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This is an author -deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID: 4505

To link to this article: DOI:10.1049/iet-com.2009.0721

URL: http://dx.doi.org/10.1049/iet-com.2009.0721

To cite this version: MONTESINOS Julien, BESSON Olivier, LARUE DE TOURNEMINE Cécile. Adaptive beamforming for large arrays in satellite communications systems with dispersed coverage. IET Communications, 2011, vol. 5, n° 3, pp. 350-361. ISSN 1751-8628

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Adaptive beamforming for large arraysin satellite communications systemswith dispersed coverageJ. Montesinos1,2 O. Besson1 C. Larue de Tournemine2

1University of Toulouse, ISAE/TeSA, 10 Avenue Edouard Belin, 31055 Toulouse, France2Thales Alenia Space, 26 Avenue J.-F. Champollion, FranceE-mail: olivier.besson@isae.fr

Abstract: Conventional multibeam satellite communications systems ensure coverage of wide areas throughmultiple fixed beams where all users inside a beam share the same bandwidth. The authors consider a newand more flexible system where each user is assigned his own beam, and the users can be verygeographically dispersed. This is achieved through the use of a large direct radiating array coupled withadaptive beamforming so as to reject interferences and to provide a maximal gain to the user of interest.New fast-converging adaptive beamforming algorithms are presented, which allow one to obtain good signalto interference and noise ratio with a number of snapshots much lower than the number of antennas in thearray. These beamformers are evaluated on reference scenarios.

1 IntroductionSatellite telecommunications systems specified today requirehigh-gain satellite antennas to supply sufficient linkbudgets towards small user terminals. In this context,satellite on-board multiple beam antennas (MBA) havebeen successfully employed either for personalcommunications, military communications or Internetapplications [1]. They allow one to insure high gain bymeans of multiple adjacent patterns covering an extendedzone. They also allow one to introduce some frequencyreuse on non-adjacent beams, thereby increasing theavailable capacity for a given bandwidth.

Standard MBA provide a fixed coverage with adjacentbeams crossing typically 3–4 dB below maximum gainlevel. They are associated to a fixed bandwidth to beamallocation, with typical 1:3 or 1:4 frequency reuse scheme,that is to say, the frequency is reused on one beam overthree or four. On classical payloads, beamforming isperformed in radio frequency domain, prior tochannelisation function (usually on intermediate frequencyor in digitised baseband). By introducing digitalbeamforming networks associated to transparent or

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regenerative processor, the interest of such a multibeamantenna can be considerably improved. As a matter of fact,digital beamforming allows one to introduce fully flexiblebeamforming. Moreover, by crossing on boardbeamforming functions with channelisation, it becomespossible to introduce some flexibility in terms of bandwidthto beam allocation. Finally, the coverage reconfigurabilityoffered by digital beamforming associated to flexiblechannelisation makes it possible to set capacity whererequired when required, thus adapting the providedresources to effective traffic requirements [2, 3].

When considering a moderate bandwidth, it is possible togeneralise this flexible approach, and go towards a ‘one beamper user’ concept within a reasonable on board processingcomplexity. In such a system, a suited capacity (i.e.bandwidth) can be attributed to each user, or to a smallgroup of users. A specific beam, pointed towards the userof interest (UOI), provides the maximum gain madeavailable by the antenna aperture, thus improves the linkbudget by up to 3 or 4 dB with regard to traditionalmultibeam coverage where edge of coverage gain must beconsidered. Furthermore, if considering several usersassociated to the same radio resource, it is possible to

adaptively form the beams associated to each of these users, inorder to reject specifically co-users interferences. Therejection of interfering signals results in a significantlyimproved signal-to-interference and noise ratio (SINR) foreach user, as compared to standard solution where antennapattern sidelobes are minimised over an extended zone.

Thanks to combined gain improvement, and isolationenhancement, the obtained SINR on each link can then bedramatically increased with regard to the performanceobtained with a standard coverage provided by anequivalent antenna aperture. The improved SINR can leadto increase effective payload throughput by using moreefficient modulation and coding scheme on user to satellitelink. Alternatively, keeping given SINR specification, thesystem capacity can be improved by increasing cumulatedprocessed bandwidth. As a matter of fact, the adaptedisolation between users makes it possible to reuse frequencyon closer users than what was made possible with standardsystems. However, to provide significant improvement byproperly pointing and shaping antenna patterns, it isnecessary to introduce some adaptive antenna beamforming.

It has been shown in previous works [4, 5] that digitalbeamforming antenna can provide flexible beams toindividual users on demand. The combination of bothadaptive processing and a suited resource allocation schemeensure the best use of available capacity. We can then referto space division multiple access (SDMA). Frequencychannels are reused as much as made possible by antennaspatial filtering capacity. The allocation process highlyimpacts final capacity, but the optimisation of suchalgorithm is outside the scope of the present paper.However in [4], the restricted area covered by the antenna(about 1000 km wide) leads one to use a focal arrayantenna including few radiating elements, thus few arraycontrols. The associated adaptive array processing thenstays within a reasonable complexity, and no specific effortwas set on considering adaptive algorithm optimisation interms of computational cost.

In this paper, we study the reception antenna of ageostationary satellite communication system, operating inKa-band, which must provide high-gain beams towardsusers located anywhere on the visible earth surface. Wepresent in the first part of the paper the constraints that areassociated with such a coverage, and show that the antennadesign leads to a radiating panel including a large numberof controls. The classical adaptive beamforming methodssuch as those considered in [4–6] are then not suited anymore, as leading to excessive complexity. The second partof the paper is then devoted to the description of specificadaptive beamforming algorithms that allow one to avoidarray elements covariance matrix inversion and work in alow-dimensional subspace. This matrix can thus beestimated with a much reduced number of samples,without affecting adaptive processing convergence leadingto a dramatic reduction of the algorithm associated

complexity and number of acquisitions. Finally, the lastpart of the paper presents simulation results illustrating theperformances expected in a reference scenario.

2 Antenna design and constraintsrelated to adaptive beamforming2.1 Coverage constraints and associatedantenna design

We here consider a ‘one beam per user’ coverage. Beams aredigitally formed by applying proper weighting, after priorchannelisation of the signal sampled after each radiatingelement. The same radio resource can be used in severalbeams as long as spatial filtering allows separatingassociated signals. Beams are pointed individually towardseach user, thus the gain provided in the direction of theuser is maximised with regard to antenna size. Users can beplaced anywhere on visible side of the Earth as seen from ageostationary satellite. Locally contiguous coverage can beneeded implying isolation capacity requirement. Isolatedusers can also be covered. Fig. 1 gives an example of thecoverage that can be needed and illustrates time-frequencyresources reuse for close users. The square represents theUOI. The UOI is assumed to share the same resource asthe co-users represented here by full circles, whereas thediamonds are the symbols of users of the system sharing adifferent resource. Finally, the cross symbolises a jammingstation for the UOI.

The system has to work, even if intentional orunintentional interferences or jammers are active in thecoverage. If users are sharing a resource (the samefrequency channel for instance), they are interfering witheach other. Intentional jamming stations may also bepresent. We consider here that jammers are acting like

Figure 1 Example of required coverage

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co-users interferences, except that their power is much higher(typ. 10 dB).

The simplest way to implement an MBA is to provide eachbeam with a single radiating element, located at the focalplane of an optical device (single or multiple reflectors).However, such an implementation cannot provide acontinuous multiple beam coverage with high efficiency. Asa matter of fact, the inter-feed spacing is driven by beamspacing requirement. The allowable feed size is then small,leading to non-optimum illumination of the reflector anddegraded performances for edge of coverage gain. Toobtain sufficient performances, one possible solution is touse three or four ‘one feed per beam’ antennas, having theirbeam lattice interleaved. The feed size is then allowed oneto increase, without compromising beam spacing. Three orfour reflector antennas must be used to provide thecomplete coverage. In addition to its bulkiness, the mainissue of such a solution lies in its lack of flexibility, since nobeam reconfigurability is possible.

Another solution to provide multiple beams, using onlyone reflector, is to use overlapping sub-arrays to increasethe effective aperture of each equivalent feed. Thefocalising device associated with the array can be a single ora double reflector. The array fed reflector (AFR) beams aregenerated by combining signals coming from clusters oftypically 7–19 active feeds per beam depending on gainand inter-beam isolation specifications. As compared to a‘one feed per beam’ solution using identical focalisingdevice, the C/I ratio (power received from the nominalbeam, divided by that coming from the ones whichinterfere in the same sub-band) can be increased by 3–5 dB. Each feed contributes to several beams, typically 1 to7, for central feeds. The total number of feeds of theantenna is mainly related to the global extension of thecoverage. Such antennas are privileged for moderatelyextended coverage (typically Europe or the contiguousUSA) or when lower-frequency reuse scheme are required.For instance, Inmarsat 4 satellite uses a seven coloursfrequency reuse scheme [7]. It is, however, difficult toprovide both high gain and rejection capacity on a widelyextended coverage as required in our application with suchan antenna.

Finally, the best antenna solution to provide multiple high-gain beams spread all over Earth coverage withoutcompromising spatial filtering capacity for close users is adirect radiating antenna (DRA). In such an antenna, eachradiating element contributes to every beam. In addition tobeing well suited to the coverage requirements, such aphased array antenna is also attractive, as it is likely toreduce the dynamics of beamforming network signal, whichis critical for DBFN implementation. Furthermore thegraceful degradation principle avoids the ‘2 per 1’redundancy, which is very expensive for antenna receivingdevices. For an example of system using a DRA, there isthe WINDS satellite which has two 128 elements DRA,

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aiming at covering the Asia-Pacific region with fourreconfigurable beams (two for transmission and two forreception) using analogue BFN [8].

Designing a DRA radiating panel mainly consists inconsidering the two following constraints [9]:

† Antenna diameter is mainly determined by minimumdirectivity level, which must be insured for all users of thecoverage, in order to comply with link budget requirements.

† Grid lattice is constrained by grating lobe rejection outsidea given domain (typically outside the Earth, for ageostationary satellite antenna considered here), to avoidlosses and possible interferences related to grating lobes.The maximum size of radiating elements is then related tocoverage global extension, defining maximum beampointing angle.

For our particular case, both high gain and extendedcoverage are required. DRA sizing for geostationary satelliteand extended coverage lead typically to a number ofradiating elements going from one to a few hundreds.Some solutions are investigated to reduce the number ofcontrols used in the beamforming process either bythinning the array, or by introducing a first stage ofanalogue beamforming. The irregular subarray divisionallows one to break lattice periodicity, thus avoids gratinglobes keeping a reduced number of digital controls [10].However, for our case, the remaining number of controlstays too important to use classical beamformingtechniques. Indeed, complexity of usual adaptive algorithmsissues become a bottleneck for SDMA-likeimplementation. We here briefly review conventional full-dimension adaptive beamforming techniques, prior topresenting the solutions that could allow for fastconvergence at a low computational cost.

2.2 Adaptive beamforming

Adaptive beamforming consists in spatially filtering thereceived array data with a view to recover a signal ofinterest (SOI) while eliminating the interferences thatcould possibly be present. This area has been extensivelystudied in recent years and numerous solutions have beenproposed, see for example [11] for a thorough review. Theoptimal beamformer amounts to solving the followingminimisation problem

minw

wH Rw, subject to wH v = 1 (1)

where v is the spatial signature of the SOI and R is thecovariance matrix of the array output. The constraint in (1)guarantees that the SOI will be undistorted afterbeamforming, and because of the minimisation of theoutput power, the beamformer will tend to place nullstowards the interferences in order to reduce theircontribution. When the array measurements contain the

interferences and noise only, R = Ri+n where Ri+n denotesthe interferences plus noise covariance matrix, and thebeamformer is usually referred to as the minimum variancedistortionless response (MVDR) beamformer. When theSOI is also present in the measurements, R = Ri+n+PvvH and the terminology minimum power distortionlessresponse (MPDR) beamformer is used [11]. Whatever thevalue of R, the solution to (1) is given by [11]

wopt =R−1v

vH R−1v(2)

When the covariance matrix is known, wopt is the samewhether R = Ri+n or R = Ri+nPvvH and the optimalbeamformer enables one to achieve the optimal SINR

SINRopt = PvH R−1i+nv (3)

where P stands for the SOI power. At this point, we wouldlike to emphasise a significant difference between the twobeamformers. The MVDR beamformer requires that data,free of the SOI, be available, which may be a majorconstraint in communications systems. Hence, specificaccess schemes need to be planned in order to obtain SOI-free measurements and to implement the MVDRbeamformer. In contrast, this requirement does not holdfor the MPDR beamformer.

In practical situations, R is not known and is estimatedfrom N available snapshots X = [x(1) x(2) · · · x(N )] as

R = N −1XX H = N −1∑N

t=1

x(t)xH (t) (4)

with the latter referred to as the sample covariance matrix(SCM). When the number of snapshots N is larger thanthe number of array elements m, that is N . m, the SCMis invertible with probability 1 and one usually substitutesR for the SCM in (2) to yield

wsmi =R−1

v

vH R−1

v(5)

where the subscript ‘smi’ stands for sample matrix inversion.The computational complexity associated with (5) dependson the number of snapshots N and the number of elementsm since one needs to invert a m × m matrix. Thereforeconsiderable effort has been devoted to design beamformersthat can provide good SINR with a limited number ofsnapshots and at a low computational cost. A commonlyadmitted criterion is the so-called convergence measure ofeffectiveness (MOE), that is the number of snapshotsrequired for a beamformer to achieve–on average–theoptimal SINR (3) within 3 dB. The MOE directly impactsthe computational load as well as the rate at which thebeamformer can be updated, in case of non-stationaryenvironments for instance. The MOE of the MVDR is

significantly lower than that of the MPDR when using theSCM. Since the landmark paper by Reed Mallett andBrennan [12], it is known that the MOE of the MVDR isapproximately 2m 2 3 snapshots. In contrast, for the sameperformance, 2m − 3 + (m − 1)SINRopt snapshots arenecessary [11, 13, 14], which is prohibitive in mostapplications. We would also like to stress that, in the caseN , m we consider in this paper, the beamformer in (5) iseven not implementable as R is not invertible, actually itsrank is N. The methods we are looking for should, onceagain, avoid the estimated covariance matrix inversion whilekeeping a good MOE.

3 Low-complexity adaptivebeamformingIn order to improve the beamformer MOE, several strategieshave been proposed in the literature. Our aim here is not toprovide an exhaustive review of them, rather to sample themost widely used and most effective techniques. A premierchoice is diagonal loading [15] which consists in adding a

scaled identity matrix to R before inversion, yielding

wdl =(R + s2

dlI )−1v

vH (R + s2dlI )−1v

(6)

The above beamformer can be implemented even for N ≤ m,its MOE is commensurate with twice the number ofinterferences J, and it is thus very effective. The drawbackof such a method lies in the fact that it is not always simpleto fix the diagonal loading level s2

dl although some ways tofix it, either adaptively or automatically have been proposed,see for example [16–18]. Additionally, the inversion of am × m covariance matrix is still required, which iscomputationally intensive for large m.

A second and important class of fast-convergingbeamformers consists of the so-called reduced-rank (RR)beamformers [19, 20] whose principle is to operate in asubspace of the measurements, hence reducing the size ofthe observations, and subsequently the sample supportrequired. They exploit the fact that the interferences occupya subspace that can be estimated by eigenvaluedecomposition (EVD) of the covariance matrix. In thiscategory, the more widely known methods are the principalcomponents (PC) method [21, 22] and the cross-spectralmetric method (CSM) [23]. Although their MOE is of theorder of 2J, and they do not require directly a matrixinversion, these methods are still computationally intensiveas they require the EVD of R. Given the scenario weconsider in this paper–large number of elements m andsmall number of snapshots N ≪ m–we are looking formethods whose MOE is of the order 2J without requiringthe inversion of a large matrix or its EVD. Hence, themethods listed above are not entirely satisfactory and wenow focus on three recently proposed approaches which canmeet our requirements.

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3.1 Conjugate gradient (CG) beamformer

The CG is a well-known technique to solve linear systems ofequations such as Rw = v [24]. It enables to obtain thesolution in m steps without any matrix inversion and isthus computationally attractive. For the sake ofconvenience, a possible implementation of the CGbeamformer is displayed in Fig. 2. As for all algorithms,the input matrix R should be considered as a “generic”input matrix, keeping in mind that it can correspond to thecovariance matrix of interferences plus noise (MVDRscenario) or the covariance matrix of the SOI, theinterferences and the noise (MPDR scenario). Moreover, inpractice all algorithms considered will use R in place of Ras input. In fact, the CG algorithm was not really used toobtain the MVDR solution until its connection with themulti-stage Wiener filter (MWF) [25] was discovered [26].The MWF is an RR filter in which the RR subspace isdetermined by maximising cross-correlations between thedesired signal and the observed data. Therefore in contrastto PC or CSM, it does not require EVD. Yet, itsperformance was shown to be generally better than that ofPC and CSM with a possibly smaller MOE. The MWFwas found to operate in the Krylov subspaceK(v, R, n) = R{v, Rv, . . . , Rn−1v} where R{·} stands forthe range space, hence its connection with the CG sincethe latter, at iteration n, is known to operate in this subspace.

In Fig. 2, the iterations should be run till r = m to solveRw = v. However, as explained now, it may berecommended to stop the iterations earlier. If thecovariance matrix contains J interferences with highinterference to noise ratio (INR), its EVD can be writtenas R = U iLiU

Hi + s 2U nU H

n with Li = diag(l1, . . . , lJ )and lk ≫ s 2. It follows that Rn ≃ U iL

ni U H

i and hence, atiteration n = J + 1, the Krylov subspace will capture theinterference subspace: the filter will then be most effective

Figure 2 CG beamformer

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in rejecting the interferences. In other words, Fig. 2 shouldnot be necessarily run until n = min(m, N ) but should bestopped at n = J + 1 where the SINR will be maximum.In fact, when using R instead of R, it has been observedthat the SINR generally decreases when n is increasedabove J + 1 and tends to that of wsmi in (5), provided thatR is invertible. The problem is that in practical situationsone does not know exactly the number of interferingsignals. On one hand, one should not stop beforen = J + 1 as all interferences will not be suppressed. Onthe other hand, a too large r will result in poorer SINR,especially if a rank deficient R is used in Fig. 2. In order toremedy this problem, we propose to include diagonalloading directly in the CG beamformer, and refer to as theCG–DL beamformer. This simply amounts to replacingR–actually R–by R + s 2

dlI in line 1 of Fig. 2. Althoughthe modification is minor, it yields considerableimprovement. Indeed, one does not need to know J, onlyan upper bound for this value. In contrast to the originalCG, choosing r . J + 1 does not result in a decrease of theSINR in a MPDR-type scenario. In fact, we observed, byvarying r, that the SINR is generally optimum at r ¼ J + 1but then tends to be rather constant. Moreover, when oneuses the SCM and the latter is rank deficient (because ofN , m), one does not encounter any numerical problem asthe inverse of R + s 2

dlI exists whatever N. This fact will beillustrated in the next section.

3.2 Auxiliary vector (AV) beamformer

The AV beamformer was introduced in [27] as a simplesolution to compute the MVDR solution without resortingto any matrix inversion. The implementation of the AVbeamformer is described by Fig. 3 where P⊥

v in line 3 ofFig. 3 stands for the orthogonal projector onto thesubspace orthogonal to v. The basic idea behind thisalgorithm is to build, at each iteration step n, an AV gn

which is orthogonal to v. This AV is able to capture the

Figure 3 AV beamformer

most (in sense of maximum magnitude cross-correlation) ofthe interferences present at the previous iteration filteroutput. In [27], it was shown that, with R a positivedefinite covariance matrix, the AV weight vector convergesto wopt in (2) as r goes to infinity. In fact, the AV weightvector comes very close to its limit for a relatively smallnumber of iterations, and a procedure to select the value ofr in Fig. 3 was presented in [28]. In finite samples, that iswhen using R instead of R, the AV beamformer wasshown to have performances commensurate with those ofthe CG beamformer [29] and is thus a method to considerfor our problem. However, the proof of convergence in[27] is based on the fact that the input matrix R is positivedefinite. When R is used this implies that N . m. In thecase of most interest to us, N ≤ m and hence R has rankN. In [30], we proved that if the input matrix R of Fig. 3is rank deficient, with EVD R = ULU H whereU = [u1 u2 · · · up] [ C

m×p is a unitary matrix, then thelimit of the weight vector wn of Fig. 3 is

limn�1

wn = U⊥U H⊥ v

vH U⊥U H⊥ v

W wAV−1 (7)

where U⊥ is an orthonormal basis for R{U }⊥.

First observe that this limit belongs to a low-rank subspace,and hence the AV beamformer asymptotically (in n) belongs tothe class of RR beamformers. Hence it should inherit theirproperties in terms of low MOE. Moreover, it enables oneto achieve effective interference cancellation. As explained in[30], in a MVDR scenario, the columns of U⊥ will mostlylie in the null space of the interferences, that is, U H

⊥Ai ≃ 0,where the columns of Ai [ Cm×J are the interferencessteering vectors. In an MPDR-type scenario, the SOI is alsopresent. In the absence of noise, the matrix X has rank J + 1for N ≥ J + 1, and its range space is R{X } = R{[ v Ai ]}.It follows that U⊥ will be orthogonal to both v and Ai,which is desirable for Ai but is to be avoided for v. In thepresence of noise, the rank of R is again N and most of theenergy of v and Ai will be confined in U. It follows that, ina MPDR scenario, the asymptotic vector wAV − 1 may notbe advocated. However, in order to balance the previouscomment, one should mention that usually the iterations arenot run until convergence, mainly because convergence maybe very slow and thus the corresponding computational costbecomes prohibitive. In fact, the transient behaviour ofFig. 3 is more interesting. Indeed, in order to keep thecomputational cost low, r is chosen relatively small.Moreover, it can be observed, see the next section, that thebeamformer obtained after a few iterations has a betterperformance than the asymptotic beamformer, especially in aMPDR scenario.

3.3 Random beamspace processing

Beamspace processing is a usual way to operate in a low-dimensional subspace [11]. It consists in applying first abeamspace transformation on the data, that is,

x � x = FH x where F is a m × r matrix and then incomputing the adaptive weight in the transformed space,that is, in solving

minw

wH Rw, subject to wH v = 1 (8)

where v = FH v and R = FH RF. The solution to the aboveproblem is obviously

w = R−1

v

vH R−1

v(9)

and the ‘equivalent’ beamformer is Fw. Note that the onlymatrix to be inverted is R whose dimension is r × r, and r istypically small. Usually, the columns of the transformationmatrix F are steering vectors, and the corresponding beamsshould let the SOI pass undistorted and capture theinterferences. When these conditions are met, such atechnique is very efficient from a computational point ofview and effective in terms of MOE. However, thedirections of the interfering signals are seldom known andfixing F a priori is a delicate issue. Of course, thebeamspace transformation can be chosen adaptively, forexample, from the largest eigenvectors of the covariancematrix, but, as said previously, we wish to avoid such EVD.An interesting alternative, suggested in [31], is to selectrandom unitary matrices F and to average the correspondingbeamformers, so as to benefit from a diversity effect.

In [31], the technique was used for direction findingproblems and the idea is to average, over a certain number ofrandom F, the output power after beamspace processing.This method can work with singular covariance matrices –which is our case herein – and avoids diagonal loading andthe problem of choosing the loading level. It was shown toperform well with a number of snapshots smaller than thenumber of array elements. In this paper, we adapt this ideato the problem of designing a beamformer. Since thetransformation is constrained to let the SOI pass throughundistorted, the matrix F can be chosen as

F = v‖v‖ f2 · · · fr

[ ](10)

where f2, . . . , fr are mutually orthonormal beams and areorthogonal to v. Let Q be a unitary matrix that rotates vinto the first component, that is,QH v = ‖v‖[ 1 0 · · · 0 ]T = ‖v‖e1. Then, F can bewritten as

F = [ Qe1 f2 · · · fr ]

= Q[ e1 QHf2 · · · QHfr ]

= Q1 0T

r−1

0m−1 C

[ ](11)

where C is an m − 1 × r − 1 unitary matrix, which isunconstrained, and chosen isotropically random. For

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instance, C can be generated from a m − 1 × r − 1 matrix Wwith independent and identically distributed Gaussian entriesas

C = W (W H W )−1/2 (12)

Once C is drawn, the beamforming transformation F isapplied to the data, and the corresponding beamspaceweight vector w is computed, as well as the equivalentweight vector Fw. The latter is averaged over multiplerealisations with different matrices C to yield the finalbeamformer, which we refer to as the unitary beamspace(UB) beamformer. The steps involved can thus besummarised by Fig. 4.

The three presented algorithms are potential solutions foran adaptive algorithm implementation on an antenna with ahigh number of elements. We are now going to assess theirperformances in a particular scenario.

4 Numerical results4.1 Antenna design and simulationscenarios

In this section, we illustrate the performances of thebeamformers described above: we compare the CGbeamformer, with possibly diagonal loading, the AVbeamformer, the UB beamformer and the asymptotic AVbeamformer, based on their output SINR, which is defined as

SINR = P|wH v|2

wH Ri+nw(13)

The latter is averaged over 1000 independent Monte-Carlotrials where, for each trial, a different data matrix X isdrawn randomly and a weight vector w is calculated as wellas the corresponding SINR in (13). In addition, we alsodisplay the worst-case (over 1000 independent Monte-Carlo trials) SINR. Each SINR is plotted in decibels. Wewill particularly investigate:

Figure 4 UB beamformer

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1. the influence of r, that is the number of iterations at whichFig. 2 –the CG beamformer– and Fig. 3 –the AVbeamformer– are stopped. It should be recalled that r alsocorresponds to the dimension of the subspace where theCG and UB weight vectors lie.

2. the influence of the number of snapshots N.

We distinguish two scenarios, a MVDR-type scenariowhere the array data contain only the interferences and thenoise and a MPDR-type scenario where, in addition tointerferences and noise, the SOI is also present. Note that aMVDR scenario can be a serious constraint for somecommunications systems, since the communication needs tobe interrupted to obtain SOI-free data matrices. The reasonsfor this distinction are that the output SINRs will besignificantly different but corresponds to different operatingmodes. Analysis show that the relative performance betweenthe beamformers might be different in both cases.

To this aim, we consider a 1.2 m wide circular DRA withm ¼ 401 non-isotropic elements, each of them of size5.7l× 4.93l, that are distributed on a triangular grid at30 GHz (reception mode), see Fig. 5. The maximumpossible gain provided by the array is 50.8 dB and the half-power beam-width is 0.438. We deliberately chose an arrayof several hundreds of radiating elements to illustrate a casewhere the computational is highly critical. The array is alsochosen to provide a high gain all over the visible earth.

We consider a scenario where the UOI is located in the(u, v)–space at u0 = v0 = 0, where u = sin(u) cos(f),v = sin(u) sin(f) and u, f stand, respectively, for theelevation and azimuth angles. We assume that J = 3interferences are present: two of them are users of thesystem sharing the same resources as the UOI and one is ajamming station. They are located at 0.28 from the UOIand their INR are respectively 0, 0 and 10 dB. The output

Figure 5 Layout of the array

SINR obtained with the conventional –non-adaptive–beamformer is 28.8 dB, which is considered as a value ofreference we would like to enhance. Towards that end, itappears a need to consider adaptive beamformers. Besides,as concerns the UB beamformer, L = 50 random matricesF are used to compute the weight vector.

4.2 MVDR scenario

In Figs. 6 and 7 we plot the average and worst-case outputSINR against r. The number of snapshots is fixed toN = 20. The CG beamformer is possibly combined withdiagonal loading, and the loading levels are either 10 or20 dB above the white noise level.

Let us consider first the average SINR in Fig. 6. Frominspection of this figure, it is clear that the CGbeamformer achieves the best performance with the lowestnumber of iterations. As we already discussed, themaximum SINR is obtained for r = J + 1 = 4 iterationsand comes very close to the optimum SINR which is equalto 37.5 dB. Moreover, once the iteration index exceedsJ + 1, the SINR remains constant. This is a very appealingfeature as one does not need to know exactly the number ofinterferences but only an upper-bound for this value inorder to decide when the iterations should be stopped.Diagonal loading is not really helpful in this scenario andcan even degrade the performance in case of a too largeloading level. Regarding the AV beamformer, the followingobservations can be made. The SINR of the asymptotic AVbeamformer wAV−1 in (7) is very close to that obtainedwith the CG beamformer. This asymptotic regime isreached for r ¼ 60 which is not very high but still largerthan the number of iterations required by the CGbeamformer. Therefore if the two beamformers have thesame steady SINR, the CG converges faster. The UBbeamformer has a lower maximum SINR than the two other

Figure 6 MVDR scenario

Average SINR against iteration index

beamformers (note that r is kept lower than N, the rank ofR, as FH RF must be invertible). Moreover, this maximumis achieved for r ≃ 12, which is larger than the actual size ofthe interference subspace. The reason for this is that thebeamspace matrix F is drawn randomly and hence a largersubspace dimension is required to include, with highprobability, the true interference subspace. If r is too small Fmay not capture the interference subspace, which results inpoorer SINR. In order to improve the UB beamformer, thematrix F should not be drawn completely at random butrandomly within a spatial sector where the interferences areknown to lie. This method, which would require some apriori information about the interferences localisation, maylead to some improvement. Since this is beyond the scope ofthe present paper, we do not elaborate further.

Regarding the worst-case SINR in Fig. 7, we can make thefollowing comments. For all algorithms, except the AVbeamformer, the worst-case SINR curves have the samecharacteristics as their average SINR counterparts, with asimple shift of about 2–3 dB, which indicates that the SINRare rather concentrated around their mean. In contrast, theworst-case SINR for the AV beamformer can be quite farfrom the average SINR, which shows a greater variability.

We now study the influence of the number of snapshots.Each algorithm is used with a ‘close to optimal’ value of rand the latter varies between the methods. As explainedabove, if J was known, one should use the CG withr = J + 1. Here we simply assume that an upper-bound ofthe number of interferences Jsup = 5 is known and the CGalgorithm is stopped at r = Jsup + 1. The AV beamformeris stopped at r = 40, that is, before convergence in order tohave a reasonable computational complexity. Finally, theUB beamformer is used with r = 10, so as to slightlyoverestimate the actual interference subspace dimension.Results are shown in Figs. 8 and 9, which confirms the

Figure 7 MVDR scenario

Worst-case SINR against iteration index

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hierarchy established earlier, viz the CG beamformer issuperior to the AV beamformer and the UB beamformer.Additionally, it can be observed that the MOE of the threebeamformers is very good. More precisely, the CG and theAV beamformers attain the optimal SINR within 3 dBwith only ten snapshots while slightly more snapshots arerequired for the UB beamformer. Finally, that the CGbeamformer enjoys the smallest difference between averageand worst-case SINR, followed by the UB beamformer andthe AV beamformer.

4.3 MPDR scenario

We now consider that the SOI is present in the arraymeasurements, in addition to the interferences and the

Figure 9 MVDR scenario

Worst-case SINR against number of snapshots

Figure 8 MVDR scenario

Average SINR against number of snapshots

8

noise. The SNR is set to 0 dB. Similarly to the previouscase, we successively investigate the influence of r and N.In Figs. 10 and 11, we display the output SINR as afunction of r, for N ¼ 20.

Several observations can be made, that contrast with thosemade for the MVDR scenario.

† The CG still achieves its maximum SINR at r = J + 1but, beyond this value, one can observe a significant dropof the SINR. Hence, in contrast to the MVDR scenario,the iterations should be stopped precisely at r = J + 1.Otherwise, a performance loss is incurred. In practice, thisrequires an accurate knowledge of the number of

Figure 10 MPDR scenario

Average SINR against iteration index

Figure 11 MPDR scenario

Worst-case SINR against iteration index

interferences, which is seldom available. Diagonal loadingcan be very helpful to alleviate this phenomenon, providedthat the loading level is chosen properly. Indeed, forsDL

2 ¼ 10 dB, one can still observe a SINR drop afterr = J + 1 but this drop is limited and the steady-stateSINR is better. With sDL

2 ¼ 20 dB, the SINR remainsconstant. Therefore this simple modification we introducedin the CG beamformer tends to be effective. However, thechoice of the loading level remains a delicate issue in practice.

† The AV beamformer has a performance equivalent to thebest CG–DL beamformer and reaches a close to maximumpossible SINR very quickly, viz r = 30 in this case.Therefore it is a very interesting method since no parametersuch as sDL

2 needs to be chosen. Moreover, it is not requiredto choose r with a high degree of accuracy –the SINR isconstant in Fig. 10 from r = 30 to r = 100– which is veryappealing. We would like to emphasise that at r = 100 theAV beamformer is still in a ‘transient’ behaviour as theSINR obtained with wAV−1 is equivalent to that of the CGbeamformer. It means that if r was increased further, theSINR of the AV beamformer would decrease and wouldconverge to the SINR of wAV−1 as r � 1. Consequently,there is a double advantage not to increase r too much, interms of SINR and computational load.

† The UB beamformer exhibits a similar behaviour as in theMVDR scenario, except that the maximum SINR is achievedfor r = 6. However, when r increases too much the SINR drops.

† The difference between the average SINR and the worst-case SINR is more pronounced than in the MVDR scenario.Indeed while the average SINR is around 8–9 dB, the worst-case SINR are around 0 dB. Again, the AV exhibits thelargest difference.

Figure 12 MPDR scenario

Average SINR against number of snapshots

Figs. 12 and 13 display the output SINR against thenumber of snapshots. One can observe that the AVbeamformer provides the highest SINR and outperformsthe CG beamformer. The performance of the latter doesnot really improve when N is increased, mainly because r isnot matched to J + 1. Introducing diagonal loading leadsto some improvement of the SINR but, again, s2

DL shouldbe chosen properly. The UB beamformer performs verywell. Even if r = 10 is not an optimal choice for the UBbeamformer, its SINR is better than that of the CG forN ≥ 20 and is close to the best SINR.

4.4 Summary about the simulationsresults

To conclude this section, we can make the following remarks:

† In a MVDR scenario, the CG appears to be the method ofchoice. It converges very quickly in r and N, enables one toachieve a close to optimal SINR and does not need to knowthe exact number of interferences (only an upper-bound).

† In a MPDR scenario, the hierarchy is not as clear. The AVbeamformer has a very good performance without requiringany user-parameter to tune. Its transient behaviour enablesto obtain the highest SINR and it is rather robust to thechoice of r. Diagonal loading is very helpful for the CGbeamformer but the issue of selecting s 2

DL is delicate.Without diagonal loading, the CG beamformer does notperform as well as the AV beamformer.

† The UB beamformer is potentially very interesting but thechoice of the ‘random’ matrices F needs to be investigatedfurther in order to obtain some improvement. In fact, thematrices F are drawn randomly and therefore they arelikely to focus on spatial sectors which do not contain all

Figure 13 MPDR scenario

Worst-case SINR against number of snapshots

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interferences. It then results that a larger number L ofmatrices should be averaged in order for all interferences tobe captured. In order to improve the UB beamformer (andreduce L), it would be interesting, if available, to use somea priori information about the location of the interferers soas to draw matrices F not isotropically but selectively inorder for the beams drawn to contain the interferers withhigher probability.

5 ConclusionsIn this paper, we showed the feasibility of a satellitecommunications system that offers a global coverage fordispersed users and that allows one to optimise SINR andradio resource utilisation efficiency. This was achieved withthe use of a large antenna array coupled with adaptivebeamforming. We proposed three iterative, computationallyefficient, and fast-converging beamformers that can providegood SINR with a very limited number of snapshots. In aMVDR scenario, we showed that the CG beamformerprovides the best performance, at least when a tight upper-bound of the number of interferences is known. In aMPDR scenario, the AV beamformer is advocated as it isrobust to the choice of the number of iterations.

6 AcknowledgmentThis work was supported by the Delegation Generale pourl’Armement (DGA) and by Thales Alenia Space.

7 References

[1] JACOMB-HOOD A., LIER E.: ‘Multibeam active phased arraysfor communications satellites’, IEEE Microw. Mag., 2000, 1,(4), pp. 40–47

[2] HAARDT C., ROUSSET D.: ‘Flexible processor solutions forsatellite communications’. Proc. ESA Workshop onAdvanced Flexible Telecom Payload, ESA/ESTEC,Noordwijk, The Netherlands, 18–20 November 2008

[3] CRAIG T., BISHOP A., LEONG C., ET AL.: ‘Flexible processorsolutions for satellite communications’. Proc. ESAWorkshop on Advanced Flexible Telecom Payload, ESA/ESTEC, Noordwijk, The Netherlands, 18–20 November 2008

[4] SOW G.: ‘Methodes d’Acces multiple a repartitionspatiale pour communications par satellite’. PhD thesis,Ecole Nationale Superieure des Telecommunications,2006

[5] SOW G., BESSON O., BOUCHERET M.-L., GUIRAUD C.:‘Beamforming for satellite communications in emergencysituations’, Eur. Trans. Telecommun., 2008, 19, (2),pp. 161–171

0

[6] GUIRAUD C., ROUSSET D., ALBERT I.: ‘Digital beamformingfor multimedia flexible antennas’. Proc. 17th Int. Conf.Applied Electromagnetics and Communications, 2003,pp. 366–369

[7] GUY R.F.E., WYLLIE C.B., BRAIN J.R.: ‘Synthesis of theinmarsat 4 multibeam mobile antenna’. Proc. 12thInt. Conf. Antennas and Propagation, 2003, vol. 1,pp. 90–93

[8] KOJIMA N., KITAO S., SHIRAMATSU K., YAJIMA M., SHIMADA M.,NAKAMURA Y.: ‘Development results of a proto flight modelof the ka-band active phased array antenna for winds’.Proc. First European Conf. on Antennas and Propagation,6–10 November 2006, pp. 1–5

[9] MAILLOUX R.J.: ‘Phased array antennas handbook’ (ArtechHouse, Inc., 1994, 2nd edn.)

[10] GUIRAUD C., CAILLOCE Y., CAILLE G.: ‘Reducing directradiating array complexity by thinning and splittingitno non-regular subarrays’. Proc. 29th ESA AntennaWorkshop on Multiple Beams and ReconfigurableAntennas, ESA/ESTEC, Noordwijk, The Netherlands, 18 –20April 2007

[11] VAN TREES H.L.: ‘Optimum array processing’ (John Wiley,New York, 2002)

[12] REED I.S., MALLETT J.D., BRENNAN L.E.: ‘Rapid convergencerate in adaptive arrays’, IEEE Trans. Aerosp. Electron. Syst.,1974, 10, (6), pp. 853–863

[13] BOROSON D.M.: ‘Sample size considerations for adaptivearrays’, IEEE Trans. Aerosp. Electron. Syst., 1980, 16, (4),p. 446–451

[14] FELDMAN D.D., GRIFFITHS L.J.: ‘A projection approach forrobust adaptive beamforming’, IEEE Trans. Signal Process.,1994, 42, (4), pp. 867–876

[15] CARLSON B.D.: ‘Covariance matrix estimation errors anddiagonal loading in adaptive arrays’, IEEE Trans. Aerosp.Electron. Syst., 1988, 24, (4), pp. 397–401

[16] LI J., STOICA P., WANG Z.: ‘On robust Capon beamformingand diagonal loading’, IEEE Trans. Signal Process., 2003,51, (7), pp. 1702–1715

[17] VOROBYOV S.A., GERSHMAN A.B., LUO Z.-Q.: ‘Robustadaptive beamforming using worst-case performanceoptimization: a solution to the signal mismatch problem’,IEEE Trans. Signal Process., 2003, 51, (2), pp. 313–324

[18] LI J., DU L., STOICA P.: ‘Fully automatic computationof diagonal loading levels for robust adaptivebeamforming’. Proc. ICASSP, Las Vegas, NV, April 2008,pp. 2325–2328

[19] GUERCI J.R., GOLDSTEIN J.S., REED I.S.: ‘Optimal and adaptivereduced-rank STAP’, IEEE Trans. Aerosp. Electron. Syst.,2000, 36, (2), pp. 647–663

[20] PECKHAM C.D., HAIMOVICH A.M., AYOUB T.F., GOLDSTEIN J.S., REED

I.S.: ‘Reduced-rank STAP performance analysis’, IEEE Trans.Aerosp. Electron. Syst., 2000, 36, (2), pp. 664–676

[21] HAIMOVICH A.M., BAR-NESS Y.: ‘An eigenanalysis interferencecanceler’, IEEE Trans. Acoust. Speech Signal Process., 1991,39, (1), pp. 76–84

[22] HAIMOVICH A.M.: ‘The eigencanceler: adaptive radar byeigenanalysis methods’, IEEE Trans. Aerosp. Electron. Syst.,1996, 32, (2), pp. 532–542

[23] GOLDSTEIN J.S., REED I.S.: ‘Subspace selection for partiallyadaptive sensor array processing’, IEEE Trans. Aerosp.Electron. Syst., 1997, 33, (2), pp. 539–544

[24] GOLUB G., VAN LOAN C.: ‘Matrix computations’ (JohnHopkins University Press, Baltimore, 1996, 3rd edn.)

[25] GOLDSTEIN J.S., REED I.S., SCHARF L.L.: ‘A multistagerepresentation of the wiener filter based on orthogonalprojections’, IEEE Trans. Inf. Theory, 1998, 44, (7),pp. 2943–2959

[26] HONIG M.L., XAO W.: ‘Performance of reduced-rank linearinterference suppression’, IEEE Trans. Inf. Theory, 2001, 47,(5), pp. 1928–1946

[27] PADOS D.A., KARYSTINOS G.N.: ‘An iterative algorithm for thecomputation of the MVDR filter’, IEEE Trans. SignalProcess., 2001, 49, (2), pp. 290–300

[28] QIAN H., BATALAMA S.N.: ‘Data record-based criteria for theselection of an auxiliary vector estimator of the MMSE/MVDR filter’, IEEE Trans. Commun., 2003, 51, (10),pp. 1700–1708

[29] PADOS D.A., KARYSTINOS G.N., BATALAMA S.N., MATYJAS J.D.:‘Short-data-record adaptive detection’. Proc. IEEE RadarConf., Boston, MA, 17–20 April 2007, pp. 357–361

[30] BESSON O., MONTESINOS J., LARUE DE TOURNEMINE C.: ‘Onconvergence of the auxiliary- vector beamformer withrank-deficient covariance matrices’, IEEE Signal Process.Lett., 2009, 16, (4), pp. 249–252

[31] MARZETTA T.L., SIMON S.H., REN H.: ‘Capon-MVDRspectral estimation from singular data covariancematrix, with no diagonal loading’. Proc. 14th AdaptiveSensor Array Processing Workshop, Lexington, MA, 6 – 7June 2006

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