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Adaptive Reduced-Rank LCMV Beamforming Algorithms Based on Joint Iterative Optimization of Filters: Design and Analysis
R. C. de Lamare, L. Wang, R. Fa
TL;DR
Large and dynamic beamforming systems make covariance estimation challenging, motivating reduced-rank LCMV methods. This paper jointly optimizes a projection matrix and reduced-rank filter, and reports improved convergence in ULA simulations with complexity comparable to existing reduced-rank RLS methods when the rank is sufficiently smaller than the array dimension.
Problem
Covariance-matrix estimation can be challenging in large or highly dynamic systems, and adaptive-filter convergence and tracking depend on sensor count and covariance eigenvalue spread.
Source: §I · p. 1, §III · p. 2Method
The method jointly optimizes a projection matrix and reduced-rank filter under the minimum-variance criterion, with stochastic-gradient and RLS implementations and automatic rank selection.
Source: §I · p. 1, A. Stochastic Gradient Algorithm · p. 3Results
ULA simulations report that the proposed scheme and algorithms outperform in convergence; with D significantly smaller than M, complexity is slightly higher than full-rank RLS and comparable to MSWF-RLS and AVF.
Source: C. Complexity of Proposed Algorithms · p. 4, §VIII · p. 8Takeaways & Limitations
When D is significantly smaller than M, the proposed reduced-rank RLS has complexity comparable to MSWF-RLS and AVF while trading off a full-rank filter for a projection matrix and reduced-rank filter.
Source: C. Complexity of Proposed Algorithms · p. 4The authors caution that their automatic rank-adaptation technique may not apply universally, although it was effective for the problems examined.
Source: D. Automatic Rank Selection · p. 5
Abstract
from arXiv · showhide
This paper presents reduced-rank linearly constrained minimum variance (LCMV) beamforming algorithms based on joint iterative optimization of filters. The proposed reduced-rank scheme is based on a constrained joint iterative optimization of filters according to the minimum variance criterion. The proposed optimization procedure adjusts the parameters of a projection matrix and an adaptive reducedrank filter that operates at the output of the bank of filters. We describe LCMV expressions for the design of the projection matrix and the reduced-rank filter. We then describe stochastic gradient and develop recursive least-squares adaptive algorithms for their efficient implementation along with automatic rank selection techniques. An analysis of the stability and the convergence properties of the proposed algorithms is presented and semi-analytical expressions are derived for predicting their mean squared error (MSE) performance. Simulations for a beamforming application show that the proposed scheme and algorithms outperform in convergence and tracking the existing full-rank and reduced-rank algorithms while requiring comparable complexity
I. INTRODUCTION
Adaptive LCMV beamforming faces covariance-estimation and computational challenges in large or dynamic systems. The paper motivates reduced-rank processing and formulates the full-rank and reduced-rank filtering problems for antenna-array beamforming.
I. INTRODUCTION
- Covariance estimation can be challenging in large, dynamic systems, while adaptive-filter convergence and tracking depend on sensor count and covariance eigenvalue spread.The paper motivates reducing dimension while retaining key signal features.Source: §I · p. 1
- Prior reduced-rank methods include eigen-decomposition, Krylov-subspace, and joint-optimization approaches, but adaptive LCMV algorithms based on joint optimization had not been considered.The paper positions its contribution as addressing this gap for antenna-array beamforming.Source: §I · p. 1
II. SYSTEM MODEL
- The system model considers K narrowband sources with unknown directions of arrival impinging on an M-element array, with K < M and zero-mean Gaussian sensor noise.The array sources are assumed to be in the far field.Source: §II · p. 2
III. PROBLEM STATEMENT
- The full-rank LCMV filter minimizes output power subject to a unit response toward the signal of interest, while its adaptive convergence and tracking depend on M and the covariance eigenvalue spread.The reduced-rank problem maps received data through an M × D projection matrix before applying a D-dimensional filter; designing that projection is the stated cost-effective challenge.Source: §III · p. 2
IV. PROPOSED REDUCED-RANK METHOD
The proposed method jointly optimizes a projection matrix and a reduced-rank filter under a minimum-variance criterion and a unity-response constraint. The projection reduces input dimension, while the reduced-rank filter forms the beam toward the signal of interest.
IV. PROPOSED REDUCED-RANK METHOD
- The M × D projection matrix maps the M-dimensional received vector to reduced-rank data, which a D-dimensional filter processes into a scalar estimate.The two components are jointly optimized under the minimum-variance criterion with a unity response toward the signal of interest.Source: §IV · pp. 2–3
- For D > 1, the full-rank filters in the projection matrix perform subspace projection, while the reduced-rank filter supplies unity gain toward the signal of interest.This division of roles is presented as exploiting the low-rank nature of typical beamforming signals.Source: §IV · p. 3
- For D = 1, the scheme becomes conventional full-rank LCMV filtering with an additional weight parameter that provides amplitude gain.Source: §IV · p. 3
V. ADAPTIVE ALGORITHMS
The paper develops stochastic-gradient and recursive-least-squares implementations of the jointly optimized reduced-rank scheme, together with automatic rank adaptation. Both implementations update the projection matrix and reduced-rank filter in an exchanging, iterative process.
A. Stochastic Gradient Algorithm
- The stochastic-gradient expressions for the projection matrix and reduced-rank filter depend on one another, so the method iterates them from initial values.Their iterative information exchange supports a simpler adaptive implementation than the MSWF and AVF approaches described in the passage.Source: A. Stochastic Gradient Algorithm · p. 3
- The stochastic-gradient scheme trades a full-rank filter for a projection matrix and a reduced-rank adaptive filter that operate simultaneously and exchange information.The projection matrix reduces input dimension, while the reduced-rank filter yields unity response toward the signal of interest.Source: A. Stochastic Gradient Algorithm · p. 3
B. Recursive Least Squares Algorithms
- The RLS implementation recursively updates full- and reduced-rank inverse covariance matrices, replacing covariance inversion at each step with recursive processes.Its reduced-rank recursion is initialized with a positive constant times the D × D identity matrix; the projection and reduced-rank filter operate simultaneously and exchange information.Source: B. Recursive Least Squares Algorithms · p. 4
C. Complexity of Proposed Algorithms
The proposed reduced-rank SG algorithm has complexity that grows linearly with DM, while the proposed RLS algorithm is quadratic in M and D. The RLS algorithm remains near full-rank RLS complexity when D is much smaller than M.
C. Complexity of Proposed Algorithms
- Proposed reduced-rank SG complexity grows linearly with DM, about D times full-rank SG complexity and significantly below MSWF-SG.The additional cost may be acceptable when D is much smaller than M if performance gains justify it.Source: C. Complexity of Proposed Algorithms · p. 4
- With D << M, proposed reduced-rank RLS complexity is slightly higher than full-rank RLS and comparable to MSWF-RLS and AVF.Its complexity is quadratic in M and D.Source: C. Complexity of Proposed Algorithms · p. 4
- Proposed reduced-rank RLS complexity is below MSWF-RLS and AVF, while proposed reduced-rank SG complexity lies between full-rank RLS and full-rank SG.These trends are shown as additions and multiplications versus input size M.Source: C. Complexity of Proposed Algorithms · p. 4
D. Automatic Rank Selection
The rank-selection method chooses D using an exponentially weighted a posteriori least-squares cost computed from the projection matrix and reduced-rank filter. The analysis section outlines stability and MSE convergence analysis for the stochastic-gradient algorithms and states that RLS algorithms are expected to converge to the optimal LCMV filter.
D. Automatic Rank Selection
- The algorithm selects D to minimize an exponentially weighted a posteriori least-squares cost computed from the projection matrix and reduced-rank filter.It updates extended quantities for D_max, then selects the best rank at each time instant; the cost uses the associated rank-D filters.Source: D. Automatic Rank Selection · pp. 4–5
- The useful rank range was D_min = 3 to D_max = 8 in the examined scenarios, and choosing an inadequate rank may degrade performance.Smaller ranks may adapt faster initially, while greater ranks usually yield better steady-state performance; rank adaptation can be relaxed once steady state is reached.Source: D. Automatic Rank Selection · p. 5
- The rank-adaptation method may not apply universally; cross-validation is another option but may increase filter lengths and complexity.The authors identify cost-effective rank selection as an area for further investigation.Source: D. Automatic Rank Selection · p. 5
VI. ANALYSIS OF ALGORITHMS
- The analysis derives stability conditions and semi-analytically determines steady-state MSE for the proposed stochastic-gradient algorithms.The RLS algorithms are expected to converge to the optimal LCMV filter, which the authors report verifying in their studies.Source: §VI · p. 5
A. Stability Analysis
The stability analysis links convergence to the spectral radius of a matrix formed from the joint filter updates. The MSE analysis treats the joint adaptation errors and uses a semi-analytical approach to predict steady-state performance.
A. Stability Analysis
- Convergence requires step sizes for which the eigenvalues of P^H P are less than one, because stability depends on the spectral radius of P.The stability terms are more involved than in most adaptive algorithms because the terms depend on each other.Source: A. Stability Analysis · p. 5
B. MSE Convergence Analysis
- The steady-state excess MSE is analyzed for the D-rank approximation filter, with the MSE trajectory assessed through the rotated tap-error covariance.The analysis jointly accounts for the errors of both adaptive filters and relates the limiting MSE to the minimum variance plus a weighted covariance-error trace.Source: B. MSE Convergence Analysis · p. 6
- Computing the joint-adaptation covariance error requires a semi-analytical approach using simulations to precompute terms across step sizes.The authors report a good match between semi-analytical and simulated curves, while noting that the joint terms are difficult to isolate.Source: B. MSE Convergence Analysis · p. 7
VII. SIMULATIONS
The simulations assess beamforming with a uniform linear array and compare proposed algorithms with full-rank and reduced-rank alternatives. Performance is evaluated using MSE and SINR across scenarios.
VII. SIMULATIONS
- The simulations compare proposed SG and RLS algorithms with full-rank LCMV-SG and LCMV-RLS, MSWF, AVF, and an optimal linear beamformer assuming known covariance.The setup uses a uniform linear array with M sensor elements.Source: §VII · p. 7
- Performance is compared using mean-squared error and signal-to-interference-plus-noise ratio.For full-rank schemes, the SINR expression uses the identity projection matrix.Source: §VII · p. 7
- Each scenario uses 200 runs, with desired-signal power set to 1 and the filters initialized with specified reduced-rank weights and projection matrix.The projection-matrix initialization includes a D by (M − D) zero block.Source: §VII · p. 7
A. MSE Analytical Performance
The semi-analytical expressions closely predict simulated steady-state MSE for both tested step-size settings. Smaller step sizes converge more slowly but reach the optimal LCMV MSE level, unlike the larger-step-size case.
A. MSE Analytical Performance
- The semi-analytical MSE curves agree with simulations for both tested step-size settings, supporting the analysis’s prediction of steady-state performance.The estimates use the data covariance matrix R and are compared with simulations under two sets of step sizes.Source: A. MSE Analytical Performance · p. 7
- Smaller step sizes converge more slowly but reach the optimal LCMV MSE level, whereas larger step sizes produce higher misadjustment.The smaller settings were µs = 0.001 and µw = 0.001; the larger settings were µs = 0.0025 and µw = 0.01.Source: A. MSE Analytical Performance · p. 7
B. SINR Performance
The simulations report that the proposed reduced-rank algorithms improve convergence and tracking over existing full-rank and reduced-rank methods at comparable complexity. Across SINR experiments, they approach full-rank performance, offer favorable convergence and steady-state trade-offs, and outperform existing algorithms under a changing environment.
B. SINR Performance
- With optimized parameters and 250 snapshots, rank D = 4 is best for the proposed scheme and is very close to optimal full-rank LCMV; the preferred rank is relatively invariant to system size.Rank adaptation is proposed to balance convergence speed, steady-state performance, and tracking.Source: B. SINR Performance · p. 7
- In the equal-SINR comparison, the proposed scheme converges much faster than full-rank SG and performs better than MSWF-RLS and AVF.The SG version of MSWF cannot approach optimal LCMV in these situations because it does not tridiagonalize its covariance matrix [17].Source: B. SINR Performance · p. 8
- Automatic rank selection converges significantly faster than fixed rank D = 8 while attaining the same steady-state performance.The comparison evaluates automatic rank selection against fixed ranks D = 3 and D = 8 for SG and RLS algorithms.Source: B. SINR Performance · p. 8
- In a non-stationary scenario with a sudden change at i = 800, the proposed reduced-rank algorithms have superior performance to existing algorithms.The algorithms use automatic rank adaptation and are tuned to converge as fast as possible to the same SINR value.Source: B. SINR Performance · p. 8
VIII. CONCLUSIONS
- The proposed algorithms outperform existing full-rank and reduced-rank methods in convergence and tracking at comparable complexity in ULA beamforming simulations.The conclusion also notes that the algorithms can be extended to other array geometries and applications.Source: §VIII · p. 8
APPENDIX
The appendices state a condition for preserving full-rank minimum variance and analyze the proposed joint optimization. They examine its convexity, solutions, initialization, and projection-matrix derivation.
APPENDIX
- The reduced-rank scheme preserves the full-rank minimum variance if and only if the full-rank optimal filter lies in the projection matrix’s range.That range is the subspace generated by SD(i).Source: APPENDIX · p. 8
- The derivation concludes that the proposed optimization problem has multiple solutions.Source: APPENDIX · p. 8
- For analysis, the scheme is rewritten with a Lagrangian and its output represented using a stacked parameter vector and block-diagonal data matrix.The representation defines a single parameter vector f(i) and a block-diagonal matrix containing the input data vector.Source: APPENDIX · p. 8
- The Hessian has both positive-semidefinite and indefinite terms, so the optimization is not classified as convex; studies indicate no local minima and multiple solutions.The fifth Hessian term depends on the constraint and is typically positive in the proposed scheme.Source: APPENDIX · p. 9
- Different initializations did not degrade performance in the reported study, but the zero projection initialization must be avoided because it annihilates the signal.Source: APPENDIX · p. 9
- A fixed scalar projection yields a convex function, whereas a time-varying projection is nonconvex; local-minimum existence remains unverified in the vector case.The vector case presents a multidimensional surface left as an open problem.Source: APPENDIX · p. 9
- The appendix derives the projection matrix SD(i) from the Lagrangian and simplifies its expression to reduce computational complexity.Source: APPENDIX · p. 9