OPARC: Optimal and Precise Array Response Control Algorithm -- Part II: Multi-points and Applications

1 OP ARC: Optimal and Precise Array Response Control Algorithm – P art II: Multi-points and Applications Xuejing Zhang, Student Member , IEEE, Zishu He, Member , IEEE, Xiang-Gen Xia, F ellow , IEEE, Bin Liao, Senior Member , IEEE, Xuepan Zhang, and Y ue Y ang, Student Member , IEEE Abstract —In this paper , the optimal and pr ecise array response control (OP ARC) algorithm proposed in Part I of this two paper series is extended from single point to multi-points. T w o com- putationally attractive parameter determination appr oaches are pro vided to maximize the array gain under certain constraints. In addition, the applications of the multi-point OP ARC algorithm to array signal processing ar e studied. It is applied to realize array pattern synthesis (including the general array case and the large array case), multi-constraint adaptiv e beamforming and quiescent patter n control, where an innovative concept of normalized cov ariance matrix loading (NCL) is pr oposed. Finally , simulation results are presented to validate the superiority and effectiveness of the multi-point OP ARC algorithm. Index T erms —Array response control, adaptiv e array theory , array pattern synthesis, adaptive beamforming, quiescent pattern control. I . I N T RO D U C T I O N I N the companion paper [1], optimal and precise array response control (OP ARC) algorithm was proposed and analyzed. OP ARC provides a new mechanism to control array responses at a giv en set of angles, by simply assigning virtual interference one-by-one. The optimality (in the sense of array gain) of OP ARC in each step is guaranteed. Nevertheless, OP ARC only controls one point per step and may be inef- ficient if multiple points are needed to be precisely adjusted. Moreov er , how to use the OP ARC algorithm in practical cases (where real data commonly exists) remains. This paper first extends the OP ARC algorithm from single point response control per step to multi-point response control per step. Note that a multi-point accurate array response control (MA 2 RC) algorithm has been recently developed in [2]. Nev ertheless, since it is built on the basis of the accurate array response control (A 2 RC) algorithm [3], the MA 2 RC suffers from the similar drawbacks to A 2 RC, i.e., a solution is empirically adopted and hence a satisfactory performance cannot be alw ays guaranteed as analyzed in details in [1]. In this paper , we first carry out a careful in vestigation on the change rule of the optimal beamformer when multiple virtual X. Zhang, Z. He and Y . Y ang are with the University of Elec- tronic Science and T echnology of China, Chengdu 611731, China (e-mail: xjzhang7@163.com; zshe@uestc.edu.cn; yueyang@std.uestc.edu.cn). X. Zhang and X.-G. Xia are with the Department of Electrical and Computer Engineering, Univ ersity of Delaware, Newark, DE 19716, USA (e-mail: xjzhang@udel.edu; xxia@ee.udel.edu). B. Liao is with College of Information Engineering, Shenzhen Univ ersity , Shenzhen 518060, China (e-mail: binliao@szu.edu.cn). X. P . Zhang is with Qian Xuesen Lab of Space T echnology , Beijing 100094, China (e-mail: zhangxuepan@qxslab .cn). interferences are simultaneously assigned. Then, a generalized methodology of the weight vector update is observed and utilized for the realization of the multi-point array response control. Similar to the OP ARC in [1], we formulate a con- strained optimization problem such that the array response lev els of multiple points can be optimally (in the sense of array gain) and precisely controlled. Then, two different solvers, by either taking advantage of the OP ARC algorithm or employing the recently dev eloped consensus alternating direction method of multipliers (C-ADMM) approach in [4], are provided to find an approximate solution of the established optimization problem. Note that since the OP ARC in [1] only optimally controls the array response at one point in each step, it has a closed-form solution, while this is not the case for the multi- point OP ARC in this paper . In other words, this paper does not cov er [1]. The differences between the proposed multi-point OP ARC and MA 2 RC are similar to those between OP ARC and A 2 RC as described in [1] in details. Meanwhile, for the proposed multi-point OP ARC, its applications to, such as, array pattern synthesis, multi-constraint adaptive beamforming and quiescent pattern control, are also presented as detailed below . Application to Array P attern Synthesis: Array pattern syn- thesis is a fundamental problem for radar , communication and remote sensing. Most of the existing pattern synthesis ap- proaches, for instance, the global optimization based methods in [5]–[7], the con vex programming (CP) methods in [8]– [10], and the adaptive array theory based method in [11], hav e no ability to precisely control the beampattern according to a given requirement. In this paper , the above shortcoming is overcome by synthesizing desirable patterns with the pro- posed multi-point OP ARC algorithm. W e start the synthesis procedure from the quiescent pattern, and iterati vely adjust the responses of multiple angles to their desired le vels. Simulation results show that it only requires a few steps of iteration to complete the syntheses of well-shaped beampatterns. In addition to the consideration for a general array , lar ge array pattern synthesis problem [12], where the existing meth- ods consume a large amount of computing resources or ev en not work at all, is particularly discussed. W e will see that the large array pattern synthesis can be readily realized with the multi-point OP ARC algorithm, in a computationally attractiv e manner . Application to Multi-constraint Adaptive Beamforming: Adaptiv e beamforming plays an important role in various 2 application areas, since it enables us to receiv e a desired signal from a particular direction while it simultaneously blocks undesirable interferences. Multi-constraint adaptiv e beamforming, i.e., designing an adaptiv e beamformer with sev eral fixed directional constraints, is a common strategy to improv e the robustness of the adaptive beamformer , see [13]– [15] for example. The existing methods may cause distorted beampatterns, due to their imperfections on model building or parameter optimization. Based on the proposed multi- point OP ARC algorithm, a new approach to multi-constraint adaptiv e beamforming is presented in this paper . W e modify the traditional adaptive beamformer to make the prescribed amplitude constraints satisfied by utilizing the multi-point OP ARC algorithm. In the proposed algorithm, the total signal- to-interference-plus-noise ratio (SINR) (taking both real inter- ferences and assigned virtual interferences into consideration) is maximized, and the real unexpected components can be well rejected without leading to any undesirable pattern distortion. Inspired by this, a new concept of normalized covariance matrix loading (NCL), which can be regarded as a general- ization of the con ventional diagonal loading (DL) in [16]– [18], is dev eloped. Moreover , NCL is also exploited to realize quiescent pattern control as introduced next. Application to Quiescent P attern Contr ol: In brief, when an adapti ve array operates in the presence of white noise only , the resultant adaptiv e beamformer is referred to as the quiescent weight vector , and the corresponding array response is termed as the quiescent pattern. As pointed out in [19], having overall low sidelobes is important to adaptiv e arrays and how to specify a quiescent response pattern is worthwhile in vestigating. Most of the existing quiescent pattern control methods [19]–[21] are established on the foundation of the linearly constrained minimum v ariance (LCMV) framework, where the unnecessary phase constraints of array response are implicitly imposed. In this paper , a simple yet effecti ve quiescent pattern control algorithm is proposed. W e synthesize a satisfactory deterministic pattern, i.e., the ultimate quiescent pattern, by adopting the multi-point OP ARC algorithm, and meanwhile, collect the resulting virtual normalized covariance matrix (VCM) for later use. Under the real data circumstance, the quiescent pattern control is completed by conducting a simple NCL operator to the existed VCM, and the weight vector can be obtained accordingly . This paper is organized as follows. The proposed multi- point OP ARC algorithm is presented in Section II. The three applications of the multi-point OP ARC are discussed in Sec- tion III. Representative experiments are carried out in Section IV and conclusions are drawn in Section V . Notations: The same as [1], we use bold upper -case and lower -case letters to represent matrices and vectors, respec- tiv ely . In particular, we use I to denote the identity matrix. j , √ − 1 . ( · ) T and ( · ) H stand for the transpose and Hermitian transpose, respectiv ely . | · | denotes the absolute value and k · k 2 denotes the l 2 norm. W e use ( g ) i to stand for the i th element of vector g . < ( · ) and = ( · ) denote the real and imaginary parts, respecti vely .  represents the element-wise division operator . W e use Diag( · ) to stand for the diagonal matrix with the components of the input vector as the diagonal elements. R and C denote the sets of all real and all complex numbers, respectiv ely . Finally , ∪ denotes the set union and card( · ) returns the number of elements in a set. I I . M U LTI - P O I N T O PA R C A L G O R I T H M T o present our multi-point OP ARC algorithm, we first make a detailed analysis on the optimal weight vector . A. Multi-interference Optimal Beamformer Consider an array with N elements. The same as [1], the optimal weight vector: w opt = T − 1 n + i a ( θ 0 ) (1) maximizes both the output signal-to-interference-plus-noise ratio (SINR) and the array gain of an array system, where SINR and array gain are defined, respectively , as [22] SINR , σ 2 s | w H a ( θ 0 ) | 2 w H R n + i w , G , | w H a ( θ 0 ) | 2 w H T n + i w (2) where a ( θ ) stands for the array steering vector: a ( θ ) = [ g 1 ( θ ) e − j ωτ 1 ( θ ) , · · · , g N ( θ ) e − j ωτ N ( θ ) ] T (3) where g n ( θ ) denotes the pattern of the n th element, τ n ( θ ) is the time-delay between the n th element and the reference point, n = 1 , · · · , N , ω denotes the operating frequency . In the abov e notations, θ 0 is the beam axis, R n + i denotes the N × N noise-plus-interference covariance matrix, T n + i stands for the normalized cov ariance matrix satisfying T n + i = R n + i σ 2 n = I + Q X l =1 β l a ( θ l ) a H ( θ l ) (4) where β l , σ 2 l /σ 2 n denotes the interference-to-noise ratio (INR), Q is the number of interferences, a ( θ l ) is the steering vector of the l th interference, σ 2 s , σ 2 n and σ 2 l stand for the powers of signal, noise and the l th interference, respecti vely . Note that G in (2) represents the amplification factor of the input signal-to-noise ratio (SNR) σ 2 s /σ 2 n , and the criterion of array gain maximization is adopted to achiev e the optimal weight vector . From (1)-(2), one can see that the optimal weight vector w opt depends on R n + i or T n + i , which is normally data- dependent. For this reason, R n + i or T n + i may not be av ail- able if we need to design a data-independent array response pattern L ( θ , θ 0 ) , | w H a ( θ ) | 2  | w H a ( θ 0 ) | 2 that satisfies some specific requirements. In this case, for a giv en response design task, the concept of virtual normalized noise-plus-interference cov ariance matrix (VCM) was introduced in [1]. Moreover , it was shown in [1] that a VCM can be constructed by assigning suitable virtual interferences one-by-one. In this paper , for a giv en response control task, we assign multiple virtual interferences (instead of a single virtual interference) at one step, and study how the optimal weight vector in (1) changes. W e use induction to describe the problem. Suppose that we have already assigned interferences for ( k − 1) times, the total number of interferences is accumulated as Q k − 1 and T k − 1 denotes the total VCM upto the ( k − 1) th step. The 3 w k,? = w k − 1 ,? − T − 1 k − 1 A k  I + Σ k,? A H k T − 1 k − 1 A k  − 1 Σ k,? A H k T − 1 k − 1 a ( θ 0 ) (15) corresponding optimal weight vector at the ( k − 1) th step is giv en by w k − 1 = T − 1 k − 1 a ( θ 0 ) (5) where the subscript ( · ) opt has been omitted for notational simplicity . Then, we carry out the k th step by assigning M k interferences from directions θ k,m with INR to be β k,m , m = 1 , · · · , M k , where θ k,m are renamed from those θ l in (4). Then, T k = T k − 1 + M k X m =1 β k,m a ( θ k,m ) a H ( θ k,m ) = T k − 1 + A k Σ k A H k (6) where A k = [ a ( θ k, 1 ) , · · · , a ( θ k,M k )] (7) Σ k = Diag([ β k, 1 , · · · , β k,M k ]) (8) and T k is the resulting VCM after implementing the k th step of the interference assigning. Clearly , if M k = 1 , (6) degenerates to Eqn. (6) of [1], and the related discussions return to our previous work in [1]. T o make the discussion meaningful, the matrix A k in this paper is assumed to hav e a full column rank. By applying the Generalized W oodbury Lemma [23] to (6), we obtain that T − 1 k = T − 1 k − 1 − T − 1 k − 1 A k  I + Σ k A H k T − 1 k − 1 A k  − 1 Σ k A H k T − 1 k − 1 . (9) Accordingly , the obtained optimal weight vector satisfies w k = T − 1 k a ( θ 0 ) = w k − 1 + T − 1 k − 1 A k h k (10) where h k ∈ C M k is h k = −  I + Σ k A H k T − 1 k − 1 A k  − 1 Σ k A H k T − 1 k − 1 a ( θ 0 ) . (11) As shown in (10), the current optimal weight w k is obtained by making a modification to the previous weight w k − 1 . Recalling the adaptive array theory , the weight w k performs optimally in maximizing the array gain G k defined as G k , | w H k a ( θ 0 ) | 2 / | w H k T k w k | (12) although the response levels at θ k,m , m = 1 , · · · , M k , may not reach their expected values. T o precisely adjust the array responses of θ k,m to their desired lev els ρ k,m , the INRs β k,m , m = 1 , · · · , M k , or equi valently the diagonal matrix Σ k , should be carefully selected. In the meantime, the array gain G k in (12) should be maximized. Note also that h k in (11) acts as a mapping of Σ k , and we can express Σ k by h k as Σ k = Diag  − h k   A H k T − 1 k − 1 ( a ( θ 0 ) + A k h k )  . (13) From (11) and (13), one can see that Σ k and h k are one-one mapping. Therefore, the multi-point optimal and precise array response control (OP ARC) can be realized by either finding a suitable Σ k or selecting an appropriate h k . B. Multi-point OP ARC Pr oblem F ormulation Let us first formulate the multi-point OP ARC by optimizing Σ k as: max Σ k G k = | w H k a ( θ 0 ) | 2 / | w H k T k w k | (14a) sub ject to L ( θ k,m , θ 0 ) = ρ k,m , m = 1 , · · · , M k (14b) w k = w k − 1 ,? + T − 1 k − 1 A k h k (14c) where w k − 1 ,? is the resultant weight vector of the ( k − 1) th step (we use the star symbol to indicate it as the ultimate selection of w k − 1 ), the vector h k is giv en by (11). Once the optimal Σ k,? has been obtained, we can express the ultimate weight vector w k,? as (15) on the top of this page. T o find the solution of problem (14), an iterative method is first provided below . C. Iterative Approac h The OP ARC algorithm, dev eloped in the companion pa- per [1], is able to optimally and precisely adjust one-point response level at a time. Thus, we may apply it to the M k - point OP ARC problem (14) as follows. For a fixed k > 0 , we apply the OP ARC algorithm for M k steps. In the m th step, OP ARC is to realize L ( θ k,m , θ 0 ) = ρ k,m , m = 1 , · · · , M k . Unfortunately , OP ARC brings inevitable pattern variations on the pre vious controlled angles as we hav e discussed in [1]. More specifically , the response le vels of θ k,i , i = 1 , · · · , m − 1 , vary after accurately controlling the response level of θ k,m to its desired level ρ k,m , 2 ≤ m ≤ M k . T o reduce the undesirable pattern variations on the pre-adjusted angles, we propose to iterati vely apply the M k -point OP ARC for a number of times, until a certain termination criterion is met. A temporary v ariable Ξ = T k − 1 and Σ k = 0 are taken as the initializations in the first iteration. Then, in each iteration, an M k -step OP ARC is carried out. More specifically , in the m th step, we adjust the response lev el of θ k,m to be ρ k,m , by calculating the INR of the ne wly assigned virtual interference at θ k,m , denoted as β k,m,? , m = 1 , · · · , M k , from Eqn. (38) of [1], and then update the associated VCM as Ξ = Ξ + β k,m,? a ( θ k,m ) a H ( θ k,m ) . Once an iteration, i.e., an M k -step OP ARC, is completed, β k,m,? is added to the m th diagonal element of Σ k , and then we set the resulting Ξ as the initial VCM in the next iteration. Note that T 0 = I . Naturally , whether the response levels of the adjusted angles θ k,m , m = 1 , · · · , M k , are close enough to their desired lev els can be a criterion to terminate the iteration of OP ARC. Howe v er , this strategy needs to calculate all the intermediate weight vectors that may be computationally inefficient. T o improv e the computational ef ficiency , we propose to terminate the iteration of OP ARC by examining whether the magnitudes of INRs of the newly assigned virtual interferences approxi- mate enough to zero, since there is no need to assign virtual interferences if their values are small enough. Finally , we summarize the above iterativ e solver of problem (14) in Algorithm 1, where β  stands for a small tolerance 4 Algorithm 1 Iterative Approach to Problem (14) 1: giv e a ( θ 0 ) , θ k,m , ρ k,m , m = 1 , · · · , M k , and A k , set β  > 0 , β MAX > β  , Ξ = T k − 1 , Σ k = 0 2: while β MAX > β  do 3: for m = 1 , · · · , M k do 4: calculate β k,m,? from Eqn. (38) of [1], by setting L ( θ k,m , θ 0 ) = ρ k,m 5: update VCM Ξ = Ξ + β k,m,? a ( θ k,m ) a H ( θ k,m ) 6: end f or 7: update Σ k as Σ k = Σ k + Diag([ β k, 1 ,? , · · · , β k,M k ,? ]) 8: obtain β MAX = max 1 ≤ m ≤ M k | β k,m,? | 9: end while 10: obtain Σ k,? = Σ k parameter . Note that β k,m,? in Algorithm 1 is calculated with Eqn. (38) of [1]. In addition, we can express the ultimate Σ k,? as Σ k,? = Diag([ ¯ β k, 1 ,? , · · · , ¯ β k,M k ,? ]) (16) where ¯ β k,m,? represents the total INR of the virtual inter- ference assigned at θ k,m in the k th step, and equals to the summation of all β k,m,? ’ s of different iterations for a fixed m = 1 , · · · , M k . As discussed earlier, once the optimal Σ k,? has been obtained, we can use Σ k,? to obtain the VCM T k by Eqn. (6), update h k in (11) and (14c), and calculate w k,? by Eqn. (15). It shall be noted that an in verse of normalized cov ariance matrix is indispensable in determining β k,m,? ’ s by Eqn. (38) of [1]. This may lead to a high cost in memory or/and computation especially for a large array , although it may not need a large number of iterations. D. C-ADMM Appr oach W e next propose another approach to solve problem (14). W e first reformulate the original problem (14) as a quadrati- cally constrained quadratic program (QCQP) problem. Then, the recently dev eloped consensus-ADMM (C-ADMM) [4] approach is employed to find its solution. 1) Pr oblem Reformulation: Since h k is a one-one mapping of Σ k , we can formulate the multi-point OP ARC, i.e., problem (14), by finding h k as max h k ∈ C M k G k = | w H k a ( θ 0 ) | 2 / | w H k T k w k | (17a) sub ject to L ( θ k,m , θ 0 ) = ρ k,m , m = 1 , · · · , M k (17b) w k = w k − 1 ,? + T − 1 k − 1 A k h k . (17c) W e substitute the constraint (17c) into G k and obtain G 2 k =   a H ( θ 0 )( w k − 1 ,? + T − 1 k − 1 A k h k )   2 = − h H k e Ch k + 2 <  e c H h k  + | a H ( θ 0 ) w k − 1 ,? | 2 (18) where w k = T − 1 k a ( θ 0 ) is used, e C and e c are defined as e C , − ( T − 1 k − 1 A k ) H a ( θ 0 ) a H ( θ 0 ) T − 1 k − 1 A k ∈ C M k × M k (19a) e c , ( T − 1 k − 1 A k ) H a ( θ 0 ) a H ( θ 0 ) w k − 1 ,? ∈ C M k . (19b) Since | a H ( θ 0 ) w k − 1 ,? | 2 is a constant, the maximization of G k is thus equiv alent to the minimization of h H k e Ch k − 2 < ( e c H h k ) . On the other hand, recalling the expression of L ( θ , θ 0 ) , we can re write the constraint (17b) as w H k S k,m w k = 0 , m = 1 , · · · , M k (20) where S k,m = a ( θ k,m ) a H ( θ k,m ) − ρ k,m a ( θ 0 ) a H ( θ 0 ) . Substi- tuting the constraint (17c) into (20), we have h H k e D m h k − 2 < ( e d H m h k ) = α m , m = 1 , · · · , M k (21) where e D m = ( T − 1 k − 1 A k ) H S k,m T − 1 k − 1 A k ∈ C M k × M k (22a) e d m = − ( T − 1 k − 1 A k ) H S k,m w k − 1 ,? ∈ C M k (22b) α m = − w H k − 1 ,? S k,m w k − 1 ,? ∈ R . (22c) Thus, problem (17) can be reformulated as min h k h H k e Ch k − 2 <  e c H h k  (23a) sub ject to h H k e D m h k − 2 < ( e d H m h k ) = α m (23b) m = 1 , · · · , M k . In the sequel, we adopt the newly dev eloped C-ADMM approach [4] to solve problem (23). 2) C-ADMM Solver: W e first conv ert (23) into its real domain as min z z T Cz − 2 c T z (24a) sub ject to z T D m z − 2 d T m z = α m (24b) m = 1 , · · · , M k where z =  < ( h T k ) = ( h T k )  T ∈ R 2 M k (25a) c =  < ( e c T ) = ( e c T )  T ∈ R 2 M k (25b) d m = h < ( e d T m ) = ( e d T m ) i T ∈ R 2 M k (25c) C = " < ( e C ) −= ( e C ) = ( e C ) < ( e C ) # ∈ R 2 M k × 2 M k (25d) D m = " < ( e D m ) −= ( e D m ) = ( e D m ) < ( e D m ) # ∈ R 2 M k × 2 M k . (25e) T o tackle (24), we introduce the auxiliary v ariable vectors p m , m = 1 , · · · , M k , and then formulate (24) as min z , { p m } M k m =1 z T Cz − 2 c T z (26a) sub ject to p m = z (26b) p T m D m p m − 2 d T m p m = α m (26c) m = 1 , · · · , M k . Note that the non-con ve x constraint in problem (26) is only imposed on p m and not related to z . Moreover , for any given m = 1 , · · · , M k , the noncon vex-constraint, i.e., (26c), is a QCQP with only one constraint (QCQP-1), which can be easily solved as pointed out in [4]. Thus, the newly formulated problem (26) simplifies the original problem (24) to solve. 5 T o see the details, we first devise the augmented Lagrangian by ignoring the constraint (26c): L η ( z , p , λ ) = z T Cz − 2 c T z + M k X i =1 λ T m ( z − p m ) + M k X i =1 η 2 k z − p m k 2 2 (27) where η > 0 is the penalty parameter , λ m ∈ R 2 M k are La- grange multiplier vectors. Note that the augmented Lagrangian (27) acts as the (unaugmented) Lagrangian associated with the following problem: min z , { p m } M k m =1 z T Cz − 2 c T z + M k X i =1 η 2 k z − p m k 2 2 (28a) sub ject to p m = z , m = 1 , · · · , M k (28b) which is equi valent to problem (26a)-(26b), since for any fea- sible z and p m , m = 1 , · · · , M k , the added term, i.e., the last term in (28a), to the objectiv e function is zero. As mentioned in [24], the augmented Lagrangian brings robustness to the dual ascent method adopted later . Since the constraints (26c) are imposed on p m and not related to z , they only play roles in finding p m , m = 1 , · · · , M k . For this reason, we don’t include (26c) in the abov e augmented Lagrangian intentionally . Instead, we take the constraints in (26c) into consideration when minimizing L η ( z , p , λ ) as shown next. The alternating direction method of multipliers (ADMM) [24], which is an operator splitting algorithm originally de- vised to solve con ve x optimization problems, has been ex- plored as a heuristic method to solv e non-con ve x problems [4]. Follo wing the decomposition-coordination procedure of ADMM in [24], we can determine { z , p m , λ } via the alterna- tiv e and iterative steps below . Step 1 : Update z z ( t + 1) = arg min z L η ( z , p ( t ) , λ ( t )) = arg min z z T ( C + η M k 2 I ) z − 2 g T ( t + 1) z = ( C + η M k 2 I ) − 1 g ( t + 1) (29) where g ( t + 1) = c − (1 / 2) M k P m =1 ( λ m ( t ) − η p m ( t )) . Step 2 : Update p For m = 1 , · · · , M k , we update the vector p m as p m ( t + 1) = arg min p m L η ( z ( t + 1) , p , λ ( t )) = arg min p m η p T m p m − 2( η z ( t + 1) + λ m ( t )) T p m = arg min p m k p m − ζ m ( t + 1) k 2 2 (30a) sub ject to p T m D m p m − 2 d T m p m = α m (30b) where ζ m ( t + 1) = z ( t + 1) + (1 /η ) λ m ( t ) . Since the abov e problem is QCQP-1 which is equiv alent to solving a polynomial as mentioned in [4], the bisection or Newtons method can be adopted to find its (approximate) solution, see [4] and [25] for reference. Step 3 : Update λ Algorithm 2 C-ADMM Approach to Problem (14) 1: giv e a ( θ 0 ) , T k − 1 , w k − 1 ,? = T − 1 k − 1 a ( θ 0 ) , θ k,m , ρ k,m , m = 1 , · · · , M k , and A k , obtain c , C , d m , D m from (25), initialize p m , m = 1 , · · · , M k , by (34), set δ MAX > δ > 0 and η > 0 2: while δ MAX > δ do 3: update z by (29) 4: update p m , m = 1 , · · · , M k , by (30) 5: update λ m , m = 1 , · · · , M k , by (31) 6: calculate δ MAX by (32) 7: end while 8: obtain z ? = z 9: obtain h k,? by (25a) For m = 1 , · · · , M k , we update the vector λ m as λ m ( t + 1) = λ m ( t ) + η ( z ( t + 1) − p m ( t + 1)) . (31) The abov e steps 1 to 3 are repeated until a stopping criterion is reached, e.g., a maximum it eration number is attained and/or δ > δ MAX , max 1 ≤ m ≤ M k k z ( t + 1) − p m ( t + 1) k 2 (32) where δ > 0 is a small tolerance parameter . 3) Initialization of C-ADMM: Note that due to the non- con ve xity of problem (26), typical con v ergence results on ADMM do not apply and the ultimate z is not guaranteed to be optimal. Nevertheless, an appropriate initialization makes the abov e iterative algorithm [4] work well and even con ver ge to a Karush-Kuhn-T ucker (KKT) point. Follo wing [4], we initialize p m as p m =  < ( e p T m ) = ( e p T m )  T , m = 1 , · · · , M k (33) where e p m = [0 , · · · , 0 | {z } m − 1 , γ m,? , 0 , · · · , 0] T ∈ C M k . (34) In (34), γ m,? is obtained by the OP ARC algorithm and satisfies   ( w k − 1 ,? + γ m,? T − 1 k − 1 a ( θ k,m )) H a ( θ k,m )   2   ( w k − 1 ,? + γ m,? T − 1 k − 1 a ( θ k,m )) H a ( θ 0 )   2 = ρ k,m . (35) It can be verified that, the constraints (26c) can be satisfied if the initial settings p m , m = 1 , · · · , M k , take (33). This makes it easier to find an approximate solution of problem (26). Once the solution z ? has been obtained, we can reconstruct h k,? by (25a) and obtain w k,? as w k,? = w k − 1 ,? + T − 1 k − 1 A k h k,? . (36) The INRs of the ne wly assigned virtual interferences can by calculated via Σ k,? = Diag  − h k,?   A H k T − 1 k − 1 ( a ( θ 0 ) + A k h k,? )  . (37) T o make the above procedure clear , we summarize the C- ADMM approach to solve problem (14) in Algorithm 2. Notice from [4] that the C-ADMM approach is memory-efficient and can be implemented in a parallelized or distributed manner . Thus, for a large array , the C-ADMM approach in Algorithm 2 may be a better choice to solve problem (14) compared to the iterativ e approach in Algorithm 1, although more iterations may be needed. 6 Algorithm 3 Multi-point OP ARC Algorithm 1: giv e a ( θ 0 ) , T k − 1 and the weight vector w k − 1 ,? = T − 1 k − 1 a ( θ 0 ) , prescribe the angle θ k,m and the correspond- ing desired level ρ k,m , m = 1 , · · · , M k 2: calculate Σ k,? or h k,? using Algorithm 1 or Algorithm 2 3: obtain T k by (38) and calculate w k,? by (15) or (36) E. Update of Covariance Matrix Similar to the OP ARC algorithm, the VCM T k also needs to be renewed so as to facilitate the next execution of multi- point OP ARC. From the above discussions, T k is updated as T k = T k − 1 + A k Σ k,? A H k . (38) Accordingly , the weight vector is w k,? = T − 1 k a ( θ 0 ) . (39) This completes the procedure of multi-point OP ARC. Finally , we describe the steps of multi-point OP ARC in Algorithm 3. Note that in our proposed multi-point OP ARC algorithm, we carry out the parameter determination in a subspace with dimension M k , not in the whole space of dimension N . The benefit is the reduced amount of calculation. In addition, one can see that at most M max = N − 1 points can be precisely controlled, due to the limited degrees of freedom in problem (14) or (17). As a remark, the differences between the recent MA 2 RC in [2] and the proposed multi-point OP ARC in this paper are similar to those between A 2 RC and OP ARC described in [1] in details. I I I . A P P L I C A T I O N S O F M U L T I - P O I N T O P A R C In this section, we present three applications of multi-point OP ARC to array signal processing. A. Array P attern Synthesis Giv en the beam axis θ 0 , the problem of array pattern synthe- sis is to find an appropriate N × 1 weight vector that makes the response L ( θ, θ 0 ) meet some specific requirements. F or simplicity , we denote the desired pattern as L d ( θ ) . Basically , the proposed algorithm herein shares a similar concept of pattern synthesis using A 2 RC in [3]. Ho we ver , it is able to significantly reduce the number of iterations and improve the performance. 1) General Case: Generally , the array pattern synthesis can be started by setting k = 0 and the initial weight as w 0 ,? = a ( θ 0 ) . For k > 0 , multiple directions are selected by comparing L k − 1 ( θ , θ 0 ) : L k − 1 ( θ , θ 0 ) , | w H k − 1 a ( θ ) | 2  | w H k − 1 a ( θ 0 ) | 2 (40) with the desired pattern L d ( θ ) as follows. These angles can be in either the sidelobe region or the mainlobe region. For sidelobe synthesis, we only choose the peak angles in the set Ω k,S =  θ   L k − 1 ( θ , θ 0 ) > L k − 1 ( θ − ε, θ 0 ) and L k − 1 ( θ , θ 0 ) > L k − 1 ( θ + ε, θ 0 ) , θ ∈ Ω S } (41) where ε is a small positiv e quantity , Ω S denotes the sidelobe sector of the desired pattern. Different from the angle selection method in A 2 RC where the chosen peak angles hav e larger response le vels than their desired values, a selected peak angle in set Ω k,S may have a less response lev el than its desired one. For mainlobe synthesis, some discrete angles where the responses deviate considerably from the desired ones are chosen, and we denote the set of selected angles in the mainlobe region as Ω k,M . Then, we take: Ω k = Ω k,S ∪ Ω k,M , { θ k, 1 , · · · , θ k,M k } (42) where M k = card(Ω k ) . The multi-point OP ARC algorithm can thus be applied to adjust the corresponding responses of angles θ k,m to their desired values ρ k,m = L d ( θ k,m ) , m = 1 , · · · , M k , and the current response pattern L k ( θ , θ 0 ) can be obtained by using the resulting weight of multi-point OP ARC. Then, set k = k + 1 and repeat the above process until the response is satisfactorily synthesized. Note that the abov e iteration procedure is different from that in Section II.C where k is fixed and an internal iteration within the k th step is conducted. T o summarize, we describe the multi-point OP ARC based array pattern synthesis algorithm in Algorithm 4. As mentioned earlier , Ω k is forced to satisfy card(Ω k ) < N . Otherwise, we can simply reduce card(Ω k ) by modifying Ω k similar to what is done next. 2) P articular Consideration for Larg e Arrays: As afore- mentioned, the proposed multi-point OP ARC algorithm op- erates in an M k -dimensional subspace of the original N - dimensional space. This provides us an ef fecti ve strategy to pattern synthesis for large arrays, where the traditional ap- proaches may not work well or require extensi ve computation due to the large dimension. More specifically , for a large array and a pre-determined angle set Ω k (whose cardinality normally approaches to N ) in (42), we construct a new angle set Θ k as Θ k =  ¯ θ k, 1 , ¯ θ k, 2 , · · · , ¯ θ k,C k  (43) where C k is a prescribed number that is much smaller than N , ¯ θ k,c , c = 1 , · · · , C k , is the c th element of the vector: Sort(Ω k ) ∈ R card(Ω k ) (44) where Sort(Ω k ) re-arranges the elements of Ω k in the fol- lowing way: the larger | L k − 1 ( ¯ θ , θ 0 ) − L d ( ¯ θ ) | for ¯ θ ∈ Ω k is, the smaller index of ¯ θ in Sort(Ω k ) is, which makes ¯ θ more likely to be chosen as an element in the angle set Θ k in (43). The reason for this is that we expect to reduce the ov erall difference between the resulting pattern and the desired one. Once the new angle set Θ k is obtained, the multi-point OP ARC algorithm can be applied to realize L k ( ¯ θ , θ 0 ) = L d ( ¯ θ ) for ¯ θ ∈ Θ k . Then, set k = k + 1 and repeat the abov e process until the response is satisfactorily synthesized, and the cardinality of set Θ k , i.e., C k , can be flexibly v aried with the iteration number k . Finally , the above-described large-array pattern synthesis can be readily realized via Algorithm 4, by simply replacing Ω k in the 4th line of Algorithm 4 with the new angle set Θ k in (43). 7 Algorithm 4 Multi-point OP ARC based Array Pattern Syn- thesis Algorithm 1: giv e L d ( θ ) , w 0 ,? = a ( θ 0 ) , set k = 1 , T 0 = I , ε > 0 2: while 1 do 3: determine Ω k from (42) 4: apply multi-point OP ARC algorithm to realize L k ( θ , θ 0 ) = L d ( θ ) ( θ ∈ Ω k ), update w k,? and T k 5: if L k ( θ , θ 0 ) meets the requirement then 6: break 7: end if 8: set k = k + 1 9: end while 10: output w k,? and L k ( θ , θ 0 ) Since the above proposed algorithm, in either the general case or the large-array scenario, iteratively adjusts the re- sponses of sidelobe peaks, it is able to make all the sidelobe peaks align with the desired values. Thus, all the sidelobe responses can be well controlled to be lower than the giv en thresholds, and a satisfactory sidelobe shape can be well main- tained. Nevertheless, array pattern synthesis works in a data- independent way , the resulting weight or its corresponding beampattern is lack of adaptivity in suppressing undesirable interference and noise, which can be well rejected by the adaptiv e beamformer as discussed next. B. Multi-constraint Adaptive Beamforming The linearly constrained minimum variance (LCMV) beam- former is commonly used to enhance the robustness of array systems [13]–[15]. In LCMV beamformer , sev eral linear con- straints are imposed when minimizing the output variance, i.e., min w w H R n + i w (45a) sub ject to C H w = g (45b) where C is the constraint matrix that consists of D spatial steering vectors corresponding to the D constrained directions θ d , d = 0 , · · · , D − 1 , i.e., C = [ a ( θ 0 ) , a ( θ 1 ) , · · · , a ( θ D − 1 )] , g is a prescribed D -dimensional vector usually satisfying ( g ) 1 = 1 . The solution of problem (45) is given by w LCMV = R − 1 n + i C ( C H R − 1 n + i C ) − 1 g . (46) From (45b), we can clearly see that both the amplitude and the phase of the array output, i.e., w H a ( θ ) , hav e been strictly constrained at θ d , d = 0 , · · · , D − 1 . As a matter of fact, a less restrictiv e quadratically constrained minimum variance (QCMV) beamformer should be formulated by removing the unnecessary phase constraints, i.e., min w w H R n + i w (47a) sub ject to | ( C H w ) d | 2 = | ( g ) d | 2 , d = 1 , · · · , D . (47b) Note that in this subsection the variable d is an index and does not mean “desired” as used previously . Comparing to the QCMV in (47), we can see that the LCMV beamformer in (45) strictly limits the optimization of the weight vector to a smaller space, although it has a closed-from solution. It, thus, may cause the output SINR of LCMV beamformer to suffer from a loss, and the resulting pattern may be distorted. W e adopt the multi-point OP ARC algorithm to solve the QCMV problem (47), in the hope that the resulting output SINR can be improv ed (comparing to LCMV). If D = 1 , i.e., one constraint | a H ( θ 0 ) w | 2 = 1 is imposed in (47b), the optimal solution of (47) is given by w = R − 1 n + i a ( θ 0 ) a H ( θ 0 ) R − 1 n + i a ( θ 0 ) . (48) If D > 1 , based on the first constraint that | a H ( θ 0 ) w | 2 = 1 , we have L ( θ d − 1 , θ 0 ) = | w H a ( θ d − 1 ) | 2 in (47b). Then, the additional ( D − 1) constraints can be taken into account by imposing the following constraints: L ( θ d − 1 , θ 0 ) = | ( g ) d | 2 , d = 2 , · · · , D . (49) Then, the problem becomes how to realize the above described multi-point response control, starting from the optimal weight vector in (48). T o apply the multi-point OP ARC algorithm, we rewrite w in (48) as w = 1 σ 2 n a H ( θ 0 ) R − 1 n + i a ( θ 0 ) T − 1 n + i a ( θ 0 ) , c w 0 (50) where c is a constant satisfying c = ( σ 2 n a H ( θ 0 ) R − 1 n + i a ( θ 0 )) − 1 , T n + i and w 0 = T − 1 n + i a ( θ 0 ) act as the initial VCM in (4) and the initial weight vector in multi-point OP ARC, respectiv ely . Then, a multi-point OP ARC procedure can be applied to fulfill the response requirement described in (49), and the ultimate weight vector of QCMV (denoted as w QC ) can be obtained accordingly . Note that in practical applications, R n + i can be estimated from data x ( t ) : ˆ R n + i = 1 T T X t =1 x ( t ) x H ( t ) (51) where T is the number of snapshots. In addition, σ 2 n can be estimated by [26] ˆ σ 2 n = 1 N − J r N X n = J +1 λ n (52) where J r is the number of interferences, λ 1 ≥ λ 2 ≥ · · · ≥ λ N are eigenv alues of ˆ R n + i . Replacing R n + i and σ 2 n with ˆ R n + i and ˆ σ 2 n , respectively , we have summarized the proposed algorithm in Algorithm 5. T o ha ve a better understanding, we denote the corresponding VCM of w QC as T QC . Recalling the property (39) of multi- point OP ARC, w QC and T QC satisfy w QC = T − 1 QC a ( θ 0 ) . (53) W e can see that the obtained weight w QC minimizes the total variance w H T QC w with the constraints (47b), rather than minimizing w H T n + i w or its equiv alent term w H R n + i w (for a fix ed σ 2 n ) in (47a). Nevertheless, we kno w from Proposition 7 of the companion paper [1] that the obtained weight of OP ARC also minimizes the variance at the previous step. Thus, w QC is the optimal solution of problem (47) for the special case when 8 w H QC T QC w QC = w H QC T n + i + D X d =2 β d − 1 a ( θ d − 1 ) a H ( θ d − 1 ) ! w QC = w H QC T n + i w QC + | w H QC a ( θ 0 ) | 2 D X d =2 β d − 1 | ( g ) d | 2 ! | {z } =0 = w H QC T n + i w QC = w H QC R n + i w QC σ 2 n (54) Algorithm 5 Multi-point OP ARC based Multi-constraint Adaptiv e Beamforming Algorithm 1: giv e interference number J r , constraint matrix C and vec- tor g , estimate ˆ R n + i and ˆ σ 2 n by (51) and (52), respectively , calculate T n + i = ˆ R n + i / ˆ σ 2 n and w 0 = T − 1 n + i a ( θ 0 ) 2: apply multi-point OP ARC algorithm to realize L ( θ d − 1 , θ 0 ) = | ( g ) d | 2 , d = 2 , · · · , D , by setting T n + i and w 0 as the initial VCM and the initial weight vector , respectiv ely , to obtain w QC D = 2 , i.e., only one extra constraint is imposed besides the constraint | a H ( θ 0 ) w | 2 = 1 . In addition, the obtained w QC offers the optimal solution of problem (47) if we impose null constraint at θ d − 1 , d = 2 , · · · , D , based on the following argument. In this case, we set | ( g ) d | 2 = 0 , d = 2 , · · · , D , and thus obtain (54) on the top of this page, where we ha ve used the fact that | w H QC a ( θ d − 1 ) | 2 | w H QC a ( θ 0 ) | 2 = | ( g ) d | 2 = 0 , d = 2 , · · · , D (55) and T QC = T n + i + D X d =2 β d − 1 a ( θ d − 1 ) a H ( θ d − 1 ) (56) with β d − 1 denoting the INR of the assigned virtual interfer- ence at θ d − 1 . From (54) we know that w QC also minimizes w H R n + i w . The optimality (in the sense of output SINR) of the proposed algorithm is guaranteed in the abo ve two scenarios. Otherwise, the proposed algorithm performs better than LCMV algorithm in most cases as we shall see from the simulations later . Moreov er , (53) and (56) indicate that the resulting weight vector w QC is obtained by making a normalized cov ariance matrix loading (NCL), which can be regarded as a generaliza- tion of the diagonal loading (DL) in [16]–[18], on the initial T n + i . The loading quantity is precisely determined by multi- point OP ARC algorithm as ∆ = D X d =2 β d − 1 a ( θ d − 1 ) a H ( θ d − 1 ) . (57) Recalling Eqn. (38) of [1], one learns in OP ARC that the INR of a ne wly assigned virtual interference depends on the previous normalized covariance matrix and also contributes to the current one. Then, re visiting Algorithm 1, where OP ARC is iterativ ely applied, and Eqn. (16), one can see that the resulting β d − 1 , d = 2 , · · · , D , depend on the initial T n + i . Thus, the loading quantity ∆ in (57) is related to the gi ven constraints in (47b) and also the real data. Note that the abov e-described multi-constraint adaptive beamforming algorithm improves the robustness of array sys- tems while blocking the unexpected interference and noise. Howe v er , different from the method in the preceding subsec- tion where the sidelobe peaks can be controlled iteratively , the algorithm in this subsection only has constraints on the re- sponse lev els of sev eral pre-assigned angles θ 0 , θ 1 , · · · , θ D − 1 . It cannot control/guarantee an overall sidelobe pattern. C. Quiescent P attern Contr ol In adaptiv e beamforming, weight vector is designed in a data-dependent manner . Howe ver , the traditional adaptiv e beamforming methods usually yield a beampattern with high sidelobes. T o obtain low sidelobes in adaptive arrays, the concept of quiescent pattern control is introduced in [19], by combining the adaptiv e beamforming and deterministic pattern synthesis techniques. In brief, when an adaptiv e array operates in the presence of white noise only , the resultant adaptiv e beamformer is named as the quiescent weight vector , and the corresponding array response is termed as the quiescent pattern. Following the concept of quiescent pattern control in [19]–[21], it is required to find a mechanism to design a beamformer having the ability to reject an interference (if it exists) and noise, and meanwhile, maintaining the desirable shape of the quiescent pattern when only white noise presents. Note that the quiescent weight v ector of LCMV beamformer in (46) is w q = C ( C H C ) − 1 g that can be readily obtained by setting R n + i = σ 2 n I . Unfortunately , for a giv en desired quiescent pattern, which usually has specific constraints on the upper lev el of sidelobes, it is not easy to have a satisfactory quiescent pattern via LCMV by specifying C and g , since LCMV only imposes constraints on a fixed set of pre-assigned finite angles as mentioned at the end of Section III.B. This is similarly true for the multi-point OP ARC algorithm presented in the preceding Section III.B. Moreov er , if we employ the iterativ e approach adopted in deterministic pattern synthesis in Section III.A to modify the shape of the obtained beam- pattern, nulls may not be always formed at the directions of unknown real interferences, and the adaptivity in suppressing undesirable components is thus not well guaranteed. In this subsection, a systematic approach to quiescent pat- tern control is proposed. A two-stage procedure is dev eloped, by taking adv antage of the deterministic pattern synthesis approach in Section III.A and also the concept of NCL mentioned in Section III.B. More specifically , giv en a desired quiescent pattern, denoted as L d ( θ ) , the multi-point OP ARC based pattern synthesis algorithm in Section III.A, see, Al- gorithm 4, is adopted in the first stage to design a desirable quiescent pattern off-line. Denote by w q , T q and L q ( θ , θ 0 ) the obtained (quiescent) weight vector , the associated VCM 9 Algorithm 6 Multi-point OP ARC based Quiescent Pattern Control Algorithm 1: giv e L d ( θ ) , synthesize a desirable quiescent pattern L q ( θ , θ 0 ) using Algorithm 4, obtain w q and T q 2: estimate ˆ R n + i and ˆ σ 2 n by (51) and (52), respectiv ely , set T n + i = ˆ R n + i / ˆ σ 2 n 3: obtain adaptiv e weight vector w a by Eqn. (60) 4: if extra constraints needed, modify w a by conducting the multi-point OP ARC algorithm 5: output the obtained weight w a and its corresponding response pattern L a ( θ , θ 0 ) and the resulting response pattern, respectively . It satisfies w q = T − 1 q a ( θ 0 ) . (58) As mentioned earlier , the resulting L q ( θ , θ 0 ) performs well in maintaining the shape of L d ( θ ) , howe ver , the abo ve weight w q has no ability to reject the potential interferences and noise. A strategy of finding weight vector is thus required in quiescent pattern control to, not only maintain the shape of L d ( θ ) if only white noise exists, but also suppress a possible real interference and noise. From the adaptive array theory , a data-dependent loading quantity ∆ needs to be added to the VCM T q , such that the potential interferences and noise can be rejected. Moreover , in the white noise only case, ∆ should be zero such that the weight w q in (58) can be retriev ed. T o do so, we carry out the second stage, by taking a real data into consideration and carrying out an NCL operator to the VCM T q via setting the associated loading quantity ∆ as ∆ = − I + T n + i (59) where T n + i = R n + i /σ 2 n . The ultimate (adaptive) weight vector is thus calculated as w a = ( T q − I + T n + i ) − 1 a ( θ 0 ) . (60) The corresponding response pattern of w a (denoted as L a ( θ , θ 0 ) ) can be obtained accordingly . One can see that there are tw o components being suppressed by w a in (60). The first one is the component of the virtual interference which corresponds to T q − I and helps to maintain the shape of L d ( θ ) . The second component is T n + i , which contains the real interference and noise that need to be rejected. In the noise only scenario, the loading quantity ∆ offsets zero automatically and the quiescent weight vector w q in (58) appears, provided that the real noise shares the same structure as the virtual noise, i.e., R n + i = σ 2 n I or T n + i = I . Therefore, we can see that the weight vector w q in (58) and its corresponding beampattern L q ( θ , θ 0 ) are exactly the quiescent weight vector and quiescent pattern, respectively . Also, we should replace the unknown R n + i and σ 2 n with ˆ R n + i in (51) and ˆ σ 2 n in (52), respectiv ely , and set T n + i = ˆ R n + i / ˆ σ 2 n in practical applications. It should be emphasized that we do not impose extra constraints (e.g., fixed null constraints considered in [19]) on the resulting response pattern L a ( θ , θ 0 ) , since such kind of constraints can be aforehand considered in the first stage of the above procedure. In addition, we can also make the fixed constraints satisfied by performing the multi-point OP ARC algorithm starting from the obtained w a in (60) and its corre- sponding normalized covariance matrix T = T q − I + T n + i . This is similar to the idea used in the preceding subsection. T o make it clear, we have summarized the multi-point OP ARC based quiescent pattern control algorithm in Algorithm 6. I V . N U M E R I C A L R E S U LTS W e next present some simulations to demonstrate the proposed multi-point OP ARC algorithm and its applications. Unless otherwise specified, we set ω = 6 π × 10 8 rad / s and consider an 11-element nonuniform spaced linear array with nonisotropic elements. Both the element locations x n and the element patterns g n ( θ ) are listed in T able I in Part I [1], and the same array configuration has been adopted in Part I [1]. The beam axis is steered to θ 0 = 20 ◦ . W e set β  = 10 − 10 in conducting the iterativ e approach, and take δ = 10 − 15 and η = 900 for the C-ADMM approach. In addition, f n is specified as the all-zero vector for the MA 2 RC algorithm in [2] for comparison, SNR is taken as 10dB when it applies. A. Illustration of Multi-point OP ARC In this subsection, we demonstrate the multi-point OP ARC algorithm. Both the iterativ e approach and the C-ADMM approach are conducted, and then compared with the MA 2 RC algorithm. For conv enience, we carry out two steps of the array response control algorithms with each step controlling two angles, i.e., M 1 = M 2 = 2 , and denote the adjusted angles and the corresponding desired lev els of the k th ( k = 1 , 2 ) step as θ k,m and ρ k,m , m = 1 , · · · , M k , respectively . F ollo wing the e valuation strategy adopted in [1], we define D m , | L 2 ( θ 1 ,m , θ 0 ) − L 1 ( θ 1 ,m , θ 0 ) | (61) to measure the response level differences between two con- secutiv e response controls at θ 1 ,m , m = 1 , · · · , M 1 , where L k ( θ , θ 0 ) represents the resultant response after finishing the k -th step of weight update, k = 1 , 2 . In addition, the de viation J : J ∆ = v u u t 1 I I X i =1   L 2 ( ϑ i , θ 0 ) − L 1 ( ϑ i , θ 0 )   2 (62) is also considered, where ϑ i stands for the i th sampling point in the angle sector, I denotes the number of sampling points. More specifically , we set θ 1 , 1 = − 45 ◦ , ρ 1 , 1 = − 40dB , θ 1 , 2 = − 5 ◦ and ρ 1 , 2 = − 30dB for the first step of the response control. Note that the same settings have been adopted in Section V .A in Part I [1], where the single-point response control is realized in sequence. In this part, we first conduct multi-point OP ARC algorithm by using the iterative method described in Algorithm 1. In the first iteration, the OP ARC algorithm in [1] is applied to control the responses of θ 1 ,m to their desired lev els ρ 1 ,m , m = 1 , 2 , one-by-one on m . W e have β 1 , 1 ,? = 1 . 5683 , β 1 , 2 ,? = 0 . 2504 , which is the same as the results obtained in Section V .A in Part I [1]. Then, we continue our multi-point OP ARC algorithm by conducting the abov e iteration procedure for a number of times. The curve of 10 1 2 3 4 5 6 7 8 Iteration Number 10 -15 10 -10 10 -5 10 0 MAX Fig. 1. Curve of β MAX versus the iteration number . 0 50 100 150 200 Iteration Number 10 -20 10 -15 10 -10 10 -5 10 0 MAX Fig. 2. Curve of δ MAX versus the iteration number . β MAX versus the iteration number is depicted in Fig. 1. Note that the parameter β MAX measures the maximal magnitudes of INRs of the ne wly assigned virtual interferences in the current iteration, as shown in the 8th line of Algorithm 1. From Fig. 1, one can see that β MAX decreases with iteration. Moreov er , observation shows that it only requires fiv e iterations to con ver ge, i.e., β MAX ≤ β  , and the result is ¯ β 1 , 1 ,? = 1 . 4700 and ¯ β 1 , 2 ,? = 0 . 2506 , which is, respectiv ely , close to β 1 , 1 ,? and β 1 , 2 ,? . Now we test the performance of the C-ADMM approach. The obtained δ MAX in (32) reduces with the iter- ation, i.e., the procedure described in (29)-(31), as shown in Fig. 2, and δ MAX ≤ δ is met after about 130 iterations. W e obtain h 1 ,? = [ − 0 . 1458 − j 0 . 0203 , − 0 . 0687 − j 0 . 0397] T . Not surprisingly , it can be checked that the results of the above two approaches correspond to the same weight vector . Hence, the same beampatterns are synthesized for these two approaches as shown in Fig. 3(a), from which one can see that the responses of the two adjusted angles hav e been precisely controlled to their desired values. Interestingly , when testing the MA 2 RC, the resulting pattern is completely the same as that of the multi-point OP ARC algorithm. W e believ e that this occurs not accidentally but with a reason that is, unfortunately , not clear yet. -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) initial pattern MA 2 RC proposed (iteration) proposed (C-ADMM) (a) The first step -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) previous step MA 2 RC proposed (iteration) proposed (C-ADMM) (b) The second step Fig. 3. Illustration of multi-point OP ARC algorithm. T ABLE I O B T A IN E D P A RA M E T ER C OM PA RI S O N MA 2 RC Multi-point OP ARC D 1 (dB) 21 . 3110 5 . 7620 D 2 (dB) 13 . 5149 10 . 0816 J 0 . 3132 0 . 1909 G 1 (dB) 10 . 0078 10 . 0078 G 2 (dB) 9 . 9192 11 . 2550 In the second step of the response control, we take θ 2 , 1 = 7 ◦ , ρ 2 , 1 = − 25dB , θ 2 , 2 = 28 ◦ and ρ 2 , 2 = 0dB . When con- ducting the multi-point OP ARC algorithm, we obtain ¯ β 2 , 1 ,? = 0 . 2555 and ¯ β 2 , 2 ,? = − 0 . 0804 for the iterativ e approach, and find h 2 ,? = [ − 0 . 1803 − j 0 . 0653 , − 0 . 5434 − j 0 . 9252] T after implementing the C-ADMM method. Again, the above two sets of results correspond to the same beampattern as sho wn in Fig. 3(b), where the resulting pattern of MA 2 RC is also displayed. From Fig. 3(b), one can see that all the adjusted angles hav e been accurately controlled as expected, for the three approaches. Howe ver , the mainlobe of the ultimate pattern of MA 2 RC is distorted and a high sidelobe lev el is resulted. For comparison purpose, we ha ve listed several parameter measurements in T able I, from which one can see that the MA 2 RC method brings large v alues on both D k 11 -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) initial pattern the 1st step -60 -40 -20 -35 -30 -25 -20 (a) Synthesized pattern at the 1st step -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) previous step the 2nd step -40 -20 0 -35 -30 -25 -20 (b) Synthesized pattern at the 2nd step -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) previous step the 3rd step -14 -12 -10 -8 -6 -4 -37 -36 -35 -34 -33 (c) Synthesized pattern at the 3rd step Fig. 4. Resultant patterns at different steps when carrying out a nonuniform sidelobe synthesis for a nonuniform linear array . -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) Philip's CP A 2 RC MA 2 RC proposed -70 -60 -50 -40 -30 -30 -25 -20 -15 Fig. 5. Resultant pattern comparison. ( k = 1 , 2 ) and J , and results a less array gain compared to the proposed multi-point OP ARC algorithm. B. Array P attern Synthesis Using Multi-point OP ARC Starting from this subsection, the applications of multi-point OP ARC are simulated and the iterative approach in Section II. C is adopted to illustrate the results. In this subsection, we focus upon the application of multi-point OP ARC to array pattern synthesis and give two representativ e examples for demonstration. 1) Nonuniform Sidelobe Synthesis: In the first example, the desired pattern has nonuniform sidelobes. Fig. 4 shows the synthesized patterns of the proposed algorithm at different steps. Clearly , in each synthesis step, all the sidelobe peaks, i.e., Ω k in (42), are first determined from the previously synthesized pattern. Notice that the response le vel of a selected sidelobe peak can be either higher or lower (see Fig. 4(a) for reference) than its desired level. It has been sho wn in Fig. 4 that it only requires 3 steps, i.e., k = 3 , to synthesize a satisfactory beampattern. For comparison, the resulting patterns of the proposed algorithm, Philip’ s method in [11], con ve x programming (CP) method in [8], A 2 RC method (after carrying out 30 steps) in [3] and MA 2 RC method (after carrying out 3 steps) in [2] are displayed in Fig. 5. As expected, we can see that the T ABLE II E X EC U T I ON T IM E C O M P A R I SO N W H E N C O ND U C T IN G A L A RG E - AR R AY P ATT E R N S Y NT H E S IS Philip’ s CP A 2 RC MA 2 RC proposed T (sec) 2.22 12.36 3.55 2.55 0.05 pattern en velopes of Philip’ s method and CP method are not aligned with the desired lev el, since they cannot control the beampattern precisely according to the required specifications. Although A 2 RC and MA 2 RC have the ability to precisely control the giv en array responses, the obtained sidelobe peaks are not aligned with the desired ones either , since only the sidelobe peaks higher than the desired levels are selected and adjusted in these two approaches. 2) Larg e Array Consideration: In this example, pattern synthesis for a large linearly half-wa velength-spaced array with N = 80 isotropic elements is considered. The desired pattern steers at θ 0 = 50 ◦ with nonuniform sidelobes. More specifically , the upper level is − 35dB in the sidelobe region [ − 90 ◦ , 50 ◦ ) and − 25dB in the rest of the sidelobe region. Fig. 6 demonstrates sev eral intermediate results of the proposed algorithm. In ev ery step, we select C k = 20 sidelobe peak angles (see Eqn. (43) and (44) for details) and then adjust their responses to the desired levels by using multi- point OP ARC algorithm. Simulation result shows that it only requires 11 steps, i.e., k = 11 , to synthesize a qualified pattern, see the ultimate pattern in Fig. 6(c) for reference. The ex ecution times of various methods are provided in T able II, where the superiority of the proposed algorithm can be clearly observed. C. Multi-constraint Adaptive Beamforming Using Multi-point OP ARC In this subsection, the multi-constraint adaptiv e beamform- ing is realized by using the multi-point OP ARC algorithm. For simplicity , a perfect knowledge of the data covariance matrix is assumed. 1) Sidelobe Constraint: In the first case, four sidelobe constraints are required. More specifically , the response levels of − 20 ◦ , − 18 ◦ , − 16 ◦ and − 14 ◦ are expected to be all − 40dB . T wo interferences are impinged from − 40 ◦ and − 28 ◦ with INRs 30dB and 25dB , respectively . 12 -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) initial pattern the 1st step 30 35 40 45 -35 -30 -25 (a) Synthesized pattern at the 1st step -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) previous step the 2nd step -18 -16 -14 -12 -38 -36 -34 (b) Synthesized pattern at the 2nd step -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) previous step the 11th step 10 12 14 -35.1 -35 -34.9 (c) Synthesized pattern at the 11th step Fig. 6. Resultant patterns at different steps when carrying out a nonuniform sidelobe synthesis for a large uniform linear array . -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) optimal LCMV proposed -20 -18 -16 -14 -42 -40 -38 (a) The first case -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) optimal LCMV proposed 16 18 20 22 24 -0.2 0 0.2 (b) The second case Fig. 7. Result comparison of multi-constraint adaptive beamforming. Fig. 7(a) displays the results of the optimal beamformer with no sidelobe constraint, the LCMV method [13] and the proposed one. Clearly , both the LCMV beamformer and the proposed algorithm are able to shape deep nulls at the directions of interferences (see the blue line). Meanwhile, the giv en sidelobe constraints are well satisfied for both. When considering the output SINR, we have SINR = 19 . 5601dB for the LCMV method and SINR = 19 . 6906dB for the proposed one. W e can see that the proposed beamformer brings an improv ement on the output SINR compared to the LCMV beamformer . 2) Mainlobe Constraint: In the second case, two constraints are imposed in the mainlobe region. The constraint angles are 19 ◦ and 21 ◦ , and both of the desired lev els are 0dB . There are three interferences coming from − 32 ◦ , 50 ◦ and 60 ◦ with an identical INR 30dB . Fig. 7(b) depicts the resultant patterns. One can see that the obtained pattern of the LCMV method is severely distorted, although the two prescribed constraints are satisfied and the three interferences are rejected. The corresponding output SINR is 11 . 1767dB . Observing the resulting pattern of the proposed algorithm, the two-point constraint is well satisfied and a flat-top mainlobe is shaped with no distortion occurred. The corresponding output SINR is 17 . 1260dB , which is much higher than that of the LCMV method. D. Quiescent P attern Contr ol Using Multi-point OP ARC In this subsection, we test the performance of the multi- point OP ARC based quiescent pattern control algorithm. The desired quiescent pattern has a nonuniform sidelobe lev el as depicted with black dash lines in Fig. 5. In our proposed algorithm, quiescent pattern synthesis and quiescent pattern control are jointly designed by the multi- point OP ARC algorithm. W e ha ve detailed the of f-line syn- thesis procedure in Section IV .B and illustrated the obtained quiescent pattern by red line in Fig. 5. Suppose that two interferences come from − 55 ◦ and − 49 ◦ with INRs 30dB . The obtained adaptiv e response pattern is shown in Fig. 8(a), where we can observe that two nulls are formed at the directions of the real interferences, and the resultant sidelobe is close to the quiescent one. The obtained output SINR is 19 . 2984dB for the proposed algorithm. For comparison purpose, the classical linearly-constraint based quiescent pattern control approach (denoted as LC-QPC method for briefness) in [19] is also demonstrated, by using the same synthesized quiescent pattern in Fig. 5. The resulting pattern of LC-QPC is displayed in Fig. 8(a), where we find that an obvious perturbation is caused in the sector [ − 15 ◦ , 0 ◦ ] and the overall shape can not be well maintained compared to the desired one. The obtained output SINR is 19 . 2161dB , which is lower than that of the proposed algorithm. 13 -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) LC-QPC proposed -30 -20 -10 0 -35 -30 -25 -20 (a) The first case -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) LC-QPC proposed 55 60 65 -42 -40 -38 -36 -34 (b) The second case Fig. 8. Result comparison of quiescent pattern control. Now we take extra fixed constraints into consideration by restricting the response lev els at directions 58 ◦ and 62 ◦ to be all − 40dB . The results of the proposed algorithm and the LC- QPC method are presented in Fig. 8(b), where we observe that both of these two methods are able to reject the undesirable interferences with the prescribed constraints being satisfied. The same as before, the proposed algorithm maintains a more desirable shape than that of the LC-QPC method. When taking the output SINR into account, the corresponding values are, respectiv ely , 19 . 2382dB (for the proposed algorithm) and 19 . 0967dB (for the LC-QPC method). The advantage of the proposed algorithm is verified again. V . C O N C L U S I O N S In this paper , the optimal and precise array response control (OP ARC) algorithm proposed in Part I [1] has been extended from a single point per step to a multi-points per step. 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