International Journal of Antennas and Propagation

Volume 2015, Article ID 765385, 7 pages

http://dx.doi.org/10.1155/2015/765385

## Robust Adaptive Beamforming Based on Worst-Case and Norm Constraint

^{1}Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China^{2}China Shipbuilding Industry Corporation, Nanjing, Jiangsu 210003, China^{3}China Academy of Space Technology, Xi’an, Shanxi 710000, China

Received 24 October 2014; Accepted 9 December 2014

Academic Editor: Giampiero Lovat

Copyright © 2015 Hongtao Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1]. However, adaptive beamforming confronts the problem of intense decrease in robustness in the cases that array steering vector (ASV) has errors, or receipt signal contains desired signal component, or the ideal covariance matrix is replaced by signal covariance matrix with finite snapshots [2]. It can be proven that the errors caused by using signal covariance matrix could be treated equally as the ASV errors at the circumstance of finite snapshots [3]. Therefore, the research of beamforming robustness primarily focuses on the ASV errors and the receipt signal containing desired signal.

In order to improve the adaptability of beamforming against those above situations, plenty of research on beamforming robustness has been carried out recently [4–16]. ESB algorithm possesses excellent robustness against ASV errors while its receipt signal must contain comparatively strong desired signal and prior information or estimation of the dimension of subspace is demanded [4]. Diagonal loading class-based adaptive beamforming algorithm possesses certain adaptability to the various situations while it is incapable of retaining the maximum gain to the actual desired signal and thus the signal to interference plus noise ratio (SINR) will, to some extent, suffer from loss with ASV errors [5–13]. Magnitude response constraints-based robust adaptive beamforming algorithm is blessed with favorable robustness against ASV errors by forming flat response in main beam while it demands prior information of main beam width and extra interference along with noise can be involved in the range of main beam [14–16].

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.

#### 2. The Signal Model

Consider a uniform linear array (ULA) with array elements separated from each other by a distance . far-field narrowband incoherent signals are received from the orientation of , and then the receiving data of array can be expressed as where , denote the observation vectors of array and signal at separately; is independent identically distributed gaussian white noise vector; is the array manifold matrix; is the steering vector of signal at orientation . Assume that the signal and noise are unrelated.

The output of array is the weight sum of the observation signals from each array element. The weight vector , where denotes the* k*th weight coefficient and the output of array is expressed as
The output power of array is presented as
where denotes the mathematical expectation, and is the covariance matrix of array snapshot.

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

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 [6]: where denotes the uncertainty factors of ASV and denotes the actual ASV.

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

Equation (8) can be transformed to SOCP problem, to be solved by internal point method (IPM) algorithm. And then this algorithm is deduced further to reduce the computational complexity by general iterative method [6].

#### 3. The Proposed Algorithm

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 (7), is to ensure the output gain of beamforming no less than unity within the error range of ASV, or in other words to form flat response within the main beam. Therefore, (9) can be transformed to where denotes the main beam width, denotes the ripple of main beam, and denotes the weight vector of expected flat response beamforming.

From the definition of , we know that

Using the 3rd constraint of (10), then

By using the constraint again, hence we have

Thus according to (12) and (13), we can get

Multiplying the constant in both sides of the 3rd constraint of (10),

Since , then

According to the property of vector norm,

After simplification,

According to the symmetry of signal in both space and frequency domains, if is treated as a “signal” with point time sequence, then the physical meaning of (14) is that the energy interval of signal is . Taking as the coefficient of -order filter, then the filter is a band-pass one with center frequency being . Since , the maximum gain of the filter is unity. Combining the physical meaning of (14) and (18), the energy interval of output signal is after the signal goes through band-pass filter, of which maximum gain is unity.

The position of peak value of “signal” in frequency domain corresponds to the position of main beam in space domain. Consider the worst case when the energy of input “signal” is while the output energy of signal through the filter is . So the maximum attenuation of signal through filter in frequency domain is . Assume that the attenuation of filter is equal to or less than , and the corresponding frequency range is , so the main peak of “signal” must drop in the range of .

Assume that the corresponding angle of ASV of real main beam in space domain is ; then the value interval of is , according to the symmetry of space domain and frequency domain. Therefore, the real range of main beam can be easily obtained by solving the arc-sin function.

So, the parameters of main beam width, and , can be achieved by solving

At last, amending (10) using the main beam width information obtained from (19), RAB-WC-NC algorithm can be easily drawn:

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

Hence, (4) can be transformed to

Thus, (20) can be rewritten as As mentioned before, (23) can be solved by IPM algorithm as well.

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

*Worst-Case and Norm Constraint-Based Robust Adaptive Beamforming Algorithm*

*Step 1. *Use (19) to calculate the main beam width [].

*Step 2. *Adopt IPM algorithm to solve (24) to obtain the RAB-WC-NV algorithm weight vector:

#### 4. Simulation

In our simulations, a ULA of array elements spaced a half wavelength apart is used. Desired signal and interference are both far-field narrow-band signal; the additive noise is modeled as a complex circularly symmetric Gaussian zero-mean spatially and temporally white process. RAB-WC-NC algorithm is compared with WCPO, norm constraint Capon beamforming (NCCB), SMI and LSMI algorithm with different snapshots, input SNR, and errors of ASV.

*Example 1 (RAB-WC-NC algorithm with different error upper bonds and sufficient snapshots). *We assume that the orientation of desired signal is , SNR of signal is 10 dB, and directions of arrival (DOA) of two interferences are , separately, the INR of which is 40 dB, while snapshots are . Figure 1(a) shows the beam patterns of RAB-WC-NC algorithm with different error upper bonds . As illustrated in Figure 1(a), RAB-WC-NC algorithm can still form effective beams when the receipt signal contains comparatively strong desired signal. With the increase of , the main beam width in the beam patterns of RAB-WC-NC algorithm widens accordingly and forms nulls in several interference points, the depth of which can be −55 dB, satisfying the requirement of interference suppression. Figure 1(b) shows the main beam of RAB-WC-NC algorithm.