Adaptive channel selection for DOA estimation in MIMO radar
David Mateos-Núñez, María A. González-Huici, Renato Simoni, Stefan Brüggenwirth
AAdaptive channel selection for DOA estimation in MIMO radar
David Mateos-N´u˜nez Mar´ıa A. Gonz´alez-Huici Renato Simoni Stefan Br¨uggenwirth
Abstract — We present adaptive strategies for antenna selectionfor Direction of Arrival (DoA) estimation of a far-field sourceusing TDM MIMO radar with linear arrays. Our treatment isformulated within a general adaptive sensing framework thatuses one-step ahead predictions of the Bayesian MSE usinga parametric family of Weiss-Weinstein bounds that dependon previous measurements. We compare in simulations ourstrategy with adaptive policies that optimize the Bobrovsky-Zaka¨ı bound and the Expected Cram´er-Rao bound, and showthe performance for different levels of measurement noise.
Index Terms — Adaptive Sensing, Antenna Selection, Array Pro-cessing, Weiss-Weinstein Bound, Bayesian Filtering, Directionof Arrival (DOA), MIMO, Cognitive Radar.
I. I
NTRODUCTION
Recent advances in millimeter-wave radar circuits makepossible low-cost and compact multi-channel radar systemsthat can be controlled by software. This motivates the designof signal generation and processing algorithms that attemptto maximize the information extracted from the scene, inwhat is considered the basis of the perception-action cycleof a cognitive radar architecture [1], [2].Algorithms for adaptive transmission typically employ aprediction of the conditional Bayesian mean-square error(BMSE) given previous observations. In the category ofadaptive strategies that attempt to optimize one-step aheadpredictions, recent works optimize parameters such as thepulse repetition frequency (PRF) in Pulse-Doppler radar ina joint framework for detection and tracking [3], [4], orthe transmitted signal autocorrelation matrix in MIMO radarfor DoA estimation [5], [6], using, respectively, the condi-tional Bayesian Cram´er-Rao bound (BCRB) and the Reuven-Messer bound (RMB) [7]. In the category of algorithmsthat consider the consequences of actions based on somelong-term reward, the work [8] schedules measurements ina tracking scenario where the target is temporarily occluded,and [9] optimizes waveform parameters using planning andreinforcement learning.Few works have considered these approaches for adaptiveantenna selection for Direction of Arrival (DoA) estimation.Accuracy of angular estimation improves with the length ofthe antenna array, and thus with the number of antenna ele-ments that need to be adequately spaced to avoid ambiguitydue to aliasing. Bigger apertures demand more Tx and Rxmodules (and hence a higher system cost) and more data tobe processed in real time. To overcome these constraints, the
The authors are with the department of Cognitive Radar atFraunhofer FHR, Wachtberg, Germany, { david.mateos-nunez,maria.gonzalez, renato.simoni,stefan.brueggenwirth } @fhr.fraunhofer.de . works [10], [11] study adaptive receiver selection algorithmsfor far-field DoA and SNR estimation with SIMO lineararrays based on optimization of the Bobrovsky-Zaka¨ı Bound(BZB), which provides better one-step ahead predictions thanthe Expected CRB (ECRB). The latter is not sensitive tosidelobe level but is related to the average mainlobe widthof the array [12], and selects the receivers that yield biggestaperture regardless of previous measurements [10]. Alterna-tively, the Weiss-Weinstein bound [13] is computationallymore expensive but predicts more accurately the contributionto the Mean-Square-Error (MSE) of sidelobe ambiguity atlow SNR [14], [15].We extend the work of [10], [11] to transmitter and receiverselection for DoA estimation in Time Domain Multiplexing(TDM) MIMO radar with linear arrays and propose a generalalgorithm for adaptive sensing using the Weiss-Weinsteinbound. Using a particle filter [16] to incorporate sequentiallythe information from measurements into the belief distribu-tion of the unknown parameter, we construct the conditionalWWB (along with the BZB and the ECRB, for reference),that lower bounds the achievable MSE. This requires adouble optimization procedure, first over the so-called test-points, to evaluate the tightest bound, and then over candidatesensing parameters. The resulting strategies are illustrated insimulations where we compare the performance of channelselection based on the WWB, the BZB, and the ECRB.The rest of the paper is organized as follows: Section IIproposes a general framework for adaptive sensing based onone-step ahead predictions of the MSE. The general strategyis then particularized in Section III to MIMO channel se-lection for DoA estimation. Finally, Section IV presents ourconclusions and ideas for future work.II. A DAPTIVE SENSING VIA W EISS -W EINSTEIN BOUND
In this section we present the general strategy to optimizesensing parameters based on a prediction of the MSE. Firstwe introduce the WWB, then we connect it to the conditionalBMSE, and finally we describe the computation of theconditional WWB involved in our algorithm.
A. Preliminaries on the Weiss-Weinstein bound
The WWB provides a lower bound on the BMSE of anyestimator and thus gives an indication of the achievableestimation performance. Namely, the expected error of anyestimator ˆ θ ( x ) , over possible pairs of observations x and one-dimensional parameter values θ modeled with probabilitydistribution p ( x, θ ) , is bounded as follows, E p ( x,θ ) (cid:2) (ˆ θ ( x ) − θ ) (cid:3) ≥ WWB( s, h ) , (1) a r X i v : . [ c s . S Y ] A p r here WWB( s, h ) is a member of the parametric family ofbounds [13], [15, eq. 76] given by WWB( s, h ) := h η ( s, h ) η (2 s, h ) + η (2 − s, − h ) − η ( s, h ) , (2)where η is the moment generating function [17, pp. 337, pp.65] defined as η ( α, β ) := (cid:90) Θ (cid:90) Ω p α ( x, θ + β ) p α − ( x, θ ) dxdθ = (cid:90) Θ (cid:16) (cid:90) Ω p α ( x | θ + β ) p α − ( x | θ ) dx (cid:17) p α ( θ + β ) p α − ( θ ) dθ, (3)where p ( x | θ ) is the probability, or likelihood , of the obser-vation x ∈ Ω ⊆ R n given the parameter value θ ∈ Θ := { θ ∈ R : p ( θ ) > } , and p ( θ ) is the prior probabilitydistribution of θ , which is considered a modeling choice.The value of the so-called test-point h ∈ (0 , ∞ ) , and theadditional degree of freedom s ∈ (0 , , determine thebound on the BMSE, the tightest bound being obtained as WWB := sup s,h
WWB( s, h ) . (For further generalizationswe refer the reader to [17].) The BZB can be obtainedfrom (2) in the limit cases s = 1 or s = 0 , BZB( h ) = WWB( s = 1 , h ) = h η (2 , h ) − . (4)The BCRB [17, pp. 72] is in turn a particular case of (4),under suitable assumptions on the differentiability of p ( θ ) ,in the limit as h → . In the next section we presentthe connection between the WWB described here and theconditional BMSE, relevant for our adaptive strategies. B. Conditional BMSE and adaptive sensing
Consider an estimation task where a sequence of obser-vations X ( k − := ( X , . . . , X k − ) of an unknown pa-rameter θ is obtained using a sequence of sensing pa-rameters G ( k ) := ( G , . . . , G k ) , in a suitable domain,according to an observation model with joint probabilitydistribution p ( X ( k ) , θ | G ( k ) ) . An adaptive sensing strategy orpolicy can be defined in general by a probability distribu-tion over sensing parameters given previous measurements, p ( G k | X ( k − , G ( k − ) . In this work, the proposed strategiesare evaluated with respect to the BMSE, which is defined, forany estimator ˆ θ ≡ ˆ θ ( X ( k ) , G ( k ) ) , as the following integrationover observations and realizations of the parameter, BMSE(ˆ θ, G ( k ) ) := E p ( X ( k ) ,θ | G ( k ) ) (cid:2) (ˆ θ − θ ) (cid:3) = E p ( X ( k − | G ( k ) ) (cid:104) E p ( X k ,θ | X ( k − ,G ( k ) ) (cid:2) (ˆ θ − θ ) (cid:3)(cid:105) . Following [5] and [11], we consider the inner expectationabove, called conditional BMSE (CBMSE),
CBMSE(ˆ θ, X ( k − , G ( k ) ) := E p ( X k ,θ | X ( k − ,G ( k ) ) (cid:2) (ˆ θ − θ ) (cid:3) , as an optimization metric for adaptive algorithms that at-tempt to find, at each step k ≥ , a policy G k that The usual convention is to consider the logarithm, but in our presentationwe choose this notation for convenience. minimizes the BMSE given any sequences of previoussensing policies G ( k − and historical observations X ( k − .Such metric is usually impossible to compute explicitly, butcan be lower-bounded in a similar fashion as the BMSEin relation (1). Motivated by this observation, we definethe parametric family of conditional WWBs, denoted by WWB( s, h ; X ( k − , G ( k ) ) , as in (2), where in the defini-tion (3) we use the likelihood function p ( X k | θ, G ( k ) ) andreplace the prior distribution by the posterior p k − ( θ ) := p ( θ | X ( k − , G ( k − ) . The moment generating function in (3)becomes then η k ( α, β ) := (cid:90) Θ (cid:90) Ω p ( x | θ + β, G ( k ) ) α p ( x | θ, G ( k ) ) α − dx p k − ( θ + β ) α p k − ( θ ) α − dθ, (5)where p ( θ ) := p ( θ ) is the prior probability. (Note that thedomain of integration in (5) is such that p k − ( θ ) > .) Proposition 2.1: (Conditional WWB and CBMSE): Con-sider the observation model p ( X ( k ) , θ | G ( k ) ) under the fol-lowing two assumptions, i) X k and X ( k − are conditionallyindependent given θ and G ( k ) , i.e., p ( X k , X ( k − | θ, G ( k ) ) = p ( X k | θ, G ( k ) ) p ( X ( k − | θ, G ( k − ) , and ii) p ( X k | θ, G ( k ) ) = p ( X k | θ, G k ) . Then CBMSE(ˆ θ, X ( k − , G ( k ) ) ≥ WWB( s, h ; X ( k − , G ( k ) ) . (The proof is standard and is omitted for lack of space.)Motivated by the above result, we define adaptive strategiesthat select at step k the sensing policy G k based on knowl-edge from previous measurements X ( k − and previoussensing policies G ( k − , as the solution of G ∗ k ∈ arg min G k sup s ∈ (0 , h ∈ (0 , ∞ ) WWB( s, h ; X ( k − , G ( k ) ) . (6)In general, the inner optimization problem in (6) over test-points is nonconvex, requiring methods for global opti-mization such as simulated annealing [18], and the outeroptimization over sensing policies can be discrete. In the nextsection we explain how to construct the parametric familyof conditional bounds WWB( s, h ; X ( k − , G ( k ) ) using thelikelihood function, the sequence of measurements X ( k − ,and previous sensing policies G ( k − . C. Computation of the conditional WWB
To evaluate
WWB( s, h ; X ( k − , G ( k ) ) we first re-write themoment generating function (5), consistently with the nota-tion in [11], as η k ( α, β ) = (cid:90) Θ D k ( θ, α, β ) φ k − ( θ, α, β ) p k − ( θ ) dθ = E p k − ( θ ) (cid:2) D k ( θ, α, β ) φ k − ( θ, α, β ) (cid:3) , (7)where D k ( θ, α, β ) contains the observation model, D k ( θ, α, β ) := (cid:90) Ω p ( x | θ + β, G k ) α p ( x | θ, G k ) α − dx, and φ k − ( θ, α, β ) contains the posterior distribution, φ k − ( θ, α, β ) := (cid:16) p k − ( θ + β ) p k − ( θ ) (cid:17) α . (8)he computation of φ k − ( θ, α, β ) requires special attention.Again by Bayes Law and using the assumptions of Proposi-tion 2.1, we can express the posterior probability as follows, p k − ( θ ) := p ( θ | X ( k − , G ( k − )= p ( θ | G ( k − ) p ( X ( k − | G ( k − ) k − (cid:89) m =1 p ( X m | θ, G m ) . (9)Next we make an observation that connects the iterativecomputation of the posterior in (9) with the approximationof the expectation in (7).Fig. 1: Basic diagram of adaptive sensing based on WWB. Remark 2.2: (Computation of the expectation (7) and depen-dence of measurements in adaptive algorithms): Suppose thatthe sensing parameters are chosen randomly without usingprevious data, so that p ( θ | G ( k − ) = p ( G ( k − | θ ) p ( G ( k − ) p ( θ ) = p ( θ ) . Under this approximation, the expectation in (7) canbe approximated numerically via Monte Carlo integrationleveraging two facts: i) the quotient in (8) can be factorized,in view of (9), as ˆ φ k − ( θ, α, β ) := (cid:16) p ( θ + β ) p ( θ ) (cid:81) k − m =1 p ( X m | θ + β, G m ) (cid:81) k − m =1 p ( X m | θ, G m ) (cid:17) α , and ii) one can sample from the posterior using Monte Carlomethods such as particle filters. • In summary, measurements depend on the sensing policies,as prescribed by the likelihood. The likelihood serves twopurposes, see Fig. 1: i) filtering in the processor, where parti-cles, i.e., guesses of the parameter, are re-sampled accordingto which ones make the measurements more likely, andii) prediction in the controller, where the WWB is computedintegrating the joint distribution that combines the likelihoodof possible observations and the current posterior.III. A
DAPTIVE CHANNEL SELECTION
In this section, we particularize the strategy for adaptivesensing in Section II to the problem of adaptive channelselection for DoA estimation with MIMO linear arrays. To our knowledge, the study of filtering performance in scenarios ofdependent measurements due to greedy adaptation of the sensing policiesis absent in the literature. For -dimensional cases, integration using the empirical PDF of theparticles might be more efficient than Monte Carlo integration. We use thisapproach for DoA estimation with and without the approximation given byfitting the posterior by a Gaussian, which lowers the computational cost. A. Problem statement
Here we consider the problem of angle of arrival estimationof a single far-field point target. For this, we use a lineararray of I omnidirectional antennas, with observation model x j,k = m k ( θ ) s j,k + n j,k , (10)where x j,k ∈ C I is the observation at snapshot j ∈{ , . . . , J } in step k ∈ { , , . . . } , m k ( θ ) ∈ C I is thesteering vector for the unknown electronic azimuth θ :=sin( φ ) , where φ ∈ ( − π, π ) is the azimuth or direction ofarrival, s j,k ∈ C is the target signal (which here we assumeis known), and n j,k is the noise, modeled by independentand identically distributed zero-mean complex Gaussianswith real and imaginary parts also independent with covari-ance σ I I (i.e., a multiple of the identity matrix). In SIMOradar (i.e., a single transmitter and multiple receivers), m k ( θ ) corresponds to the receive steering vector a Rx ( θ ) ∈ C N ,which is defined, for a far-field source and N receiverslocated at positions d Rx := [ d Rx , · · · , d Rx N ] ∈ R N , as a Rx ( θ ) := e jk d Rx θ , where k = 2 π/λ is the wavenumber and λ isthe received wavelength. To incorporate into the model theselection of Rx elements, we define the receive switchingmatrix G Rx k ∈ { , } I Rx × N , for a total of I Rx active receivers,such that the i th row contains a nonzero element only incolumn n i , and each column has at most a nonzero element.The switched receive steering vector is then defined as m SIMO k ( θ ) := G Rx k a Rx ( θ ) = (cid:104) e jk d Rx n θ , · · · , e jk d Rx nI Rx θ (cid:105) (cid:62) = e jk G Rx k d Rx θ . (11)Similarly, for DoA estimation using MIMO arrays, we definethe switched TDM MIMO steering vector as m MIMO k ( θ ) := ( γ ( f D ) (cid:12) ( G Tx k a Tx ( θ ))) ⊗ ( G Rx k a Rx ( θ )) , (12)where a Tx ( θ ) := e jk d Tx θ ∈ C M is the transmitsteering vector for M transmitters located at positions d Tx := [ d Tx , · · · , d Tx M ] ; the transmit switching matrix G Tx k ∈{ , } I Tx × M , for I Tx active transmitters, is such that the i throw contains only a nonzero element in column m i (andeach column has at most a nonzero element); and γ ( f D ) := e j πT [1 , ··· ,I Tx ] f D ∈ C I Tx contains the Doppler frequency shift f D ∈ R (that we assume is known here), typical in a TDMscheme. The latter term results from the sequence of pulsesfrom the active transmitters with inter-pulse duration T > .With this notation, (12) can be written as m MIMO k ( θ ) = (cid:104) e j πf D T e jk d Tx m θ , · · · , e j πf D T I Tx e jk d Tx mI Tx θ (cid:105) (cid:62) ⊗ (cid:104) e jk d Rx n θ , · · · , e jk d Rx nI Rx θ (cid:105) (cid:62) = e j ( k d Virt k θ + d TDM f D ) , (13)where d TDM := 2 πT [1 , · · · , I Tx ] ⊗ I Rx , and d Virt k := I Tx ⊗ ( G Rx k d Rx ) + ( G Tx k d Tx ) ⊗ I Rx . The goal is to choose a total of I = I Tx + I Rx activetransmitters and receivers, specified by G k = { G Tx k , G Rx k } at each step k ∈ { , , . . . } , that help extract the maximummount of information about the angle of arrival accordingto (6). The only part of the adaptive sensing strategy ofSection II that needs to be particularized is the likelihoodfunction, which naturally depends on the observation modelabove, cf. Fig. 1. Using the corresponding likelihood functionfor DoA estimation in SIMO and MIMO radar, in the nextsection we construct the WWB associated to these problems. B. Conditional WWB for DoA estimation
To apply the general strategy of Section II to the problem ofantenna selection, we need to use the likelihood functionassociated to the observation model (10), see Fig.1. Thelikelihood function of J snapshots X k = [ x ,k , . . . , x J,k ] ,given θ and sensing parameters G k = { G Tx k , G Rx k } , isdistributed as a product of complex Gaussian distributionsbecause snapshots are assumed independent, i.e., p ( X k | θ, G k ) = J (cid:89) j =1 (cid:16) πσ ) I e − σ (cid:107) x j,k − m k ( θ ) s j,k (cid:107) (cid:17) . From the computation in [15, eq. (137)], one has D k ( θ, α, β ) = J (cid:89) j =1 (cid:90) R I p α ( x j,k | θ + β ) p α − ( x j,k | θ ) dx j,k = e s k α ( α − σ (cid:107) m k ( θ + β ) − m k ( θ ) (cid:107) , where s k := (cid:80) Jj =1 | s j,k | . (Note that the model withunknown stochastic target signals, called unconditional , re-quires a different calculation, cf. [11], [19].) In the SIMOcase, using the definition (11), we get (cid:107) m SIMO k ( θ + β ) − m SIMO k ( θ ) (cid:107) = (cid:107) m SIMO k ( θ + β ) (cid:107) + (cid:107) m SIMO k ( θ ) (cid:107) − { m SIMO k ( θ + β ) H m SIMO k ( θ ) } = 2 I Rx − { I Rx (cid:88) i =1 m SIMO k ( β ) i } , which is related to the ambiguity surface (cf. [17, pp. 269,eq. 4.229]) for the selected receivers. Therefore, D SIMO k ( α, β ) := e α ( α − s kσ (cid:0) I Rx − (cid:80) I Rx i =1 cos( k ( G Rx k d Rx ) i β ) (cid:1) . Similarly, for the MIMO case, using (13), we obtain D MIMO k ( α, β ) := e α ( α − s kσ (cid:0) I Tx I Rx − (cid:80) I Tx I Rx i =1 cos( k ( d Virt k ) i β ) (cid:1) . Equipped with the functions D SIMO k ( α, β ) and D MIMO k ( α, β ) (which incidentally do not depend on θ ), the parametricfamily of conditional bounds WWB( s, h ; X ( k − , G ( k ) ) canbe expressed in terms of (7) according to (2). Note that theposterior can be approximated following Remark 2.2. We canthen evaluate candidate sets of channels specified by G k = { G Tx k , G Rx k } , and select the optimal ones according to (6).Next we present simulations with synthetic measurements. C. Simulations
Here we compare in simulations the performance of channelselection policies that optimize the WWB, the BZB, and theECRB for SIMO and MIMO arrays. The separation between The optimization domain for the WWB is ( s, h ) ∈ [ . , . × [10 − , ;for the BZB we use s = 0 . , numerically more stable than s = 1 ; and weuse the ECRB instead of the BCRB because they yield equivalent policies. Fig. 2: Optimal channel choices in a typical execution in theSIMO case, with SNR = − , where Rx is always fixed. Fig. 3: Optimal channel choices in the MIMO case for eachpolicy, under a Gaussian approximation of the posterior ateach step, where Tx and Rx are always fixed. (Overlap-ping virtual elements are represented with concentric circles.)adjacent transmitters and receivers is . λ/ , the number ofsnapshots is J = 2 , the target signal s j,k is assumed knownand equal to , and we assume an initial prior distribution forthe electronic azimuth uniform in [ − , . The target is static, f D = 0 , and therefore the order of transmitter activations isirrelevant for any given subset of them. We perform the inneroptimization in (6) using simulated annealing [18] with a cooling speed of intermediate temperatures when theSNR is less than , and otherwise, and the posterioris sequentially updated using a particle filter with residualresampling [20] and particles.The channel choices for the SIMO and MIMO cases areshown in Figs. 2 and 3 for a single execution of our algorithmwith SNR = − . These choices depend on the posteriordistribution updated by each strategy and thus on the uniquehistory of previous measurements and channel selections. Inthe SIMO case, we observe a qualitative behavior for thepolicies that optimize the WWB and BZB analogous to thesimulations in [10], [11], where during the first measure-ments receivers tend to be chosen closer together to avoidambiguity in the estimation, and in subsequent measurementsare selected farther apart to increase resolution. A similarbehavior can be seen in the MIMO case.We analyze the performance using the MSE of the condi-tional mean estimator ˆ θ that results from each sensing policy. Matlab code, by H´ector Corte, available in MathWorks File Exchange. Matlab code “Resampling methods for particle filtering,” by J.-L. BlancoClaraco, available in MathWorks File Exchange. his is computed at each measurement step with respectto the true parameter value θ = sin( φ ) = 0 . using Monte Carlo realizations of each snapshot. In the SIMO case,Fig. 4 (top) shows the same single execution as in Fig. 2. Inthe MIMO case, we have simulated a computationally fasterversion of the adaptive policies where the expectation in (7)is approximated replacing the posterior given by the particlesby a Gaussian distribution with the same mean and variance.Using the result in [15, eqs. (138), (152)], this allows us toobtain a close form for (7). With this approximation, Fig. 4(bottom) compares the average MSE, over algorithmtrajectories for each SNR, of the conditional mean estimatorat measurement step . We observe that optimizing theWWB yields slightly better performance than using the BZBfor low SNR values. In addition, these adaptive policiesoutperform ad hoc strategies with the same number of activeantennas, including the SIMO “stair” switch and the fixedMIMO uniform virtual array. The evaluation of the WWBwith s = 0 . yields comparable performance to (but notalways the same channel choices as) the WWB even thoughit uses a single test-point as the BZB. The discussion of thecomputational complexity depends on the cooling speed thatis required for the optimization of each Bayesian bound andwill be included in future work. -5 -4 -3 -2 -1 -15 -10 -5 0 5 10 1510 -6 -5 -4 -3 -2 -1 Fig. 4: MSE of the conditional mean for each policy. In theSIMO case (top) we depict a single execution over time withSNR = − . In the MIMO case (bottom) we plot the averageMSE at step , over executions, for each SNR.IV. C ONCLUSSIONS AND FUTURE WORK
Adaptive strategies based on the Weiss-Weinstein boundoutperform some common channel selections for DoA es-timation. The biggest concern is the computational time of Some preliminary analyses show this might be due to a “more stable”behavior of the closed-loop that combines the optimal channel selection andthe particle filter with regards to aliasing. policy evaluation at the controller, which for DoA estimationof a single target can be greatly reduced by fitting the outputof the particle filter by a Gaussian, and also the numberof candidate subsets of channels. Future work also includestarget dynamics and estimation of model parameters such asthe SNR or the Doppler frequency, and employing multi-stepahead predictions. R
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