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Hybrid Precoding for Multi-User Millimeter Wave Massive MIMO Systems: A Deep Learning Approach

Ahmet M. Elbir, Anastasios Papazafeiropoulos

arXiv:1911.04239v1eess.SPcs.ITcs.LGmath.IT

TL;DR

Multi-user mmWave hybrid precoding must balance achievable sum-rate with computational cost, while conventional methods may be complex or depend on channel quality. CNN-MIMO learns precoder and combiner outputs from channel matrices using exhaustive-search labels, and the authors report competitive sum-rate performance with reduced computation time. Its network requires retraining when antenna parameters change.

  • Problem

    Hybrid precoding methods based on optimization or greedy search can be computationally complex or suboptimal, and their performance depends on channel-data quality.

    Source: §I · p. 1, A. Motivation · p. 2

  • Method

    CNN-MIMO is trained on channel matrices paired with exhaustive-search precoder and combiner labels, then predicts hybrid beamformers from channel data.

    Source: B. Contribution · p. 2, §IV · p. 4

  • Results

    CNN-MIMO follows the MO algorithm’s performance and has the highest sum-rate among the other algorithms, while MO performs slightly better and CNN-MIMO computes beamformers faster.

    Source: §VI · pp. 8–10

  • Takeaways & Limitations

    The authors conclude CNN-MIMO offers robust performance with less computation time than state-of-the-art hybrid-precoding approaches.

    Source: §VII · p. 10

    The CNN structure must be retrained when antenna parameters such as NT, NR, or N_RF change.

    Source: C. Training · p. 7

Abstract

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In multi-user millimeter wave (mmWave) multiple-input-multiple-output (MIMO) systems, hybrid precoding is a crucial task to lower the complexity and cost while achieving a sufficient sum-rate. Previous works on hybrid precoding were usually based on optimization or greedy approaches. These methods either provide higher complexity or have sub-optimum performance. Moreover, the performance of these methods mostly relies on the quality of the channel data. In this work, we propose a deep learning (DL) framework to improve the performance and provide less computation time as compared to conventional techniques. In fact, we design a convolutional neural network for MIMO (CNN-MIMO) that accepts as input an imperfect channel matrix and gives the analog precoder and combiners at the output. The procedure includes two main stages. First, we develop an exhaustive search algorithm to select the analog precoder and combiners from a predefined codebook maximizing the achievable sum-rate. Then, the selected precoder and combiners are used as output labels in the training stage of CNN-MIMO where the input-output pairs are obtained. We evaluate the performance of the proposed method through numerous and extensive simulations and show that the proposed DL framework outperforms conventional techniques. Overall, CNN-MIMO provides a robust hybrid precoding scheme in the presence of imperfections regarding the channel matrix. On top of this, the proposed approach exhibits less computation time with comparison to the optimization and codebook based approaches.

I. INTRODUCTION

MmWave systems use hybrid analog and digital processing to reduce the cost of large antenna arrays, while prior hybrid-precoding approaches face complexity or channel-quality constraints. This work proposes CNN-MIMO for multi-user hybrid precoding with imperfect channel data.

I. INTRODUCTION

  • Hybrid analog and digital processing reduces the cost of mmWave systems with many antennas, while phase shifters steer beams using discrete phases.The analog and digital precoders share signal processing, rather than assigning dedicated RF hardware to every antenna element.Source: §I · p. 1
  • Earlier multi-user hybrid-precoding methods assumed perfect channel state information, and prior robust CNN approaches addressed only single-user settings.Multi-user design must account for interference from other users.Source: §I · pp. 1–2

B. Contribution

  • The paper proposes CNN-MIMO, which maps users’ channel matrices to hybrid precoder weights and uses exhaustive-search solutions as training labels.Array-response sets F and W are needed for label generation but not for prediction.Source: B. Contribution · p. 2
  • CNN-MIMO is proposed for multi-user massive-MIMO mmWave precoding to handle channel deviations, with performance compared against greedy and codebook-based methods –.The approach estimates precoder and combiner weights from channel data.Source: B. Contribution · p. 2
  • The framework estimates precoders by feeding in a corrupted channel matrix, avoiding optimization or greedy searches during precoding.The training data is generated offline and is described as almost half a million samples.Source: B. Contribution · p. 2

C. Notation

  • The notation defines matrix and vector indexing, the Frobenius norm, the Moore–Penrose pseudoinverse, and the phase-quantization operator Q(a).Q(a) maps a continuous phase to one sampled from 2^B quantized angles.Source: C. Notation · pp. 2–3

II. SYSTEM MODEL

The downlink model serves K users from a multi-antenna base station using hybrid precoding and analog-only combining at the users. It assumes one stream per user and at least K base-station RF chains.

II. SYSTEM MODEL

  • The system has a base station with NT antennas and K users, each with NR antennas; the base station uses hybrid precoding and users use analog-only combining.The figure depicts analog and baseband precoding at the base station and analog-only combining at the K users.Source: §II · p. 3, Figure 1 · p. 3
  • Each user receives a single stream, and the base station has at least K RF chains for simultaneously served users.The single-stream assumption means no baseband precoding is applied at the receivers.Source: §II · p. 3
  • The base station applies baseband precoding to the user signals before RF precoding conveys them through phase-shifter hardware.The baseband precoder is normalized to satisfy the stated transmit-power constraint.Source: §II · p. 3
  • The model assumes equal power allocation across users and constant equal-norm elements for the RF precoder and user combiners.The combiner elements have squared magnitude 1/NR; the RF precoder elements have squared magnitude 1/NT.Source: §II · p. 3

A. Channel Model

The channel model represents each user’s narrowband mmWave channel through a limited set of scattering paths and a uniform planar array. It defines the achievable rate under Gaussian signaling and uses the sum of users’ rates as the system measure.

A. Channel Model

  • Each user’s mmWave channel is modeled as a sum of L scattering paths for a two-dimensional uniform planar array.The path representation includes angular parameters, complex path gains, antenna-element gains, and array steering vectors.Source: A. Channel Model · p. 3
  • The receive and transmit steering vectors represent the array responses at the user and base station, respectively.The model specifies antenna positions and wavelength when defining steering-vector elements.Source: A. Channel Model · p. 3
  • Under Gaussian signaling, the model defines each user’s achievable rate and the system’s achievable sum-rate.The supplied sum-rate passage is partial and does not provide its full expression.Source: A. Channel Model · p. 3

III. PROBLEM FORMULATION

The paper seeks hybrid precoders and combiners that maximize sum-rate with imperfect channel data. It generates training labels by exhaustive search, then trains a network to predict beamformers from channel matrices.

III. PROBLEM FORMULATION

  • The design objective is to choose baseband and analog precoders and user combiners that maximize sum-rate with imperfect channel data.The network maps channel matrices to analog beamformers.Source: §III · p. 4
  • The label-generation process depends on a channel model, but the authors state that the model is not required for updating network parameters and that other mmWave channel models may be used.This work uses a block-fading channel model; applying DL to other channel models is left to ongoing research.Source: §III · p. 4
  • Training uses channel realizations corrupted by synthetic noise, so prediction does not necessarily require perfect CSI and is evaluated for achievable sum-rate under corrupted channels.The authors identify channel estimation as challenging in large-antenna systems and channel coherence time as short in mmWave massive MIMO.Source: §III · p. 4
  • An offline exhaustive search over feasible precoder and combiner sets produces the sum-rate-maximizing labels for training samples.The search visits all combinations in F and W; the network later predicts precoders from channel data.Source: §III · p. 4

IV. HYBRID PRECODING DESIGN IN MULTI-USER MIMO SYSTEMS

The section formulates multi-user hybrid precoding as a sum-rate maximization over feasible analog precoder and combiner sets, then uses exhaustive search to select the training labels.

IV. HYBRID PRECODING DESIGN IN MULTI-USER MIMO SYSTEMS

  • The exhaustive search evaluates every feasible precoder–combiner combination and selects the pair that maximizes achievable sum-rate.The selected analog weights are later used as training labels; the search is performed offline.Source: §IV · pp. 4–5
  • The feasible analog sets comprise combinations of quantized array-response steering vectors, with one candidate choice for each user.The precoder candidates span QF = L̄^K combinations when each user has L̄ possible paths.Source: §IV · p. 4
  • For each candidate pair, the algorithm forms each user's effective channel, computes and normalizes its digital precoder, then evaluates the sum-rate.The algorithm returns the analog precoder and combiner associated with the maximum-rate candidate.Source: §IV · pp. 4–5
  • The search visits QFQW candidate pairs, and the paper says its broader combination search improves results over.The learning-based approach is proposed to reduce the computation time and avoid requiring array responses at prediction.Source: §IV · pp. 4–5

V. LEARNING-BASED HYBRID PRECODING

CNN-MIMO maps a three-channel representation of a user's channel matrix to a vector output. The section expresses the network as a composition of layer operations and describes its fully connected and convolutional computations.

V. LEARNING-BASED HYBRID PRECODING

  • CNN-MIMO maps an NR × NT × 3 input to a K(NT + NR) × 1 output through a composition of layer operations.The architecture is represented as a function from R^(NR×NT×3) to R^(K(NR+NT)).Source: §V · p. 5
  • A fully connected layer computes each output element as an inner product of its input vector with a column of the layer weights.The passage defines the input, output, weight matrix, and output-column vector for this operation.Source: §V · p. 5
  • A convolutional layer produces feature-map responses by combining kernel weights with input features across spatial positions.The text defines the spatial positions, feature maps, and kernel-weight vectors used in the response.Source: §V · p. 5

A. Training Data Generation

Training examples pair channel-matrix inputs with analog precoder and combiner outputs. The inputs include synthetic noise realizations, while the analog weights remain associated with the underlying channel.

A. Training Data Generation

  • The training data uses N channel realizations for K users, with synthetic noise added to each matrix for G realizations.The noise variance is associated with the training-data SNR; the resulting dataset has T = NGK inputs.Source: A. Training Data Generation · pp. 5–6
  • Each channel-matrix input has three channels containing entry magnitudes, real parts, and imaginary parts.The network input represents a complex channel matrix through these three real-valued components.Source: A. Training Data Generation · p. 5
  • Each input is paired with an output vector representing the analog precoder and combiners, and the same analog weights are used across noisy realizations.The passage states that this reuse introduces synthetic noise in the inputs to make the network robust to corrupted channel data.Source: A. Training Data Generation · p. 6

B. Network Architecture

The ten-layer CNN uses convolutional feature extraction, fully connected layers, dropout, and a regression output. The authors report tuning architectural hyperparameters and describe where computation and memory costs arise.

B. Network Architecture

  • The network has ten layers, including two 256-filter 2 × 2 convolutional layers and two 2048-unit fully connected layers.Normalization follows each convolutional layer, and 50% dropout follows each fully connected layer.Source: B. Network Architecture · p. 6
  • The regression output has K(NR + NT) units and includes the phase information of the analog precoders.The input layer accepts channel data of size NR × NT × 3.Source: B. Network Architecture · p. 6
  • The authors tuned layer count, filter count, and kernel size; the 2 × 2 kernel was one sufficiently performing, lower-complexity option among similar-performing structures.The passage notes that different kernel sizes can also be used for the problem.Source: B. Network Architecture · pp. 6–7
  • Convolutional layers account for more than 90% of inference time, while fully connected layers require approximately 90% of memory.The paper states that CNN complexity is directly proportional to parameter and layer counts, and reports 41481 parameters for CNN-MIMO.Source: B. Network Architecture · p. 7

C. Training

Training uses exhaustively generated precoder and combiner labels for noisy multi-user channels, with multiple training SNRs and varied propagation scenarios. The network’s prediction is fast after one-time training, but changes to antenna or RF-chain dimensions require retraining.

C. Training

  • The CNN is trained in MATLAB using stochastic gradient descent with momentum 0.9, learning rate 0.005, batches of 500, and 100 epochs.Mean squared error is used as the loss function.Source: C. Training · p. 7
  • The dataset uses 500 three-user scenarios, 1500 channel realizations, and AWGN at training SNRs of 15, 20, and 25 dB.The data are split 80% for training and 20% for validation; validation is used for hyperparameter tuning and simulations over 100 Monte Carlo trials.Source: C. Training · p. 7
  • Training scenarios use 10 propagation paths, angular grids of 60 azimuth and 20 elevation points, and randomly selected user directions in sectorized angular intervals.Antenna gains are set to unity within the selected angular ranges and zero outside them.Source: C. Training · p. 7
  • Training takes about 5 hours for 450000 samples, while prediction takes milliseconds; training is performed only once.The reported prediction time is demonstrated in Table I.Source: C. Training · p. 7
  • The network can handle changes in user count, path count, and training or test SNR, but changes to antenna or RF-chain dimensions require retraining.Those dimensions determine the network’s input and output dimensions; performance also depends on the chosen angular interval and sectorized antenna gains.Source: C. Training · p. 7

D. Prediction

At prediction time, corrupted channel data are fed to the trained network to predict analog precoders. The predicted phases are quantized to discrete values for hardware-efficient realization.

D. Prediction

  • For prediction, synthetic noise is added to test channels, which are fed to the network to predict analog precoders.Users are randomly selected from the validation data, and the test SNR is used to reduce similarity between test and training data.Source: D. Prediction · p. 7
  • Predicted phases are quantized over [0, 2π] using 2^B discrete points for hardware-efficient realization of analog precoders and combiners.The quantized phases belong to the set of evenly indexed discrete phase values specified in the passage.Source: D. Prediction · p. 7

VI. NUMERICAL SIMULATIONS

The simulations compare CNN-MIMO with optimization, codebook, and learning baselines across SNR, corrupted feedback, phase resolution, user counts, and antenna counts. CNN-MIMO performs strongly across these settings, with a computation-time advantage over manifold optimization despite slightly lower performance.

VI. NUMERICAL SIMULATIONS

  • CNN-MIMO follows manifold optimization’s performance and exceeds the other compared algorithms in the SNR experiment, while manifold optimization is slightly better overall.The comparison uses NT = 36, NR = 9, K = 3, B = 3, and SNRTEST = 20 dB.Source: §VI · pp. 8, 10, Figure 3 · p. 8
  • With corrupted channel and array-response data, CNN-MIMO is more robust than the other methods, and its prediction stage does not require feasible response sets F and W.SOMP and TS-HB require both the channel matrix and response sets as inputs; Algorithm 1 uses perfect CSI for its benchmark.Source: Figure 4 · p. 8, §VI · p. 9
  • CNN-MIMO achieves higher sum-rate than the other algorithms starting at one-bit phase quantization, while the other algorithms converge after four bits.The analog precoders use discrete phase shifters with constant modulus.Source: §VI · p. 9, Figure 5 · p. 9
  • For K = 2 to 8 users with L = 10 fixed, CNN-MIMO outperforms the other algorithms and closely follows Algorithm 1; its gap from the no-interference benchmark widens as K increases.With L selected uniformly from [1, 10], CNN-MIMO also achieves satisfactory performance similar to the fixed-path setting.Source: §VI · p. 9, Figure 6 · p. 10
  • When path counts vary independently across users, with L drawn uniformly from [1, 10], CNN-MIMO retains satisfactory performance similar to the fixed-path setting.The authors attribute reduced similarity between users’ channel data to the varying path counts.Source: §VI · p. 9, Figure 6 · p. 10
  • Across varying BS antenna counts, CNN-MIMO performs better than the other algorithms, and CNN-MIMO and MLP have the least computation time among the compared methods.Manifold optimization is slightly better-performing than CNN-MIMO, but its computation time is significantly higher and its complexity grows faster.Source: §VI · pp. 9–10, Table I · p. 9, Figure 7 · p. 10

VII. CONCLUSIONS

CNN-MIMO uses exhaustive-search precoder and combiner labels to train a network for multi-user mmWave MIMO. It achieves better performance than state-of-the-art hybrid precoding approaches with less computation time, and is robust to channel and array-response imperfections.

VII. CONCLUSIONS

  • CNN-MIMO achieves better performance than state-of-the-art hybrid precoding approaches while requiring less computation time.The framework was compared with optimization-based, greedy-based, and deep-learning approaches.Source: §VII · p. 10
  • Exhaustive search supplies the best analog precoders and combiners used as labels in the network's training stage.Source: §VII · p. 10
  • Training on nearly half a million samples provides robust performance against deviations in channel data.Source: §VII · p. 10
  • CNN-MIMO achieves more robust results when the channel matrix and array responses contain imperfections.Source: §VII · p. 10
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