This Letter solves steering vector estimation under mismatch for adaptive beamforming. The proposed beamformer implements a stepwise estimation of steering vector, and zone orthogonal constraint is added first based on adaptive constraint framework from Khabbazibasmenj [IEEE Trans. Signal Process. 60(6), 2974–2987 (2012)], which ensures that the estimated steering vector does not converge to an interference steering vector outside the predefined sector, even if the sector deviates from the central observation area of arrays. Then uncertainty set error constraint is used to improve the estimation accuracy. The numerical simulation and experimental results verify the effectiveness of the proposed method.

Topics
Optical signal processing, Signal-to-noise ratio, Newton Raphson method, Numerical linear algebra, Mathematical optimization, Optimization problems, Calculus of variations, Regression analysis, Descriptive statistics, Covariance and correlation

Robust adaptive beamforming is a popular research direction in wireless communications, radar, sonar, and other fields.1 The conventional scheme for achieving adaptive beamforming assumes that the steering vector of the desired signal is precisely known, such as the minimum variance distortionless response (MVDR) algorithm2 and diagonal loading (DL) algorithm.3 However, the traditional adaptive beamforming cannot provide enough robustness to maintain high performance when array signal modal errors arise due to array perturbations and steering vector error.4,5

Various robust adaptive beamforming techniques have been developed to solve the problem, including the projection beamforming algorithm6 and the eigenspace-based beamforming algorithm.7 However, these algorithms require the signal subspace to be known and operated under a high signal-to-noise ratio (SNR). Another adaptive beamforming approach converts the steering vector mismatch problem into convex optimization via imposing constraints. Reference 8 applies an uncertainty set upper bound to the steering vector by the norm of the mismatch vector. However, the upper bound is difficult to determine in practical applications, affecting the robustness of the method. Sequential quadratic programming (SQP) beamformer9 is proposed to improve the robustness by iteratively solving a quadratic convex optimization, which constructs two orthogonal subspaces that one contains desired signal vector and the other contains steering vectors of all interfering signals. Then norm constraints are used to search the actual steering vector. However, it needs enough prior information, including preset steering vectors of the signal of interest (SOI) and the array structure. In Ref. 10, a robust adaptive beamforming based on steering vector estimation (SV-RAB) is presented to overcome the mismatches. It guarantees that the steering vector does not converge to interference steering vectors by maximizing the beamformer output power and reduces the requirement for presumed steering vector. Then, other quadratic constraints are proposed based on the methods in Refs. 11 and 12, respectively. The shortcoming of the SV-RAB method is that the constraint factor mismatches the rules of constraint terms in different observation sectors, which means that the estimated steering vector will deviate from the real steering vector when the preset observation sector deviates from the central observation area of the array.

In this Letter, an improved beamformer is proposed using zone orthogonal constraint and steering vector estimation to overcome the constraint term mismatch problem, and the corresponding direction of arrival (DOA) estimation result can also be obtained from the steering vector. This Letter is outlined below. Section 2 reviews the problem formulation. The proposed method is presented in Sec. 3. Simulation and experimental results are obtained and analyzed in Sec. 4. Finally, Sec. 5 concludes this work.

Considering a uniform linear array (ULA) of sensors, the corresponding optimization problem proposed by the MVDR beamformer2 is
min w w H R xx w ,         s . t .     w H a ( θ 0 ) = 1 ,
(1)
where a ( θ 0 ) = [ 1 , e − j π   cos   θ 0 , … , e − j ( M − 1 ) π   cos   θ 0 ] T is the steering vector of a desired signal impinged from θ 0⁠, and argument θ 0 is omitted below unless necessary. w is the MVDR weight vector to be designed, R x x = E [ x x H ] is the covariance matrix, and x is observed data. The solution for Eq. (1) can be obtained via Lagrange multiplier method,13 
w o = R − 1 x x a a H R − 1 x x a .
(2)
The actual R x x is not accessible in practical applications, and the sample covariance matrix R ̂ x x is used to replace R x x by
R ̂ x x = 1 N ∑ n = 1 N x ( n ) x H ( n ) ,
(3)
where N is the number of snapshots.

MVDR algorithm has excellent performance when the steering vector is accurately known. However, its performance degrades severely once there is a mismatch, which means a ̂ = a ¯ + e ,   a ̂ ,   a ¯ , and e denote the real steering vector, assumed steering vector, and error steering vector, respectively.

Reference 10 proposes the SV-RAB method to overcome the mismatch problem. It constructs two orthogonal subspaces: one contains the steering vector of the desired signal, while all the steering vectors of the interference signals are in the other subspace. In order to ensure that the estimated steering vector does not converge to the interference steering vectors outside the observation sector, the observation sector is constrained by a quadratic constraint. This optimization problem can be modelled as
min a ̂   a ̂ H R ̂ x x − 1 a ̂ s . t .       | | a ̂ | | 2 = M ,   a ̂ H C ¯ a ̂ ≤ ζ ,
(4)
where C ¯ = ∫ Θ ¯ a ¯ ( θ ) a ¯ ( θ ) H d θ ,   Θ ¯ denotes the complementary interval of the interested observation sector Θ = ( θ 1   ,   θ 2 ) , ζ = Δ   max θ ∈ Θ a ¯ ( θ ) H C ¯ a ¯ ( θ ) is constraint factor. It is guaranteed that Θ contains the direction of the desired signal and all other interference signal impinges on the ULA from Θ ¯⁠.
Semi-definite programming (SDP) relaxation is used to provide a solution to the non-convex issue in Eq. (4). Let A   = Δ a ̂ a ̂ H⁠, then Eq. (4) is rewritten as
min A   T r ( R ̂ x x − 1 A ) s . t .       T r ( A ) = M ,   T r ( C ¯ A ) ≤ ζ ,     A ≥ 0   ,
(5)
where T r ( · ) is trace operation.

The kernel constraint of the SV-RAB method is a ̂ H C ¯ a ̂ ≤ ζ . It can be found that the constraint value is only related to the observation sector Θ. The values of the quadratic term a ( θ ) H C ¯ a ̂ ( θ ) under different azimuths are shown in Fig. 1, where Fig. 1(a) shows the constraint results in the case of Θ = ( 80 ° ,   110 ° )⁠, the black line indicates the range of the observation sector Θ = ( 80 ° ,   110 ° )⁠, and the red line indicates the constraint range under ζ = Δ max θ ∈ Θ a ¯ ( θ ) H C ¯ a ¯ ( θ ) . It can be seen that the estimated steering vector must be searched within the predefined observation sector. However, when the observation sector deviates from the centre of the array (⁠ Θ = ( 120 ° ,   150 ° )⁠), as shown in Fig. 1(b), the search range of the constraint a ̂ H C ¯ a ̂ ≤ ζ is encapsulated in part of the interval of the complementary sector Θ ¯⁠, i.e., θ ∈ ( 30 ° ,   120 ° )⁠, which means that the estimation of SV-RAB method may converge to the interference vectors outside the predefined sector.

FIG. 1.
Values of the a ̂ H ( θ ) C ¯ a ̂ ( θ ) for different observation sector Θ. (a) Extreme value constraint in Θ = ( 80 ° ,   110 ° ), (b) Extreme value constraint in Θ = ( 120 ° ,   150 ° ).
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Values of the a ̂ H ( θ ) C ¯ a ̂ ( θ ) for different observation sector Θ. (a) Extreme value constraint in Θ = ( 80 ° ,   110 ° )⁠, (b) Extreme value constraint in Θ = ( 120 ° ,   150 ° )⁠.

FIG. 1.
Values of the a ̂ H ( θ ) C ¯ a ̂ ( θ ) for different observation sector Θ. (a) Extreme value constraint in Θ = ( 80 ° ,   110 ° ), (b) Extreme value constraint in Θ = ( 120 ° ,   150 ° ).
View largeDownload slide

Values of the a ̂ H ( θ ) C ¯ a ̂ ( θ ) for different observation sector Θ. (a) Extreme value constraint in Θ = ( 80 ° ,   110 ° )⁠, (b) Extreme value constraint in Θ = ( 120 ° ,   150 ° )⁠.

Close modal

To improve the robustness of the SV-RAB method, this Letter proposes a robust beamforming method based on zone orthogonal constraint and steering vector estimation (ZSV-RAB).

The zone orthogonal constraint is imposed to replace the extreme constraint T r ( C ¯ A ) ≤ ζ in the optimization problem Eq. (5), which can be written as
VA = 0 ,
(6)
and
V = I − U U H ,
(7)
C = ∫ θ 1 θ 2 a ¯ ( θ ) a ¯ ( θ ) H d θ ,
(8)
where V is an orthogonal projection onto the C⁠, and U is a set of dominant eigenvectors of C⁠. According to the definitions of A and C ,   V is the projective subspace orthogonal to the semi-positive definite matrix A containing the actual steering vectors, thus the zone orthogonal constraint can force the estimation to always converge to the steering vector within the observation sector Θ = ( θ 1 ,   θ 2 ) .

The strength of constraint Eq. (6) can be adjusted with the integral limit, and the orthogonality is always valid regardless of the choice of the observation sector, which means that it is more flexible and robust than constraint T r ( C ¯ A ) ≤ ζ . The new optimized strategy is still convex, and the relaxed solution a ̂ c can be solved efficiently with the CVX toolbox.14 

It is worth noting that the relaxed problem of Eq. (5) may not have an optimal solution because of the removal of rank-one constraint rank ( A ) = 1⁠. Fortunately, even if A is not rank-one, the rank-one solution can be found using the rank reduction technique.10 Considering that the rank-one solution a ̂ c obtained by Eq. (5) may deviate from the optimal solution, the uncertainty set error constraint is imposed to improve the estimation accuracy
min e ̂   ( a ̂ c + e ̂ ) H R ̂ x x − 1 ( a ̂ c + e ̂ )   s . t .   | | e ̂ | | 2 = κ ,
(9)
where κ is the upper bound of error and is used to constrain the range of uncertain sets in which the actual steering vector is located. The high bound of the constraint error κ increases the tolerance range of the uncertainty set, resulting in the modified steering vector converging to the interference steering vector. In this Letter, the optimized strategy preliminarily corrected the steering vector error. Therefore, the upper bound does not need to be too large, κ ∈ ( 0 ,   1.5 ) is considered a proper value boundary and κ = 1 is chosen in the simulations. Equation (9) can be solved via Lagrange multiplier methodology,
f ( e ̂ , λ ) = ( a ̂ c + e ̂ ) H R ̂ x x − 1 ( a ̂ c + e ̂ ) + λ ( | | e ̂ | | 2 − κ ) ,
(10)
where λ ≥ 0 is the Lagrange multiplier, the partial differential concerning e ̂ can be obtained as
∇ f ( e ̂ , λ ) = ∂ f ( e ̂ , λ ) ∂ e ̂ = R ̂ x x − 1 ( a ̂ c + e ̂ ) + λ e ̂ .
(11)
Then the estimation error e ̂ can be expressed via setting Eq. (10) equal to 0,
e ̂ = − ( R ̂ x x − 1 + λ I ) − 1 R ̂ x x − 1 a ̂ c ,
(12)
where the value of λ can be obtained via Newton's method.15 
Combining Eqs. (5), (6), and (12), the final steering vector estimation result a ̂ f n can be obtained as
a ̂ f n = a ̂ c + e ̂ .
(13)

In our simulations, a 10-element ULA receives the narrowband signal, and two random interference signals are arrived at from 70° and 150°, respectively. The corresponding interference-to-noise ratio (INR) is 40 dB. Observation sector is set as Θ = ( 120 ° , 150 ° )⁠, the presumed target direction is 125°, and the actual target direction is 130° with SNR = 0 dB and N = 100. The proposed method is compared with the MVDR,2 DL,3 and SV-RAB10 method, the load factor μ = 10 is used for DL, The parameter ζ of SV-RAB is set automatically by the selection rule and κ = 1 is set for the proposed method.

Figure 2 illustrates the beam patterns and output SINR vs SNR in the case of Θ = ( 120 ° , 150 ° )⁠. From Fig. 2(a), MVDR and DL algorithms failed to align the actual direction, SV-RAB method points to 70° of the interference signal outside the observation area. This result is consistent with the analysis of the constraint range in Fig. 1(b). The constraint term of SV-RAB is disabled to search the steering vector of SOI, causing the estimation converges to the strong interference steering vector impinged from 70°. In contrast, the ZSV-RAB method proposed in this Letter can still align the estimated result with the real orientation by adding zone orthogonal constraint. In Fig. 2(b), the ZSV-RAB method exhibits better robustness than other methods, where 100 Monte Carlo trials are operated. Notably, the SINR of the SV-RAB method decreases seriously, which is caused by the deviation of the estimated steering vector from the SOI. SV-RAB method still performs well when the observation sector is in the centre of arrays.

FIG. 2.
Comparison of the beam pattern and output SINR in the case of Θ = ( 120 ° , 150 ° ). (a) Beam pattern, (b) Output SINR versus SNR.
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Comparison of the beam pattern and output SINR in the case of Θ = ( 120 ° , 150 ° )⁠. (a) Beam pattern, (b) Output SINR versus SNR.

FIG. 2.
Comparison of the beam pattern and output SINR in the case of Θ = ( 120 ° , 150 ° ). (a) Beam pattern, (b) Output SINR versus SNR.
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Comparison of the beam pattern and output SINR in the case of Θ = ( 120 ° , 150 ° )⁠. (a) Beam pattern, (b) Output SINR versus SNR.

Close modal
In order to analyze the influence of observation sector Θ, root mean square error (RMSE) against different observation sectors is presented in Fig. 3. The corresponding parameters are set as follows: SNR is set to be 0 dB, presumed target direction θ s varies from 25° to 155°, actual target direction is ( θ s + 5 ° ) , and the range of observation sector is ( θ s − 15 ° , θ s + 15 ° )⁠, the other parameter settings remain unchanged, where 100 independent Monte Carlo trials are performed. The RMSE16 is
RMSE = 1 S T ∑ s = 0 S − 1 ∑ t = 1 T | θ s − θ ̂ s , t | 2 ,
(14)
where S is target number, T is number of Monte-Carlo trails, θ s is the actual direction for the sth target, and θ ̂ s , t is the corresponding DOA estimation at the tth trail.
FIG. 3.
RMSE against different observation sectors.
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RMSE against different observation sectors.

FIG. 3.
RMSE against different observation sectors.
View largeDownload slide

RMSE against different observation sectors.

Close modal

From Fig. 3, MVDR and DL methods cannot correct the steering vector in the presence of a mismatch. The SV-RAB method is able to correct the steering vector in the central observation area of the array to achieve an effective estimation, but the performance decreases as Θ deviates from the centre of the array. The ZSV-RAB method proposed in this Letter can keep excellent estimation accuracy in any observation sector.

To verify the reliability of the proposed method, we conducted sea trials in the South China Sea in 2020. The seabed array is a 16-element ULA with a spacing of 4 m between the array elements. The received data are processed in the frequency domain with the frequency band of f  = (90 187) Hz, and the corresponding cross-spectral density matrix consists of 16 snapshots. For the performance comparison, Fig. 4(a) shows the azimuth result of the MVDR algorithm. There are four targets in the entire observation sector: two strong targets near 40° and 80° and two weak targets near 90° and 140°, SV-RAB and ZSV-RAB are used to process the observation sector where the weak target is located.

FIG. 4.
Experimental results for different methods. (a) MVDR result within the entire observation sector, (b) SV-RAB result within Θ = ( 85 ° , 115 ° ), (c) ZSV-RAB result within Θ = ( 85 ° , 115 ° ), (d) SV-RAB result within Θ = ( 120 ° , 150 ° ), (e) ZSV-RAB result within Θ = ( 120 ° , 150 ° ).
View largeDownload slide

Experimental results for different methods. (a) MVDR result within the entire observation sector, (b) SV-RAB result within Θ = ( 85 ° , 115 ° )⁠, (c) ZSV-RAB result within Θ = ( 85 ° , 115 ° )⁠, (d) SV-RAB result within Θ = ( 120 ° , 150 ° )⁠, (e) ZSV-RAB result within Θ = ( 120 ° , 150 ° )⁠.

FIG. 4.
Experimental results for different methods. (a) MVDR result within the entire observation sector, (b) SV-RAB result within Θ = ( 85 ° , 115 ° ), (c) ZSV-RAB result within Θ = ( 85 ° , 115 ° ), (d) SV-RAB result within Θ = ( 120 ° , 150 ° ), (e) ZSV-RAB result within Θ = ( 120 ° , 150 ° ).
View largeDownload slide

Experimental results for different methods. (a) MVDR result within the entire observation sector, (b) SV-RAB result within Θ = ( 85 ° , 115 ° )⁠, (c) ZSV-RAB result within Θ = ( 85 ° , 115 ° )⁠, (d) SV-RAB result within Θ = ( 120 ° , 150 ° )⁠, (e) ZSV-RAB result within Θ = ( 120 ° , 150 ° )⁠.

Close modal

Figures 4(b) and 4(c) are the experimental results in the Θ = ( 85 ° , 115 ° )⁠, it can be seen that estimated target orientation of the two methods is consistent with the MVDR result, both methods can effectively constrain the estimated steering vector within the observation sector, even if there is strong interference (80°) near the observation sector. Figures 4(d) and 4(e) are the results in the Θ = ( 120 ° , 150 ° )⁠, while the observation sector offsets from the central area of the array, and the estimation result of SV-RAB converges to the strong interference steering vector (80°) because of invalid constraint, which is also consistent with the theoretical analysis in Fig. 1(b). As shown in Fig. 4(e), ZSV-RAB method avoids this problem through zone orthogonal constraint and uncertain set error constraint, and can still accurately estimate the weak target trajectory within Θ = ( 120 ° , 150 ° )⁠.

This Letter proposes the ZSV-RAB method to achieve DOA estimation via steering vector estimation, and zone orthogonal constraint and uncertain set error constraint are utilized to guarantee the steering vector does not converge to interference steering vectors. Simulations and experimental results verify the effectiveness of the proposed method.

This work was supported by the National Natural Science Foundation of China (Grant No. 52071111), and the Open fund of the National Key Laboratory of Science and Technology on Underwater Acoustic Antagonizing (JCKY2022207CH12).

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