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Energy Efficiency in Cell-Free Massive MIMO with Zero-Forcing Precoding Design
L. D. Nguyen, T. Q. Duong, H. Q. Ngo, K. Tourki
TL;DR
Cell-free massive MIMO offers coordinated service through many distributed APs, but its energy efficiency is challenged by AP, circuit, and backhaul power consumption. The paper develops low-complexity ZF power-control algorithms that account for QoS, per-AP power limits, backhaul consumption, and imperfect CSI. Numerical results show that the proposed power control outperforms equal or no power allocation, with EE depending non-monotonically on AP count and saturating at high transmit power.
Problem
Energy efficiency in bits/Joule has been neglected in much previous cell-free-network research, despite substantial AP transmission, circuit, and backhaul power consumption.
Method
The paper formulates constrained EE maximization for ZF precoding and solves perfect- and imperfect-CSI cases using low-complexity path-following algorithms.
Results
The proposed power-control algorithms outperform equal or no power allocation in EE, while additional APs beyond the optimum do not improve EE and high transmit power eventually saturates performance.
Takeaways & Limitations
EE optimization in cell-free massive MIMO should jointly account for AP count, transmit power, QoS, per-AP power limits, backhaul consumption, and CSI quality.
Abstract
from arXiv · showhide
We consider the downlink of a cell-free massive multiple-input multiple-output (MIMO) network where numerous distributed access points (APs) serve a smaller number of users under time division duplex operation. An important issue in deploying cell-free networks is high power consumption, which is proportional to the number of APs. This issue has raised the question as to their suitability for green communications in terms of the total energy efficiency (bits/Joule). To tackle this, we develop a novel low-complexity power control technique with zero-forcing precoding design to maximize the energy efficiency of cell-free massive MIMO taking into account the backhaul power consumption and the imperfect channel state information.
I. INTRODUCTION
Cell-free massive MIMO can provide uniformly good service through distributed APs, but its energy efficiency is a major neglected concern. The paper proposes low-complexity ZF-based power allocation algorithms to maximize EE under practical constraints.
- Motivation: Cell-free massive MIMO coherently serves randomly located users with numerous distributed, low-power APs.The technology can use simple linear processing, including conjugate beamforming and zero-forcing.
- Motivation: Energy efficiency in bits/Joule has been neglected in much previous cell-free-network research despite the large number of APs.Total energy includes AP transmission, circuit, and backhaul power, while users must satisfy minimum spectral-efficiency QoS.
- Contributions: The paper develops a low-complexity ZF precoding design that handles inter-user interference and formulates EE maximization as a constrained power allocation problem.The formulation accounts for practical energy-efficiency constraints described in the introduction.
- Contributions: Simple path-following algorithms require only a few iterations to converge to a locally optimal solution.The proposed process is intended to reduce the complexity of the EE maximization problem.
A. System Model
The system is a TDD cell-free massive MIMO downlink in which randomly deployed single-antenna APs jointly serve single-antenna users. A CPU coordinates the APs through a backhaul network carrying channel and precoding information.
- Network configuration: K single-antenna users are served by M randomly deployed single-antenna APs using the same time-frequency resource.The model assumes M ≫ K.
- Network configuration: A CPU connects to all APs through a backhaul network to exchange channel estimates, precoding vectors, and power-control coefficients.The backhaul provides the information needed for coordinated transmission.
- Channel model: The reciprocal channel between AP m and user k is modeled with large-scale fading β_mk and small-scale fading h_mk.The channel matrix between all APs and users is denoted by G.
- TDD operation: Each coherence interval uses TDD with uplink training followed by downlink payload transmission based on AP channel estimates.Users send pilots synchronously, after which APs estimate channels and precode the downlink data.
1) Uplink training:
During uplink training, users synchronously transmit pilot sequences and APs use the received signals for MMSE channel estimation. Orthogonal pilots require the training length to be at least the number of users.
- Pilot signaling: The kth user is assigned a pilot sequence ϕ_k of length τ_u samples.The training slot length and pilot vectors define the uplink-training signaling structure.
- Pilot signaling: Pilot sequences are mutually orthonormal, requiring τ_u ≥ K.Orthogonality is expressed by zero cross-inner-products and unit pilot norms.
- Received pilots: The mth AP receives the aggregate pilot signal from all K users plus additive Gaussian noise.The normalized uplink power is ρ_r, and the noise covariance is I_τu.
- Channel estimation: APs apply MMSE estimation to the received pilots to obtain channel estimates and corresponding estimation errors.Under MMSE estimation, the channel estimate and estimation error are independent.
2) Downlink Payload Data Transmission:
In the downlink payload phase, each AP transmits precoded user symbols under a per-AP power constraint. The received signal includes the intended symbol, interference, and noise.
- Transmission model: The mth AP transmits a precoded combination of user symbols using downlink power ρ_f.The precoding coefficients satisfy E{||x_m||^2} ≤ ρ_f, while each intended symbol has unit average power.
- Reception model: The received downlink signal at each user contains the transmitted user data together with additive noise.The receiver noise is modeled as a circularly symmetric complex Gaussian variable with unit variance.
3) Zero-Forcing Precoding Design:
The downlink uses zero-forcing precoding with power-control coefficients, while modeling transmit, circuit, and backhaul power consumption under channel-estimation error.
- Zero-forcing precoding can eliminate inter-user interference when the APs have perfect channel state information.
- The precoding coefficients combine user power-control coefficients η_k with elements b_mk derived from the channel estimate matrix.
- With imperfect channel estimation, the received signal contains desired-signal and interference terms caused by channel-estimation error.
- The user and sum spectral efficiencies account for channel-estimation overhead and the selected power-control vector.
- Total downlink power includes static circuit power, AP transmit-related circuit and amplifier power, and backhaul-link consumption.
B. Formulation problem
The paper formulates energy-efficiency maximization as a constrained ratio of sum throughput to total power consumption, including user QoS and positive power-control constraints.
- The optimization includes QoS requirements for each user and positivity constraints on all power-control coefficients.
- The objective is the ratio of sum throughput to total power consumption, representing energy efficiency in bits/Joule.
- Maximizing energy efficiency is reformulated through an equivalent objective while retaining the original constraints.
- The solution considers separate cases for perfect and imperfect channel estimation at the APs.
III. MAXIMIZING EE WITH PERFECT CHANNEL ESTIMATION (PCE)
Under perfect channel estimation, the interference term caused by estimation error disappears, and the resulting fractional optimization can be solved with Dinkelbach’s algorithm.
- With perfect channel estimation, the estimation-error term is removed from the received-signal expression.
- The perfect-estimation energy-efficiency problem is reformulated before applying fractional-programming methods.
- Because the objective is a ratio of concave and affine functions with convex constraints, Dinkelbach’s algorithm solves the problem through a convex program.
- The algorithm iteratively solves the associated convex program to identify λ > 0 for the fractional optimum.
IV. MAXIMIZING EE WITH IMPERFECT CHANNEL ESTIMATION (IPCE)
With imperfect channel estimation, the original energy-efficiency numerator is no longer concave, so the paper replaces direct fractional optimization with an efficient procedure based on quadratic convex programs.
- MMSE channel estimation introduces estimation-error statistics into the imperfect-estimation formulation.
- The imperfect-estimation energy-efficiency problem is expressed as a constrained optimization with nonnegative power-control coefficients.
- Because the objective numerator is no longer concave, Dinkelbach’s algorithm cannot be applied directly.
- The proposed procedure requires solving only a few quadratic convex programs.
- The procedure uses convexity-based inequalities and feasible-point updates to construct successive optimization steps.
- An initial feasible point can be determined because the constraints in the quadratic formulation are convex.
V. NUMERICAL RESULTS
Numerical evaluations consider randomly deployed APs and users under specified propagation, power, and noise settings. The proposed power-control scheme improves energy efficiency over equal power allocation, with optimal AP counts and transmit-power saturation identified.
- Simulation setup: Each AP and user has maximum transmit powers of 200 mW and 100 mW, respectively, with τ = 200 and τ_u = K training samples.The model also includes amplifier, circuit, and static circuit power parameters.
- EE versus AP count: The proposed scheme outperforms equal power allocation in EE for both perfect-CSI and imperfect-CSI cases.The comparison includes a no-power-control case with equal power coefficients.
- EE versus AP count: The optimal AP count is M = 80 without power control, versus M = 40 for perfect CSI and M = 60 for imperfect CSI with the proposed algorithms.Adding APs beyond these points does not improve EE because power consumption also increases with AP count.
- EE versus transmit power: EE increases noticeably below 1 W of AP transmit power and saturates when transmit power exceeds 1 W.The transmit-power range evaluated is 0.2 to 2.2 W per AP, with M = 100 and K = 16.
- EE versus transmit power: At high transmit power, interference-limited operation and transmission-power domination prevent further EE improvement from simply increasing transmitted power.This explains the saturation observed in the transmit-power comparison.
VI. CONCLUSION
The paper proposes low-complexity algorithms for energy-efficient zero-forcing precoding in cell-free massive MIMO under QoS and per-AP power constraints. The algorithms are reported effective against no power control and more tractable for imperfect CSI than the Dinkelbach approach.
- VI. CONCLUSION: The proposed algorithms maximize zero-forcing downlink energy efficiency subject to per-user QoS and per-AP transmit-power constraints.They are designed for cell-free massive MIMO and have low complexity.
- VI. CONCLUSION: For imperfect CSI, the path-following algorithm is presented as more tractable and applicable than the Dinkelbach approach.The conclusion states that Dinkelbach's approach is suitable only for perfect CSI.
- VI. CONCLUSION: Numerical results demonstrate the effectiveness of the power-control algorithms compared with no power control.The conclusion summarizes the reported numerical validation.