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Power Control in Cellular Massive MIMO with Varying User Activity: A Deep Learning Solution

Trinh Van Chien, Thuong Nguyen Canh, Emil Björnson, Erik G. Larsson

arXiv:1901.03620v3cs.IT

TL;DR

The paper addresses joint pilot and data power control for maximizing sum SE in multi-cell Massive MIMO with varying user activity. It develops a stationary-point iterative algorithm and PowerNet, a single neural network using large-scale fading to predict both powers; PowerNet achieves less than 1% sum-SE loss in one reported symmetric setting and operates far below 1 ms.

  • Problem

    Jointly optimizing pilot and data powers is important for addressing pilot contamination in multi-cell Massive MIMO, but varying user activity creates power-control functions with different input and output sizes.

  • Method

    The paper derives a weighted-MMSE-inspired iterative algorithm for the non-convex problem and designs PowerNet to predict pilot and data powers from large-scale fading with one network handling varying active-user counts.

  • Results

    PowerNet leads to less than 1% loss in sum SE in a symmetric multi-cell system serving 90 users, while its runtime is far below 1 ms.

  • Takeaways & Limitations

    A single large-scale-fading-based neural network can support real-time power control with varying numbers of active users in Massive MIMO systems.

  • Takeaways & Limitations

    The sum-SE algorithm can assign zero power to active users with small but non-zero large-scale fading coefficients, effectively rejecting them.

Abstract

from arXiv · show

This paper considers the sum spectral efficiency (SE) optimization problem in multi-cell Massive MIMO systems with a varying number of active users. This is formulated as a joint pilot and data power control problem. Since the problem is non-convex, we first derive a novel iterative algorithm that obtains a stationary point in polynomial time. To enable real-time implementation, we also develop a deep learning solution. The proposed neural network, PowerNet, only uses the large-scale fading information to predict both the pilot and data powers. The main novelty is that we exploit the problem structure to design a single neural network that can handle a dynamically varying number of active users; hence, PowerNet is simultaneously approximating many different power control functions with varying number inputs and outputs. This is not the case in prior works and thus makes PowerNet an important step towards a practically useful solution. Numerical results demonstrate that PowerNet only loses $2\%$ in sum SE, compared to the iterative algorithm, in a nine-cell system with up to $90$ active users per in each coherence interval, and the runtime was only $0.03$ ms on a graphics processing unit (GPU). When good data labels are selected for the training phase, PowerNet can yield better sum SE than by solving the optimization problem with one initial point.

I. INTRODUCTION

The paper studies joint pilot and data power control for cellular Massive MIMO with varying user activity, and proposes an iterative optimization method plus PowerNet for low-complexity prediction. The system model uses large-scale fading and supports dynamically varying active-user subsets.

  • Massive MIMO uses many antennas to serve tens of users on the same time-frequency resource, improving spectral and energy efficiency through channel decorrelation and array gains.
  • Joint pilot and data power optimization is important because pilot contamination contributes to inter-user interference in multi-cell systems.
  • Prior deep-learning power-control methods may require many neural networks because input and output dimensions vary with the number of active users.For L cells and up to Kmax users per cell, one prior approach would require L2Kmax networks; a small L = 4, Kmax = 5 setup requires 128 networks.
  • The paper formulates joint pilot and data power control for maximum sum SE when cells may contain different numbers of active users.
  • PowerNet is a residual, densely connected CNN that uses large-scale fading coefficients to predict powers and handles any active-user count up to a design maximum.
  • The dynamic system model represents each cell as serving a random active subset within coherence intervals, while the channel model uses i.i.d. Rayleigh fading with large-scale coefficients.

B. Uplink Data Transmission Phase

During uplink data transmission, active users send symbols across cells, and each base station uses maximum-ratio combining to detect them. The resulting closed-form ergodic SE accounts for array gain, pilot contamination, power, and the coherence-interval pre-log factor.

  • Each active user transmits a unit-power data symbol with a controllable data power, while each base station receives the superposition of signals from users across cells.
  • Base stations use maximum-ratio combining and a use-and-then-forget bound to obtain a closed-form lower bound on uplink ergodic SE.
  • The SINR denominator includes pilot contamination, while the numerator contains array gain proportional to the number of antennas at the serving base station.
  • Increasing the number of pilots reduces the data-transmission pre-log factor, equal to 1 − Kmax/τc.

III. JOINT PILOT AND DATA POWER CONTROL FOR SUM SPECTRAL EFFICIENCY OPTIMIZATION

The paper formulates non-convex joint pilot and data power control to maximize sum SE under per-symbol power constraints. An equivalent element-wise convex formulation enables an alternating algorithm with closed-form updates and lower implementation complexity.

  • The objective is to maximize the sum SE of all active users subject to pilot and data power constraints for each transmitted symbol.
  • The optimization is relevant for rapidly changing scheduling decisions and user mobility, but global optimization is generally computationally expensive.Branch-and-bound can obtain a global solution, but its complexity grows exponentially with the number of users.
  • An equivalent problem introduces auxiliary variables and has a convex structure when any one variable set is optimized while the others remain fixed.
  • Closed-form solutions for each block make the proposed iterative method particularly efficient while jointly accounting for channel-estimation errors and pilot contamination.
  • The alternating optimization procedure updates the auxiliary, weight, pilot-power, and data-power variables successively.

B. Iterative Algorithm

The iterative algorithm alternates closed-form updates until consecutive solutions change by less than a tolerance, producing a stationary point of the original power-control problem. Zero large-scale fading provides a structural way to represent inactive users for a single neural network.

  • The algorithm alternates updates of u, w, pilot powers, and data powers to obtain a stationary point.
  • Each update has a closed-form solution, and the equivalence of the reformulated and original problems transfers the stationary-point result to the original formulation.
  • The process stops when the variation between consecutive iterations satisfies a prescribed accuracy tolerance.
  • A user with zero large-scale fading receives zero pilot and data power, so inactive users can be represented by zero fading coefficients.
  • Sum-SE optimization may assign zero power to active users with small non-zero fading coefficients, unlike fairness-oriented controls that allocate non-zero power to all users.
  • This zero-power structure enables one neural network to mimic the algorithm for different numbers of active users.

IV. A LOW-COMPLEXITY SOLUTION WITH CONVOLUTIONAL NEURAL NETWORK

This section introduces a supervised deep learning framework that mimics the power control produced by Algorithm 1 for joint pilot and data power allocation in dynamic cellular Massive MIMO systems.

  • PowerNet uses supervised learning to mimic Algorithm 1’s joint pilot and data power allocation.The framework targets low-complexity implementation for a non-convex optimization problem.

A. Conditions on Large-Scale Fading Model

The model generates dynamic cellular Massive MIMO realizations with randomly active users and large-scale fading coefficients drawn under explicit distributional assumptions.

  • Each cell contains Kmax i.i.d. users, with each user independently active with probability p.The active-user set therefore varies across network realizations.
  • Active users’ large-scale fading vectors are i.i.d. realizations from a cell-specific PDF fl(β).The model assumes each user’s strongest channel is from its serving BS.
  • Each cell uses a common user-distribution function over its coverage area, while that function may differ between cells.The maximum number of users and activity probability are shared across cells for notational convenience but can be generalized.
  • The framework assumes large-scale fading coefficients lie in [0, 1] and reruns shadow-fading realizations when specified conditions are violated.Shadow fading is modeled with zero mean and 7 dB standard deviation.
  • The simulation model follows the 3GPP LTE Rayleigh-lognormal fading standard for non-line-of-sight conditions.Users are uniformly distributed in serving cells beyond 35 m, with activity probability p = 2/3.

B. Existence of a Single Neural Network for Joint Pilot and Data Power Control

The paper exploits the optimization structure and cellular geometry to construct one CNN that handles varying user activity through fixed-size, zero-padded large-scale fading inputs.

  • A single neural network handles activity patterns by representing inactive users with zero-valued large-scale fading coefficients and zero power.This removes the need to expose active-user counts and pilot assignments as separate inputs.
  • PowerNet uses fixed numbers of inputs and outputs regardless of user activity, allowing one network to approximate the varying power-control mappings.Its input is large-scale fading information, and its outputs are pilot and data powers.
  • The zero-insertion construction relies on the sum-SE optimization structure and does not directly apply to maximum product-SINR or max-min fairness objectives.Under those alternative metrics, zero insertion would assign zero powers to everyone.
  • A CNN processes the L × L × Kmax fading tensor because users are identically distributed within cells and the deployment creates spatial structure.Convolutional kernels are reused across users and can learn recurrent patterns tied to the BS geometry.
  • The CNN design is evaluated for symmetric deployments, while learning analogous patterns for asymmetric BS locations is left for future work.The cited figure uses a square-grid deployment to make the spatial pattern visible.

1) The forward propagation:

PowerNet forward propagation transforms the large-scale fading tensor through convolutional feature extraction and residual-dense blocks, then predicts pilot and data powers within their budgets.

  • The first forward-propagation stage applies convolutional layers to the large-scale fading tensor I.The initial convolutions extract large-scale fading features while preserving spatial dimensions through stride 1 and zero padding.
  • Sequential ResDense blocks use multiple convolutional operators and residual-dense connections to extract propagation features.The first three modules apply ReLU activations to their feature maps.
  • Horizontal and vertical one-dimensional convolutions align the network’s input and output sizes and exploit correlations in both directions.A transpose layer produces the L × 1 × Kmax prediction format used for pilot and data powers.
  • Sigmoid activations restrict feature maps to [0, 1], after which predicted pilot and data powers are scaled by users’ maximum power budgets.The same forward propagation is used during training and testing.

2) The back propagation:

The section describes how PowerNet is trained with supervised labels generated by the alternating-optimization algorithm, while noting that label quality and training-data requirements affect performance.

  • The supervised-learning loss uses the Frobenius norm and is optimized over the network’s convolution kernels and biases.
  • The supervised loss makes PowerNet perform, on average, no better than the continuous mapping, while future work could reduce training samples through semi-supervised learning.
  • PowerNet uses stochastic gradient descent with momentum and learning-rate updates to obtain a local solution for its parameters.
  • Because Algorithm 1 can converge to different stationary points, multiple initializations can provide higher-quality labels, increasing the likelihood of finding the global optimum.
  • PowerNet is trained using user realizations paired with jointly optimized pilot and data-power outputs generated by Algorithm 1.
  • PowerNet’s forward propagation requires 9KmaxL^2Q + 28Q^2L^2N + 4L^2QKmax + 4LKmax arithmetic operations, matching testing complexity because each input passes once without backpropagation.

V. NUMERICAL RESULTS

The numerical results show that Algorithm 1 converges near its stationary point in roughly 300–400 iterations, while PowerNet closely matches optimization-based performance and scales to larger systems.

  • Convergence: At iteration 300, Algorithm 1 reaches 99% of its stationary-point sum SE per cell for both four-cell and nine-cell systems.
  • Convergence: The average convergence time is 400 iterations at activity probability 1/3 and 300 iterations when all users are active.
  • Sum spectral efficiency: Using more random initializations yields only tiny gains, with the average improvement remaining below 1% when increasing from one to five initializations.
  • Power consumption: JPDPO allocates 127 mW per data symbol and 150 mW per pilot symbol on average, using 18% extra pilot power to improve channel estimation.
  • Power consumption: JPDPO uses 25% and 36% less power than full power, while DPOO allocates 124 mW per data symbol.
  • Predicted performance of PowerNet: In the four-cell case, PowerNet is 1.5% below Algorithm 1 in sum SE and about 1% below JPDPO in per-user SE.
  • Predicted performance of PowerNet: In the nine-cell case, PowerNet improves sum SE over full power by up to 16.3%, improves per-user SE by 12.87%, and loses only 2% relative to JPDPO.
  • Predicted performance of PowerNet: PowerNet’s low prediction error at the larger optimization scale supports its scalability, with performance attributed to spatial-feature extraction and residual dense connections.

C. Varying User Activity, Channel Models, and Data Label Effect

PowerNet remains close to the iterative joint pilot-and-data power-control solution under changing activity patterns, channel models, antenna counts, and training labels, while supporting rapid implementation.

  • Varying User Activity: 99% of JPDPO’s sum SE is achieved when user activity changes from probability 2/3 during training to 1/3 during testing.The setting uses L = 4, Kmax = 10, and M = 200.
  • Varying User Activity: 98.4% of Algorithm 1’s performance is achieved when testing activity probability increases to 5/6.In this scenario, JPDPO exceeds FDNN by 15%, while the gap between full power and JPDPO is 30%.
  • Varying User Activity: 19.12 b/s/Hz per cell is obtained by JPDPO, compared with 15.84 b/s/Hz for FP, while PowerNet loses only 1.34% relative to JPDPO under uniformly distributed activity probabilities.DPOO loses 1.32% relative to JPDPO in the same experiment.
  • Channel and Antenna Variation: 1.30% is PowerNet’s SE loss when the testing phase uses a different number of BS antennas than training.The result supports application when antennas are switched on or off for energy-efficiency purposes.
  • Channel Models and Data Labels: Less than 2% prediction loss is observed when the large-scale fading distributions differ between training and testing.Using labels from 40 initializations improves performance by about 2.5% at the 95%-likely point and by about 0.5% over JPDPO based on one initialization.
  • Implementation: PowerNet predicts both pilot and data powers using only large-scale fading and runs far below 1 ms, supporting real-time power control.The framework uses specialized CNN hardware, while classical optimization requires dedicated hardware circuits.

APPENDIX

The appendix derives the signal model, decoding error, and optimization substitutions used to characterize the power-control problem and obtain closed-form conditional updates.

  • Signal Model: The detected desired signal is formed with a real beamforming coefficient and Gaussian noise.The desired symbol and noise model are introduced before computing the decoding MSE.
  • MSE Derivation: The decoding MSE is substituted into the signal model to obtain the expression used in the optimization problem.This connects the detection rule to the objective formulation.
  • Conditional Optimization: The optimal decoder coefficient is obtained by differentiating the MSE with respect to the beamforming coefficient and setting the derivative to zero.The resulting solution is identified as uopt_l,k.

B. Proof of Theorem 2

The proof shows that alternating optimization converges to a fixed point through bounded monotonic updates and then establishes equivalence between the reformulated and original stationary conditions.

  • Feasible Set and Optimal Updates: Nonnegative pilot and data powers define the feasible set, with Lagrange multipliers enforcing the corresponding constraints.Complementary slackness relates each multiplier to its pilot-power variable.
  • Feasible Set and Optimal Updates: Closed-form pilot-power updates follow from solving the stationarity and complementary-slackness conditions, with data-power optimization obtained similarly.The conditional data-power solution is described as a global optimum for fixed remaining variables.
  • Convergence: Algorithm 1 converges to a fixed point because each one-variable quadratic subproblem is optimized over a bounded convex feasible set.The objective is monotonically non-increasing across iterations.
  • Stationary-Point Equivalence: The proof verifies stationary-point equivalence by matching derivatives of the original and reformulated objectives with respect to data and pilot powers.The data-power derivative equality is established directly, with the pilot-power case handled analogously.
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