Adaptive beamforming with multiple-beam constraints in the presence of coherent jammers

(1)

Adaptive beamforming with multiple-beam constraints in

the presence of coherent jammers

夽

Ju-Hong Lee, Ti-Fei Hsu*

Room 517, Building 2, Department of Electrical Engineering, National Taiwan University, Taipei, 106 Taiwan, ROC Received 28 June 1999; received in revised form 27 March 2000

Abstract

This paper deals with the problem of adaptive beamforming using a uniform linear array to provide multiple-beam constraints and suppress coherent jammers simultaneously. By constructing 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 can achieve an adaptive array beamformer with multiple-beam constraints. For coping with the performance degradation due to the presence of coherent jammers, a matrix reconstruction scheme in conjunction with an iterative algorithm is presented to alleviate the coherent jamming e!ect. Simulation examples for illustration and comparison are also given. 2000 Elsevier Science B.V. All rights reserved.

Zusammenfassung

Die Arbeit beschaKftigt sich mit dem Problem der adaptiven Strahlerzeugung durch eine regelmaK{ig angeordnete Antennenanordnung, um gleichzeitig Randbedingungen fuKr mehrere Strahlrichtungen zu ermoKglichen und kohaKrente StoKrer zu unterdruKcken. Durch die Konstruktion einer Steuermatrix, bei der jedem Spaltenvektor ein Steuervektor eines selektiven Strahls entspricht, und einem Bedingungsvektor, dessen Komponenten den VerstaKrkungen eines selektiven Strahls entsprechen, koKnnen wir eine adaptive Strahlerzeugung mit Bedingungen fuKr Mehrfachstrahlen erreichen. Die moKgliche Leistungs-beeintraKchtigung durch die Anwesenheit kohaKrenter StoKrer wird durch ein Verfahren der Matrixumfor-mung umgegangen, wobei ein iterativer Algorithmus vorgestellt wird, der den E!ekt kohaKrenter StoKrer vermeidet. Es werden Simulationsbeispiele zur ErlaKuterung und fuKr Vergleiche angegeben. 2000 Elsevier Science B.V. All rights reserved. Re2sume2

Cer article traite du proble`me du formatage de voie adaptatif sur un reHseau lineHaire uniforme dans l'optique de simultaneHment fournir des contraintes sur des faiscaux multiples et de supprimer des brouilleurs coheHrents. Par construction d'une matrice directionnelle dont chaque colonne correspond au vecteur directionnel d'un faisceau seHlectif et d'un vecteur de contrainte dont chaque composante est eHgale au gain d'un faisceau seHlectif, nous pouvons obtenir un formateur de reHseau adaptatif avec des contraintes de faisceaux multiples. Pour combattre la deHgradation de perfor-mances due a` la preHsence de brouilleurs coheHrents, nous preHsentons une technique de reconstruction de matrice en conjonction avec un algorithme iteHratif pour diminuer l'e!et du brouillage coheHrent. Des exemples de simulation a` des "ns d'illustration et de comparaison sont eHgalement donneHs. 2000 Elsevier Science B.V. All rights reserved.

Keywords: Beamforming; Adaptive array; Multiple beams

夽_{This work was supported by the National Science Council under Grant NSC88-2218-E002-027.}

* Corresponding author.

(2)

1. Introduction

In many applications, such as satellite commun-ications [4], an antenna array must possess beam-forming capability to receive more than one signal with speci"ed gain requirements while suppressing all jammers. This purpose can be e!ectively achieved by using an antenna array with multiple-beam pattern [4,8]. In [8], 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. However, the resulting problem to be solved is a non-linear optimization problem and, hence, solving it requires a sophisti-cated procedure as shown in [8]. Moreover, the existing methods for synthesizing an antenna array with multiple-beam pattern cannot deal with the situation where there exists coherence between a desired signal and the jammers.

In this paper, we present a technique for adaptive beamforming with capability of providing mul-tiple-beam constraints (MBC) in addition to coher-ent jamming suppression. To satisfy the "rst goal, we formulate the problem as "nding such an array weight vector that the array output power is mini-mized subject to MBC. It is shown that an analyti-cal solution for the resulting optimization problem can be easily obtained. To achieve the second goal, a matrix reconstruction scheme in conjunction with an iterative algorithm is presented to cure the array performance deterioration due to coherent jam-mers. Simulation results demonstrate the e!ec-tiveness of the proposed technique.

2. Adaptive beamforming with multiple-beam constraints

In this section, we formulate the problem for adaptive beamforming with capability of receiving more than one desired signals. Consider a uniform linear array (ULA) with M sensors and interele-ment spacing equal toj/2, where j 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 hG, i"1,2,2,K o! broadside. The signal received at the mth array

sensor can be expressed as

xK(t)" )

G sG(t)aK(hG)#nK(t), m"1,2,2, M, (1)

where aK(hG)"exp(j(2pdK sinhG)/j) and dK is the distance between the mth and the "rst array sen-sors, sG(t) is the complex waveform of the ith signal, and nK(t) is the spatially white noise with mean zero and variancepL received at the mth array sensor. The corresponding data vector received by the ar-ray can be written as

x(t)"As(t)#n(t), (2)

where A"[a(h) a(h)2a(h))] with the direction vector of the ith signal given by a(hG)"

[a(hG) a(hG)2a+(hG)]2, the signal source vector is

s(t)"[s(t) s(t)2s)(t)]2, and the noise vector is n(t)"[n(t) n(t)2n+(t)]2. The superscript T

de-notes 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

RV"[RGH]"[R(i!j)]"E+x(t)x&(t),

"ASA&#pLI, (3)

where the superscript H denotes the complex conjugate transpose. S"E+s(t)s&(t), has rank (K!J#1), where J denotes the number of coher-ent signals.

Consider the application in a communication system where more than one signal reception is required. Based on the Frost's algorithm [2], we can utilize an adaptive antenna array which pos-sesses the capability to provide selective gain/null arrangements for di!erent signal beams while sup-pressing all jammers. Let the ULA uses a weight vector w"[w w2w+]2 for processing the re-ceived data vector x(t). Then the signal at the array output is given by y(t)"w&x(t). Assume that the selective gain/null requirements are speci"ed by assigning a gain cH at the direction vector a(hH) for

j"1,2,2P, where P denotes the number of signals

with gain/null constraint. Consequently, the prob-lem can be formulated by the following constrained optimization problem:

minimize E+"y(t)","w&RVw

(3)

where G"[a(h) a(h)2a(h.)] denotes the con-straint matrix and c"[c c2c.]2 the corre-sponding gain vector. According to the theory of Compton [1], the optimal solution, i.e., the optimal weight vector for (4) can be obtained as follows:

w"R\V G(G&R\V G)\c. (5)

It is shown by simulations that the proposed adap-tive array beamformer with MBC demonstrates the capability of receiving multiple signals as well as suppressing incoherent jammers.

However, the e!ectiveness of the above proposed adaptive beamformer will be deteriorated when there exists coherence between a desired signal and jammers. This is due to the fact that the eigenstruc-ture of the correlation matrix RV will be destroyed because the rank of RV is a!ected by the coherence between the desired signals and jammers.

3. An iterative matrix reconstruction scheme To deal with the coherent jamming problem, it is appropriate to consider a possible manner for res-toring the desired eigenstructure of RV. It is well known in the literature that the spatial smoothing (SS) algorithm developed by [6] is e!ective in cur-ing the coherent problem for bearcur-ing estimation. The Toeplitz approximation approach of [3] is originally developed for bearing estimation in the coherent source environment. According to the SS algorithm, we can modify the expression of the optimal weight vector by replacing RV in (5) with the spatial smoothed correlation matrix RM V which is given by RM V"(1/J) (H RHV, where RHV denotes the ensemble correlation matrix of the data vector re-ceived by the jth subarray from the partition of the original array. However, the resolution capability of the array is reduced due to the e!ective aperture size decreases as the size of the subarrays decreases. Therefore, we resort to the following matrix recon-struction scheme without sacri"cing the e!ective aperture size of the original array. After computing the correlation matrix RV from (3), a reconstructed

M;M matrix is given by RK V"[RKGH]"[RK(i!j)], (6) where R_{K (!m)"} 1 M!m +\K G RGG>K, 0)m(M, (7) R_{K (m)"RKH(!m),}

where the superscript * represents conjugate opera-tion. Eq. (7) reveals that the resulting matrix RK V is also Hermitian with size M;M. In fact, the recon-struction scheme is similar to the original Toeplitz approximation approach of [3].

In general, the reconstructed matrix RK V would not have the desired eigenstructure property that its minimum eigenvalue has a multiplicity of (M!K) unless the array size M is in"nite. There-fore, we propose an iterative algorithm to make the reconstructed matrix possess both the Toeplitz} Hermitian and the desired eigenstructure proper-ties. First, the problem of reconstructing the desired eigenstructure from the estimated Toeplitz matrix

RK V is solved by performing the following

minimiz-ation problem: minimize

R_{I VZ1}_# "RIV!RKV",

(8) where S# denotes the set of matrices that their (M!K) smallest eigenvalues are positive and equal. The notation "Q" used in (8) represents "Q""( +_{G +}_{H "qGH") with the M;M matrix}

Q"[qGH]. The optimal solution for (8), denoted as RI VM, in minimum metric distance sense is given

by [7] RI VM" ) I jIeIe&I#j  + I)> eIe&I, (9) wherej*j*2j+ and eK, m"1,2,2,M are the eigenvalues and the corresponding eigenvectors of RK V, respectively, and j is the average of j.>,j.>,2,j+. 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

RI VQ" . I jIeIe&I#j  + I.> eIe&I (10) for (8). Moreover, we note that the non-linear op-erations performed by (9) and (10) cannot guaran-tee to produce a Toeplitz matrix. On the other

(4)

Fig. 1. The beampatterns for Example 1.

hand, the above matrix reconstruction scheme for obtaining R_{K V from RV may alter the eigenstructure} of a matrix. Therefore, it cannot be ensured that the reconstructed matrix RI VQ possesses both the Toeplitz}Hermitian and the desired eigenstructure properties. However, the goal can be achieved us-ing an iterative algorithm in which operations for obtaining RK V and RIVQ are performed alternatively. Consequently, we summarize the proposed iter-ative matrix reconstruction scheme step-by-step as follows:

Step 1. Estimate RV from the received signals. Then

let the iteration number i"0 and R_I

VQ"

RV.

Step 2. Compute the matrix RK G>_V from R_{I G}

VQ by

using the operation of (6).

Step 3. Compute the matrix RI G>_VQ from R_{K G>}

V by

using the operation of (10).

Step 4. If the matrix norm "RIG>

VQ !_RK G>

V "'e,

where e 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 resulting RI G>

VQ to replace RV in

obtaining the optimal weight vector given by (5). Finally, the proof regarding the convergence of the proposed iterative scheme is presented in the appendix.

4. Simulation results

In this section, several simulation examples per-formed on a PC with Pentium-II CPU using MAT-LAB programming language are presented for illustration and comparison. For all simulation examples, we use a ULA with 10 array sensors and the interelement spacing equal to the half of the minimum wavelength of the signals with speci"ed gain/null requirements. These are three signals im-pinging on the array from !303, 03, and 303, respectively, o! array broadside. Moreover, the "rst two signals are assumed to be the desired signals with gains all equal to one and the third signal is the jammer which is coherent with the "rst desired signal. All simulation results presented are obtained by averaging 50 independent runs with independent noise samples for each run. The value of e for terminating the iterative process is set to 10\. In practice, the ensemble correlation matrix

RV is not available. We resort to using the "nite

sample-size estimate R_{M V (or called the sample} cor-relation matrix) to replace RV and performing the iterative scheme proposed in Section 3 on RM V in-stead of RV for simulations. 1200 data snapshots are used for computing the necessary sample correla-tion matrices related to the ensemble correlacorrela-tion matrices. For each example, the beam pattern shown by dash curve represents the result based on the optimal weight vector given by (5).

Example 1. Here, the three signals have signal-to-noise power ratio (SNR) all equal to 0 dB. The constraint vector c is set to [11]2. Hence, P is set to 2. Fig. 1 shows the beam patterns with and without using the proposed technique. Clearly, the pro-posed technique can cope with the problem of coherent jamming situation. The result of using the SS algorithm instead of the proposed technique is also plotted in Fig. 1 for comparison. For perform-ing the SS algorithm, three subarrays with 8 sensors for each are used. Although the SS algorithm is able to alleviate the e!ect of coherent jammers, the pro-posed technique shows the advantages of deeper null in the coherent jammer direction and narrower beam widths over the SS algorithm.

Example 2. Here, the simulations of Example 1 are repeated except that the three signals have SNR equal to 0,10, and 20 dB, respectively. The resulting

(5)

beam patterns with and without using the proposed technique are shown in Fig. 2, while the result of using the SS algorithm is also depicted in Fig. 2. From Fig. 2, we observe again that the proposed technique outperforms the SS algorithm.

Example 3. In this example, we repeat the simula-tions of Example 1 except that the constraint vector

c is set to [1 1 0]2. Thus, P is equal to 3 in this case.

Fig. 3 shows that the resulting beam patterns with and without using the proposed technique, and the result of using the SS algorithm. Although the two techniques provide very deep null in the coherent

jammer direction, the result of using proposed tech-nique possesses narrower beam widths and lower sidelobes than those of using the SS algorithm.

5. Conclusion

This paper has presented a technique for adap-tive beamforming using a uniform linear array with multiple signal gain/null speci"cations in addition to jammer suppression. An iterative matrix recon-struction scheme has further been proposed to incorporate the technique for dealing with the situation of coherent jammers. The convergence property of the proposed iterative scheme has also been analyzed. Simulation results have shown that the proposed technique can e!ectively cure the problem of adaptive beamforming with multiple-beam constraints in the presence of coherent jam-mers.

Appendix A

Here, we prove the convergence of the iterative scheme proposed in Section 3. Given an arbitrary matrix as the initial point, denoted as RI VQ, the proposed iterative scheme generates a matrix sequence _{S0"+RIVQ, RK}

V , RI VQ, RK V ,2, RK GV, RI GVQ, RK G>

V , RI G>VQ ,2, by the following recursive

man-ner: First, obtain RK G>_V from RI G_VQ by using the operation shown by (6). Second, obtain RI G>_VQ from

RK G>

V by using the operation shown by (10), for i"0,1,2,2 . Moreover, we observe that the

opera-tions shown by (7) and (10) are norm-reduced and constant trace operations because

"RK(!m)") 1 M!m +\K G "RGG>K", 0)m(M, (A.1) hence RK (!m)") 1 M!m +\K G "RGG>K", 0)m(M, (A.2)

(6)

and trace[RK V]"MRK(0)" + GRGH"trace[RV], (A.3) and "RIVQ"" . IjI#(M!P)j) +

IjI""RV",_(A.4)

and trace[RI VQ]" . IjI#(M!P)j  " + IjI"trace[RV]. (A.5) Since RK G>

VQ is the optimal solution of (6) when RV"RIGVQ, we have

"RKGV!RIGVQ"*"RKG>V !_RI G

VQ". (A.6)

Similarly, RI G>

VQ is obtained by the norm-reduced

operation of (10) when RV"RKG>V , we have

"RKG>_V !RI G

VQ"*"RKG>

V !RI G>

VQ ". (A.7)

From (A.6) and (A.7), it follows that "RKGV!RIGVQ"*"RKG>V !RI G>

VQ ". (A.8)

Next, de"ne a real non-negative sequence+dG, as

dG""RKGV!RIGVQ" (A.9)

with i"1,2,2 . From (A.8) and (A.9), we note that the descending sequence +dG, must converge to some non-negative constant c [5]. If c"0, then

RK G V"_RI G

VQ as i approaches R. This leads to the

result that the matrix sequence S0 converges. On the other hand, if c'0, then we have from (A.6)}(A.9) that "RKGV!RIGVQ"""RKG>V !RI G VQ"""RKG> V !RI G> VQ " (A.10) as i approaches R. It follows from (A.10) that

RK G and RKG> are the solution of (5) when RV"RIGVQ.

Hence, RK G

V"RK G>

V since the solution for the

min-imization problem of (5) is unique. Therefore, the

matrix subsequence+RKGV, converges. Similarly, the matrix subsequence +RIGVQ, also converges because

RI G

VQ is obtained from (10) when RV"RKGV"RKG>V

and, hence, RI G

VQ"_RI G>

VQ . As a result, we would

expect that the two subsequences converge to two di!erent matrices since c'0. Moreover, based on the facts that RI G

VQORK G>

V and both the operations

for obtaining RK V and RIVQ are norm-reduced opera-tions, we have

"RIGVQ"'"RKG>V "'"RIG>VQ ". (A.11)

This leads to the result that"RIGVQ"'"RIG>_{VQ "}. How-ever, this contradicts the results R_{K G}

V"RK G>

V

ob-tained from (A.10). Consequently, c must be zero. Both the operations for obtaining R_{K V and RIVQ are} also constant trace operations, we can thus "nd that"R1"'0 for any R13S0. Therefore, the pro-posed iterative matrix reconstruction scheme will not converge to the trivial solution. i.e., the null matrix. This completes the necessary proof. References

[1] R.T. Compton Jr., Adaptive Antennas: Concepts and Per-formance, Prentice-Hall, Englewood Cli!s, New York, 1988, pp. 361}376.

[2] O.L. Frost, An algorithm for constrained adaptive array processing, Proc. IEEE 60 (August 1972) 926}935. [3] S.Y. Kung, C.K. Lo, R. Foka, A Toeplitz approximation

approach to coherent source direction "nding, Proceedings of the ICASSP, Tokyo, Japan, April 1986, pp. 193}196. [4] J.T. Mayhan, Area coverage adaptive nulling from

geosyn-chronous satellites: phased arrays versus multiple-beam an-tennas, IEEE Trans. Antennas Propag. AP-34 (March 1986) 410}419.

[5] M.H. Protter, C.B. Morrey, A First Course in Real Analysis, Springer, New York, 1977.

[6] T.J. Shan, M. Wax, T. Kailath, On spatial smoothing for direction-of-arrival estimation of coherent signals, IEEE Trans. Acoust. Speech Signal Process. ASSP-33 (August 1985) 806}811.

[7] M. Wax, T. Kailath, Detection of signals by information theoretic criteria, IEEE Trans. Acoust. Speech Signal Pro-cess. ASSP-33 (April 1985) 387}392.

[8] K.-B. Yu, Adaptive beamforming for satellite communica-tion with selective earth coverage and jammer nulling capa-bility, IEEE Trans. Signal Process. 44 (December 1996) 3162}3166.

Adaptive beamforming with multiple-beam constraints in the presence of coherent jammers