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Iterative Channel Estimation Using LSE and Sparse Message Passing for MmWave MIMO Systems

Chongwen Huang, Lei Liu, Chau Yuen, Sumei Sun

arXiv:1611.05653v2cs.IT

TL;DR

mmWave MIMO channel estimation must exploit channels that are sparse because only a few propagation components contribute significant coefficients. The paper proposes an iterative LSE-SMP estimator that detects support with SMP and estimates values with LSE, achieving strong performance and near-CRLB behavior under partial channel prior knowledge.

  • Problem

    MmWave channels are approximately sparse, but existing estimators require differing prior information and may be poorly matched to hybrid mmWave architectures.

  • Method

    The proposed LSE-SMP algorithm models channel coefficients as Bernoulli-Gaussian and alternates SMP support detection with LSE value estimation, using Gaussian approximation and EXIT analysis.

  • Results

    The estimator achieves the best tested NMSE, reducing it by 3.5dB relative to EM-BG-AMP and 1.5dB relative to BP-MF SBL.

  • Takeaways & Limitations

    The algorithm is reported as an MVUE with partial channel prior knowledge and typically needs only five turbo iterations while approaching the relevant CRLB.

Abstract

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We propose an iterative channel estimation algorithm based on the Least Square Estimation (LSE) and Sparse Message Passing (SMP) algorithm for the Millimeter Wave (mmWave) MIMO systems. The channel coefficients of the mmWave MIMO are approximately modeled as a Bernoulli-Gaussian distribution and the channel matrix is sparse with only a few non-zero entries. By leveraging the advantage of sparseness, we propose an algorithm that iteratively detects the exact locations and values of non-zero entries of the sparse channel matrix. At each iteration, the locations are detected by the SMP, and values are estimated with the LSE. We also analyze the Cramér-Rao Lower Bound (CLRB), and show that the proposed algorithm is a minimum variance unbiased estimator under the assumption that we have the partial priori knowledge of the channel. Furthermore, we employ the Gaussian approximation for message densities under density evolution to simplify the analysis of the algorithm, which provides a simple method to predict the performance of the proposed algorithm. Numerical experiments show that the proposed algorithm has much better performance than the existing sparse estimators, especially when the channel is sparse. In addition, our proposed algorithm converges to the CRLB of the genie-aided estimation of sparse channels with only five turbo iterations.

I. INTRODUCTION

mmWave systems offer abundant spectrum but suffer severe path loss, motivating large arrays and sparse-channel estimation methods tailored to hybrid analog-digital architectures. The paper proposes an LSE-, EM-, and SMP-based iterative estimator and analyzes its performance and convergence.

  • 40 dB worse path loss at 28 GHz than at 2.8 GHz motivates large antenna arrays for mmWave systems.
  • Only 3-4 scattering clusters are typically observed in dense-urban non-line-of-sight mmWave channels, yielding approximately sparse channels.
  • Conventional MIMO iterative estimators are unsuitable because mmWave systems combine sparse channel characteristics with hybrid analog-digital architectures.
  • The proposed estimator combines beamspace modeling, Bernoulli-Gaussian channel coefficients, LSE, EM, and SMP for large-array mmWave MIMO systems.
  • Gaussian-approximated message densities and EXIT-chart analysis are used to study iteration evolution and optimize design parameters.
  • The algorithm is reported to outperform classical LSE and existing sparse estimators and approximately achieve the CRLB with fast convergence.

III. SPARSE CHANNEL ESTIMATION

LSE-SMP first obtains a coarse channel estimate, then iteratively detects non-zero positions with SMP and refines coefficient values with LSE until a stopping condition is met.

  • The proposed LSE-SMP procedure has four phases: coarse LSE estimation, SMP detection, updates for LSE estimation and sparsity ratios, and decision output.
  • Because no prior knowledge of the channel vector is assumed initially, LSE provides the coarse estimate used to start the iterations.
  • The coarse estimate is obtained by solving a least-squares problem that minimizes the mean-squared error of the channel estimate.

1) Factor Graph Representation of the mmWave Channel:

The factor-graph formulation represents the sparse channel through Bernoulli position variables and coefficient values, enabling SMP to pass messages between sum and variable nodes for support detection.

  • The channel vector is decomposed into a diagonal coefficient matrix and a binary position vector whose entries indicate non-zero coefficients.
  • The resulting observation model is written as the training matrix multiplied by the coefficient and position representation, plus noise.
  • The factor graph contains sum nodes for observations and variable nodes for channel coefficients, constructed from the governing equations.
  • SMP estimates non-zero positions by passing extrinsic messages along edges, analogously to belief-propagation decoding.
  • The sparsity ratio is η = L/(NrNt), and its estimated value is updated during the algorithm while the initial value is set to 0.5.
  • Under a Gaussian approximation, messages from other variable nodes are treated as equivalent Gaussian noise when computing position probabilities.

3) Message Update at Variable Nodes :

At variable nodes, message updates combine information from connected sum nodes to estimate Bernoulli activity variables. The SMP uses extrinsic messages for subsequent updates and full-information estimates to support coefficient estimation, with Gaussian approximation enabling near-optimal performance at lower complexity.

  • Message Update at Variable Nodes: Variable-node updates treat messages as a broadcast process and produce probability messages for each Bernoulli activity variable at the next iteration.The estimated activity variable is obtained from the updated message probabilities.
  • Message Update at Variable Nodes: Extrinsic information is computed from messages on the other edges connected to the same node and reused in the next sum-node update.This separates messages passed onward from estimates formed using all incoming information.
  • Message Update at Variable Nodes: The full-information estimate of each Bernoulli variable is used in the subsequent coefficient-estimation step.Unlike extrinsic messages, the estimate incorporates information from all connected sum nodes.
  • Message Update at Variable Nodes: Gaussian approximation of the sum-node term can achieve near-optimal performance without heavy computational complexity when the number of transmit antennas is large.The approximation is justified by the law of large numbers and is presented as specialized to mmWave systems.

C. Update for LSE Estimation and Sparsity Ratio

The LSE-SMP procedure alternates sparse-support detection with coefficient estimation, while LLR-based message updates reduce numerical and computational burdens. The resulting mutual refinement improves convergence, particularly for sparser channels.

  • LSE Fine Estimation: After SMP estimates the non-zero positions, LSE estimates the coefficient values by reformulating the model to exploit the sparse activity vector.The strategy swaps the positions of the channel and activity variables before solving the least-squares problem.
  • LSE Fine Estimation: The updated coefficient estimates and variances replace the previous iteration’s values when calculating the equivalent Gaussian-noise statistics.The update is performed at each iteration using the estimated matrix and its variance.
  • Sparsity-Ratio Update: The EM procedure learns the sparsity ratio η under the Bernoulli-Gaussian channel model.The update uses estimated parameters for the sparsity ratio and the mean and variance of non-zero entries.
  • Decision and Output of LSE-SMP: LSE and SMP mutually improve one another across iterations, accelerating convergence and yielding a larger advantage as the channel becomes sparser.The process stops when the MSE reaches the required level or the iteration limit is reached, after which the final estimate combines SMP and LSE fine estimation.
  • LLR Message Updates: LLRs replace direct probability products during message updates, preventing overflow and reducing computational complexity.The LLR representation also facilitates later performance analysis and prediction.

F. LSE-SMP in the Matrix Form

The LSE-SMP analysis formulates iterative sparse estimation in matrix form and establishes its statistical guarantees under partial knowledge of non-zero channel positions. Under this assumption, the estimator is unbiased, achieves the CRLB, and has no larger MSE than conventional LSE.

  • F. LSE-SMP in the Matrix Form: The matrix-form implementation defines low-dimensional decompositions to reduce the complexity of diagonal and block-diagonal matrix operations.The formulation introduces indexed matrix quantities and a matrix-product notation for the LSE-SMP updates.
  • F. LSE-SMP in the Matrix Form: Algorithm 1 summarizes the detailed LLR-matrix process and iteratively outputs the estimated channel vectors.The procedure initializes a coarse LSE estimate, performs iterative updates, and stops using convergence or iteration limits.
  • A. Cram´er-Rao Low Bound Of LSE-SMP: The CRLB analysis assumes a deterministic channel and prior knowledge of the non-zero positions b.This assumption separates estimation of the channel values from estimation of their support.
  • A. Cram´er-Rao Low Bound Of LSE-SMP: The LSE and LSE-SMP have different CRLBs because LSE-SMP estimates the sparse support while LSE does not.The distinction reflects the different prior knowledge available to the two estimators.
  • A. Cram´er-Rao Low Bound Of LSE-SMP: The proposed LSE-SMP estimator is the MVUE and can achieve the CRLB under the stated support-knowledge assumption.The proof verifies unbiasedness, computes the Fisher information, and applies the CRLB equality condition.
  • A. Cram´er-Rao Low Bound Of LSE-SMP: The singular Fisher information case remains well-posed because the required constraint holds, giving the estimator finite variance.The analysis uses the support matrix and generalized CRLB treatment for the sparse channel.
  • A. Cram´er-Rao Low Bound Of LSE-SMP: The proposed estimator satisfies the CRLB equality condition, confirming its minimum-variance unbiased status under the assumption.The conclusion follows by substituting the signal model, Fisher information, and estimator expressions into the equality condition.
  • A. Cram´er-Rao Low Bound Of LSE-SMP: MSE_LSE-SMP ≤ MSE_LSE for deterministic sparse channels, so the proposed estimator is bounded above by conventional LSE.The comparison follows from the eigenvalue relationship between the full and support-restricted covariance matrices.

B. Analysis of Iterative Evolution of LSE-SMP

The iterative-evolution analysis approximates message densities as Gaussian and uses closed-form updates and EXIT charts to study SMP convergence. It links larger variable-node convergence variance to lower BER and better estimation performance, while numerical results identify LSE-SMP as the strongest tested estimator.

  • B. Analysis of Iterative Evolution of LSE-SMP: The analysis treats SMP convergence as the core determinant of the iterative LSE-SMP behavior because direct analysis of the full loop is difficult.The full loop includes LSE estimation, Gaussian-function solving, SMP estimation, and EM learning.
  • B. Analysis of Iterative Evolution of LSE-SMP: Under Assumption 2, the channel support is known, channel moments are specified, and the training signal is Gaussian.The assumption provides the conditions used for the iterative evolution analysis.
  • B. Analysis of Iterative Evolution of LSE-SMP: Gaussian message-density approximations and the symmetry condition allow variable-node updates to be represented using the message mean.The symmetry condition relates message variance to its mean, reducing the quantities required for updating.
  • B. Analysis of Iterative Evolution of LSE-SMP: EXIT-chart simulations use variable-node variance as extrinsic information to predict the performance of the SMP component.This analysis adapts EXIT-chart methods to the variance behavior of variable nodes.
  • B. Analysis of Iterative Evolution of LSE-SMP: Larger convergence variance σ2_v yields better converged LSE-SMP estimation performance.The analysis connects convergence variance to BER, which decreases as σ2_v increases.
  • B. Analysis of Iterative Evolution of LSE-SMP: Theorem 2 and Corollary 2 connect convergence behavior to training length, SNR, channel variation, and variable-node variance.These relationships provide criteria used to interpret the numerical experiments.
  • B. Analysis of Iterative Evolution of LSE-SMP: The proposed LSE-SMP estimator exhibits the best NMSE performance among the tested algorithms in the reported setting.The cited setting uses h = 10 and training-sequence length T = 64.

V. NUMERICAL RESULTS

The numerical study evaluates channel estimation in a 32 × 64 mmWave MIMO system using Monte Carlo simulations across SNR values. Performance is measured by average NMSE over 500 randomly generated channel realizations.

  • V. NUMERICAL RESULTS: The performance metric is Normalized Mean Square Error (NMSE), averaged over 500 different channel realizations.The reported results use the average NMSE across the simulated channels.

A. Performance Comparison

The LSE-SMP estimator is compared with LSE and CRLB references across SNR, sparsity ratios, and runtime. Its reported gap from the bound is partly attributed to non-zero-position detection errors and coarse initialization.

  • Performance Comparison: Fig. 6 compares LSE-SMP NMSE and its CRLB versus SNR under different turbo iterations.The supplied figure caption specifies Nt = 32, Nr = 64, η = 0.031, and β = 10.
  • Performance Comparison: Fig. 7 compares LSE-SMP, LSE, and their CRLBs across SNR for sparsity ratios η ∈ {0.007, 0.125, 0.50, 0.80}.The experiment uses Nt = 32, Nr = 64, T = 64, and β = 10.
  • Performance Comparison: The reported LSE-SMP CRLB is the lower bound for LSE-SMP, while the LSE CRLB is the upper bound.
  • Performance Comparison: 1.8dB is the reported remaining gap, partly due to non-zero-position detection errors and reliance on a coarse initial channel estimate.

B. Effect of Iterations

Increasing turbo iterations lowers LSE-SMP NMSE, but improvements diminish as the algorithm approaches its analyzed CRLB. After five iterations, further improvement is not significant.

  • Effect of Iterations: After the fifth turbo iteration, LSE-SMP NMSE is very close to the analyzed CRLB with no significant further improvement.The passage characterizes the convergence as fast, requiring five iterations.
  • Effect of Iterations: Increasing turbo iterations lowers LSE-SMP NMSE, while adjacent-iteration gains become smaller at higher iteration counts.The passage attributes this trend to increasingly accurate estimates of η and the non-zero-position vector b.

C. Effect of Sparsity Ratios

The experiments examine sparsity, training length, channel dispersion, runtime, and convergence. LSE-SMP benefits from sparser channels and reaches the CRLB approximately after five turbo iterations, while design trade-offs remain between accuracy and computation.

  • Effect of Sparsity Ratios: Sparser channels produce better LSE-SMP performance, bringing it closer to the lower CRLB and demonstrating exploitation of channel sparsity.
  • Effect of Training Sequence Length: Longer training sequences improve estimation performance and reduce required iterations, but increase overall computational time.The passage links the time cost to exponential per-iteration complexity in matrix dimension.
  • Effect of Training Sequence Length: T = 16 is the shortest training sequence reported to make LSE-SMP work and converge under the tested setting.EXIT analysis is also used to predict training length for a given BER.
  • Effect of the Coefficient of Variation of the Channel: More dispersed channel conditions lower system performance but require less time to reach convergence.The EXIT analysis reports higher convergent variance points as the coefficient of variation increases.
  • Performance Comparison: LSE-SMP provides the best reported performance-complexity trade-off when NMSE is less than −80dB, although it may require more time.ITD-SE, LASSO, and BP-MF SBL are reported as best in higher-NMSE ranges.
  • Conclusion: The proposed algorithm typically needs only five turbo iterations to approximately achieve the CRLB.The conclusion notes that adjacent beam-domain channel correlation was ignored.

APPENDIX

The appendix develops Gaussian-approximation updates for the iterative analysis, using symmetry and moment calculations to obtain recursive expressions. These derivations support the algorithm’s density-evolution and convergence analysis.

  • APPENDIX: The analysis defines an update expression after substituting prior equations into the recursion.
  • APPENDIX: Transmit and receive antenna symmetry permits omission of the variable-node subscript in the analysis.
  • APPENDIX: Expected signal and noise terms are calculated and substituted into the variance expression.
  • APPENDIX: The signal-to-noise ratio and coefficient of variation are introduced to simplify the symmetric variance expression.
  • APPENDIX: With Gaussian lv, the expectation definition yields the recursion used to state Theorem 2.
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