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High Dimensional Channel Estimation Using Deep Generative Networks

Eren Balevi, Akash Doshi, Ajil Jalal, Alexandros Dimakis, Jeffrey G. Andrews

arXiv:2006.13494v1eess.SP

TL;DR

High-dimensional channel estimation faces scaling pressure from pilot overhead and computationally intensive sparse recovery, while the best sparsifying basis may be unknown. This paper estimates channels by optimizing the input to a learned generative model, including for one-bit pilots. It reports gains over sparse-recovery methods and lower pilot needs, with execution time approximately independent of pilot count when that count is small.

  • Problem

    High-dimensional channel estimation faces pilot-overhead and computational scaling challenges, and real channels may not be exactly sparse in a known basis.

    Source: A. Motivation · p. 2

  • Method

    The method learns a generative channel prior and optimizes its input to estimate channels from compressive pilots, including one-bit measurements.

    Source: C. Contributions · pp. 5–6

  • Results

    The method significantly outperforms prior sparse-recovery techniques on CDL channel models, with at least 5 dB NMSE improvement over EM-GM-AMP at SNR -10 dB for α = 0.2.

    Source: A. Full Resolution Channel Estimation · p. 20, §VI · p. 27

  • Takeaways & Limitations

    The approach avoids requiring the channel’s sparsifying basis and reduces pilots needed for the same NMSE as Lasso, OMP, or EM-GM-AMP, including with one-bit pilots.

    Source: §VI · p. 27

    The GAN was trained on simulated channel realizations because realistic channel data were unavailable, and its ability to learn from noisy channel estimates remains untested.

    Source: D. Explanations & Caveats · p. 25

Abstract

from arXiv · show

This paper presents a novel compressed sensing (CS) approach to high dimensional wireless channel estimation by optimizing the input to a deep generative network. Channel estimation using generative networks relies on the assumption that the reconstructed channel lies in the range of a generative model. Channel reconstruction using generative priors outperforms conventional CS techniques and requires fewer pilots. It also eliminates the need of a priori knowledge of the sparsifying basis, instead using the structure captured by the deep generative model as a prior. Using this prior, we also perform channel estimation from one-bit quantized pilot measurements, and propose a novel optimization objective function that attempts to maximize the correlation between the received signal and the generator's channel estimate while minimizing the rank of the channel estimate. Our approach significantly outperforms sparse signal recovery methods such as Orthogonal Matching Pursuit (OMP) and Approximate Message Passing (AMP) algorithms such as EM-GM-AMP for narrowband mmWave channel reconstruction, and its execution time is not noticeably affected by the increase in the number of received pilot symbols.

I. INTRODUCTION

The introduction motivates generative channel estimation as a response to scaling limits in high-dimensional systems and gaps in sparse and discriminative methods. It then outlines a GAN-based estimator for full-resolution and one-bit pilot measurements.

A. Motivation

  • Future high-dimensional systems face channel-estimation approaches whose pilot overhead and per-interval complexity do not scale well.The paper motivates stronger structural priors as a way to address these demands.Source: A. Motivation · p. 2

B. Related Work

  • Sparse recovery depends on a suitable channel basis, while many learned estimators are discriminative or require extensive transceiver changes.Real-world channels may not be exactly sparse in the DFT basis, and the most sparse basis is not known.Source: B. Related Work · pp. 3–4, C. Contributions · p. 5

C. Contributions

  • The paper trains a Wasserstein GAN on simulated channel realizations to learn the spatial-channel distribution.The distribution is described as complex and analytically intractable.Source: C. Contributions · p. 5
  • For full-resolution estimation, the method optimizes the generator input to estimate a channel from a small number of noisy pilots.The optimization is in a low-dimensional subspace whose dimension is independent of the number of received pilots.Source: C. Contributions · pp. 5–6
  • For one-bit pilots, a custom loss seeks a generator-range channel estimate with low rank and high correlation to the measurements.The paper compares the estimate with CS techniques and evaluates its use in hybrid-precoded transmission.Source: C. Contributions · p. 6

II. SYSTEM MODEL

The system model considers training-based channel estimation for a narrowband downlink MIMO channel, with received pilots determined by the channel, training beams, and noise. When pilots are fewer than transmit antennas, the resulting system does not uniquely determine the channel.

A. Training based channel estimation

  • The model is a point-to-point downlink MIMO setup with hybrid beamformers and combiners, described for one narrowband channel and extendable to multiple subcarriers.The transmitted symbol is set to 1 in the experiments.Source: A. Training based channel estimation · pp. 6–7
  • Each training beam produces N_r UE measurements, and the total is M = N_rN_p while the channel remains constant over the N_p training slots.The setup assumes channel coherence time greater than N_pT.Source: A. Training based channel estimation · p. 7
  • With N_p < N_t, the pilot system has no unique solution, so least-squares estimation is not directly usable in the low-pilot regime.The model also extends to quantized received signals.Source: A. Training based channel estimation · pp. 7–8

B. Hybrid Precoding for Data Transmission

The paper describes hybrid precoding for data transmission and a Wasserstein GAN for learning spatial channel matrices. GAN training uses a generator-discriminator game and a Wasserstein objective intended to reduce training difficulty and mode collapse.

B. Hybrid Precoding for Data Transmission

  • In data transmission, the UE conveys its channel estimate to the BS, which uses OMP-based spatially sparse precoding to select RF and baseband precoders.The precoders are optimized for throughput while incorporating hardware and power constraints.Source: B. Hybrid Precoding for Data… · p. 8

A. Training a GAN to learn the channel distribution

  • A GAN trains a generator and discriminator in an iterative minimax game, with the generator mapping latent noise to spatial channel matrices.The Wasserstein GAN uses the Wasserstein-1 distance as its loss; the paper states this avoids careful training balance and drastically reduces mode collapse.Source: A. Training a GAN to learn the channel… · pp. 9–10
  • GANs can suffer mode collapse, in which they characterize only a few distribution modes; the paper therefore employs a Wasserstein GAN for channel-distribution learning.Its training procedure is outlined in an algorithm and depicted in Fig. 1.Source: A. Training a GAN to learn the channel… · pp. 9–10, Figure 1 · p. 11

B. CS-based channel estimation using generative networks

Generative compressed sensing estimates a channel by optimizing a low-dimensional generator input against pilot measurements. The framework uses separate objectives for full-resolution and one-bit pilots, with the latter balancing measurement correlation and low rank.

B. CS-based channel estimation using generative networks

  • The generative compressed-sensing approach optimizes a low-dimensional input so that the generated reconstruction has small measurement error.Gradient descent was found empirically to work well despite the non-convex objective; under sub-Gaussian sensing matrices, gives a guarantee relative to the closest point in the generator range.Source: B. CS-based channel estimation using… · p. 11, §2 · p. 11

2. Here, we minimize

  • The Generative Channel Estimator (GCE) trains a Wasserstein GAN on realistic channel realizations, then optimizes its input using pilot-specific losses.The estimate is the generator output G(z*).Source: §2 · p. 12
  • For full-resolution pilots, the method replaces the sensing matrix with P^T ⊗ I_Nr and regularizes the latent input during non-convex optimization.The reconstructed estimate is G(z*), and λ_reg is the regularization parameter.Source: §2 · p. 12
  • For one-bit pilots, the proposed objective maximizes measurement correlation while minimizing the generator output’s nuclear norm.The authors describe this loss as heuristically designed.Source: §2 · p. 13

C. Beamforming using the Generative Channel Estimator

The channel estimate’s optimal generator input provides a compressed channel representation. After conveying the generator to the base station, the system can feed back that input to support precoder design.

C. Beamforming using the Generative Channel Estimator

IV. SIMULATION DETAILS & BENCHMARKS

This section defines normalized mean square error (NMSE) as the performance metric for channel estimation, using the actual and estimated channel taps in the frequency domain across antennas.

IV. SIMULATION DETAILS & BENCHMARKS

  • NMSE is the stated performance metric for evaluating channel estimates.Source: §IV · p. 14
  • The metric compares H, the actual channel taps, with H-hat, the estimated channel taps.Source: §IV · p. 14
  • The channel taps used in the metric are specified in the frequency domain over all antennas.Source: §IV · p. 14

A. Data Generation

The simulations use generated channel realizations and focus on their spatial structure. Preprocessing separates real and imaginary components and uses training-set statistics during testing.

A. Data Generation

  • Channel realizations are generated in MATLAB with the 5G Toolbox according to 3GPP TR 38.901.Source: A. Data Generation · p. 14
  • The simulations assume λ/10 antenna spacing and extract the first-subcarrier, first-OFDM-symbol channel matrix to focus on spatial structure.The full generated realization has dimensions (Nf, 12, Nr, Nt).Source: A. Data Generation · p. 15

B. Data Pre-processing

  • Preprocessing splits the channel into real and imaginary parts for GAN training, while the generator output is reshaped as a complex-valued matrix for optimization.The generator output has dimensions (Nt, Nr, 2), with the last dimension representing real and imaginary parts.Source: B. Data Pre-processing · p. 15
  • Testing reuses training-set means and variances, and a simulation found the resulting artifact negligible.Normalization is used because the GAN was empirically unable to learn mean-shifted distributions.Source: B. Data Pre-processing · pp. 15–16

C. NN Architecture of Generator

The generator uses a fine-tuned deep convolutional architecture to produce channel estimates, and the trained model is used to optimize the latent input for each test channel. The paper describes two optimization approaches for this input search, depending on the framework used to train the generator.

C. NN Architecture of Generator

  • The generator G takes z ∈ Rd as input and passes it through a dense layer with output size 128NtNr/16; it was fine-tuned to improve learning of the underlying probability distribution.Source: C. NN Architecture of Generator · p. 16
  • The generator upsamples and applies 2D convolutions to model local spatial correlations, then outputs an Nr × Nt channel estimate.Its two upsampling stages use a 2 × 2 factor and kernel size 4; oversampling by 4 or more prevents the generator from learning the channel distribution.Source: C. NN Architecture of Generator · p. 16

D. GAN Training Details & GCE

  • For the Keras-trained generator, Powell’s conjugate direction method minimizes the GCE loss because Keras does not differentiate it with respect to z.The optimization uses a relative error tolerance of 10^-5.Source: D. GAN Training Details & GCE · p. 17
  • For the PyTorch-trained generator, Adam minimizes the GCE loss using automatic differentiation, with learning rate 10^-2 and 100 iterations.The paper also identifies Powell’s method as a derivative-free approach for minimizing the loss.Source: D. GAN Training Details & GCE · p. 17

E. Compressed Sensing Based DL Channel Estimation

The paper evaluates generative channel estimation against three sparse-recovery baselines: OMP, Lasso, and EM-GM-AMP. These baselines use a sparse representation, while the comparison notes that EM-GM-AMP does not exploit correlation in the virtual-channel coefficients.

E. Compressed Sensing Based DL Channel Estimation

  • OMP solves the sparse-recovery problem directly and stops when the residual energy falls below a threshold chosen as σ2.The stated maximum is 100 iterations; allowing more can make OMP fit noise at low SNR.Source: E. Compressed Sensing Based DL Channel… · p. 18
  • The Lasso baseline applies an L1 convex relaxation and solves the resulting complex-valued problem as a second-order conic program.The paper notes that standard convex solvers such as CVXPY can solve this formulation.Source: E. Compressed Sensing Based DL Channel… · p. 18
  • EM-GM-AMP recovers a virtual-channel estimate from y and the sensing matrix, but does not exploit correlation among its coefficients.The paper leaves comparison with structured-sparsity methods such as EMturboGAMP for future work.Source: E. Compressed Sensing Based DL Channel… · pp. 18–19
  • The three sparse-recovery baselines—OMP, Lasso, and EM-GM-AMP—require knowledge of the sparsifying basis, which GCE does not use.For the narrowband clustered channel model, the baselines use a virtual-channel representation in the DFT basis.Source: E. Compressed Sensing Based DL Channel… · pp. 17, 19

V. RESULTS

The results examine latent dimension and pilot density, then compare GCE with sparse-recovery methods for full-resolution and one-bit measurements. GCE achieves substantial NMSE gains over EM-GM-AMP in the reported full-resolution comparison, and the paper also reports an improvement for one-bit estimation.

V. RESULTS

  • At pilot ratio α = Np/Nt = 0.4, d = 35 appears sufficient; increasing Np/Nt beyond 0.4 has no measurable impact on NMSE, indicating additional measurements would not improve channel prediction accuracy.It uses under 40 parameters to represent a 16 × 64 channel matrix realization (2048 real values), without requiring prior knowledge that the channel is sparse in the DFT basis.Source: §V · p. 20

A. Full Resolution Channel Estimation

B. One-bit Quantized Channel Estimation

C. Hybrid Precoding for Quantized Channel Estimation

With one-bit pilot measurements, GCE-based channel estimates support higher-capacity hybrid-precoded transmission than EM-GM-AMP. The discussion identifies high spatial correlation and simulated training data as important boundaries on the approach.

C. Hybrid Precoding for Quantized Channel Estimation

  • GCE channel estimates enable hybrid precoders that support higher capacity data transmissions than EM-GM-AMP.The comparison uses OMP-based RF and baseband precoders and includes a perfect-CSI curve.Source: C. Hybrid Precoding for Quantized… · p. 22

D. Explanations & Caveats

  • At low pilot density and low SNR, GCE outperforms CS-based methods; its advantage is especially evident with one-bit measurements.The methods approach similar performance as the number of pilot symbols reaches or exceeds the number of transmit antennas.Source: D. Explanations & Caveats · p. 23
  • GCE required λ/10 rather than λ/2 antenna spacing to learn the channel distribution successfully.At λ/2, lower spatial correlation produces a higher effective rank, and the trained generator fails to learn the distribution, with poor performance reported at α = 0.75.Source: D. Explanations & Caveats · pp. 23–24, Figure 7 · p. 24
  • The authors have not quantified whether the generative prior is more informative than the sparsifying basis used by conventional CS methods.GCE does not use beamspace sparsity, whereas conventional CS methods exploit it.Source: D. Explanations & Caveats · p. 25
  • The GAN was trained on simulated channel realizations because realistic channel data were unavailable, leaving its ability to learn from noisy measured estimates untested.The authors identify this as an open question.Source: D. Explanations & Caveats · p. 25

E. Timing Analysis

GCE's execution time is not noticeably affected by increasing pilot symbols, unlike the CS baselines. The conclusion also emphasizes reduced pilot requirements, no need to know the sparsifying basis, and performance gains on CDL channel models.

E. Timing Analysis

  • GPU acceleration speeds up backpropagation through the neural network, which is used to compute z* in (8).The timing comparison evaluates GCE on CPU and GPU.Source: E. Timing Analysis · pp. 25–26
  • GCE execution time is not noticeably affected by increasing pilot symbols, while CS-baseline execution time increases with α.The comparison concerns execution time per iteration; the table caption specifies milliseconds and a single channel realization at -10 dB.Source: E. Timing Analysis · p. 26, Table III · p. 26
  • GCE's latent-vector gradient computation uses matrix multiplications, while its large generator makes that computation costly and runtime comparable to OMP at low pilot density.The authors expect pilot-count effects on GCE runtime only at much higher Np than for CS-based algorithms.Source: E. Timing Analysis · p. 26

VI. Conclusion

  • On CDL channel models, GCE achieves significant performance gains without requiring knowledge of the channel's sparsifying basis and reduces the pilots needed to match Lasso, OMP, or EM-GM-AMP NMSE.The conclusion also reports approximately pilot-independent execution time when Np is small.Source: §VI · p. 27
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