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Performance of Cell-Free Massive MIMO Systems with MMSE and LSFD Receivers
Elina Nayebi, Alexei Ashikhmin, Thomas L. Marzetta, Bhaskar D. Rao
TL;DR
Cell-free massive MIMO requires uplink receiver analysis for distributed APs serving users without cells. This paper derives MMSE and LSFD rate bounds and a large-scale-fading-based MMSE approximation, finding strong gains over MF at 5%-outage.
Problem
The paper investigates uplink performance in cell-free systems where distributed APs jointly serve users, including receiver designs based on channel information and large scale fading.
Method
The paper derives capacity lower bounds for MMSE and LSFD, introduces partial MMSE, and develops an MMSE SINR approximation using large scale fading coefficients.
Results
MMSE and LSFD provide 5.1-fold and 2.6-fold gains over MF, respectively, in 5%-outage rate.
Takeaways & Limitations
The asymptotic approximation remains accurate with small numbers of APs and users, while LSFD reduces overall complexity relative to MMSE.
Abstract
from arXiv · showhide
Cell-Free Massive MIMO comprises a large number of distributed single-antenna access points (APs) serving a much smaller number of users. There is no partitioning into cells and each user is served by all APs. In this paper, the uplink performance of cell-free systems with minimum mean squared error (MMSE) and large scale fading decoding (LSFD) receivers is investigated. The main idea of LSFD receiver is to maximize achievable throughput using only large scale fading coefficients between APs and users. Capacity lower bounds for MMSE and LSFD receivers are derived. An asymptotic approximation for signal-to-interference-plus-noise ratio (SINR) of MMSE receiver is derived as a function of large scale fading coefficients only. The obtained approximation is accurate even for a small number of antennas. MMSE and LSFD receivers demonstrate five-fold and two-fold gains respectively over matched filter (MF) receiver in terms of 5%-outage rate.
I. INTRODUCTION
Cell-free massive MIMO distributes single-antenna APs without cell partitioning, serving each user through all APs. The paper develops uplink MMSE, partial MMSE, and LSFD approaches, with asymptotic analysis and numerical comparisons against MF.
- Cell-free mMIMO uses randomly located single-antenna APs, no geographical cell partitioning, and simultaneous service of each user by all APs.
- The paper studies uplink MMSE, partial MMSE, and LSFD receivers, extending prior cell-free uplink work focused on MF reception.
- LSFD generalizes pilot-contamination postcoding to cell-free systems using only large scale fading coefficients and yields an SINR expression in those coefficients.
- With a fixed user count and unbounded AP count, MF performance is limited by coherent interference from users sharing pilot sequences.
- Numerical experiments under independent and correlated shadow fading show MMSE and LSFD outperform MF, while MMSE outperforms LSFD at higher complexity.
II. SYSTEM MODEL AND CHANNEL ESTIMATION
The system contains many distributed APs and fewer users, with channels modeled through large and small scale fading. TDD pilots enable channel estimation, but short coherence intervals require pilot reuse and create contamination.
- The model has M randomly distributed single-antenna APs and K single-antenna users with K ≪ M, connected through a network controller.
- Each channel combines large scale fading for path loss and shadow fading with i.i.d. small scale fading that remains constant over a coherence interval.
- TDD pilots are transmitted synchronously so APs can estimate channel coefficients and forward them to the network controller.
- Short coherence intervals make the pilot length small relative to the user count, forcing pilot reuse and causing pilot contamination.
- Users sharing a pilot have correlated channel estimates, allowing each AP to send only one representative estimate per pilot group to the controller.
III. UPLINK DATA TRANSMISSION
During uplink data transmission, APs receive users’ symbols and noise, while the network controller combines channel estimates through linear postcoding. Achievable rates account for estimation error and pilot contamination.
- In the data phase, users transmit symbols and each AP receives the resulting superposition with additive noise.
- The network controller uses channel estimates to form postcoding vectors and estimate each user’s transmitted data symbol.
- The kth user’s achievable uplink rate is expressed as E(log2(1 + SINR_k)) using a worst-case uncorrelated additive-noise bound.
- The achievable SINR incorporates both channel-estimation error and pilot-contamination effects.
A. MMSE Receiver
The MMSE receiver maximizes each user’s SINR using channel estimates from all users, but correlated estimates complicate asymptotic analysis. A partial MMSE receiver is introduced to retain near-MMSE performance while enabling approximation.
- MMSE Receiver: The MMSE receiver maximizes each user’s SINR by using channel estimates of all users in the network.
- MMSE Receiver: Computing MMSE rates through Monte Carlo simulation requires long averaging over small scale fading, motivating an approximation based only on large scale fading coefficients.
- MMSE Receiver: Correlated channel estimates prevent direct use of random matrix theory tools for the desired approximation.
- MMSE Receiver: The partial MMSE receiver is designed to overcome this analytical obstacle while achieving performance very close to the full MMSE receiver.
B. Partial MMSE Receiver
The partial MMSE receiver reduces the full MMSE computation by selecting a subset of user channel vectors, prioritizing coherent interferers and nearby users. Its asymptotic SINR approximation depends only on large scale fading coefficients and remains numerically tractable for large systems.
- Partial MMSE construction: The partial MMSE vector for user k is defined using a selected subset of user channel vectors.The selection rule determines which users are included in the partial receiver.
- User selection: Coherent interferers are included because they dominate performance limitations as the antenna count grows.The selected set contains all users sharing user k’s pilot sequence.
- User selection: Randomly selecting one user from each non-coherent interference group leads to poor performance.The paper therefore motivates a smarter user-selection rule.
- User selection: The selection rule chooses a user from each relevant interference group who is in the close vicinity of user k.The choice is based on the large scale fading vectors β_i.
- Asymptotic approximation: The asymptotic approximation is derived as M and K grow with a finite M/K ratio and is used for finite system dimensions.The derivation applies random matrix theory and follows approximation approaches used in related systems.
- Asymptotic approximation: The partial MMSE SINR approximation is a function only of large scale fading coefficients and can be calculated numerically for large M and K.The formulation is long but remains computationally usable at large system sizes.
C. Large Scale Fading Decoding
LSFD uses slowly varying large scale fading coefficients to coordinate APs while reducing backhaul traffic. The section derives its achievable SINR, power-allocation formulation, and an asymptotic MF benchmark highlighting pilot-contamination limits.
- LSFD design: LSFD transmits only large scale fading coefficients from APs to the network controller, reducing backhaul traffic because these coefficients change about 40 times more slowly than small scale fading.The coefficients are independent of frequency and vary slowly over time.
- LSFD design: The LSFD estimate combines received signals using postcoding and power coefficients computed solely from large scale fading coefficients.The network controller forms the coefficients and estimates each user’s data symbol through a linear combination of received signals.
- SINR analysis: The paper derives an achievable SINR expression for the kth user with LSFD and identifies the corresponding SINR-maximizing receiver.The section also presents the associated SINR and the optimal LSFD vector.
- Power allocation: Power coefficients can be obtained through a max-min power-allocation problem with per-user transmit-power constraints, solvable by bisection because the objective is quasiconcave and the constraints are convex.The formulation imposes η_i ≤ 1 for each user.
- MF asymptotics: With infinitely many APs and a fixed number of users, MF SINR is limited by coherent pilot-contamination interference, while cell-free SINR becomes constant in the limit.The asymptotic MF result assumes independent large scale fading coefficients.
IV. NUMERICAL RESULTS
Numerical experiments compare MMSE, partial MMSE, asymptotic MMSE, LSFD, and MF receivers under independent and correlated large scale fading. The asymptotic MMSE approximation is tight, while MMSE and LSFD substantially outperform MF in 5%-outage rate.
- 5.1-fold and 2.6-fold gains over MF are achieved by MMSE and LSFD, respectively, in 5%-outage rate.
- The asymptotic MMSE approximation is very tight relative to the MMSE receiver in the per-user rate CDFs.
- MMSE, partial MMSE, and the asymptotic approximation produce closely matching rates across the evaluated scenarios.The partial MMSE is reported as virtually optimal, and the approximation remains very accurate even with few APs and users.
- Shadow-fading correlation significantly affects system performance in the evaluated independent and correlated models.
- 5%-outage rates for MMSE and partial MMSE increase as the network size increases when M/K = 8 and K/τ = 4.
V. CONCLUSION
The paper studies uplink cell-free systems with MMSE and LSFD receivers and introduces tractable partial-MMSE and LSFD approaches. MMSE-related rates closely match the asymptotic approximation, while both MMSE and LSFD outperform MF, with a considerable gap between them.
- The asymptotic approximation remains very accurate even for a small number of APs and users.
- MMSE, partial MMSE, and the asymptotic approximation achieve rates that are very close.
- MMSE and LSFD provide significant gains over MF, while a considerable performance gap remains between MMSE and LSFD.