GSC-based adaptive beamforming with multiple-beam
constraints under random array position errors
Ju-Hong Lee
a;∗, Ching-Lun Cho
aaDepartment of Electrical Engineering, National Taiwan University, Room 517, Building 2, Taipei 106, Taiwan Received 17 January 2003; received in revised form 24 September 2003
Abstract
This paper deals with the problem of adaptive beamforming under random array position errors to provide multiple-beam constraints and suppress jammers simultaneously. Using a steering matrix with each column vector corresponding to the steering vector of a selective beam and a constraint vector with each entry equal to the gain of a selective beam, we construct the quiescent weight vector and the blocking matrix required by a generalized sidelobe canceller (GSC) to achieve a GSC-based adaptive array beamformer with multiple-beam constraints. For coping with the performance degradation due to random perturbations in array sensor positions, an iterative matrix reconstruction scheme in conjunction with derivative constraints is presented to alleviate the e6ect of random position errors. Simulation results show the e6ectiveness of the proposed technique.
? 2003 Elsevier B.V. All rights reserved.
Keywords: Generalized sidelobe canceller; Adaptive array; Multiple beams; Random position error
1. Introduction
In many applications, such as satellite communica-tions [10], an antenna array must possess beamforming capability to receive more than one signal with speci-?ed gain requirements while suppressing all jammers. This purpose can be e6ectively achieved by using an antenna array with multiple-beam pattern [10,19]. In [19], an adaptive algorithm was proposed to ?nd an adaptive weight close to a desired quiescent beam pattern under a unit norm constraint on the weight.
This work was supported by the National Science Council under Grant NSC91-2219-E002-037.
∗Corresponding author. Fax: 886-2-23638247.
E-mail address:[email protected](J.-H. Lee).
However, the resulting problem to be solved is a non-linear optimization problem and, hence, solving it re-quires a sophisticated procedure as shown in [19]. To tackle this problem, a technique based on Frost’s al-gorithm [3] was recently presented in [8] for adaptive beamforming with multiple-beam constraints (MBC). Nevertheless, the existing methods for synthesizing an antenna array with multiple-beam pattern cannot deal with the situation where there exists random perturba-tions in array sensor posiperturba-tions.
Adaptive beamforming based on a generalized side-lobe canceller (GSC) has been widely considered be-cause of its e6ectiveness and simplicity for achieving multiple linearly constrained beamforming [1,4] and partially adaptive beamforming [16–18,20]. However, many reports show that the GSC-based adaptive beam-formers are usually very sensitive to the mismatches in 0165-1684/$ - see front matter ? 2003 Elsevier B.V. All rights reserved.
steering angle [5] and weight vectors [21]. Recently, the problem of adaptive array signal processing under the situation with random perturbation in array sensor positions has been widely investigated in the literature [12–15].
In this paper, we present a technique for GSC-based adaptive beamforming with the capability of provid-ing MBC in addition to jammprovid-ing suppression. To sat-isfy the ?rst goal, we formulate the problem as ?nding such a quiescent weight vector and a blocking ma-trix for a GSC that the array output power is mini-mized subject to MBC. It is shown that an analytical solution for the resulting optimization problem can be easily obtained. To achieve the second goal, an itera-tive matrix reconstruction scheme in conjunction with derivative constraints is presented to cure the perfor-mance deterioration due to random perturbations in array sensor positions. Simulation results demonstrate the e6ectiveness of the proposed technique.
This paper is organized as follows. In Section 2, the theory of adaptive array beamforming based on a GSC is brieGy discussed. Section 3 presents the technique for GSC-based adaptive array beamform-ing with multiple-beam constraints. In Section4, we present an iterative matrix reconstruction scheme in conjunction with derivative constraints to tackle the problem due to using an array with random sensor position errors. Several simulation examples are pro-vided in Section 5 for showing the e6ectiveness of the proposed technique. A conclusion for the paper is given in Section6.
2. Adaptive beamforming based on GSC
Consider a uniform linear array (ULA) with M sensors and interelement spacing equal to =2, where is the smallest signal wavelength of the signals with speci?ed gain/null arrangements. Assume that K narrow-band and far-?eld signals are impinging on the array from direction angles i, i = 1; 2; : : : ; K, o6
broadside. The signal received at the mth array sensor can be expressed as xm(t) = K i=1 si(t)am(i) + nm(t); m = 1; 2; : : : ; M; (1)
Fig. 1. The generalized sidelobe canceller (GSC) structure. X(t) denotes the received array data vector.
where am(i) = exp(j2dmsin i=) and dmis the
dis-tance between the mth and the ?rst 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 2
n received at the mth array sensor. The
correspond-ing data vector received by the array can be written as
x(t) = As(t) + n(t); (2)
where A = [a(1) a(2) · · · a(K)] with the
di-rection vector of the ith signal given by a(i) =
[a1(i) a2(i) · · · aM(i)]T, the signal source vector
is s(t) = [s1(t) s2(t) · · · sK(t)]T, and the noise vector
is n(t) = [n1(t) n2(t) · · · nM(t)]T. The superscript T
denotes the transpose operation. Assume that s(t) and n(t) are uncorrelated. Then the ensemble correlation matrix of x(t) is Toeplitz–Hermitian with size M ×M and given by Rx= [Rij] = [R(i − j)] = E{x(t)xH(t)} = ASAH+ 2 nI = K i=1 E{|si(t)|2}a(i)aH(i) + 2nI; (3)
where the superscript H denotes the complex conju-gate transpose. S = E{s(t)sH(t)} has rank K since the
K signals are not jointly correlated, i.e. they do not exhibit coherence among themselves.
Using the GSC structure as shown in Fig.1, we can realize an adaptive array beamformer with linear con-straints. The GSC-based adaptive array beamformer was proposed and shown to be e6ective in [4]. The operation performed by B is referred to as the signal blocking operation for removing the desired signals from the received array data. The advantages of the GSC structure are the easy implementation and the assessment of the performance degradation caused by steering or gain errors in array sensors. The output
signal d(t) of the upper branch is given by d(t) = wH
qx(t). The quiescent weight vector wqis utilized to
realize the constrained weight subspace and is cho-sen such that the output signal power E[|d(t)|2] is
minimized subject to a set of L linear constraints. Accordingly, wq can be found from the following
minimization problem: Minimize E[|d(t)|2]
Subject to CHw
q= f; (4)
where C denotes the constraint matrix with size M ×L and f is an L × 1 response vector. Assume that the received data are composed of white noise only. Then wqis given by
wq= C(CHC)−1f: (5)
The sidelobe cancelling branch is utilized for the re-alization of the unconstrained weight subspace which is complementarily orthogonal to the column space of the constraint matrix C. The adaptive weight vector wa
can be determined as follows. Since the output signal y(t) = wH
qx(t) − wHaBHx(t), where B is the so-called
signal blocking matrix for blocking the desired signal from x(t), it follows that B must be chosen so that CHB = 0. In this context, w
a is the optimal solution
for the following minimization problem:
Minimize E[|y(t)|2]: (6)
The solution of (6) can be easily determined as wa= (BHRxB)−1BHRxwq: (7)
Accordingly, the overall weight vector for the GSC-based adaptive beamformer is given by w = wq− Bwa.
3. GSC-based adaptive beamforming with MBC Consider the application in a communication sys-tem where a plurality of signals must be received simultaneously. Based on the considered GSC struc-ture, we can utilize an adaptive antenna array which possesses the capability to provide selective gain/null arrangements for di6erent signal beams while suppressing all jammers. Let the ULA use a weight vector w = [w1 w2· · · wM]Tfor processing the
received data vector x(t). Then the signal at the ar-ray output is given by y(t) = wHx(t). Assume that
the selective gain/null requirements are speci?ed by assigning a gain cj at the direction vector a(j) for
j = 1; 2; : : : ; P, where P denotes the number of signals with gain/null constraint. Consequently, the problem can be formulated by the following constrained opti-mization problem:
Minimize E{|y(t)|2} = wHR xw
Subject to GHw = c; (8)
where G = [a(1) a(2) · · · a(P)] denotes the M × P
steering matrix and c=[c1 c2· · · cP]Tthe
correspond-ing P × 1 gain vector. From the theory of GSC-based adaptive beamforming described above, we can sub-stitute w = wq− Bwa into (8) and reformulate (8) as
the following equivalent optimization problem: Minimize (wq− Bwa)HRx(wq− Bwa); (9)
where the blocking matrix B must satisfy GHB = 0.
Then the solution for (9) can be found as follows: wq= G(GHG)−1c and
wa= (BHRxB)−1BHRxwq: (10)
It is shown by simulations that the proposed GSC-based adaptive array beamformer with MBC does possess the capability of receiving multiple signals as well as suppressing incoherent jammers.
However, the e6ectiveness of the proposed GSC-based adaptive beamformer will be deteriorated when there exists random perturbation in sensor posi-tions. This is due to the fact that the eigenstructure of the correlation matrix Rxis destroyed by the random
perturbation. To show the e6ect of random pertur-bations in sensor positions, let ˆum= [xm; ym] be the
location of the mth array sensor as shown in Fig. 2
with
ˆum= um+ Kum
= [(m − 1)dx; 0] + [Kxm; Kym]; (11)
where um=[(m−1)dx; 0] and Kum=[Kxm; Kym], Kxm
and Kymare the position perturbations with the same
variance 2
Fig. 2. Geometrical illustration of 1-D array with sensor position errors. becomes xm(t) = K i=1 si(t) ˆam(i) + nm(t); m = 1; 2; : : : ; M; (12)
where ˆam(i) = exp(j(2=) ˆuTmi) and i =
[sin i; cos i]T. Then, the resulting correlation matrix
Rx can be expressed as Rx= E{x(t)xH(t)} = K i=1 E{|si(t)|2}Pia(i)aH(i)PHi + 2nI; (13) where Pi= diag exp j2 KuT 1i ; exp j2 KuT 2i ; : : : ; exp j2 KuT Mi : (14)
Comparing (3) and (13), we note that the Piin (14)
represents the e6ect of sensor position perturbations. Moreover, (14) reveals that this e6ect depends on the source bearing i and [Kxm; Kym]; m = 1; 2; : : : ; M.
We also note from (3) that the correlation matrix Rx
exhibits a Toeplitz structure and has K signi?cant eigenvalues greater than 2
n when the K signals
are uncorrelated. However, the correlation matrix given by (13) loses this Toeplitz structure and its signal subspace is spanned by Pia(i) instead of
a(i); i=1; 2; : : : ; K. As a result, the array performance
will be degraded in the presence of sensor position perturbations.
4. Solution to the random perturbation problem 4.1. An iterative matrix reconstruction scheme (IMRS)
To deal with the problem of random position perturbations, we shall consider an appropriate man-ner for restoring the desired eigenstructure of Rx.
After computing the correlation matrix Rxfrom (3), a
reconstructed M × M matrix is given by
ˆRx= [ ˆRij] = [ ˆR(i − j)]; (15) where ˆR(−m) =M − m1 M−m i=1 Ri(i+m); 0 6 m ¡ M; ˆR(m) = ˆR∗(−m); (16)
where the superscript ∗ represents conjugate opera-tion. (15) reveals that the resulting matrix ˆRxis also
Hermitian with size M ×M. In fact, the reconstruction scheme is similar to the Toeplitz approximation ap-proach of [6] which was originally developed for bear-ing estimation in the coherent source environment.
In general, the reconstructed matrix ˆRx would not
have the desired eigenstructure property that its mini-mum eigenvalue has a multiplicity of (M − K) unless the array size M is in?nite. Therefore, we propose an iterative algorithm to make the reconstructed matrix possess both the Toeplitz–Hermitian and the desired eigenstructure properties. First, the problem of recon-structing the desired eigenstructure from the estimated Toeplitz matrix ˆRx is solved by performing the
fol-lowing minimization problem:
where SE denotes the set of matrices that their
(M − K) smallest eigenvalues are positive and equal. The notation |Q| used in (17) represents |Q| = M
i=1
M
j=1 |qij|2
1=2
with the M × M matrix Q = [qij]. The optimal solution for (17), denoted as ˜Rxo,
in the minimum metric distance sense is given by [9] ˜Rxo= K k=1 kekeHk + av M k=K+1 ekeHk; (18) where 1¿ 2¿ · · · M and em; m = 1; 2; : : : ; M,
are the eigenvalues and the corresponding eigenvec-tors of ˆRx, respectively, and av is the average of
P+1; P+2; : : : ; M. In practice, it is generally the case
that the total number K of signal sources is unknown for adaptive beamforming. Hence, we resort to a suboptimal solution given by
˜Rxs= P k=1 kekeHk + av M k=P+1 ekeHk (19)
for (17). Moreover, we note that the nonlinear oper-ations performed by (18) and (19) cannot guarantee a resulting matrix with a Toeplitz structure. On the other hand, the previous reconstruction scheme (Eqs. (15) and (16)) for obtaining ˆRxfrom Rxmay alter the
eigenstructure of a matrix. Therefore, it cannot be en-sured that the reconstructed matrix ˜Rxspossesses both
the Toeplitz–Hermitian and the desired eigenstructure properties. However, the goal can be achieved using an iterative algorithm in which operations for obtain-ing ˆRx and ˜Rxs are performed alternatively.
Conse-quently, we summarize the proposed IMRS step by step as follows:
Step 1: Estimate Rxfrom the received signals. Then
let the iteration number i = 0 and ˜R(0)xs = Rx.
Step 2: Compute the matrix ˆR(i+1)x from ˜R(i)xs by
using the operation of (15).
Step 3: Compute the matrix ˜R(i+1)xs from ˆR(i+1)x by
using the operation of (19).
Step 4: If the matrix norm | ˜R(i+1)xs − ˆR(i+1)x | ¿ ,
where is a preset positive real number, then let i=i+1 and go to Step 2. Otherwise, go to the next step.
Step 5: Use the ˜R(i+1)xs to replace Rx.
Finally, the proof regarding the convergence of the proposed iterative scheme is presented in the Appendix.
4.2. Derivative constraints
The concept of derivative constraints is proposed in [1] for dealing with the problem of adaptive beam-forming in the presence of steering angle error. In general, steered-beam adaptive arrays are very sensi-tive to the mismatch between the direction vector of the desired signal and the steering vector [2,7]. By using derivative constraints on the main beam, the adaptive arrays can perform in a satisfactory man-ner without producing nulls within the main lobe re-gion. From the direction vector of the ith signal given by a(i) = [a1(i) a2(i) · · · aM(i)]T with am(i) =
exp(j2dmsin i=) and dmequal to the distance
be-tween the mth and the ?rst array sensors, we let !i=
2d sin i=, where d denotes the interelement spacing
of the array sensors. With this new parameterization, the direction vector a(i) can be re-expressed as
a(!i) = [exp(j(1 − r)!i); exp(j(2 − r)!i); : : : ;
exp(j(M − r)!i)]T; (20)
where r represents the location of the phase origin. Accordingly, the lth derivative constraint on the di-rection vector a(!i) is implemented by taking the
corresponding derivative vector as follows [1]: al(!i) = @ @j!i l a(!i) = [(1 − r)lexp(j(1 − r)! i); (2 − r)l ×exp(j(2 − r)!i); : : : ; (M − r)l ×exp(j(M − r)!i)]T (21)
and setting the product of wHal(!i) to a gain value.
For example, the gain value is set to zero for providing Gatter beam/null response in the direction vector a(!i)
so that the array can possess the robust capabilities against steering errors.
By incorporating the IMRS and the derivative con-straints, GSC-based adaptive beamforming with MBC and robustness to random perturbations in array posi-tions can be achieved as follows. First, we perform the IMRS on the computed Rxto obtain ˜R(i+1)xs from Step
5 to replace Rx. Then, a new M × P(l + 1) constraint
matrix Gd is constructed from the steering matrix G
of the desired signal directions as follows: Gd= [a(!1) a(!2) : : : a(!P) a1(!1) a1(!2) : : :
a1(!P) a2(!1)a2(!2) : : : a2(!P) : : :
al(!1) al(!2) : : : al(!P)]: (22)
Consequently, the P(l + 1) × 1 gain vector cd
corre-sponding to Gdis formed as follows:
cd= [c1 c2: : : cP 0 0 : : : 0]T: (23)
The required signal blocking matrix Bdmust therefore
satisfy CH dBd= 0.
5. Simulation results
In this section, several simulation examples performed on a PC with Pentium-IV CPU using Matlab programming language are presented for illustration and comparison. For all simulation ex-amples, we use a ULA with 13 array sensors and the interelement spacing equal to half of the min-imum wavelength of the signals with speci?ed gain/null requirements. There are four signals im-pinging on the array from 1, 2, 3, and 4
de-grees, respectively, o6 array broadside. Moreover, the ?rst two signals are assumed to be the de-sired signals with gains all equal to one. The other two signals are the jammers. All simulation results presented are obtained by averaging 25 indepen-dent runs with indepenindepen-dent noise samples for each run. The value of for terminating the iterative process is set to 10−6. The array performance is
evaluated in terms of the resulting beam pattern and output signal-to-interference plus noise ratios (SINR). In practice, the ensemble correlation ma-trix Rx is not available. We resort to using the ?nite
sample-size estimate PRx (also called the sample
cor-relation matrix) to replace Rx and performing the
iterative scheme proposed in Section 4 on PRx
in-stead of Rx for simulations. 6000 data snapshots
are used for computing the necessary sample corre-lation matrices related to the ensemble correcorre-lation matrices.
Example 1. Here, the desired signals and jammers with [1; 2; 3; 4] = [5; 25; 15; 40] degrees have a
signal-to-noise power ratio (SNR) equal to 5 and 10 dB, respectively. The gain vector c is set to [1 1]T. Hence, P is set to 2. The variance 2
e of
the random position perturbations is set to 0:052.
Fig. 3 shows the array performance of using the proposed GSC-based adaptive beamformer with MBC under the situations with (shown by the leg-end of error) and without (shown by the legleg-end of no error) the random perturbations. Fig. 3 also de-picts the array performance of using the proposed GSC-based adaptive beamformer with MBC and the proposed technique to deal with random per-turbations (shown by the legend of proposed). The corresponding array output SINR for each of the three cases is plotted in Fig. 4. After utilizing 6000 data snapshots, the array output SINRs obtained for the cases without and with random perturbations are 15.77 and 2:59 dB, respectively. The array output SINR becomes 14:85 dB when applying the pro-posed technique. From these results, we note that the proposed technique can e6ectively deal with the problem of random position perturbations as well as speed up the convergence behavior of the array response.
Example 2. Here, the simulations of Example 1
are repeated except that the SNRs for the desired signals with [1; 2] = [ − 25; 25] degrees are set
to 3 dB. Both the jammers with [3; 4] = [0; 40]
degrees have SNR equal to 10 dB. The resulting array beam pattern and output SINR for each of the three cases are shown in Figs. 5 and 6, re-spectively. The array output SINRs obtained after using 6000 data snapshots are 13.32 and 2:69 dB for the cases without and with random pertur-bations, respectively. By applying the proposed technique, the array output SINR is increased to 13:12 dB. Again, the proposed technique provides e6ective capability for dealing with the considered problem.
6. Conclusion
This paper has presented a technique for adap-tive beamforming using the generalized sidelobe canceller (GSC) structure with multiple signal gain/null speci?cations in addition to jammer
Fig. 3. The resulting array beam patterns for Example1.
Fig. 4. The corresponding array output SINR versus number of snapshots for Example1.
suppression. An iterative matrix reconstruction scheme in conjunction with derivative constraints has further been proposed to incorporating with
the technique for dealing with the problem due to the random position perturbations of array sen-sors. The convergence property of the proposed
Fig. 5. The resulting array beam patterns for Example2.
Fig. 6. The corresponding array output SINR versus number of snapshots for Example2.
iterative scheme has also been provided. Simula-tion results have shown that the proposed technique can e6ectively cure the problem of GSC-based
adaptive beamforming with multiple-beam constraints when random perturbations in array sensor positions exist.
Appendix [8]
Here, we prove the convergence of the IMRS. Given an arbitrary matrix as the initial point, denoted as ˜R(0)xs,
the proposed iterative scheme generates a matrix se-quence SR={ ˜R(0)xs; ˆR(1)x ; ˜R(1)xs; ˆR(2)x ; : : : ; ˆR(i)x ; ˜R(i)xs; ˆR(i+1)x ;
˜R(i+1)
xs ; : : :} in the following recursive manner: First,
obtain ˆR(i+1)x from ˜R(i)xs by using the operation shown
by (15). Second, obtain ˜R(i+1)xs from ˆR(i+1)x by
us-ing the operation shown by (19), for i = 0; 1; 2; : : :. Moreover, we observe that the operations shown by (16) and (19) are norm-reduced and constant trace operations because | ˆR(−m)| 6M − m1 M−m i=1 |Ri(i+m)|; 0 6 m ¡ M; (A.1) hence, | ˆR(−m)|2 6M − m1 M−m i=1 |Ri(i+m)|2; 0 6 m ¡ M (A.2) and trace[ ˆRx] = M ˆR(0) = M i=1 Rij= trace[Rx] (A.3) and | ˜Rxs|2= P k=1 2 k+ (M − P)2av 6M k=1 2 k= |Rx|2 (A.4) and trace[ ˜Rxs] = P k=1 k+ (M − P)av =M k=1 k= trace[Rx]: (A.5)
Since ˆR(i+1)x is the optimal solution of (17) when Rx=
˜R(i)
xs, we have | ˆR(i)
x − ˜R(i)xs| ¿ | ˆR(i+1)x − ˜R(i)xs|: (A.6)
Similarly, ˜R(i+1)xs is obtained by the norm-reduced
op-eration of (19) when Rx= ˆR(i+1)x , we have | ˆR(i+1)
x − ˜R(i)xs| ¿ | ˆR(i+1)x − ˜R(i+1)xs |: (A.7)
From (A.6) and (A.7), it follows that
| ˆR(i)
x − ˜R(i)xs| ¿ | ˆR(i+1)x − ˜R(i+1)xs |: (A.8)
Next, de?ne a real nonnegative sequence {di} as
di= | ˆR(i)x − ˜R(i)xs| (A.9)
with i = 1; 2; : : :. From (A.8) and (A.9), we note that the descending sequence {di} must converge to some
nonnegative constant c [11]. If c = 0, then ˆR(i)x = ˜R(i)xs
as i approaches ∞. This leads to the result that the matrix sequence SR converges. On the other hand, if
c ¿ 0, then we have from (A.6)–(A.9) that
| ˆR(i)
x − ˜R(i)xs| = | ˆR(i+1)x − ˜R(i)xs|
= | ˆR(i+1)
x − ˜R(i+1)xs | (A.10)
as i approaches ∞. It follows from (A.10) that ˆR(i)
and ˆR(i+1) are the solution of (17) when R
x= ˜R(i)xs.
Hence, ˆR(i)x = ˆR(i+1)x since the solution for the
min-imization problem of (17) is unique. Therefore, the matrix subsequence { ˆR(i)x } converges. Similarly, the
matrix subsequence { ˜R(i)xs} also converges because
˜R(i)
xs is obtained from (19) when Rx= ˆR(i)x = ˆR(i+1)x and,
hence, ˜R(i)xs= ˜R(i+1)xs . As a result, we would expect that
the two subsequences converge to two di6erent ma-trices since c ¿ 0. Moreover, based on the facts that
˜R(i)
xs = ˆR(i+1)x and both the operations for obtaining ˆRx
and ˜Rxsare norm-reduced operations, we have | ˜R(i)
xs| ¿ | ˆR(i+1)x | ¿ | ˜R(i+1)xs |: (A.11)
This leads to the result that | ˜R(i)xs| ¿ | ˜R(i+1)xs |. However,
this contradicts the result ˆR(i)x = ˆR(i+1)x obtained from
(A.10). Consequently, c must be zero. Both the op-erations for obtaining ˆRx and ˜Rxs are constant trace
operations; therefore, we ?nd that |RS| ¿ 0 for any
RS∈ SR. Therefore, the proposed iterative matrix
re-construction scheme will not converge to the trivial solution, i.e. the null matrix. This completes the nec-essary proof.
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