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Sparse Channel Estimation and Hybrid Precoding Using Deep Learning for Millimeter Wave Massive MIMO

Wenyan Ma, Chenhao Qi, Zaichen Zhang, Julian Cheng

arXiv:2002.06380v1eess.SP

TL;DR

The paper tackles channel estimation and hybrid precoding in multi-user mmWave massive MIMO, including low-resolution phase shifters. It proposes DLCS and DLQP neural-network methods, reporting better channel-estimation performance and higher spectral efficiency than comparison methods, including at low phase-shifter resolution.

  • Problem

    Multi-user mmWave massive MIMO requires sparse channel estimation and hybrid precoding under limited phase-shifter resolution.

  • Method

    DLCS predicts beamspace channel amplitudes offline and reconstructs channels from dominant indices, while DLQP uses quantized-phase neural networks for analog precoding and zero-forcing for digital precoding.

  • Results

    The DLCS scheme outperforms existing schemes in NMSE and spectral efficiency, while DLQP achieves better spectral efficiency than comparison methods with low-resolution phase shifters.

  • Takeaways & Limitations

    Deep learning provides the paper’s channel-estimation and hybrid-precoding approach for improving performance under sparse channels and low phase-shifter resolution.

Abstract

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Channel estimation and hybrid precoding are considered for multi-user millimeter wave massive multi-input multi-output system. A deep learning compressed sensing (DLCS) channel estimation scheme is proposed. The channel estimation neural network for the DLCS scheme is trained offline using simulated environments to predict the beamspace channel amplitude. Then the channel is reconstructed based on the obtained indices of dominant beamspace channel entries. A deep learning quantized phase (DLQP) hybrid precoder design method is developed after channel estimation. The training hybrid precoding neural network for the DLQP method is obtained offline considering the approximate phase quantization. Then the deployment hybrid precoding neural network (DHPNN) is obtained by replacing the approximate phase quantization with ideal phase quantization and the output of the DHPNN is the analog precoding vector. Finally, the analog precoding matrix is obtained by stacking the analog precoding vectors and the digital precoding matrix is calculated by zero-forcing. Simulation results demonstrate that the DLCS channel estimation scheme outperforms the existing schemes in terms of the normalized mean-squared error and the spectral efficiency, while the DLQP hybrid precoder design method has better spectral efficiency performance than other methods with low phase shifter resolution.

I. INTRODUCTION

The paper addresses sparse channel estimation and hybrid precoding for multi-user mmWave massive MIMO under limited phase-shifter resolution. It proposes deep-learning methods to estimate dominant beamspace entries simultaneously and design quantized-phase hybrid precoders.

  • Motivation: MmWave systems use large antenna arrays and directional beamforming to address high path loss at high frequencies.Short wavelengths allow large arrays in compact form factors.
  • Motivation: Large antenna arrays make channel estimation costly in time slots, while mmWave channels exhibit beamspace sparsity that can be exploited.Existing work includes distributed greedy recovery of dominant channel entries.
  • Motivation: Low-resolution phase shifters restrict codebook-based analog precoding because randomly distributed arrival angles may not align with available quantized steering vectors.Prior methods include beamsteering, DFT-codebook, and quantized-angle linear-search designs.
  • Contributions: The DLCS scheme predicts beamspace channel amplitudes with an offline-trained neural network and reconstructs the channel from dominant-entry indices.Its online input is formed from correlations between received signal vectors and the measurement matrix.
  • Contributions: DLCS estimates dominant beamspace entries simultaneously rather than sequentially, targeting better channel-estimation performance than existing sequential methods.The paper contrasts this approach with OMP- and DGMP-based sequential estimation.
  • Contributions: The DLQP method designs an analog precoder with a neural network under approximate phase quantization, then obtains the digital precoder by zero-forcing.Deployment replaces approximate quantization with ideal quantization and stacks user analog vectors into the analog matrix.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system is a downlink multi-user mmWave massive MIMO link with a ULA base station, single-antenna users, and hybrid precoding. Its operating assumptions include many more antennas than RF chains and no more active users than RF chains.

  • A. System Model: The downlink system contains a base station and U single-antenna users, with the base station equipped with a uniform linear array.The method is stated to generalize to other array structures.
  • A. System Model: The base station uses hybrid precoding because the antenna count is much larger than the number of RF chains, NA ≫ NR.This combines RF analog and baseband digital precoding.
  • A. System Model: The number of simultaneously active users satisfies U ≤ NR under orthogonal multiple access.When U < NR, only U RF chains are activated to save base-station power.
  • A. System Model: Downlink transmission uses an analog precoder F_R and a digital precoder F_B to process the transmitted signal.The received signal includes additive white Gaussian noise.
  • A. System Model: The channel follows the Saleh–Valenzuela mmWave model with one line-of-sight path and remaining non-line-of-sight paths.Each path has a complex gain and steering vector.
  • A. System Model: Each path’s angle of arrival is uniformly distributed over [−π, π).The steering-vector parameter is defined from the path angle.

B. Problem Formulation

The paper formulates uplink channel estimation using orthogonal pilots and multiple hybrid-precoding measurements, then represents the channel in a sparse beamspace dictionary. Limited beamspace resolution can create leakage and weaken exact sparsity.

  • B. Problem Formulation: Downlink precoder design requires estimating the channel, which is obtained through uplink estimation using channel reciprocity.The neural network is trained and used at the base station, which is assumed to have greater computing power than users.
  • B. Problem Formulation: Users transmit mutually orthogonal pilot sequences for K repetitions, each using distinct analog and digital precoding matrices.The received pilot measurements are processed using pilot-sequence orthogonality.
  • B. Problem Formulation: The repeated measurements are stacked into a matrix, with NA > NRK because the antenna array is large and training uses few time slots.The resulting formulation is applied separately to each user’s measurement vector.
  • B. Problem Formulation: A dictionary of steering vectors quantizes the angle-of-arrival range into G grid points to form the beamspace channel representation.The dictionary matrix contains G column steering vectors.
  • B. Problem Formulation: The beamspace formulation is a sparse recovery problem because mmWave channels are sparse in the beamspace domain.This motivates applying compressed-sensing techniques.
  • B. Problem Formulation: Limited beamspace resolution causes channel-power leakage, so the beamspace channel is not perfectly sparse and contains many small nonzero entries.This complicates sparse channel estimation.
  • B. Problem Formulation: OMP and DGMP estimate dominant beamspace entries sequentially and greedily, but cannot guarantee global optimality.The paper therefore proposes simultaneous dominant-entry estimation with DLCS.

III. DLCS CHANNEL ESTIMATION

DLCS combines beamspace channel-amplitude estimation with channel reconstruction. It uses an offline-trained CENN to rank estimated amplitudes and reconstruct the channel from the selected dominant-entry indices.

  • III. DLCS CHANNEL ESTIMATION: DLCS first estimates beamspace channel amplitudes, then selects dominant-entry indices and reconstructs the channel.The procedure is summarized in Algorithm 1 and illustrated in Fig. 2.
  • III. DLCS CHANNEL ESTIMATION: The CENN is trained offline and deployed online as the core of beamspace amplitude estimation.The online stage uses the trained network to predict amplitudes before ranking them.

A. Beamspace Channel Amplitude Estimation

The DLCS beamspace amplitude estimator uses an offline-trained CENN to predict beamspace channel amplitudes from received-signal and measurement-matrix correlations. Its architecture processes a real-valued representation of the correlation vector through fully connected and batch-normalized layers.

  • CENN input and output: The CENN receives the correlation vector c_u between the measurement matrix and received signal vectors to estimate beamspace channel amplitudes.The correlation vector is formed from the measurement matrix Φ and received signal vector r_u, then supplied to the offline-trained network.
  • Algorithm flow: Algorithm 1 obtains c_u, feeds it to the offline-trained CENN, and produces the predicted amplitude vector ˆg_u before channel reconstruction.The algorithm separates beamspace channel amplitude estimation from subsequent channel reconstruction.
  • Offline training and online deployment: The DLCS amplitude-estimation stage first trains the CENN offline and then deploys it online for amplitude prediction.Training data are generated from simulated mmWave channels, while deployment uses real measurements to compute c_u before prediction.
  • Network architecture: The CENN contains three hidden layers and a fully connected output layer, with hidden-layer widths of 1,024, 512, and 256 neurons.Each hidden layer combines a fully connected layer with batch normalization, and the fully connected layers use ReLU activation.
  • Training objective: CENN training minimizes mean-square error between the predicted amplitude vector ˆg_u and target vector g_u using the Adam optimizer.The network is trained for 1,000 epochs with 50 mini-batches per epoch and a learning rate reduced fivefold every 400 epochs.

B. Channel Reconstruction

Channel reconstruction selects predicted dominant beamspace entries and estimates their values with least squares before recovering each user’s channel vector. This avoids the sequential greedy search used by conventional sparse channel estimators.

  • Channel Reconstruction: Limited beamspace resolution causes channel power leakage, so the beamspace channel is not perfectly sparse and includes small nonzero entries.This nonideal sparsity is an explicit limitation of the beamspace representation.
  • Channel Reconstruction: The predicted amplitude vector ˆg_u is sorted by absolute value, and the first J entries identify the estimated dominant beamspace indices Γ.J denotes the beamspace channel sparse level, or number of dominant entries in g_u.
  • Channel Reconstruction: After selecting Γ, the remaining beamspace channel entries are set to zero and the selected entries are estimated by least squares using the corresponding columns Φ_Γ.The reconstructed beamspace channel is then mapped to the estimated channel vector using the dictionary relation involving A^H A.
  • Channel Reconstruction: The DLCS method estimates dominant beamspace entries simultaneously rather than selecting atoms sequentially through greedy search.This design is motivated by the lack of guaranteed global optimality in sequential sparse channel estimation schemes.

IV. DLQP HYBRID PRECODER DESIGN

The DLQP method designs analog and digital precoders after channel estimation under limited phase-shifter resolution. It trains a neural network with approximate quantization, deploys an ideal-quantization version for analog precoding, and uses zero-forcing digitally.

  • IV. DLQP HYBRID PRECODER DESIGN: DLQP first trains a training hybrid precoding neural network using estimated channels while incorporating approximate phase quantization.The approximate quantization enables neural-network training before deployment.
  • IV. DLQP HYBRID PRECODER DESIGN: The deployment hybrid precoding neural network replaces approximate phase quantization with ideal phase quantization and outputs analog precoding vectors.The estimated channel vectors are fed into the DHPNN during deployment.
  • IV. DLQP HYBRID PRECODER DESIGN: The analog precoding matrix is formed by stacking the user-specific analog precoding vectors, and the digital precoding matrix is calculated by zero-forcing.The method therefore performs analog design before digital precoder computation.

A. Analog Precoder Design

The analog precoder design uses separate training and deployment neural networks to handle quantized phase shifts. It targets settings where phase-shifter resolution is lower than the number of antennas.

  • A. Analog Precoder Design: The THPNN maps an estimated channel vector to an approximate analog precoder, while the DHPNN maps it to the deployed analog precoder.The networks differ in their phase-quantization operation: approximate during training and ideal during deployment.
  • A. Analog Precoder Design: Each phase-shifter RF phase is quantized into Q = 2^B discrete values, where B is the phase-shifter quantization bit number.Each analog-precoder entry is drawn from the corresponding discrete complex phase set.
  • A. Analog Precoder Design: Beamsteering-codebook methods require Q ≥ N_A for high beamforming gain, motivating neural-network analog-precoder design when Q < N_A.The stated requirement links phase-shifter resolution to the number of antennas in those codebook-based schemes.

B F Online Deployment

The DLQP hybrid precoder uses offline THPNN training with differentiable approximate phase quantization, then deploys a DHPNN with ideal quantization to produce practical analog precoders.

  • Offline Training and Online Deployment: The hybrid precoder design has offline THPNN training followed by online DHPNN deployment, with the DHPNN formed by replacing one THPNN layer.Both networks take the estimated channel vector as input; the DHPNN outputs the analog precoder vector, whereas the THPNN outputs an approximate analog precoder vector.
  • Phase Quantization: Ideal phase quantization is nondifferentiable, so the THPNN uses differentiable approximate quantization during offline training.The approximate quantizer Γ(x) becomes more accurate as η increases and is differentiable over [0, 2π).
  • Offline Training: The THPNN is trained using estimated and real channel vectors to maximize beamforming gain through the inner product of the approximate precoder and channel vector.The training loss is the opposite of this inner product because the neural network minimizes loss.
  • Online Deployment: During deployment, the DHPNN replaces the approximate quantization layer with ideal quantization, ensuring RF phases use the discrete hardware-supported set.Each analog precoder entry is formed as fu,n = e^jψn after quantizing the phase vector.
  • Online Deployment: The analog precoding matrix is constructed by stacking the deployed analog precoder vectors for all users.The resulting matrix is represented as F_R = [f_1, f_2, ..., f_U].
  • Phase Quantization: The training-to-deployment quantization replacement preserves trainability offline while enforcing limited phase-shifter resolution online.The neural-network output is an analog precoder vector, while forming the analog precoding matrix from these vectors is not differentiable and is therefore handled outside training.

B. Digital Precoder Design

The digital precoder is designed after analog precoding using zero-forcing to mitigate interference among multiple data streams, then normalized to satisfy the total power constraint.

  • Digital Precoder Design: Zero-forcing digital precoding is used after analog precoding to mitigate interference among multiple data streams.Analog precoding forms directional beams through the phase-shifter network, while digital precoding handles multi-stream interference.
  • Power Constraint: Each column of the digital precoder is normalized to satisfy the total power constraint.The normalized columns are denoted by f_B,u for users u = 1, 2, ..., U.

V. SIMULATION RESULTS

Simulations evaluate DLCS channel estimation and DLQP hybrid precoding in a multi-user mmWave massive MIMO system. DLCS improves estimation and spectral efficiency over OMP and DGMP, while DLQP provides higher spectral efficiency than Exhaustion and QALS under limited phase-shifter resolution.

  • Simulation setup: The simulations use a BS with 4 RF chains and 64 antennas serving 3 single-antenna users.The beamspace dictionary size is G = 128, and the channel contains two multiple paths per user.
  • DLCS channel estimation: At SNR = 10 dB, DLCS with J = 6 improves channel-estimation performance by 51.7% over OMP and 65.8% over DGMP.With J = 7, the corresponding improvements are 51.3% and 65.5%.
  • DLCS channel estimation: With K = 7 pilot slots, DLCS with J = 6 improves channel-estimation performance by 56.1% over OMP and 85.2% over DGMP.The corresponding improvements for J = 7 are 56.0% and 84.3%.
  • DLCS spectral efficiency: When channel-training slots exceed eight, DLCS spectral efficiency remains constant, indicating that K = 8 suffices to obtain full channel state information.Spectral efficiency is less sensitive than NMSE to sparse-recovery success because it depends on beamforming gain.
  • DLQP hybrid precoding: At SNR = 10 dB, DLQP with J = 7 improves spectral efficiency by 62.0% over Exhaustion, 86.5% over QALS with Q = 16, and 3.6% over QALS with Q = 64.For J = 6, the corresponding improvements are 59.9%, 83.6%, and 3.5%.
  • DLQP hybrid precoding: With K = 7 pilot slots, DLQP with J = 7 improves spectral efficiency by 51.0% over Exhaustion, 71.0% over QALS with Q = 16, and 2.7% over QALS with Q = 64.The corresponding improvements for J = 6 are 49.3%, 70.1%, and 2.9%.

VI. CONCLUSIONS

The paper proposes DLCS channel estimation and DLQP hybrid precoding for multi-user mmWave massive MIMO systems. Simulations report better channel estimation for DLCS and high spectral efficiency for DLQP with low-resolution phase shifters.

  • DLCS and DLQP are proposed for channel estimation and hybrid precoder design in multi-user mmWave massive MIMO systems.
  • Simulations show that DLCS has better channel-estimation performance than existing schemes and DLQP achieves high spectral efficiency with low phase-shifter resolution.
  • Future work includes channel estimation and hybrid precoding for wideband multi-user mmWave massive MIMO transmission using deep learning.
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