Yang Zhang, Xinmin Wang School of Automation, Northwestern Polytechnical University, Xi an , China

Transcription

1 oi:1.1311/ Robust Aaptive Beamforming ith Null Broaening Yang Zhang, inmin ang School of Automation, Northestern Polytechnical University, i an 717, China Abstract Aaptive beamformers are sensitive to mismatch of the array steering vector of the esire signal an motional interference. In particular, hen the esire signal is present in training snapshots, even a small error can lea to a serious egraation in performance. In this paper, a ne robust aaptive beamforming algorithm via null broaening is investigate, hich is obtaine by reconstructing an imizing the interference-plus-noise covariance matrix, an estimating the true steering vector to improve the robustness against array vector errors an motional interference. Simulation results emonstrate that the performance of propose algorithm outperforms the existing robust aaptive beamformers, an the interference-plus-noise ratio (SINR) is close to imal values. Key ors: Aaptive Beamformer, Array Vector Errors, Null Broaening, otional Interference 1. INTRODUCTION Aaptive beamforming is a ubiquitous task in array signal processing an has been iely use in sonar, raar, acoustics, seismology, communications an so on (Jian Li,3; Elnashar,6; Bucris,1). It is ell knon that even in the ieal case here the signal steering vector is precisely knon at the receiving sensor array, the presence of esire signal in the training snapshots can lea to severe performance egraation of aaptive beamforming algorithm (Du, 1; Gu, 1). In aition, a major cause of performance egraation of the algorithm is ue to motional interferences an vibration of array platforms, especially hen interferences move fast, the eight vector of aaptive beamforming algorithm can't aapt to nonstationary signals quickly, resulting in a sharp performance egraation of the algorithm. Therefore, robust aaptive beamforming has been an intensive research topic, an various robust techniques have been propose in recent years (Arash, 1; Yang, 13). In general, these robust methos can be classifie into to categories. The first category is mainly imizing the interference-plus-noise covariance matrix, hich is because the exact interference-plus-noise covariance matrix is usually unknon at the receiving sensor array. In this category, the most popular one is iagonal loaing technique (Elnashar, 5),here a scale ientity matrix is ae to the sample covariance matrix, an Robust Capon beamforming (RCB) propose in(jian Li,3) can be the classical representative. Recently, a ne metho of interference-plus-noise covariance matrix reconstruction (Gu, 1) is propose, here it attempts to reconstruct the interference-plus-noise covariance matrix instea of searching for the imal iagonal loaing factor for the sample covariance matrix, an this metho is shon to be more robust than using the sample covariance matrix. oever, these approaches are unavailable for interference ith fast movement. Therefore, another category of approaches simply processes motional interference. An efficient ay to suppress motional interference is null broaening, hich makes DOA of motional interference alays into the null in the beam pattern. ailloux (ailloux, 1995) an Zatman (Zatman, 1995) have inepenently evelope techniques that impart robustness into aapte patterns by juicious choice of null placement an ith, an Guerci (Guerci, 1999) unifie the to approaches through an introuction of the concept of a Covariance atrix Taper (CT). The CT technique is realize by moifying the interference covariance matrix (usually a sample covariance) via a aamar prouct ith a positive (semi) efinite matrix. oever, there are still to isavantages of this technique hich are higher sielobe gain of beam pattern an shalloer null epth. To aress the problems of CT, e propose a ne robust aaptive beamforming algorithm ith null broaening for motional interference in this paper. Instea of CT, the propose metho is performe by expaning the stanar Capon spatial spectrum, then reconstructing the interference-plus-noise covariance matrix, an finally estimating the true steering vector to improve the robustness against array steering vector errors. Simulation results emonstrate the correctness an effectiveness of the propose algorithm.. SIGNAL ODEL Consier that one esire signal t () an J narroban interferers i ( t), j 1,, J in the far fiel, impinge on a uniform linear array (ULA) consisting of sensor elements ith Directions-Of-Arrival (DOA) an respectively. n () t is the aitive hite noise on the array element k. j k j 414

2 Therefore, the receive signal on the array can be moele as x ( t) ( t) n ( t) 1 1 x ( t) i ( t) n ( t) [ a( ), a( ),, a( )] 1 J x ( t) i ( t) n ( t) J here a( ) an a( ),( j 1,, J) are steering vectors for esire signal an interference signals. The sample j covariance matrix R of receive signal array is given by here ( t) x ( t) x ( t) x ( t) 1 T is the sample matrix of receive signals. R = E( ( t) ( t)) () The common formulation of Stanar Capon beamforming (SCB) problem can be expresse as the folloing imization moel here is the eight vector. The imal solution min R st. a ( ) 1 (3) to the above imization problem is given by (1) a 1 Ra( ) 1 ( ) R a( ) (4) 3. TE PROPOSED ALGORIT It is often esirable to ien the null ith associate ith an interferer in an aapte pattern to impart robustness into the realize aapte pattern. The CT null broaening approach is realize by the folloing moification of the original covariance (sample covariance) matrix(guerci,1999) Rˆ Rˆ T (5) here R ˆ is the sample covariance, onates aamar prouct, an T is the extension matrix. Obviously, T etermines the null ith an epth of the beam pattern. oever, the T must vary ith ifferent array geometries an interference environments, an the imension of T increases ith the increasement of array elements, hich leas to more an more complex operation in (5). oreover, the CT metho has the isavantages of a loer array gain an shalloer null epth. ence, this metho ill not be use in our formulation. In this section, a ne aaptive algorithm ith null broaening for motional interference is evelope. The basic iea of the propose algorithm is to construct the extension Capon spatial spectrum first, hich is realize by the convolution beteen the stanar Capon spatial spectrum an a specific ino function, subsequently reconstruct interference-plus-noise covariance matrix ith the extension Capon spatial spectrum, hich makes the esire signal filtere out effectively, an finally utilize RCB algorithm to estimate the actual steering vector to improve the robustness against mismatch of the array steering vector of the esire signal The Novel Null Broaening Techniques As is knon to all, Capon spatial spectrum reflects the spatial response characteristics of aaptive beamformers, an has sharp irectional response pattern in the irection of interference. Therefore, if e just a the metho of interference covariance matrix reconstruction propose by(gu, 1), the beam pattern can only form a narro null in the irection of interference. Thus, in orer to achieve the ie null in the beam pattern, e propose to construct the extension spatial spectrum P ( ) first, an P ( ) is given by P ( ) P( ) ( ) (6) 1 P( ) (7) a R a 1 ( ) ( ) here P( ) is the stanar Capon spatial spectrum given by (7), a( ) is the steering vector associate ith a hypothetical irection base on the knon array structure, ( ) is the ino function, an onates 415

3 convolution. ere e a Blackman ino function to broa the Capon spatial spectrum, an ( ) is escribe as ( ).4.5cos ( ) +.8cos (4 ) (8) here,, is the ino ith, hich also onates the null ith. The null ith ajustment can be achieve by setting ifferent values of. Fig.1 shos simulation results of ifferent P ( ), hen varies. e can see that the stanar Capon spatial spectrum P( ) forms sharply narro peaks in the signal irections, hile the extension spatial spectrum P ( ) forms ie peaks in the same irections. oreover, the peak values of P ( ) are coincient ith P( ), hich means that the null epth of the beam pattern ith this metho ill be consistent ith that of stanar Capon beamformer. After constructing P ( ), e ill use (9) to reconstruct the interference-plus-noise covariance matrix R in (Yujie Gu,1). R P ( ) ( ) ( ) in a a (9) here is the sielobe omain of beam pattern. In practical applications, the choice of is not critical but only ensure that DOAs of all interferences are locate except the esire signal. ere, assuming is the mainlobe angle sector of the beam pattern, in hich the esire signal is present. Thus, covers the hole spatial omain, an is empty. Figure 1. The extension Capon spatial spectrum 3.. Optimal Steering Vector Estimation To improve the robustness of the propose algorithm against mismatch of the array steering vector of the esire signal, e utilize RCB metho to estimate the actual steering vector. RCB is base on the uncertainty set of the steering vector to achieve the imal steering vector estimation, an the metho can be escribe as the folloing imization problem (Jian Li,3) -1 min a R a a (1) st.. a a = here a an a are the imal steering vector an esire steering vector separately, is a constant value, an.the imal solution of RCB is calculate by a Lagrange multiplier methoology, an the imal steering vector solution can be expresse as(jian Li,3) R a 1 I a here is the Lagrange multiplier (Jian Li,3). Then, the key to this solution is to get the Lagrange multiplier. For this purpose, e take (11) into the secon equality of (1). Thus, the is obtaine as folloings 1 g( ) ( ) (11) I R a (1) 416

4 Then, let s make eigenvalue ecomposition of the covariance matrix R R UΛU (13) here the columns of U contain the eigenvectors of R, an Λ iag(,, ) 1, are the 1 corresponing eigenvalues. Let Then, (1) can be ritten as z U a (14) zm g( ) (15) m1 (1 ) here z m is the m th element of z. Then, the solution to (15) can be solve by a Neton-like metho. Thus, the term is use in (11) to obtain the imal steering vector a, an the imization problem of (1) is resolve. Up to no, both the reconstructe interference-plus-noise covariance matrix R an the imal steering vector a have been obtaine. Substituting them into (4), the imal eight vector m 1 in R a 1 a R a in in can be expresse as To summarize, The propose algorithm is summarize as follos. step1: Compute the extension Capon spatial spectrum P ( ), an use P ( ) to reconstruct the interferenceplus-noise covariance matrix R -1. in step: Estimate the imal steering vector a ith RCB metho. step3: Use (16) to calculate the eight vector base on the reconstructe interference-plus-noise covariance matrix 4. SIULATIONS -1 R an the imal steering vector a. in e provie numerical simulation results in this section. In our simulations, a ULA ith =5 elements space a half avelength is consiere. To interfering sources are assume to have DOAs of -6 an 4, ith corresponing interference-to-noise ratio (INR) are 4B an 5B respectively. The presume esire signal impinges on the array from irection =. In the first example, the propose algorithm is compare ith SCB, RCB an CT methos. In the propose algorithm, the general angular location of the esire signal is set to be [ 3 3 ] an the corresponing sielobe omain is assume to be ithin the interval [ 9 3 ] [3 9 ], an is equal to (Jian Li,3). In this simulation, the sample covariance matrix is compute base on N 5 ata snapshots, an the input signal-to-noise ratio (SNR) is set to be B. Fig. shos beam patterns in the presence of a irection mismatch, hich means that the actual esire signal comes from a irection of. It can been foun that the beam pattern ith propose algorithm an RCB metho form peaks in the actual esire signal DOA of, hile CT an SCB methos form nulls in the same irection. This is because the steering vector estimation is obtaine in our metho an RCB, hile that oes not occur in the other to methos. In aition, from the perspective of null epth an sielobe gain, the propose algorithm has eeper nulls an much loer sielobe gain than CT. Next, e compare beam patterns versus input SNR. ere e employ a fixe snapshots value N 5. The Comparison of beam patterns at ifferent SNR is shon in Fig.3. It can be seen that the sielobe of the beam patterns ith our metho keeps a stable value less than -4B, an peaks are alays forme in the main angle sector. That is to say, our metho ill scarcely be affecte by the changes of input SNR. hile the CT metho forms nulls in the esire signal irection, ith the increasement of SNR. In aition to the beam pattern analysis above, e conuct quantitative analysis mainly from the perspective of mathematical statistics. In the next examples, the propose beamformer is compare to SCB, iagonal loaing (LSI), RCB, interference-plus-noise construction (Reconstruction), an CT. In the propose algorithm an Reconstruction In LSI metho, the iagonal metho, the sielobe angular sector is set to be (16) 417

5 loaing factor is 1, here 1is the noise variance. In CT, the is set to be.1 (ailloux,1995). All of the folloing results are calculate base on 3 onte Carlo experiments. The output SINR versus input SNR is shon in Fig.4.(a). In the simulation, e employ N 5 snapshots an consier a fixe irection mismatch of. It can be observe that the output SINR of 6 methos are consistent at lo SNR, meanhile SCB, LSI, RCB an CT suffer from saturation effects ith the increase of SNR. hile the Reconstruction an propose algorithm can maintain goo SINR performances an achieve near the imal values, since the reconstructe interference-plus-noise covariance matrix provie a quasisignal-free environment. In the SCB, LSI an CT methos, because of the presence of the esire signal an irection mismatch, the actual signal is consiere as an interference to be suppresse, hich results in the signal cancellation phenomenon. In aition, ith the increasement of SNR, the output SINR performance of RCB is better than SCB, LSI an CT ue to its imal steering vector estimation against the irection mismatch. (a) (b) Figure.. (a) beam pattern ith a irection mismatch of ; (b) zoome-in beam patterns of (a) (a) (b) (c) Figure.3. (a) Comparison of beam pattern ith SNR=-B; (b) Comparision of beam pattern ith SNR=B; (c) Comparision of beam pattern ith SNR=B; 418

6 (a) (b) (c) Figure.4. (a) output SINR versus the input SNR ith a irection mismatch of ; (b) output SINR versus number of snapshots; (c) output SINR versus angle vibration. (The presume irection of the esire signal is ) In Fig.4.(b), the output SINR versus number of snapshots for the input SNR fixe at 5B is shon. e can see that the performance of the propose algorithm an Reconstruction metho almost remain a constant value close to the imal value, hereas that is much loer ith the other four methos. Therefore, the SINR performance of SCB, LSI, SCB an CT methos is unsatisfactory uner the conition of ifferent snapshots. Furthermore, the propose algorithm an Reconstruction metho enjoys much faster convergence rate than other methos, hen the number of snapshots is poor. Fig.4.(c) illustrates the output SINR of the six methos hen the DOAs of the interferers vibrates. Take the DOA of interference as the center an 6 as the maximum offset angle. Simulation emonstrates that the propose algorithm an CT outperform all the other methos ith the increase of angle offset. oreover, the propose metho has better SINR performance than CT, hich is because the latter has isavantages of a loer array gain an shalloer null epth. Base on the simulation results, it has been foun that the output SINR performance of propose algorithm outperforms the other methos. 5. CONCLUSIONS In this paper, e evelop a novel aaptive beamforming algorithm ith null broaening. In the propose metho, the Capon spatial spectrum is expane firstly by convolving ith Blackman ino function to achieve the null iening. Subsequently, the expane spatial spectrum is use to reconstruct the interferenceplus-noise covariance matrix, hich makes the esire signal filtere out effectively. An finally e make use of RCB metho of imal steering vector estimation to improve the robustness against mismatch of the array steering vector of the esire signal. Simulation results emonstrate that the propose algorithm has excellent output SINR performance an goo robustness against irection mismatch an angle vibration. REFERENCES Jian Li, P.Stocia, Z.ang (3) On Robust Capon Beamforming an Diagonal Loaing, IEEE International Conference on Acoustics Speech an Signal Processing, pp A.Elnashar, Sai. Elnoubi, an ami A.(6) Further Stuy on Robust Aaptive Beamforming ith Optimum Diagonal Loaing, IEEE Transaction on Antennas an Propagation, 54(1), pp Yaakov Bucris,iriam A.(1) Bayesian Focusing for Coherent ieban Beamforming, IEEE Transaction on Auio Speech an Language Processing, (4), pp Lin.D, J Li.(1) Fully Automatic Computation of Diagonal Loaing Levels for Robust Aaptive Beamforming, IEEE Transaction on Aerospace an Electronic Systems, 46(1), pp Y. Gu, A. Leshem.(1) Robust Aaptive Beamforming Base on Interference Covariance atrix Reconstruction an Steering Vector Estimation, IEEE Transaction on Signal Processing, 6(1), pp

7 Arash K an A. Vorobyov (1) Robust Aaptive Beamforming Base on Steering Vector Estimation ith as Little as Possible Prior Information, IEEE Transaction on Signal Processing, 6(6), pp T. Yang, T. Su,.T. Zhu an L. Zhang (13) Robust Aaptive Beamforming using Beamspace Steering Vector Estimation, Electronics Letters, 49(19), pp A.Elnashar, S.Elnoubi an.elmikati (5) Robust Aaptive Beamforming ith Variable Diagonal Loaing, IEE international Conference on 3G&Beyon, pp.1-5. R J. ailloux (1995) Covariance atrix Augmentation to Prouce Aaptive Array Pattern Troughs, Electronic Letters, 31(1), pp Zatman (1995) Prouction of Aaptive Array Troughs by Dispersion Synthesis, Electronic Letters, 31(5), pp J R. Guerci (1999) Theory an Application of Covariance atrix Tapers for Robust Aaptive Beamforming, IEEE Transaction on Signal Processing, 47(4), pp