^{1}

^{*}

^{2}

^{3}

^{1}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

To overcome the performance degradation in the presence of steering vector mismatches, strict restrictions on the number of available snapshots, and numerous interferences, a novel beamforming approach based on nonlinear least-square support vector regression machine (LS-SVR) is derived in this paper. In this approach, the conventional linearly constrained minimum variance cost function used by minimum variance distortionless response (MVDR) beamformer is replaced by a squared-loss function to increase robustness in complex scenarios and provide additional control over the sidelobe level. Gaussian kernels are also used to obtain better generalization capacity. This novel approach has two highlights, one is a recursive regression procedure to estimate the weight vectors on real-time, the other is a sparse model with novelty criterion to reduce the final size of the beamformer. The analysis and simulation tests show that the proposed approach offers better noise suppression capability and achieve near optimal signal-to-interference-and-noise ratio (SINR) with a low computational burden, as compared to other recently proposed robust beamforming techniques.

As one important branch of modern array signal processing, the beamforming technique has been widely studied and applied in the radar, wireless communication, sonar, medical imaging, as well as astronomy domains. The standard beamforming approach, such as the minimum variance distortionless response (MVDR) beamformer [

During the last decades, in order to resist the model mismatches and possible environment changes, the robust beamforming approach have been largely studied [

In order to reject jamming signals, poor array calibration, signal wave-front distortions, the minimum-variance-distortionless-response (MVDR) beamforming is modified by the means of incorporating multiple linear constrains [

Support Vector Machines (SVM), introduced by Vapnik [

This paper presents a new LS-SVR-based approach to address the robust beamforming issue. This approach alleviates the array output SINR degradation in the presence of steering vector mismatches, strict restrictions on the number of available snapshots, and numerous interferences by replacing the conventional linearly constrained minimum variance cost function with a squared-loss function, and achieves better generalization capacity by applying Gaussian kernels to the array observations. We also present a fast recursive procedure to estimate the weight vectors on real-time, and a novelty criterion to perform model reduction. The paper is organized as follows. The signal model, also the minimum mean square error (MMSE) and the MVDR-beamformer solutions are presented in Section 2. The basic principle of LS-SVR-based beamforming method is introduced in Section 3. In Section 4, a recursive procedure to calculate the regression parameters is provided. And a sparse mode is presented in Section 5. The simulation tests under different mismatch scenarios are illustrated in Section 6. A summary conclusion is given at the last of this paper.

Consider a linear array of M sensors receives signals from D narrowband source. The vector of array observations x(^{M}^{×1} at time _{1}, _{2},…, _{D}^{T} ∈ R^{D}^{×1} is the vector with the directions of arrival (DOA) and (·)^{T}_{1}), _{2})…_{D}^{M}^{×D} is the matrix containing the array steering vectors _{i}^{−}^{j}^{2}^{π}^{sin(}^{θi}^{)}^{d}^{/}^{λ}^{−}^{j}^{2}^{π}^{sin(}^{θi}^{)}^{d}^{/}^{λ}^{T}_{1}(_{2}(_{D}^{T}^{M}^{×1}. The vector ^{M}^{×1} is the sensor noise, and it is assumed as complex Gaussian with zero-mean:

The output of the beamformer is defined as:
_{1}, …, _{M}^{T} ∈ C^{M}^{×1} is the complex vector of beamformer weights, (·)^{T} denotes transpose, and (·)^{H} denotes Hermitian transpose.

If certain observations are known during the procedure of training parameters, then, according to the MMSE criterion, the complex vector of beamformer weights

The classical MVDR beamformer minimizes the array output energy, and the weights subject to a constraint of unity array response on the desired array steering vectors, that is:

The constraint ^{H}_{1}) = 1 prevents the gain at the look direction from being reduced, and the solution of

In practice, it is not feasible to calculate the exact covariance matrix

The performance of MVDR beamformer in

Consider a set of snapshots _{i}_{i}_{i}_{i}

The parameter set _{i}_{n}_{n}

The weight vector

Instead of the inequality constrains in standard SVM algorithm, the equality ones are taken in LS-SVR, and the linear equation of the

The array observations of the beamformer are complex, whereas the variables in the objective function of SVM are real. So, it is necessary to rewrite the complex variables as real variables. For this reason, the array observations _{i}_{i}_{t}

The result of the quadratic optimization problem of _{t}_{1}, _{2}, … _{2}_{N}^{T}, _{i} > 0 is Lagrange multipliers, defined as regression parameters in this paper.

According to the Karush-Kuhn-Tucker (KKT) conditions, differentiating the above function with respect to the Lagrange multipliers _{t}_{i,t}_{t}_{i,t}

The system obtained from the KKT conditions is linear. Its result is obtained by solving the linear system which is expressed as following matrix:
_{t}_{1}, _{2} … _{2}_{N}^{T}, ^{T}, _{i}_{i}_{i,j}_{i,j}_{i,j}_{t}_{i}_{j}_{t}_{i}_{j}

The outputs of the nonlinear LS-SVR beamformer are:

From _{t}_{t}_{t}_{t}^{−1} = (_{t}^{−1}^{−1}, the result of LS-SVR (

Then, we have:

As the number of snapshots increases, the dimension of Gramm matrix _{i}_{t}_{t}_{i}

At time step _{i}_{i}

As time run to _{t}_{+1} and the corresponding desired array output _{t}_{+1} are added to the current training set. So _{t}_{+1} and _{t}_{+1} can be represented as:

Comparing the elements of _{t}_{t}_{+1}, the matrix _{t}_{+1} could be reconstructed by the matrix _{t}_{t}_{+1} = [_{1}, (_{2(}_{N}_{+1)}), … _{2}_{N}_{2(}_{N}_{+1)})]^{T}, _{t}_{+1} = _{2(}_{N}_{+1)}), _{2(}_{N}_{+1)}) + 1/

According to the theorem of inverting block matrix, the inverse of _{t}_{+1} can be expressed by the inverse of _{t}_{t}_{+1} as:
_{t}_{+1},which is equal to _{t}_{+1}, can be calculated from the inverse of _{t}_{t}_{t}

The crucial drawback of LS-SVR beamformer is that it deals with high-dimension matrix, which is equal to the number of the snapshots due to the use of a quadratic constraint function. This would bring a big implementation problem to the proposed beamforming method since it is required to increase memory and computational resources as time evolves. Several methods have been proposed to cope with these problems [

Step 1: Initialing an empty center set C_{0};

Step 2: Calculating the distance between the new snapshot x_{t}_{c}_{k}_{∈ C}_{i}_{t}_{k}

Step 3: If the distance obtained from Step 2 is smaller than the preset threshold _{1}, x_{t}_{i}_{i}_{i}

Step 4: if |e|_{i}_{2}, x_{t}_{i}_{i}_{+1}, otherwise go to Step 2.

Increasing _{1} and _{2}, the final size of the LS-SVR beamformer will be decreased. But this will result to performance degradation. In practical applications, _{1} is set to around one tenth of the kernel bandwidth, and _{2} is around the square root of the steady-state mean square error (MSE). Cross-validation also can be used to select these appropriate thresholds.

Applying the above sparsification procedure, the computation complexity of the proposed beamformer will be reduced from ^{2}) to ^{2}), where

To evaluate the performance of the proposed LS-SVR-based beamformer, simulation tests are carried out. A 10 elements uniform linear array with half-wavelength spacing is taken into account. The desired signal comes from a presumed direction _{2} = −32° and _{3} = 17° respectively. The additive noise is assumed to be a 0-dB complex white Gaussian distributed random variable. For comparison purpose, the conventional MVDR, the diagonal loading MVDR (MVDR-DL), the ES [_{1} and _{2}, are chosen as 1.0, 0.1 and 0.08 respectively. The load value of MVDR-DL beamformer is set to (_{e}_{e}

The first simulation aims to compare the performance of these beamformers when steering vector mismatch is presented. From

The covariance matrix would be inaccurately estimated owing to insufficient snapshots, DOA mismatch of desired signal and array calibration errors. This kind of inaccuracy may result in the degradation of array response. Hence, both the errors of insufficient snapshots and DOA mismatch are considered to verify the proposed beamformer in our second simulation tests.

The performance of the proposed beamformer in the scenario with multiple interferences is demonstrated in the third test. The steering vector mismatch is also presented. As it can be seen from

The corresponding beampatterns of the beamformers are demonstrated in _{i}

To show the computation complexity of the novel approach, the dictionary size growth with the input samples is given in

We present a novel nonlinear LS-SVR-based beamforming approach in this paper. This approach first uses a squared-loss function to replace the conventional linearly constrained minimum variance cost function, which can significantly increase robustness against mismatch problems and provide additional control over the sidelobe level. The method also applies Gaussian kernels to the array observations to improve the generalization capacity. Finally, the method uses a recursive regression procedure to estimate the weight vectors on real-time and performs mode reduction to reduce the final size of the beamformer.

The simulation tests, with steering vector mismatch, numerous interferences and limited available snapshots, are carried out to verify the performance of the proposed beamforming algorithm in comparison with other recently proposed ones. The test results show that the proposed beamforming method significantly outperforms many other recently proposed linear robust beamforming techniques in terms of signal distortion in the desired signal and noise reduction in scenarios with DOA mismatch, limited observation samples, and numerous interferences.

This research was supported by the National Natural Science Foundation of China (Grant No.61071191) and Natural Science Foundation of Chongqing (CSTC 2011BB2048).

Scenario with only DOA mismatch (

Scenario with limited snapshots and two interferences.

Scenario with DOA mismatch and multiple interferences (

Dictionary size