PaperlayerView exact revision on arXiv

Source-linked AI summary

Max-Min Rate of Cell-Free Massive MIMO Uplink with Optimal Uniform Quantization

Manijeh Bashar, Kanapathippillai Cumanan, Alister G. Burr, Hien Quoc Ngo, Merouane Debbah, Pei Xiao

arXiv:1907.03262v1cs.ITmath.IT

TL;DR

Limited-capacity fronthaul is a key issue for cell-free Massive MIMO, motivating comparison of different quantized signals sent from APs to the CPU. The paper develops an iterative max-min rate solution and a fronthaul-capacity-based user assignment algorithm; its numerical results report substantial minimum-rate improvements over existing algorithms.

  • Problem

    Limited-capacity fronthaul links between APs and the CPU remain a major issue for cell-free Massive MIMO, where infinite-capacity fronthaul is not realistic.

    Source: §I · p. 1

  • Method

    The paper models quantization with Bussgang decomposition and iteratively alternates receiver-filter design and power allocation, while proposing user assignment based on fronthaul capacity.

    Source: §I · p. 2, A. Equivalent Max-Min Uplink Problem · p. 8, §VII · p. 9

  • Results

    The proposed max-min rate algorithm increases the median of the CDF of minimum uplink user rate by more than two times over existing algorithms, and user assignment achieves more than three times improvement in minimum user rate.

    Source: §IX · p. 12

  • Takeaways & Limitations

    The reported results show that the proposed max-min scheme and user assignment improve minimum-rate performance in cell-free Massive MIMO with limited-capacity fronthaul.

    Source: §IX · p. 12

    The numerical evaluation averages 300 random realizations and uses wireless microwave fronthaul links with C_fh = 100 Mbits/s unless otherwise indicated.

    Source: §VIII · p. 10

Abstract

from arXiv · show

Cell-free Massive multiple-input multiple-output (MIMO) is considered, where distributed access points (APs) multiply the received signal by the conjugate of the estimated channel, and send back a quantized version of this weighted signal to a central processing unit (CPU). For the first time, we present a performance comparison between the case of perfect fronthaul links, the case when the quantized version of the estimated channel and the quantized signal are available at the CPU, and the case when only the quantized weighted signal is available at the CPU. The Bussgang decomposition is used to model the effect of quantization. The max-min problem is studied, where the minimum rate is maximized with the power and fronthaul capacity constraints. To deal with the non-convex problem, the original problem is decomposed into two sub-problems (referred to as receiver filter design and power allocation). Geometric programming (GP) is exploited to solve the power allocation problem whereas a generalized eigenvalue problem is solved to design the receiver filter. An iterative scheme is developed and the optimality of the proposed algorithm is proved through uplink-downlink duality. A user assignment algorithm is proposed which significantly improves the performance. Numerical results demonstrate the superiority of the proposed schemes.

I. INTRODUCTION

The paper addresses limited-capacity fronthaul in cell-free Massive MIMO, comparing quantized channel-and-signal forwarding with quantized weighted-signal forwarding. It develops quantization and max-min rate methods for this setting, alongside a fronthaul-aware user assignment algorithm.

I. INTRODUCTION

  • Two fronthaul schemes are analyzed: forwarding quantized channel estimates and received signals, or forwarding quantized signals weighted by conjugate channel estimates.The comparison at equal fronthaul capacity depends on AP antennas, AP count, and channel coherence time; uniform quantization is used for simplicity and scalability.Source: §I · pp. 1–2
  • The max-min rate problem jointly optimizes receiver-filter coefficients and power allocation under limited fronthaul, using an iterative scheme validated by uplink-downlink duality.The receiver-filter subproblem is solved through a generalized eigenvalue problem, and power allocation through geometric programming.Source: §I · p. 2
  • The work extends an earlier iterative design to multiple-antenna APs and limited-capacity fronthaul, and adds user assignment.The earlier work considered single-antenna APs and perfect fronthaul links.Source: §I · p. 2

B. Optimal Quantization Model

  • The paper models quantization with Bussgang decomposition and presents an analytical solution for the optimal uniform-quantizer step size.For any quantization resolution, the paper states that the resulting step size and distortion power can be calculated.Source: §I · p. 2, B. Optimal Quantization Model · p. 3

II. SYSTEM MODEL

  • The channel vector is modeled using large-scale fading and small-scale fading, with the latter represented by i.i.d. CN(0, 1) elements.The large-scale fading is assumed known, and the paper notes that realistic COST channel models are future work.Source: §II · pp. 2–3, A. Uplink Channel Estimation · p. 3

A. Uplink Channel Estimation

  • The MMSE channel estimate is formed from user pilot sequences collected in a matrix and processed by de-spreading.The estimate is used in the paper's channel-estimation model.Source: A. Uplink Channel Estimation · p. 3

C. Uplink Transmission

In uplink transmission, users send symbols with powers set by their respective power coefficients, and each AP receives the combined signal with noise.

C. Uplink Transmission

  • Each user's transmitted symbol is scaled by the square root of its power coefficient and the normalized uplink SNR.The transmitted signal is x_k = √(ρq_k)s_k, with unit-power symbol s_k.Source: C. Uplink Transmission · p. 4
  • The received signal at each AP includes noise whose elements are modeled as CN(0, 1).Source: C. Uplink Transmission · p. 4

III. PERFORMANCE ANALYSIS

The section derives achievable uplink rates for two fronthaul architectures using the UaF bound, accounting for channel-estimation and quantization errors. The first sends quantized channel estimates and received signals to the CPU; the second sends quantized weighted signals.

III. PERFORMANCE ANALYSIS

  • Both architectures use the UaF bounding technique to derive achievable rates while accounting for channel-estimation and quantization errors.The rate analysis uses the bound to obtain expressions for the two quantization cases.Source: §III · p. 4

N2 ÍK

  • Case 1 quantizes the real and imaginary parts of both the received signal and channel estimates before CPU-side MRC detection.Theorem 1 gives the resulting closed-form achievable rate.Source: §III · p. 4, N2 ÍK · p. 5

III. PERFORMANCE ANALYSIS

  • The rate analysis separates desired signal, beamforming uncertainty, inter-user interference, noise, and quantization-error terms, which Proposition 1 identifies as mutually uncorrelated.The UaF-derived bound is described as tight for cell-free Massive MIMO in.Source: §III · p. 4, N2 ÍK · p. 5

N2 KÍ

  • Case 2 has each AP quantize the channel-weighted received signal and forward it to the CPU, where Theorem 2 gives the achievable rate with MRC detection.The weighted input is approximated as Gaussian to use quantizer values derived for Gaussian inputs.Source: N2 ÍK · p. 5, N2 KÍ · p. 6

A. Required Fronthaul Capacity

The section sets fronthaul requirements for the two quantization cases and formulates a constrained uplink max-min rate problem. It then fixes quantization levels at the fronthaul capacity and separates the non-convex design into receiver-coefficient and power-allocation subproblems.

A. Required Fronthaul Capacity

  • Case 1 requires 2αm,1(NK + Nτf) bits per AP per coherence interval, while Case 2 requires 2αm,2(Kτf) bits.The comparison holds total fronthaul bits equal across cases: 2(NK + Nτf)αm,1 = 2(Kτf)αm,2.Source: A. Required Fronthaul Capacity · p. 6

IV. PROPOSED MAX-MIN RATE SCHEME

  • The max-min uplink problem maximizes the minimum user rate subject to per-user transmit-power and AP-to-CPU fronthaul-capacity constraints.The formulation is presented for Case 2, with the same approach applicable to Case 1.Source: §IV · p. 6
  • Because rates increase monotonically with fronthaul capacity, the optimal quantization allocation uses each AP link's full capacity, fixing the quantization-bit values.The resulting problem remains non-convex jointly in receiver coefficients and user powers, so it is split into two subproblems.Source: §IV · pp. 6–7

A. Receiver Filter Coefficient Design

The scheme alternates between designing receiver coefficients for fixed powers and allocating power for fixed receiver coefficients. Receiver design is a generalized eigenvalue problem, while power allocation is formulated as a geometric program.

A. Receiver Filter Coefficient Design

  • For fixed user powers and quantization levels, each user's receiver coefficients are chosen to maximize that user's uplink SINR.The resulting receiver-design problem is a generalized eigenvalue problem solved using the eigenvector corresponding to the maximum generalized eigenvalue.Source: A. Receiver Filter Coefficient Design · p. 7, N2 ÍK · p. 7

B. Power Allocation

  • For fixed receiver coefficients and quantization levels, the power-allocation subproblem is formulated as a standard geometric program.The GP formulation allows the subproblem to be solved with convex optimization software.Source: B. Power Allocation · p. 7, N2 ÍK · p. 7

N2 ÍK

  • The algorithm alternates between receiver-coefficient design and power allocation until the stopping condition is met.Each iteration updates receiver coefficients using the preceding powers, then computes powers for the resulting coefficients.Source: N2 ÍK · p. 7

V. CONVERGENCE

The paper establishes that the proposed algorithm converges and is optimal by relating its max-min uplink problem to an equivalent virtual downlink problem. The proof uses equivalent power constraints, matching SINRs, and quasi-convex optimization.

V. CONVERGENCE

  • The algorithm's achievable max-min rate increases monotonically across iterations and is bounded above by the per-user power and fronthaul constraints, so it converges to a specific solution.Each subproblem is solved optimally with the other design variable fixed.Source: §V · pp. 7–8

VI. OPTIMALITY OF THE PROPOSED MAX-MIN RATE

  • Uplink-downlink duality guarantees the algorithm's optimality because the equivalent uplink and virtual downlink achieve the same SINRs, and the virtual downlink is quasi-convex.The downlink problem can therefore be solved optimally by bisection.Source: §VI · p. 8

A. Equivalent Max-Min Uplink Problem

  • The original per-user-power problem and the equivalent uplink problem with a total-power constraint have the same optimal solution.The equivalent problem is also handled by alternating receiver design and power allocation.Source: A. Equivalent Max-Min Uplink Problem · p. 8

B. Uplink-Downlink Duality for Cell-free Massive MIMO

  • With equivalent total-power constraints, the uplink and virtual downlink can realize the same SINRs for all users using the same filter coefficients and different power allocations.The stated duality condition applies to uplink MRC detection and virtual-downlink conjugate beamforming.Source: B. Uplink-Downlink Duality for… · p. 8

C. Equivalent Max-Min Downlink Problem

  • The equivalent downlink problem is made tractable by decoupling filter coefficients and power allocations, then solving a quasi-convex formulation through bisection.Its power-minimization subproblem can be reformulated as a second-order cone program.Source: N2 ÍK · p. 9

D. Prove of Optimality of Algorithm 1

The paper establishes Algorithm 1's optimality through equivalence between uplink and virtual-downlink formulations, then develops an AP user-assignment scheme and evaluates the system under several transmission cases.

D. Prove of Optimality of Algorithm 1

  • Problems P2, P6, and P7 are equivalent and achieve the same minimum user rate, supporting Algorithm 1's optimality through uplink-downlink duality.The virtual-downlink problem is optimally solved by bisection search.Source: D. Prove of Optimality of Algorithm 1 · p. 9

VII. USER ASSIGNMENT

  • The assignment method selects each AP's users by channel gain while choosing active-set sizes to use limited fronthaul capacity more efficiently.Users not selected by any AP are assigned to their best-link AP, with a low-gain active user displaced when needed.Source: §VII · p. 9

VIII. NUMERICAL RESULTS AND DISCUSSION

  • Simulations average system rates over 300 random realizations and use Cfh = 100 Mbits/s unless otherwise specified.The APs and users are randomly placed, and the stated setup uses 200 mW pilot and uplink data powers.Source: §VIII · p. 10

1) Performance of Different Cases of Uplink Transmission:

2) Performance of the Proposed User Max-Min Rate Algorithm:

The proposed max-min scheme improves minimum-rate performance over existing algorithms, and user assignment can further improve it. Simulations also support convergence and uplink-downlink equivalence, while showing how fronthaul capacity and transmission cases affect rates.

2) Performance of the Proposed User Max-Min Rate Algorithm:

3) Convergence:

  • Algorithm 1 converges after a few iterations, while the users' minimum rate increases with iteration number.The convergence simulations cover different random realizations and pilot settings.Source: 3) Convergence · pp. 10–11

4) Uplink-Downlink Duality in Cell-Free Massive MIMO System:

  • The original uplink and equivalent uplink and downlink formulations achieve the same minimum user rate, supporting their equivalence and the optimality proof.The comparison is reported for systems with 30 or 70 APs and both orthogonal and random pilots.Source: 4) Uplink-Downlink Duality in… · p. 11

5) Performance of the Proposed User Assignment Algorithm 2:

6) Effect of the Capacity of Fronthaul Links:

IX. CONCLUSIONS

  • The proposed max-min algorithm increases the median of the CDF of minimum uplink rates by more than two times versus existing algorithms; user assignment yields more than three times improvement.These are the conclusion's reported comparisons.Source: §IX · p. 12

APPENDIX A: PROOF OF LEMMA 1

The appendix derives uniform-quantizer parameters and uses properties of the Gaussian quantizer input in its proof.

APPENDIX A: PROOF OF LEMMA 1

  • The proof uses equations (3) and (6) to find a and b for the uniform quantizer.The supplied passage does not further specify these parameters' roles.Source: Appendix A · p. 12
  • The quantizer steps a1 and a2 follow from the Gaussian input distribution and l′ = l + 1.These are the stated properties used in the derivation.Source: Appendix A · p. 13
  • The proof concludes by establishing that the listed signal and quantization terms are mutually uncorrelated.The passage identifies this as completing Proposition 1's proof.Source: Appendix A · p. 13

APPENDIX C: PROOF OF THEOREM 2

The appendix derives the kth user's SINR by evaluating signal and noise terms, including quantization error, and substituting the intermediate expressions into the SINR equation.

APPENDIX C: PROOF OF THEOREM 2

  • The appendix separately derives the total noise for user k.The passage introduces this as the total noise term.Source: Appendix C · p. 14
  • Quantization-error power for user k is derived using the error covariance and the covariance of the quantizer input.The passage names these matrices as the covariance of the quantization error and of the input, respectively.Source: Appendix C · p. 14
  • The appendix obtains the kth user's SINR by substituting intermediate expressions into equation (22).This completes the proof of Theorem 2.Source: Appendix C · p. 15

APPENDIX D: PROOF OF PROPOSITION 2

The appendices establish that the power-allocation problem can be cast as a geometric program and that total- and per-user-power formulations have the same optimal solution. The latter result relies on a unique positive eigenvector and constraint-specific scaling.

APPENDIX D: PROOF OF PROPOSITION 2

  • A transformation makes the SINR constraint posynomial, establishing that power-allocation Problem P4 is a geometric program.Its objective and constraints are monomial and posynomial, respectively.Source: Appendix D · p. 15

APPENDIX E: PROOF OF LEMMA 2

  • For fixed receiver coefficients, the optimal power allocation is obtained from the eigenvector associated with the eigensystem’s unique positive eigenvalue.The allocation is formed by scaling that eigenvector to satisfy the applicable power constraint.Source: Appendix E · p. 15
  • The per-user-constrained iteration updates its total-power budget from the previous allocation and converges with equal uplink SINR for every user.At convergence, the per-user power constraints are satisfied.Source: Appendix E · p. 16
  • Problems P2 and P6 converge to the same optimal solution because they scale elements of the same unique positive eigenvector to meet their respective power constraints.P6 scales to meet the total-power constraint, while P2 scales to meet the per-user constraint.Source: Appendix E · p. 16

APPENDIX F: PROOF OF THEOREM 4

The proof relates uplink and downlink SINR matching to a condition on their total transmit powers. Satisfying that condition realizes the same SINRs for all users.

APPENDIX F: PROOF OF THEOREM 4

  • Matching all users’ uplink and downlink SINRs requires a condition between their total transmit powers.The condition is obtained by substituting the uplink and downlink SINRs and summing the resulting equations.Source: Appendix F · p. 16
Loading 1907.03262v1…