Adaptive array beamforming with robust
capabilities under random sensor position errors
J.-H. Lee and C.-C. Wang
Abstract: The problem of adaptive array beamforming with multiple-beam constraints in the presence of steering error caused by random sensor position errors is considered. First the statistical relationship between the random sensor position errors and the induced random phase perturbation is derived. Based on the result, a cost function consisting of terms which utilise a posteriori information owing to the received array data and a priori information owing to the probabilistic distribution of the resulting phase perturbation, respectively, is constructed. Then, an appropriate estimate of the actual phase angle vector associated with each of the desired signals can be obtained by performing a nonlinear optimisation problem. An implementation algorithm is further presented to solve iteratively the problem. Theoretical analysis regarding the convergence property of the iterative procedure is also investigated. Finally, several computer simulation examples are provided for demonstrating the effectiveness of the proposed technique.
1 Introduction
An adaptive array beamformer is a spatial filter designed for automatically preserving the desired signals while cancel-ling the interference and noise. The only a priori knowledge for a main-beam or a multiple-beam constrained beam-former is the actual direction vectors of the desired signals. A direction vector of a desired signal can be obtained from the information of the array sensor locations, signal impinging directions, and the propagation characteristics. However, the information may not be perfectly known in practice. This results in a mismatch between the presumed steering vectors and the actual direction vectors. Many reports have shown that the performance of a steered beam adaptive array beamformer is very sensitive to the mismatch [1 – 5]. Particularly, it has been shown in [6] that the amplitude and phase errors owing to array imper-fections produce the effect of reducing interelement corre-lation, leading to a lowering of main lobe gain and decreased ability to discriminate against interferers.
To cure the problem of array performance degradation owing to the above mismatch, most robust techniques propose to impose additional constraints such as multiple linear constraints, derivative constraints, and norm con-straints on the array weight vector [5, 7 – 16]. However, imposing additional constraints deteriorates the array capability in suppressing interference and noise. In contrast, the authors of[17]presented a robust approach based on the worst-case performance optimisation for curing the problem of array performance degradation owing to the signal covariance matrix with some fixed error. The authors of [18] proposed a diagonal loading approach for the beam-forming problem of the desired signal with non-random
steering vector error. Recently, based on the assumption that the steering vector error is an additive Gaussian random vector, two methods have been presented in [19] to find two appropriate closed-form solutions for estimating the optimal steering constraint vector. To deal with the case of steering vector errors owing to phase perturbation, the techniques of[19]resorted to an iterative algorithm to esti-mate the actual phase angle vector because of the resulting nonlinear programming problem. Nevertheless, the conver-gence of the modified techniques is not guaranteed. Moreover, all of the above mentioned techniques [5, 7 – 19] are developed under the situation of adaptive beam-forming with main-beam constraint. In many applications, such as satellite communications [20], an antenna array must possess beamforming capability to receive more than one signal with specified gain requirements while sup-pressing all jammers. This purpose can be achieved effec-tively by using an antenna array with multiple-beam pattern [20, 21]. Recently, a technique for adaptive beam-forming with capability of providing multiple-beam con-straints (MBC) has been presented in[22]. In addition, the theoretical result was extended to deal with adaptive beam-forming using the well-known generalised sidelobe cancel-ler (GSC) in the presence of random array position errors [23]. A technique based on the use of an iterative Toeplitz approximation scheme in conjunction with derivative con-straints was also presented to tackle the problem owing to random array position errors.
In this paper, we consider the problem of adaptive beam-forming with MBC under the random errors in array sensor locations. A robust method in conjunction with an iterative procedure is presented for coping with the considered problem. We first develop the statistical relationship between the sensor position errors and the induced array phase perturbations. As a result, the considered problem can be viewed as a problem associated with the random phase angle errors. To find the optimal phase angle vector, we construct a cost function consisting of the squared norm of the projection of the steering vector on the noise subspace and a constraint related to a likelihood function asso-ciated with the random phase error vector. Minimising the #IEE, 2005
IEE Proceedings online no. 20045018 doi:10.1049/ip-rsn:20045018
Paper first received 4th June 2004 and in final revised form 26th April 2005 The authors are with the Department of Electrical Engineering, National Taiwan University, Taipei, 10617, Taiwan, Republic of China
squared norm of the projection of the steering vector on the noise subspace is equivalent to maximising the squared norm of the projection of the steering vector on the signal plus interference subspace. The constraint related to a like-lihood function associated with the random phase error vector is utilised to prevent that the obtained optimal phase angle vector for each desired signal becomes one of the interference phase angle vectors. Since the resulting minimisation problem is highly nonlinear, we use a gradient method to iteratively search the solution. During the itera-tion process, the algorithm developed by [19]is extended to update each beam position in turn for the desired signals. It is shown that using the constraint related to a like-lihood function of the random phase error vector provides the advantage of properly adjusting the step size during the gradient search procedure. The analysis regarding the investigation of the convergence property of the proposed method is also presented. Several computer simulation examples provide the illustration and comparison.
This paper is organised as follows. Section 2 formulates the problem of adaptive beamforming with MBC in the presence of random sensor position errors and derives the statistical relationship between the random sensor position errors and the resulting random phase angle errors. Then, a robust technique is presented in Section 3 for dealing with the considered problem. In Section 4, we present a theoretical analysis to provide a proof regarding the conver-gence property of the proposed technique. Section 5 shows several simulation examples to show the effectiveness of the proposed technique. Finally, we conclude the paper in Section 6.
2 Problem formulation
Consider a uniform linear array (ULA) with M sensors and interelement spacing equal to half of the smallest signal wavelength l of the signals with specified gain/null arrangements. Let K narrow-band and far-field signals be impinging on the array from direction angles ui, i ¼ 1, 2, . . . , K, off broadside. The signal received at the mth array sensor can be expressed as
zmðtÞ ¼ XK
i¼1
siðtÞamðuiÞ þnmðtÞ; m ¼ 1; 2; . . . ; M ð1Þ where am(ui) ¼ exp( j(2pdmsinui)/l) and dm¼l(m 2 1)/2 is the distance between the mth and the first array sensors, si(t) is the complex waveform of the ith signal, and nm(t) is the spatially white noise with mean zero and variance sn2 received at the mth array sensor. In matrix form, we can write the data vector received by the ULA as follows
zðtÞ ¼ AsðtÞ þ nðtÞ ð2Þ
where the matrix A ¼ [a(u1) a(u2) . . . a(uK)] with the direction vector of the ith signal given by a(ui) ¼ [a1(ui) a2(ui) . . . aM(ui)]T, the signal source vector is s(t) ¼ [s1(t) s2(t) . . . sK(t)]T, and the noise vector is n(t) ¼ [n1(t) n2(t) . . . nM(t)]T. The superscript T denotes transpose operation. Under the assumption that s(t) and n(t) are uncorrelated, the M M ensemble correlation matrix of z(t) is Toeplitz-Hermitian and given by
Rz¼ ½Rkl ¼ ½Rðk lÞ ¼ EfzðtÞzðtÞHg ¼ASAHþs2nI ð3Þ where the superscript H denotes the complex conjugate transpose. S ¼ Efs(t)s(t)Hg has rank K if the K signals are uncorrelated.
Let the ULA use a weight vector w ¼ [w1w2. . . wM] for processing the received data vector z(t) to produce the array output signal u(t) ¼ wHz(t). Assume that the selective gain/ null requirements are specified by assigning a gain cjat the direction vector a(up) for p ¼ 1, 2, . . . , P, where P denotes the number of signals with gain/null constraint. In the case without errors, the steering constraint vectors sp, p ¼ 1, 2, . . . , P, required for adaptive beamforming are set to the direction vectors, i.e., sp¼ a(up), p ¼ 1, 2, . . . , P. Then, the optimal weight vector for the adaptive array can be found from the following constrained optimisation problem[20]
Minimise EfjuðtÞj2g ¼wHRzw Subject to GHw ¼ c
ð4Þ where the matrix G ¼ [a(u1) a(u2) . . . a(uP)] denotes the constraint matrix and c ¼ [c1c2. . . cP]T denotes the gain vector. Accordingly, the optimal weight vector is given by wo¼R1z GðG HR1 z GÞ 1c ð5Þ Substituting (5) into Efju(t)j2g ¼wHRzw yields the corre-sponding array output power equal to
EfjuðtÞj2g ¼wHRzw ¼ cHðGHR1z GÞ 1c
ð6Þ
2.1 Random errors in sensor positions
In the presence of random sensor position errors, the steering constraint vectors sican not be set to the direction vectors a(ui) ¼ [a1(ui) a2(ui) . . . aM(ui)]T, i ¼ 1, 2, . . . , P, because the actual direction vectors a(ui) ¼ [a1(ui) a2(ui) . . . aM(ui)]T, i ¼ 1, 2, . . . , P, are not exactly known owing to the errors. We assume that uˆm¼ [xm, ym] is the location of the mth array sensor as shown inFig. 1 with
^um¼umþDum¼ ½lðm 1Þ=2; 0 þ ½Dxm; Dym ð7Þ where um¼ [l(m 2 1)/2, 0] and Dum¼ [Dxm, Dym], Dxm
Fig. 1 Geometrical illustration of 1-D array with sensor position errors
and Dymare the associated independent Gaussian random position errors with zero mean and the same variancese2. The signal received at the mth array sensor becomes
zmðtÞ ¼ XK
i ¼1
siðtÞ ^amðuiÞ þnmðtÞ; m ¼ 1; 2; . . . ; M ð8Þ where aˆm(ui) ¼ exp( j2puˆmTQi/l) andQi¼ [sinui, cosui]T. As a result, the correlation matrix becomes
Rz¼EfzðtÞzðtÞHg
¼X
K
i¼1
EfjsiðtÞj2gBiaðuiÞaðuiÞHBHi þs 2
nI ð9Þ where
Bi¼diagfexpð j2pDu1Qi=lÞ; expð j2pDu2Qi=lÞ;. . . ;
expð j2pDuMQi=lÞg ð10Þ
We note from (3) and (10) that the Biin (10) represents the effect induced by sensor position errors. In addition, Bi depends on the source bearinguiand Dum, m ¼ 1, 2, . . . , M. As a result, each of the Bi matrices is a function of the position error vectors Dum, m ¼ 1, 2, . . . , M. The structure of the correlation matrix given by (10) reveals that the Toeplitz property possessed by (3) is destroyed by the exist-ence of Biand the resulting signal subspace is spanned by Bia(ui) instead of a(ui), i ¼ 1, 2, . . . , K. Consequently, the array performance will be deteriorated owing to random array position errors.
2.2 The resulting random phase angle errors The phase observed at the mth array sensor owing to the signal with directional angle ui impinging on the array is given by am(ui) ¼ exp( j(2pdmsinui)/l). The correspond-ing phase angle fm¼ (2pdmsinui)/l) is proportional to the distance dm between the first and the mth array sensors. Hence, the random position error Dum¼ [Dxm, Dym] will cause a random error in fm. Substituting (7) into aˆm(ui) ¼ exp( j2puˆm
T
Qi/l) and performing the necess-ary algebraic manipulations provides
^amðuiÞ ¼expð j2p^uTmQi=lÞ ¼expð j2p=l½ðm 1Þd sinui þj2p½dxmsinuiþdymcosuiÞ ð11Þ where dxm¼ Dxm/l and dym¼ Dym/l. From (11), the resulting phase angle error fem corresponding to the random position error Dum¼ [Dxm, Dym] is also Gaussian and is given by
fem¼2pðdxmsinuiþdymcosuiÞ ð12Þ Since the Gaussian random position errors Dxmand Dymare independent with zero mean and the same variancese
2 for m ¼ 1, 2, . . . , M, the random phase angle error fem has zero mean and variance given by
Varffemg ¼Var f2pðdxmsinuiþdymcosuiÞg
¼ ð2pÞ2½VarfdxmgðsinuiÞ2þVarfdymðcosuiÞ2
¼ ð4p2s2eÞ=l2 ð13Þ
Moreover, the covariance Effemfengbetweenfemandfen is zero for m = n. As a result, the M M covariance matrix Ci associated with the random phase angle error vector Qei¼ [fe1,fe2, . . . ,feM]T becomes a diagonal matrix given by 4p2se2I, where I denotes the identity matrix.
2.3 The resulting likelihood function
Following the beamforming criterion established by (4), we consider here that the phase angle error vector for the signal with direction angle up, p ¼ 1, 2, . . . , P, owing to the random position errors is given by
Qep¼QpQdp ð14Þ
whereQpandQdp denote the phase angle vectors associ-ated with actual direction vector a(up) and the presumed direction vector ad(up), respectively. Without loss of generality, let the mth entry of the direction vector a(up) be expressed as am(up) ¼ exp( jypm) and the corresponding phase angle vector be constructed as Qp¼ [yp1yp2. . . ypM]
T
. Similarly, let the mth entry of the direction vector ad(up) be expressed as adm(up) ¼ exp( jydpm) and the corre-sponding phase angle vector be constructed asQdp¼ [ydp1 ydp2. . .ydpM]T. According to the discussion presented in Section 2.2, the phase angle error vector Qep shown by (14) is a real Gaussian random vector with mean zero and covariance matrix Cp. Hence, the probability density func-tion (PDF) forQepis given by
PDFðQepÞ ¼ ½ð2pÞM detðCpÞ1=2expfðQTepC1p QepÞ=2g ð15Þ Accordingly, the likelihood function regarding the phase angle vector error for the signal with gain cp can be defined as
LF ¼ expfðQTepC1p QepÞ=2g
¼expf½ðQpQdpÞTC1p ðQpQdpÞ=2g ð16Þ To deal with the problem of array beamforming with MBC in the presence of random position errors as described above, we present a robust method in the next Section.
3 A robust method
From the property of a gain constrained array beamformer, it is well known that the output power of the beamformer will achieve its maximum when each presumed direction vector ad(up) of the constraint matrix G coincides with the actual direction vector a(up), p ¼ 1, 2, . . . , P. Moreover, from the eigendecomposition of Rz, we can express
Rz¼X M
i ¼1
lieieHi ð17Þ
where l1^l2^ . . . ^lJþP^lJþPþ1¼ . . . ¼lM¼sn2, are the eigenvalues of Rz in the descending order, and ei are the corresponding eigenvectors. J is the number of interferers. The eigenvectors associated with the mini-mum eigenvalue sn2 are orthogonal to the direction vectors of the signals with specified gain/null constraints and interferers. Therefore, the subspaces spanned by En¼ feJþPþ1, . . . , eMg (called the noise subspace) and Es¼ fe1, e2, . . . , eJþPg(called the signal plus interference subspace) are orthogonal. Consequently, we can rewrite Rz as follows
Rz¼ XM i ¼1
lieieHi ¼EsLsEHs þEnLnEHn ð18Þ where Ls¼ diagfl1,l2, . . . ,lJþPg and Ln¼sn2I, where I denotes the identity matrix with appropriate size.
Based on (6), (16), and (18), an appropriate cost function regarding the phase angle error for the signal with gain cp is defined as
J ðQspÞ ¼ ðspÞHEnEHnðspÞ
kexpf½ðQspQdpÞTC1p ðQspQdpÞ=2g ð19Þ where sp¼ [sp1sp2. . . spM]T¼ [exp( jnp1) exp( jnp2) . . . exp( jnpM)]T, and Qsp¼ [np1np2. . . npM]T. The first term of (19) represents the squared norm of the projection of the steering constraint vectors sp, p ¼ 1, 2, . . . , P, onto the noise subspace spanned by En.The second term is the likelihood function related constraint. k denotes a posi-tive weighting parameter providing the relaposi-tive weight between these terms. As a result, the optimal solution Qspoof minimising (19) can then be used as an appropriate estimate of the actual phase angle vectorsQp, p ¼ 1, 2, . . . , P, for array beamforming. However, the cost function of (19) is a highly nonlinear function of the phase angle vectors Qsp, p ¼ 1, 2, . . . , P. Thus, a closed form solution for the optimal solution cannot exist. We resort to an iterative pro-cedure to solve this problem as follows. First, the gradient vector of J(Qsp) is computed according to
rFJ ðQspÞ ¼ ½2 Ref jfðEnEHnspÞ spgg T
þk½expf½ðQspQdpÞTC1p ðQspQdpÞ=2g
C1p ðQspQdpÞ ð20Þ
where Refxg denotes the real part of x and the Hadamard (or elementwise) product. Then, we update the phase angle vectorQspand the corresponding steering con-straint vector spas follows
Qkþ1sp ¼Qksp1rFJ ðQkspÞ ð21Þ
skþ1pm ¼expð jnkþ1pm Þ; m ¼ 1; 2; . . . ; M ð22Þ where the superscript k denotes the kth iteration and 1 the preset positive step size. From (20), we note that the second term includes the factor of likelihood func-tion related to the phase angle vector error Qsp2Qdp, p ¼ 1, 2, . . . , P, at the kth iteration. Hence, it would be expected that the resulting gradient approach for finding the optimal solution ofQspcan provide a more appropriate estimate ofQspsince the resulting step size becomes vari-able according to the exponential term as shown in (20). After finding the optimal estimate for the actual phase angle vector Qp, we substitute the optimal estimate into the cost function and repeat the above iteration process to find the optimal estimate for another actual phase angle vector Qq, q ¼ 1, 2, . . . , P but q = p. From (19) and (20), we note that Qdp stays constant throughout the proposed iteration process to provide a weighting to stop the obtained solution moving too far and, hence, the obtained solution can provide satisfactory beamforming performance.
Consider the required computational complexity. For practical implementation, we compute the sample data cor-relation matrix Rz(i) using i data snapshots as follows
RzðiÞ ¼ 1 1 i Rzði 1Þ þ 1 izðiÞ zðiÞ H ð23Þ which is used as the estimate of the required Rz, where z(i) denotes the ith data snapshot sampled from the received array data vector z(t). After computing the sample data cor-relation matrix Rz(i) using i data snapshots, we perform the iterative process by computing the gradient vector shown by
(20) and then updating the steering constraint vectors shown by (22) to iteratively search the optimal estimate of the steering constraint vectors required for adaptive beamforming. Accordingly, the proposed robust method finds the solution after the iterative procedure is completed for every new snapshot. More than one iteration may be required. According to the ergodic property, the sample data corre-lation matrix Rz(i) computed by (23) gradually approaches the ensemble data correlation matrix Rz given by (3) as the number of snapshots increases. This will lead to the result that the array performance gradually approaches its steady state as expected. The required computational com-plexity is O(M2). Then, using a conventional approach to perform the eigendecomposition of Rz(i) provides the corre-sponding basis matrix En(i) and needs O(M3) in compu-tational complexity. To construct the matrix En(i)En(i)H requires O(M3). Therefore, the required computational complexity is about O(M3) þ O(M2) þ M2þM in order to obtain the first term of the right-hand side of (20). Moreover, it is easy to show that the computational com-plexity for obtaining the second term of the right-hand side of (20) is about 4M. As a result, the computational complexity required for computing the gradient of J(F) when receiving i data snapshots is about O(M3) þ O(M2) þ M2þ5M.
Next, we present an appropriate scheme for choosing the initial guess for each of the steering constraint vectors sp, p ¼ 1, 2, . . . , P, to initiate the iterative process of the pro-posed robust method. According to the optimal weight vector given by (5), the output of the adaptive array is approximately given by
uðtÞ ¼ wHozðtÞ X P
p ¼1
spðtÞgpþwHonðtÞ ð24Þ based on the assumptions that M . K and the interference signals are suppressed enough, where gp; woHa(up) denotes the array gain for the pth specified signal. Equation (24) reveals that the output of the adaptive array can be used as a reference signal to find the actual phase angle vector Qp. Consider the cross-correlation between z(t) and u(t). We have
EfzðtÞuðtÞg ¼EfzðtÞzðtÞHgwo ¼Rzwo
X
P
p¼1
ppgpaðupÞ þsn2wo ð25Þ whereppdenotes the power associated with the pth speci-fied signal. In practice, the noise power sn2 is unknown. However, it can be obtained by setting the average of the (M 2 P) smallest eigenvalues of the autocorrelation matrix as the estimate ofsn2. From (25), we can therefore adopt the following vector for choosing the initial guess for each of sp, p ¼ 1, 2, . . . , P
v ¼ Rzwos2
nwo ð26Þ
Based on (26), it is clear that the direction vector a(up) is approximately proportional to v with a proportional constant equal to ppgp if we let the gain vector c have entries equal to zero except cp¼ 1 in finding the optimal weight vector wo from (5). Consequently, an appropriate initial guess spofor spcan be set to the following normalised vector
sop ¼ ðvp1Þ1vp ð27Þ
and vpthe result given by (19) with the gain vector c having entries equal to zero except cp¼ 1, the superscript ‘o’ rep-resents the initial guess.
4 Convergence of the proposed method
In this Section, the convergence property of the proposed method is evaluated. For sake of simplicity, we let the vector A represent the second term of (21), i.e., A ¼ – 1rFJ(Qspk) ¼ [Ap1Ap2. . . ApM]T, p ¼ 1, 2, . . . , P. Assume that A is a nonzero real vector with a norm small enough at the kth iteration. Then, ATA . 0, i.e.
ATf21 Ref jEnEHnðskpÞ ðskpÞ g 1kexpf½ðQspQdpÞTC1p ðQspQdpÞ=2g C1p ðQspQdpÞg. 0 Hence, ATf2 Ref jEnEHnðs k pÞ ðs k pÞ gg .kexpf½ðQspQdpÞTC1p ðQspQdpÞ=2g ATC1p ðQspQdpÞg ð28Þ
and expf jAg 1 þ jA, where expf jAg ; [expf jAp1g expf jAp2g . . .expf jApMg]Tand 1 denotes an M 1 vector with all entries equal to one. Therefore, the objective function after the (k þ 1)th iteration is given by
J ðQkþ1sp Þ ¼ ðskþ1p ÞHEnEHnðskþ1p Þ kexpf½ðQkþ1sp QdpÞT C1p ðQspkþ1QdpÞ=2g ð29Þ According to the above definition, (21), and (22), we have
skþ1p ¼expf jQspkþ1g ¼expf jðQspk þAÞg ¼expf jAg expf jQspkg
ð1 þjAÞ expf jQspkg ð30Þ
Substituting the approximation expf jAg 1 þ jA and (30) into (29) and performing the necessary algebraic manipula-tions yields
J ðQkþ1sp Þ ½ð1 þjAÞ expf jQkspgHEnEHnð1 þjAÞ expf jQkspg kexpf½ðQkspþA QdpÞT C1p ðQkspþA QdpÞ=2g ðskpÞ HE nEHnðs k pÞ k expf½ðQkspQdpÞTC1p ðQkspQdpÞ=2g þ ðskpÞH ½jAEnEHnðs k pÞ þ ðs k pÞ HE nEHn½jA ðskpÞ þ ðskpÞH ½jAEnEHn½jA ðs k pÞ þ fðQkspQdpÞTC1p A þ A TC1 p A=2g kexpf½ðQkspQdpÞTC1p ðQ k spQdpÞ=2g J ðQkspÞ þ2 RefðskpÞHEnEH n½jA ðskpÞg þ ðQkspQdpÞTC1p Ak expf½ðQ k spQdpÞT C1p ðQkspQdpÞ=2g ð31Þ
since the norm of A is small enough and we neglect the terms ATCp21A and (spk)H[2jA]EnEnH[ jA] (spk).
From (31), it is clear that we have to show 2 RefðskpÞHEnEHn½jA ðskpÞg þ ðQ k spQdpÞTC1p Ak expf½ðQkspQdpÞTC1p ðQ k spQdpÞ=2g % 0 ð32Þ for any k in order to ensure the convergence property of the proposed method. Based on (28), the condition of (31) can be reformulated as follows
2 RefðskpÞHEnEHn½jA ðskpÞg þAT2 Ref j½EnEHns k p ðs k pÞ g% 0 ð33Þ Next, we investigate the left hand side of (33) as follows 2RefðskpÞHEnEHn½jAðskpÞgþA
T
2 Ref½jEnEHnskpðskpÞ g ¼2RefðskpÞHEnEHn½jAðskpÞþA
T
j½EnEHnskpðskpÞ g ¼2RefðskpÞHEnEHn½jAðskpÞþ½EnEHnskpT½jAðskpÞg ¼2RefðskpÞHEnEHn½jAðskpÞþfðspkÞH½EnEHnH½jAðskpÞgg ¼2RefðskpÞHEnEHn½jAðskpÞfðskpÞ
H
½EnEHn½jAðskpÞg g ¼2Ref2j Im fðskpÞHEnEHn½jAðskpÞgg ¼0 ð34Þ Hence, the result given by the left hand side of (33) is always equal to zero, i.e.
2 RefðskpÞHEnEH n½jA ðs k pÞg þA T 2 Ref j½EnEHns k p ðs k pÞ g ¼0 ð35Þ Consequently, we obtain 2 RefðskpÞHEnEHn½jA ðskpÞg þ ðQkspQdpÞTC1p Ak expf½ðQkspQdpÞTC1p ðQ k spQdpÞ=2g , 0 ð36Þ It follows from (31) and (36) that
J ðQspkþ1Þ, J ðQspkÞ ð37Þ The result shown by (37) ensures the convergence of the proposed method.
5 Computer simulation examples
In this Section, we present several simulation examples for illustration and comparison. For all simulation examples, we use a ULA with interelement spacing equal to the half of the minimum wavelengthlof the signals with specified gain/null requirements. Let the random position error for each of Dxm and Dym have zero mean and variance of 0.01l2. All simulation results presented are obtained by averaging 50 independent runs with independent noise samples for each run. The iterative procedure is completed for every new snapshot. More than one iteration may be required. The parameters 1 and kused for simulations are set to 0.1 and 0.0001, respectively. The dash curve in each of the Figures represents the simulation result obtained from a constrained beamformer with no correction for sensor position errors.
Example 1: In this example, three signal sources with signal-to-noise (SNR) equal to 5, 10, and 10 dB, respect-ively, are impinging on the array with size M ¼ 12 from direction angles 08, 608, and 808, respectively. Assume that the specified signal is the first signal with c1¼ 1 and the others are the jammers. Figure 2 plots the simulation
results in terms of the array beam patterns and the corre-sponding array output signal-to-interference-plus-noise ratio (SINR) with and without utilising the proposed method. For comparison, the results of using the diagonal loading method [18]with loading factor of 2000 which is chosen optimally by experiment, the method of[23], and without random position errors are also shown. The output SINRs obtained by using 15,000 data snapshots for the results of using the proposed method after 143 iter-ations, the diagonal loading method, the method of [23], and the result without random position errors are 15.25, 6.90, 6.77, and 15.50 dB, respectively. We observe from the simulation results that the proposed method can cope effectively with the performance degradation owing to the random array position errors.
Example 2: Here, we consider the case of three signals with SNR equal to 3, 4, and 10 dB, respectively, are impinging on the array with size M ¼ 12 from direction angles 178, 2518, and 678, respectively. Assume that the specified signals are the first two signals with c1¼ c2¼ 1 and the third one is the jammer. Figure 3 depicts the simulation results in terms of the array beampatterns and the corre-sponding array output SINR with and without utilising the proposed method. For comparison, the results of using the
diagonal loading method[18] with loading factor of 2000 which is chosen optimally by experiment, the method of [23], and without random position errors are also shown. The output SINRs obtained by using 15,000 data snapshots for the results of using the proposed method after 223 iterations, the diagonal loading method, the method of [23], and the result without random position errors are 14.18, 9.55, 9.65, and 14.52 dB, respectively. For this case with multiple-beam constraints, we observe from the simulation results that the proposed method can effectively cure the performance degradation owing to the random array position errors.
Example 3: Here, the case of four signals with SNR equal to 5, 6, 7, and 10 dB, respectively, are impinging on the array with size M ¼ 15 from direction angles 208, 2208, 2608, and 708, respectively. Assume that the specified signals are the first three signals with c1¼ c2¼ c3¼ 1 and the fourth one is the jammer.Figure 4depicts the simu-lation results in terms of the array beam patterns and the corresponding array output SINR with and without utilising the proposed method. For comparison, the results of using the diagonal loading method [18] with loading factor of 2000 which is chosen optimally by experiment, the method of [23], and without random position errors are ideal random error method of [23] proposed method diagonal loading 15 10 5 0 -5 -10 output SINR, dB number of snapshots 0 500 1000 1500 2000 2500 3000 b 10 5 0 -5 -10 -15 -20 -25 -30 -35 -40 po w er ga in, dB angle, deg -100 -80 -60 -40 -20 0 20 40 60 80 100 a ideal random error method of [23] proposed method diagonal loading
Fig. 2 Array beam patterns and SINR against number of snapshots for example 1
a Array beam patterns
b Output SINR against number of snapshots
15 10 5 0 -5 -10 output SINR, dB number of snapshots 0 500 1000 1500 2000 2500 3000 b 10 5 0 -5 -10 -15 -20 -25 -30 -35 -40 po w er gain, dB angle, deg -100 -80 -60 -40 -20 0 20 40 60 80 100 a ideal random error method of [23] proposed method diagonal loading ideal random error method of [23] proposed method diagonal loading
Fig. 3 Array beam patterns and SINR against number of snapshots for example 2
a Array beam patterns
also shown. The output SINRs obtained by using 15,000 data snapshots for the results of using the proposed method after 268 iterations, the diagonal loading method, the method of[23], and the result without random position errors are 17.11, 12.17, 11.50, and 17.25 dB, respectively. Again, we observe from the simulation results that the pro-posed method can provide very satisfactory performance in this case with multiple-beam constraints.
6 Conclusion
This paper has presented an efficient method for multiple-beam adaptive multiple-beamforming in the presence of random array position errors. We have illustrated that the performance degradation of an adaptive beamformer with multiple-beam constraints owing to random array position errors is significant. According to the proposed method, the equivalent random phase angle errors owing to random array position errors is first developed. Then, we construct a cost function consisting of the squared norm of the projection of the steering vector on the noise subspace and a constraint related to a likelihood function associated with the resulting random phase error vector. The resulting minimisation problem is highly nonlinear but can be solved
through the use of an iterative procedure. In conjunction with a steepest-descent algorithm, the estimates for the phase angles of the signals with specified gain constraints can be obtained simultaneously. The corresponding constraint matrix required for finding the optimal weight vector is then constructed. The convergence property regarding the proposed method has been investi-gated. Several simulation examples have shown the effectiveness of the proposed method in dealing with multiple-beam adaptive beamforming under random array position errors. In general, the proposed robust method would not work with any amplitude errors owing to array imperfections. It is worth further investi-gating the improvement of the proposed robust method to deal with array performance degradation due to any amplitude errors.
7 Acknowledgments
This work was supported by the National Science Council under Grant NSC92-2213-E002-033.
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