- Email: [email protected]

Transformation-based adaptive array beamforming Jung-Lang Yu!,*, Maw-Lin Leou" !Department of Electronic Engineering, Tung-Nan Junior College of Technology, Taipei, Taiwan, ROC "Department of Electronic Engineering, Nan-Jeon Junior College of Technology, Chai-Yi, Taiwan, ROC Received 13 November 1998

Abstract Recently, the generalized eigenspace-based beamformer (GEIB) has been proposed to combat the pointing errors and to enhance the convergence speed. The weight vector of the GEIB is generated by projecting the weight vector of the linearly constrained minimum variance beamformer (LCMVB) onto a modi"ed signal subspace. Unfortunately, numerical instability and high computational complexity have prohibited the GEIB from practical applications. In the paper, we propose the transformation-based adaptive array beamforming to overcome those problems. With the introduction of the transformation matrix, we "rst present an equivalent structure of the LCMVB. Based on the proposed LCMVB structure, the transformation-based GEIB is further developed without computing the modi"ed signal subspace. With the removing of the computation of the modi"ed signal subspace, the transformation-based GEIB becomes numerically stable and computationally e$cient. Computer simulations are also given to demonstrate the correctness and usefulness of the transformation-based adaptive array beamforming. ( 2000 Elsevier Science B.V. All rights reserved. Zusammenfassung Von kurzem wurde ein verallgemeinerter, auf einer Eigenraumdarstellung basierender Beamformer (GEIB) vorgeschlagen, um Richtungsfehler zu bekaK mpfen und die Konvergenzgeschwindigkeit zu erhoK hen. Der GEIB berechnet den Gewichtsvektor durch die Projektion eines anderen Gewichtsvektors auf einen modi"zierten Signalraum. Der urspruK ngliche Gewichtsvektor resultiert aus einem Beamformer mit minimaler Varianz und linearen Nebenbedingungen (LCMVB). Die numerische InstabilitaK t und der gro{e Rechenufwand des GEIB verhinderten dessen Einsatz im Zusammenhang mit praktischen Anwendungen. Um diese Probleme zu umgehen, schlagen wir in diesem Artikel den auf einer Transformation basierenden adaptiven Beamformer vor. ZunaK chst stellen wir durch die Einfuhrung der Transformationsmatrix eine aK quivalente Struktur des LCMVB vor. Ausgehend von der Struktur des LCMVB wird der auf einer Transformation basierende GEIB weiterentwickelt, um die Berechnung des modi"zierten Signalunterraums zu umgehen. Da der modi"zierte Signalunterraum nicht berechnet werden muss, wird der auf einer Transformation basierende GEIB numerisch stabil und e$zient in Bezug auf den Rechenaufwand. Rechnersimulationen demonstrieren die Richtigkeit und die NuK tzlichkeit des auf einer Transformation basierenden adaptiven Beamformers. ( 2000 Elsevier Science B.V. All rights reserved. Re2 sume2 ReH cemment, le formateur de voies baseH sur un espace propre geH neH raliseH (FVEPG) a eH teH proposeH pour combattre les erreurs de pointage et pour ameH liorer la vitesse de convergence. Le vecteur de poids du FVEPG est geH neH reH en projetant le

* Corresponding author. Fax: #886-2-266-29-592. E-mail address: [email protected] (J.-L. Yu) 0165-1684/99/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 9 9 ) 0 0 1 2 5 - 5

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vecteur de poids du formateur de voies a` vari ance minimale contrait lineH airement (FVVMCL) sur un sous-espace des signaux modi"eH . Malheureusement, une instabiliteH numeH rique et une complexiteH de calcul eH leveH e ont interdit l'utilisation du FVEPG dans des applications pratiques. Dans cet article, nous proposons une formation de voies a` reH seaux adaptatifs baseH e sur une transformation pour deH passer ces proble`mes. Avec l'introduction de la matrice de transformation, nous preH sentons tout d'abord une structure eH quivalente au FVVMCL. Sur la base de cette structure, le FVEPG baseH sur une transformation est ensuite deH veloppeH sans calculer le sous-espace des signaux modi"eH . Avec cette suppression, le FVEPG baseH sur une transformation devient numeH riquement stable et e$cace quant a` sa complexiteH de calcul. Des simulations sur ordinateur sont eH galement preH senteH es qui deH montrent l'exactitude et l'utiliteH du formateur de voies a` reH seau adaptatif baseH sur une transformation. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: LCMVB; Generalized eigenspace-based beamformer (GEIB); Signal subspace; Noise subspace; Gram}Schmidt orthonormalization (GSO); LMS; DMI

1. Introduction Eigensubspace-based methods have been widely investigated in the applications of array beam forming. Based on the eigendecomposition of the correlation matrix of the array input data vector, the eigenspace-based beamformer [4,7] and generalized eigenspace-based beamformer (GEIB) [20,21] have been shown to get better convergence properties and to be less sensitive to the array imperfections than the conventional beamformers. In the paper, we propose the transformationbased array beamforming to implement the GEIB. The GEIB combines the features of the eigenspace-based beamformer [4,7] and the linearly constrained minimum variance beamformer (LCMVB) [8,9]. These constraints in the LCMVB can be chosen to be preserved or not preserved when computing the GEIB weight vector. To preserve these chosen constraints, the GEIB has to compute a modi"ed signal subspace by performing the Gram}Schmidt orthonormalization (GSO) on the signal subspace and the columns of the constraint matrix. However, the calculation of modi"ed signal subspace may result in numerical instability and high computational complexity. To overcome these problems, the transformation-based adaptive array beamforming is proposed. We "rst introduce a transformation matrix to construct an equivalent structure of the LCMVB which generates the optimal weight vector with the transformation matrix followed by the transformed LCMVB weight vector. Based on the proposed

LCMVB structure, the transformation-based GEIB is further developed without computing the modi"ed signal subspace. The transformationbased GEIB is generates the optimal weight vector with the transformation matrix followed by the transformed GEIB weight vector to preserve chosen constraints. The transformed GEIB weight vector is computed by projecting the transformed LCMVB weight vector onto the signal subspace of the transformation domain. Since the computation of the modi"ed signal subspace is removed, the transformation-based GEIB becomes numerically stable and computationally e$cient. This chapter is organized as follows. In Section 2, the GEIB is reviewed brie#y. In Section 3, the concept of the transformation-based LCMVB is proposed. Then the transformation-based GEIB is presented in Section 4. Some discussions about the transformation-based adaptive array beamforming are given in Section 5. Simulation results are presented in Section 6. Finally, Section 7 contains the conclusion.

2. Review of GEIB Consider an M-element narrowband antenna array illuminated by a desired signal and J uncorrelated interferers. The input data vector to the array sensors can be written as J x(t)"s (t)a # + s (t)a #n(t), $ $ i i i/1

(1)

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where s (t) and a are the waveform and the phase $ $ vector of the desired signal, s (t) and a , i"1,2, J i i represent those of the J interferers and n(t) is the background noise which is assumed to be spatially white and uncorrelated with s (t), i"d, 1,2, J. i The correlation matrix of the array input data vector is J R"E[x(t)x(t)H]"p2 a aH# + p2a aH#p2 I, (2) / i i i $ $ $ i/1 where E[ ) ] and H denote expectation and complex conjugate transpose, and p2, p2 and p2 are the / $ i input powers of the desired signal, ith interferer and white noise, respectively. With the assumption that the received source number is less than the array element number, the correlation matrix R can be eigendecomposed as M R" + j e eH"E K EH#E K EH, i i i 4 4 4 / / / i/1

(3)

where j *j *2*j "2"j "p2 are / 1 2 J`2 M eigenvalues in the descending order, e , i"1,2, M, i are the corresponding orthonormal eigenvectors, and E "[e 2e ], 4 1 J`1

(4)

E "[e e ], / J`2 2 M

(5)

K "diag[j 2j ], 4 1 J`1

(6)

K "diat[j j ]. / J`2 2 M

(7)

It is noted that the columns of E span the same 4 subspace as that spanned by a , a ,2, a , which is $ 1 J called signal subspace, and the columns of E are / orthogonal to the signal subspace and span the noise subspace [10,16]. The GEIB combines the features of the LCMVB and the eigensubspace-based beamformer. With the array input data vector x(t), the LCMVB that minimizes the array output power subject to the linear constraints CHw"f generates a weight vector [8] w "R~1C(CHR~1C)~1f, #

(8)

where C and f are the constraint matrix and response vector, respectively. In order to enhance the

233

performance of the LCMVB, the GEIB generates a weight vector by projecting w onto a vector # subspace containing the signal subspace and the columns of the constraint matrix. By properly choosing the projection matrix, each of the linear constraints can be either preserved or not preserved by the GEIB at one's desire. The output SINR decreases with the increase in the number of the linear constraints preserved which are employed to maintain the desired responses in the interested region. Generally, the null constraints [6,12] can be preserved by the GEIB to specify zero responses in the spatial or frequency domains of interest, and linear constraints for model parameter mismatches [2,17] would not be preserved by the GEIB without a!ecting the protection of the desired signal. With a view to facilitating discussion, the linear constraints CHw"f can be partitioned into the form [C C ]Hw"[ f H f H]H p u p u

(9)

where C is an M]p matrix, C is an M]u matrix, p u CHw"f and CHw"f denote the preserved and u u p p unpreserved constraints chosen by the GEIB, respectively. It is noted that the look-direction gain constraint should be included in the unpreserved ones to improve the output SINR [20]. Without loss of generality, f can be assumed to be a zero p vector, i.e. f "0. Then, the weight vector of the p GEIB that preserves the linear constraints CHw"0 p can be expressed by w "EK EK Hw , g 4 4 #

(10)

where EK contains the basis of the subspace 4 spanned by a , a ,2, a and the columns of $ 1 J C , c ,2, c , which is called the modi"ed signal p 1 p subspace. Performing the GSO on e ,2, e 1 J`1 c ,2, c , we can obtain the matrix EK as 1 p 4 EK "[e 2e c( c( ], 4 1 J`1 1 2 p{

(11)

where c( , i"1, 22, [email protected] are in the noise subspace i and [email protected])p. For convenience of discussions, the beamformers described in (8) and (10) are called the direct-form-based LCMVB and the direct-formbased GEIB, respectively.

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3. Transformation-based LCMVB Considering the real-time implementation, one has to compute the matrix EK in (11) to get the 4 weight vector w in (10) for every iteration. Howg ever, this is not an e$cient of performing GSO operations on E and C . Especially, the calculation 4 p of EK is numerically unstable when the constraint 4 vectors are close to the signal subspace and the computational complexity becomes heavy when the number of the constraints preserved by the GEIB is increasing. Therefore, we propose transformation-based adaptive array beamforming technique to overcome these problems. In the following, "rst, the transformation-based LCMVB is given and then the transformation-based GEIB is presented in the next section. The structure of the transformation-based LCMVB is shown in Fig. 1 where the optimum array weight vector is given by w T "Tw6 , (12) # # where T is an arbitrary M](M!p) transformation matrix with column vectors containing the basis of the orthogonal complement of the subspace spanned by C , and w6 is the solution of the p # following minimization problem: minimize w6 HRM w6 subject to

CM Hw6 "f , u u

(13)

where RM "THRT,

(14)

CM "THC . u u The solution of (13) is given by

(15)

(16) w6 "RM ~1CM (CM HRM ~1CM )~1f . u u # u u The weight vector w6 of (16) is called transformed # LCMVB weight vector. From (8), (12) and (16), we have the following theorem. Theorem 1. Let T be an arbitrary M](M!p) transformation matrix with column vectors containing the basis of the orthogonal complement of the subspace spanned by C , and w6 be the transformed p #

Fig. 1. Transformation-based LCMVB.

LCMVB weight vector dexned by (16). Then, the optimum weight vector of the transformation-based LCMVB, w T or Tw6 , is equal to w computed by (8). # # # Proof. Let w be an arbitrary M]1 vector which 1 satis"es the constraints CHw"0. Since T spans the p orthogonal complement of the subspace spanned by C , w can be expressed by the linear combinap 1 tion of the columns of T: w "Tw6 , 1 1

(17)

where w6 is the coe$cient vector. Furthermore, if 1 w also satis"es the other constraints in (9), 1 CH"f , we have from (15) and (17) that u u CM Hw6 "f . u 1 u

(18)

From (17) and (18), one can see that when w satis1 "es the constraints of CHw"f, there exists a vector w6 such that w "Tw6 and w6 satis"es the con1 1 1 1 straints of CM Hw6 "f . Conversely, when w6 satis"es u u 1 the constraints of CM Hw6 "f , the weight vector w in u u 1 (17) will satisfy the constraints of CHw"f. It follows from the above discussion that given w6 , w T satis"es all the constraints CHw"f. Assum# # ing that w T is not the solution of the minimization # problem of the LCMVB, there exists a w which 1

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(14), RM can be rewritten as

satis"es the constraints CHw"f such that wHRw (wHT Rw T . # 1 1 #

(19)

Since w "Tw6 and w T "Tw6 , we have # 1 1 # w6 HRM w6 (w6 HRM w6 . # # 1 1

(20)

However, from (16), w6 is the weight vector that # makes w6 HRw6 minimal while satisfying the conalso satis"es the straints CM Hw6 "f . Since w6 u u 1 constraints CM Hw6 "f , the inequality in (20) is not u u true and the assumption is a contradiction. Therefore, w T is the solution of the minimization prob# lem of the LCMVB. Because the solution of the minimization problem is unique, this concludes the proof that w T "w . # #

h

235

(21)

From Theorem 1, it is shown that the weight vector of the LCMVB can be generated by using the transformation matrix followed by the transformed LCMVB weight vector. Furthermore, the structure of the transformation-based LCMVB in Fig. 1 is much similar to that of the partially adaptive array beamformer [5]. However, the partially adaptive array beamformer is designed to choose the transformation matrix to result in faster convergence speed and is not equivalent to the LCMVB. In addition, [18] had proposed the partially adaptive eigenspace-based GSC beamformer. Unfortunately, the partially adaptive eigenspace-based GSC beamformer can only achieve nearly the same performance as the fully adaptive beamformer under steady-state conditions. Nevertheless, using the proposed transformation-based LCMVB, we can further develop the transformation-based GEIB which results in the same performance as the direct-form-based GEIB.

J RM "p2THa aHT# + p2THa aHT#p2THT. (22) / i i i $ $ $ i/1 Since T contains the basis of the orthogonal complement of the subspace spanned by C , we have p THT"I. Then (22) can be eigendecomposed by (23) RM "EM KM EM H#EM KM EM H, / / / 4 4 4 where EM "[e6 2e6 ] is an (M!p)]r matrix with 4 1 r Me6 Nr being the eigenvectors spanning the signal i i/1 subspace of RM and the number r being the number of signals in the transformation domain, which is equal to or less than J#1, KM is a diagonal matrix 4 whose diagonal elements are the corresponding eigenvalues of the eigenvectors in EM , EM contains 4 / those eigenvectors which span the noise subspace of RM , and KM is a diagonal matrix whose diagonal / elements are equal to p2 . From (16) and (23), one / can generate a weight vector, named the transformed GEIB weight vector, by projecting w6 onto # the signal subspace of the transformation domain (24) w6 "EM EM Hw6 . u 4 4 # Using (24), the weight vector of the transformation-based GEIB is given by w "Tw6 "TEM EM Hw6 . (25) gT u 4 4 # In the following theorem, we show that the weight vector of the transformation-based GEIB in (25) is equal to that of the direct-form-based GEIB in (10). Theorem 2. Let T be an arbitrary M](M!p) transformation matrix with column vectors containing the basis of the orthogonal complement of the subspace spanned by C . Then the two weight vectors p w "EK EK Hw and w "TEM EM Hw are the same, gT 4 4 # g 4 4 # where w , EK and w6 are computed by (8), (11) and # 4 # (16), respectively, and EM contains those eigenvectors 4 that span the signal subspace of RM .

4. Transformation-based GEIB Based on the transformation-based LCMVB shown in the last section, the transformation-based GEIB is developed as follows. Substituting (2) into

Proof. As stated in Section 2 that EK contains the 4 basis of the subspace spanned by [E C ], EK can be 4 p 4 expressed as [E c( 2c( ] in (11). In similar way, 4 1 p{ one can construct the matrix EK by performing the 4

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GSO on the columns of C and E sequentially. Let p 4 Q be an orthonormal matrix resulting from the p GSO of C . The EK can be rewritten as p 4 EK "[Q [email protected] ], 4 p 4

(26)

where [email protected] contains the basis of the subspace span4 ned by (I!Q QH)E (or by TTHE ). It is worth 4 p p 4 noting that the two matrices in (11) and (26) are not the same but their columns span the same subspace. From (26), we have EK EK H"Q QH#[email protected] [email protected] 4 4 p p 4 4

(27)

Substituting (27) into (10) and using the fact that QHw "0, w can be expressed by p # g w "[email protected] [email protected] . 4 4 # g

(28)

In the following, we will shown that w "TEM EM Hw6 also has the form of (28). Since gT 4 4 # [email protected] is generated from the GSO of TTHE , [email protected] and 4 4 4 TTHE also span the same subspace. By permultip4 lying TH with [email protected] and TTHE , one can see that [email protected] 4 4 4 spans the subspace same as THE . Further, from 4 (22) and (23), a lemma is given to show that EM spans the subspace same as THE . 4 4 Lemma 1. Let T be an arbitrary orthonormal matrix and RM "THRT. The columns of E and EM span the 4 4 signal subspaces of R and RM , respectively. Thus, EM span the same subspace same as THE . 4 4 Proof. See the appendix. From the above discussion that [email protected] spans the 4 subspace same as THE and the result of Lemma 4 1 that EM spans the subspace same as THE , we can 4 4 see that [email protected] and EM span the same subspace. In 4 4 addition, we show that [email protected] is a full column rank 4 matrix as follows. Since [T Q ] is a unitary matrix, p [T Q ][email protected] and [email protected] have the same column rank. 4 4 p Because [email protected] is orthogonal to Q shown in (26), we 4 p have [T Q ][email protected] " 4 p

T [email protected] 4 . 0

A B

(29)

From (29), [email protected] and [email protected] have the same column 4 4 number and column rank. Since [email protected] is a full column 4 rank matrix, so is [email protected] . Consequently, with [email protected] 4 4 and EM spanning the same subspace, EM EM H can be 4 4 4 represented by EM EM H"[email protected] ([email protected]@ )[email protected] 4 4 4 4 4 4 Substituting (30) into (25), we have

(30)

w T "[email protected] ([email protected]@ )[email protected] . (31) g 4 # 4 4 4 Since [email protected] is orthogonal to Q , the columns of 4 p [email protected] have fallen onto the subspace spanned by T. 4 Therefore, we have [email protected] "[email protected] and (31) becomes 4 4 w T "[email protected] ([email protected]@ )[email protected] . (32) g 4 # 4 4 4 Because of [email protected]@ "I and w "w T "Tw6 shown # # 4 4 # in (12) and (21), we conclude from (32) and (28) that w T "w . g g

h

(33)

From Theorem 2, the GEIB weight vector can be iteratively computed by updating EM and w6 instead 4 # of EK and w . The numerical stability and computa4 # tional complexity of the GEIB is therefore improved considerably.

5. Discussion With the array input vector x(t), the objective of the array beamforming is to adapt the weight vector to generate the optimal SINR in the array output z(t). In this section, we discuss the di!erence of the direct-form-based GEIB and the transformation-based GEIB from the point of view of deriving the array output as follows. In the case of the direct-form-based GEIB, the block diagram to get the array output is shown in Fig. 2. Given x(t), w and E can be calculated, respectively. Then, # 4 with the constraint matrix C , EK is also computed. p 4 Based on EK and w , the weight vector of the di4 # rect-form-based GEIB w is derived and so also the g array output z(t)"wHx(t). One can observe that g EK is computed only after E is derived. 4 4 On the other hand, the transformation-based GEIB transmit x(t) into transformation matrix T and derives the transformed signal x6 (t). Since the

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237

Fig. 2. Direct-form-based GEIB. Fig. 3. Transformation-based GEIB.

transformation matrix appears as a "xed component in the beamforming, it can be implemented in special purpose high-speed hardware and the transformed signal x6 (t) is derived as soon as the array input vector x(t) is given. Based on the transformed signal, the array output of the transformationbased GEIB is computed in Fig. 3. One can "nd that the transformed GEIB weight vector w6 is u calculated when EM and w6 are given, and the array 4 # output is then computed only by w6 Hx6 (t) without u computing the corresponding GEIB weight vector w T . In a word, the transformation-based GEIB g computes the matrix EM instead of matrix EK and 4 4 adjusts the weight vector with a smaller dimension than direct-form-based GEIB. Therefore, the transformation-based GEIB is more e$cient in deriving the array output than the direct-form-based GEIB. In the following, the computational complexities, which are in terms of the number of complex multiplications, for the direct-form-based and the transformation-based GEIBs are also compared. The weight vector of the direct from-based GEIB is EK EK Hw . To obtain E , one has to perform the 4 4 4 # adaptive signal subspace estimation [13,19] at the expense of O(MJ) complexity. Then performing

the Gram}Schmidt orthonormalization on E and 4 C to get EK requires Mp(p#2J#3) complex mulp 4 tiplications. The weight vector w can be computed # by the constrained least mean square (CLMS) [8,9] or the recursive least-squares (RLS) [3,14] algorithms, which require O(M2) complexity. To obtain EK EK Hw , one requires additional 2M(p#J#1) 4 4 # complex multiplications if the columns of E and 4 C are mutually linearly independent. Therefore, p the total amount of the complex multiplications required to "nd w is g CM "O(MJ)#Mp(p#2J#3) $*3%#5 #O(M2)#2M(p#J#1) "[O(MJ)#O(M2)#2M(J#1)] #Mp(p#2J#5).

(34)

One can observe from (34) that the computational complexity of w is divided into two parts. g The one in the bracket represents the complex multiplications for the GEIB weight vector with all the linear constraints unpreserved and the other

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represents the additional complex multiplications to preserve the linear constraints. On the other hand, the computational complexity of the transformation-based GEIB is examined as follows. Since the transformation matrix T appears as "xed components in the beamforming, they can be implemented in special purpose highspeed hardware. Thus, the computational complexity of the transformed signal x6 (t) is ignored. We only count the number of complex multiplications of EM , w6 and w6 , which are given by O((M!p)r), 4 # u O((M!p)2) and 2(M!p)r, respectively. Therefore, the total number of complex multiplications required to "nd w6 is u CM "O((M!p)r)#O((M!p)2) 53!/4 #2(M!p)r. (35) One can observe from (34) and (35) that the number of complex multiplications saved by the transformation-based GEIB is primarily the second term in (34), which is Mp(p#2J#5). Besides, the computation of these terms in the transformationbased GEIB also saves a lot of complex multiplications since the dimension is reduced to M!p.

Fig. 4. Output SINRs of the transformation-based GEIB and LCMVB. Scenario consists of a desired signal with h "33, $ SNR"15 dB and two interferers with h "423, INR"20 dB 1 and h "!253, INR"25 dB, respectively. (**) transforma2 tion-based GEIB, (oooo) direct-form-based GEIB, ( ) ) ) ) ) ) ) transformation-based LCMVB, and (****) direct-form-based LCMVB.

6. Simulation results A 12 element uniform linear array with halfwavelength spacing is used for simulations. The simulation results are plotted by averaging from 100 independent Monte Carlo trials with the steering direction at the broadside of the array. We verify "rst the correctness and the usefulness of the transformation-based adaptive array beamforming techniques. The array is illuminated by the desired signal and two uncorrelated interferers with angles of arrival h "03, h "423 and h "!253, re$ 1 2 spectively. There are two types of linear constraints in Fig. 4, the "rst-order derivative constraints and the null constraints. The null constraints, which are closely spaced at angles 263, 273 and 283, are chosen to be preserved while the derivative constraints are not preserved by the GEIB. The LCMVB weight vector is solved by the direct-matrix-inversion (DMI) method [11] and the signal subspace is computed from the sample correlation matrix by

Fig. 5. Output SINRs of the transformation-based GEIB and LCMVB. Scenario is the same as that in Fig. 4 except for h "283. (**) transformation-based GEIB, (oooo) direct1 form-based GEIB, ( ) ) ) ) ) ) ) transformation-based LCMVB, and (****) direct-form-based LCMVB.

the function of eig in the PC-MATLAB. One can "nd that the curves for the transformation-based GEIB and the direct-form-based GEIB are nearly identical. At the same time, these two curves for the transformation-based LCMVB and the directform-based LCMVB are also identical. Those plots

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239

Fig. 6. Output SINRs of the transformation-based GEIB. Scenario is the same as that in Fig. 4 except for h "27.53. (a) Eigenvalue 1 distribution of RM : (**) the largest eigenvalue, (- ) - ) - ) ) the second largest eigenvalue, (- - - -) the third largest eigenvalue, and ( ) ) ) ) ) ) ) ) the fourth largest eigenvalue. (b) Output SINRs: (**) transformation-based GEIB, (- - - - -) direct-form-based GEIB with null constraints preserved, and ( ) ) ) ) ) ) ) direct-form-based GEIB with null constraints unpreserved. (c) Bean-patterns: (**) transformation-based GEIB, (- - - - -) direct-form-based GEIB with null constraints preserved, and ( ) ) ) ) ) ) ) direct-form-based GEIB with null constraints unpreserved.

show the correctness of the proposed transformation-based adaptive array beamforming. Fig. 5 presents the stability property of the transformation-based adaptive array beamforming. As one interferer arrives exactly from the direction of 283, Fig. 5 shows that both the transformationbased LCMVB and the direct-form-based LCMVB still have the same performance. However, the transformation-based GEIB converges faster and better than the direct-form-based GEIB. This is due to the fact that null constraint vectors of 263 and 273 are close to the phase vector of the interferer from 283. Thus, these two null constraint vectors

are almost in the signal subspace of R. Therefore, when computing the modi"ed signal subspace of (11), the matrix [E c (263) c (273)] is nearly 4 1 2 numerically rank de"cient, which causes the output noise power of the direct-form-based GEIB to increase considerably. On the other hand, the transformation-based GEIB preserves the null constraints by using the transformation matrix and removes the computation of the modi"ed signal subspace. Therefore, the transformation-based GEIB is more numerically stable than the directform-based GEIB. The phenomenon is especially evident when one interferer arrives from in the

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direction of 27.53 as shown in Fig. 6. In the case of direct-form-based GEIB, the modi"ed signal subspace may be computed from (11), whose rank is equal to 6. However, since these null constraint vectors are very close to signal subspace, we may omit them from (11) as if they were falling on the signal subspace. Then the modi"ed signal subspace is degenerated to the signal subspace of R. As for the transformation-based GEIB, though all the signals are passed through the transformation matrix, only one power of the interferer from 27.53 is left in the transformation domain. Therefore, the rank of EM , r, is determined as 2 from the eigenvalue distri4 bution shown in Fig. 6a by the method of adaptive rank FST [13] or other criteria [1,15]. With the above descriptions, we plot the output SINRs of these di!erent GEIBs in Fig. 6b, which shows that the transformation-based GEIB outperforms the direct-from-based GEIB with either the null constraints preserved or not preserved. Fig. 6c plots the beam pattern of the GEIB at the 2000th sample number. One can observe from Fig. 6c that the direct-from-based GEIB with null constraints preserved generates higher sidelobe level than the others. Further, the null constraints are destroyed when they are not preserved. Nevertheless, the transformation-based GEIB successfully protects the desired signal and generates deep nulls in directions from 263 and 283.

7. Conclusion In this paper, we proposed the techniques of the transformation-based adaptive array beamforming to combat the drawbacks of the GEIB. When deriving the GEIB weight vector, one has to compute a modi"ed signal subspace. However, the computation of the modi"ed signal subspace may make the GEIB numerically unstable and computationally ine$cient. To increase the numerical stability and reduce the computational complexity, the transformation-based adaptive array beamforming is developed. With the introduction of the transformation matrix, we present an equivalent structure of the LCMVB "rst. Based on the proposed LCMVB structure, the transformation-based GEIB is further developed without computing the

modi"ed signal subspace. Since the computation of the modi"ed signal subspace is removed, the transformation-based GEIB becomes numerically stable and computationally e$cient. Computer simulations are also give to demonstrate the correctness and usefulness of the transformation-based adaptive array beamforming.

Appendix A In the appendix, we show that EM spans the 4 subspace same as THE . One can observe "rst from 4 (22) and (23) that EM and THA span the same sub4 space since T is an orthonormal matrix where A"[a a 2a ]. Note that THA may not be a full $ 1 J column rank matrix, i.e. the column number of THA may be larger than that of EM . In addition, because 4 of A and E spanning the same subspace shown in 4 Section II, the columns of A can be written as a linear combination of the columns of E 4 J`1 a"+ et , j i ij i/1

j"d12J,

(A.1)

where t is the coe$cient. We rewrite (A.1) in the ij matrix form as A"E W, 4

(A.2)

where W is a square matrix with column vectors being linearly independent. Premultiplying (A.2) by TH, we have THA"(THE )W. 4

(A.3)

From (A.3), THA is also a linear combination of THE , which means THA and THE span the same 4 4 subspace. Since EM and THA also span the same 4 subspace, we conclude that EM spans the same sub4 space as THE . 4 Acknowledgements This work was supported by the National Science Council of the Republic of China under Grant NSC lg-2213-E-236-001.

J.-L. Yu, M.-L. Leou / Signal Processing 80 (2000) 231}241

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