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Robust Adaptive Beamforming Based on Low-Complexity Shrinkage-Based Mismatch Estimation
Hang Ruan, Rodrigo C. de Lamare
TL;DR
Adaptive beamforming can suffer when data are limited or the desired steering vector is mismatched, motivating robust methods with lower computational cost. The paper proposes LOCSME, which uses OAS shrinkage to estimate covariance quantities and the steering-vector mismatch from limited prior information. Simulations report that LOCSME outperforms previously reported RAB algorithms and performs close to the optimum.
Problem
Adaptive beamformers can degrade because of short data records, desired-signal contamination in training data, or imprecise desired-signal steering vectors.
Method
LOCSME estimates the mismatched steering vector and INC matrix using OAS-based shrinkage, requiring the desired signal’s angular sector and avoiding an optimization program.
Results
LOCSME outperforms previously reported RAB algorithms in coherent and incoherent local-scattering simulations and is close to optimum SINR in the coherent case.
Takeaways & Limitations
LOCSME provides a lower-cost RAB approach that requires only the desired signal’s angular sector and avoids direction finding for all interferers.
Takeaways & Limitations
Performance degrades with larger antenna arrays and with an inappropriate assumed angular sector.
Abstract
from arXiv · showhide
In this work, we propose a low-complexity robust adaptive beamforming (RAB) technique which estimates the steering vector using a Low-Complexity Shrinkage-Based Mismatch Estimation (LOCSME) algorithm. The proposed LOCSME algorithm estimates the covariance matrix of the input data and the interference-plus-noise covariance (INC) matrix by using the Oracle Approximating Shrinkage (OAS) method. LOCSME only requires prior knowledge of the angular sector in which the actual steering vector is located and the antenna array geometry. LOCSME does not require a costly optimization algorithm and does not need to know extra information from the interferers, which avoids direction finding for all interferers. Simulations show that LOCSME outperforms previously reported RAB algorithms and has a performance very close to the optimum.
I. INTRODUCTION
Adaptive beamformers can degrade under limited data, desired-signal contamination, or steering-vector mismatch, motivating robust adaptive beamforming. The paper proposes LOCSME as a lower-complexity alternative requiring little prior information.
- Motivation: Short data records, desired-signal contamination, and steering-vector imprecision can degrade adaptive-beamformer performance.RAB techniques address performance degradation caused by steering-vector mismatches.
- Prior approaches: Recent RAB methods jointly estimate the mismatched steering vector and interference-plus-noise covariance matrix using shrinkage or covariance reconstruction.These approaches combine multiple design principles to improve robustness.
- Prior approaches: Covariance reconstruction can provide outstanding performance but requires costly matrix reconstruction.The cited approach uses SQP for steering-vector estimation and covariance reconstruction for the INC matrix.
- Proposed approach: LOCSME estimates the steering vector and covariance quantities with OAS-based shrinkage, using only the desired signal’s angular sector as prior knowledge.The method avoids an optimization program and computes the sector subspace projection matrix in simple steps.
- Paper organization: The paper is organized around the system model, the LOCSME algorithm, simulation results, and conclusions.These topics are presented in Sections II through V after the introduction.
II. SYSTEM MODEL
The system model represents array snapshots as mixtures of steering-vector-weighted source signals and complex Gaussian noise. It formulates beamforming through SINR maximization and identifies SMI’s snapshot and steering-mismatch sensitivities.
- Signal model: A linear array with M sensors receives K narrowband signals at each snapshot.The model uses the received snapshot x(i) and a steering matrix associated with the signal directions.
- Signal model: The steering matrix contains one steering vector per direction of arrival, while e denotes the desired-signal steering-vector mismatch.The additive noise is modeled as complex Gaussian with zero mean and variance σ2_n.
- Beamforming formulation: The beamformer output is formed from the received data using a weight vector whose Hermitian transpose is denoted by (·)^H.The weight vector has M entries corresponding to the array sensors.
- Beamforming formulation: The optimum beamformer is obtained by maximizing the signal-to-interference-plus-noise ratio under a precisely known desired steering vector.The desired signal power is σ2_1 and R_i+n denotes the interference-plus-noise covariance matrix.
- Sample covariance limitations: The MVDR beamformer uses the optimum weight solution, but its required INC matrix is usually estimated from the received-data sample covariance matrix.The SMI beamformer requires many snapshots and is sensitive to steering-vector mismatches.
III. PROPOSED LOCSME ALGORITHM
LOCSME separately estimates the steering vector and the interference-plus-noise covariance matrix. It projects an iteratively shrinkage-estimated cross-correlation vector onto a predefined subspace and obtains the INC matrix by covariance subtraction.
- Steering-vector estimation: LOCSME estimates the steering vector by projecting an iteratively shrinkage-estimated cross-correlation vector onto a predefined subspace.The subspace is associated with the assumed angular sector of the desired signal.
- INC-matrix estimation: LOCSME obtains the INC matrix by subtracting the desired-signal covariance matrix from the OAS-estimated data covariance matrix.The steering-vector and INC-matrix estimations are performed separately, following previous approaches.
A. Steering Vector Estimation using LOCSME
LOCSME estimates the desired steering vector by shrinkage-processing a cross-correlation vector and projecting it onto a subspace defined by the known angular sector. Its iterative OAS-based estimation is designed to improve correlation-vector accuracy before producing the final steering-vector estimate.
- Steering-vector estimation: LOCSME estimates the steering vector from an iteratively shrinkage-estimated cross-correlation vector between the beamformer output and array observation.The shrinkage procedure targets a more accurate correlation-vector estimate before steering-vector recovery.
- Subspace projection: The estimated correlation vector is projected onto a predefined subspace to remove unwanted components and recover the desired steering-vector information.The projection uses a matrix formed from principal eigenvectors of a matrix built over the specified angular sector.
- Prior information: The only prior information used for the subspace projection is an angular sector containing the desired signal, represented as [θ1 − θe, θ1 + θe].The projection matrix is constructed from p principal eigenvectors associated with that sector.
- OAS shrinkage: The correlation-vector estimate is obtained by shrinking the sample correlation vector toward a structured target using an MSE-based shrinkage coefficient.The method defines the sample correlation vector and selects the coefficient to reduce the stated mean square error.
- Convergence: The iterative shrinkage process is guaranteed to converge when its initial shrinkage value lies between 0 and 1, after which the final steering-vector estimate is obtained.The convergence condition applies to the iterative process used to obtain the correlation vector.
B. Interference-Plus-Noise Covariance Matrix Estimation
LOCSME estimates the interference-plus-noise covariance matrix by shrinkage-processing the data covariance matrix, estimating desired-signal power, and subtracting the desired-signal covariance. This avoids direction finding and has lower stated complexity than several earlier RAB methods.
- INC estimation: LOCSME applies OAS shrinkage to the data covariance matrix before estimating and removing the desired-signal covariance.The data covariance matrix initially contains the desired signal, so it is refined before INC estimation.
- Desired-signal removal: The desired-signal power is estimated from the received data using the estimated steering vector and assumptions about noise and interferer correlations.The resulting estimate is used to form the desired-signal covariance matrix.
- INC construction: The INC matrix is obtained by subtracting the estimated desired-signal covariance matrix from the shrinkage-estimated data covariance matrix.This subtraction removes the desired-signal contribution from the covariance estimate.
- Complexity and information requirements: The INC estimation step does not require direction finding, unlike SMI and existing methods described in the paper.The stated advantage is avoiding the need to identify interferer directions.
- Complexity and information requirements: LOCSME has complexity O(M^3), compared with O(M^3.5) or higher for the cited previous RAB algorithms.The stated LOCSME steps include covariance shrinkage and norm computations.
IV. SIMULATIONS
The simulations evaluate output SINR using a 12-sensor half-wavelength ULA, fixed source geometry, and comparisons against existing beamformers across snapshots and SNR. The study uses repeated trials and a shared angular-sector setting for the compared methods.
- Simulation setup: The simulation uses a ULA with M = 12 omnidirectional sensors spaced at half wavelength, one desired source, and two interferers.The desired signal arrives from θ1 = 10°, while interferers arrive from θ2 = 50° and θ3 = 90°.
- Simulation setup: The SIR is fixed at 20 dB, with i = 50 snapshots and 100 repetitions used to obtain each plotted curve point.Only one iteration is performed per snapshot.
- Evaluation: The beamformers are compared using output SINR as a function of snapshots and SNR.The comparison includes LOCSME and beamformers from references,,, and.
- Evaluation: All compared methods use the angular sector [θ1 − 5°, θ1 + 5°], while p = 8 principal eigenvectors are used for the subspace projection.CVX is used for the compared beamformers that require optimization.
A. Mismatch due to Coherent Local Scattering
The coherent-local-scattering experiment models the desired steering vector with a direct path and randomly varying scattered paths. Under this mismatch model, LOCSME outperforms the other algorithms and remains close to the optimum SINR.
- Mismatch model: The desired steering vector is modeled with one direct path and four scattered paths to represent coherent local scattering.The scattered-path angles are independently randomized for each simulation run.
- Mismatch model: The scattered-path angles have mean 10° and standard deviation 2°, while their azimuthal phases are uniformly drawn from [0, 2π].The angles and phases change between trials but remain constant over snapshots.
- Results: LOCSME outperforms the other algorithms and is close to the optimum SINR under coherent local scattering.The comparison is reported for SINR versus snapshots and SNR.
B. Mismatch due to Incoherent Local Scattering
Under incoherent local scattering, all algorithms degrade relative to coherent scattering, but LOCSME outperforms the other robust beamformers across a wide input-SNR range. Larger arrays and an incorrectly chosen angular sector further reduce performance.
- B. Mismatch due to Incoherent Local Scattering: Incoherent local scattering causes performance degradation for all algorithms compared with coherent scattering.The desired signal has a time-varying signature, with scattering coefficients changing across runs and snapshots.
- B. Mismatch due to Incoherent Local Scattering: LOCSME outperforms the remaining robust beamformers over a wide range of input SNR.Figures 1(b) and 2(b) show SINR versus snapshots and SNR, respectively.
- B. Mismatch due to Incoherent Local Scattering: The improved LOCSME performance is attributed to accurate estimates of both the interference-plus-noise covariance matrix and steering-vector mismatch.
- B. Mismatch due to Incoherent Local Scattering: Around 2 dB of LOCSME degradation occurs when the array size reaches M = 60.Testing with larger antenna arrays indicates performance degradation for all algorithms.
- B. Mismatch due to Incoherent Local Scattering: An inappropriate angular sector for the desired signal causes obvious performance degradation.
V. CONCLUSION
The paper concludes that LOCSME requires only prior knowledge of the desired signal’s angular sector, costs less than existing methods, and outperforms prior RAB algorithms in both scattering settings.
- V. CONCLUSION: LOCSME requires only prior knowledge of the desired signal’s angular sector and is less costly than existing methods.
- V. CONCLUSION: LOCSME outperforms prior-art RAB algorithms in both coherent and incoherent local scattering cases.
APPENDIX
The appendix derives the shrinkage intensity by rewriting the shrinkage estimator, formulating an optimization, and solving sequentially for its target and intensity under Gaussian assumptions.
- APPENDIX: The shrinkage estimator is rewritten as a weighted combination of ˆF(i) and ˆS(i), with ˆρ(i) as the shrinkage intensity.
- APPENDIX: The optimization objective is rewritten using E[ˆS(i)] = E[ˆF(i − 1)].
- APPENDIX: The optimal ˆν(i) is obtained from a problem independent of ˆρ(i).
- APPENDIX: Differentiating the objective with respect to ˆν(i) and then ˆρ(i) yields the shrinkage intensity solution.
- APPENDIX: Under Gaussian assumptions, replacing ˆF(i − 1) with ˆD(i) and the sample number n with snapshot index i gives equation (13).