchronous reception errors, though it may assist synchronization calibration.Small clusters reduce arrival-time differences among geographically distributed APs, but those offsets remain large relative to the wavelength.Source: Figure 7 · p. 24, §V · p. 25
• For LN = 80, increasing AP phase variance from −50 dB to −20 dB causes 6.4 bit/s/Hz sum-SE loss, while the same UE increase causes 11.3 bit/s/Hz loss; the negative influence grows with antenna count.Source: §V · p. 25
• As transmit power increases, conventional UatF sum SE saturates under imperfect CSI, while RS avoids saturation but grows slowly; perfect-CSI rates without RS also saturate under asynchronous interference.RS gains diminish in the high-power regime, and residual beamforming-gain uncertainty causes saturation in the UatF bound.Source: Figure 9 · p. 26, §V · pp. 26–27
• With random AP deployment, simple-common RS gains 1.5 bit/s/Hz over no RS, and robust precoding adds 0.2; under UE oscillator mismatch, their gains are 0.1 and 1 bit/s/Hz.In nonuniform coverage, the simple-precoding gain is less than half the robust-precoding gain; uniform coverage makes simple common precoding suitable for large gains.Source: Figure 10 · p. 27, §V · pp. 27–28
• With DF-MR private precoding and unknown delay phase, robust common precoding gains 1.6 bit/s/Hz, while simple common precoding gains no more than 0.1 bit/s/Hz.The results support robust common precoding when RS is used without delay-phase awareness.Source: Figure 11 · p. 27, §V · p. 28

VI. CONCLUSION

• Asynchronous reception destroys pilot orthogonality and coherent data transmission, and accurate delay-phase estimation is important for coherent CF massive MIMO.The derived closed-form SE expressions account for coherent and non-coherent transmission, phase-asynchronization, and pilot contamination.Source: §VI · p. 28
• UE oscillator phase affects SE more than AP oscillator phase, while increasing antenna counts significantly reduces only the AP oscillator-phase effect.Source: §VI · p. 28
• RS significantly improves CF massive MIMO performance but remains seriously affected by asynchronous phases, especially delay phase and UE oscillator phase.Source: §VI · p. 28
• Robust common-message precoding maximizes minimum individual SE and performs well under serious oscillator mismatch or unknown delay phase.Future work will study scalable RS in user/cell-centric CF networks and imperfect SIC.Source: §VI · pp. 28–29

APPENDIX A PROOF OF THEOREM 1

• The proof derives the use-and-then-forget capacity bound using SINR_p.It assumes DU-MR precoding v_kl = θ_kl ĥ_kl[λ] and derives its normalization parameter.Source: Appendix A · p. 29
• The desired signal term DSk is part of INTk and is subtracted to obtain beamforming-gain uncertainty as residual interference.Source: Appendix A · p. 29
• The derivation uses the MMSE-estimation property that ĥ_kl[λ] and h̃_kl[λ] are independent.Source: Appendix A · p. 30

APPENDIX B PROOF OF THEOREM 3

• The proof uses the UatF bound to derive the common-message SINR, including residual interference from beamforming-gain uncertainty and interference from the private signal.The private-signal interference is given by (69).Source: Appendix B · p. 31
• The derivation assumes DU-MR private precoding v_kl = θ_klĥ_kl[λ] and linear common precoding v_c,l = Σ_i a_il v_il.The common precoding is specified as the sum of the private precoders weighted by a_il.Source: Appendix B · p. 31
• The proof derives the precoding normalization parameter, expands the relevant quantity, obtains Υ8 by similar steps, and concludes that the result follows.It also invokes (66) and (70) in deriving an intermediate expression.Source: Appendix B · pp. 31–32