Robust Waveform Optimization for MIMO Radar to Improve the Worst-Case Detection Performance

Transcription

1 Journl of Communictions Vol., No., December 5 obust Wveform Optimiztion for MIMO dr to Improve the Worst-Cse Detection Performnce ongfeng Wng nd ongn Wng chool of Computer cience nd echnolog, Zhoukou Norml Universit, Zhoukou, 466, Chin College of Informtion Engineering, Dlin Universit, Dlin, 66, Chin Emil: {cnhfwng, gglongs}@63.com Abstrct Wveform optimiztion for Multi-Input Multi- Output (MIMO) rdr, which usull depends on the initil prmeter estimtes (i.e., some prior informtion on the trget of interest nd scenrio), is often sensitive to estimtion errors nd uncertint in the prmeters. obust wveform design ttempts to sstemticll llevite this sensitivit b eplicitl incorporting prmeter uncertint model in the optimiztion problem. In this pper, we consider the robust wveform optimiztion to improve the worst-cse detection performnce over conve uncertint model. An itertive lgorithm is proposed to optimize the Wveform Covrince Mtri (WCM) for mimizing the worst-cse output ignl-interference- Noise-tio (IN) such tht the worst-cse detection performnce cn be improved. Ech itertion step in the proposed lgorithm cn be reformulted s emidefinite Progrmg (DP) problem, which cn be solved ver efficientl. Numericl results show tht the worst-cse detection performnce cn be improved considerbl b the proposed method compred to those of the non-robust method nd uncorrelted wveforms. Inde erms Multi-Input Multi-Output (MIMO) rdr, robust wveform optimiztion, conve optimiztion, trget detection, emidefinite Progrmg (DP) I. INODUCION Multiple-Input Multiple-Output (MIMO) rdr is n emerging technolog tht hs ttrcted gret from both the cdemic nd industril communities for lmost decde []. MIMO rdr cn emplo multiple trnsmitting elements to trnsmit rbitrr wveforms other thn coherent wveforms in trditionl phsed-rr rdrs, which is the so-clled wveform diversit. wo ctegories of MIMO rdr sstems cn be clssified b the configurtion of the trnsmitting nd receiving ntenns: ) MIMO rdr with widel seprted ntenns, nd ) MIMO rdr with colocted ntenns []. For MIMO rdr with widel seprted ntenns, the trnsmitting nd receiving elements re widel spced such tht ech views different spect of the trget. imilr to the multipth diversit concept in wireless communiction over fding chnnels, this tpe of MIMO rdr cn eploit the wveform diversit to overcome performnce degrdtions cused b trget scintilltions. Mnuscript received April 3, 5; revised December, 5. his work ws supported b NFC under Grnt Corresponding uthor emil: cnhfwng@63.com. doi:.7/jcm In contrst, MIMO rdr with colocted ntenns, whose elements in trnsmitting nd receiving rrs re close enough such tht the trget dr Cross ections (Cs) observed b MIMO rdr re identicl cn utilize the wveform diversit to increse the virtul perture of the receiving rr. Accordingl, it hs severl dvntges, for emple, more fleibilit for trnsmit bempttern design (e.g., []). For both tpes of MIMO rdr, prticulrl criticl issue is the wveform optimiztion (see, e.g., [] for more detils). According to the trget model used in the problem of wveform design, the current design methods cn be divided into two ctegories: ) point trget bsed design []-[4], nd ) etended trget bsed design [5]- [7]. Focusing on point trgets, the corresponding methods optimize the Wveform Covrince Mtri (WCM) []. he methods optimizing the WCM onl consider the sptil do chrcteristics of the trnsmitted signls, while the pproches optimizing the rdr mbiguit function tret the sptil, rnge, nd Doppler do chrcteristics jointl. In the cse of etended trgets, the prior informtion bout the trget nd noise re used to design the trnsmitted wveforms. o improve the detection performnce of MIMO rdr, one w is detector design which ws investigted in [8]. In [8], Awis K. et l. derived detector sttistic for nullspce projected nd orthogonl wveform, nd studied the performnce for both wveforms. Another w to improve the detection performnce of MIMO rdr is wveform optimiztion, which hs been studied in [3]-[7]. In [3],. Wng et l. designed the trnsmitted wveforms to improve the detection performnce of MIMO pce-ime Adptive Processing (AP) b eploiting Digonl Loding (DL) method with perfect trget nd clutter prior knowledge. owever, the informtion bout trget nd clutter used in the wveform optimiztion must be estimted with error in prctice. herefore, the robust wveform optimiztion for MIMO-AP in the cse of imperfect spce-time steering vector prior knowledge is considered to improve the worst-cse detection performnce in [4]. Bo Jiu et l. investigted the problem of knowledge-bsed MIMO rdr wveform design for trget detection in heterogeneous clutter zone [5], while B. ng et l. considered the wveform design problem for MIMO rdr in order to improve the detection performnce b 5 Journl of Communictions 983

2 Journl of Communictions Vol., No., December 5 emploing reltive entrop [6]. In [7], n itertive lgorithm is proposed to mimize the output signl-tointerference-plus-noise rtio (IN) for improving the detection performnce for etended trget. It is known tht wveform optimiztion for improving the performnce of MIMO rdr usull depends on the initil prmeter estimte (e.g., the trget loction, reflection coefficients, etc.) [3]-[7]. As sequence, the optimized wveforms depend on these pre-ssigned vlues. In prctice, these prmeters re estimted with errors, nd hence the re uncertin. As illustrted b numericl emples in [3], the resultnt output IN, i.e., detection probbilit, is sensitive to these estimtion errors nd uncertint in prmeters, which is similr to tht in the cse of wveform design for improving the prmeter estimtion performnce []. It mens tht the optimized wveforms bsed on certin prmeter estimte cn give ver low detection performnce for nother resonble estimte. In order to improve the worst-cse detection performnce, the problem of robust wveform design is ddressed in this pper, which ttempts to sstemticll llevite the sensitivit b eplicitl incorporting prmeter uncertint model in the optimiztion issue. he WCM is optimized to obtin the best worst-cse detection performnce over conve uncertint set. An itertive lgorithm is proposed to solve the optimiztion problem such tht the worst-cse performnce cn be improved. Ech step in the proposed lgorithm cn be reformulted s semidefinite progrmg (DP) problem [9], nd hence it cn be solved efficientl. he rest of this pper is orgnized s follows. ection II introduces the MIMO rdr model, nd formultes the wveform optimiztion problem incorporting the conve uncertint set of prmeters. ection III proposes n itertion lgorithm to improve the worst-cse detection performnce. ection IV shows the effectiveness of the proposed method vi numericl emples. Finll, ection V concludes this pper. hroughout the pper, mtrices nd vectors re denoted b boldfce uppercse nd lowercse letters, respectivel. We use, *, nd to represent the trnspose, conjugte, nd conjugte trnspose, respectivel. he smbol indictes the Kronecker product, I denotes the identit mtri, nd vec is the vectoriztion opertor stcking the columns of mtri on top of ech other. he nottion A stnds for the Frobenius norm of the mtri. Denote b F tr, e nd Im the trce, the rel nd imginr prt of mtri, respectivel. Finll, the nottion A B mens tht B - A is positive semidefinite. II. POBLEM FOMULAION Consider MIMO rdr sstem with M t trnsmitting elements nd M r receiving elements. Let Mt L [ s, s,, sm ] be the trnsmitted t L wveform mtri, s i, i,,, Mt denotes the discrete-time bsebnd signl of the ith trnsmitting element with L being the number of snpshots. Under the ssumption tht the probing signls re nrrowbnd nd the propgtion is non-dispersive, the signls received b MIMO rdr cn be epressed s (see, e.g., []): Y v () ( ) ( ) the columns of M r L Y re the collected dt snpshots, is the comple mplitudes proportionl to the C of the trget of interest, nd denotes those loction prmeters. he prmeters nd need to be estimted from the received signl Y. he term is the noise plus interference, whose columns cn be ssumed to be independent nd identicll distributed circulrl smmetric comple Gussin rndom vectors with men zero nd n unknown covrince denoting b Q []. Also, ( ) nd v( ) denote, respectivel, the receiving nd trnsmitting steering vectors for the trget locted t, which cn be described s ( ) [e,e,,e ] j f ( ) j f j ( ) fm( ) r v( ) [e,e,,e ] j f ( ) j f j ( ) fm( ) t f represents the crrier frequenc. m( ), m,, Mr is the propgtion time from the trget locted t to the mth receiving element, nd n( ), n,, Mt is the propgtion time from the nth trnsmitting element to the trget. / B emploing ( ) s the mtched-filter bnk t the receiver to get the sufficient sttistics for trget detection, nd the output of the filter cn be stcked in MM vector s r t () h vec( Z ) (3) / / vec( Y ( ) ), ( ) IM, r /,. h = ( ) b( ) Z Z ( ) Considering the problem of detecting trget in the cell under test, with the model in (3), this issue cn be formulted in terms of the following binr hpotheses test : vec( Z) : h vec( Z) Under the ssumption of Z bove, the joint probbilit densit function (pdf) of the received vectors conditioned on the phse of nd hpotheses cn be written s f (, ) MM r t e IQ ( IQ) (4) (5) 5 Journl of Communictions 984

3 Journl of Communictions Vol., No., December 5 f (, ) MM r t e IQ ( -h) ( IQ) ( -h) If is uniforml distributed in [, ] [], with (5) nd (6), the generlized likelihood test rtio function cn be epressed s f (, )d f (, )d is the detection threshold. After clcultion, the generlized likelihood rtio test (GL) detector for the binr hpotheses test cn be given b ' (6) (7) ( I Q ) h (8) ' is the detection threshold set ccording to desired vlue of the flse lrm probbilit ( P f ). In the cse of nonfluctuting trget, n nlticl epression of the detection probbilit ( P d ), for given vlue of P f, cn be epressed s []: d h I Q h f P Q( tr(( ) ( ) ), ln P ) (9) Q (, ) denotes the Mrcum Q function of order. B using some mtri mnipultions, (9) cn be rewritten s d Q f P Q( tr( ), ln P ) () v ( ) ( ) is the trget chnnel mtri t the considered rnge bin. It is noted tht the output signl-noise rtio (N) cn be obtined s (which cn be derived in Appendi A): N tr( Q ) () Given P f, it cn be seen from () nd () tht P d is n incresing function of the output N. As sequence, mimiztion of P d is tntmount to mimiztion of the output N. It cn be seen from () tht N is function of the loction s well s the noise plus interference. In prctice, these prmeters re estimted with errors nd so the re uncertin. ence, the optimized wveforms bsed on the N emploing prmeter estimte cn give ver low ccurc for nother resonble estimte, which cn be seen from numericl emples in [], [3]. ere, we ssume tht the trget chnnel mtri cn be modeled s: () nd denote, respectivel, the ctul nd corresponding presumed chnnel, nd is the error of, which belongs to the set (3) he robust wveform optimiztion for trget detection cn now be briefl stted s follows: Optimize the WCM to mimize the worst-cse N over the conve set under the constrints bout the WCM. Bsed on () nd (3), this optimiztion problem cn be illustrted s m N s.t., tr( ) LP, (4) P denotes the totl trnsmitted power. he third constrint holds due to the power trnsmitted b n trnsmitting element is more thn or equl to zero in prctice. III. POPOED IEAIVE MEOD In this section, we will show how to solve the robust optimiztion problem to obtin the better worst-cse detection performnce. We now tret the inner optimiztion problem firstl. It cn be seen the optimiztion vribles in the problem bove re comple. o fcilitte the solution of (4), we cn convert it to rel-vlued form in Appendi B, which is shown s nd s.t., tr( Q, F ) (5),, nd re defined in (3), in (3), Q -Q Q = (6) Q Q Q e( Q), Q Im( Q ) (7) It is obvious tht the term conve with respect to lrgest eigenvlue of equivlentl represented s λ m tr( Q ) is, λ the []. Denoting b m Q, s.t. Q λ I, m F, (5) cn be (8) / =,, i.e., is the squre root of, []. he constrints in (8) cn be reformulted s liner mtri inequlities (LMI) with respect to, reling on the following lemm []: 5 Journl of Communictions 985

4 Journl of Communictions Vol., No., December 5 A B B C be ermitin mtri with C, then Z if nd onl if C, C is the chur complement of C in Z nd is given b C A - BC B. We use the following two MIMO rdr sstems with the following ntenn configurtion: MIMO rdr (.5,.5), the prmeters specifing ech rdr sstem re the inter-element spcing of the trnsmitter nd receiver (in units of wvelengths), respectivel. he number of snpshots is L 56. he rr signl-to-noise rtio (AN) vring from -5 to db in the following emples is defined s PM t M r / W, P stnds Lemm (chur s Complement): Let Z B using Lemm, the problem cn be recst s n DP shown s for the totl trnsmitted power, nd W denotes the vrince of the dditive white therml noise. here is strong jmmer t with n rr-interference-to-noise rtio (AIN) defined s the product of the incident, equl to 6 interference power nd M r divided b W db. In the following numericl emples, there is onl one trget with unit mplitude t in the considered rnge bin. It is known from ection II tht the output N must be estimted using the initil loction prmeter estimte. here re mn methods for estimting this prmeter (see, e.g., [4] nd the reference therein for more detils). In the following emples, we ee the effectiveness of the proposed method in two cses, i.e., one is tht onl the initil ngle estimte error is considered, nd the other is tht onl the clibrtion error in both the trnsmitting nd receiving rrs is considered. In the former cse, it is ssumed tht the initil ngle estimte hs n uncertint [, ]. In the ltter cse, both the trnsmitting nd receiving rrs re ssumed to hve clibrtion errors (the sensor mplitude nd phse error s well s position error). Ech element of the trnsmitting nd receiving rr steering vectors is perturbed with zero-men circulrl smmetric comple Gussin rndom vrible with vrince denoting b e.. λm I s.t. I (9) λm I Q ubstituting obtined from (9) into (4), cn be solved b n DP,t s.t. t tr Q s t tr ( ) LP () t is n uilir vrible. o fr, we know how to solve for fied, nd for fied. We cn itertivel optimize nd. imilr to Algorithm 3 proposed in [7], n itertive lgorithm is proposed to improve the worst-cse detection performnce, which is shown s follows. Algorithm: Given n initil vlue of the WCM, nd cn be optimized b repeting the following steps:. olve (9) to obtin the optimum.. olve () to obtin. 3. Go to step until the N increse becomes insignificnt. Using mn well-known lgorithms (see, e.g., [9]) for solving DP problems, the problems (9), () cn be solved ver efficientl. In the following numericl emple, the optimiztion toolbo in [9] is used for these problems. It is noticed tht the proposed method cn onl obtin the WCM other thn the ultimte trnsmitted wveforms. In prctice, the ultimte wveforms cn be smptoticll snthesized b using the method in [3]. A. Uncertint in Initil Angle Estimtion Fig. shows the optiml trnsmit bempttern with AN= db. It cn be seen tht the pek of the trnsmit bempttern is plced round the trget loction, which mens tht the worst-cse detection performnce in the conve uncertint cn be improved b the proposed method. Moreover, it cn be seen tht notch is plced lmost t the jmmer loction. Bempttern (db) -5 IV. NUMEICAL EXAMPLE In this section, we ssess the performnce of the proposed method compred to the non-robust method tht cn be clculted b (), nd uncorrelted wveforms tht cn be generted b using dmrd codes [3]. Consider MIMO rdr sstem with M t 4 trnsmitting elements nd M r 4 receiving elements. 5 Journl of Communictions Angle (deg) Fig. Optiml trnsmit bempttern with AN= db in the cse of the initil ngle estimte error. 986

5 Journl of Communictions Vol., No., December 5 In Fig., the worst-cse detection probbilit obtined b the proposed method versus AN in the cse of the initil ngle estimte error is showed, s compred to tht of uncorrelted wveforms nd the non-robust method, in the cse of P = f. It is obvious tht the detection probbilities obtined b three methods increse s the increse of AN. Moreover, one cn observe tht the trnsmitted wveforms obtined b the proposed method hve better worst-cse detection performnce s compred to those proposed b the non-robust method nd uncorrelted wveforms. he Worst-Cse Detection Probbilit he Proposed Method he Non-obust Method Uncorrelted Wveforms B. Clibrtion Error in the rnsmitting nd eceiving Arrs Fig. 3 displs the optiml trnsmit bem pttern with AN= db. From Fig. 3, we cn drw conclusion similr to tht in Fig.. he worst-cse detection probbilit obtined b the proposed method versus AN in the cse of the rr clibrtion error with f P = is depicted in Fig. 4, s well s those of the nonrobust method nd uncorrelted wveforms. he conclusion obtined from Fig. 4 is similr to tht of Fig... Fig. he worst-cse detection probbilit for. P cse of the initil ngle estimte error N db f =. versus AN, s well s tht of uncorrelted wveforms nd the non-robust method, in the - -4 Bempttern (db) Angle (deg) Fig. 3 Optiml trnsmit bemptterns with AN= db in the cse of the rr clibrtion error. he Worst-Cse Detection Probbilit he Proposed Method he Non-obust Method Uncorrelted Wveforms N db Fig. 4 he worst-cse detection probbilit for cse of the rr clibrtion error. = P versus AN, s well s tht of uncorrelted wveforms nd the non-robust method, in the f V. CONCLUION In this pper, we hve investigted the problem of robust wveform optimiztion for trget detection in MIMO rdr b eplicitl incorporting the uncertint in prmeters into the optimiztion model. An itertive lgorithm hs been proposed to improve the worst-cse detection performnce. Ech step in the proposed 5 Journl of Communictions 987

6 Journl of Communictions Vol., No., December 5 lgorithm is solved resorting to conve reltion tht belongs to the DP clss, nd hence it cn be solved efficientl. Numericl emples hve shown tht the proposed itertive lgorithm improves the worst-cse detection probbilit ver obviousl compred to uncorrelted wveforms s well s the non-robust method. herefore, the trnsmitted wveforms obtined b the proposed lgorithm cn improve the overll detection performnce for MIMO rdr. APPENDIX A DEIVAION OF () he output N cn be obtined b using the imum vrince distortion response (MVD) filter [5]. he problem cn be stted s follows: m w It cn be recst s w h E[ w vec( Z) ] vec( )vec( ) w s.t. w Z Z w w h C () () C is constnt. It is well known tht () is MVD problem, nd the solution is ( ) w I Q h (3) is constnt number relted to the steering vector. ubstituting (3) into (), () cn be obtined immeditel. APPENDIX A DEIVAION OF (5) o simplif nottion, we define e( ), Im( ), e( ), Im( ) e( ), Im( ), e( ), Im( ), Z e( Z), Z Im( Z), Z Q. hen Q ( j ) ( Z j Z )( j ) ( j ),, ( Z j Z j Z Z j Z Z,,,, (4) Z j Z )( j ) (5) Z Z Z - Z Z Z,, Z Z,,,, nd Let, hen we hve Z, -,, -, - Z -Z Z =,, Z Z,, - Z -Z Z Z - -,,,, ( Z,, Z Z,, Z Z,, Z - Z Z,, herefore, (5) holds immeditel. Besides, becuse We cn obtin ) ( 6) ( 7) (8) (9) (3) F (3) (3) ACKNOWLEDGMEN his work is sponsored in prt b NFC under Grnt 6358, the development project of enn Provincil Deprtment of science nd technolog under Grnt 5338, the Eduction Deprtment of enn province universit innovtion tlent support progrm under Grnt AI3. EFEENCE [] D. rchi, F. Oliveri, nd P. F. mmrtino, MIMO rdr nd ground-bsed A imging sstems equivlent pproches for remote sensing, IEEE rns. on Geoscience nd emote ensing, vol. 5, no., pp , 3. [] J. Li, L. Xu, P. toic, K. W. Forsthe, nd D. W. Bliss, nge compression nd wveform optimiztion for MIMO rdr: A crmer-ro bound bsed stud, IEEE rns. on ignl Processing, vol. 56, no., pp. 8-3, Jn Journl of Communictions 988

7 Journl of Communictions Vol., No., December 5 [3]. Y. Wng, G.. Lio, J. Li, nd. Lv, Wveform optimiztion for MIMO-AP to improve the detection performnce, ignl Processing, vol. 9, no., pp. 6996,. [4]. Y. Wng, G.. Lio, J. Li, nd W. M. Guo, obust wveform design for MIMO AP to improve the worst-cse detection performnce, EUAIP J. Adv. ignl Processing, vol. 5, no., pp. -3, Mr. 3. [5] B. Jiu,. W. Liu, X. Wng, L. Zhng, Y.. Wng, nd B. Chen, Knowledge-Bsed sptil-temporl hierrchicl MIMO rdr wveform design method for trget detection in heterogeneous clutter zone, IEEE rns. on ignl Processing, vol. 63, no. 3, pp , Feb. 5. [6] B. ng, M. M. Nghsh, nd J. ng, eltive entrop-bsed wveform design for MIMO rdr detection in the presence of clutter nd interference, IEEE rns. on ignl Processing, vol. 65, no. 99, pp. -5, 5. [7] C. Y. Chen nd P. P. Vidnthn, MIMO rdr wveform optimiztion with prior informtion of the etended trget nd clutter, IEEE rns. on ignl Processing, vol. 57, no. 9, pp , 9. [8] A. Khwr, A. Abdelhdi, nd. Clnc, rget detection performnce of spectrum shring MIMO rdrs, IEEE ensors Journl, vol. 5, no. 8, pp. -, 5. [9] M. Grntnd nd. Bod. (Mr. 4). CVX: Mtlb softwre for disciplined conve progrmg, version.. [Online]. Avilble: com/cv [] J.. Goldstein, I.. eed, nd P. A. Zulch, Multistge prtill dptive AP CFA detection lgorithm, IEEE rns. Aerosp. Electron. st., vol. 35, no., pp , Apr []. J. Kim, A. Mgnni, A. Mutpcic,. P. Bod, nd Z. Q. Luo, obust bemforg vi worst-cse IN mimiztion, IEEE rns. on ignl Processing, vol. 56, no. 4, pp , Apr. 8. []. A. orn nd C.. Johnson, Mtri Anlsis, Cmbridge, U.K.: Cmbridge Univ. Press, 985, pp. 47. [3]. Ahmed nd M.. Alouini, MIMO rdr trnsmit bempttern design without snthesising the covrince mtri, IEEE rns. on ignl Processing, vol. 6, no. 9, pp , 4. [4] D. Oh, Y. C. Li, J. Khodje, J. W. Chong, nd J.. Lee, Joint estimtion of direction of deprture nd direction of rrivl for multiple-input multiple-output rdr bsed on improved joint EPI method, IE dr onr Nvig., vol. 9, no. 3, pp , 5. [5]. L. V. rees, Detection, Estimtion, nd Modultion heor,optimum Arr Processing (Prt IV), Wile-Interscience, April,. ongfeng Wng received the M.E. degree in computer ppliction technolog from uzhong Universit of cience nd echnolog of Chin in. e joined chool of Computer cience nd echnolog, Zhoukou Norml Universit in 4, he is currentl lecture. is reserch interests re l in spce-time dptive processing, signl processing for rdr. ongn Wng, received the Ph.D. degree from Xidin Universit in. Dr. Wng joined the College of Informtion Engineering, Dlin Universit in 3, he is currentl lecturer. is reserch interests re l in MIMO rdr wveform optimiztion, MIMO communiction, spcetime dptive processing, nd rr signl processing. 5 Journl of Communictions 989