OPARC: Optimal and Precise Array Response Control Algorithm -- Part I: Fundamentals

1 OP ARC: Optimal and Precise Array Response Control Algorithm – P art I: Fundamentals 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 pr oblem of how to optimally and precisely control array response levels is addressed. By using the concept of the optimal weight vector from the adaptive array theory and adding virtual interferences one by one, the change rule of the optimal weight vector is f ound and a new formulation of the weight vector update is thus de vised. Then, the issue of how to precisely control the response le vel of one single direction is inv estigated. More specifically , we assign a virtual interference to a direction such that the r esponse level can be pr ecisely controlled. Moreov er , the parameters, such as, the interference- to-noise ratio (INR), can be figured out according to the desired level. Additionally , the parameter optimization is carried out to obtain the maximal array gain. The resulting scheme is called optimal and pr ecise array r esponse control (OP ARC) in this paper . T o understand it better , its properties are given, and its comparison with the existing accurate array response control (A 2 RC) algorithm is provided. Finally , simulation results are presented to verify the effectiveness and superiority of the proposed OP ARC. Index T erms —Array response control, adaptive array theory , array pattern synthesis, array signal processing . I . I N T R O D U C T IO N A RRA Y antenna has been extensiv ely applied in many fields, such as, radar , navig ation and wireless commu- nications [1]. It is known that the array pattern design is of significant importance to enhance system performance. For instance, in radar systems, it is desirable to mitig ate returns from interfering signals, by designing a scheme which results in nulls at directions of interferences. In some communication systems, it is critical to shape multiple-beam patterns for multi-user reception. Additionally , synthesizing a pattern with broad mainlobe is beneficial to extend monitoring areas in satellite remote sensing. Generally speaking, array pattern can be designed either adaptiv ely or non-adaptiv ely . Determining the comple x weights for array elements so as to achie v e a desired beampattern is known as array pattern synthesis. W ith regard to this problem, it is expected to find weights that satisfy a set of specifications on a gi ven beampattern, in a data-independent or nonadaptiv e manner . 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, University of Delaware, Newark, DE 19716, USA (e-mail: xjzhang@udel.edu; xxia@ee.udel.edu). B. Liao is with College of Information Engineering, Shenzhen University , 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). Over the past se veral decades, a great number of pattern synthesis approaches have been proposed, see, e.g., [2]–[14]. Particularly , in [12], an iterative sampling method is utilized to make the sidelobe peaks conform to a specified shape within a given tolerance. For sidelobe control in cylindrical arrays, an artificially created noise source en vironment can be utilized [13]. For a circular ring array , a symmetrical pattern with low sidelobes is achiev ed in [14] by adopting a field- synthesis technique. Note that although the solutions in [12]– [14] are able to control sidelobes for arrays with some partic- ular configurations, they cannot be straightforwardly extended to general geometries. On the contrary , the accurate array response control (A 2 RC) approach [15] provides a simple and effecti v e manner to accurately control array response level of arbitrary arrays. Since the A 2 RC approach deals with the response control of a single direction, more recently , a multi- point accurate array response control (MA 2 RC) method has been de veloped in [16] to flexibly adjust array responses of multiple points. Ho wev er , a satisfactory performance cannot be always guaranteed in either [15] or [16] due to the empirical solution adopted in this kind of approaches. These shortcomings of the existing approaches motiv ate us to hav e an innovati ve method to precisely , flexibly and opti- mally control the response lev el. T o do so, we first in vestigate how the optimal weight vector in the adapti ve array theory changes along with the increase of the number of interferences. Then, a new scheme for weight vector update is developed and further exploited to realize the precise array response control. Furthermore, a parameter optimization mechanism is proposed by maximizing the array gain [17]. It is sho wn that, the proposed optimal and precise array response control (OP ARC) approach is capable of precisely and flexibly controlling the response lev el of an arbitrary array . Furthermore, its optimality (in the sense of array gain) can be well guaranteed. In this paper , the proposed OP ARC scheme is de veloped by e xploiting the interference-to-noise ratio (INR). It should be mentioned that this paper focuses on the main concepts and fundamentals of the OP ARC scheme, while the extensions and applications (such as pattern synthesis and quiescent pattern control [18]) will be carried out in the companion paper [19]. The rest of the paper is organized as follows. Our proposed OP ARC algorithm is presented in Section II. Further insights into OP ARC are presented in Section III to provide more useful and interesting properties. In Section IV , comparisons between OP ARC and the existing A 2 RC are presented. Representative simulations are conducted in Section V and conclusions are drawn in Section VI. 2 Notations: W e use bold upper-case and lower -case letters to represent matrices and vectors, respectively . In particular , we use I , 1 and 0 to denote the identity matrix, the all-one vector and the all-zero vector , respectively . j , √ − 1 . ( · ) T , ( · ) ∗ and ( · ) H stand for the transpose, complex conjugate and Hermitian transpose, respectively . | · | denotes the absolute value and k · k 2 denotes the l 2 norm. W e use H ( i, l ) to stand for the element at the i th row and l th column of matrix H . < ( · ) and = ( · ) denote the real and imaginary parts, respectively . det( · ) is the determinant of a matrix. The sign function is denoted by sign( · ) .  represents the element-wise division operator . W e use diag( · ) to return a column vector composed of the diagonal elements of a matrix, and use Diag( · ) to stand for the diagonal matrix with the components of the input vector as the diagonal elements. Finally , R and C denote the sets of all real and all complex numbers, respectively , and S N ++ denotes the set of N × N positi ve definite matrices. I I . O P A R C A L G O R I T H M In order to present our proposed OP ARC algorithm, we first briefly recall the adaptiv e array theory . A. Adaptive Array Theory Consider an array of N elements and assume that the noise is white and the interferences are independent with each other . T o suppress the unwanted interferences and noise, the optimal adaptiv e beamformer weight vector w steering to the direction θ 0 can be obtained by maximizing the output signal- to-interference-plus-noise ratio (SINR) defined as SINR = σ 2 s | w H a ( θ 0 ) | 2 w H R n + i w (1) where σ 2 s stands for the signal po wer , R n + i denotes the N × N noise-plus-interference covariance matrix and a ( θ 0 ) represents the signal steering vector . More exactly , for a giv en θ , we hav e a ( θ ) = [ g 1 ( θ ) e − j ωτ 1 ( θ ) , · · · , g N ( θ ) e − j ωτ N ( θ ) ] T (2) 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 . It is known that the optimal weight vector w opt , which maximizes the SINR, is gi ven by [17] w opt = α R − 1 n + i a ( θ 0 ) (3) where α is a normalization factor and does not affect the output SINR, and hence, will be omitted in the sequel. Note that the above SINR can be expressed as G · σ 2 s /σ 2 n , where G is defined as G = | w H a ( θ 0 ) | 2 w H T n + i w (4) with T n + i , R n + i /σ 2 n standing for the normalized noise- plus-interference covariance matrix, i.e., T n + i = R n + i σ 2 n = I + k X ` =1 β ` a ( θ ` ) a H ( θ ` ) (5) where β ` , σ 2 ` /σ 2 n denotes the interference-to-noise ratio (INR), k is the number of interferences, a ( θ ` ) is the steering vector of the ` th interference, σ 2 n and σ 2 ` stand for the noise and interference powers, respectiv ely . Note that G represents the amplification factor of the input signal-to-noise ratio (SNR) σ 2 s /σ 2 n , and therefore, is termed as the array gain [17]. As a result, the criterion of array gain G maximization is adopted to achiev e the optimal weight vector . B. Update of the Optimal W eight V ector It can be seen from (3)–(5) that the optimal weight vector w opt depends on R n + i or T n + i , which is not available for the follo wing data-independent array response control: for a giv en steering vector a ( θ ) in (2) and a beam axis θ 0 , design a weight vector w such that the normalized array response L ( θ , θ 0 ) , | w H a ( θ ) | 2  | w H a ( θ 0 ) | 2 meets some specific re- quirements. In this paper, we are interested in the requirements of array response le vels, i.e., finding weight vectors such that the array responses at a given set of angles are equal to a set of predescribed values. Our basic idea is to construct a virtual normalized noise-plus-interference cov ariance matrix (VCM), denoted as T k , to achie ve the given response control task. Note that since the VCM T k to be determined is not produced by real data, it may not hav e any physical meaning. Moreov er , it can be neither positiv e definite nor Hermitian (its rationality will be discussed later). By making use of the VCM, the data-dependent adapti ve array theory can be applied to the data-independent situation considered in this paper . This allows us to optimally update the weight v ector w k − 1 , opt = T − 1 k − 1 a ( θ 0 ) to w k, opt such that a desired response lev el ρ k at θ k can be achie ved by assigning an appropriate virtual interference. Thus, the problem we concern here is to figure out the characteristics, e.g., INR, of the virtual interference. W e use induction to describe the problem and the algorithm below . Suppose that the response le vels of the k − 1 directions hav e been successiv ely controlled by adding k − 1 virtual interferences. Meanwhile, the corresponding VCM is denoted as T k − 1 . For a giv en θ k and its desired level ρ k , we can assign the k th virtual interference coming from θ k by designing its INR (i.e., β k ). T o find out β k , from (5) we notice that the VCM can be updated as T k = T k − 1 + β k a ( θ k ) a H ( θ k ) . (6) Using the W oodbury Lemma [20], we have T − 1 k = T − 1 k − 1 − β k T − 1 k − 1 a ( θ k ) a H ( θ k ) T − 1 k − 1 1 + β k a H ( θ k ) T − 1 k − 1 a ( θ k ) . (7) Accordingly , the optimal weight vector is giv en by w k, opt = T − 1 k a ( θ 0 ) . Recalling (3) and (7), we can express w k, opt as w k, opt = w k − 1 , opt + γ k T − 1 k − 1 a ( θ k ) (8) where w k − 1 , opt = T − 1 k − 1 a ( θ 0 ) denotes the previous optimal weight vector and γ k is given by γ k = − β k a H ( θ k ) T − 1 k − 1 a ( θ 0 ) 1 + β k a H ( θ k ) T − 1 k − 1 a ( θ k ) , Ψ k ( β k ) (9) 3 with Ψ k ( · ) denoting a mapping from β k to γ k . Note that the solution in (8)–(9) gi ves the optimal solution for maximizing the SINR, which may not meet the response lev el ρ k at θ k . In order to meet this response lev el requirement, we next consider the following questions first. Given the previous weight vector w k − 1 , opt = T − 1 k − 1 a ( θ 0 ) , does there exist γ k (or equiv alently β k ) such that the response lev el at θ k is precisely ρ k ? and what value it should be if it exists? T o do so, we reformulate the weight vector as w k = w k − 1 + γ k v k (10) where the subscript ( · ) opt is omitted for notational simplicity and v k is defined as v k , T − 1 k − 1 a ( θ k ) . (11) Mathematically , the problem of finding γ k such that the array response lev el at θ k is ρ k can be written as L ( θ k , θ 0 ) = | w H k a ( θ k ) | 2  | w H k a ( θ 0 ) | 2 = ρ k (12) where the desired array response lev el satisfies ρ k ≤ 1 . The combination of (10) and (12) yields z H k H k z k = 0 (13) where z k and H k are, respectively , defined as z k , [1 γ k ] T H k , [ w k − 1 v k ] H  a ( θ k ) a H ( θ k ) − ρ k a ( θ 0 ) a H ( θ 0 )  [ w k − 1 v k ] . (14) By expanding (13) and (14), we immediately hav e the follow- ing proposition. Pr oposition 1: Suppose that γ k (i.e., the second entry of z k ) satisfies (13), if H k (2 , 2) = 0 , it can be deriv ed that the trajectory of  < ( γ k ) = ( γ k )  T is a line as < [ H k (1 , 2)] < ( γ k ) − = [ H k (1 , 2)] = ( γ k ) = − H k (1 , 1) / 2 . If H k (2 , 2) 6 = 0 , the trajectory of [ < ( γ k ) = ( γ k )] T is a circle, denoted by C γ : C γ = n [ < ( γ k ) = ( γ k )] T      [ < ( γ k ) = ( γ k )] T − c γ   2 = R γ o with the center c γ = 1 H k (2 , 2)  −< [ H k (1 , 2)] = [ H k (1 , 2)]  (15) and the radius R γ = p − det( H k )  | H k (2 , 2) | . (16) From this proposition, it is known that, giv en the previous weight vector w k − 1 = T − 1 k − 1 a ( θ 0 ) , there exist infinitely many solutions of γ k to achieve a response level of ρ k at θ k . This implies that the response lev el at a certain direction can be precisely adjusted by assigning a virtual interference with properly designed INR parameter γ k . It is clear that H k (2 , 2) = 0 is equi valent to ρ k =   ( T − 1 k − 1 a ( θ k )) H a ( θ k )   2   ( T − 1 k − 1 a ( θ k )) H a ( θ 0 )   2 . (17) In this case ρ k is equal to the normalized power response at θ k when the weight vector is T − 1 k − 1 a ( θ k ) , i.e., when the beampattern steers to the beam axis θ k . T ypically , a beam- pattern reaches its maximum at the beam axis, i.e., we have ρ k > 1 if θ k 6 = θ 0 . This would contradict with the fact that ρ k ≤ 1 . Hence, H k (2 , 2) = 0 usually will not occur and in the sequel we only focus on the case of H k (2 , 2) 6 = 0 , and from Proposition 2, the trajectory of [ < ( γ k ) = ( γ k )] T is a circle, as illustrated in Fig. 1. Then, the remaining question is among all these v alid solutions of γ k (or β k ), to meet the response lev el requirement, which one is to maximize the SINR or beam gain? W e will study this question below . C. Selection of γ k and Update of the W eight V ector The preceding problem can be formulated as the follow- ing constrained optimal and precise array response control (OP ARC) problem: maximize γ k G k , | w H k a ( θ 0 ) | 2 / | w H k T k w k | (18a) sub ject to L ( θ k , θ 0 ) = ρ k (18b) w k = w k − 1 + γ k T − 1 k − 1 a ( θ k ) . (18c) From (18a), one can see that the desired weight vector is expected to provide the maximum array gain with some additional constraints. Moreover , in the above de vised OP ARC scheme, apart from the response lev el constraint (18b), we hav e also imposed constraint (18c) to make the resultant w k be a particular optimal weight vector in correspondence to assigning another interference at θ k to the existing k − 1 interferences at directions θ 1 , . . . , θ k − 1 . When γ k satisfies (9), (18c) leads to w k = T − 1 k a ( θ 0 ) . In or- der to solve problem (18), we first substitute w k = T − 1 k a ( θ 0 ) into the objective function and get G k = | a H ( θ 0 ) T − H k a ( θ 0 ) | . Then, recalling (7) and (9), we can rewrite G k as G k =   a H ( θ 0 ) T − 1 k − 1 a ( θ 0 ) + γ k a H ( θ 0 ) T − 1 k − 1 a ( θ k )   = | e ξ c | · | ξ 0 / e ξ c + γ k | (19) where ξ 0 , ξ k , ξ c and e ξ c are defined as ξ 0 , a H ( θ 0 ) T − 1 k − 1 a ( θ 0 ) (20a) ξ k , a H ( θ k ) T − 1 k − 1 a ( θ k ) (20b) ξ c , a H ( θ k ) T − 1 k − 1 a ( θ 0 ) (20c) e ξ c , a H ( θ 0 ) T − 1 k − 1 a ( θ k ) . (20d) Then, from Proposition 1, problem (18) can be expressed as maximize γ k | ξ 0 / e ξ c + γ k | (21a) sub ject to  < ( γ k ) = ( γ k )  T ∈ C γ . (21b) Although the problem (21) is non-con ve x, it will be shown that it can be analytically solved as follows. Pr oposition 2: Denote the intersections of the circle and the line connecting the origin O = [0 , 0] T and the center c γ in (15) as F a , [ < ( γ k,a ) = ( γ k,a )] T and F b , [ < ( γ k,b ) = ( γ k,b )] T , respectiv ely , and assume that | γ k,a | < | γ k,b | . If T k − 1 is Hermitian (note that the Hermitian property of T k − 1 has 4 < ( " ) -0.5 0 0.5 1 1.5 2 2.5 3 = ( " ) -0.5 0 0.5 1 1.5 2 2.5 O C . F a F b R . Fig. 1. Geometric distribution of γ k . not been guaranteed as we have mentioned earlier), then the optimal solution of (21) satisfies γ k,? =  γ k,a , if ζ > 0 γ k,b , otherwise (22) where ζ , sign[ c γ (1)] · sign[ < ( d ) − c γ (1)] (23) and d , − ξ 0  ξ ∗ c . (24) In addition, γ k,a and γ k,b in (22) are calculated as γ k,a = − ( k c γ k 2 − R γ ) χξ c k c γ k 2 H k (2 , 2) , γ k,b = − ( k c γ k 2 + R γ ) χξ c k c γ k 2 H k (2 , 2) where χ = ξ k − ρ k ξ 0 ∈ R , c γ and R γ are defined in Proposition 1. Pr oof: See Appendix A. T o hav e a better understanding, the locations of γ k,a and γ k,b hav e been illustrated in Fig. 1. Obviously , once the optimal γ k,? has been obtained, we can update the weight vector as w k = w k − 1 + γ k,? v k . (25) This completes the update of weight vector of the k th step. D. Update of the In version of VCM Since the calculation of γ k,? requires the inv ersion of VCM (i.e., T − 1 k − 1 ) which is assumed to be Hermitian in Proposition 2, in the sequel we shall discuss how to update T − 1 k (in order to make the ne xt step of response control feasible) and how to guarantee the Hermitian property . T o address these two problems, we first assume that T k − 1 is Hermitian and the optimal γ k,? has been obtained with the aid of Proposition 2. Then, from (9) we hav e − β k,? / (1 + β k,? ξ k ) = γ k,? /ξ c (26) Algorithm 1 OP ARC Algorithm 1: give the initial weight vector w 0 = a ( θ 0 ) and set T 0 = I , prescribe the direction θ k and the corresponding desired lev el ρ k , k = 1 , 2 , · · · 2: for k = 1 , 2 , · · · , do 3: calculate γ k,? from (22) and obtain v k from (11) 4: update w k as w k = w k − 1 + γ k,? v k 5: update T − 1 k as T − 1 k = T − 1 k − 1 + γ k,? v k v H k  ξ c 6: end for 7: output w k directly where β k,? = Ψ − 1 k ( γ k,? ) denotes the INR corresponding to γ k,? in the k th step. Ob viously , the combination of (7) and (26) yields T − 1 k = T − 1 k − 1 + γ k,? v k v H k ξ c . (27) Therefore, T − 1 k can be calculated out straightforwardly (with- out the calculation of T k and its inv ersion), once the optimal γ k,? obtained. T o further explore the Hermitian property of T k , let us revisit Proposition 2 and get γ k,? ξ c = − ( k c γ k 2 ± R γ ) χ k c γ k 2 H k (2 , 2) ∈ R (28) which is a real-valued number . Thus, it is kno wn that T k is Hermitian as long as T k − 1 is Hermitian. Accordingly , it can be readily concluded that if we set T 0 = I , then T i is Hermitian for i = 1 , 2 , · · · , k . Now , it is seen that the response le vels can be successi vely adjusted by assigning virtual interferences. Therefore, we can utilize the abov e update procedures iterati vely to fulfill the prescribed response control requirement. Finally , the proposed OP ARC method is summarized in Algorithm 1. I I I . S O M E P R O P E RT I E S O F O P A R C In the pre vious section, we have shown that the response lev el at a certain direction can be optimally and flexibly adjusted by assigning a virtual interference. Instead of deter- mining β k (i.e., the INR of the virtue interference assigned in the k th step), an alternativ e parameter γ k , mapping of β k , is chosen to facilitate the algorithm deriv ation. Ho we ver , since INR has a physical meaning and is alw ays a non-ne gativ e v alue in a real data case, it is worth examining the direct relationship between the response lev el and β k . T o do so, in this section we continue the analysis on the OP ARC scheme, mainly focus on the selection of INR β k rather than its mapping γ k . A. Geometrical Distribution of β k As shown in (9), γ k is a mapping of β k , and β k can be expressed with respect to γ k as β k = − γ k  ( ξ c + γ k ξ k ) = Ψ − 1 k ( γ k ) (29) where Ψ − 1 k ( · ) is the in verse function of Ψ k ( · ) in (9). Thus, β k can be calculated once γ k is av ailable. For the trajectory 5 of β k when the array response le vel ρ k is satisfied at angle θ k , let us recall Proposition 1 and express γ k as γ k = c γ (1) + j c γ (2) + R γ e j ϕ (30) where c γ and R γ are the center and the radius of the circle giv en in Proposition 1, ϕ can be any real-valued number . Substituting (30) into (29), one gets β k =  p 1 + p 2 e j ϕ  /  q 1 + q 2 e j ϕ  (31) where p l and q l ( l = 1 , 2 ) are complex numbers satisfying p 1 = − c γ (1) − j c γ (2) , p 2 = − R γ (32a) q 1 = ξ c + ( c γ (1) + j c γ (2)) ξ k , q 2 = R γ ξ k . (32b) W ith some calculation, it is not dif ficult to hav e the follow- ing proposition. Pr oposition 3: The trajectory of [ < ( β k ) = ( β k )] T with β k satisfying (31) is a circle C β with the center c β =  ξ 0 / ( | ξ c | 2 − ξ 0 ξ k ) , 0  T (33) and the radius R β = | ξ c |   √ ρ k ·   | ξ c | 2 − ξ 0 ξ k    (34) i.e., β k = c β (1) + j c β (2) + R β e j φ (35) where φ can be any real-valued number . Similar to γ k , all the β k on the above circle can be used to precisely adjust the response lev el at θ k to its desired level ρ k . An interesting difference with γ k is that the calculations of c β and R β do not require the knowledge of w k − 1 . This implies that all β k ’ s (including the optimal one later) can be obtained without knowing any weight vectors. On the contrary , the determination of γ k relies on the av ailability of the previous weight vector w k − 1 . In addition, Proposition 3 implies that the center of the trajectory of [ < ( β k ) = ( β k )] T is located on the real axis, and is independent of the desired level ρ k . B. Determination of the Optimal β k Among all the valid β k for the control of the array response lev el ρ k at θ k , the optimal one is the one that maximizes the array gain. Therefore, the follo wing constrained optimization problem can be formulated to select the optimal β k : maximize β k G k = | w H k a ( θ 0 ) | 2 / | w H k T k w k | (36a) sub ject to L ( θ k , θ 0 ) = ρ k (36b) w k = w k − 1 + Ψ k ( β k ) T − 1 k − 1 a ( θ k ) (36c) where the parameter γ k has been replaced in (36c) by Ψ k ( β k ) . Clearly , the above optimization problem (36) is equiv alent to problem (18). Therefore, the optimal solution (denoted as β k,? ) of (36) can be readily obtained by utilizing the mapping as β k,? = Ψ − 1 k ( γ k,? ) . (37) Combining the result of γ k,? in (22) with some calculation, we can deriv e Proposition 4 below . Algorithm 2 OP ARC Algorithm (an Equiv alent V ariant) 1: give a ( θ 0 ) and set T 0 = I , specify the direction θ k and the corresponding desired lev el ρ k with k = 1 , 2 , · · · 2: for k = 1 , 2 , · · · , do 3: calculate β k,? from (38) 4: update T k as T k = T k − 1 + β k,? a ( θ k ) a H ( θ k ) 5: end for 6: calculate w k = T − 1 k a ( θ 0 ) Pr oposition 4: The optimal solution of (36) is given by β k,? =  β k,r , if − 1 /ξ k > ξ 0 /  | ξ c | 2 − ξ 0 ξ k  β k,l , otherwise (38) where β k,r and β k,l are the intersections of circle C β and the real axis = ( · ) = 0 : β k,r = R β + ξ 0 /  | ξ c | 2 − ξ 0 ξ k  (39) β k,l = − R β + ξ 0 /  | ξ c | 2 − ξ 0 ξ k  . (40) It is not hard to see from (38) that the optimal β k,? is a real-valued number , while the v alid β k in Proposition 3 for the array response control may be complex valued. Howe ver , as mentioned earlier, the physical meaning of β k is the INR as it is used in (5) and it cannot be negati ve. From (38)-(40), the solved optimal β k,? may be negati ve, which might be because there is no assumption of the used VCM T k − 1 being non- negati ve definite. This will be studied together with the update of the VCM below . On the other hand, if T k − 1 is Hermitian, then T k = T k − 1 + β k,? a ( θ k ) a H ( θ k ) is also Hermitian, since β k,? is real. This is consistent with the inference obtained in the paragraph below Eqn. (28). Finally , it is obvious that the optimal β k in (38) does not depend on the knowledge of the weight vectors in the previous steps. Once the optimal β k has been obtained, we can express the VCM at the current stage as T k = T k − 1 + β k,? a ( θ k ) a H ( θ k ) = I + A k Σ k A H k (41) where A k , [ a ( θ 1 ) , · · · , a ( θ k )] and Σ k is a diagonal matrix containing all β ’ s of virtual interferences, i.e., Σ k = Diag ([ β 1 ,? , β 2 ,? , · · · , β k,? ]) . (42) Accordingly , we have w k = T − 1 k a ( θ 0 ) . T o make it clear , the variant of the OP ARC method is summarized in Algorithm 2. Note that the calculation of intermediate weight vectors is av oided, due to the fact that neither the calculation of β k,? nor T k relies on weight vectors. Therefore, the procedure of array response control is simplified. Before proceeding, it is interesting to provide a deep insight and a geometrical perspectiv e on the relationship between γ k and β k . It is not hard to see that the condition − 1 /ξ k > ξ 0 /  | ξ c | 2 − ξ 0 ξ k  in (38) is equiv alent to the condition ζ > 0 in (22), if and only if ρ k ξ 0 < ξ k . It implies that the conditions for selecting β k,? between β k,l and β k,r and selecting γ k,? between γ k,a and γ k,b may be different and are the same under a certain condition, i.e., ρ k ξ 0 < ξ k . Thus, we hav e  γ k,a = Ψ k ( β k,r ) , γ k,b = Ψ k ( β k,l ) , if ρ k ξ 0 < ξ k γ k,a = Ψ k ( β k,l ) , γ k,b = Ψ k ( β k,r ) , otherwise . (43) 6 1 2 3 ∑ M M - 1 ∑ ∑ ∑ ∑ ∑ ∑ O u t p u t o f C a l i b r a t e d S e n s o r O u t p u t o f U n c a l i b r a t e d S e n s o r O u t p u t o f R e f e r e n c e S e n s o r ∑ S u m o f O u t p u t f r o m D i f f e r e n t S e n s o r s X 0 f l i p u d ( X 0 ) X 2 X 1 2 X 1 1 2 M  1 2 M  2 v 1 v 0 v   0  a   0 j    a   0 j   a   0  a   j  a   0 j    a   0 j   a w     j s p a n   a       00 a f f , jj    aa ' w 1  1  2  2  X Y Z       0 1, , l l    aa   l  a 1, l w 0 w 1, l  w 1, l w O   1, l l   a   0  a O k w   1 1 1 k k k k        a w w 1 k c  w   1 k   a   1 1 k k     a 1 k       1 0 1 , kk     A w   11 kk    Vw 1 , 1 k  w 1 , 2 k  w   0  a   1 , 1 k   a   1 , 2 k   a 11 k  ， 11 k  ， 1 , 1 1 , 2 k k    O     1 2 1 2 kk  Vw ， ，     1 1 1 1 kk  Vw ， ， Fig. 2. Illustration of the mapping Ψ k ( · ) . T o hav e an intuitiv e perspectiv e on Ψ k ( · ) , a geometrical illustration is giv en in Fig. 2, where J r , J l , F a and F b stand for the points [ < ( β k,r ) = ( β k,r )] T , [ < ( β k,l ) = ( β k,l )] T , [ < ( γ k,a ) = ( γ k,a )] T and [ < ( γ k,b ) = ( γ k,b )] T , respectively . C. P ositive Definite V irtual Covariance Matrices Firstly , the following conclusion, which simplifies the se- lection of β k,? , can be obtained. Pr oposition 5: If T k − 1 ∈ S N ++ , we have β k,? = β k,r = ( | ξ c | − √ ρ k ξ 0 ) / [ √ ρ k ( ξ 0 ξ k − | ξ c | 2 )] . (44) Furthermore, if T k − 1 ∈ S N ++ , then T k ∈ S N ++ if and only if ρ k < ξ 2 k / | ξ c | 2 . Pr oof: See Appendix B. Similar to the argument in the paragraph below Eqn. (17), ξ 2 k / | ξ c | 2 is in general greater than 1 and it is assumed ρ k ≤ 1 . Thus, we have ρ k < ξ 2 k / | ξ c | 2 . As a consequence, in each step of weight vector update, we hav e T k ∈ S N ++ and β k,? = β k,r , as long as T k − 1 ∈ S N ++ . Since in our algorithm T 0 = I is taken as the initial VCM, we have T k ∈ S N ++ and β k,? = β k,r . Pr oposition 6: If T k − 1 ∈ S N ++ , then β k,? ≥ 0 ⇔ | ξ c | ≥ √ ρ k ξ 0 (45a) β k,? < 0 ⇔ | ξ c | < √ ρ k ξ 0 . (45b) Pr oof: From (68) in the proof of Proposition 5 in Ap- pendix B, we ha ve ξ 0 ξ k − | ξ c | 2 > 0 pro vided that T k − 1 ∈ S N ++ . Then, from (44), the proof of (45) is completed. Substituting the definitions of ξ c and ξ 0 into (45) and using w k − 1 = T − 1 k − 1 a ( θ 0 ) , we have β k,? ≥ 0 ⇔ ρ k ≤   w H k − 1 a ( θ k )   2 /   w H k − 1 a ( θ 0 )   2 (46a) β k,? < 0 ⇔ ρ k >   w H k − 1 a ( θ k )   2 /   w H k − 1 a ( θ 0 )   2 . (46b) Notice that   w H k − 1 a ( θ k )   2 /   w H k − 1 a ( θ 0 )   2 abov e represents the normalized response at θ k , of the previous weight vector w k − 1 . Clearly , (46) shows that the resultant β k,? is non- negati ve if the desired lev el ρ k is lower than the response lev el at θ k of the previous weight vector w k − 1 . Otherwise, a negati ve β k,? is obtained if it is required to ele v ate the pre vious response level of θ k . W e can see that the negati ve β k,? is still meaningful in our discussion of array response control using virtual interferences, although it cannot occur in a real data cov ariance matrix with real interferences. In addition to the abov e two propositions, the following result can be obtained. Pr oposition 7: If T k − 1 ∈ S N ++ , then problem (36) has the same optimal solution as that of the following one maximize β k | w H k a ( θ 0 ) | 2 w H k T k − 1 w k (47a) sub ject to L ( θ k , θ 0 ) = ρ k (47b) w k = w k − 1 + Ψ k ( β k ) v k . (47c) Pr oof: See Appendix C. Interestingly , from Proposition 7, it is kno wn that under the same constraints (i.e., (36b) and (36c)), the optimal β k to (36) also maximizes the previous array gain, in which only T k − 1 (but not T k ) is taken into consideration. For instance, consider the case when T k − 1 is a real nor- malized noise-plus-interference cov ariance matrix (i.e., T k − 1 is calculated from real data that contains both noise and interference), and one applies the OP ARC scheme to realize a specific array response control task in (47b) by assigning a virtual interference at θ k . Then, it is seen from Proposition 7 that the optimal β k,? of problem (36) also maximizes the real output SINR (not taking the virtual interference into consideration) of beamformer . This property will be further exploited in the companion paper [19] to design an adaptiv e beamformer with specific constraint. I V . C O M PA R I S O N W I T H A 2 R C In the above sections, the optimal v alues of γ k and β k of the virtual interference assigned in the k th step are specified. Meanwhile, useful conclusions are obtained and two versions of OP ARC are described. In this section, comparisons will be carried out to elaborate the differences between the recent A 2 RC algorithm [15] and the abov e OP ARC algorithm from two perspectiv es. A. Comparison on the F ormula Updating In the A 2 RC method, the weight vector is updated as w k = w k − 1 + µ k a ( θ k ) (48) where µ k is the hyperparameter to be optimized. T o minimize the deviation between the resultant responses of adjacent two steps, and meanwhile, a void the computationally inefficient global search, µ k is empirically selected in [15] as µ k,a , which is the solution to the following problem: minimize µ k | µ k | (49a) sub ject to  < ( µ k ) = ( µ k )  T ∈ C µ (49b) where C µ is the following circle: C µ = n [ < ( µ k ) = ( µ k )] T      [ < ( µ k ) = ( µ k )] T − c µ   2 = R µ o 7 with the center c µ = 1 Q k (2 , 2)  −< [ Q k (1 , 2)] = [ Q k (1 , 2)]  (50) and the radius R µ = p − det( Q k )  | Q k (2 , 2) | (51) where the matrix Q k satisfies Q k = [ w k − 1 a ( θ k )] H  a ( θ k ) a H ( θ k ) − ρ k a ( θ 0 ) a H ( θ 0 )  [ w k − 1 a ( θ k )] . Note that such an empirical selection may not perform well under all circumstances. As a matter of fact, this scheme may ev en lead to severe pattern distortion, as we will show later in simulations in Section V . In the OP ARC algorithm, the weight vector is updated via (18c). It is seen that, different from the A 2 RC algorithm, a scaling of T − 1 k − 1 a ( θ k ) is added to w k − 1 , and γ k T − 1 k − 1 a ( θ k ) makes the resultant w k be an optimal weight vector . Additionally , in the proposed OP ARC algorithm, we opti- mize the parameter γ k by maximizing the array gain when the preassigned response lev el is satisfied. T o hav e a similar weight form with A 2 RC, it is shown in Appendix D that we can reformulate (18c) as w k = w k − 1 + γ k A ( θ k , · · · , θ 1 ) d k = w k − 1 + γ k a ( θ k ) + γ k A ( θ k − 1 , · · · , θ 1 ) ¯ d k (52) where A ( θ k , · · · , θ 1 ) , [ a ( θ k ) , · · · , a ( θ 1 )] , with θ i ( 1 ≤ i ≤ k − 1 ) denoting the angles of interferences that assigned previously , d k is a k × 1 vector with its first element 1, ¯ d k is a ( k − 1) × 1 vector obtained by removing the first element from d k . From (52), it is observed that the added component to the previous weight v ector w k − 1 in w k is a linear combination of the steering vectors of all interferences (including both the current a ( θ k ) and the pre vious a ( θ 1 ) , · · · , a ( θ k − 1 ) ). On the contrary , in the A 2 RC algorithm, the added component is a scaling of the steering vector of the single interference to be assigned (i.e., a ( θ k ) ). Furthermore, we can obtain the follo wing corollary of Proposition 2, which describes a similarity between A 2 RC and OP ARC. Cor ollary 1: In the first step of weight update (i.e, w 0 = a ( θ 0 ) , T 0 = I ), if ρ 1 ≤ k a ( θ 1 ) k 2 2 / k a ( θ 0 ) k 2 2 , then µ 1 ,? = γ 1 ,? , otherwise, µ 1 ,? = γ 1 , × , where γ 1 , × = { γ 1 ,a , γ 1 ,b } \ γ 1 ,? . Pr oof: See Appendix E. From Corollary 1, it is kno wn that in the first step of the weight vector update, A 2 RC will lead to the same result as OP ARC, provided that ρ 1 ≤ k a ( θ 1 ) k 2 2 / k a ( θ 0 ) k 2 2 . Otherwise, the inferior parameter γ 1 , × (in the sense of array gain) will be adopted by A 2 RC. B. Comparison on INRs of V irtual Interfer ences W e next compare the INRs of virtual interferences to sho w an essential difference between A 2 RC and OP ARC. T o begin with, we express the weight vector of A 2 RC in the k th step as w k = a ( θ 0 ) + µ 1 a ( θ 1 ) + · · · + µ k a ( θ k ) = a ( θ 0 ) + A k b k (53) where b k , [ µ 1 , µ 2 , · · · , µ k ] T . Note from the above that the update of the weight vector w k in A 2 RC does not depend on any VCM and no VCM update is needed. Howe ver , in order to compare the INRs, we need to associate it to a VCM that may be implicit/virtual. T o do so, we rewrite the weight vector as w k = ˘ T − 1 k a ( θ 0 ) = a ( θ 0 ) − A k  I + ˘ Σ k A H k A k  − 1 ˘ Σ k A H k a ( θ 0 ) (54) where ˘ T k = I + A k ˘ Σ k A H k denotes a VCM, ˘ Σ k = Diag([ ˘ β k, 1 , ˘ β k, 2 , · · · , ˘ β k,k ]) specifies the INR of the interfer- ence at θ i ( i = 1 , · · · , k ) when completing the current k th step of the weight vector update. Note that no interference was assigned at the current θ k in the previous k − 1 steps of the response control. It is shown from Appendix F that in the k th step of the weight update of the A 2 RC algorithm, the INR of the virtual interference assigned at θ k is ˘ β k,k = − µ k a H ( θ k ) ˘ w k − 1 + µ k k a ( θ k ) k 2 2 . (55) In addition, k − 1 ne w interferences are additi vely assigned at directions θ 1 , · · · , θ k − 1 of A 2 RC to the pre vious ( k − 1) th step. Denote the INRs of these ne w interferences assigned at θ i in the k th step of the weight update as ˘ ∆ k,i ( 1 ≤ i ≤ k − 1 ). Clearly they satisfy ˘ ∆ k,i = ˘ β k,i − ˘ β k − 1 ,i . (56) It can be further deri ved (see Appendix F) that ˘ ∆ k,i = µ k a H ( θ i ) a ( θ k ) ˘ β 2 k − 1 ,i µ i − µ k a H ( θ i ) a ( θ k ) ˘ β k − 1 ,i . (57) Generally speaking, in the k th step of A 2 RC, the INRs of the newly-assigned interferences (including both ˘ β k,k and ˘ ∆ k,i ( 1 ≤ i ≤ k − 1 )) are complex-v alued numbers. This is a difference between A 2 RC and OP ARC. Moreov er , the abov e analysis sho ws that there are k − 1 additional inter- ferences assigned to the pre viously controlled angles (i.e., θ 1 , · · · , θ k − 1 ) in the k th step of A 2 RC, while in OP ARC only a single interference is assigned (at θ k ). Since our aim is to control the array response le vel at θ k , the newly-assigned virtual interferences at θ 1 , · · · , θ k − 1 actually bring undesirable array response v ariations at these adjusted angles. According to the above notations, the VCM of A 2 RC satisfies implicitly: ˘ T k = ˘ T k − 1 + A k Diag([ ˘ ∆ k, 1 , · · · , ˘ ∆ k,k − 1 , ˘ β k,k ]) A H k (58) which is different from that of OP ARC in (41). T o summarize, the main differences between the proposed OP ARC algorithm and the existing A 2 RC algorithm include: • Different formulas of the weight update are employed in OP ARC and A 2 RC. • The resultant weight of OP ARC can be guaranteed to be an optimal beamformer , while A 2 RC method does not. • The array gain is introduced to the parameter optimization of OP ARC, while A 2 RC is not. 8 • The update of VCM is necessary for OP ARC, while A 2 RC is free of this procedure although its VCM is implicitly updated by (58). • T wo different strategies of virtual interference assign- ing are adopted in these tw o approaches. The INRs of OP ARC are always real, but they may not be in A 2 RC. V . S I M U L A T I O N R E S U LT S W e next present some simulations to verify the ef fectiv e- ness of our proposed OP ARC. T o validate the superiority of OP ARC, we also test another precise array response control (P ARC) scheme, in which we adopt the following non-optimal parameter intentionally: γ k = γ k, × , { γ k,a , γ k,b } \ γ k,? (59) and use the same remaining procedure as OP ARC. Denote β k, × = Ψ − 1 k ( γ k, × ) . Note that γ 1 , × is the same as that in Corollary 1. Clearly , P ARC can precisely control array re- sponse le vel as well, while its parameter γ k is not optimally se- lected as in OP ARC. Besides OP ARC and P ARC, the A 2 RC al- gorithm in [15] is also compared. W e set ω = 6 π × 10 8 rad / s , which corresponds to a wavelength λ = 2 π c/ω = 1m with the light speed c . Consider a 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, from which the τ n ( θ ) in (2) can be specified as τ n ( θ ) = x n sin( θ ) /c . Additionally , we take the quiescent weight vector a ( θ 0 ) as the initial weight and fix the beam axis at θ 0 = 20 ◦ for all experiments conducted. For con v enience, we carry out two steps of the array response control algorithms and denote the two adjusted angles as θ 1 and θ 2 , respectively . T o measure the performances of different methods, we introduce two cost functions. The first one is defined as D , | L 2 ( θ 1 , θ 0 ) − L 1 ( θ 1 , θ 0 ) | (60) where L k ( θ , θ 0 ) represents the resultant response after finish- ing the k -th step of weight update. It is seen that D mea- sures the response le vel difference between tw o consecutiv e response controls at θ 1 . The second cost function is defined as J ∆ = v u u t 1 I I X i =1   L 2 ( ϑ i , θ 0 ) − L 1 ( ϑ i , θ 0 )   2 (61) where ϑ i stands for the i th sampling point in the angle sector , I denotes the number of sampling points. J measures the deviation between tw o response patterns L 2 and L 1 . W e will uniformly sample the region [ − 90 ◦ , 90] ◦ ev ery 0 . 2 ◦ and hence obtain I = 901 discrete points. Besides D and J abov e, we also test the obtained array gains of different methods, and consider pattern variation and pattern distortion for performance comparison. A. P attern V ariation In the first example, we test the performances of different approaches for sidelobe response control. More specifically , the normalized responses at θ 1 = − 45 ◦ and θ 2 = − 5 ◦ are T ABLE I E L EM E N T L O CAT IO N S A N D E L EM E N T P A TT E R N S O F T HE N ON U N I FO R M L I NE A R A R R A Y n x n g n ( θ ) n x n g n ( θ ) 1 0.00 1 . 00cos(1 . 00 θ ) 7 3.05 1 . 02cos(1 . 00 θ ) 2 0.45 0 . 98cos(0 . 85 θ ) 8 3.65 1 . 08cos(0 . 90 θ ) 3 1.00 1 . 05cos(0 . 98 θ ) 9 4.03 0 . 96cos(0 . 75 θ ) 4 1.55 1 . 10cos(0 . 70 θ ) 10 4.60 1 . 09cos(0 . 92 θ ) 5 2.10 0 . 90cos(0 . 85 θ ) 11 5.00 1 . 02cos(0 . 80 θ ) 6 2.60 0 . 93cos(0 . 69 θ ) expected to be successiv ely adjusted to ρ 1 = − 40dB and ρ 2 = − 30dB . In the first step of response control, we can figure out that c γ = [ − 0 . 1704 , − 0 . 0315] T , d = − 8 . 5231 − j 1 . 5766 , γ 1 ,a = − 0 . 1559 − j 0 . 0288 and γ 1 ,b = − 0 . 1849 − j 0 . 0342 . On this basis, we obtain that ζ = 1 > 0 and hence choose γ 1 ,? = γ 1 ,a for OP ARC and select γ 1 , × = γ 1 ,b for P ARC, according to (22) and (59), respectiv ely . Additionally , it can be figured out that c β = [ − 0 . 1488 , 0] T and R β = 1 . 7171 . W e adopt β 1 ,? = β 1 ,r = 1 . 5683 for OP ARC and take β 1 , × = β 1 ,l = − 1 . 8659 for P ARC. For A 2 RC, it is found that µ 1 = γ 1 ,? = − 0 . 1559 − j 0 . 0288 , which coincides with the result of Corollary 1. As predicted, one also obtains that ˘ β 1 , 1 = β 1 ,? = 1 . 5683 . Fig. 3(a) illustrates the resultant response patterns of different schemes. As we can see, all these three approaches are capable of precisely controlling the array response levels as expected. Notice also that the result of A 2 RC is exactly the same as that of OP ARC. In the second step, with the same manner we found out that γ 2 ,? = − 0 . 0685 − j 0 . 0399 , β 2 ,? = 0 . 2504 , γ 2 , × = − 0 . 1148 − j 0 . 0695 , β 2 , × = − 0 . 4277 and µ 2 = − 0 . 0674 − j 0 . 0393 . Fig. 3(b) depicts the results of different methods. It is seen that all methods can adjust L ( θ 2 , θ 0 ) to ρ 2 . For the proposed OP ARC scheme, it can be checked that β k,r is the ultimate selection of β k,? ( k = 1 , 2 ). In f act, this is consistent with the conclusion of Proposition 5. One can see that both β 1 ,? and β 2 ,? are positiv e in this case. This coincides with the theoretical prediction of Proposition 6, since it is required to lower the response lev els in either step. T o further examine the performance, we ev aluate D and J as defined earlier and list their measurements in T able II. It is observed that the proposed OP ARC scheme minimizes both D and J among the three methods. From T able II and Fig. 3(b), it is found that A 2 RC causes serious perturbation (about 5dB ) at the pre vious point θ 1 . In f act, besides the virtual interference assigned at θ 2 , another one is also assigned at the previously adjusted direction (i.e., θ 1 ), for the existing A 2 RC algorithm. W e can calculate that ˘ β 2 , 2 = 0 . 2465 + j 0 . 0001 and ˘ ∆ 2 , 1 = − 0 . 4120 + j 2 . 5879 (INR of the additional interference assigned at θ 1 in the second step of response control). Notice that both ˘ β 2 , 2 and ˘ ∆ 2 , 1 are complex numbers, which is different from that of OP ARC. Finally , we have listed the obtained array gains of these approaches at both steps in T able II. Clearly , it is seen that OP ARC outperforms the other two methods. Since both D and J depend on the desired level at θ 2 (i.e., 9 -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) initial pattern A 2 RC PARC OPARC -48 -46 -44 -42 -44 -42 -40 -38 (a) Comparison of synthesized patterns 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) initial pattern A 2 RC PARC OPARC -47 -46 -45 -44 -43 -44 -42 -40 -10 -5 -32 -30 -28 (b) Comparison of synthesized patterns at the 2nd step Fig. 3. Resultant pattern comparison (the first example). -50 -45 -40 -35 -30 -25 -20 2 (dB) 0 1 2 3 4 5 6 D (dB) A 2 RC PARC OPARC (a) Curves of D versus ρ 2 -50 -45 -40 -35 -30 -25 -20 2 (dB) 0 0.002 0.004 0.006 0.008 0.01 0.012 J A 2 RC PARC OPARC -40 -38 -36 -34 5.4 5.6 5.8 10 -3 (b) Curves of J versus ρ 2 Fig. 4. Pattern variation comparison (the first example). T ABLE II O B T A IN E D P A RA M E T ER C OM PA RI S O N ( T HE F IR S T E X A MP L E ) A 2 RC P ARC OP ARC D (dB) 5 . 05 0 . 86 0 . 51 J 4 . 72 e − 3 7 . 84 e − 3 4 . 69 e − 3 G 1 (dB) 10 . 0482 10 . 0331 10 . 0482 G 2 (dB) 10 . 0026 9 . 9653 10 . 0074 ρ 2 ), we vary ρ 2 from − 50dB to − 20dB and recalculate D and J with the other settings unchanged. Fig. 4(a) and Fig. 4(b) plot the curves of D versus ρ 2 and J v ersus ρ 2 , respectively . It can be clearly observed that the proposed OP ARC algorithm performs the best on both D and J . The existing A 2 RC performs well on J , ho wev er , it causes a large deviation to the response lev el at θ 1 as displayed in Fig. 4(a). B. P attern Distortion In this part, we shall further sho w the advantages of the OP ARC. For conv enience, we set θ 1 and its desired lev el ρ 1 the same as the first example, and then conduct the second step of the response control by taking θ 2 = 23 ◦ and ρ 2 = 0dB . Notice that θ 2 is in the mainlobe region in this case, and it is required to elev ate the response lev el there. Clearly , the obtained parameters of the second step are renewed for all methods tested, while the results of the first step keep unaltered compared to the pre vious e xample. Here, it can be obtained with the A 2 RC algorithm that µ 2 = − 0 . 5931 + j 0 . 8040 , ˘ β 2 , 2 = − 0 . 3923 − j 0 . 4011 and ˘ ∆ 2 , 1 = − 1 . 8001 + j 0 . 0334 . For the P ARC algorithm, we obtain γ 2 , × = − 0 . 7108 + j 0 . 7171 and β 2 , × = − 0 . 8522 . While for OP ARC, its parameter satisfies γ 2 ,? = γ 2 ,b = 0 . 8352 − j 0 . 8438 and β 2 ,? = β 2 ,r = − 0 . 0577 . It is w orth noting that we hav e selected γ b , which is different from that at Step 1 (where γ a is selected), to obtain the final γ 2 ,? . In 10 -80 -60 -40 -20 0 20 40 60 80 Angle of Arrival (degrees) -50 -40 -30 -20 -10 0 10 Beampattern (dB) initial pattern A 2 RC PARC OPARC -46 -45 -42 -40 -38 Fig. 5. Comparison of synthesized patterns at the 2nd step of the 2nd e xample. T ABLE III O B T A IN E D P A RA M E T ER C OM PA RI S O N ( T HE S EC O N D E X AM P L E ) A 2 RC P ARC OP ARC D (dB) 31 . 6001 10 . 7083 1 . 2595 J 1 . 0685 2 . 5149 0 . 0624 G 2 (dB) 2 . 5060 0 . 7366 13 . 1370 fact, this flexible mechanism of parameter determination in OP ARC enables us to avoid certain pattern distortion, which is inevitable in A 2 RC or P ARC. T o see this clearer, we have depicted the synthesized patterns in Fig. 5. It can be found that all the response lev els at θ 2 still meet the requirement as before. Howe ver , it shows clearly that the patterns of A 2 RC and P ARC are se verely distorted. The obtained mainlobes are split and the resultant sidelobe levels are raised for both A 2 RC and P ARC. For the proposed OP ARC, none of the above undesirable phenomena happens and a well-shaped pattern has been obtained. By the way , notice that β 2 ,? is negati v e in this scenario. This is consistent with the conclusion of Proposition 6, since the response lev el needs to be lifted in this second step of response control. The details of D , J and the obtained array gains have been specified in T able III, from which the merits of the proposed OP ARC algorithm are clearly observed. Note that the array gain G 1 is not listed in T able III since it has been reported in T able II. Again, to further examine the performance, we vary ρ 2 from − 20dB to 0dB , and depict the curve of D versus ρ 2 in Fig. 6(a) and curve of J v ersus ρ 2 in Fig. 6(b), respectiv ely . As illustrated in these two figures, A 2 RC causes a great perturbation on θ 1 (i.e., high value of D ) and P ARC performs poor on the average deviation J . On the other hand, OP ARC performs the best when measuring either D or J . V I . C O N C L U S I O N S In this paper , a nov el algorithm of optimal and precise array response control (OP ARC) has been proposed. This algorithm origins from the adaptiv e array theory and the change rule of -20 -15 -10 -5 0 2 (dB) 0 5 10 15 20 25 30 35 D (dB) A 2 RC PARC OPARC (a) Curves of D versus ρ 2 -20 -15 -10 -5 0 2 (dB) 0 0.5 1 1.5 2 2.5 3 J A 2 RC PARC OPARC -18 -16 -14 -12 -10 0.3 0.4 0.5 (b) Curves of J versus ρ 2 Fig. 6. Pattern variation comparison (the second example). the optimal weight vector , when adding interferences one by one, has been found. Then, the parameter selection mechanism has been carried out to maximize the array gain with the constraint that the response level at one direction is precisely adjusted. Some properties of OP ARC hav e been presented and OP ARC is compared in details with A 2 RC. Finally , simulation results hav e been shown to illustrate the effecti veness of the proposed OP ARC method. Based on the fundamentals dev eloped in this paper , a further extension of OP ARC to multi-point array response control and its applications to, for example, pattern synthesis, multi-constraint adaptiv e beam- forming and quiescent pattern control will be considered in [19]. A P P E N D I X A P RO O F O F P R O P O S I T I O N 2 Since T k − 1 is assumed to be Hermitian, both ξ 0 and ξ k are real-valued and one also gets e ξ c = ξ ∗ c . According to (14), we hav e H k (1 , 2) = w H k − 1 [ a ( θ k ) a H ( θ k ) − ρ k a ( θ 0 ) a H ( θ 0 )] v k = χξ ∗ c 11 < ( " ) -0.5 0 0.5 1 1.5 2 2.5 3 3.5 = ( " ) -0.5 0 0.5 1 1.5 2 2.5 3 D F a F b F 0 C O (a) < ( d ) > c γ (1) ≥ 0 < ( " ) -3 -2 -1 0 = ( " ) -0.5 0 0.5 1 1.5 2 2.5 3 D F b F 0 C F a O (b) < ( d ) < c γ (1) < 0 < ( " ) -0.5 0 0.5 1 1.5 2 2.5 3 3.5 = ( " ) -0.5 0 0.5 1 1.5 2 2.5 3 O C D F 0 F a F b (c) c γ (1) ≥ 0 and < ( d ) < c γ (1) < ( " ) -3 -2 -1 0 = ( " ) -0.5 0 0.5 1 1.5 2 2.5 3 O C D F 0 F a F b (d) c γ (1) < 0 and < ( d ) > c γ (1) Fig. 7. Geometrical illustration of different cases when maximizing array gain. where χ = ξ k − ρ k ξ 0 ∈ R . From (15), we hav e c γ (1) + j c γ (2) = − χξ c / H k (2 , 2) . (62) Thus, γ k,a = − ( k c γ k 2 − R γ ) χξ c k c γ k 2 H k (2 , 2) , γ k,b = − ( k c γ k 2 + R γ ) χξ c k c γ k 2 H k (2 , 2) . From (62) and both χ and H k (2 , 2) are real, we have c γ (2)  c γ (1) = = ( ξ c )  < ( ξ c ) . (63) For the array gain G k in (19) we hav e G k = | ξ ∗ c | · | ξ 0 /ξ ∗ c + γ k | = | ξ ∗ c | ·   γ k − d   = | ξ ∗ c | ·    < ( γ k ) = ( γ k )  T −  < ( d ) = ( d )  T   2 (64) where d = − ξ 0 /ξ ∗ c . Also, = ( d )  < ( d ) = = ( ξ c )  < ( ξ c ) = c γ (2)  c γ (1) . (65) This shows that the origin O , the center c γ and D = [ < ( d ) = ( d )] are co-linear on the plane as shown in Fig. 7. Note that in Fig. 7 we hav e denoted C as the center of the circle. From (64) it can be observed that G k is a scaling of the Euclidean distance between D and a point F = [ < ( γ k ) , = ( γ k )] T located on the circle C γ . From this observation, the optimal solution to (18) can thus be obtained in a geometrical approach belo w . W ithout loss of generality , we first assume that < ( d ) 6 = c γ (1) , otherwise, all γ k on circle C k will have the same G k . In the case of < ( d ) > c γ (1) ≥ 0 or < ( d ) < c γ (1) < 0 , it can be deriv ed that γ k,? = γ k,a , i.e., when F = F a , G k is maximized. In fact, these two cases can be geometrically illustrated by Fig. 7(a) and Fig. 7(b), respectiv ely . Similarly , in the case of c γ (1) ≥ 0 , < ( d ) < c γ (1) (as shown in Fig. 7(c)), or c γ (1) < 0 , < ( d ) > c γ (1) (as shown in Fig. 7(d)), the two points O and D are located on the same sides (right or left) of C . As a result, γ k,? = γ k,b , i.e., when F = F b , G k is maximized in these two cases. In summary , it can be concluded that if ζ > 0 , γ k,? = γ k,a , otherwise, γ k,? = γ k,b , where ζ has been defined in (23). This completes the proof. A P P E N D I X B P RO O F O F P R O P O S I T I O N 5 It is easy to see that − 1 /ξ k > ξ 0 /  | ξ c | 2 − ξ 0 ξ k  in (38) is actually equiv alent to ( | ξ c | 2 − ξ 0 ξ k ) ξ k < 0 . (66) When T k − 1 ∈ S N ++ , we have T − 1 k − 1 ∈ S N ++ . Thus, from (20b) ξ k > 0 . (67) Let the Cholesky decomposition of T − 1 k − 1 be T − 1 k − 1 = ΞΞ H , where Ξ is an in vertible matrix. If a ( θ 0 ) 6 = % a ( θ k ) for ∀ % ∈ C that al ways holds in array antenna theory , we ha ve Ξ H a ( θ 0 ) 6 = % Ξ H a ( θ k ) for ∀ % ∈ C . Then, from the Cauchy- Schwarz inequity we hav e | ξ c | 2 − ξ 0 ξ k =   ( Ξ H a ( θ k )) H ( Ξ H a ( θ 0 ))   2 −   Ξ H a ( θ 0 )   2 2 ·   Ξ H a ( θ k )   2 2 < 0 . (68) Hence, from Proposition 4, we have β k,? = β k,r . From (34), (39) and the fact that β k,? = β k,r , we have β k,? = | ξ c | − √ ρ k ξ 0 √ ρ k ( ξ 0 ξ k − | ξ c | 2 ) . (69) This completes the proof of (44). Furthermore, if T k − 1 ∈ S N ++ , one learns from T k = T k − 1 + β k,? a ( θ k ) a H ( θ k ) that T k ∈ S N ++ ⇔ T k − 1 + β k,? a ( θ k ) a H ( θ k ) ∈ S N ++ (70a) ⇔ I + β k,? T − 1 / 2 k − 1 a ( θ k ) a H ( θ k ) T − 1 / 2 k − 1 ∈ S N ++ (70b) ⇔ 1 + β k,? a H ( θ k ) T − 1 k − 1 a ( θ k ) > 0 (70c) ⇔ β k,? > − 1 /ξ k (70d) ⇔ | ξ c | ξ k − √ ρ k | ξ c | 2 √ ρ k ( ξ 0 ξ k − | ξ c | 2 ) ξ k > 0 (70e) ⇔ ρ k < ξ 2 k / | ξ c | 2 . (70f) This completes the proof of Proposition 5. A P P E N D I X C P RO O F O F P R O P O S I T I O N 7 From the update procedure of weight vector , one gets w k = w k − 1 + γ k v k = T − 1 k − 1 a ( θ 0 ) − β k ξ c T − 1 k − 1 a ( θ k ) / (1 + β k ξ k ) . (71) Then we can obtain a H ( θ 0 ) w k = ξ 0 − β k | ξ c | 2 / (1 + β k ξ k ) and | w H k a ( θ 0 ) | 2 = | ( ξ 0 ξ k − | ξ c | 2 ) β k + ξ 0 | 2  | 1 + β k ξ k | 2 . (72) 12 On the other hand, from (71) we hav e w H k T k − 1 = a H ( θ 0 ) − β ∗ k ξ ∗ c a H ( θ k ) / (1 + β ∗ k ξ k ) and further get w H k T k − 1 w k =  a H ( θ 0 ) − β ∗ k ξ ∗ c a H ( θ k ) / (1 + β ∗ k ξ k )  ·  T − 1 k − 1 a ( θ 0 ) − β k ξ c T − 1 k − 1 a ( θ k ) / (1 + β k ξ k )  = ( ξ 0 ξ k − | ξ c | 2 ) ξ k | β k + 1 ξ k | 2 + | ξ c | 2 ξ k | 1 + β k ξ k | 2 . (73) Combining (72) and (73), we get | w H k a ( θ 0 ) | 2 w H k T k − 1 w k =  ξ 0 ξ k −| ξ c | 2 ξ k  · R 2 β   β k + 1 ξ k   2 + | ξ c | 2 ( ξ 0 ξ k −| ξ c | 2 ) ξ 2 k (74) where we hav e utilized the fact that ξ 0 ξ k − | ξ c | 2 > 0 (see (68) when T − 1 k − 1 ∈ S N ++ ) and   β k + ξ 0 ξ 0 ξ k −| ξ c | 2   = R β . Obviously , the maximization of | w H k a ( θ 0 ) | 2 / ( w H k T k − 1 w k ) is equi valent to minimizing   β k + 1 ξ k   . Define f ,  − 1 /ξ k 0  T (75) then we can reformulate problem (47) as minimize β k k [ < ( β k ) = ( β k )] T − f k 2 (76a) sub ject to [ < ( β k ) = ( β k )] T ∈ C β . (76b) On the other hand, substituting the constraint (36c) into G k and recalling the conclusion of Proposition 3, the array gain satisfies G k =   a H ( θ 0 ) T − 1 k a ( θ 0 )   (77a) =    ξ 0 − β k | ξ c | 2 1 + β k ξ k    (77b) =    ξ 0 ξ k − | ξ c | 2 ξ k    ·    β k − ξ 0 /  | ξ c | 2 − ξ 0 ξ k  β k − ( − 1 /ξ k )    (77c) =    ξ 0 ξ k − | ξ c | 2 ξ k    · k [ < ( β k ) = ( β k )] T − c β k 2 k [ < ( β k ) = ( β k )] T − f k 2 (77d) =    ξ 0 ξ k − | ξ c | 2  · R β /ξ k   k [ < ( β k ) = ( β k )] T − f k 2 . (77e) Note that (77a) comes from the intermediate result of (19), whereas (77d) is obtained from the result of Proposition 3. Since    ξ 0 ξ k − | ξ c | 2  · R β /ξ k   is a constant, from (77e), we can also reformulate problem (36) as (76). Consequently , if T k − 1 ∈ S N ++ , problem (36) has the same optimal solution as the problem (47). This completes the proof. A P P E N D I X D D E R I V AT I O N O F (52) W e first show T − 1 k − 1 a ( θ t ) = A ( θ t , θ k − 1 , · · · , θ 1 ) d k ( θ t ) , ∀ θ t ∈ R (78) where the first component of d k ( θ t ) is 1. W e use induction to prove (78). When k = 1 , since T 0 = I , (78) is obvious, where d 1 ( θ t ) degenerates to the scalar 1 . Suppose (78) is true when k = p , i.e., T − 1 p − 1 a ( θ s ) = A ( θ s , θ p − 1 , · · · , θ 1 ) d p ( θ s ) , ∀ θ s ∈ R (79) where d p ( θ s ) is a p × 1 vector with its first entry 1. When k = p + 1 , we want to show T − 1 p a ( θ r ) = A ( θ r , θ p , · · · , θ 1 ) d p +1 ( θ r ) , ∀ θ r ∈ R (80) where d p +1 ( θ r ) is a ( p + 1) × 1 vector with its first entry 1. T o see (80), one recalls (7) with k = p , and (79), and obtains (81) on the top of next page, where ν = − β p a H ( θ p ) T − 1 p − 1 a ( θ r ) / [1 + β p a H ( θ p ) T − 1 p − 1 a ( θ p )] (82) d p +1 ( θ r ) is a ( p + 1) × 1 vector as sho wn in (83), where d p ( θ r ) i stands for the i th element of d p ( θ r ) . Then, (52) can be seen by substituting (78) with θ t = θ k into (18c). A P P E N D I X E P RO O F O F C O R O L L A RY 1 In the first step of the weight vector update in the OP ARC, we hav e C γ = C µ due to the fact that T 0 = I and hence H 1 = Q 1 . Therefore, one gets µ 1 ,? = γ 1 ,a since [ < ( γ 1 ,a ) = ( γ 1 ,a )] T has the minimum module among the elements in the set C γ as shown in Fig. 1. On the other hand, substituting v 1 = a ( θ 1 ) and w 0 = a ( θ 0 ) into (14) yields H 1 (1 , 2) = ( k a ( θ 1 ) k 2 2 − ρ 1 k a ( θ 0 ) k 2 2 ) · a H ( θ 0 ) a ( θ 1 ) and H 1 (2 , 2) = k a ( θ 1 ) k 4 2 − ρ 1 | a H ( θ 1 ) a ( θ 0 ) | 2 . Recalling (15), we obtain c γ (1) = −< [ a H ( θ 0 ) a ( θ 1 )] ( k a ( θ 1 ) k 2 2 − ρ 1 k a ( θ 0 ) k 2 2 ) k a ( θ 1 ) k 4 2 − ρ 1 | a H ( θ 1 ) a ( θ 0 ) | 2 . Meanwhile, since T 0 = I , one obtains from (23) that ζ = sign( k a ( θ 1 ) k 2 2 − ρ 1 k a ( θ 0 ) k 2 2 ) . Finally , from (22), if ρ 1 ≤ k a ( θ 1 ) k 2 2 / k a ( θ 0 ) k 2 2 , we have γ 1 ,? = γ 1 ,a , otherwise, we obtain γ 1 ,? = γ 1 ,b . Recalling µ 1 ,? = γ 1 ,a , we complete the proof. A P P E N D I X F P RO O F O F (55) A N D (57) From the equiv alence of (53) and (54), one gets b k = − ( I + ˘ Σ k A H k A k ) − 1 ˘ Σ k A H k a ( θ 0 ) . Multiplying by I + ˘ Σ k A H k A k to both sides from the left of this equality yields ˘ Σ k A H k ( a ( θ 0 ) + A k b k ) = − b k . Since ˘ Σ k is a diagonal matrix and ˘ w k = a ( θ 0 ) + A k b k , we obtain ˘ Σ k = Diag  − b k  ( A H k ˘ w k )  . (84) Furthermore, as w k = w k − 1 + µ k a ( θ k ) , b k = [ b T k − 1 µ k ] T and A k = [ A k − 1 a ( θ k )] , we can re write (84) as (85) on the top of next page. Consequently , the following formulation can be obtained: 1 ˘ β k,i =            1 ˘ β k − 1 ,i − µ k a H ( θ i ) a ( θ k ) µ i , 1 ≤ i ≤ k − 1 − a H ( θ k ) ˘ w k − 1 µ k − k a ( θ k ) k 2 2 , i = k . (86) From (86), the po wers of interferences can be clearly observed. After some calculation, either (55) or (57) can be derived. This completes the proof. 13 T − 1 p a ( θ r ) = T − 1 p − 1 a ( θ r ) + ν T − 1 p − 1 a ( θ p ) = A ( θ r , θ p − 1 , · · · , θ 1 ) d p ( θ r ) + ν A ( θ p , θ p − 1 , · · · , θ 1 ) d p ( θ p ) = A ( θ r , θ p , · · · , θ 1 ) d p +1 ( θ r ) , ∀ θ r ∈ R (81) d p +1 ( θ r ) = [ d p ( θ r ) 1 , 0 , d p ( θ r ) 2 , · · · , d p ( θ r ) p ] T + ν [0 , d p ( θ p ) p , d p ( θ r ) p − 1 , · · · , d p ( θ r ) 1 ] T (83) 1  diag( ˘ Σ k ) = − ( A H k ˘ w k )  b k = −  A H k − 1 ˘ w k a H ( θ k ) ˘ w k    b k − 1 µ k  = −  ( A H k − 1 ˘ w k − 1 )  b k − 1 + ( µ k A H k − 1 a ( θ k ))  b k − 1 a H ( θ k ) ˘ w k − 1 /µ k + k a ( θ k ) k 2 2  =  1  diag( ˘ Σ k − 1 ) − ( µ k A H k − 1 a ( θ k ))  b k − 1 − a H ( θ k ) ˘ w k − 1 /µ k − k a ( θ k ) k 2 2  . (85) R E F E R E N C E S [1] R. J. Mailous, Phased Array Antenna Handbook . Norwood, MA: Artech House, 1994. [2] C. L. 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