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Low-Rank Tensor Decomposition-Aided Channel Estimation for Millimeter Wave MIMO-OFDM Systems

Zhou Zhou, Jun Fang, Linxiao Yang, Hongbin Li, Zhi Chen, Rick S. Blum

arXiv:1609.03355v2cs.IT

TL;DR

The paper tackles downlink channel estimation for wideband, frequency-selective mmWave MIMO-OFDM systems with hybrid beamforming and limited channel access. It represents received signals as a low-rank CP-decomposition tensor to estimate channel parameters, and reports near-CRB errors with higher accuracy and lower complexity than compressed sensing.

  • Problem

    Hybrid beamforming and large antenna arrays make practical CSI acquisition difficult, while wideband frequency-selective mmWave channel estimation is less addressed than narrowband estimation.

  • Method

    The method organizes received signals into a third-order tensor and extracts angles, delays, fading coefficients, and channels from its low-rank CP factor matrices.

  • Results

    Simulation mean square errors are close to corresponding CRBs, while the proposed method achieves higher estimation accuracy and comparable low complexity relative to compressed sensing baselines.

  • Takeaways & Limitations

    CP decomposition provides a gridless, multidimensional channel-estimation approach with potential for substantial training-overhead reduction.

Abstract

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We consider the problem of downlink channel estimation for millimeter wave (mmWave) MIMO-OFDM systems, where both the base station (BS) and the mobile station (MS) employ large antenna arrays for directional precoding/beamforming. Hybrid analog and digital beamforming structures are employed in order to offer a compromise between hardware complexity and system performance. Different from most existing studies that are concerned with narrowband channels, we consider estimation of wideband mmWave channels with frequency selectivity, which is more appropriate for mmWave MIMO-OFDM systems. By exploiting the sparse scattering nature of mmWave channels, we propose a CANDECOMP/PARAFAC (CP) decomposition-based method for channel parameter estimation (including angles of arrival/departure, time delays, and fading coefficients). In our proposed method, the received signal at the BS is expressed as a third-order tensor. We show that the tensor has the form of a low-rank CP decomposition, and the channel parameters can be estimated from the associated factor matrices. Our analysis reveals that the uniqueness of the CP decomposition can be guaranteed even when the size of the tensor is small. Hence the proposed method has the potential to achieve substantial training overhead reduction. We also develop Cramer-Rao bound (CRB) results for channel parameters, and compare our proposed method with a compressed sensing-based method. Simulation results show that the proposed method attains mean square errors that are very close to their associated CRBs, and presents a clear advantage over the compressed sensing-based method in terms of both estimation accuracy and computational complexity.

I. INTRODUCTION

The paper addresses wideband mmWave MIMO-OFDM channel estimation under hybrid beamforming, where limited channel access and large antenna arrays make CSI acquisition difficult. It proposes a low-rank CP decomposition approach that estimates channel parameters while reducing training overhead.

  • Motivation: Wideband mmWave channels require accurate estimation to support directional precoding and beamforming with large BS and MS antenna arrays.Large bandwidth enables high data rates, while high-frequency attenuation motivates large-array beamforming gains.
  • Motivation: Hybrid precoding limits digital-baseband access to the full channel dimension, creating a channel subspace sampling limitation during practical channel coherence times.This makes acquiring useful CSI difficult with large antenna arrays.
  • Related approaches: Compressed sensing exploits sparse scattering to reduce training overhead, but it formulates channel estimation using discretized parameter representations.Prior work includes adaptive and distributed compressed sensing for mmWave channels.
  • Contribution: The proposed method organizes received signals into a third-order tensor whose low CP rank enables estimation of angles, delays, and fading coefficients from factor matrices.The low rank follows from the small number of scattering paths relative to tensor dimensions.
  • Contribution: CP uniqueness can be guaranteed for small tensors, implying substantial training overhead reduction and providing conditions for determining the exact overhead needed for unique decomposition.The method also develops CRB results and analyzes computational complexity against compressed sensing methods.
  • System model: The system uses hybrid analog/digital precoding and combining, with common digital precoders and pilot symbols across subcarriers to facilitate tensor-factorization-based CSI extraction.The objective is to estimate all subcarrier channel matrices using as few measurements as possible.

IV. PROPOSED CP DECOMPOSITION-BASED METHOD

The proposed method converts multicarrier received signals into a third-order tensor and models it through a CP decomposition. Sparse multipath structure makes the decomposition low rank and identifiable, allowing channel parameters and subcarrier channels to be recovered from factor matrices.

  • Tensor construction: With identical digital precoding matrices and pilot symbols across subcarriers, each subcarrier’s received signal is represented as Y_k = Q^T H_k P + W_k.Here p(t) is formed from the common RF precoder and pilot vector.
  • Tensor construction: Stacking time frames, sub-frames, and subcarriers produces a third-order tensor whose entries are the received signals y_k,m(t).The three modes correspond respectively to time frame, sub-frame, and subcarrier.
  • CP model: Each tensor slice is a weighted sum of common rank-one outer products, yielding a CP representation with path-dependent factors.The factors include frequency-dependent path gains and projected MS and BS array responses.
  • CP model: Sparse scattering makes the number of paths small relative to tensor dimensions, giving the received tensor an intrinsic low-rank structure.This structure supports unique CP decomposition up to scaling and permutation ambiguities.
  • Parameter extraction: The channel gains, angles of arrival and departure, delays, and subcarrier channel matrices are obtained from the decomposed factor matrices.The factor matrices correspond to a noiseless version of the received tensor.

A. CP Decomposition

The CP decomposition can be computed with alternating least squares when the path count is known, or with sparsity-promoting model-order estimation when it is unknown. These procedures recover low-rank factor matrices while controlling data-fitting error.

  • Known rank: When the number of paths L is known or estimated, CP decomposition is formulated as an optimization problem over the factor matrices.The optimization seeks a CP representation of the received tensor.
  • Known rank: Alternating least squares efficiently solves the CP optimization by minimizing data-fitting error for one factor matrix while fixing the other two.The procedure alternates across the factor matrices.
  • Unknown rank: When L is unavailable, sparsity-promoting CP techniques jointly estimate model order and factor matrices using an overestimated rank.Negligible rank-one components are removed after convergence to estimate the true rank.
  • Unknown rank: The overestimated factor matrices have dimensions M×L_hat, T×L_hat, and K×L_hat, while regularization controls the tradeoff between low-rankness and data-fitting error.The true CP rank is obtained by removing negligible components.

B. Channel Estimation

The method estimates mmWave channel parameters from CP factor matrices, with uniqueness enabling reliable recovery under small training dimensions. It uses factor-matrix relationships to estimate angles, delays, fading coefficients, and reconstruct channel matrices.

  • Factor-matrix estimation: The estimated factor matrices are related to the true factors through scaling, permutation, and estimation-error ambiguities.The diagonal scaling matrices satisfy Λ1Λ2Λ3 = I, while the common permutation can be ignored.
  • Parameter estimation: Angles of arrival and departure are estimated from correlations between estimated factor-matrix columns and array responses.The angle-of-arrival correlation estimator is maximum likelihood under i.i.d. circularly symmetric Gaussian factor-matrix errors.
  • Parameter estimation: Time delays are estimated from columns of the third factor matrix, whose columns combine fading coefficients with delay-dependent vectors.After estimating angles, the scaling matrices and fading coefficients are recovered using the factor-matrix relationships.
  • Uniqueness: The CP decomposition is essentially unique under Kruskal’s condition, up to a common permutation and compensating diagonal scalings.The theorem states that alternative CP solutions have the same factors under these ambiguities.
  • Uniqueness: With K ≥ L, training can satisfy Kruskal’s condition using either T = L, M = 2 or M = L, T = 2.Random unit-circle beamforming and combining matrices support the stated k-rank conditions with probability one.
  • Training overhead: The method needs T = L or T = 2 time frames and M = 2 or M = L sub-frames for reliable parameter estimation, reducing training overhead.Observation noise and estimation errors may require slightly larger T and M for accurate channel estimates.

V. CRB

The paper develops CRB benchmarks for channel-parameter estimation and characterizes a two-step CP-based estimator. It also formulates a compressed sensing alternative, whose grid mismatch creates accuracy and complexity trade-offs.

  • CRB analysis: The CRB provides a lower bound on the variance of unbiased estimators and benchmarks the proposed channel-parameter estimator.The analysis assumes complex circularly symmetric i.i.d. Gaussian observation noise.
  • Proposed estimator: The proposed estimator first applies ALS for CP decomposition, then uses correlation-based processing on factor matrices to estimate channel parameters.Under stated Gaussian assumptions and a reached global minimum, both stages have maximum-likelihood interpretations.
  • Compressed sensing: Compressed sensing reformulates channel estimation as sparse recovery using an unknown sparse vector and overcomplete angle and delay dictionaries.The angle and delay domains are discretized into grids, with dictionary sizes substantially exceeding the number of paths.
  • Compressed sensing: Grid mismatch deteriorates compressed-sensing performance when true angles or delays do not lie on the discretized grids.Finer grids increase computational complexity and can cause numerical instability through highly coherent dictionary columns.
  • Compressed sensing: Off-grid compressed sensing mitigates discretization errors but typically requires iterative joint dictionary refinement and sparse estimation, increasing computational complexity.This establishes the principal accuracy-versus-complexity trade-off for the compressed-sensing alternative.

VII. COMPUTATIONAL COMPLEXITY ANALYSIS

The proposed CP method has computational complexity that scales linearly with the observed tensor size when the CP rank is small, while compressed sensing methods can be substantially more costly. Simulations report comparable runtime to OMP alongside higher estimation accuracy.

  • CP decomposition complexity: O(MT K) is the dominant computational complexity of the CP method when the CP rank L is small.The full least-squares update has order O(MT KL + MKL^2 + L^3), and the other two updates also require O(MT K) flops.
  • Compressed sensing complexity: OMP and FISTA solve a sparse recovery problem whose complexity is higher than the CP method when measurements are far fewer than the sparse-signal dimension.OMP has lower complexity but barely satisfactory recovery accuracy, whereas ℓ1-minimization methods such as FISTA achieve better performance at higher computational cost.
  • Compressed sensing complexity: O(10^4) is the signal dimension for a typical compressed-sensing grid with N1 = 32, N2 = 64, and N3 = 32.The dimension equals N1N2N3, where the factors discretize the AoA, AoD, and time-delay domains.
  • Runtime comparison: The CP method achieves higher estimation accuracy than OMP while taking similar run times to OMP with the coarser grid.FISTA has prohibitive computational complexity for this channel-estimation problem, whereas CP has complexity as low as OMP.

IX. CONCLUSIONS

The paper develops a low-rank CP decomposition method for wideband mmWave MIMO-OFDM channel estimation and reports advantages over compressed sensing in accuracy and complexity.

  • The method estimates wideband frequency-selective channel parameters by representing the BS received signal as a third-order tensor with a low-rank CP decomposition.Channel parameters are extracted from the decomposed factor matrices.
  • CP decomposition uniqueness can be guaranteed with a small number of measurements, enabling substantial training overhead reduction.
  • Simulation results show a clear advantage over compressed sensing in both estimation accuracy and computational complexity.

APPENDIX A

This appendix derives an estimation procedure by simplifying the likelihood after optimizing one unknown parameter and applying the Cauchy-Schwarz inequality.

  • The appendix assumes circularly symmetric complex Gaussian errors with zero mean and covariance matrix ǫ2I, then forms the log-likelihood function.
  • For fixed θl, the optimal λl is obtained by differentiating the log-likelihood with respect to λl and setting the derivative to zero.
  • After substituting the optimized parameter, maximizing the log-likelihood over θl is reformulated using the Cauchy-Schwarz inequality.

APPENDIX B

This appendix characterizes the random measurement matrix induced by a uniform linear array and establishes conditions under which its columns are full k-rank.

  • For a uniform linear array, the steering vector is expressed using the signal wavelength, antenna spacing, and sine of the arrival angle.
  • The entries of the analog combining matrix are generated with uniformly distributed phases and scaling 1/NMS.
  • With sufficiently many MS antennas, steering vectors for distinct angles become mutually quasiorthogonal, making the matrix entries uncorrelated.
  • By the central limit theorem, the matrix entries are approximately i.i.d. Gaussian with zero mean and variance 1/NMS.
  • The k-rank of the resulting matrix equals the smaller of its number of columns and rows with probability one.

APPENDIX C THE DERIVATION OF CRAM´ER RAO LOWER BOUND

This appendix sets up the likelihood and Fisher information calculations for deriving Cramér-Rao lower bounds on the channel parameters.

  • The derivation considers the M × T × K observation tensor Y and defines the real-part operator and canonical vectors used in the model.
  • The unknown channel parameters are path gains, arrival and departure angles, and time delays, collected into the vector p.
  • The noise matrix W is modeled with independent zero-mean circularly symmetric Gaussian entries of variance σ2.
  • The likelihood is expressed through factor matrices A, B, and C, which depend on the parameter vector p.
  • The complex Fisher information matrix is used to obtain the information needed for the CRB calculation.
  • The appendix begins the derivative calculation by obtaining the partial derivative of the log-likelihood with respect to each arrival angle θl.

B. Calculation of Fisher Information Matrix

The Fisher information calculation derives covariance-related quantities from tensor unfoldings and their vectorizations, then evaluates entries of the matrix Ω(p), including principal and off-principal minors.

  • Matrix-entry evaluation: The derivation evaluates principal and off-principal minor entries of Ω(p), using indexed entries of N_a and related quantities.Diagonal-index definitions and products involving N_a and N_b appear in the minor calculations.
  • Covariance calculations: The entries of W are mapped across tensor unfoldings, vectorized unfoldings, and the tensor itself to identify corresponding indices.These mappings support the computation of Cw1,w2, Cw1,w3, and Cw2,w3.
  • Noise statistics: Because W has i.i.d. entries, vec(W^H(1)) is modeled as a zero-mean circularly symmetric complex Gaussian vector with covariance σ^2I.The transformed noise vector n_a consequently also follows a circularly symmetric complex Gaussian distribution.

C. Cram´er Rao bound

After the Fisher information matrix is obtained, the Cramér–Rao bound for the parameters p is calculated using the stated reference formula.

  • CRB calculation: The CRB for parameters p is calculated after obtaining the Fisher information matrix.The calculation is attributed to reference.
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