MIMO Radar Transmit Beampattern Design with PAR Constraint in a Spectrally Crowded Environment

Transcription

1 nternationa Conference on Coputer etwors and Counication Technoogy (CCT6) O Radar Transit Beapattern Design with PAR Constraint in a Spectray Crowded Environent Qing-in ZA*, Yu-xi WAG,* and Wei Science and Technoogy on Autoatic Target Recognition aboratory, ationa University of Defense Technoogy, Changsha 473, China Schoo of nforation and avigation, Air Force Engineering University, Xi an 777, China *Corresponding author Keywords: Transit beapattern design, PAR, Spectra copatibiity, SDR. Abstract. Considering the fact that the eectroagnetic spectru is becoing ore and ore crowded due to the increasing deand for both iitary and civiian wireess services. A nove O radar transit beapattern design schee with PAR constraint in a spectray crowded environent is proposed. Different with the existing beapattern design ethods, the PAR and spectra constraints are considered in the optiization of the space-tie transit wavefor covariance atrix. Seidefinite reaxation (SDR) is adopted to tace the resuting P-hard optiization probe. With the optiized covariance atrix, the fina transit wavefors which satisfy the practica constraints can be synthesized directy via randoization. Soe nuerica siuation resuts are shown to deonstrate the effectiveness of the new technique to devise O radar transit wavefors in beapattern design copying with the PAR and spectra coexistence requireents. ntroduction utipe-input utipe-output (O) radar as the next generation radar syste has received a ot of interest in the ast decade. Copared with the traditiona phased array radar, O radar can eit different wavefors fro different transit antennas, which can provide extra degrees-of-freedo (DOF) to iprove syste perforance. Thus the fexibe wavefor design for transit beapttern is of interest for coocated O radars []. n order to focus the transit power into a range of interesting anges whie iniizing the transit power for other directions, there are usuay two steps to achieve this goa. n the first step, a wavefor covariance atrix R [] shoud be designed according to the desired transit beapattern. Various criterion such as iniu ean square error (SE) [-4], iniu difference (D) [5], iniu pea side-obe eve (PS) [6] and iniu integrated side-obe eve (S) [7, 8] can be used in this step to optiize the wavefor covariance atrix. Besides, the covariance atrix can aso be paraeterized using the coordinates of a hypershere as shown in [4] and [9]. n the second step, the actua wavefors are synthesized to reaize the optiized covariance atrix with a pea-to-average power ratio (PAR) or constant enveope constraint []. owever, it was noted in [4, ] that this step is not easy to achieve. A ore efficient technique proposed in [] introduces a weighting atrix to severa orthogona wavefors and optiizes the covariance atrix of weighting atrix instead of the covariance atrix of transitted wavefors. Once the covariance atrix Copyright 7, the Authors. Pubished by Atantis Press. This is an open access artice under the CC BY-C icense ( 369

2 is obtained, the weighting atrix can be easiy derived using randoization procedure []. Athough any soutions and discussions have been ade upon this probe, no open ecture has studied the case of O radar transit beapattern design in a spectray crowded environent, which is an iportant and chaenging probe due to the increasing deand for spectru fro both iitary sensing appications and civiian wireess services. The design of advanced radar wavefors to iprove the spectra copatibiity with other overaid icensed radiators is of priary concern [3]. Recenty, any papers have considered radar wavefor design with a suitabe frequency ocation [4] to itigate interference [5] or set the signa-to-interference pus noise ratio (SR) as the figure of erit to optiize the transitted wavefor by constraining the overa aount of interference energy on crowded frequency bands [6, 7]. n this etter, a nove optiization technique to design O radar transit beapattern whie ensuring the PAR constraint of the transit wavefors and the spectra coexistence with other overaid radiators is proposed. Unie the existing O radar transit beapattern design ethods [-], through atheatica anipuations, the covariance atrix of the space-tie transit wavefor can be optiized directy with the practica constraints of the axiu radiated energy on the shared bandwidth and the PAR of each antenna s transit sequence. The resuting optiization probe is a nonconvex quadraticay constrained quadratic prograing (QCQP) probe which is a P-hard probe due to the presence of constraints of the spectra copatibiity and the PAR. ence, the seidefinite reaxation (SDR) is used to sove this nonconvex QCQP probe in poynoia tie. Then, with the optiized covariance atrix, the fina transit wavefors which satisfy those practica constraints can be synthesized directy through the randoization procedure [8]. Finay, soe interesting case studies are anayzed to assess the perforance of the proposed agorith. otations: T, and denote the transpose, conjugate and conjugate transpose, respectivey., and denote absoute operate, Eucidean nor and the Kronecer product. eans identity atrix. vec( ) and tr( ) denote vectorization operator and trace operator, respectivey. Syste ode Suppose a coocated O radar syste with transitters and the transit array is assued to be unifor inear array (UA) with haf-a-waveength eeent-separation. et s [ s (), s (),..., s ( )] T () denote the transit sequence of the th antenna and is the nuber of sapes of each transit signa puse. So the transit wavefor atrix of the O radar is S [ s, s,..., s ] T C () The steering vector of the array towards direction is denoted j sin( ) j( ) sin( ) T as a ( ), e,, e. The transit beapattern can be defined as P( ) a ( ) Ra ( ) (3) where R SS is the approxiated covariance atrix using a finite nuber of sapes. 37

3 Due to the iited dynaic range of the radio frequency apifier, the transit wavefor of each antenna needs to have a ow PAR, which can be foruated as foows: ax s ( n) n PAR( ), [, ] s ( n) n s (4) where is the given upper bound to the PAR. f, the transit wavefor is constant-enveope and if, it eans that there is no requireent on wavefor s apitude. As to the icensed radiators coexisting with O radar, suppose that each of the wors over a frequency band [ f, f],,,...,,where f and f are the ower and upper noraized frequencies for the th radiator, respectivey. To ensure the spectra copatibiity, the transit wavefor has to be shaped to anage the aount of interfering energy produced on the shared bandwidth between icensed radiators and O radar. Fro an anaytica point of view, the energy transitted on the th bandwidth fro the th antenna is f f where S ( f ) df s R s (5) S( f ) s( n) e n j fn is the energy spectra density (ESD) of the code s and f f p q j ( pq) f j ( pq) f R ( p, q) e e ( p, q) {,..., } (6) j ( p q) p q Transit Beapattern Design and Wavefor Synthesis n this section, a transit beapattern design ethod that optiize the covariance atrix of the transit space-tie wavefor with practica constraints of spectra coexistence and PAR is introduced. To this end, et the desired beapattern denoted as Pd ( ), the desired power eve within the focusing ange range isand that outside this range is. The tota ange interva [, ] can be divided into grid of points denoted e.t.,,,...,. The beapattern design cost function can be defined as d a SS a (7) J(, S) P ( ) ( ) ( ) where is a scaar factor. So the design probe can be atheaticay expressed as: in Pd ( ) a ( ) SS a( ), S s.t. E s R s P ax s( n) n,,,..., s( n) n tr( SS ) E (8) 37

4 where R w R, w is the weight coefficient associated with the th radiator, E is the tota transit power and E denotes the axiu aowed interference in spectru that can be toerated by other radio systes. n order to sove this optiization probe, we need to do soe atheatica anipuations. With the hep of properties T vec( ABC) ( C A) vec( B ) and (A B)(C D) = (AC) ( BD), the transit beapattern (3) can be transfored into P( ) a ( ) Ra( ) vec( a ( ) S) vec( a ( ) S) ( a ( )vec( S)) ( a ( )vec( S)) vec( S) a( ) a ( )vec( S) x ( a( ) a ( )) x x V( ) x tr( V( ) X) where x vec( S), ( ) ( ( ) ( )) X=xx V a a and. et e denote as a coun vector of ength with a eeents zero except the th eeent equa to one, so the transit power of the th antenna can be expressed as (9) T T T s n ess e es es n ( ) vec( ) vec( ) T x e e x x ( Diag( e )) x = tr( B X) where B = Diag( e ) C, Diag( e ) indicates the diagona atrix fored by the coponents of the vector e. So the PAR constraint of the wavefor transitted by the th antenna (4) can be equivaenty expressed as ax( diag( BX)) tr( BX ) () where diag( BX ) indicates a vector fored by the diagona eeents of the atrix BX. As to the spectra constraint of O radar transit wavefor, it can be expressed as () s R s x R x = tr( R X) E () where R R. Finay, with the expressions (9),(),(), the optiization probe P can be equivaenty recast as in Pd ( ) tr( V( ) X), X s.t. tr( RX ) E tr( X) E X ran( X) = P ax( diag( BX)) tr( BX),,,..., Obviousy, probe P is a nonconvex probe and its nonconvexity is copetey confined in the ast ran constraint. So we resort the SDR fraewor and the SDR of probe P, obtained by dropping the ran-one constraint, shares the foowing for (3) 37

5 in Pd ( ) tr( V( ) X), X P s.t. tr( RX ) E ax( diag( BX)) tr( BX),,,..., tr( X) E, X After reaxation, probe P is a seidefinite prograing (SDP) convex probe and its optia soution X can be obtained with CVX toobox in poynoia tie. owever the optiized covariance atrix X coud not be feasibe to P, because it is not ran-one. ence, in the foowing, a randoization procedure is used to synthesize the fina high-quaity approxiate transit wavefor x fro X. f X is ran-one, the transit wavefor x can be obtained by the eigen decoposition of X = x ( x ). n this case, the ran-one reaxation in probe P is tight and the soution x is optia. Otherwise, a suboptia procedure can be adopted foowing the proposed agorith in [8]. nterested readers can refer to [8] for ore detais, we wi present the synthesis agorith in the seque to ae the etter sef-contained. f ran( X ), in order to guarantee the PAR of the transit wavefor, we can define vectors d C and d C, whose ith entry are d( i) X ( i, i) and (),if di () ( ) d i d i, if di ( ), i,...,, respectivey. With these two vectors d and d, two diagona atrix can be fored as D Diag( d ), D Diag( d ). Draw K rando vectors x ( =,...,K ) fro the copex nora distribution C (, C), where j arg( x g ) C D ( X ( t D D)) D. Define vector x d e, for each vector x, we copute the cost function (4) Pd ( ) x V( ) x (5) and choose the inia vaue over {,,..., K }, the fina suboptia soution is x arg in Pd ( ) x V( ) x (6) x t worth pointing out that in the optiization stage we have aready taen the spectra copatibiity and the PAR constraints into account, so we don t need to synthesize the transit wavefor which has to satisfy the practica PAR constraint according to the given covariance atrix R as the conventiona ethods do. On the contrary, a high-quaity transit wavefor with PAR constraint can be obtained directy through randoization procedure, which is ore convenient. The nuber of randoizations K aows to iprove the approxiation quaity and the approxiation bound is guaranteed [9]. As to the coputationa copexity, the coputationa burden is coposed by two parts. The first is the copexity of soving the SDP 4.5 probe P, which is of order O(( ) ), and the second part is the copexity of the randoization step to synthesize fina wavefors, which is of order ( ( ) ) O K. Siuation Resuts We assue a UA of transit antenna eeents spaced haf a waveength apart fro each other. The carrier frequency and bandwidth of the transitted wavefors are f Gz and B z. The code ength is 6 and the tota transit energy 373

6 is E. K 5 is the nuber of randoization and the interest ange sector is [, ]. t is noted that, according to the expression (9), we have a ( )Ra( ) tr(v( )X) (7) which eans that the optia beapattern is deterined by the freedos of the transit array and the PAR and spectra constraints can t infuence the shape of the optia transit beapattern. So the optia beapatterns fored by X are sae with those fored by R introduced in [] as shown in Fig.. And the beapatterns fored by X with the sae eeent nuber under different PAR constraints are a in fu accord as shown in Fig.. Fro Fig. we can aso observe that the beapattern obtained by the synthesized wavefor S through randoization under different PAR constraints is cose to the desired one. 4 Optia R, =6 Optia R, = 3.5 Optia X, =6, = Optia X, =, = Beapattern Ange(degree) Fig.. Transit Beapatterns generated by the optia covariance atrix X and R with different eeent nubers. 4 4 Synthesized S, = Ange(degree) 4 6 Optia X, = Desired 3.5 Beapattern Beapattern Synthesized S, = Optia X, = Desired (a) - Ange(degree) (b) Fig.. Beapattern atching design with sae eeent nuber and different PAR constraints. According to probe P, we now that the optiized X satisfies the PAR constraint, so the synthesized S via randoization aso satisfies the PAR constraint as shown in Fig. 3. t is interested to observe that the PAR of the synthesized wavefor through the proposed ethod is sa even under the unconstrained case. ence, it is reasonabe to beieve that the PAR of the idea wavefor whose covariance is R shoud aso be sa. 374

7 =5 = =.5.4 Unconstrained.3 PAR ndex of Transit Antenna 8 9 Fig. 3. Actua PAR vaues for the synthesized wavefors with different PAR constraints. n order to assess the perforance of the randoization procedure, we et Pd ( ) ( x ) V( ) x (8) denote the difference between the beapattern fored by the synthesized wavefor and the desired one. For the case of, the average vaue of obtained by onte Caro siuation runs is shown in Fig. 4 as a function of the randoization nuber K. Fro Fig. 4, we can see that the average vaue of decreases as K increases and when K 5, the quaity of synthesized wavefors can be guaranteed δ Randoization nuber K Fig. 4. Average vaue of with, as a function of the randoization nuber K. Finay, we extend the proposed agorith to accoodate the spectra constraint. Suppose there are two coexisting icensed radiators and their noraized frequency bands are [ f, f ] [.4,.5], [ f, f ] [.8,.9], respectivey. Without oss of generaity suppose w w, the axiu aowed interference is E 5 4 and the PAR constraint is =. We define the tota energy spectra density of O radar as S ( f ) S ( f ) (9) The spectra coexisting perforance of the proposed ethod is shown in Fig. 5. t is obvious that the synthesized wavefors can contro the aount of energy produced on the woring frequency bands of the radiators. As a resut, the spectra coexistence of the O radar with other wireess syste is iproved. 375

8 ESD(dB) oaized frequency Fig. 5. ESD of the synthesized wavefors. Concusion We have considered the probe of O radar transit beapattern design with a PAR constraint in a spectray crowded environent. Different with the existing beapattern design ethods, we tae the practica constraints of PAR and spectra coexistence into account during the optiization of the space-tie wavefor covariance atrix X. After anipuation, the new optiization probe is a QCQP nonconvex probe. n order to tace this P-hard probe, seidefinite reaxation fraewor is adopted and the optia covariance atrix X can be achieved. Because the optia X satisfies these constraints, the synthesized wavefors through randoization procedure aso have a satisfying perforance in PAR and spectra copatibiity. Finay, a nuerica siuation resuts have been provided to deonstrate the effectiveness of the proposed agorith. References [] J. i, and P. Stoica, O radar with coocated antennas: Review of soe recent wor, EEE Signa Process. ag., vo.4, no.5, pp.6-4, Sep. 7. [] P. Stoica, J. i, and Y. Xie, On probing signa design for O radar, EEE Trans. Signa Process., vo.9, no., pp , Aug. 7. [3] D. R. Fuhrann, G. San Antonio, Transit beaforing for O radar systes using signa cross-correation, EEE Trans. Aerosp. Eectron. Syst., vo.44, no., pp.7-86, Jan.8. [4] S. Ahed, J. S. Thopson, Y. R. Petiot, and B. ugrew, Finite aphabet constant-evveope wavefor design for O radar, EEE Trans. Signa Process., vo.59, no., pp , ov.. [5] A. Khabbazibasenj, A. assanien, S. A. Vorobyov, and. W. orency, Efficient transit beaspace design for search-free based DOA estiation in O radar, EEE Trans. Signa Process., vo.6, no.6, pp.49-5, ar.4. [6] P. Gong, Z. Shao, G. Tu, and Q. Chen, Transit beapattern design based on convex optiization for O radar systes, Signa Process., vo.94, pp.95-, Jan. 4. [7]. Xu, J. Wang, J. Yuan, and X. Shan, O radar transit beapattern synthesis 376

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