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Design of Robust Adaptive Beamforming Algorithms Based on Low-Rank and Cross-Correlation Techniques

H. Ruan, R. C. de Lamare

arXiv:1606.01313v1cs.CEcs.ITmath.IT

TL;DR

Robust adaptive beamforming must handle steering-vector mismatch and the computational burden of large sensor arrays. This paper develops cross-correlation-based low-rank mismatch estimation and adaptive weight-update algorithms. The authors report excellent output SINR performance, particularly at medium-high input SNR.

  • Problem

    Adaptive beamformers are sensitive to steering-vector mismatch, and existing RAB methods can impose high computational cost for large sensor arrays.

    Source: §I · p. 1, A. Prior and Related Work · p. 1

  • Method

    The paper estimates mismatch by projecting observation/output cross-correlation onto an orthogonal Krylov subspace and develops SG and CG-based adaptive weight updates.

    Source: B. Contributions · p. 2

  • Results

    The proposed algorithms show robustness to user-defined parameter choices and environmental effects, with excellent output SINR performance especially at medium-high input SNR.

    Source: §VII · p. 13

  • Takeaways & Limitations

    The proposed low-rank techniques are presented as cost-effective for high-dimensional subspaces and large sensor arrays.

    Source: §I · p. 1, §VII · p. 13

Abstract

from arXiv · show

This work presents cost-effective low-rank techniques for designing robust adaptive beamforming (RAB) algorithms. The proposed algorithms are based on the exploitation of the cross-correlation between the array observation data and the output of the beamformer. Firstly, we construct a general linear equation considered in large dimensions whose solution yields the steering vector mismatch. Then, we employ the idea of the full orthogonalization method (FOM), an orthogonal Krylov subspace based method, to iteratively estimate the steering vector mismatch in a reduced-dimensional subspace, resulting in the proposed orthogonal Krylov subspace projection mismatch estimation (OKSPME) method. We also devise adaptive algorithms based on stochastic gradient (SG) and conjugate gradient (CG) techniques to update the beamforming weights with low complexity and avoid any costly matrix inversion. The main advantages of the proposed low-rank and mismatch estimation techniques are their cost-effectiveness when dealing with high dimension subspaces or large sensor arrays. Simulations results show excellent performance in terms of the output signal-to-interference-plus-noise ratio (SINR) of the beamformer among all the compared RAB methods.

I. INTRODUCTION

Adaptive beamformers are vulnerable to steering-vector mismatch, while large arrays increase computational cost and slow convergence. The paper proposes low-rank, cross-correlation-based RAB methods and frames the design around robustness and computational efficiency.

A. Prior and Related Work

  • Large arrays make existing RAB methods costly, while conventional beamformers are sensitive to environmental uncertainty and steering-vector mismatch.The paper identifies cubic-or-greater computational cost for some existing methods and increased complexity with slower convergence in large arrays.Source: A. Prior and Related Work · p. 1, §I · p. 1
  • Existing dimensionality-reduction approaches include beamspace, Krylov-subspace, and joint iterative optimization methods, each with different design challenges.Krylov methods face model-order selection, while beamspace uses constrained optimization to determine its projection matrix.Source: A. Prior and Related Work · p. 1

B. Contributions

  • The proposed RAB design targets robustness to steering-vector uncertainty while retaining low complexity for large sensor arrays.The paper proposes low-rank and cross-correlation techniques and uses presumed angular-sector knowledge to estimate mismatch.Source: B. Contributions · pp. 1–2, §II · p. 3
  • OKSPME estimates steering-vector mismatch by projecting cross-correlation information onto an orthogonal Krylov subspace, while SG and CG variants update weights without costly matrix inversion.The paper also analyzes steering-vector MSE and computational complexity.Source: B. Contributions · p. 2

II. SYSTEM MODEL AND PROBLEM STATEMENT

  • The design problem includes the SMI beamformer's sensitivity to steering-vector mismatch and its need for many snapshots to converge.The paper seeks cost-efficient algorithms robust to uncertainty and suitable for large arrays.Source: §II · p. 3

III. PROPOSED OKSPME METHOD

The method estimates desired-signal power and uses cross-correlation between array observations and beamformer output as part of mismatch estimation. Its power-estimation expression is low-complexity under stated estimation conditions.

III. PROPOSED OKSPME METHOD

  • The OKSPME method estimates steering-vector mismatch by projecting an estimated observation/output cross-correlation vector onto an orthogonal Krylov subspace.The cross-correlation is defined as d = E[xy*].Source: §III · p. 3

A. Desired Signal Power Estimation

  • The power-estimation derivation assumes interferer steering vectors are orthogonal or approximately orthogonal to the desired steering vector.Under that condition, the interferer contribution can be neglected in the stated expression.Source: A. Desired Signal Power Estimation · p. 3
  • The desired-signal-power subsection presents an iterative estimation method and notes that its expression has O(M) complexity when steering vector and noise level are accurately estimated.The passage contrasts this approach with maximum-likelihood and minimum-variance estimators, which it describes as higher complexity.Source: A. Desired Signal Power Estimation · p. 3

B. Orthogonal Krylov Subspace Approach for Steering Vector Mismatch Estimation

The mismatch estimator formulates a linear system whose solution is the steering-vector mismatch, then uses an orthogonal Krylov subspace to solve it at reduced cost. Its model order is selected using stopping and minimum-rank criteria.

B. Orthogonal Krylov Subspace Approach for Steering Vector Mismatch Estimation

  • The method constructs a high-dimensional linear system whose solution represents steering-vector mismatch, then solves it in a reduced-order Krylov subspace.The system is expressed using available statistics or estimated parameters.Source: B. Orthogonal Krylov Subspace Approach… · pp. 3–4
  • The method applies Arnoldi-modified Gram-Schmidt within FOM to compute an orthogonal Krylov subspace and uses a residue vector to represent estimation error.The residue is defined for the current snapshot.Source: B. Orthogonal Krylov Subspace Approach… · p. 4
  • Krylov model order stops when the Arnoldi execute-and-stop criterion is met or the minimum sufficient rank for capturing the signal of interest and interferers is reached.The minimum-rank condition is j ≥ K + 1, where K is the number of signal sources.Source: B. Orthogonal Krylov Subspace Approach… · p. 4
  • The method projects the sample cross-correlation vector onto the Krylov subspace and adds the resulting estimation error to the current steering-vector estimate.The projection matrix is computed from the Krylov subspace bases.Source: B. Orthogonal Krylov Subspace Approach… · p. 4

C. INC Matrix and Beamformer Weight Vector Computation

The section relates the interference-plus-noise covariance to the sample covariance and desired-signal covariance, then proceeds to beamformer-weight computation.

C. INC Matrix and Beamformer Weight Vector Computation

IV. PROPOSED ADAPTIVE ALGORITHMS

The section introduces adaptive OKSPME algorithms that retain the steering-vector and signal-power recursions while using SG- and CG-based weight updates. The SG variant avoids matrix inversion and reduces computational complexity relative to OKSPME.

IV. PROPOSED ADAPTIVE ALGORITHMS

  • Adaptive OKSPME uses SG, conventional CG, and modified CG recursions to update beamforming weights, while retaining OKSPME’s signal-power and steering-vector estimation recursions.The weight updates are derived from a reformulated optimization problem.Source: §IV · pp. 4–5

A. OKSPME-SG Adaptive Algorithm

  • The SG weight update uses a stochastic-gradient optimization problem and recursion, with µ as the step size.The recursion enforces the beamformer constraint w^H(i + 1)â_1(i + 1) = 1.Source: A. OKSPME-SG Adaptive Algorithm · p. 5
  • OKSPME-SG avoids matrix inversion and reduces complexity from O(M^3) in OKSPME to O(M^2).The step size µ must satisfy 0 < µ < 1.Source: A. OKSPME-SG Adaptive Algorithm · p. 5

B. OKSPME-CCG Adaptive Algorithm

The CCG and MCG variants use conjugate-gradient updates for the steering-vector estimate and beamforming weights. CCG performs multiple iterations per snapshot; MCG uses one, with the choice depending on system stationarity.

B. OKSPME-CCG Adaptive Algorithm

  • OKSPME-CCG performs N iterations per snapshot and has complexity O(NM^2), higher than OKSPME-SG because of its inner iterations.After updating the steering-vector estimate and CG-based vector for N iterations, the beamforming weights are computed.Source: B. OKSPME-CCG Adaptive Algorithm · pp. 5–6
  • CCG updates the steering-vector estimate and CG-based weight vector using conjugate-gradient iterations, with direction vectors and negative gradients defined for both.The scaling parameters minimize the cost function, while β parameters provide conjugacy for the direction vectors.Source: B. OKSPME-CCG Adaptive Algorithm · pp. 5–6

C. OKSPME-MCG Adaptive Algorithm

  • OKSPME-MCG performs one iteration per snapshot, reducing complexity relative to CCG while achieving similar performance in most studied scenarios.MCG has complexity O(M^2).Source: C. OKSPME-MCG Adaptive Algorithm · pp. 6–7
  • CCG is suggested for stationary systems, whereas MCG is suggested for non-stationary scenarios where one update per snapshot can track environmental variations.MCG estimates α(i) using forgetting factor λ and constant η, so performance may depend on their choice.Source: C. OKSPME-MCG Adaptive Algorithm · p. 7

V. ANALYSIS

The analysis covers steering-vector estimation error, comparisons with SQP, and computational complexity for proposed and existing algorithms.

V. ANALYSIS

  • The analysis considers MSE between estimated and actual steering vectors for the general approach using a presumed angular sector.Source: §V · p. 7
  • It analyzes MSE for OKSPME and SQP and examines their relationships and differences.Source: §V · p. 7
  • It also analyzes the computational complexity of proposed and existing algorithms.Source: §V · p. 7

A. MSE analysis

The MSE analysis derives bounds for steering-vector estimation under an unbiased angular-sector assumption and compares the proposed OKSPME method with SQP. Both MSE bounds for OKSPME are lower than SQP’s, provided the presumed sector spread is below π/2.

  • For the norm analysis, the unbiased case places the true steering vector at the sector center, making E[ε] = 0 by symmetry.This removes the non-analytical last term in the MSE expression.Source: A. MSE analysis · p. 8
  • SQP’s accumulated error satisfies θ_i||a1|| < ||ê(i)|| ≤ ||a1||tan θ_i, with its lower bound unattainable in finite snapshots.Source: A. MSE analysis · p. 9
  • OKSPME projects cross-correlation-derived error information into a Krylov subspace, with ||a1||sin θ_i ≤ ||ê(i)|| < θ_i||a1||.Source: A. MSE analysis · pp. 9–10
  • OKSPME’s update error can lie inside the sector, a case the authors associate with faster convergence and smaller estimation error.Source: A. MSE analysis · p. 10
  • Both MSE bounds for OKSPME are lower than SQP’s and depend only on ||a1|| and sector spread 2θ, which must be below π/2.Source: A. MSE analysis · p. 10

B. Complexity Analysis

The complexity analysis compares per-snapshot additions and multiplications as sensor count grows. It reports that the proposed OKSPME-SG and OKSPME-MCG have lower complexity than the other algorithms.

B. Complexity Analysis

  • OKSPME-SG and OKSPME-MCG have lower complexity than the other algorithms in the comparison.The analysis counts additions and multiplications per snapshot as M varies from 10 to 100; Fig. 4 presents the complexity comparison.Source: B. Complexity Analysis · p. 11
  • The proposed algorithms’ complexity depends on Krylov subspace order m, which is no larger than K + 1.For the plotted comparison, m = K + 1 with K = 3.Source: B. Complexity Analysis · p. 11
  • The SQP method’s complexity varies substantially across snapshots and is usually O(M^3.5).Its variation is attributed to the online optimization program’s random choices of presumed steering vector.Source: B. Complexity Analysis · p. 11

VI. SIMULATIONS

The simulations evaluate beamformer output SINR under coherent and incoherent local-scattering mismatch, varying snapshots or input SNR. The paper also includes a complexity comparison.

VI. SIMULATIONS

  • The simulations measure output SINR versus training snapshots or input SNR under coherent and incoherent local-scattering mismatch.The input SNR spans −10 dB to 30 dB; the setup uses 100 repetitions per curve point and at most 300 snapshots.Source: §VI · p. 11
  • The baseline simulation setup uses omnidirectional uniform linear arrays with half-wavelength sensor spacing.The desired signal arrives at 10°, SIR is fixed at 0 dB, and the assumed angular sector is [5°, 15°].Source: §VI · p. 11
  • Fig. 4 presents a complexity comparison.The supplied passage identifies the figure’s subject but does not provide its plotted values or outcome.Source: Figure 4 · p. 11

A. Mismatch due to Coherent Local Scattering

For coherent local scattering, the proposed OKSPME methods perform similarly to or better than LOCSME in the reported comparisons. The section also examines larger arrays and a time-varying interferer scenario.

A. Mismatch due to Coherent Local Scattering

  • The coherent-scattering model uses scattered paths whose angles and phases vary across simulation runs but remain constant over snapshots.The scattered-path angles are drawn with mean 10° and standard deviation 2°; the figure caption identifies the M = 10, K = 3 snapshot comparison at INR = 20 dB.Source: A. Mismatch due to Coherent Local… · p. 11, Figure 5 · p. 12
  • The initial comparison tests M = 10, K = 3, INR = 20 dB, and up to 100 snapshots against classical robust adaptive beamformers.The comparison includes fixed diagonal loading, RCB, and SQP; the interferers are at 30° and 50°.Source: A. Mismatch due to Coherent Local… · p. 11, Figure 5 · p. 12
  • OKSPME performs similarly to or slightly better than LOCSME, and both have the best performance in the M = 12 comparisons.OKSPME-CCG and OKSPME-MCG also achieve performance very close to OKSPME; Figs. 6 and 7 compare SINR versus snapshots and input SNR.Source: A. Mismatch due to Coherent Local… · pp. 11–12
  • The comparisons include a time-varying scenario with a halfway redistribution of interferers at M = 12.The interferer details are listed in Table VI.Source: A. Mismatch due to Coherent Local… · p. 12, Table VI · p. 12
  • At M = 40, the proposed OKSPME-based algorithms perform better than the beamspace approach, with some degradation relative to M = 12.Figs. 9 and 10 show SINR versus snapshots and input SNR, respectively.Source: A. Mismatch due to Coherent Local… · p. 12

B. Mismatch due to Incoherent Local Scattering

Incoherent local scattering is associated with degraded SINR across the studied algorithms. The conclusion summarizes the proposed mismatch-estimation and adaptive-weight methods and reports robustness and strong SINR, particularly at medium-high input SNR.

B. Mismatch due to Incoherent Local Scattering

VII. CONCLUSION

  • The conclusion reports robustness to user-defined parameter choices and environmental effects, with excellent output SINR especially at medium-high input SNR.The conclusion also describes the proposed OKSPME mismatch estimator and SG, CCG, and MCG weight updates without matrix inversions.Source: §VII · p. 13
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