Robust Adaptive Beamforming Based on Worst-Case and Norm Constraint

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 covariancematrix 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.


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][5][6][7][8][9][10][11][12][13][14][15][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][6][7][8][9][10][11][12][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][15][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.

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 ( 1 ,  2 , . . .,   ), and then the receiving data of array x() can be expressed as where x() = [ The output power of array is presented as where {⋅} denotes the mathematical expectation, and R  = {x()x  ()} 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, min where a( 0 ) 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  0 denotes the uncertainty factors of ASV and ã( 0 ) 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].

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 min 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 min 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 a(  ), 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 International Journal of Antennas and Propagation 3 Multiplying the constant ‖a  ( 0 )‖ 2 in both sides of the 3rd constraint of (10), Since ‖a  ( 0 )‖ 2 = 1, then According to the property of vector norm, After simplification, According to the symmetry of signal in both space and frequency domains, if ã( 0 ) is treated as a "signal" with  point time sequence, then the physical meaning of ( 14) is that the energy interval of signal ã( 0 ) is [1− 0 , 1+ 0 ].Taking a( 0 ) as the coefficient of -order filter, then the filter is a bandpass one with center frequency being  0 .Since ‖a  ( 0 )‖ 2 = 1, the maximum gain of the filter is unity.Combining the physical meaning of ( 14) and (18), the energy interval of output signal is [1 −  0 , 1 +  0 ] after the signal ã( 0 ) goes through band-pass filter, of which maximum gain is unity.
The position of peak value of "signal" ã( 0 ) in frequency domain corresponds to the position of main beam in space domain.Consider the worst case when the energy of input "signal" ã( 0 ) is 1 +  0 while the output energy of signal through the filter is 1 −  0 .So the maximum attenuation of signal through filter in frequency domain is (1 −  0 )/(1 +  0 ).Assume that the attenuation of filter is equal to or less than (1 −  0 )/(1 +  0 ), and the corresponding frequency range is [  ,   ], so the main peak of "signal" ã( 0 ) must drop in the range of [  ,   ].
Assume that the corresponding angle of ASV of real main beam in space domain is θ0 ; then the value interval of sin( θ0 ) 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 [  ,   ].

Simulation
In modeled as a complex circularly symmetric Gaussian zeromean 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 0 ∘ , SNR of signal is 10 dB, and directions of arrival (DOA) of two interferences are −20 ∘ , 10 ∘ separately, the INR of which is 40 dB, while snapshots are 20 ×  = 640.Figure 1(a) shows the beam patterns of RAB-WC-NC algorithm with different error upper bonds  0 .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  0 , 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.
Example 2 (RAB-WC-NC algorithm with different error upper bonds and finite snapshots).As illustrated in Figure 2(a), with the decrease of snapshots, nulls produced by RAB-WC-NC algorithm with different main beam widths become slightly shallower while the depth of them still meets the requirement of interference suppression.Figure 2(b) shows the main beam of RAB-WC-NC algorithm.
Example 3 (beam patterns with high SNR).We assume that the orientation of desired signal is 0 ∘ , orientation of actual desired signal is 0. , it can be seen that the depth of nulls of RAB-WC-NC algorithm with finite snapshots tends to be slightly smaller than it with sufficient snapshots, which nevertheless can still meet the requirement of interference suppression; NCCB algorithm, SMI algorithm, and LSMI algorithm all share the phenomena that the depth of nulls becomes smaller and the sidelobe tends to be higher, leading to the degradation of performance.
Example 4 (output SINR with different error upper bonds).We assume that DOAs of two interferences are −20 ∘ , 10 ∘ separately, the INR of which is 50 dB, SNR of signal is 10 dB, the snapshots are 20 ×  = 640, and 500 times of Monte Carlo experiments have been done.Figure 4 shows the output SINR versus different error upper bond  0 , of RAB-WC-NC algorithm, WCPO algorithm, NCCB algorithm, SMI algorithm, and LSMI algorithm.From Figure 4, it can be seen that the output SINR of RAB-WC-NC algorithm remains basically unchanged with the increase of error upper bond  0 , which means that RAB-WC-NC algorithm possesses outstanding adaptability against the ASV errors.The output  SINR of NCCB algorithm and WCPO algorithm slightly decrease with the increase of error upper bond  0 because they cannot retain the maximum gain to desired signal.Due to the appearance of target signal cancellation, the output SINR of SMI algorithm and LSMI algorithm decrease drastically with the increase of error upper bond  0 .Due to the diagonal loading, the performance of LSMI algorithm slightly exceeds that of SMI.
Example 5 (output SINR with different input SNR).We assume that DOAs of two interferences are −10 ∘ , 20 ∘ separately, the INR of which is 50 dB, the DOA of desired signal is 1 ∘ , the snapshots are 20 ×  = 640, and 500 times of Monte Carlo experiments have been done.seen that the performances of the proposed beamformers are always close to the optimal SINR in a large range from −20 dB to 40 dB.Furthermore, the proposed algorithms enjoy much faster convergence rates than others.

Conclusion
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.
3 ∘ , SNR of signal is 30 dB, DOAs of two interferences are −20 ∘ , 10 ∘ separately, the INR of which is 40 dB, the error upper bond  0 = 0.3, and LNR of LSMI algorithm is 10 dB.Figure 3(a) demonstrates the comparison among the beam patterns of RAB-WC-NC algorithm, NCCB algorithm, SMI algorithm, and LSMI algorithm at the snapshots of 2 ×  = 64.As shown in Figure 3(b), the array gain of RAB-WC-NC algorithm at 0.3 ∘ loses 0.02 dB compared to it at 0 ∘ , but the gain of desired signal suffers almost no loss.In contrast, the array gain of NCCB algorithm in the direction of 0.3 ∘ loses more than 1.5 dB compared to the maximum one and the SMI algorithm and the LMSI algorithm both produce nulls in the direction of 0.3 ∘ .Figure 3(b) shows the comparison among the beam patterns of RAB-WC-NC algorithm, NCCB algorithm, SMI algorithm, and LSMI algorithm at the snapshots 2 ×  = 64.From Figure 3(b)