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A novel robust adaptive beamforming based on worst-case and norm constraint (RAB-WC-NC) is presented. The proposed beamforming possesses superior robustness against array steering vector (ASV) error with finite snapshots by using the norm constraint and worst-case performance optimization (WCPO) techniques. Simulation results demonstrate the validity and superiority of the proposed algorithm.

As an important branch of array signal processing, adaptive beamforming technique has achieved a wide range of applications in the fields of radar, sonar, wireless communication, radio astronomy, and so forth [

In order to improve the adaptability of beamforming against those above situations, plenty of research on beamforming robustness has been carried out recently [

To solve these problems, in this paper, a robust adaptive beamforming algorithm based on the worst-case and norm constraint (RAB-WC-NC) is proposed. RAB-WC-NC algorithm forms the flat response in the main beam width determined by the uncertainty set of ASV and improves the performance of beamforming by adopting norm constraint under the circumstance of finite snapshots. The proposed algorithm can improve the robustness of beamforming and suppress interference with finite snapshots.

Consider a uniform linear array (ULA) with

The output of array is the weight sum of the observation signals from each array element. The weight vector

It is well known that MVDR beamforming minimizes array output power while constraining the desired signal response to be unity. That is,

The weight vector of MVDR beamforming algorithm can be derived from Lagrange multiplier method:

However, the presumed steering vector always deviates from the actual one. In this case, the performance of the MVDR beamformer is severely limited by target signal cancellation. To maintain a fairly stable gain in the region of interest, the following inequality constraints on the steering vector are imposed [

WCPO algorithm can be achieved by analyzing constraint condition and making the optimal performance of beamforming on the worst case:

Equation (

In this section, we propose a robust beamformer with the worst case performance optimization and the norm constraint. We can formulate the constrained robust problem as

Meanwhile, the physical meaning of the constraint in WCPO algorithm, demonstrated in (

From the definition of

Using the 3rd constraint of (

By using the constraint again, hence we have

Thus according to (

Multiplying the constant

Since

According to the property of vector norm,

After simplification,

According to the symmetry of signal in both space and frequency domains, if

The position of peak value of “signal”

Assume that the corresponding angle of ASV of real main beam in space domain is

So, the parameters of main beam width,

At last, amending (

Using the Cholesky decomposition, covariance matrix of array snapshot can be given as

Hence, (

Thus, (

In conclusion, the step of RAB-WC-NC algorithm can be generalized as follows.

Use (

Adopt IPM algorithm to solve (

In our simulations, a ULA of

We assume that the orientation of desired signal is

Various beamformers' beam patterns with

Beam patterns

Main beams

As illustrated in Figure

Various beamformers’ beam patterns with

Beam patterns

Main beams

We assume that the orientation of desired signal is

Various beamformers’ beam patterns with high SNR.

We assume that DOAs of two interferences are

Output SINR versus

We assume that DOAs of two interferences are

Output SINR versus input SNR.

RAB-WC-NC algorithm is deduced specifically to handle the situations where ASV has errors, and the receipt data contains desired signal with finite snapshots. The uncertain set of ASV is adopted to determine the main beam width, within which the flat response is formed, increasing the robustness of beamforming against ASV error; and the performance of the proposed algorithm in the situations that receipt data contains desired signal with finite snapshots is improved by utilizing norm constraint. To sum up, RAB-WC-NC algorithm is blessed with certain adaptability to any kind of errors and effectively increases the output SINR under the circumstance of various errors, consequently proven to be a robust adaptive beamforming algorithm.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China under Grant 61401204.