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Joint Pilot Design and Uplink Power Allocation in Multi-Cell Massive MIMO Systems

Trinh Van Chien, Emil Björnson, Erik G. Larsson

arXiv:1707.03072v2cs.ITmath.IT

TL;DR

In multi-cell Massive MIMO, pilot reuse causes contamination, and prior work leaves joint pilot design and power control unaddressed. This paper treats pilot signals as continuous variables, derives spectral-efficiency expressions, and formulates max-min optimization. Reported comparisons find the proposed design performs best among related algorithms.

  • Problem

    Pilot reuse causes channel-estimation degradation and mutual interference, while prior work had not analyzed joint pilot design and power control in Massive MIMO.

    Source: §I · p. 2

  • Method

    The paper represents pilots with continuous optimization variables, derives spectral-efficiency expressions for Rayleigh fading with maximum-ratio detection, and formulates joint max-min optimization of pilots and powers.

    Source: §I · p. 2

  • Results

    The proposed pilot design yields the highest performance among the related algorithms in the reported comparisons as user and pilot counts vary.

    Source: §VII · p. 12

  • Takeaways & Limitations

    Continuous pilot variables provide a framework that includes pilot structures beyond predefined combinatorial assignments and supports joint pilot and power optimization.

    Source: §I · p. 2

Abstract

from arXiv · show

This paper considers pilot design to mitigate pilot contamination and provide good service for everyone in multi-cell Massive multiple input multiple output (MIMO) systems. Instead of modeling the pilot design as a combinatorial assignment problem, as in prior works, we express the pilot signals using a pilot basis and treat the associated power coefficients as continuous optimization variables. We compute a lower bound on the uplink capacity for Rayleigh fading channels with maximum ratio detection that applies with arbitrary pilot signals. We further formulate the max-min fairness problem under power budget constraints, with the pilot signals and data powers as optimization variables. Because this optimization problem is non-deterministic polynomial-time hard due to signomial constraints, we then propose an algorithm to obtain a local optimum with polynomial complexity. Our framework serves as a benchmark for pilot design in scenarios with either ideal or non-ideal hardware. Numerical results manifest that the proposed optimization algorithms are close to the optimal solution obtained by exhaustive search for different pilot assignments and the new pilot structure and optimization bring large gains over the state-of-the-art suboptimal pilot design.

I. INTRODUCTION

Pilot reuse is necessary in multi-cell Massive MIMO, but users sharing pilots contaminate channel estimates and increase interference. The paper proposes continuous pilot design and joint power optimization to address this problem under max-min fairness.

I. INTRODUCTION

  • Pilot reuse is needed because orthogonal pilots would make overhead proportional to all network users, but reuse lets same-pilot users degrade channel estimation and cause interference.The paper identifies pilot design as important for mitigating this contamination.Source: §I · p. 1
  • Prior coordinated assignment methods face a combinatorial problem, while existing algorithms assume fixed pilot and data powers.The paper identifies joint pilot design and power control as an unaddressed problem.Source: §I · pp. 1–2
  • The paper derives closed-form spectral-efficiency expressions for Rayleigh fading with maximum-ratio detection, for ideal hardware and hardware impairments.The expressions show how mutual interference, noise, and pilot contamination affect spectral efficiency.Source: §I · p. 2
  • The max-min fairness formulation jointly optimizes pilot signals, pilot powers, and data powers; the paper proposes a polynomial-time local-optimum algorithm for the NP-hard signomial problem.It also investigates exhaustive search over pilot assignments for comparison.Source: §I · p. 2
  • Numerical evaluations with ideal hardware and hardware impairments find the local solution close to exhaustive search and show significant improvements over prior heuristic algorithms.The paper positions its proposed algorithms against both an exhaustive-search benchmark and prior heuristic methods.Source: §I · p. 2

A. Proposed Pilot Design

  • Each pilot is represented using mutually orthonormal basis vectors, with users assigning power to the basis vectors to form their pilot signals.The construction can produce orthogonal pilots and many non-orthogonal sets, providing additional degrees of freedom to reduce network interference.Source: A. Proposed Pilot Design · p. 2

B. Other Pilot Designs

The section describes pilot structures that combine assignment and power choices, then uses reuse sets to characterize contamination and channel-estimation quality. Exhaustive assignment search requires a dictionary of network-wide possibilities that grows rapidly with system size.

B. Other Pilot Designs

  • Prior assignment methods use orthogonal pilots with equal fixed power, while other cited work optimizes pilot powers under single-cell or fixed-assignment assumptions.The proposed combination of pilot structure and assignment permits joint optimization of power allocation and assignment.Source: B. Other Pilot Designs · p. 3
  • The modified pilot design is a special case of the proposed structure and defines a reuse set containing users whose pilots contaminate a given user.The section assumes τp = K and orthogonal pilots within each cell for exhaustive search.Source: B. Other Pilot Designs · p. 3
  • Exhaustive search enumerates pilot assignments in a dictionary of network-wide combinations, whose deduplicated size still grows rapidly with K and L.The dictionary is used to seek the assignment that maximizes spectral efficiency.Source: B. Other Pilot Designs · p. 3
  • Channel-estimation quality depends on pilot power and the reuse set, whose selection can mitigate estimation errors and reduce coherent interference during data transmission.The section relates the estimate and estimation error to the pilot structure and its pilot powers and inner products.Source: B. Other Pilot Designs · p. 4

B. Uplink Data Transmission

The section derives an uplink ergodic-capacity lower bound for maximum-ratio detection and arbitrary pilot structures. Its effective SINR reflects pilot design, channel-estimation quality, contamination, and data power, including coherent interference that persists as antenna count grows.

B. Uplink Data Transmission

  • The paper computes a closed-form expression for the uplink ergodic-capacity lower bound with maximum-ratio detection and arbitrary pilots.The bound uses the use-and-then-forget technique; the authors describe it as believed to be close to actual capacity.Source: B. Uplink Data Transmission · p. 5
  • As the number of base-station antennas grows, the signal numerator and coherent interference from pilot contamination both scale linearly with M.The denominator's non-coherent interference term is independent of M.Source: B. Uplink Data Transmission · p. 5
  • Optimizing pilot and data power terms can improve spectral efficiency by enhancing channel estimation and reducing coherent interference.The limiting spectral efficiency depends only on pilot design, through pilot inner products, and data power.Source: B. Uplink Data Transmission · p. 5
  • The lower-bound SINR jointly reflects pilot design, channel-estimation quality, pilot contamination, and data-power control, making it the metric used for max-min fairness optimization.The passage contrasts this joint accounting with MSE, which cannot distinguish pilot contamination from noise.Source: B. Uplink Data Transmission · p. 5

IV. MAX-MIN FAIRNESS OPTIMIZATION

The section frames joint pilot and data optimization as NP-hard and presents both exhaustive search for a global solution under one pilot design and a polynomial-time local approach under the new design.

IV. MAX-MIN FAIRNESS OPTIMIZATION

  • The fairness problems are NP-hard; exhaustive search over pilot assignments provides a globally optimal solution for pilot structure (12).The section also presents a polynomial-time algorithm for a locally optimal solution with pilot structure (7).Source: §IV · p. 5
  • For pilot structure (7), the proposed alternative to global search finds a locally optimal solution in polynomial time.The section first uses the spectral-efficiency expressions to formulate joint pilot and data optimization problems.Source: §IV · p. 5
  • The section compares the global-search route for pilot structure (12) with the local-solution route for pilot structure (7).The former searches all pilot assignments; the latter avoids seeking the global optimum.Source: §IV · p. 5

A. Problem Formulation

The max-min fairness formulation jointly chooses pilot signals and pilot and data powers under per-user power limits. The paper contrasts exhaustive global search over pilot assignments with a locally optimal continuous-design approach, while noting the coordination and computational costs of the optimization.

A. Problem Formulation

  • The max-min fairness problem jointly optimizes pilot signals and pilot and data powers under power budgets, including a maximum data power for each user.The optimization jointly generates the pilots and performs pilot and data power control.Source: A. Problem Formulation · p. 6
  • Signomial SINR constraints make the max-min problem NP-hard, although the power constraints create a compact feasible domain where an optimum exists.The SINRs are continuous functions of the optimization variables.Source: A. Problem Formulation · p. 6
  • The global optimization requires coordination among cells; distributed dual/primal decomposition is identified as a possible response to practical signaling, delay, and scalability limits.The paper's main target is to measure the improvement possible with joint pilot design and uplink power control.Source: A. Problem Formulation · p. 6
  • Exhaustively checking each pilot-assignment instance and solving its geometric program obtains the global optimum, but the search grows rapidly with K and L.Each iteration solves a geometric program, and the exhaustive algorithm is computationally heavy as the number of users and cells increases.Source: A. Problem Formulation · p. 6
  • A benchmark fixes data powers at their maximum and optimizes only pilot powers, following the approach of the authors' previous work.Source: A. Problem Formulation · p. 7

B. Local Optimality Algorithm

The local algorithm approximates the signomial SINR constraints with monomial bounds and repeatedly solves geometric programs. Its successive updates converge to a KKT local point, with convergence reported after only a few iterations.

B. Local Optimality Algorithm

  • Weighted arithmetic mean-geometric mean bounds convert the signomial SINR constraints into monomial constraints for geometric-program optimization.The monomial approximation is selected using weights based on the previous iterate's optimal pilot powers.Source: B. Local Optimality Algorithm · p. 7
  • The algorithm iteratively updates approximation weights and solves a geometric program, converging to a KKT local point of the max-min problem.The initial powers must lie in the feasible domain; the process repeats until convergence.Source: B. Local Optimality Algorithm · p. 7
  • The method has KL(τp+1)+1 optimization variables and 3KL constraints, and its complexity scaling is characterized as a general estimate rather than an exact runtime.The source notes exact complexity and runtime depend on computer configuration and implementation.Source: B. Local Optimality Algorithm · p. 7
  • Although each iteration is more costly than in the exhaustive-search algorithm, the successive-approximation method converges after only a few iterations.Source: B. Local Optimality Algorithm · p. 8

V. PILOT OPTIMIZATION FOR CELLULAR MASSIVE MIMO SYSTEMS WITH HARDWARE IMPAIRMENTS

User hardware distortions create coherent self-interference that can be viewed as pilot contamination from the user itself. This changes the pilot-optimization setting because sufficiently weaker contamination from distant users can be neglected.

V. PILOT OPTIMIZATION FOR CELLULAR MASSIVE MIMO SYSTEMS WITH HARDWARE IMPAIRMENTS

  • User hardware distortion creates coherent self-interference, which can be viewed as pilot contamination caused by the user itself.Source: §V · p. 8
  • Unlike the BS impairments, which can be neglected because independent antenna distortions combine non-coherently, single-antenna user distortion affects spectral efficiency substantially.Source: §V · p. 8
  • With user self-interference accounted for, pilot contamination from distant users can be neglected when it is substantially weaker than that self-interference.Source: §V · p. 8

A. Channel Estimation under Hardware Impairments

The hardware-impairment model attenuates the user signal and adds Gaussian distortion with power equal to the lost signal power. The resulting channel estimate uses LMMSE because the MMSE estimator is intractable, and estimation-error variance grows with impairment level.

A. Channel Estimation under Hardware Impairments

B. UL Data Transmission and Max-min Fairness Optimization under Hardware Impairments

For user hardware impairments, the section gives an MR-detection spectral-efficiency bound and characterizes coherent self-interference. For correlated Rayleigh fading, it identifies which interference term changes and adapts the max-min fairness solution methods.

B. UL Data Transmission and Max-min Fairness Optimization under Hardware Impairments

  • Assuming Gaussian user distortion, Theorem 2 gives an uplink ergodic-capacity lower bound for the stated pilot structure with MR detection.The hardware-impairment SE expression is obtained for user k in cell l.Source: B. UL Data Transmission and Max-min… · p. 9
  • Hardware impairments reduce coherent SINR gain by (1 −ϵ2)2 and add coherent self-interference; as M→∞, only the signal and self/coherent-interference terms remain.The self-interference can be relatively large because the user generally has a stronger channel than pilot-contaminating interferers.Source: B. UL Data Transmission and Max-min… · p. 9
  • The hardware-impairment fairness problem generalizes the ideal-hardware problem; successive approximation yields a polynomial-time local optimum, while exhaustive search gives the combinatorial-pilot global solution.The ideal-hardware algorithms are stated to extend to this setting.Source: B. UL Data Transmission and Max-min… · p. 9

VI. GENERALIZATION TO CORRELATED RAYLEIGH FADING

  • The correlated-Rayleigh model assumes covariance matrices with equal diagonal elements for co-located BS antennas, while nonzero off-diagonal elements represent spatial correlation.The channel estimate uses the pilot transmission model and element-wise MMSE estimation.Source: §VI · p. 9
  • Channel correlation affects non-coherent interference but leaves coherent interference as in uncorrelated Rayleigh fading; the fairness problem can reuse the proposed local and global algorithms.The local solution uses Algorithm 2, while the global solution modifies Algorithm 1 for the combinatorial pilot design.Source: §VI · p. 10

VII. NUMERICAL RESULTS

Numerical results show that joint pilot and data power optimization improves max-min spectral efficiency over the compared pilot designs and approaches exhaustive-search performance. Gains persist across user counts, pilot lengths, antenna counts, hardware impairment levels, and correlated fading, while using less transmit power in the reported comparisons.

VII. NUMERICAL RESULTS

  • At M = 300 and K = τp = 2, joint pilot/data design adds 1.53× average max-min SE over smart assignment and performs similarly to exhaustive search.The joint approach averages 84% of optimal SE with substantially lower computational complexity.Source: Figure 2 · p. 10, §VII · p. 11
  • Pilot-power exhaustive search consumes 111 mW on average for pilot signaling; proposed pilot-only design uses 50 mW per pilot symbol, below the 200 mW full-power schemes.The proposed structure allows non-orthogonal pilots, unlike prior designs that assign power to only one basis vector per user.Source: Figure 3 · p. 10, §VII · p. 11
  • Across K = τp = 2–8, the proposed design improves max-min SE 2.39×–4.58× over random assignment and gains 1.72× over smart assignment at K = τp = 8 with fixed data power.The reported gains over random assignment rise with the tested user and pilot counts.Source: Figure 4 · p. 11, §VII · p. 12
  • With 100 antennas, joint pilot/data optimization reaches 1.18 b/s/Hz; with 900 antennas it reaches 2.12 b/s/Hz, and the gap over smart assignment grows with antenna count.The proposed joint design's SE is observed to saturate at about 2.10 b/s/Hz due to pilot contamination.Source: Figure 5 · p. 11, §VII · p. 12
  • Increasing pilot length from 1 to 19 yields up to 12.17× gain with joint pilot/data optimization; lengths above τp = KL add SE, but not significantly.The paper identifies τp = KL as a good choice when sufficient time-frequency resources are available.Source: Figure 6 · p. 11, §VII · p. 12
  • At impairment level ε = 0.1, SE reductions range from 3.61% to 7.27%; correlated fading has little SE difference from uncorrelated fading, while the proposed design remains higher than alternatives.At ε = 0.2, reductions are 15.04%, 16.51%, 18.88%, and 28.87% for random assignment, smart assignment, proposed pilot design with full data power, and proposed joint optimization, respectively.Source: Figure 7 · p. 11, §VII · pp. 12–13, Figure 8 · p. 12
  • Joint pilot/data power optimization yields 1.86× higher average SE than pilot-only optimization and averages 84% of optimal SE at substantially lower complexity.The comparison is reported for the K = τp = 2 case.Source: §VII · p. 11
  • As users and pilots increase from 2 to 8, joint pilot/data optimization gains 1.87×–4.01× over pilot-only optimization, while max-min SE declines as mutual interference strengthens.The different designs perform about the same for large user counts.Source: §VII · p. 12

VIII. CONCLUSION

The paper unifies pilot design and uplink power allocation for cellular Massive MIMO, treating pilots as continuous optimization variables. Its algorithm finds a local optimum that outperforms prior state-of-the-art pilot-assignment algorithms and is close to exhaustive search.

VIII. CONCLUSION

  • The proposed algorithm finds a local optimum that outperforms prior state-of-the-art pilot-assignment algorithms and is close to exhaustive search.The globally optimal solution is NP-hard to find.Source: §VIII · p. 13
  • The framework jointly optimizes pilot signals and uplink powers, treating pilot signals as continuous variables rather than combinatorial assignments.It formulates a general max-min SE optimization problem.Source: §VIII · p. 13
  • The paper derives an uplink ergodic SE result for Rayleigh fading channels with MR detection, for ideal hardware and hardware impairments.Source: §VIII · p. 13

APPENDIX

The appendix derives the SINR expressions used in the paper and establishes convergence of the iterative optimization procedure to a KKT local point.

A. Proof of Lemma 2

  • The proof evaluates the SINR theorem’s expectations using independence properties and a fourth-order channel moment, then substitutes them to obtain the SINR expression.Source: A. Proof of Lemma 2 · p. 13

B. Proof of Theorem 1

  • The iterative geometric program has a nondecreasing, bounded objective and converges to a KKT local point of the signomial problem.The proof uses the approximation properties and matching conditions for the constraint functions.Source: B. Proof of Theorem 1 · p. 14

C. Proof of Theorem 2

  • For the hardware-impairment SINR, the proof computes expectations using uncorrelation properties despite channel estimates and errors being neither Gaussian nor independent.The resulting expectations are substituted into the effective SINR expression.Source: C. Proof of Theorem 2 · p. 14
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