We propose doubly constrained robust least-squares constant modulus algorithm (LSCMA) to solve the problem of signal steering vector mismatches via the Bayesian method and worst-case performance optimization, which is based on the mismatches between the actual and presumed steering vectors. The weight vector is iteratively updated with penalty for the worst-case signal steering vector by the partial Taylor-series expansion and Lagrange multiplier method, in which the Lagrange multipliers can be optimally derived and incorporated at each step. A theoretical analysis for our proposed algorithm in terms of complexity cost, convergence performance, and SINR performance is presented in this paper. In contrast to the linearly constrained LSCMA, the proposed algorithm provides better robustness against the signal steering vector mismatches, yields higher signal captive performance, improves greater array output SINR, and has a lower computational cost. The simulation results confirm the superiority of the proposed algorithm on beampattern control and output SINR enhancement.
Adaptive beamforming, as an attractive solution to signal detection and estimation in harsh environments, has received considerable attention in the fields of radar, sonar, seismology, radio astronomy, medical imaging, artificial intelligence, and neural network [
In this paper, robust LSCMA based on double constraints is proposed via the worst-case performance optimization. The quadratic constraint on the weight vector can improve robustness to the signal steering vector mismatches. In order for LSCMA to provide improved performance, the updating weight vector subject to the constraints of distortionless array response is derived by the partial Taylor-series expansion and Lagrange multiplier method, in which the multipliers can be optimally derived and incorporated at each step. The implementation of the proposed algorithm based on iterative minimization eliminates the covariance matrix inversion estimation, so it has a low computational load. Compared with the linearly constrained LSCMA, the proposed algorithm suffers the least distortion from the direction near the desired steering angle, yields better signal captive performance, and has superior performance on SINR improvement. The theoretical analysis and simulation results have been carried out to demonstrate effectiveness and superiority of the proposed algorithm in the signal steering vector mismatches. So the proposed algorithm can be an appealing technique and be implemented in digital system to improve the receiver performance.
We assume that there are
In the array signal processing, the objective of adaptive beamforming is to enhance the desired signal and suppress the noise and interference signals, which improves the array output SINR. In the adaptive array antenna system, the output SINR achieved the optimal one by regulating the weight vector.
The linearly constrained LSCMA that is an effective solution to the problem of interference capture can be used for equalization, blind adaptive beamforming, and other similar applications when the desired signal has a constant envelope [
We define
From (
To overcome the above-mentioned problem, robust constrained LSCMA is proposed, which provides excellent robustness against signal steering vector mismatches, suppresses the interference signals effectively, and enhances the array output SINR. In practical applications, the array beampattern error is formulated as
The cost function of robust constrained LSCMA can be written as
The quadratic constraint is adjoined to the cost function by the Lagrange multiplier
From (
In order to detect the desired signal under directional uncertainty, we can impose another constraint on an average steering vector via the Bayesian approach. We assume that the direction of arrival (DOA) is a discrete random variable with known a priori probability density function (pdf)
When interferers are present, the a posteriori probability density function
At low SNR, it will be relatively flat over all DOAs and revert to the a priori pdf. At high SNR, the a posteriori probability of the true DOA will approach one, whereas the posteriori probability of the other DOAs will approach zero.
Based on (
Gauss’s method updates
Using the Lagrange multiplier method, the optimal solution to (
The gradient of
By inserting the multiplier
The complexity cost of the conventional LSCMA and the proposed robust LSCMA can be shown in Tables
The complexity cost of the conventional LSCMA.
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The complexity cost of the proposed LSCMA.
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The proposed robust constrained LSCMA is globally stable and convergent via Agee’s inequalities. The first input stream is successfully extracted by establishing the following inequalities, given that
To extract the second input stream, we begin with the convergence of
For
The output signal of the proposed beamformer can be expressed as
Assuming that the initial beamformer SINR is known, the SINR of the normalized output signal
In this section, Matlab software is used to evaluate the performance of the proposed algorithm. The sampling frequency is
The simulation scenario diagram.
The SNR is equal to 10 dB. The aforementioned algorithms are simulated by using a mismatched steering vector of the desired signal, where the practical angle of incidence equals 6°. This corresponds to a
Array beampattern (in no mismatch case).
Array beampattern (in the mismatch case).
In the second experiment, the SINR performance of the aforesaid algorithms for the fixed
Output SINR versus
Output SINR versus
In this experiment, we evaluate the SINR performance versus input SNR with DOA error for the fixed sample data size
Output SINR versus SNR (in no mismatch case).
Output SINR versus SNR (in the mismatch case).
In this paper, a novel robust LSCMA algorithm based on double constraints is proposed via the Bayesian approach and worst-case performance optimization. To improve robustness, the weight vector is optimized to involve minimization of the objective function with penalty for the worst-case signal steering vector by the partial Taylor-series expansion and Lagrange multiplier method, in which the parameters can be precisely derived at each iterative step. Moreover, the online implementation of the proposed algorithm eliminates the covariance matrix inversion estimation, which has a low computational load. The proposed robust constrained LSCMA has a faster convergence rate, provides better robustness against the signal steering vector mismatches, and yields improved array output performance compared with the linearly constrained LSCMA. The theoretical analysis and simulation experiments have been carried out to illustrate the significant performance improvement of the proposed method for the signal steering vector mismatches.
The authors would like to thank the anonymous reviewers for their insightful comments that helped improve the quality of this paper. This work was supported by the Program for New Century Excellent Talents in University no. NCET-12-0103, the Natural Science Foundation of Liaoning province under Grant no. 201102057, and the Natural Science Foundation of Hebei province under Grant no. F2012501044.