A constrained optimisation approach for null broadening adaptive beamforming is proposed. This approach improves the robustness of the traditional MVDR beamformer by broadening nulls for interference direction and the mainlobe for the desired direction. This optimisation is efficiently solved by semidefinite programming. The proposed approach, when applied to high altitude platform communications using a vertical linear antenna array, provides significantly better coverage performance than a previously reported null broadening technique

Capon

Lobo

Lorenz

Song

Y. Zakharov

Z. Xu

English

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Modified null broadening adaptive beamforming: constrained optimisation approach

Xu, Z.

Zakharov, Y.

White Rose Research Online

This is a repository copy of Modified null broadening adaptive beamforming: constrained optimisation approach.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/3422/Article:Xu, Z. and Zakharov, Y. orcid.org/0000-0002-2193-4334 (2007) Modified null broadening adaptive beamforming: constrained optimisation approach. Electronics Letters. pp. 145-146. ISSN 0013-5194 https://doi.org/10.1049/el:20073561eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. promoting access to White Rose research papers      Universities of Leeds, Sheffield and York http://eprints.whiterose.ac.uk/    This is an author produced version of a paper to be/subsequently published in Electronics Letters. (This paper has been peer-reviewed but does not include final publisher proof-corrections or journal pagination.)    White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/3422    Published paper Xu, Zhengyi and Zakharov, Yuriy (2007) Modified null broadening adaptive beamforming: a constrained optimisation approach. Electronics Letters, 43 (3). 145-146.   White Rose Research Online eprints@whiterose.ac.uk  1Modified Null Broadening AdaptiveBeamforming: A Constrained OptimisationApproachZhengyi Xu and Yuriy ZakharovAbstractA constrained optimisation approach for null broadening adaptive beamforming is proposed.This approach improves the robustness of the traditional MVDR beamformer by broadening nullsfor interference direction and the mainlobe for the desired direction. This optimisation is efficientlysolved by semidefinite programming. The proposed approach, when applied to high altitude platform(HAP) communications using a vertical linear antenna array, provides significantly better coverageperformance than a previously reported null broadening technique.Introduction: Antenna beampattern optimisation is widely used to improve the system capacityin wireless communications. Adaptive beamformers are well known for their high resolutionand sidelobe suppression [1]. Among them, the Minimum Variance Distortionless Response(MVDR) beamformer [2] is one of the most traditional techniques. However, the performanceof the MVDR beamformer significantly degrades if the steering information is inaccurate [3].The most common technique to improve the robustness of the MVDR beamformer is thediagonal loading [3], [4]. The drawback is that there is currently no reliable method to selectthe optimal loading factor [5]. Null broadening [6], [7] is another robust approach. In [6],a cluster of equal-strength incoherent sources are artificially distributed around each originalsource in order to generate a trough like pattern. However, the mainlobe is not broadened.This may result in a poor coverage performance when the system is suffering high steeringerrors while the mainlobe of the beampattern is narrow. Furthermore, arranging equal numberof additive sources for each interferer evidently limits the coverage performance.In this letter, we propose an improved null broadening method. In particular, the followingrefinements are introduced to improve the robustness of null broadening techniques: 1) anew constrained optimisation problem is formulated to broaden both nulls and mainlobe; 2)the optimisation problem is solved using semidefinite programming (SDP) [8], [9]; 3) non-equal number of additive sources are assigned to different users to reduce the total number ofoptimisation parameters, which makes the SDP solution more efficient. The proposed approachis compared with the conventional null broadening technique proposed in [6] for downlinkcommunications from high altitude platforms (HAPs) [10]. The performance improvementdue to the modified null broadening approach is significant.Null broadening techniques: MVDR adaptive beamforming places nulls in the direction ofinterferers whilst assigning unit power to the desired user. The N × 1 MVDR weight vectorsfor the mth desired user is given by [2]wm =R−1u(θm)uH(θm)R−1u(θm), m = 1 · · ·M, (1)where: [·]H denotes the Hermitian transpose; M is the total number of users; u(θm) is theN × 1 steering vector pointing θm degrees from the broadside; N is the number of antennaelements; R is an N × N covariance matrix whose entries can be written in an analyticalform as [7]Rqp = Npδqp +M∑m=1Pmejk0(xq−xp) sin θm , q, p = 1 · · ·N, (2)2where Np is the noise variance, δqp is the Kronecker delta function, Pm is the receivedpower of the mth user, j =√−1, k0 = 2π/λ is the wave number, λ the wavelength, and xqrepresents the element location. In order to produce a trough pattern in each of the interferencedirection, Mailloux proposed to distribute U equal-strength incoherent sources around eachoriginal interference source [6]. Fig.1 illustrates the null broadening approach in the scenarioof HAP communications [10] with a vertical linear antenna array mounted on a HAP at analtitude H , providing the coverage over a circular area of radius r. In this case, the modifiedcovariance matrix R̃ has entries [6]R̃qp =sin(UΛqp)U sin(Λqp)Rqp, q, p = 1 · · ·N, (3)where Λqp = π(xq − xp)ξ/λ, ξ = ε/(U − 1) and ε represents the trough width, shown inFig.1.An improvement of Mailloux’s method can be achieved by broadening the mainlobe inaddition to broadening the nulls. Let’s denoteΘ = [Θ1, Θ2, · · · , ΘM ] (4)andΘm = [θ(−U−12)m , θ(−U−12+1)m , · · · , θ(0)m , · · · , θ(U−12+1)m , θ(U−12)m ], m = 1, · · · ,M, (5)where θ(0)m = θm represents the original user position. The set Θ contains all user positionsincluding the additive sources around each original user. We denote Θm as the complementaryset of Θm with respect to Θ, that is Θm ∪Θm = Θ. The improved null broadening approachis based on minimising the total power of all original interferers and their respective additivesources, while assigning unity received power to the desired user and its additive sources.Then, the beamformer weight vector wm = [w(1)m , · · · , w(N)m ]T providing a beampattern forthe mth user can be found by solving the following constrained optimisation problemwm = arg minw∑θi∈Θm∣∣wHu(θi)∣∣2 subject to∑θn∈Θm∣∣wHu(θn)∣∣2 = 1, (6)where [·]T denotes the transpose, i = 1, · · · , (M − 1)U, n = 1, · · · , U . Different fromassigning a unit power to the desired user, as in the case of the MVDR beamformer [2] andthe null broadening method in [6], the problem (6) sets a constraint to the total receivedpower of a cluster of ’desired users’ in order to broaden the mainlobe. The problem (6) canbe described as a minimisation problem, consisting of an objective function and one equalityconstraint function. Such optimisation problem is especially appropriate for an SDP solver (byusing the interior point method) [11]. SDP creates a smooth convex nonlinear barrier functionfor the constraints and makes it practical to solve convex problems of a large size [8], [9].However, the performance of SDP is affected by the total number of optimisation parameters,that is M ×U in the problem (6). Therefore, in order to obtain more efficient SDP solutions,the total number of optimisation parameters should be reduced and this can be achieved byassigning non-equal number of additive sources for different users. In particular, the numberof additive sources required can be made inversely proportional to the beamwidth of thebeampattern projected to the desired user. For a vertical linear array, it can be found that thenumber of additive sources for the mth original user Um can be made proportional to sin(θm).We now rewrite (4) asΘ = [Θ(U1)1 , Θ(U2)2 , · · · , Θ(UM )M ], (7)andUm = [a sin(θm) + b], (8)where, [·] denotes the rounding integer. For a given number of antenna elements, the coeffi-3cients a and b can be found by computer simulations to maximize the signal to interferenceratio (SIR), which is defined below.Numerical results: Assume that M users are uniformly distributed over the coverage area. Themth user receives signals of powers {P1, · · · , PM} from all M beams of the HAP antenna.Downlink SIR for the mth user is given by [10]ηm =Pm∑Mu=1,u 6=m Pu. (9)The coverage performance is represented by the probability distribution function C(γ) =Pr{ηm > γ}. Fig.2 compares the coverage performance of the following four adaptivebeamforming methods in the downlink HAPs communication scenario: 1) MVDR beamformer(1); 2) null broadening method in [6]; 3) proposed null broadening method using equal numberof additive sources (6) for all users; and 4) proposed null broadening method using non-equalnumber of additive sources.The communication scenario is defined as: r = 32.7 km; H = 20 km; N = 171; M = 60.The users position errors are random with uniform distribution within the interval [-60m,+60m]. Coefficients in (8) are set to a = 5 and b = 3. The number of additive sourcesin method 2 is Û = 1M∑Mm=1 Um. We assume a noise free scenario for simplicity and useSIR (9) to estimate the coverage performance. Fig.2 shows the comparison results. It canbe seen that the MVDR beamformer has a poor performance in such a scenario. The nullbroadening method in [6] significantly improves the performance when compared to theMVDR beamformer. Method 3, using constrained optimisation, however, can achieve bettercoverage performance. Further improvement is obtained by method 4, using non-equal numberof additive sources; in total, a 4.5 dB SIR improvement is achieved for 95% coverage ,compared with the Mailloux’s null broadening method proposed in [6].Conclusions: A constrained optimisation approach is proposed to improve the performanceof null broadening adaptive beamforming. This approach broadens the pattern for the desireduser as well as the interferers. The constrained optimisation problem is solved by semidefiniteprogramming and its efficiency and performance are further improved when non-equal numberof additive sources are distributed among users to reduce the total number of optimisationparameters. Simulation results show that the proposed approach achieves significant improve-ment in the coverage performance, compared with the previous null broadening method andthe MVDR beamformer.REFERENCES[1] C. G. Lal, “Application of antenna arrays to mobile communications, part II: beam-forming and direction-of-arrivalconsiderations,” IEEE Proceeding., vol. 85, no. 8, pp. 1195–1245, Aug 1997.[2] J. Capon, “High-relolusion frequency-wavenumber spectrum analysis,” Proc. IEEE., vol. 57, pp. 1408–1418, 1969.[3] J. Li, P. Stoica, and Z. Wang, “On robust Capon beamforming and diagonal loading,” IEEE Trans. Signal Processing.,vol. 51, no. 7, pp. 1702–1715, July 2003.[4] R.G. Lorenz and S.P. Boyd, “Robust minimum variance beamforming,” IEEE Trans. Signal Processing., vol. 53, no.5, pp. 1648–1696, May 2005.[5] S.A. Vorobyov, A.B. Gershman, and Z.Q. Luo, “Robust adaptive beamforming using worst-case performanceoptimization: a solution to the signal mismatch problem,” IEEE Trans. Signal Processing., vol. 51, no. 2, pp. 313–324,Feb. 2003.[6] R. J. Mailloux, “Covariance matrix augmentation to produce adaptive array pattern troughs,” Electron. Lett., vol. 31,1995.[7] H. Song, W. A. Kuperman, P. Gerstoft, and J. S. Kim, “Null broadening with snapshot-deficient covariance matricesin passive sonar,” IEEE Oceanic Eng., vol. 28, no. 2, pp. 250–261, 2003.[8] M. Lobo, “Applications of second-order cone programming,” Linear Algebra Applicat., pp. 193–228, Nov 1998.[9] L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Review., , no. 38(1), pp. 49–95, Mar 1996.[10] Z. Xu, Y. Zakharov, and G. White, “Vertical antenna array and spectral reuse for ring-shaped cellular coverage fromhigh altitude platform,” Loughborough Antennas and Propagation Conference (LAPC2006), pp. 209–212, April 2006.[11] Z. Luo and W. Yu, “An introduction to convex optimization for communications and signal processing,” IEEE SelectedArea Comms., vol. 24, no. 8, pp. 1426–1438, Aug 2006.4Authors’ affiliations:Zhengyi Xu and Yuriy Zakharov are with the Communication Research Group, Department of Electronics,University of York, York YO10 5DD, UK, (email: zx500, yz1@ohm.york.ac.uk).Figure captionsFig.1 The use of null broadening approach for HAPs communications.Fig.2 Coverage performance comparison of MVDR beamformer, null broadening method [6] and the improvednull broadening method in the HAPs communications scenario: H = 20 km; r = 32.7 km; 0.06 km maximumuser position error; M = 60.dotted: MVDR beamformer;dashed: null broadening approach proposed in [6];dash-dotted: improved null broadening approach using constrained optimisation, uniform distribution for thenumber of additive sources;solid: improved null broadening method using constrained optimisation, non-uniform distribution for the numberof additive sources.5User 1 User m User MθmVertical antenna arrayHXYrHXArtificialequal-strengthsourcesrFig. 1.−20 −15 −10 −5 0 5 10 150.90.920.940.960.981SIR (dB)CoverageFig. 2.

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Modified null broadening adaptive beamforming: constrained optimisation approach

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