Sparse Array Transceiver Design for Enhanced Adaptive Beamforming in MIMO Radar
Syed A. Hamza, Weitong Zhai, Xiangrong Wang, Moeness G. Amin
SSparse Array Transceiver Design for EnhancedAdaptive Beamforming in MIMO Radar
Syed A. Hamza , Weitong Zhai , Xiangrong Wang and Moeness G. Amin School of Engineering, Widener University, Chester, PA 19013, USA School of Electronic and Information Engineering, Beihang University, Beijing, 100191, China Center for Advanced Communications, Villanova University, Villanova, PA 19085, USAEmails: { shamza } @widener.edu, { wtzhai, xrwang } @buaa.edu.cn, { moeness.amin } @villanova.edu Abstract —Sparse array design aided by emerging fast sensorswitching technologies can lower the overall system overheadby reducing the number of expensive transceiver chains. Inthis paper, we examine the active sparse array design enablingthe maximum signal to interference plus noise ratio (MaxS-INR) beamforming at the MIMO radar receiver. The proposedapproach entails an entwined design, i.e., jointly selecting theoptimum transmit and receive sensor locations for accomplishingMaxSINR receive beamforming. Specifically, we consider a co-located multiple-input multiple-output (MIMO) radar platformwith orthogonal transmitted waveforms, and examine antennaselections at the transmit and receive arrays. The optimumactive sparse array transceiver design problem is formulatedas successive convex approximation (SCA) alongside the two-dimensional group sparsity promoting regularization. Severalexamples are provided to demonstrate the effectiveness of theproposed approach in utilizing the given transmit/receive arrayaperture and degrees of freedom for achieving MaxSINR beam-forming.
Keywords — MIMO radar, sparse array transceiver, adaptivebeamforming, group sparsity, successive convex approximation
I. I
NTRODUCTION
Non-uniform sparse arrays have emerged as an effectivetechnology which can be deployed in various active and pas-sive sensing modalities, such as acoustics and radio frequency(RF) applications, including GPS [1]. Fundamentally, sparsearray design seeks optimum system performance, under bothnoise and interference, while being cognizant of the limitationson cost and aperture. Within the RF beamforming paradigm,maximizing the signal to interference plus noise ratio (SINR)amounts to concurrently optimizing the array configurationand beamforming weights. Equivalently, the problem can becast as continuously selecting the best antenna locations todeliver MaxSINR under time-varying environment. In lieu ofconstantly moving antennas to new positions, a more practicaland feasible approach is to have a uniform full array andthen switch among antennas with a fixed number of front-endchains, yielding the MaxSINR sparse beamformer for a givenenvironment.The above environment-dependent approach differs fromthe environment-independent sparse array design that seeks toincreasing the spatial autocorrelation lags and maximizing thecontiguous portion of the coarray aperture for a limited number
The work by W Zhai and X Wang is supported by National Natural ScienceFoundation of China under Grant No. 62071021 and No. 61827901. of sensors. The main task, therein, is to enable DOA estimationinvolving more sources than the physical sensors [2]–[6]. Forbeamforming applications, the environment-independent de-sign criteria typically strives to achieve desirable beampatterncharacteristics such as broad main lobe and minimum side-lobe levels and frequency invariant beampattern for widebanddesign [7]–[12].Environment-dependent sparse receive beamformer strivingto achieve MaxSINR can potentially provide sparse configu-rations that improve target detection and estimation accuracy[13]–[21]. In this paper, we consider MaxSINR receive beam-former design for MIMO radar operation. The beamformerdesign, in this case, jointly selects the optimum transmitand receive sensor locations for implementing an efficientreceive beamformer. We assume that transmit sensors emitorthogonal waveforms. The MIMO radar implementing sensor-based orthogonal transmit waveforms does not benefit fromcoherent transmit processing gain achieved when using direc-tional beamforming. In this context, the proposed approach isfundamentally different from the parallel sparse array MIMObeamforming designs, which primarily rely on optimizing thetransmit sensor locations and the corresponding correlationmatrix of the transmit waveform sequence [22]–[26]. Theproposed approaches, therein, maximize the transmitted signalpower towards the perspective target locations while mini-mizing the cross-correlation of the target returns to achieveefficient receive beamforming characteristics. Moreover, ex-isting design schemes essentially pursue an uncoupled trans-mit/receive design which is in contrast to the proposed ap-proach that incorporates sparse receiver design in conjunctionwith transmit array optimization.The proposed transceiver beamforming design problem,in essence, translates to configuring the transmit/receive ar-ray with the corresponding optimum receiver beamformingweights. To this end, we pursue a data dependent approachthat jointly optimizes the beamforming sensor positions andweights. The optimization only requires the knowledge of theperspective desired source locations and does not assume anyknowledge of the interfering sources. We optimally select M t from M transmitters and N r from N receivers. Such selectionis a binary optimization problem, and is NP-hard. In order toavoid extensive computations associated with enumeration ofall possible array configurations, we apply convex relaxation.The design problem is posed as SCA with two-dimensionalreweighted mixed l , -norm penalization to jointly invoke a r X i v : . [ ee ss . SP ] F e b parsity in the transmit and receive dimensions.The rest of the paper is organized as follows: In thenext section, we state the problem formulation for the MIMObeamformer design in the case of a pre-specified array con-figuration. Section III elaborates on the sparse array designby semidefinite relaxation to jointly find the optimum sparsetransmit/receive array geometry. In the subsequent section,simulations are presented to demonstrate the offerings of theproposed sparse array transmit design. The paper ends withconcluding remarks.II. P ROBLEM F ORMULATION
We consider a MIMO radar with M transmitters and N receivers illuminating the scene through omni-directionalbeampattern. This is typically achieved through emissions oforthogonal waveforms across the transmit array. Specifically,we pursue a co-located MIMO arrangement with the transmitand receive arrays in close vicinity. As such, a far-field sourceis presented by equal angles of departure and arrival. Consider K target sources arriving from { θ s,k } . The received basebanddata x ( n ) ∈ C MN × after matched filtering at the N elementuniformly spaced receiver array at time instant n is given by, x ( n ) = K (cid:88) k =1 s k ( n ) b ( θ s,k )+ L c (cid:88) l =1 c l ( n ) b ( θ j,l )+ Q (cid:88) q =1 i q ( n )+ v ( n ) , (1)where, s k ( n ) ∈ C is the k th reflected target signal. Theextended steering vector b ( θ ) of the virtual array is b ( θ ) = a t ( θ ) ⊗ a r ( θ ) . In the case of uniform transmit and receivelinear arrays with a respective inter-element spacing of d t and d r , the transmit and receive steering vectors are given by, a t ( θ ) = [1 e j π ( d t /λ ) cosθ . . . e j π ( M − d t /λ ) cosθ ] T , (2)and a r ( θ ) = [1 e j π ( d r /λ ) cosθ . . . e j π ( N − d r /λ ) cosθ ] T . (3)The variance of additive Gaussian noise v ( n ) ∈ C MN × is σ v at the receiver output. There are L c interferences c l ( n ) mimicking target reflected signal and Q narrowbandinterferences i q ( n ) ∈ C MN × . The latter is defined as theKronecker product of the receiver steering vector a r ( θ i,q ) andthe matched filtering output of the interference j q ( n ) such that i q ( n ) = j q ( n ) ⊗ a r ( θ i,q ) . The received data vector x ( n ) is thenlinearly combined to maximize the output SINR. The outputsignal y ( n ) of the optimum beamformer for MaxSINR is givenby [27], y ( n ) = w H x ( n ) (4)where w is the beamformer weight at the receiver. The optimalsolution w o can be obtained by solving the optimizationproblem that seeks to minimize the interference power atthe receiver output while preserving the desired signal. Theconstraint minimization problem can be cast as,minimize w ∈ C MN w H R x w , s.t. w H R s w = 1 , (5)where the source correlation matrix is R s = (cid:80) Kk =1 σ s,k b ( θ s,k ) b H ( θ s,k ) , with σ s,k = E { s k ( n ) s Hk ( n ) } denoting theaverage received power from the k th target return. The data correlation matrix, R x ≈ (1 /T ) x ( n ) x ( n ) H , is directly esti-mated from the T received data snapshots. The solution to theoptimum weights in (5) is given by w o = P { R − x R s } , withthe operator P { . } representing the principal eigenvector ofthe input matrix. This optimum solution yields the MaxSINR,SINR o , given by [27],SINR o = Λ max { R − n R s } , (6)which is the MaxSINR is the maximum eigenvalue ( Λ max )of the product of the two matrices, the inverse of interferenceplus noise correlation matrix and the desired source correlationmatrix. It is clear that the resulting solution holds irrespectiveof the array configuration, whether the array is uniform orsparse. For the latter, the performance of MaxSINR beam-former is intrinsically tied to the array configuration. Thesparse optimization of the above formulation is explained inthe next section.III. S PARSE ARRAY DESIGN
The separate sensor selection problem for a joint transmitand receive design is a combinatorial optimization problemand can’t be solved in polynomial time. We formulate thesparse array design problem by exploiting the structure ofthe received signal model and solve it by applying sequentialconvex approximation. To exploit the sparse structure of thejoint transmit and receive sensor selection, we introduce a twodimensional l , -mixed norm regularization to recover groupsparse solutions. One dimension pertains to the transmittersparsity, whereas the other sparsity pattern is associated withthe receiver side. Moreover, this two-dimensional sparsitypattern is entwined and coupled, meaning that when onetransmitter is discarded, all N receiving data pertaining to thistransmitter is zero. Similarly, the receiver is discarded onlywhen its received data corresponding to all the M transmittersis zero. The structure of the optimal sparse beamformingweight vector is elucidated in (7), where (cid:88) denotes a sensorlocation activated, or selected, and × denotes a sensor notactivated. w (1 ,N ) (cid:122) (cid:125)(cid:124) (cid:123) (cid:88) ×× (cid:88) ... × (cid:124) (cid:123)(cid:122) (cid:125) (Tx 1 active) w ( N +1 , N ) (cid:122) (cid:125)(cid:124) (cid:123) (cid:88) ×× (cid:88) ... × (cid:124) (cid:123)(cid:122) (cid:125) ( Tx 2 active) w (2 N +1 , N ) (cid:122) (cid:125)(cid:124) (cid:123) ×××× ... × (cid:124) (cid:123)(cid:122) (cid:125) ( Tx 3 inactive) . . . w ( N ( M − ,MN ) (cid:122) (cid:125)(cid:124) (cid:123) (cid:88) ×× (cid:88) ... × (cid:124) (cid:123)(cid:122) (cid:125) ( Tx M active) (7)In (7), each column vector denotes the receive beamformerweights for a fixed transmit location. It is noted that the optimalsparse beamformer, corresponding to the active transmit andreceive locations, follows a group sparse structure along thetransmit and receive dimensions (the consecutive N sensorsare discarded vertically or the consecutive M sensors arediscarded horizontally. It is evident that the missing transmitsensor at position , for example, translates to the sparsityalong all the corresponding N entries of w ( N consecutive × vertically in (7)). Similarly, the group sparsity is also invokedacross the received signals corresponding to all transmitters( M consecutive × horizontally in (7)). . Group Sparse solutions through SCA The problem in (5) can equivalently be rewritten by swap-ping the objective and constraint functions as follows,maximize w ∈ C MN w H ¯ R s w s.t. w H R x w ≤ (8)where ¯ R s = − R s . The sparse MIMO configuration ofuniformly spaced receivers and transmitters with a respectiveinter-element spacing of d r and d t = N d r is employed fordata collection. The covariance matrix R x of the full receivevirtual array can be then obtained by the matrix completionmethod proposed in [17].The beamforming weight vectors are generally complexvalued, whereas the quadratic functions are real. This obser-vation allows expressing the problem with only real variableswhich is typically accomplished by replacing the correlationmatrix ¯ R s by ˜ R s and concatenating the beamforming weightvector accordingly [28], ˜ R s = (cid:20) real ( ¯ R s ) − imag ( ¯ R s ) imag ( ¯ R s ) real ( ¯ R s ) (cid:21) , ˜ w = (cid:20) real ( w ) imag ( w ) (cid:21) (9)Similarly, the received data correlation matrix R x is replacedby ˜R x . The problem in (8) then becomes,minimize ˜w ∈ R MN ˜w (cid:48) ˜ R s ˜w , s.t. ˜w (cid:48) ˜R x ˜w ≤ , (10)where (cid:48) denotes transpose operation. After expressing theconstraint in terms of real variables, we convexify the objectivefunction by utilizing the first order approximation iteratively,minimize ˜w ∈ R MN m ( k ) (cid:48) ˜w + b ( k ) , s.t. ˜w (cid:48) ˜R x ˜w ≤ , (11)where, m ( k ) and b ( k ) , updated at the k + 1 iteration, aregiven by m ( k +1) = 2 ˜ R s ˜w ( k ) , b ( k +1) = − ˜w ( k ) (cid:48) ˜ R s ˜w ( k ) ,respectively. Finally, to invoke sparsity in the beamformingweight vector, the re-weighted mixed l , norm is adoptedprimarily for promoting group sparsity,minimize ˜w , c , r m ( k ) (cid:48) ˜w + b ( k ) + α ( p (cid:48) c ) + β ( q (cid:48) r ) (12)s.t. ˜w (cid:48) ˜R x ˜w ≤ , (12a) || P i (cid:12) ˜w || ≤ c i , (12b) ≤ c i ≤ , i = 1 , ..., M (12c) || Q j (cid:12) ˜w || ≤ r j , (12d) ≤ r j ≤ , j = 1 , ..., N (12e) (cid:48) M c = M t , (12f) (cid:48) N r = N r , (12g)and P i = [ N elements st group (cid:122) (cid:125)(cid:124) (cid:123) ... ... ... N elementsith group (cid:122) (cid:125)(cid:124) (cid:123) ... ... ... N elements ( M + i ) th group (cid:122) (cid:125)(cid:124) (cid:123) ... ... ... N elements Mth group (cid:122) (cid:125)(cid:124) (cid:123) ... ... (cid:48) , (13) Q j = [ N elements ofthe st group (cid:122) (cid:125)(cid:124) (cid:123) ... (cid:124) (cid:123)(cid:122) (cid:125) ( j −
1) 0 s ... . . . N elements ofthe Mth group (cid:122) (cid:125)(cid:124) (cid:123) ... (cid:124) (cid:123)(cid:122) (cid:125) ( j −
1) 0 s ... (cid:48) . (14) Algorithm 1
SCA-Sparse Transmit/Receive Beamformer De-sign
Input: M , N , M t , N r , α = β = 0 . , α = β = 1 , targetdirection θ s , auto-correlation matrix R s and R x Output:
Optimal weight w , location and number of receiversand transmitters. Initialization:
Initialize w , p , q , m as all ones matrix. Set α = 1 and β = 1. Initialize the tradeoff parameters α and β accordingto the sparsity requirement. while || ˜w ( k +1) − ˜w ( k ) || ≥ − do a) Convert w , R s and R x to the real domain to get ˜w , ˜R s and ˜R x according to Eq.(9).b) Update p ( k +1) and q ( k +1) according to Eq.(13)c) Update ˜w ( k +1) according to Eq.(12).d) Convert ˜w ( k +1) back into the complex solution w ( k +1) by w ( k +1) ( i ) = ˜w ( k +1) ( i )+ j ˜w ( k +1) ( i + M N ) . end whilereturn w Here, (cid:12) means the element-wise product, c and r are twoauxiliary binary selection vectors, P i ∈ { , } MN in (12b)is the transmission selection matrix, which is used to selectthe real and imaginary parts of the N weights correspondingto the i th transmitter, as shown in (13). Eqs. (12c) and (12f)indicate that we can select M t transmitters at most. Similarly, Q j in (12d) is the receiver selection matrix, as shown in (14).Eqs. (12e) and (12g) indicate that we can select N r receiversat most. The two parameters α and β are used to controlthe sparsity of the transmitters and the receivers, p and q are the reweighting coefficient vectors for the transmitters andreceivers respectively and the detailed update method is givenbelow. B. Reweighting Update
A common method of updating the reweighting coefficientis to take the reciprocal of | w | [29]. However, this can’t controlthe number of elements to be selected. Thus, similar to [30],in order to make the number of selected antennas controllable,we update the weights using the following formula, p ( k +1) i = 1 − c ( k ) i − e − β c ( k ) i + (cid:15) − ( 1 (cid:15) )( c ( k ) i ) α (15)Here, α and β are two parameters that control theshape of the curve, and the parameter (cid:15) avoids the unwantedexplosive case. The essence of this re-weight update methodis to take a large positive penalty for a entry close to zeroand a small negative reward for the entry close to 1. As aresult, the entries of the two selection vectors c and r tendto be either 0 or 1. The update of the re-weight vector q follows the same rule. Through iterative regression, we can find M t transmitters and N r receivers. In addition, by controllingthe values of α and β , we can obtain different sparsities oftransmitters and receivers [31]. The proposed algorithm forjoint transmit/receive beamformer design is elaborated furtherin Algorithm 1.V. S IMULATIONS
In this section, we demonstrate the effectiveness of ourproposed sparse MIMO radar from the perspective of outputSINR. We compare the performance of the optimal MIMOarray transceiver configured according to the proposed algo-rithm with randomly configured MIMO arrays. In practice,the interference may be caused by co-existence in the samebandwidth or being deliberately positioned at some anglestransmitting the same waveform as targets of interest. Weconsider both kinds of interferences.
A. Example 1
In this example, we fixed the direction of the interferences,and change the arrival angle of the target from ◦ to ◦ .We consider a full uniform linear MIMO array consistingof 8 transmitters (M = 8) and 8 receivers (N = 8). Weselect M t = 4 transmitters and N r = 4 receivers. For thereceiver, we set the minimum spacing of the sensors to λ/ ,while for the transmitter, we set the minimum spacing of thesensors to N λ/ . Suppose there are two interferences with aninterference to noise-ratio (INR) of 13dB and arrival anglesof θ q = [40 ◦ , ◦ ] . The SNR of the desired signal is fixedat 20dB. We simulate co-existing interferences and deliberateinterferences. The output SINR versus target angle is shownin Fig.1. In this figure, case 1 corresponds to two deliber-ate interferences, whereas case 2 represents two co-existinginterferences. It can be seen that, in both cases, the optimalsparse MIMO arrays exhibit better performance than randomlyselected sparse arrays. Fig.2, depicts the configuration of theoptimal MIMO array for a target at ◦ . B. Example 2
In this example, we consider the scenario where the in-terferences are spatially close to the target which causes agreat adverse impact on the array performance. We change theangle of the target from ◦ to ◦ . For each target angle, twointerferences are generated from the proximity of ± ◦ awayfrom the target. We consider a full uniform linear MIMO arrayconsisting of 20 transmitters (M = 20) and 20 receivers (N =20). We select 5 transmitters and 5 receivers to compose thesparse MIMO array. The other simulation parameters remainthe same as those in example 1. Again, we plot the outputSINR versus target angle, as shown in Fig. 3. In case 1,the two interferences are deliberate, whereas in case 2, bothinterferences are co-existing. It can be observed that when theinterferences are spatially close to the target, the proposedoptimal MIMO sparse array exhibits more superiority thanrandomly configured sparse MIMO arrays.V. C ONCLUSION
In this paper, the problem of sparse transceiver designfor MIMO radar was considered. For a given number oftransmitting and receiving sensors, a sparse MIMO arraystructure in terms of MaxSINR was jointly designed. A mixed-norm reweighted regularization was utilized to promote two-dimensional sparsity. It was shown by simulation examplesthat the proposed sparse MIMO transceiver selection algorithmprovides superior interference suppression performance to ran-domly designed array.
Azimuth ( ° ) O u t pu t S I NR ( d B ) Mt=4,Nr=4 (case 1):Optimal transceiverMt=4,Nr=4 (case 1):Random transceiverMt=4,Nr=4 (case 2):Optimal transceiverMt=4,Nr=4 (case 2):Random transceiver
Fig. 1. Relationship between output SINR and target angle with the directionsof interferences being fixed.Fig. 2. Optimal MIMO transceiver configuration when the target is from ◦ . Azimuth ( ° ) O u t pu t S I NR ( d B ) M t =5,N r =5 (case 1): Optimal transceiverM t =5,N r =5 (case 1): Random transceiverM t =5,N r =5 (case 2): Optimal transceiverM t =5,N r =5 (case 2): Random transceiver Fig. 3. Relationship between the output SINR and the target angle when theinterferences are closed to the target.
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