Robust adaptive beamforming is a key issue in array applications where there exist uncertainties about the steering vector of interest. Diagonal loading is one of the most popular techniques to improve robustness. Recently, worst-case approaches which consist of protecting the array's response in an ellipsoid centered around the nominal steering vector have been proposed. They amount to generalized (i.e. non necessarily diagonal) loading of the covariance matrix. In this paper, we present a theoretical analysis of the signal to interference plus noise ratio (SINR) for this class of robust beamformers, in the presence of random steering vector errors. A closed-form expression for the SINR is derived which is shown to accurately predict the SINR obtained in simulations. This theoretical formula is valid for any loading matrix. It provides insights into the influence of the loading matrix and can serve as a helpful guide to select it. Finally, the analysis enables us to predict the level of uncertainties up to which robust beamformers are effective and then depart from the optimal SINR

Besson, Olivier

Vincent, François

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Performance analysis for a class of robust adaptive beamformers

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PERFORMANCE ANALYSIS FOR A CLASS OF ROBUST ADAPTIVE BEAMFORMERSOlivier Besson and Franc¸ois VincentENSICA, Department of Avionics and Systems1 Place Emile Blouin, 31056 Toulouse, FranceEmail: besson,vincent@ensica.fr.ABSTRACTRobust adaptive beamforming is a key issue in array applicationswhere there exist uncertainties about the steering vector of inter-est. Diagonal loading is one of the most popular techniques to im-prove robustness. Recently, worst-case approaches which consistof protecting the array’s response in an ellipsoid centered aroundthe nominal steering vector have been proposed. They amountto generalized (i.e. non necessarily diagonal) loading of the co-variance matrix. In this paper, we present a theoretical analysisof the signal to interference plus noise ratio (SINR) for this classof robust beamformers, in the presence of random steering vectorerrors. A closed-form expression for the SINR is derived whichis shown to accurately predict the SINR obtained in simulations.This theoretical formula is valid for any loading matrix. It providesinsights into the influence of the loading matrix and can serve asa helpful guide to select it. Finally, the analysis enables us to pre-dict the level of uncertainties up to which robust beamformers areeffective and then depart from the optimal SINR.1. INTRODUCTIONAdaptive beamforming is an essential task in most systems usingan array of sensors. When the signal of interest (SOI) is present inthe measurements, the optimal beamformer is the minimum powerdistortionless response (MPDR) beamformer [1] and is given, upto a scaling factor, byR−1a. In the previous equation, R is the to-tal covariance matrix (including the SOI and possibly interferencesand noise) while a stands for the (presumed) steering vector of thesignal of interest. In order to implement the optimal beamformer,a needs to be known precisely, a property that is most often notencountered in practice where there are unavoidably mismatchesbetween the actual steering vector and the presumed steering vec-tor. The reasons for that are numerous: local scattering, uncertain-ties about the direction of arrival, propagation through an inhomo-geneous medium, fading, uncalibrated arrays, displaced sensors,etc. These mismatches are especially detrimental when the SOI ispresent in the data as the latter is considered as an interference andthus tends to be eliminated, see e.g. [2].Therefore robust adaptive beamforming has emerged as a nec-essary constituent of most systems using an array of sensors. Werefer the reader to [1, Chapter 6] for a comprehensive overview.The most widely used method, due to its simplicity and effec-tiveness, is diagonal loading [3] which consists of adding a scaledidentity matrix to the covariance matrix prior to inversion. Inter-estingly enough, generalized loading turns out to be the solutionto worst-case approaches recently proposed in [4–6]. In the latterreferences, the beamformer is designed so as to minimize the out-put power subject to the constraint that the beamformer’s responsebe above some level for all the steering vectors which lie in an el-lipsoid centered around the nominal or presumed steering vectorof interest. This guarantees that the signal of interest, whose steer-ing vector is expected to lie in the ellipsoid, will not be eliminatedand hence robustness is improved. The solution to this problemis given by (R + Q)−1 a where a denotes the nominal steeringvector (in the absence of any uncertainty) and Q stands for theloading matrix. When the ellipsoid is a sphere the solution boilsdown to diagonal loading i.e. Q ∝ I .Through numerical simulations, this robust adaptive beam-former was shown to perform well, at least when the size of theellipsoid does not grow large. However, no theoretical analysiswas provided and assessing its performance remains an open prob-lem. The finite-sample SINR analysis of the minimum variancedistortionless beamformer (MVDR) was presented in [7] wherethe probability density function (pdf) of the SINR loss (comparedto perfectly known interference plus noise covariance matrix) isderived. Similar finite-sample SINR’s pdf analysis for the MPDRbeamformer are reported in [8], see also [9,10]. In contrast, analy-sis of diagonally loaded versions of the MPDR is scarce. A large-sample analysis of the weight vector and powers at the output of adiagonally loaded MPDR can be found in [11, 12] while [13] de-rives the pdf of the beam response when diagonal loading is used.In this paper, we present a theoretical analysis of the SINR ob-tained with a general loading matrix and in the presence of randomsteering vector uncertainties. The formulas obtained are quite sim-ple and are valid for any loading matrixQ (including non diagonalor non invertible). Additionally, they are shown to predict well theperformances obtained via numerical simulations. Finally, theycan serve to provide insights into the choice of the loading matrix.2. DATA MODELWe consider an array composed of m sensors and assume that thearray’s output can be written asxt = ast + nt t = 1, · · · , N (1)where a is the actual steering vector of the source of interest, andst is the corresponding emitted signal. We assume that a is acomplex-valued, circularly symmetric random vector with meanE {a} = a and covariance matrix E{(a − a) (a − a)H}=Ca. a corresponds to the steering vector without any perturba-tion (e.g. for a perfectly calibrated array) while Ca captures theeffects of all possible errors affecting the steering vector. st is azero-mean random process with power P = E{|st|2}. nt is thenoise contribution, including interferences and thermal noise. nt,,,((( ,&$663is assumed to be drawn from a zero-mean complex-valued Gaus-sian distribution with covariance matrix C .The robust adaptive beamformers of [4–6] (although their for-mulations are different) are obtained as the solution to the follow-ing minimization problemminwwHRw subject to∣∣∣wHa∣∣∣ ≥ 1 ∀a = a + Bu; ‖u‖ ≤ 1(2)whereR is the covariance matrix andB is am×r matrix with fullcolumn rank which defines an ellipsoid centered around a. Undermild assumptions, the solution to (2) is given by (up to a scalingfactor which does not affect the SINR) w = (R + Q)−1 a withQ = λBBH and where the Lagrange multiplier λ is obtained asthe solution of a secular equation which involves the eigenvaluedecomposition of BHR−1B, see [5] for details. Note that whenB = I the ellipsoid becomes a sphere and the solution amountsto conventional diagonal loading.3. SINR ANALYSISIn this section, we provide a theoretical expression for the robustadaptive beamformer’s average SINR. We proceed as follows. In afirst step, we assume that a is given and derive the correspondingSINR. Then, we invoke the conditional expectation rule to com-pute the average SINR asSINR = Ea{SINR|a} (3)where Ea {.} denotes the expectation with respect to (w.r.t.) theprobability density function of a and SINR|a corresponds to theSINR for a given a. In order to obtain SINR|a , note that theweight vector, for a given a, is given byw|a =(R|a + Q)−1a (4)whereR|a denotes the covariance matrix for a given a. Herein wedo not consider finite-sample effects, i.e. we assume that the truecovariance matrix R|a is available. In order to introduce finite-sample effects (and thereby to combine them with steering vectorerrors), one needs to assume that the two errors are of the sameorder of magnitude. In our case, this would amount to assumingthat Ca = Ca/N where Ca is fixed and N denotes the num-ber of snapshots. We refer the reader to [14] for a detailed andcomprehensive discussion on this issue. However, this assumptionmay seem arbitrary since the errors are not likely to depend onN . Therefore, herein we consider that N is large enough so thatthe steering vector errors dominate. We will however illustrate thefinite-sample behavior of (4) in the numerical simulations.Using (4), the conditional SINR isSINR|a =P∣∣wH|a a∣∣2wH|a Cw|a=P∣∣∣aH (R|a + Q)−1 a∣∣∣2aH(R|a + Q)−1C(R|a + Q)−1a(5)However, using (1) along with the assumptions made, one hasR|a = PaaH + C (6)Inserting (6) in (5) and after some straightforward algebraic ma-nipulations, it can be shown thatSINR|a =1P (a − γ(a)a)H Z (a − γ(a)a)(7)withγ(a) =1 + PaHQ˜−1aPaHQ˜−1a(8)and where Q˜ = Q+C and Z = Q˜−1CQ˜−1. The average SINRis thus given bySINR =∫p(a)P (a − γ(a)a)H Z (a − γ(a)a)da (9)where p(a) is the probability density function of a. Obtaininga closed-form expression for the previous integral appears to bean intractable task. Hence, we prefer to approximate this integral.Towards this end, it can be shown that for any scalar function f(a)and assuming that a is circularly symmetric∫f(a)p(a) da  f(a) + Tr{∂2f∂a∂aH∣∣∣∣aCa}(10)where Tr {.} stands for the trace of the matrix between braces. Itshould be pointed out that the previous approximation does not re-quire complete knowledge of the pdf of a but only of its mean andcovariance matrix, which is an appealing feature. We now applythe result (10) to SINR|a . However, in the purpose of simpli-fying subsequent derivations, we first simplify the expression ofSINR|a by approximating γ(a). More precisely, we propose toapproximate γ(a) by its statistical mean. Observing that∂γ∂a=Q˜−1(a − γa)aHQ˜−1a(11a)∂2γ∂a∂aH=Q˜−1aHQ˜−1a(I −aaHQ˜−1aHQ˜−1a)(11b)it follows, using (10) thatEa {γ(a)} 1 + PaHQ˜−1aPaHQ˜−1a+Tr{Q˜−1Ca}aHQ˜−1a−aHQ˜−1CaQ˜−1a(aHQ˜−1a)2  γ0 (12)We propose to replace γ(a) by γ0 in (7) so thatSINR|a 1P (a − γ0a)HZ (a − γ0a)(13)We would like to point out that using γ0 in lieu of γ(a) in (10) doesnot result in a less accurate expression for the average SINR, aswill be illustrated below. The average SINR is thus approximatedbySINR ∫p(a)P (a − γ0a)HZ (a − γ0a)da (14)Let f(a) = (a − γ0a)H Z (a − γ0a) so that SINR|a  [Pf(a)]−1.First, observe that∂1/f∂a=−1f2∂f∂a∂21/f∂a∂aH=−1f2∂2f∂a∂aH+2f3∂f∂a∂f∂aH(15),,Next, using∂f∂a= Z (a − γ0a) ;∂2f∂a∂aH= Z (16)along with (10), (14) and (15) yieldsSINR 1Pf(a)−Tr {ZCa}Pf2(a)+2 |1− γ0|2aHZCaZaPf3(a)f(a) = |1− γ0|2aHZa (17)The previous equation provides a closed-form and compact ex-pression for the average SINR. We stress the fact that it holds fora large class of robust adaptive beamformers as it holds for anyloading matrix Q and any steering vector error covariance matrixCa. As will be illustrated in the next section, it predicts very ac-curately the average SINR obtained through Monte-Carlo simula-tions. Hence, it can serve as a useful tool to obtain rapid insightsinto the choice of the loading matrix Q without resorting to exten-sive simulations.4. NUMERICAL ILLUSTRATIONS AND CONCLUSIONSThe aim of this section is threefold. Firstly, we assess the valid-ity of the theoretical formula (17) by comparing it with the actualSINR obtained through Monte-Carlo simulations. Secondly, weprovide illustrations of up to which level of uncertainty general-ized loading can compensate for steering vector errors and stillprovide a performance close to optimum. Thirdly, we providerules of thumb for selecting the shape and the size of the loadingmatrix. In all simulations, we consider a uniform linear array ofm = 10 sensors spaced a half-wavelength apart. The signal of in-terest impinges from broadside and thus a =[1 1 · · · 1]T.The noise component consists of a white noise contribution withpower σ2n and two interferences whose DOAs are −20◦, 30◦ andwhose powers are 20dB and 30dB above the white noise level, re-spectively. We define the uncertainty ratio (UR) and the signalto noise ratio (SNR) as UR = 10 log10(Tr{Ca}aHa)and SNR =10 log10(P(aHa+Tr{Ca})σ2n), respectively. In all simulations, theSINR is evaluated as follows. Nr = 500 Monte-Carlo simulationsare run with a different random a and, for a given weight vectorw, the average SINR is computed asSINR(w) =1NrNr∑n=1P∣∣wHa(n)∣∣2wHCw(18)The robust adaptive beamformer (4) -which is referred to as RB inthe figures-, will be compared to the following beamformers:• the MVDR beamformer which is given bywMVDR = P{C−1(aaH + Ca)}(19)where P {.} stands for the principal eigenvector of the ma-trix between braces.• a (hypothetical) clairvoyant optimum beamformer whichmaximizes the SINR for any given a and is thus given bywopt|a = C−1a (20)• the sample covariance matrix (SCM) version of (4), i.e.wSCM =(R̂ + Q)−1a; R̂ =1NN∑t=1xtxHt (21)The performance of the SCM robust beamformer will beevaluated by (18). It will enable us to take into accountfinite-sample effects.In a first series of simulation, we consider the case where thesteering vector errors are drawn from a zero-mean complex-valuedGaussian distribution with covariance matrix Ca = σ2aI . Westudy the performance of the beamformers versus the uncertaintyratio. The loading matrix Q = λI and we define the loading level(LL) as LL = 10 log10(λσ2n). Note that it corresponds to a load-ing level relative to the white noise power. In the simulations, LLis chosen as LL = 5dB. The results are displayed in Figure 1.The following observations are in order. The theoretical formula(17) is seen to predict very accurately the SINR obtained in sim-ulations, to within 0.2dB for UR ≤ −2dB. Hence (17) providesa very good picture of the robust adaptive beamformer’s perfor-mance in most situations. Moreover, the finite-sample behavior ofthe robust beamformer is also close to the theoretical formula. Therobust beamformer has a performance rather close to that of theMVDR, at least for UR’s below−6dB. For higher UR, the robustbeamformer can no longer compensate for the uncertainties; henceone must turn to other solutions. It should be pointed out that, forUR ≥ −6dB, the MVDR also becomes less performant than theclairvoyant beamformer, which makes use of the actual steeringvector. This suggests that for high uncertainties the remedy wouldbe to obtain additional information about the actual steering vec-tor, for instance by estimating it, rather than to protect the array’sresponse over a larger and larger ellipsoid.Ŧ20 Ŧ18 Ŧ16 Ŧ14 Ŧ12 Ŧ10 Ŧ8 Ŧ6 Ŧ4 Ŧ2 0Ŧ2Ŧ1.5Ŧ1Ŧ0.500.511.522.53 Output SINR Uncertainty Ratio (dB) dBSINRoptSINRMVDRSINRRBSINRSCMSINRthFig. 1. Output SINR versus uncertainty ratio. SNR = 3dB,LL = 5dB and N = 200. Ca = σ2aI , Q ∝ I .Next, the influence of the loading level is studied in Figure 2.In this figure, the loading matrix is still proportional to the identitymatrix and UR = −6dB. Again, it can be seen that the theoreticalSINR is very close to the practical SINR. Also, it can be observed,,0 2 4 6 8 10 12 14 16 18 2000.511.522.53 Output SINR Relative loading level (dB) dBSINRoptSINRMVDRSINRRBSINRSCMSINRthFig. 2. Output SINR versus loading level. SNR = 3dB, UR =−6dB and N = 200. Ca = σ2aI , Q ∝ I .that, despite LL has an influence onto the final SINR, there exist alarge range of values for LL which provide a similar performance.Finally, we study the influence of the shape of the loading ma-trix which is directly related to the form of the ellipsoid in (2).Intuitively, since E{(a − a) (a − a)H}= Ca, it follows that acan be written as a = a+C1/2a u with C1/2a a square-root of Ca.This suggests that B should be related to C1/2a , or equivalentlythat Q ∝ Ca. Hence, we check whether this intuitive hypothesisresults in a better performance than using diagonal loading only.Towards this end, we consider the case of local scattering for whichthe steering vector cas be written as a = a + 1√L∑Lk=1 gka(θ˜k)where gk are zero-mean, independent and identically distributedrandom variables with power σ2g and θ˜k are independent randomvariables with pdf p(θ˜). The covariance matrix of the errors isgiven byCa = σ2g∫a(θ˜)aH(θ˜)p(θ˜) dθ˜ = σ2gC˘a (22)In the simulations presented below, we assume a Gaussian dis-tribution for the scatterers with standard deviation (referred to asangular spread in the literature) σθ = 15◦. For each value of UR,we consider both Q = λI and Q = λC˘a (note that Tr{C˘a}=Tr {I}) and we look for the value of λ that results in the opti-mal average SINR in (17). We plot in Figure 3 the obtained op-timum SINR. Examination of this figure reveals that the perfor-mance obtained with Q ∝ Ca is always inferior to that obtainedwith Q ∝ I . So, even if the steering vector covariance matrix isnot a scaled identity matrix, there is no gain in using Q ∝ Cainstead of diagonal loading. Note also, as discussed in [5], thatchoosing Q ∝ Ca implies that one has a strong a priori on theshape of the errors, which is seldom the case. Hence, robustnessmay be endangered if Q is chosen as Q ∝ Ca whereas the truecovariance matrix of the steering vector errors is not Ca. Thisprovides an additional argument in favor of diagonal loading.Ŧ20 Ŧ18 Ŧ16 Ŧ14 Ŧ12 Ŧ10 Ŧ8 Ŧ6 Ŧ4 Ŧ2 0Ŧ2Ŧ1.5Ŧ1Ŧ0.500.511.522.53 Output SINR Uncertainty Ratio (dB) dBSINRoptSINRMVDRSINRQ=ISINRQ=CaFig. 3. Coherent local scattering. Optimum SINR obtained withthe robust beamformer versus uncertainty level. SNR = 3dB.5. REFERENCES[1] H. V. Trees, Optimum Array Processing. New York: John Wiley, 2002.[2] N. 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Analysis of the combined effects of ﬁnite samples and model errors on array processing performance,”

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Robust adaptive beamforming using worst-case performance optimization: A solution to the signal mismatch problem,”

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Performance analysis for a class of robust adaptive beamformers

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