Robust adaptive beamforming against large steering vector mismatch based on minimum dispersion

Digital Signal Processing 99 (2020) 102676

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Digital Signal Processing www.elsevier.com/locate/dsp

Robust adaptive beamforming against large steering vector mismatch based on minimum dispersion Yang Feng a,∗ , Guisheng Liao b , Jingwei Xu b a b

Nanjing Research Institute of Electronics Technology, Nanjing 210039, China National Laboratory of Radar Signal Processing, Xidian University, Xi’an 710071, China

a r t i c l e

i n f o

Article history: Available online 22 January 2020 Keywords: Robust adaptive beamforming Minimum dispersion Worst-case performance optimization Large steering vector mismatch Non-Gaussian signals

a b s t r a c t Many efforts have been recently devoted to adaptive beamformers based on the minimum dispersion (MD) criterion for non-Gaussian signals, which results in improved signal-to-noise ratio performance. However, large steering vector mismatch and inappropriate choice of uncertainty set will cause performance degradation to the existing MD-based methods. To address these issues, we develop a robust MD-based beamformer against large steering vector mismatch, which utilizes multiple small uncertainty sets to cover the whole uncertainty region. The  p − norm (p ≥ 1) minimization problem turns to be nonlinear and nonconvex because of these multi-uncertainty-set constraints. To solve the problem, a projected gradient algorithm is utilized to transform the original problem into a nonconvex quadratically constrained quadratic programming (QCQP) problem. The semidefinite programming relaxation technique is then employed in each iteration to deal with the nonconvex QCQP problem. Numerical results demonstrate that the proposed beamformer offers a significant performance improvement in case of large steering vector mismatch for non-Gaussian signals compared with the conventional schemes. © 2020 Elsevier Inc. All rights reserved.

1. Introduction Adaptive beamforming is an important task in array signal processing which has been widely used in radar, sonar, wireless communications, audio signal processing and many other areas [1–4]. The minimum variance distortionless response (MVDR) beamformer [5] is a classical data-dependent beamformer, which minimizes the variance of the array output while keeping the magnitude response of the desired signal to be unity. However, it is well known that the MVDR beamformer is quite sensitive to array imperfections such as look direction mismatch, signal source movement, imperfect array calibration, and distorted antenna shape. What is more, in practical applications, the presence of the desired signal component in the training data will lead to performance degradation, even the steering vector of the desired signal is exactly known [6]. Furthermore, the performance degradation will result from the difference between the actual and estimated covariance matrices because of the small training sample size [7]. To improve the robustness of the MVDR beamformer, many solutions have been developed. For direction-of-arrival (DOA) mis-

*

Corresponding author. E-mail address: [email protected] (Y. Feng).

https://doi.org/10.1016/j.dsp.2020.102676 1051-2004/© 2020 Elsevier Inc. All rights reserved.

match, the linearly constrained minimum variance (LCMV) beamformer [8–11], which adds extra linear constraints to the MVDR beamformer to broaden the main beam of the beampattern, was proposed. However, this method cannot handle arbitrary steering vector mismatch. The diagonal loading approach [7] is a widely used method to improve the robustness for any mismatch types. However, its performance depends on the amount of the loading factor which is difficult to determine [12]. The eigenspace-based approach [13,14] is a powerful method, however, the performance will degrade severely when the number of interferences increases and the signal-to-noise ratio (SNR) decreases. In [15–19], beamformers based on worst-case performance optimization, which are robust against arbitrary steering vector mismatch, have been proposed. In [15], the beamformer is based on quadratically constrained minimum variance (QCMV) and the constraint is used to force the magnitude responses of the steering vector in a sphere uncertainty set to exceed unity. In [16,17], an ellipsoidal uncertainty set was utilized. In [18], an arbitrary convex set was considered. The probabilistically constrained robust beamforming method proposed in [19] has been proved to be a variant of the worstcase-based approach. In recent years, beamformers for specific scenarios were proposed [20–24]. In [20], the robust beamformer based on steering vector estimation with as little as possible prior information was proposed, which prevents the presumed steering vector from converging to interferences or their linear combina-

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Y. Feng et al. / Digital Signal Processing 99 (2020) 102676

tions through imposing a quadratic constraint. In [21], the beamformer based on reconstructing the interference-plus-noise covariance matrix via the spatial spectrum distribution was proposed but it is only robust against the DOA mismatch. In [22,23], two improved beamformers based on reconstructing the interferenceplus-noise covariance matrix were proposed, which can be used for the arbitrary steering vector mismatch. In [24], a robust beamformer which can control the main beam magnitude response precisely was presented. In addition, [25] developed a doubly constrained robust Capon beamformer, which only exploits principal eigenpairs. In [26], a signal-blocking based robust adaptive beamforming method in the presence of pointing errors which rejects coherent and incoherent interferences without requiring any information on interference directions was proposed. In [27], a robust beamformer based on part of the array output power associated with a subspace spanned by eigenvectors corresponding to some largest eigenvalues of the received signal covariance matrix was developed. It is worth noting that all of the above-mentioned robust beamformers are based on the minimum variance (MV) criterion which is statistically optimal under Gaussian signal assumption. However, many signals encountered in practice are usually non-Gaussian, such as speech signals as well as digitally modulated signals in radar, sonar and wireless communications [28–30]. According to the kurtosis of a distribution, zero mean random signals can be categorized into three classes, i.e., super-Gaussian, Gaussian and sub-Gaussian [31]. To address all these cases, the minimum dispersion distortionless response (MDDR) beamformer is proposed in [32], which minimizes the  p − norm (p ≥ 1) of the array output while requiring the magnitude response of the desired signal to be unity. The general guideline is that the optimal choice of p is case-dependent [32]. It relies on the statistical properties of the signals, that is, p > 2 for sub-Gaussian signals, p < 2 for superGaussian signals and p =2 for Gaussian signals. This is because that the second-order statistics cannot describe the probability density function of the non-Gaussian signals completely. The power of these signals may be contained in the lower or the higher order statistics. Particularly, for super-Gaussian signals, i.e., p < 2, the smaller value of p leads to a better performance. In other words, the more impulsive the signals are, the smaller value of p is preferred. In [33], a nonconvex linear regression based minimum dispersion (MD) beamformer has been proposed to achieve significant performance improvement for super-Gaussian signals. In [34], a widely linear MD beamforming method for the sub-Gaussian noncircular signals was proposed. To improve the robustness of the MDDR beamformer, the linearly constrained minimum dispersion (LCMD) beamformer [32] and quadratically constrained minimum dispersion (QCMD) beamformer [35] have been proposed. In [36], a phase-only robust MD beamformer was devised for nonGaussian signals, which employs a constant-modulus constraint on the weight vector. In [37,38] robust beamformers against fastmoving interferences based on the MD criterion were proposed, which can eliminate the power of the interference from the output through constraining the average power of the interference’s dynamic angular sector to zero. What is more, in [39], adopting the MD criterion at the receiver of multiple-input multiple-output (MIMO) radar, a MD based beamformer was developed, which implicitly exploits non-Gaussianity and hence improves the performance of the beamformer. In practice, there are some situations that the presumed target steering vector is within a very large uncertainty set. This can be due to maneuvering of targets or non-consecutive tracking when phased array radar detects and tracks the non-cooperative targets. If significant steering vector mismatch happens, as we have described in [40], these methods require a large uncertainty set to cover the possible area of the steering vector. However, the

Euclidean norm of the uncertainty set is upper-bounded, the robustness cannot be guaranteed if the mismatch vector exceeds the uncertainty set. In addition, the increasing set may lead to a trivial solution. This motivates us to develop a beamformer which is robust against large steering vector mismatch for non-Gaussian signals. In [40], we have proposed a method to handle large steering vector mismatch problem for Gaussian signals. To solve the nonconvex problem, two approaches, namely, iterative semidefinite programming-based robust adaptive beamforming (ISDP-RAB) and iterative linearization-based robust beamforming (IL-RAB) methods were devised. In this paper, we consider the large steering vector mismatch and the non-Gaussian distribution case, and a novel beamformer based on the minimum dispersion (MD) criterion is devised. The main contributions of our work can be summarized as follows. i) The problem of robust MD-based beamformer against large steering vector mismatch using multiple uncertainty sets is formulated. This method utilizes multiple small uncertainty sets to cover the whole feasible region instead of a single one. The ideal is novel and the beamformer can be seen as an extended form of the LCMD beamformer. Our proposed beamformer outperforms the MDDR, LCMD, and QCMD beamformers in the presence of significant steering vector mismatch, while maintaining similar performance in small mismatch. ii) The projected gradient algorithm is utilized to transform the nonconvex  p − norm minimization problem into a nonconvex quadratically constrained quadratic programming (QCQP) problem. In each iteration, we first introduce an auxiliary variable, recast the problem in high dimension, and then use the semidefinite programming (SDP) relaxation technique to deal with the transformed nonconvex quadratic programming problem. iii) We present a criterion for selecting the number of the uncertainty sets and their sizes which are two crucial parameters of our approach. Generally, the size of each small uncertainty set is optimally chosen based on the performance metric of worst-case performance optimization, while the number of sets is determined based on the performance optimization multiple-constraint beamformer. The rest of this paper is organized as follows. In Section 2, we present the system model and a brief introduction to the MVDR, MDDR, LCMD and QCMD beamformers. In Section 3, a robust beamformer with multiple small uncertainty sets based on the MD criterion to address the large steering vector uncertainty problem is proposed. In Section 4, a projected gradient algorithm is developed for efficient implementation of the beamformer. Simulation results are presented in Section 5. Finally, conclusions are drawn in Section 6. 2. Problem formulation 2.1. Signal model Consider an array consisting of N omnidirectional sensors. The signal vector received by the antenna array at the tth time instant can be mathematically expressed as

x (t ) = s (t ) + i (t ) + n (t )

(1)

where s (t ), i (t ) and n (t ) represent the N × 1 complex vector of target, interference, and additive noise, respectively, which are assumed to be statistically independent. Assume the DOA of the signal-of-interest (SOI) is θt , the desired signal received by the array can be expressed as

s (t ) = s (t ) a (θt )

(2)

Y. Feng et al. / Digital Signal Processing 99 (2020) 102676

where s (t ) is the desired signal waveform and a (θt ) ∈ C N ×1 is the associated steering vector. In addition, depending on the array configuration, the steering vector has different forms. The beamformer output is given as H

y (t ) = w x (t )

(3)

N ×1

where w ∈ C is the weight vector of the beamformer, (•) stands for the Hermitian transpose operation. The goal of the datadependent beamforming is that the output can preserve the desired signal and meanwhile suppress the interferences and noise. H

2.2. MVDR beamformer The beamformer output signal-to-interference-plus-noise ratio (SINR) is given as

SINR =

H

w Rs w wH Ri+n w

=





2 H 2 s w a (θt ) w H Ri+n w

σ

where Rs ∈ C N × N is the target covariance matrix, Ri+n ∈ C N × N 





is the interference-plus-noise covariance matrix, σs2 = E |s (t )|2 is the target power and E {•} denotes the statistical expectation. The MVDR beamformer is obtained by minimizing the variance of the interference and noise at the output of the beamformer while constraining the target response to be unity, which can be formulated as

min wH Ri+n w

s.t.

w

wH a (θt ) = 1.

(5)

However, Ri+n is unavailable in practice. To implement the MVDR beamformer, Ri+n is replaced by the following data sample covariance matrix

ˆ= R

1 L

XXH

(6) N ×L

where X = [x1 , · · · , x L ] ∈ C is the N × L array data matrix and L is the number of training samples which may be contaminated by the desired signal. Thus, it yields the so-called sampled matrix inverse (SMI) based MVDR beamformer

ˆ min wH Rw w

s.t.

wH a (θt ) = 1.

(7)

It is worth noticing that the covariance matrix Ri+n cannot be estimated accurately if the number of training samples L is small. Problem (7) can be solved by using Lagrange multiplier methodology, i.e., minimizing the following function



ˆ + μ 1 − wH a (θt ) H (w, μ) = wH Rw

Different from the MV-based beamformers, the MDDR beamformer is expressed as





p min E wH x (t )



wH a (θt ) = 1

(10)

where p ≥ 1. Obviously, the MVDR beamformer is a special case of the MDDR beamformer if p = 2. Note that the term





p E wH x (t )

  = E | y (t )| p is known as dispersion of y (t ) in

statistics, which is a generalization of variance [32]. Like the MVDR beamformer, the MDDR beamformer is not robust against steering vector mismatch. To address the DOA mismatch, the LCMD beamformer proposed in [32] uses a set of linear constraints to broaden the mainbeam of the beampattern. This beamformer can be formulated as





p min E wH x (t )

s.t. CH w = f

(11)



 

where, the constrained matrix C= a (θ1 ) , a (θ2 ) , · · · , a θ J ∈ C N × J contains the array responses in the J constrained directions, θ1 , θ2 , · · · , θ J are the constrained directions, which are around the estimated DOA of the SOI, and f ∈ R J ×1 is the response vector whose entries are usually equal to one. The LCMD beamformer inherits all the drawbacks from the LCMV method, e.g., it is only robust against the DOA mismatch. To address the arbitrary steering vector mismatch, the QCMD beamformer is proposed in [35], where the design principle is based on modeling the actual target steering vector a as a sum of a presumed steering vector p and an unknown mismatch vector e. It is assumed that the mismatch vector is in an uncertainty set. The mismatch vector e describes the effect of the target steering vector distortions and is assumed to belong to an uncertainty set. In [35], a sphere uncertainty set is exploited, that is,

E = {e |e ≤ ε }

(12)

where ε is the radius of the sphere and • denotes the Euclidean norm. Therefore, by minimizing the dispersion of the output y (t ) and meanwhile imposing a constraint that the amplitude response of the array response is no smaller than one for all e ∈ E , the QCMD beamformer can be formulated as

min w

s.t.





p E wH x (t )

  H w (p + e) ≥ 1, f or all e ∈ E .

(13)

3. Robust adaptive beamforming against large steering vector mismatch based on minimum dispersion

(8)

where μ ≥ 0 is the Lagrange multiplier. Taking the gradient of (8) with respect to wH and equating it to zero, the closed-form solution to problem (7) can be obtained

ˆ −1 a (θt ) w = μR

s.t.

w

w

(4)

3

(9) −1 ˆ −1 a (θt ) where (•)−1 represents matrix inverse and μ = aH (θt ) R



is the normalization constant. 2.3. MDDR, LCMD and QCMD beamformers To improve the performance of the beamformer when the signals are non-Gaussian distributed, the MD-based beamformers are proposed including the MDDR, LCMD and QCMD beamformer. The MDDR beamformer utilizes the  p − norm of the training samples.

The performance of the robust beamformers based on one uncertainty set decreases dramatically if the mismatch vector e is large and exceeds the uncertainty region [40]. This situation can happen in practical applications. Particularly, considering the following case that large DOA mismatch and other small errors exist simultaneously, a large uncertainty set is required for conventional robust beamformers based on uncertainty set constraint. Large DOA mismatch can be caused by maneuvering of targets or non-consecutive tracking for radar non-cooperative target detection. These small errors can contain, for example, array calibration errors, time varying environment and imperfections of propagation medium. The robustness of these beamformers can be obtained but the ability to suppress the noise and interferences is degraded. To solve such problems, similar to the approach in [40], multiple inequality constraints are imposed to control the array magnitude response, that is

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Y. Feng et al. / Digital Signal Processing 99 (2020) 102676

Fig. 1. Region of multiple uncertainty sets.

 H  w am  ≥ 1, f or all am ∈ Am m = 1, . . . , M

(14)

where Am denotes the mth constraint region and can be described as 

Am = {am |am = pm + em , em  ≤ εm } , m = 1, . . . , M

(15)

where, pm , em and εm represent the presumed steering vector, mismatch vector and norm size of the mth uncertainty set, respectively. M is the number of the uncertainty sets. Fig. 1 shows the union described by (15). According to the definition of the convex set, the union is nonconvex. In addition, beamformers based on the MV criterion are not statistically optimal for non-Gaussian signals. Consequently, according to [32], we utilize the MD criterion, i.e., minimizing the dispersion of the array output. The formulation of the beamformer is expressed as

min w

s.t.





p E wH x (t )

 H  w am  ≥ 1, f or all am ∈ Am m = 1, . . . , M .

(16)

applications, the dispersion of the array output In practical p

H   E w x (t ) is unavailable. Thus, the sample mean is utilized to replace the expectation in (16), that is

min w

s.t.

 H p X w

 H p w am  ≥ 1, f or all am ∈ Am m = 1, . . . , M .

(17)

Let y = [ y (1) , · · · , y ( L )]T be the data vector of the array output, then X H w represents the conjugate of y, and the  p − norm of y is defined as

y p =

 L 

1 / p | y (l)|

p

.

(18)

l =1

To solve the nonconvex problem, following the strategy of [15], applying the triangle inequality and Cauchy-Schwartz inequality, the constraints of problem (17) can be tightened and the problem is converted to

min w

s.t.

 H p X w

 H p w pm  − εm w ≥ 1 m = 1, . . . , M

(19)

It should be pointed out that (19) is a tighter version of (17). Note that the selection of p depends on the distribution of the signals. For Gaussian signals, p is chosen to be 2. For sub-Gaussian signals, the performance of beamformers is better if p > 2. For super-Gaussian signals, 0 < p < 2 should be used. Particularly, for super-Gaussian signals, the smaller value of p leads to a better performance, i.e., the more impulsive the signals are, the smaller value of p is preferred. However, the objective function of (19) is nonconvex if 0 < p < 1, and problem (19) is strongly NP-hard. Because of the mathematical difficulty, we only consider p ≥ 1 in this work. In addition, in our proposed beamformer, it is critical to answer the following questions: i) how to select the number of the uncertainty sets and ii) how to determine their uncertainty bounds. Similar to the choice of constraint matrix in the LCMD beamformer, the number of required uncertainty sets is determined by the uncertainty range of DOA. The size of each set depends on other types of mismatch such as waveform distortion, imperfectly calibrated arrays and distorted antenna shape, which is similar to the QCMD beamformer. Thus, our proposed beamformer can be seen as a combination of the LCMD and QCMD methods. Actually, the number of the constraints should be selected as small as possible due to the fact that the beamformer loses more degrees of freedom to suppress interference and noise if more uncertainty sets are added. The empirically optimal value of the number equals 2 or 3 for the small scale array. Thus, for example, we can utilize the centers of each sets being  three uncertainty    sets with  a θp − 2θ , a θp , a θp + 2θ respectively, where θp is the estimated DOA of the SOI and θ is the DOA mismatch. In addition, as shown in Fig. 4, the requirement for the selection of an appropriate size of the uncertainty sets is quite mild, which is the advantage of our proposed method compared to other single-uncertainty-based beamformers. Therefore, we can choose the size of each set to be 0.3-0.5 for the common phenomenon. Furthermore,   (19) is still a nonconvex problem because of the terms wH pm  , (m = 1, . . . , M ). Note that only if the imaginary part of wH pm equals zero and the real part of wH pm is nonnegative can the absolute operations be discarded. Due to the fact that the objective function of (19) is phase-invariant with respect to w, assuming w0 is the optimal solution to (19) and M equals one, we can rotate the phase of w0 so that wH real 0 p1 is a nonnegative    can be number. Therefore, the absolute operation of term wH p 1 0 ignored and one can say the vector w0 and p1 are in the same direction, i.e., w0 and p1 are collinear. However, if M > 1, although it is easy to rotate w0 so that w0 and p1 are collinear, it is hard to find a determined phase so that w0 can be collinear with pm , (m = 1, . . . , M i.e., it fails to find a same  ) simultaneously,  w which satisfies Im w H pm = 0, (m = 1, . . . , M ) simultaneously, where Im {•} represents the imaginary part of a complex number. This can be explained geometrically, that is, for M non-collinear presumed steering vector pm , (m = 1, . . . , M ), it is hard to find a vector w making them and w be collinear vectors. To solve problem (19), in the next section, we utilize the projected gradient algorithm. 4. Solution to beamforming problem 4.1. Framework of projected gradient algorithm Assume that the feasible region of problem (19) is C , which is denoted by

    C = w wH pm  − εm w ≥ 1, m = 1, . . . , M . Then, problem (19) can be expressed as

(20)

Y. Feng et al. / Digital Signal Processing 99 (2020) 102676

5

Take the kth iteration as an illustration. First, in the absence of the constraint in (21), the steepest descent direction of the objective function in (21) is ∇ f (wk ). Suppose the step size is μk , we can obtain the value of the weight vector in the next iteration, that is

uk+1 = wk − μk ∇ f (wk ) .

Note that the step size μk in the kth iteration can be obtained by utilizing the backtracking line search [41]. Then, take the constraints in (21) into consideration, the obtained vector uk+1 in (24) needs to be projected onto the feasible region C , and we have the final value of the weight vector in the (k + 1) th iteration, that is

Fig. 2. Iterative procedure for finding optimal weight vector.

p



min

f (w) = XH w p

s.t.

w ∈ C.

w



(21)

L L  H p   X w = | y (l)| p = | y (l)| p −2 | y (l)|2 p

l =1

··· .. .

.. .

p −2 2

· · · | y ( L )| =  (w) y2 = wH XD (w) XH w = wH R˜ (w) w



where,  = diag | y (1)|

 p −2

p −2 2

⎤2 | y (1)|   ⎥ ⎢ . ⎥ ⎥ ⎣ . ⎦  . ⎦  | y ( L )|  ⎤⎡

0

, · · · , | y ( L )|

p −2 2



w0 = (22)



, D = diag | y (1)| p −2 ,

˜ = XDXH . Note that , D and R˜ are the and R functions of the weight vector w. Equation (22) means that the  p − norm minimization problem in (21) can be rewritten as a 2 − norm minimization one, and therefore a quadratic form can ˜ (w) w has be obtained. It is worth pointing out that although wH R the similar form encountered in the MV-based beamformers, we cannot utilize the quadratic-based methods directly since R˜ is the function of w. In the following, a iterative scheme has be employed to solve the original  p − norm minimization problem. According to (22), the gradient of the objective function f (w) with respect to the complex vector w is defined as · · · , | y ( L )|

∇ f ( w) =

df (w) dw∗

=

1 2



 ∂ f ( w) ∂ f (w) +j . ∂ Re (w) ∂ Im (w)

(23)

By this definition, as a result, the gradient of f (w) is expressed as  p ∂ XH w p ∂ wH XD(w)XH w ∇ f (w) = ∂ w∗ = ∂ w∗ wH X∂ D(w)XH w H =XD (w) X w+ +0 N ×1 ∂ w∗



⎢ ⎢ ×⎢ ⎢ ⎣

∂ wH x1  ∂ w∗

2

H

.. .

··· .. .

0

···

= XD (w) XH w +

H

0

.. .

  p −2 ∂ wH x L  2 ∂ w∗



(26)

p0

(27)

p0 2

where p0 represents the presumed steering vector of the desired signal. It is worth noting that the projected gradient algorithm converges with high efficiency. In what follows, we show this property. According to the   propertyof the projected gradient method [42], Re ∇ f (wk )H wk+1 − wk < 0 can be verified if the problem is convex. Although problem (21) is nonconvex, the nonconvexity is mainly caused by the nonconvex constraints if p ≥ 1. We convert these nonconvex constraints into convex ones which will be shown in Section 4.2. Therefore, the converted problem turns to be convex, and the value of the objective function decreases in each iteration. What is more, it is easy to find the objective p function f (w) = XH w p is no smaller than zero. Thus, combining the above two results, we get the conclusion that the projected gradient algorithm converges. In addition, it has been proved in [43,44] that the algorithm converges to the global minimum for convex problem. Due to the fact that we have loosened the nonconvex constraints of our original problem, our obtained solution is suboptimal. It should be pointed out that the projected gradient algorithm is a general algorithm for constrained optimization problems. However, the key step of this technique is how to compute the projection operation which will be discussed in the next subsection. 4.2. Projection onto feasible region The feasible region of problem (21) is C . For any vector s that is not belonging to C , the essence of the projection onto C is to determine a vector w located at C that the distance between s and w is shortest. This can be described by the following formulation



=XD (w) X w+ w x1 , · · · , w x L ⎤ ⎡   p −2 H

wk+1 = PC uk+1

where PC (•) denotes the projection operation, which will be calculated in the next subsection. Assume the initial value is taken as the data-independent beamformer

Note that the cost function in (21) is equivalently written as

l =1 ⎡  | y (1)| p−2 2  ⎢ .. ⎢ = ⎣ .   0

(25)

⎥ ⎥ H ⎥ x w, · · · , xH w +0 N ×1 L ⎥ 1 ⎦

min w

s.t.

p −2 XD (w) XH w+0 N ×1 2

= 2p XD (w) XH w. (24) In order to solve problem (21), we utilize the following projected gradient algorithm. The geometrical interpretation of this iterative procedure is shown in Fig. 2.

w − s2  H  w pm  − εm w ≥ 1, m = 1, . . . , M .

(28)

Now, the remaining crucial problem is how to solve (28). Obviously, it is still nonconvex because of the nonconvex constraints. To address (28), taking square on both sides of the constraints, we have

  H  w pm  − εm w 2 =





H 2 w2 − 2εm wH pm  w ≥ 1 wH pm pm w + εm m = 1, . . . , M .

(29)

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Y. Feng et al. / Digital Signal Processing 99 (2020) 102676

˜m = Q 

Applying the Cauchy-Schwarz inequality yields

 H  w pm  ≤ w pm  .

(30)



min

H 2 w2 − 2εm w2 pm  ≥ wH pm pm w + εm   H 2 = wH pm pm w + εm − 2εm pm  w2   2 H = wH pm pm w + wH εm − 2εm pm  IN w m = 1, . . . , M .

(31)

Consequently, the constraints in problem (28) can be transformed to





εm2 − 2εm pm  IN w ≥ 1, m = 1, . . . , M . (32)

Substituting (32) back into problem (28), we have

min

w − s2

s.t.

wH Qm w ≥ 1, m = 1, . . . , M

w



(34)





2

 2  if only giving wH pm  − εm w ≥ 1. However, the norm size of the mth uncertainty set εm are less than 1, which can finally avoid the optimal weight  vector obtained by problem (33) satisfying the inequality wH pm  − εm w ≤ −1. It is observed that (33) is still a nonconvex QCQP problem. To solve it, we introduce an auxiliary variable t ∈ R, and t has the following property

t 2 = 1.

w − ts2

H

w Qm w ≥ 1, m = 1, . . . , M t 2 = 1.

s.t.

(36)

It is worth noting that problem (36) is equivalent to (33) in the following sense: suppose (w , t  ) being the optimal solution to problem (36), then w (respectively −w ) is the optimal solution to (33) if t  = 1 (respectively t  = −1). Furthermore, problem (36) is rewritten as











H

(37) Define 



˜ = w S˜ = 



˜ ≥ 1, m = 1, . . . , M ˜ H Q˜ m w w ˜ H ˜Iw ˜ = 1. w



min ˜ w

(41)

(42)



˜ S˜ tr W

⎧   ⎪ ˜ Q˜ m ≥ 1, m = 1, . . . , M tr W ⎪ ⎪ ⎪   ⎪ ⎨ ˜ ˜I = 1 tr W ⎪ ˜ 0 W ⎪ ⎪   ⎪ ⎪ ⎩ rank W ˜ =1

(43)

˜ is defined as W ˜ = where, tr (•) represents the trace operator, W ˜ is a Hermitian positive semidefinite matrix, ˜ H . Apparently, W ˜w w

w t

˜ = 1. ˜  0, and the rank of W ˜ is one, i.e., rank W i.e., W

˜ Apparently, problem (43) is linear with respect to the matrix W. However, (43) is still nonconvex because of the rank-one constraint

˜ = 1. To proceed, we employ the semidefinite relaxation rank W technique, i.e., dropping the rank-one constraint, to reformulate the above problem approximately:



min ˜ w

s.t.



IN −sH

∈ C ( N +1)×1 −s s2



∈ C ( N +1)×( N +1)

(38) (39)



˜ S˜ tr W

⎧   ˜ ˜ ⎪ ⎪ ⎨ tr WQm ≥ 1, m = 1, . . . , M ˜ ˜I = 1 tr W ⎪ ⎪ ⎩ ˜ W  0.

(44)

Now, (44) is convex, and can be solved using the standard and highly efficient interior point method software tools. It is noteworthy that the solution to problem (44) solves (42) approximately because of dropping the rank-one constraint. Therefore, the solution obtained by solving (44) may not be exactly rank-one. To obtain the rank-one solution, a common approach is the rank reduction technique [45]. For more information about the SDP and rank relaxation approaches, the reader that is interested can refer to [46–52]. An efficient method to obtain the rank-one approximation is the eigenvalue-decomposition. Thus, the weight vector of the robust beamformer can be obtained as

˜= w

IN −s w w t 2 H t     − s s ⎧  Qm 0 N ×1  H w ⎪ ⎪ ≥ 1, m = 1, . . . , M ⎨ w t 0 0 1 × N   t   0 N × N 0 N ×1 w ⎪ ⎪ = 1. ⎩ wH t t 01× N 1

s.t.

∈ R( N +1)×( N +1) .

(40)

Note that problem (42) is still nonconvex. To solve (42), we convert it to a higher dimension subspace as a SDP problem, that is

(35)

Thus, problem (33) can be reformulated as

w,t

∈ C ( N +1)×( N +1)

˜ H S˜ w ˜ w

s.t.

(33)

It should be pointed out that inequality wH pm  − εm w ≥ 1 in (29) are not equivalent   to the constraints in (28). This is because the inequality wH pm  − εm w ≤ −1 can also be obtained

min

˜ w

s.t.

 H 2 Qm = pm pm + εm − 2εm pm  IN .

w,t

0 N ×1 1







where

min

 0N ×N ˜I =

0 N ×1 0

With these notations, (37) can be further transformed as:

H 2 w2 − 2εm wH pm  w wH pm pm w + εm

H wH pm pm w + wH



Qm 01× N

01× N

Substituting (30) into the left-hand side of (29) results in





!

˜1 λ1 w

(45)

˜ 1 represent the principal eigenvalue and the corwhere λ1 and w responding eigenvector, respectively. The proposed projected gradient algorithm is summarized in Table 1. It should be pointed out that μ in Step 2 is the initial step size to calculate the initial weight vector uk0+1 in the (k + 1) th iteration. The value of μ has a great influence on the convergence rate of the algorithm. To obtain the optimal step size μk in the kth iteration, the backtracking line search is utilized in Step 3. Then, uk+1 and wk+1 can be obtained  through computing uk+1 = wk − μk ∇ f (wk ) and wk+1 = PC uk+1 respectively in Step 4. It is easy to observe that the dominant cost of our proposed method is solving problem (44), which mainly depends on the

Y. Feng et al. / Digital Signal Processing 99 (2020) 102676

7

Table 1 Summary of Algorithm 1. Step 1:

Step 2: Step 3:

Given error tolerance ς , maximum iteration number K max and the typical algorithmic parameters used in backtracking line search μ = 1, γ = 0.5, δ = 0.1, calculate w0 by (27), k = 0. Obtain ∇ f (wk ) by (24) and compute uk0+1 by uk0+1 = wk − μ∇ f (wk ). Calculate μk by the following for-loop for i = 1, 2, . . . do   Compute wk0+1 = PC uk0+1

by Section 4.2

Break if    

f wk0+1 < f (wk ) + δ 2Re ∇ f (wk )H wk0+1 − wk

μ := γ μ Step 4: Step 5: Step 6:

end for return μk = μ Compute uk+1 by uk+1 =  wk − μk ∇ f (wk ), and obtain wk+1 =PC uk+1 according to Section 4.2.  If wk+1 − wk  ≤ ς or k > K max holds, go to Step 6. Otherwise, let k := k + 1, go to Step 2. wopt := wk , stop iteration.

number of sensors N and the number  of the  constraints nc . Thus, its computational complexity is O N 4 nc2.5 per iteration [24]. Although the complexity of our proposed beamformer is slightly higher than other beamformers, it provides a superior performance when the signals are non-Gaussian with large steering vector mismatch, which will be shown in Section 5. 5. Simulation results Simulations are conducted to demonstrate the performance of our proposed beamformer. A uniform linear array of 10 omnidirectional sensors with a half-wavelength spacing (i.e., d = λ/2) is considered. In all examples, we consider applications in pulse radar and the received signals have been pulse compressed. Due to the fact that a pulsed radar transmits and receives a train of pulses, the outputs of the target are only present in several certain sample cells in the received sample data. In addition, because of the kurtosis of pulse signal being small than three, this signal is a kind of super-Gaussian signal. Therefore, the target signal can be seen as a super-Gaussian signal according to [32]. We further assume that the additive noise is spatially independent complex white Gaussian with unit variance. There are two equal power interfering sources impinging on the antenna array from the directions −20◦ and 30◦ . In all simulations, the interference-to-noise ratio (INR) in a single sensor is 30 dB and the signals are always present in the training data cell. The maximum DOA estimation mismatch θ is 6◦ in the following several examples. Similar to the LCMD beamformer, adding too much constraints will consume degrees of freedom of the beamformer which can be otherwise used to suppress the interferences. Therefore, the number of the constraints should be selected as small as possible and the empirically optimal value of the number of it equals 2 or 3 for the small scale array. Thus, we utilize three uncertainty sets and the constrained directions are θp − 2θ , θp , θp + 2θ respectively, where θp is the estimated DOA of the SOI. Besides, when plotting the SINR curves, 200 Monte Carlo simulations are carried out for their computation. The proposed beamformer is compared with the following eight methods: i) MVDR beamformer; ii) MDDR beamformer; iii) LCMD beamformer; iv) QCMV beamformer; v) QCMD beamformer; vi) robust adaptive beamformer based on steering vector estimation with as little as possible prior information [20] denoted as RABLPPI; vii) the robust adaptive beamformer based on reconstructing the interference-plus-noise covariance matrix [21] denoted as RABRCM; viii) robust adaptive beamformer with precise main beam control [24] denoted as RAB-MBC.

5.1. Example 1: Selection of parameter p In this subsection, we compare the performances of the MDDR beamformer of different values of p. Assume the desired signal impinging on the antenna array from θt = 5◦ . The DOA mismatch is θ = 3◦ . Fig. 3(a) shows the beampatterns for various p of standard MDDR beamformer when the SNR is 15 dB. We can see from Fig. 3(a) that the MDDR beamformer of p = 1 is able to provide deeper nulls at the directions −20◦ and 30◦ , which can suppress the strong interference. In addition, the mainlobe of it is nearest to the actual direction of target and the sidelobe is quite low. Fig. 3(b) shows the performance of the MDDR beamformer of different p versus SNR of the desired signal. It is seen that the MDDR beamformer of p < 2 leads to an improved performance compared with the MDDR beamformer for radar signals. This result is in good agreement with the conclusion in [32]. Thus, in the following simulations, we choose p to be 1. 5.2. Example 2: Output SINR versus ε The uncertainty set size is a very important parameter of the QCMV and QCMD beamformers. In this example, we consider the influence of the uncertainty set size on the output SINR. The direction of the presumed target is 10◦ , i.e., the DOA mismatch θ is 5◦ , and the input SNR is fixed to 25 dB. Note that to simplify this problem, we suppose the size of these three uncertainty sets of our proposed beamformer are the same. Fig. 4 plots the output SINR of the QCMV, QCMD and proposed beamformers with respect to the uncertainty set size ε . We can observe from Fig. 4 that the performance of the QCMV beamformer is quite sensitive to the uncertainty set size ε . To attain a satisfactory performance, ε needs to be chosen in the interval (0.75, 0.85) for the QCMV beamformer. On the contrary, the QCMD and proposed beamformers have a better performance in a wider interval. However, the uncertainty set size ε still has a great influence on the performance of the QCMD beamformer. Nevertheless, the proposed beamformer is insensitive to ε , the requirement for the selection of an appropriate ε is quite mild, the SINR fluctuation is less than 0.2 dB when ε varies from 0 to 0.68. In addition, it is noteworthy that if the size of each uncertainty set used in our proposed method is increased too much, the performance will degrade because the increasing set may lead to a trivial solution. This phenomenon is also present in the other beamformers. Actually, there is no need to increase the size of each uncertainty set unlimitedly. If the uncertainty region is too large, the ability to suppress the noise and interferences is also degraded.

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Y. Feng et al. / Digital Signal Processing 99 (2020) 102676

Fig. 3. Performance comparison of standard MDDR beamformer for p=1,2,4 and 8. (a) Beampatterns for different p. (b) Output SINR versus input SNR for different p.

the beamformer output. It should be clarified that all the beampatterns are normalized through scaling the weight vectors of the beamformers. Actually, in radar signal processing, the principal focus we are concerned about is the ratio of main lobe and side lobe of the beampattern. To compare the performances, considering that the noise gains of all beamformers should be the same, we scale all the beampatterns. It is noteworthy that the norm of the weight vector does not change the output SINR of the beamformer due to the definition of SINR in (4). 5.4. Example 4: Output SINR versus input SNR

Fig. 4. Comparison of output SINR versus formers.

ε of QCMV, QCMD and proposed beam-

5.3. Example 3: Beampattern comparison In the third example, we compare the beampatterns of these methods. Assume the desired signal impinge on the antenna array from θt = 5◦ . In Fig. 5(a), SNR=5 dB and the DOA mismatch is θ = 3◦ . In Fig. 5(b), SNR=25 dB and the DOA mismatch is θ = 5◦ . The constraint points are selected through the way in the LCMV method which will be used in the LCMD and proposed beamformers. The vertical dotted lines denote the DOAs of the desired signal and interferences. It is observed that all of the methods have nulls at the directions of interferences. However, as shown in Fig. 5(b), when SNR is 25 dB and DOA mismatch is 5◦ , the mainlobes of the MVDR and MDDR beamformers also form a null at the actual direction of the desired signal. In addition, although the mainlobes of beampattern of remaining beamformers are maintained well, the sidelobes are higher than that of our proposed beamformer. Note that the LCMD beamformer improves the robustness against the DOA mismatch through broadening the main lobe of the beampattern, which inevitably includes more noise in

In this example, we consider the case when the SOI power varies. For QCMV and QCMD beamformers, the radius of the uncertainty set is chosen to be 0.8, which can obtain a satisfactory performance. For our proposed method, it is unnecessary to utilize large sets, we select the radius of each uncertainty set to be 0.5. Fig. 6(a)(c) and Fig. 6(b)(d) show the output SINR versus SNR of the desired signal when the DOA mismatches are 3◦ and 6◦ . Note that Fig. 6(c) and Fig. 6(d) are the enlarged drawings in low SNR region of Fig. 6(a) and Fig. 6(b), respectively. In addition, the upper bound on the SINR is plotted as a benchmark. According to the definition of SINR in Section 2.2 and the generalized Rayleigh quotient, the upper bound on the SINR is obtained by calculating the maximum   generalized eigenvalue of the matrix pair σs2 a (θt ) aH , Ri+n . We observe from Fig. 6(a) that the MVDR beamformer is quite sensitive to the mismatch and the remaining beamformers have good performance at low SNR region. It is worth noting that the MDDR beamformer is also robust against the steering vector mismatch if the mismatch is not too large. What is more, the remaining beamformers except the MVDR beamformer outperform the LCMD beamformer. The reason is, similar to the LCMV beamformer, the equality constraints of LCMD beamformer are too strong, and the broadened mainlobe is mismatched with the standard mainbeam. However, from Fig. 6(b), when the mismatch angle increases to 6◦ , the performance of most beamformers decreases at high SNR region. Note that the MDDR beamformer has the similar behavior to the MVDR beamformer in this case. On the contrary, the proposed beamformer has a better performance than others’. We can find from Fig. 6(c) and Fig. 6(d) that if SNR is less than 0 dB, all of the beamformers have the similar performance. However, our proposed beamformer still enjoys a better performance compared to others, especially the DOA mismatch is large.

Y. Feng et al. / Digital Signal Processing 99 (2020) 102676

Fig. 5. Beampatterns of different methods. (a) SNR=5 dB, θ = 3◦ . (b) SNR=25 dB, θ = 5◦ .

Fig. 6. Output SINR versus input SNR (a)(c) mismatch angle θ = 3◦ and (b)(d) mismatch angle θ = 6◦ .

9

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Y. Feng et al. / Digital Signal Processing 99 (2020) 102676

5.5. Example 5: Output SINR versus mismatch angle In this example, we consider the steering vector mismatch induced by DOA mismatch error. The SNR of the desired signal is 25 dB and the single uncertainty set size of our proposed method is 0.5. The maximal possible DOA mismatch angle for all beamformers is 6◦ . We observe from Fig. 7 that the mismatch angle has a great influence on the performances of the MVDR and MDDR beamformers while the remaining beamformers are more robust against the DOA mismatch. It is worth noting that although the LCMD beamformer is robust against the DOA mismatch, similar to the LCMV beamformer, because of utilizing the strong equality constraints, the ability to suppress the interferences and noise is declined. Furthermore, only when the mismatch angle lies within a limited region the performances of the other beamformers can obtain a satisfactory level. On the contrary, the performance degradation of the proposed beamformer over the whole interval is less than 3 dB.

5.6. Example 6: Output SINR versus number of snapshots The output SINR versus the number of snapshots for SNR=20 dB and the mismatch angles being 3◦ and 6◦ are shown in Fig. 8(a) and (b), respectively. It is seen that the output SINR increases with the number of snapshots. The MVDR, MDDR and QCMV beamformers are sensitive to the number of snapshots while the other beamformers are relatively robust against the number of snapshots. The performance and the convergence rate of the QCMD, RAB-RCM and proposed beamformers are nearly the same. However, as indicated in Fig. 8 (b), when the mismatch angle is 6◦ , our proposed beamformer outperforms other beamformers. 5.7. Example 7: Output SINR versus input SNR when amplitude and phase errors appear in array elements In this example, we consider the scenario in the presence of the amplitude and phase errors of the array elements. Suppose the DOA mismatch angle is 3◦ . The amplitude error and the phase error of each array element are independently drawn from a zeromean Gaussian random generator, and the amplitude error and the phase error is less than ±5% and ±2.5◦ , respectively. Fig. 9 plots the output SINRs of nine aforementioned beamformers versus the input SNR. Compared to Fig. 6(a), we easily find that the performance of the RAM-RCM method degrades severely in the presence of the amplitude and phase errors. This is because the beamformer of [21] is ineffective when array calibration error happens due to the fact the spectrum integral used only considers the DOA mismatch. In addition, the performance of the LCMD beamformer slightly degrades. On the contrary, the performances of the remaining beamformers almostly remain unchanged. 6. Conclusion

Fig. 7. Output SINR versus mismatch angle for SNR=25 dB.

Many signals encountered in practical applications are nonGaussian. The MV-based beamformers have been proved to be suboptimal for non-Gaussian signals. On the other hand, large steering vector mismatch and inappropriate choice of uncertainty set will cause performance degradation. In this paper, we minimize the dispersion of the output and utilize multiple small uncertainty sets to address this issue. To solve the nonconvex problem, a projected gradient algorithm is devised. In each iteration, the nonconvex

Fig. 8. Output SINR versus number of snapshots (a) mismatch angle θ = 3◦ and (b) mismatch angle θ = 6◦ .

Y. Feng et al. / Digital Signal Processing 99 (2020) 102676

Fig. 9. Output SINR versus input SNR with amplitude and phase errors in array elements.

constraints are transformed into convex ones. We first introduce an auxiliary variable, and recast the problem in high dimension, then utilize the SDP relaxation technique by dropping the rank-one constraint to reformulate the problem. Numerical examples confirm that the proposed method improves the SINR performance significantly in case of large steering vector mismatch for nonGaussian signals compared with the other beamformers based on both the MV and the MD criteria. In addition, the performance of our method is not sensitive to the sizes of the spherical uncertainty sets. Declaration of competing interest We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. References [1] A.B. Gershman, V.I. Turchin, V.A. Zverev, Experimental results of localization of moving underwater signal by adaptive beamforming, IEEE Trans. Signal Process. 43 (10) (Oct. 1995) 2249–2257. [2] E.Y. Gorodetskaya, A.I. Malekhanov, A.G. Sazontov, N.K. Vdovicheva, Deep-water acoustic coherence at long ranges: theoretical prediction and effects on largearray signal processing, IEEE J. Ocean. Eng. 24 (2) (Apr. 1999) 156–171. [3] A.B. Gershman, E. Nemeth, J.F. Bohme, Experimental performance of adaptive beamforming in a sonar environment with a towed array and moving interfering sources, IEEE Trans. Signal Process. 48 (1) (Jan. 2000) 246–250. [4] J. Li, P. Stoica, Robust Adaptive Beamforming, Wiley Online Library, 2006. [5] J. Capon, High-resolution frequency-wavenumber spectrum analysis, Proc. IEEE 57 (8) (Aug. 1969) 1408–1418. [6] D.D. Feldman, L.J. Griffiths, A projection approach to robust adaptive beamforming, IEEE Trans. Signal Process. 42 (4) (Apr. 1994) 867–876. [7] B.D. Carlson, Covariance matrix estimation errors and diagonal loading in adaptive arrays, IEEE Trans. Aerosp. Electron. Syst. 24 (4) (Jul. 1988) 397–401. [8] O.L. Frost III, An algorithm for linearly constrained adaptive array processing, Proc. IEEE 60 (8) (Aug. 1972) 926–935. [9] M.H. Er, A. Cantoni, Derivative constraints for broad-band element space antenna array processors, IEEE Trans. Acoust. Speech Signal Process. ASSP-31 (6) (Dec. 1983) 1378–1393. [10] K.M. Buckley, L.J. Griffiths, An adaptive generalized sidelobe canceller with derivative constraints, IEEE Trans. Antennas Propag. 34 (3) (Mar. 1986) 311–319. [11] S. Zhang, I.L. Thng, Robust presteering derivative constraints for broadband antenna arrays, IEEE Trans. Signal Process. 50 (1) (Jan. 2002) 1–10. [12] J. Li, P. Stoica, Z. Wang, Doubly constrained robust capon beamformer, IEEE Trans. Signal Process. 52 (9) (Sept. 2004) 2407–2423.

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Yang Feng was born in Jiangsu, China, in 1991. He received the B.S. degree in electronic engineering and the Ph.D. degree in signal and information processing from Xidian University, China, in 2014 and 2019, respectively. He is currently an engineer with the Nanjing Research Institute of Electronics Technology. His research interests include adaptive beamforming, array signal processing and space-time adaptive processing. Guisheng Liao was born in Guangxi, China, in 1963. He received the B.S. degree from Guangxi University, Guangxi, China, and the M.S. and

Ph.D. degrees from Xidian University, Xian, China, in 1985, 1990, and 1992, respectively. From 1999 to 2000, he was a Senior Visiting Scholar with The Chinese University of Hong Kong, Hong Kong. Since 2006, he has been serving as the Panelist for the medium- and long-term development plan in high-resolution and remote sensing systems. Since 2007, he has been the Lead of the Yangtze River Scholars Innovative Team and devoted in advanced techniques in signal and information processing. Since 2009, he has been the Evaluation Expert for the international cooperation project of Ministry of Science and Technology in China. He is currently a Yangtze River Scholars Distinguished Professor with the National Laboratory of Radar Signal Processing and also serves as the Dean of the School of Electronic Engineering, Xidian University. He has authored or coauthored several books and more than 200 publications. His research interests include array signal processing, space–time adaptive processing, radar waveform design, and airborne/space surveillance and warning radar systems. Jingwei Xu received the B.S. degree in electronic engineering and the Ph.D. degree in signal and information processing from Xidian University, China, in 2010 and 2015, respectively. From 2015 to 2018, he was a Lecturer and also a Postdoctoral Researcher with the National Laboratory of Radar Signal Processing, Xidian University, where he has been an Associate Professor, since 2018. He is currently a Postdoctoral Fellow under the Hong Kong Scholar Program at the City University of Hong Kong. His research interests include radar system modeling, multi-sensor array signal processing, space–time adaptive processing, multiple-input-multipleoutput radar, and array radar using waveform diversity (especially on frequency diverse array and space–time coding array).