Robust adaptive array beamforming under steering angle mismatch

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Signal Processing 86 (2006) 296–309

Robust adaptive array beamforming under steering

angle mismatch

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Ju-Hong Lee

, Kuang-Peng Cheng, Chih-Chang Wang

Room 517, Building 2, Graduate Institute of Communication Engineering, Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan

Received 1 March 2004; received in revised form 8 February 2005; accepted 19 May 2005 Available online 5 July 2005

Abstract

The problem of adaptive array beamforming with multiple-beam constraints in the presence of steering angle error is considered. We ﬁrst construct a cost function consisting of terms that utilize a posteriori information due to the received array data and an exponential constraint associated with steering angle error, respectively. Then, an appropriate estimate of the actual phase angle vector associated with each of the desired signals can be obtained by performing nonlinear optimization based on the cost function. An implementation algorithm is further presented to iteratively solve the problem. Theoretical analysis regarding the convergence property of the iterative procedure is also investigated. Finally, several computer simulation examples are provided for illustration and comparison.

Keywords: Adaptive array beamforming; Steering angle error

1. Introduction

An adaptive array beamformer is a spatial ﬁlter designed to extract the desired signal(s) while canceling interference and noise through spatial discrimination. The only a priori knowledge for a main-beam or a multiple-beam constrained beamformer is the actual direction vectors of the desired signals. The direction vector of a desired signal can be obtained from knowledge of the array sensor locations, signal directions-of-arrival, and 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 such mismatch[1–5].

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This work was supported by the National Science Council of TAIWAN under Grant NSC92-2213-E002-033. Corresponding author. Fax: +886 2 23671909.

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To cure the problem of array performance degradation due to the above mismatch, a variety of robust techniques propose to impose additional constraints such as multiple linear constraints, derivative constraints, and norm constraints on the array weight vector [5–15]. However, imposing additional constraints deteriorates the array’s capability to suppress interference and noise, i.e. it consumes some of the array’s degrees-of-freedom. In contrast, the authors of[16]presented a robust approach based on the worst-case performance optimization for dealing with the problem of array performance degradation due to the signal covariance matrix with some fixed error. On the other hand, the problem of the steering vector error due to signal look direction mismatch is considered in[17]to formulate a constrained minimization problem which can be solved by convex optimization-based implementation using second-order cone programming. However, this method provides array performance very similar to that of using the simple so-called diagonal loading of the sample matrix inversion (LSMI) algorithm[18]and comparable with that of using eigenspace-based approaches [19] when the signal-to-noise ratio (SNR) is reasonably high. Recently, the authors of [20] have considered the beamforming problem when the desired signal has a nonrandom steering vector error using the Capon beamformer. A robust method for determining the diagonal loading value is also proposed. All of the above-mentioned techniques[5–20]are developed in the case of adaptive beamforming with main-beam constraint. In many applications, such as satellite communications [21], an antenna array must possess beamforming capability to receive more than one signal with specified gain requirements while suppressing all jammers. This purpose can be effectively achieved by using an antenna array with multiple-beam pattern[21,22]. Recently, a technique for adaptive beamforming with the capability of providing multiple-beam constraints (MBC) has been presented in[23]. In this paper, the problem of adaptive beamforming with MBC in the presence of steering angle error is considered. Instead of directly dealing with the estimation of the actual steering angle, we handle an equivalent problem where the corresponding steering phase angle vector is to be found. A robust method in conjunction with an iterative procedure is presented for coping with the considered problem. 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 an exponential function of the squared norm of the resulting phase error vector. The proposed cost function uses an exponential constraint instead of a non-exponential constraint proposed by a recent research work[24]which deals only with main-beam adaptive beamforming. Minimizing the squared norm of the projection of the steering vector on the noise subspace is equivalent to maximizing the squared norm of the projection of the steering vector on the signal plus interference subspace. The constraint related to an exponential function of the squared norm of the resulting phase error vector is utilized to prevent the obtained optimal phase angle vector for each desired signal from becoming one of the interference phase angle vectors. Since the resulting minimization problem is highly nonlinear, we use a gradient method to iteratively find the solution. It is shown that using the exponential constraint 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 show the effectiveness of the proposed method.

This paper is organized as follows. Section 2 formulates the problem of adaptive beamforming with MBC in the presence of steering angle error. Then, a robust method 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 convergence property of the proposed method. Section 5 shows several simulation examples to illustrate the effectiveness of the proposed method. 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 wavelength of the signals. Let K uncorrelated narrow-band and far-ﬁeld signals impinge on the

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array from direction angles yi, i ¼ 1; 2; . . . ; K, with respect to array broadside. The signal received at the

mth array sensor can be expressed as xmðtÞ ¼

XK i¼1

siðtÞamðyiÞ þnmðtÞ; m ¼ 1; 2; . . . ; M, (1)

where amðyiÞ ¼expðjð2pdmsinyiÞ=liÞ, liis the wavelength of the ith signal, and dmis 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 s2_nreceived at the mth array sensor. In matrix form, we can write the data vector received by the ULA as follows:

xðtÞ ¼ BsðtÞ þ nðtÞ, (2)

where the matrix B ¼ ½aðy1Þaðy2Þ. . . aðyKÞ with the direction vector of the ith signal being given by

aðyiÞ ¼ ½a1ðyiÞa2ðyiÞ. . . aMðyiÞT, the signal source vector is sðtÞ ¼ ½s1ðtÞ s2ðtÞ sKðtÞT, and the noise vector

is nðtÞ ¼ ½n1ðtÞ n2ðtÞ . . . nMðtÞT. The superscript T denotes the transpose operation. Under the assumption

that s(t) and n(t) are uncorrelated, the M M ensemble correlation matrix of x(t) is Toeplitz–Hermitian and given by

Rx¼½Rkl ¼½Rðk lÞ ¼E xðtÞxðtÞ H¼BRsBHþs2nI , (3)

where the superscript H denotes the complex conjugate transpose. Rs¼EfsðtÞsðtÞHghas rank K if the K

signals are uncorrelated.

Let the ULA use a weight vector w ¼ ½w1w2 . . . wM for processing the received data vector x(t) to

produce the array output signal yðtÞ ¼ wH_{xðtÞ. Assume that the selective gain/null requirements are}

speciﬁed by assigning a gain cpat the direction vector a(yp) for p ¼ 1; 2;. . . ; P, where P denotes the number

of signals with gain/null constraint. Then, the optimal weight vector for the adaptive array can be found from the following constrained optimization problem[21]:

Minimize EnyðtÞ2o¼wHRxw

Subject to GHw ¼ c, ð4Þ

where the matrix G ¼ ½aðy1Þaðy2Þ aðyPÞdenotes the constraint matrix and c ¼ ½c1c2 . . . cPdenotes the

gain vector. Accordingly, the optimal weight vector is given by wo¼R1x G G

H

R1_x G

1

c. (5)

Substituting (5) into Efj yðtÞj2g ¼wHRxw yields the corresponding array output power equal to

EnyðtÞ2o¼wHRxw ¼ cH GHR1x G

1

c. (6)

In the presence of a steering angle error, let the error vector associated with the direction angles be he¼ ½Dy1Dy2 . . . DyPT. We consider that the phase angle error vector for the signal with gain cpdue to the

direction angle error Dyp, p ¼ 1; 2;. . . ; P, is given by

Hep¼HpHdp, (7)

where Hp and Hdp denote the phase angle vectors associated with actual direction vector a(yp) and the

presumed direction vector ad(yp), respectively. Without loss of generality, let the mth entry of the actual

direction vector a(yp) be expressed as amðypÞ ¼expðjupmÞ and the corresponding phase angle vector be

constructed as Hp¼ ½up1up2 . . . upMT. Similarly, let the mth entry of the presumed direction vector ad(yp)

be expressed as admðypÞ ¼expðjudpmÞ and the corresponding phase angle vector be constructed as

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corresponding PM 1 phase angle error vector G for the P uncorrelated signals as follows:

W ¼ H½ e1He2. . . HeP, (8)

C ¼ H T_e1HT_e2. . . HT_eP. (9)

To deal with the problem of array beamforming with MBC in the presence of steering angle 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(yp) of the constraint matrix

G coincides with the actual direction vector a(yp), p ¼ 1; 2;. . . ; P. Moreover, from the eigendecomposition

of Rx, we can express Rx¼PMi¼1lieieHi , where l1Xl2X. . . XlJþPXlJþPþ1¼. . . ¼ lM ¼s2n, are the

eigenvalues of Rx in the descending order, ei are the corresponding eigenvectors, and J is the number of

interferers. The eigenvectors associated with the minimum eigenvalue s2

n 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 Rxas follows:

Rx¼

XM i¼1

lieieHi ¼EsKsEHs þEnKnEHn, (10)

where Ks¼diagfl1; l2; . . . ; lJþPgand Kn ¼s2nI , where I denotes the identity matrix with appropriate size.

Based on (6) and (10), we create an appropriate cost function regarding the phase angle errors as JðUÞ ¼X P p¼1 sp H EnEHn sp k exp X P p¼1 HspHdp T HspHdp h i, 2 ( ) , (11)

where sp¼ ½sp1sp2 . . . spMT¼ ½expðjvp1Þexpðjvp2Þ. . . expðjvpMÞT, Hsp¼ ½vp1vp2. . . vpMT, and U ¼

½HT_s1HT_s2 . . . HT_sPT. The ﬁrst term of (11) represents the squared norm of the projection of the constraint vectors sp, p ¼ 1; 2;. . . ; P, on the noise subspace spanned by En; the second term is the constraint

related to the squared norm of the phase angle error, and k denotes a positive weighting parameter providing the relative weight between these terms. According to the theoretical analysis presented in [25], we would expect that the proposed method shows better capabilities against the ﬁnite sample effect because of using a projection scheme to estimate the phase angle vector as shown by the ﬁrst term of Eq. (11).

As a result, the optimal solution Uoin minimizing (11) can then be used as an appropriate estimate of

UP¼ ½HT1HT2 . . . HTPT formed by the actual phase angle vectors Hp, p ¼ 1; 2;. . . ; P, for array

beamforming. However, the cost function of (11) is a highly nonlinear function of the phase angle vectors Hp, p ¼ 1; 2;. . . ; P. Thus, a closed-form solution for the optimal solution cannot exist. We resort to an

iterative procedure to solve this problem as follows. First, we rewrite (11) as follows: JðUÞ ¼ ðSÞHWðSÞ k exp X P p¼1 HspHdp T HspHdp h i, 2 ( ) , (12)

where the MP 1 vector S ¼ ½sT₁sT₂ . . . sT_PT and W is a PM PM block diagonal matrix with the pth M M diagonal block matrix given by EnEnH. Then, the gradient vector of JðUÞ can be computed

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according to rFJðUÞ ¼ 2 Re j EnEHns1 sn 1 2 Re j EnEHns2 sn 2 2 Re j EnEHnsP sn P T þk exp X P p¼1 HspHdp T HspHdp h i =2 ( ) U Ud ð Þ " # , ð13Þ

where Re{x} denotes the real part of x, the superscriptn_{the complex conjugate, and the Hadamard (or}

elementwise) product [26]. Ud ¼ ½HTd1HTd2. . . YTdPT. Then, we update the phase angle vector U and the

corresponding steering constraint vector spas follows:

Uðkþ1Þ¼ Hðkþ1Þ_s1 THðkþ1Þ_s2 T . . . H ðkþ1Þ_sP T

T

¼UðkÞrFJ U ðkÞ, (14)

sðkþ1Þ_pm ¼exp jn ðkþ1Þ_pm ; p ¼ 1; 2; . . . ; P; m ¼ 1; 2; . . . ; M, (15)

where the superscript k in bracket denotes the kth iteration and e the preset positive step size. From (14), we note that the second term includes the factor of the squared norm related to each of the phase angle vector errors HspHdp, p ¼ 1; 2;. . . ; P, at the kth iteration. Hence, it would be expected that the resulting

gradient approach for ﬁnding the optimal U can provide a more appropriate estimate of U since the resulting step size becomes variable due to the exponential term as shown in (13).

Next, we present an appropriate scheme for choosing the initial estimate for the MP 1 vector S ¼ ½sT

1sT2 . . . sTP

T _{in order to initiate the iterative process of the proposed robust method. According to the}

optimal weight vector given by (5) under the assumption that P ¼ 1 and the desired signal with direction vector a(yp), the output of the adaptive array is approximately given by

y_pðtÞ ¼ wH_opxðtÞ spðtÞgpþwHopnðtÞ (16)

based on the assumptions that M4K and the interference signals are suppressed enough, where g_p wH

opaðypÞdenotes the array gain for the speciﬁed signal, where subscript p represents the results obtained by

using the desired signal with direction vector a(yp), p ¼ 1; 2;. . . ; P. Eq. (16) reveals that the output of the

adaptive array can be used as a reference signal to ﬁnd the actual phase angle vector Hp. Consider the

cross-correlation between x(t) and yp(t). We have

E xðtÞypðtÞ n n o ¼E xðtÞxðtÞ Hwop ¼Rxwopppg n pa y p þs2nwop, (17)

where pp denotes the power associated with the speciﬁed signal. In practice, the noise power sn2 is

unknown. However, it can be estimated by taking the average of the MK smallest eigenvalues of the autocorrelation matrix as the estimate of sn2. This raises the issue of estimating the number K of signals/

jammers, which is in itself a difﬁcult signal processing problem. Nevertheless, the techniques based on AIC/ MDL criteria proposed by Wax and Kailath[27]and Fiscler et al.[28] or the bootstrap-based technique proposed by Brcich et al.[29]can be utilized to deal with this difﬁculty. From (17), we can therefore adopt the following vector as the initial estimates for each of sp:

vp¼Rxwops2nwop. (18)

From (18), we note that the direction vector a(yp) is approximately proportional to vpwith a proportional

constant ppgp. Hence, an appropriate initial estimate sp(0)for spcan be formed as follows:

up¼ up1; up2; . . . ; upM T ¼ vp1 1 vp; sð0Þpm¼ upm 1 upm; m ¼ 1; 2; . . . ; M, (19)

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s_pð0Þ¼hsð0Þ_p1; sð0Þ_p2; . . . ; sð0Þ_pMiT (20) for p ¼ 1; 2;. . . ; P, where vp1 denotes the ﬁrst entry of vp. The superscript ‘‘0’’ in bracket identiﬁes the

initial estimate. In other words, we keep only the phase portion of vpand then take the phase referencing to

the ﬁrst element of vp to form the initial estimate sp (0)

. Finally, we construct an initial estimate Sð0Þ¼ ½ðsð0Þ₁ ÞTðsð0Þ₂ ÞT. . . ðsð0Þ_P ÞTT of S for carrying out the proposed iterative process.

4. Convergence of the proposed method

In this section, the convergence property of the proposed method is evaluated. To ensure convergence, we have to show that the cost function to be minimized as given by (12) possesses the property of JðUðkþ1ÞÞoJðUðkÞÞ. For the sake of simplicity, we let the vector A(k) represent the second term of (14), i.e., AðkÞ¼ rFJðUðkÞÞ ¼ ½ðAðkÞ1 Þ

T

ðAðkÞ₂ ÞT. . . ðAðkÞ_P ÞTT and AðkÞ_p ¼ ½AðkÞ_p1AðkÞ_p2 . . . AðkÞ_pMT, p ¼ 1; 2; . . . ; P. Assume that A(k) is a nonzero real vector with a norm small enough at the kth iteration. Then, ðAðkÞÞTAðkÞ40, i.e., AðkÞ T 2 Re j W Sn ðkÞSðkÞno n k exp U ðkÞUd T UðkÞUd h i. 2 n o UðkÞUd o 40. Hence, AðkÞ T 2 Re j W Sn ðkÞSðkÞno n o 4k exp UðkÞUd T UðkÞUd h i. 2 n o AðkÞ T UðkÞUd (21) and the objective function after the (k+1)th iteration is given by

J U ðkþ1Þ¼Sðkþ1ÞHW S ðkþ1Þk exp U ðkþ1ÞUd T Uðkþ1ÞUd h i. 2 n o . (22)

According to the above deﬁnition, (14), and (15), we have

sðkþ1Þ_pm ¼ exp jv ðkþ1Þ_pm ¼exp j vn ðkÞ_pmþA_pmðkÞ o1 þ jAðkÞ_pm exp jv ðkÞ_pm ,

p ¼ 1; 2; . . . ; P; m ¼ 1; 2; . . . ; M. ð23Þ

Based on the result of (23) and the deﬁnition of (20), we can express the MP 1 vector S after the (k+1)th iteration as follows:

Sðkþ1ÞSðkÞþjAðkÞSðkÞ (24)

Substituting (24) into (22) and performing the necessary algebraic manipulations yields J U ðkþ1ÞSðkÞþjAðkÞSðkÞHW S ðkÞþjAðkÞSðkÞ k exp U ðkÞþAðkÞUd T UðkÞþAðkÞUd h i. 2 n o SðkÞHW S ðkÞk exp UðkÞUd T UðkÞUd h i. 2 n o þSðkÞH jA ðkÞW S ðkÞ

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þSðkÞHW jA ðkÞSðkÞþSðkÞH jA ðkÞW jA ðkÞSðkÞ þ UðkÞUd T AðkÞþAðkÞTAðkÞ=2 n o k exp U ðkÞUd T UðkÞUd h i. 2 n o J U ðkÞþ2RenSðkÞHW jA ðkÞSðkÞo þUðkÞUd T AðkÞk exp U ðkÞUd T UðkÞUd h i. 2 n o ð25Þ since the norm of A(k)is small enough, we neglect the terms (A(k))TA(k)and (S(k))H[–jA(k)]W[jA(k)] (S(k)). From (25), it is clear that we have to show

2RenSðkÞHW jA ðkÞSðkÞoþUðkÞUd T AðkÞk exp U ðkÞUd T UðkÞUd h i. 2 n o p0 (26) for any k in order to ensure convergence. Based on (21), condition (26) can be reformulated as follows:

2 RenSðkÞHW jA ðkÞSðkÞoþAðkÞT2 Re j WSn ðkÞSðkÞnop0 (27) which can be reformulated as follows:

XP p¼1 2 Re SðkÞ_p HEnEHn jAðkÞp h i SðkÞ_p þX P p¼1 AðkÞ T 2 Re jEnEHnSðkÞp h i SðkÞ_p n p0. (28)

Next, we manipulate the left-hand side of (28) as follows: XP p¼1 2 Re SðkÞ_p HEnEHn jAðkÞp h i SðkÞ_p þX P p¼1 AðkÞ T 2 Re jEnEHnSðkÞp h i SðkÞ_p n ¼X P p¼1 2 Re SðkÞ_p HEnEHn jA ðkÞ p h i SðkÞ_p þAðkÞTj EnEHnS ðkÞ p h i SðkÞ_p n ¼X P p¼1 2 Re SðkÞ_p H EnEHn jAðkÞp h i SðkÞ_p þ EnEHnSðkÞp h iT jAðkÞ_p h i SðkÞ_p n ¼X P p¼1 2 Re SðkÞ_p HEnEHn jAðkÞp h i SðkÞ_p þ SðkÞ_p H EnEHn H jAðkÞ_p h i SðkÞ_p n ¼X P p¼1 2 Re SðkÞ_p HEnEHn jA ðkÞ p h i SðkÞ_p SðkÞ_p H EnEHn H jAðkÞ_p h i SðkÞ_p n ¼X P p¼1 2 Re 2j Im SðkÞ_p H EnEHn jAðkÞp h i SðkÞ_p ¼0. ð29Þ

Hence, the result given by the left-hand side of (27) is always equal to zero, i.e.

2 RenSðkÞHW jA ðkÞSðkÞoþAðkÞT2 Re j WSn ðkÞSðkÞno¼0. (30) Consequently, we obtain 2 RenSðkÞHW jA ðkÞSðkÞoþUðkÞUd T AðkÞk exp U ðkÞUd T UðkÞUd h i. 2 n o o0. (31)

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It follows from (25) and (31) that

J U ðkþ1ÞoJ U ðkÞ. (32)

The result shown by (32) ensures the convergence of the proposed method.

5. Computer simulation results

In this section, several simulation examples are presented for showing the effectiveness of the proposed method. For all simulation examples, we use a ULA with interelement spacing equal to half of the minimum wavelength of the signals with speciﬁed gain/null requirements. All simulation results presented are obtained by averaging L ¼ 50 independent runs with independent noise samples for each run. Moreover, the values of e and k used by the proposed method for all examples are 0.001 and 0.0001, respectively, which were empirically found to be appropriate. Unless otherwise noted, all simulation results are based on a sample set of 15,000 snapshots. For each of the simulation examples, the results of using the proposed method are obtained after the iteration procedure is terminated. The stopping criterion for terminating the iteration procedure is that the norm of the gradient vector rFJðUÞ is not greater than 0.01.

Example 1. Here, three signal sources with (SNRs) equal to 10, 20, and 20 dB, respectively, are impinging on the array with size M ¼ 8 from direction angles 01, 601, and 801, respectively. Consider the case of main-beam constraint. Let the speciﬁed signal be the ﬁrst one with c1¼1 and the others be the interference. The

steering angle is set to 71, i.e. the steering angle error equals 71. Fig. 1 depicts the simulation results including the array beam patterns, the corresponding array output signal-to-interference-plus-noise ratio (SINR), and the array output SINR versus the SNR of the speciﬁed signal with and without utilizing the proposed method. For comparison, the results of using the diagonal loading technique of[20]and without steering angle error (i.e., the ideal case) are also shown. The output SINRs obtained by using the proposed method, the diagonal loading method, and the ideal case (beamforming with no steering angle error) are 16.42, 7.15, and 18.40 dB, respectively. We observe from these results that the proposed method can effectively cope with the performance degradation due to the steering angle error.

Example 2. In this example, three signal sources with SNRs equal to 5, 4, and 2 dB, respectively, are impinging on the array with size M ¼ 8 from direction angles 171, 511, and 691, respectively. The speciﬁed signals are the ﬁrst two signals with c1¼c2¼1 and the third one is the jammer. Let both direction angle

errors Dy1and Dy2be equal to 71.Fig. 2plots the simulation results in terms of the array beam patterns, the

corresponding array output SINR, and the array output SINR versus the SNR of the specified signal at 171 with and without utilizing the proposed method. For comparison, the results of using the diagonal loading technique of[20]with a loading factor of 2000 (which was empirically found to be optimal), and the ideal case are also shown. The output SINRs obtained by using the proposed method, the diagonal loading, and the ideal errorless beamforming are 13.57, 9.90, and 13.58 dB, respectively. We observe from the results that the proposed method can effectively cope with the performance degradation due to steering angle errors and provide array performance very close to that of the ideal case. From the array output SINR versus the number of snapshots, we note that the proposed method shows better capabilities against the finite sample effect. However, the ideal scenario demonstrates better performance than the proposed method as expected when the number of data snapshots is sufficiently large in this case.

Example 3. Here, we consider the case of four signals with SNRs equal to 5, 6, 7, and 5 dB, respectively, impinging on an array of size M ¼ 15 from direction angles 251, 251, 501, and 01, respectively. Assume that the speciﬁed signals are the ﬁrst three signals with c1¼c2 ¼c3¼1 and the fourth one is a jammer. Let

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array beam patterns using 30,000 data snapshots, the corresponding array output SINR, and the array output SINR versus the SNR of the speciﬁed signal at 251 with and without utilizing the proposed method. For comparison, the results of using the diagonal loading technique of[20] with an empirically optimal

Angles in Degree Power Gain in dB -100 -80 -60 -40 -20 0 20 40 60 80 100 -70 -60 -50 -40 -30 -20 -10 0 10 0 500 1000 1500 2000 2500 3000 -20 -15 -10 -5 0 5 10 15 20 Number of Snapshots Output SINR in dB -10 -5 0 5 10 15 20 -30 -20 -10 0 10 20 30 SNR(dB) Output SINR (dB) (a) (b) (c)

Fig. 1. (a) The array beam patterns for Example 1, (b) the output SINR versus the number of snapshots for Example 1, (c) the output SINR versus the SNR of the desired signal for Example 1.

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loading factor of 2000, and the ideal beamforming case are also shown. The output SINRs obtained by utilizing 30,000 data snapshots for the results of using the proposed method, the diagonal loading, and the ideal case are 17.76, 10.87, and 17.86 dB, respectively. Again, we observe from the simulation results that the proposed method performs very satisfactorily in the presence of steering angle errors for the

multiple--100 -80 -60 -40 -20 0 20 40 60 80 100 -60 -50 -40 -30 -20 -10 0 10 Angles in Degree Power Gain in dB 0 500 1000 1500 2000 2500 3000 -15 -10 -5 0 5 10 15 Number of Snapshots Output SINR in dB -10 -5 0 5 10 15 20 -30 -20 -10 0 10 20 30 SNR (dB) Output SINR (dB) (a) (b) (c)

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beam case. From the array output SINR versus the number of snapshots, we note that the proposed method shows better capabilities against the ﬁnite sample effect. However, the ideal scenario demonstrates better performance than the proposed method as expected when the number of data snapshots is sufﬁciently large in this case.

-100 -80 -60 -40 -20 0 20 40 60 80 100 -60 -50 -40 -30 -20 -10 0 10 Angles in Degree Power Gain in dB 0 500 1000 1500 2000 2500 3000 -20 -15 -10 -5 0 5 10 15 20 Number of Snapshots Output SINR in dB -10 -5 0 5 10 15 20 -40 -30 -20 -10 0 10 20 30 SNR (dB) Output SINR (dB) (a) (b) (c)

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In order to better highlight the contribution, the comparison between the proposed method, the method of [24], and using the quadratic constraint kPP_p¼1½ðspadðypÞÞTðspadðypÞÞ to replace the exponential

constraint–k exp PP_p¼1½ðHspHdpÞTðHspHdpÞ=2

n o

is also performed in terms of array beam patterns

-100 -80 -60 -40 -20 0 20 40 60 80 100 -60 -50 -40 -30 -20 -10 0 10 Angles in Degree Power Gain in dB 0 500 1000 1500 2000 2500 3000 -15 -10 -5 0 5 10 15 Number of Snapshots Output SINR in dB (a) (b)

Fig. 4. (a) The array beam patterns with different constraints for Example 2, (b) the SINR versus number of snapshots under different constraints for Example 2.

-4 -3 -2 -1 0 1 2 -15 -10 -5 0 5 10 15 Output SINR (dB) -4 -3 -2 -1 0 1 2 124 126 128 130 132 log₁₀Kappa Number of Iterations

(13)

and output SINR versus the number of snapshots in the case of Example 2 as shown byFig. 4. The output SINRs obtained by using the method of[24]and the quadratic constraint after using 15,000 data snapshots are 10.18 and 9.70 dB, respectively, which are significantly less than that (13.57 dB) obtained by using the proposed method. These results show that the proposed method indeed outperforms the method of[24]and the method using a quadratic constraint. Finally, we present figures to illustrate how sensitive the array performance is with respect to the values of k and e.Figs. 5 and 6show the array output SINRs and the numbers of iterations versus k and e, respectively, for Example 2. We note that the proposed method provides array performance with robust capabilities not very sensitive to the choice of k and e in the ranges shown by the figures. However, the number of iterations for obtaining the convergent results decreases in general as the value of e increases.

6. Conclusion

This paper has presented an efficient method for multiple-beam adaptive beamforming in the presence of steering angle errors. We have illustrated that the performance degradation of an adaptive beamformer with multiple-beam constraints due to steering angle errors is significant. The proposed method constructs a cost function consisting of the squared norm of the projection of the steering vector on the noise subspace and a constraint related to an exponential function of the squared norm of the corresponding phase error vector. The resulting minimization problem is highly nonlinear but can be solved through the use of an iterative procedure. In conjunction with a steepest-descent algorithm, the phase angle estimates for all of the signals with specified gain constraints can be obtained simultaneously. The convergence property of the proposed method has been investigated. Several simulation examples have shown the effectiveness of the proposed method in dealing with adaptive beamforming under steering angle errors.

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