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UL-DL duality for cell-free massive MIMO with per-AP power and information constraints

Lorenzo Miretti, Renato L. G. Cavalcante, Emil Björnson, Sławomir Stańczak

TL;DR

This work develops a novel uplink-downlink duality for cell-free massive MIMO under per-AP power and information constraints, enabling ergodic rate optimization via a dual uplink problem. By extending MMSE-based precoding to incorporate per-AP information constraints, the authors derive a structured, scalable solution that can be implemented in distributed fashion or centralized with long-term parameter tuning. The key result is that optimal joint precoders are realized as parametrized MMSE solutions with parameters $(\mathbf{p},\boldsymbol{\sigma})$ derived from a dual uplink formulation, and that a practical fixed-point based power control can recover the downlink performance. Numerical examples illustrate feasibility and performance in user-centric deployments, highlighting the benefits of local CSI with per-AP constraints and the trade-offs relative to centralized schemes. Overall, the paper provides a rigorous, tractable framework for distributed downlink precoding in cell-free networks under realistic information-sharing constraints.

Abstract

We derive a novel uplink-downlink duality principle for optimal joint precoding design under per-transmitter power and information constraints in fading channels. The information constraints model limited sharing of channel state information and data bearing signals across the transmitters. The main application is to cell-free networks, where each access point (AP) must typically satisfy an individual power constraint and form its transmit signal using limited cooperation capabilities. Our duality principle applies to ergodic achievable rates given by the popular hardening bound, and it can be interpreted as a nontrivial generalization of a previous result by Yu and Lan for deterministic channels. This generalization allows us to study involved information constraints going beyond the simple case of cluster-wise centralized precoding covered by previous techniques. Specifically, we show that the optimal joint precoders are, in general, given by an extension of the recently developed team minimum mean-square error method. As a particular yet practical example, we then solve the problem of optimal local precoding design in user-centric cell-free massive MIMO networks subject to per-AP power constraints.

UL-DL duality for cell-free massive MIMO with per-AP power and information constraints

TL;DR

This work develops a novel uplink-downlink duality for cell-free massive MIMO under per-AP power and information constraints, enabling ergodic rate optimization via a dual uplink problem. By extending MMSE-based precoding to incorporate per-AP information constraints, the authors derive a structured, scalable solution that can be implemented in distributed fashion or centralized with long-term parameter tuning. The key result is that optimal joint precoders are realized as parametrized MMSE solutions with parameters derived from a dual uplink formulation, and that a practical fixed-point based power control can recover the downlink performance. Numerical examples illustrate feasibility and performance in user-centric deployments, highlighting the benefits of local CSI with per-AP constraints and the trade-offs relative to centralized schemes. Overall, the paper provides a rigorous, tractable framework for distributed downlink precoding in cell-free networks under realistic information-sharing constraints.

Abstract

We derive a novel uplink-downlink duality principle for optimal joint precoding design under per-transmitter power and information constraints in fading channels. The information constraints model limited sharing of channel state information and data bearing signals across the transmitters. The main application is to cell-free networks, where each access point (AP) must typically satisfy an individual power constraint and form its transmit signal using limited cooperation capabilities. Our duality principle applies to ergodic achievable rates given by the popular hardening bound, and it can be interpreted as a nontrivial generalization of a previous result by Yu and Lan for deterministic channels. This generalization allows us to study involved information constraints going beyond the simple case of cluster-wise centralized precoding covered by previous techniques. Specifically, we show that the optimal joint precoders are, in general, given by an extension of the recently developed team minimum mean-square error method. As a particular yet practical example, we then solve the problem of optimal local precoding design in user-centric cell-free massive MIMO networks subject to per-AP power constraints.
Paper Structure (29 sections, 18 theorems, 60 equations, 3 figures, 1 algorithm)

This paper contains 29 sections, 18 theorems, 60 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Notation and definitions
  4. Lagrangian duality in general vector spaces
  5. System model
  6. Downlink achievable rates
  7. Per-AP power and information constraints
  8. Dual uplink rates under arbitrary noise powers
  9. Problem statement and Lagrangian duality
  10. Lagrangian dual problems
  11. Primal-dual solution methods
  12. Uplink-downlink duality
  13. Joint precoding optimization over a dual uplink channel
  14. Dual uplink power control with implicit optimal combining
  15. Optimal joint precoding structure
  16. ...and 14 more sections

Key Result

Proposition 1

Consider the functions $f: \mathcal{X} \to \mathbbmss{R}$ and $\bm{g}: \mathcal{X} \to \mathbbmss{R}^N$, where $\mathcal{X}$ is a real vector space, and the optimization problem Define the primal optimum $p^\star := \inf\{f(X)~|~\bm{g}(X)\leq \bm{0}, X\in\mathcal{X}\}$, and the dual optimum $d^\star := \sup\{d(\bm{\lambda})~|~\bm{\lambda}\in \mathbbmss{R}^N_+\}$, where $d(\bm{\lambda}):=\inf\{f(X

Figures (3)

  • Figure 1: Pictorial representation of the simulated setup: $K=16$ UEs uniformly distributed within a squared service area of size $1\times 1~\text{km}^2$, and $L=16$ regularly spaced APs with $N=4$ antennas each. Each UE is jointly served by a cluster of $Q=4$ APs offering the strongest channel gains.
  • Figure 2: Probability of feasibility of different minimum rate requirements, under different information constraints and a per-AP power constraint $P_l= 30$ dBm. Remarkably, our feasibility analysis covers optimal joint precoding under user-centric network clustering and either local CSI (local precoding) or centralized CSI (centralized precoding). The performance of the corresponding solutions under a sum power constraint $\sum_{l=1}^L P_l$ are also evaluated. As expected, due to the more restrictive information constraint, local precoding offers worse performance than centralized precoding. Similarly, a per-AP power constraints offers worse performance than a sum power constraint. Nevertheless, we observe that the local precoding with per-AP power constraint still robustly supports fairly high rates around $2.5$ b/s/Hz to all UEs.
  • Figure 3: Example of convergence behavior of the proposed algorithm for different step size constants $\alpha$. We consider an arbitrary UE drop, local precoding, and a minimum rate requirement of $3.5$ b/s/Hz. For each iteration $i\in \mathbbmss{N}$ of the outer loop, we plot (a) the dual objective $\tilde{d}(\bm{\lambda}^{(i)})$, and (b) the maximum transmit power over all APs $\max_{l\in\mathcal{L}}g_l(\bm{\lambda}^{(i)})+P_l$. The non-monotonic convergence is a common feature of projected subgradient methods. We observe that, despite a seemingly slower convergence in the first iterations, the more aggressive step size choice $\alpha = 17$ produces a feasible solution satisfying the per-AP power constraint $(\forall l\in\mathcal{L})~P_l = 30$ dBm after $20$ iterations only. This is enough to declare feasibility, since, for each outer iteration, the inner loops ensure that the SINR constraints are always satisfied.

Theorems & Definitions (41)

  • Proposition 1
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 31 more