Adaptive Design for Distributed MIMO Radar Using Sparse Modeling

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1 FOR STUDENT PAPER COMPETITION Adptive Design for Distributed MIMO Rdr Using Sprse Modeling Sndeep Gogineni Student Member IEEE nd Arye Nehori Fellow IEEE Deprtment of Electricl nd Systems Engineering Wshington University in St. Louis One rookings Drive St. Louis MO 6 USA Emil: {sgogineninehori}@ese.wustl.edu Phone: Fx: Abstrct Multiple Input Multiple Output (MIMO) rdr systems with widely seprted ntenns provide sptil diversity gin by viewing the trgets from different ngles. In this pper we propose n pproch to ccurtely estimte the properties (position velocity) of multiple trgets using such systems by employing sprse modeling. We lso propose new metric to nlyze the performnce of the rdr system. We develop n dptive mechnism for optiml energy lloction t different trnsmitters. We show tht this dptive mechnism outperforms MIMO rdr systems tht trnsmit fixed equl energy cross ll the ntenns. Index Terms Adptive Multiple Input Multiple Output (MIMO) rdr sprse modeling widely seprted ntenns multiple trgets I. INTRODUCTION Multiple Input Multiple Output (MIMO) rdr 5 hs ttrcted lot of ttention in recent times due to the improvement in performnce it offers over conventionl single ntenn systems. MIMO rdr is typiclly used in two configurtions nmely distributed (widely seprted) nd colocted. In distributed MIMO rdr the ntenns re widely seprted. This enbles viewing the trget from different ngles. Hence if the trget returns between prticulr trnsmitter nd receiver re wek then it is highly likely tht they will be compensted by the returns between other ntenn pirs. While distributed MIMO rdr exploits sptil diversity colocted MIMO rdr 4 exploits the wveform diversity. In this configurtion ll the ntenns re closely spced nd hence the trget Rdr Cross Section (RCS) vlues re the sme for ll trnsmitter-receiver pirs. In this pper we study MIMO rdr with widely seprted ntenns in the context of sprse modeling for estimting the positions nd velocities of multiple trgets. Since the number of trgets in rdr scene is limited we cn use sprse modeling to represent the rdr dt. Sprse modeling nd compressive sensing hve been pplied to the field of rdr 6 8. So fr it hs been used for MIMO rdr only in the context of closely spced ntenns 7 9. To the best of our knowledge the importnt configurtion This work ws supported by the Deprtment of Defense under the Air Force Office of Scientific Reserch MURI Grnt FA nd ONR Grnt N Corresponding uthor of distributed MIMO rdr hs not been pproched from sprse modeling perspective. This configurtion is very importnt since it provides sptil diversity. In 8 the uthors cll their system distributed MIMO rdr even though they re ctully using colocted MIMO rdr. This is evident from the fct tht they use the sme RCS vlue for ll trnsmitterreceiver pirs. In this pper we del with this importnt spect. Adptive rdr design hs been hot topic for number of yers. Implementing the rdr system in closed loop by dptively choosing the properties of the trnsmitted wveforms bsed on the knowledge of the environment gives significnt improvement in performnce. References demonstrte the dvntges of dptive design under different configurtions. In dptive polriztion design is considered in the context of Single Input single Output (SISO) rdr systems.references discuss the problem of dptive wveform design for MIMO rdr. In this pper (see lso ) we propose n optiml dptive energy lloction mechnism for distributed MIMO rdr using the reconstructed sprse vectors. We lso introduce new metric to nlyze the performnce of the rdr system. Using this metric we show tht this dptive mechnism outperforms MIMO rdr systems tht trnsmit fixed equl energy cross ll the ntenns. In section II we derive the signl model for MIMO rdr using sprse representtion. In section III we present different lgorithms for sprse support recovery to infer bout the trgets (position velocity). In section IV we propose the optiml energy lloction mechnism. In section V we use numericl results to show the performnce of the sprse recovery lgorithms in estimting the trget positions nd velocities. We demonstrte the improvement offered by dptive energy lloction. In section VI we conclude this pper. II. SIGNAL MODEL We ssume tht there re M T trnsmitters M R receivers nd K trgets. Further we ssume tht ll the trgets re moving in dimensionl plne. This cn be extended without loss of generlity to the dimensionl cse. The k th trget is locted t p k = p k xp k y nd it moves with velocity v k = v k xv k y.thei th trnsmitter nd j th receiver re locted //$6. IEEE

2 FOR STUDENT PAPER COMPETITION t t i = t ix t iy nd rj = r jx r jy respectively. We trnsmit orthonorml wveforms from the different trnsmitters. Let w i (t) be the bsebnd wveform trnsmitted from the i th trnsmitter. Further we ssume tht the cross correltions between these wveforms is close to zero for different delys. Define k ij (t) s the RCS of the kth trget between the i th trnsmitter nd the j th receiver. Then the bndpss signl rriving t the j th receiver cn be expressed s { K y j (t) = Re k ( ) ij(t)w i t τ k ij k= i= e jπ (f k D ij (t τ k ij)+f c(t τ k ij) )} where f c is the crrier frequency τij k nd f D k ij re the dely nd Doppler shift corresponding to the k th trget. τij k = ( p k t i + p k ) r j () c f k D ij = f c c ( v k u k r j v k ) u k t i () dimensionl digonl mtrix Ψ l j(n). s l j(n) = s l j(n)...s l M Tj(n) T (4) y j (n) = y j (n)...y MTj(n) T (5) e j (n) = e j (n)...e MTj(n) T (6) Ψ l j(n) = dig { ψ l j(n)...ψ l M Tj(n) } (7) where dig{ } refers to digonl mtrix whose entries re given by { } nd T denotes the trnspose of. Further we rrnge { s l j (n)} M R { y j= j (n) } M R nd {e j= j(n)} MR j= into M T M R dimensionl column vectors s l (n) y(n) nd e(n) { } respectively nd Ψ l MR j(n) into (M T M R ) (M T M R ) j= dimensionl digonl mtrix Ψ l (n). s l (n) = (s l (n) ) T (... s l (n)) T T MR (8) y(n) = (y (n)) T... ( y MR (n) ) T T (9) e(n) = (e (n)) T...(e MR (n)) T T () { ( ) T ( ) } T Ψ l (n) = dig Ψ l (n)... Ψ l M R (n). () where u k t i u k r j denote the unit vector from the i th trnsmitter to the k th trget nd the unit vector from the trget to the k th receiver respectively; is the inner product opertor nd c is the speed of propgtion of the wve in the medium. We define the trget stte vector ζ =p x p y v x v y T.The gol is to estimte ζ for ll the K trgets. The received signls t ech receiver re first down converted from the rdio frequency nd then pssed through bnk of M T mtched filters ech of which corresponds to prticulr trnsmitter. Under the orthogonlity ssumption the smpled outputs of the i th mtched filter t the j th receiver is given s y ij (n) = ( ) k ij(n)e jπ f k D ij (n τij) f k cτ k ij + e ij (n) () k K where e ij (n) is the dditive noise t the output of the i th mtched filter of the j th receiver K represents set contining ll the trgets tht contribute to the mtched filter output t time n. In order to rrive t the bove expression we lso ssume tht the trget RCS vlues do not vry within pulse durtion nd the Doppler shift is smll. Hence k ij (t)ejπfd ij t vries slowly when compred with w i (t).now we { discretize the trget } stte spce into grid of L vlues ζ l l =...L. Hence ech of the trgets is ssocited with stte vector belonging to this grid. If the presence of trget t ζ l contributes to the mtched filter output t n define ) f l D ij (n τij) f l cτ l ij ( ψij l (n) =ejπ elseψij l (n) =. Also if ζl is the stte vector of the k th trget we define s l ij (n) =k ij (n) else s l ij (n) =. For ech j we stck sl ij (n) y ij(n) nd e ij (n) corresponding to different trnsmitters to obtin M T dimensionl column vectors s l j (n) y j(n) nd e j (n) respectively. Similrly we rrnge ψij l (n) into (M T) (M T ) Finlly stcking { s l (n) } L {Ψ l= nd l (n) into LM TM R l= dimensionl column vector nd (M T M R ) (LM T M R ) dimensionl mtrix respectively we obtin } L s(n) = s (n)...s L (n) T () Ψ(n) = Ψ (n)...ψ L (n). () Therefore we cn express the received vector t the n th time snpshot s y(n) =Ψ(n)s(n)+e(n) (4) where s(n) is sprse vector with KM T M R non-zero entries. We hve expressed our observed dt using sprse representtion. We ssume tht the trget RCS vlues do not vry over period of N time snpshots. Now we stck {y(n)} N n= {e(n)} N n= nd {Ψ(n)}N n= into y (NMTM R) () = (y()) T...(y(N)) T T e (NMTM R) () = (e()) T...(e(N)) T T Ψ (NMTM R) (LM TM R) = (Ψ()) T...(Ψ(N)) T T to obtin y = Ψs + e. (5) Ψ is known nd only s depends on the ctuls trgets. III. SPARSE SUPPORT RECOVERY In order to find the properties of the trgets (position velocity) we need to recover the sprse vector s from the mesurements y. The two most populr pproches for sprse signl recovery re Mtching Pursuit (MP) nd sis Pursuit 4 (P) //$6. IEEE 4

3 FOR STUDENT PAPER COMPETITION A. Mtching Pursuit (MP) Mtching pursuit is n itertive lgorithm tht cn be used for sprse signl recovery. Since ll the columns of Ψ re not necessrily independent there re infinitely mny solutions for s even when there is no noise. In MP we first initilize the reconstructed vector s () = nd the residul r () = y. In ech subsequent itertion k we project the residul vector r (k ) onto ll the columns of Ψ nd pick the column ψ (k ) tht hs the highest correltion with the residul. We updte the estimted reconstructed vector s (k ) = s (k ) ) + r(k ψ (k ) ) ψ (k ) ψ (k ) ψ(k. We finlly updte the residul s r (k ) = r () s (k ). After sufficient number of itertions the residul pproches zero.. sis Pursuit (P) sis pursuit is n optimiztion principle. In MP we re not optimizing specific objective function. We just proceed heuristiclly to obtin the solution. However in P we minimize n objective function 4 nd we re gurnteed to solve the optimiztion problem. P is presented under two scenrios; in the bsence of noise nd in the presence of noise. ) Absence of Noise: In the bsence of noise P ims t minimizing s under the constrint y = Ψs. Since usully N L there re mny different vectors s tht stisfy the constrint. We choose the solution tht hs the lest l norm. This optimiztion problem cn be modeled s liner progrm 4. There re mny existing lgorithms to solve this problem. One such lgorithm is the P-Simplex lgorithm. In MP we begin with n empty bsis nd we keep dding columns to the bsis in ech itertion by computing the column tht gives the mximum correltion with the residue. However in P we begin with bsis set Ψ () tht contins KM TM R linerly independent columns of Ψ such tht y lies in the column spce of Ψ (). The rest of the columns form nother set Ψ(). We recover the initil estimte s () by using the bsis Ψ (). In every subsequent itertion k we choose the swp of column between Ψ (k ) nd Ψ (k ) such tht the l norm of the reconstructed vector s (k ) corresponding to this swp is lower thn tht corresponding to ny other swp. We updte the bsis sets ccording to this swp. Once we rech the optiml solution there will be no swp tht will further reduce the l norm. This lgorithm lwys gurntees convergence to the optiml solution in the bsence of noise. ) Presence of Noise: Clerly the bove pproch of bsis pursuit will fil in the presence of noise. Hence in 4 the uthors propose sis Pursuit De-Noising (PDN). This is n unconstrined minimiztion problem min y Ψs + λ s. Typiclly λ = σ log(lm T M R ) where σ represents the noise level. To solve this problem we used CVX pckge for specifying nd solving convex progrms 5 6. We present the results in section V. IV. OPTIMAL ADAPTIVE ENERGY ALLOCATION efore we propose the energy lloction mechnism we shll first define n importnt performnce metric. As mentioned erlier the LM T M R length vector s hs only KM T M R non-zero entries. Let the reconstructed vector be denoted by ˆs. We would like to hve the most significnt KM T M R entries of ˆs correspond to the sme indices s the non-zero entries of s. If this is not the cse then we will wrongly mp the trget sttes for one or more trgets. We define L length vector s M R s(l) = ˆs(M T M R (l ) + M R (j ) + i). i= j= Further define s s K length vector which contins the vlues tht s crries t the correct K indices. Similrly we define s s L length vector tht tkes vlue of t the correct K indices nd tkes the sme vlues s s t every other index. It is cler tht the non-zero entries of s correspond to the non-trget sttes. We define the metric = min s mx s. If this metric hs vlue greter thn then ll the ctul correct trget indices dominte the other indices in s. This is desirble. Otherwise the estimtes re not good enough. Hence the higher the vlue of the better the performnce of the system. In section V we use this metric to nlyze the results. Let E i be the energy of the wveform trnsmitted from the i th trnsmitter. So fr we hve ssumed tht ll the trnsmitters send out orthonorml wveforms. Hence the trnsmitting energies for ll the trnsmitters E i =nd the totl energy trnsmitted M T i= E i = M T. At the receiver side we use the lgorithms mentioned in the previous section to obtin the reconstructed sprse vector ˆs. This vector contins estimtes of the ttenutions k ij. Since the different ntenn pirs view the trgets from different ngles these ttenutions will be different from ech other. Hence equl energy lloction to ll the trnsmitters does not necessrily give the best performnce. We hve the over ll energy constrint M T i= E i = M T.We initilize the system by trnsmitting equl energies E i = from ll the trnsmitters nd estimte the RCS vlues ˆ k ij from the reconstructed vector ˆs. We im to optimize the performnce of the system for the next processing intervl by picking the trnsmit energy lloction tht mximizes the minimum trget returns. Hence the gol is to solve the following optimiztion problem nd find the optiml E i such tht M T i= E i = M T mx E i min k M R E i ˆ k ij. i= j= Since the opertors mx{ } nd min{ } cn be represented s the l nd l norms respectively we cn solve the //$6. IEEE 5

4 FOR STUDENT PAPER COMPETITION 4 bove optimiztion problem using CVX 5 6. y this optimiztion criterion we re mximizing the numertor in the expression for for the next processing intervl. We shll show in section VI tht this optiml choice of wveform energies gives significnt improvement in performnce..5 V. NUMERICAL RESULTS We simulted MIMO rdr system. We denote the positions of ll the trnsmitters trgets nd receivers on common Crtesin coordinte system. The trnsmitters re locted t m nd m. The receiver loctions re m nd m. The crrier frequency of the trnsmitted wveforms is f c =GHz. Within ech processing intervl we consider three pulses tht re trnsmitted.ms prt. We choose N = 4 for the simultion results. We choose N = 4. Therefore y hs 97 entries. We divide the trget position spce into 9 9 grid points nd the trget velocity spce into 5 5 grid points. Therefore the totl number of possible trget sttes L = 5. We considered the presence of trgets. The positions nd the velocities of the trgets re given s p = 8 m v = m/s p = 8 8 m v = m/s p = 6 m v = m/s. The ttenutions for the trgets re =.ε.ε.7ε.8ε =.4ε.5ε.ε.ε =.4ε.5ε.8ε.7ε where ε = +. The entries of e re generted independently from Gussin distribution. We ssume ech of these smples hs the sme vrince σ. We define the signl ( to noise rtio ) (SNR) for the MIMO rdr system s Ψs log d. First we compre the performnces of E( e ) the two lgorithms mtching pursuit nd bsis pursuit denoising. We perform these simultions t n SNR of.d nd for PDN we chose λ = σ log(lm T M R ).ForMP we used itertions. Since it is not possible to plot the position nd velocity on the sme plot we plotted the estimtes of position nd velocity seprtely. For computing the estimte t prticulr grid point on the position plot we verge over ll 5 5 velocity grid points corresponding to tht position grid point. Similrly we verge over ll the 9 9 position grid points in order to obtin the velocity plot. We do this only to be ble to plot position nd velocity estimtes seprtely. From Fig. nd Fig. we cn see tht both the lgorithms re ble to estimte the positions nd velocities of the trgets. However it is importnt for us to nlyze the performnces of the two lgorithms by evluting the performnce metric p x v x p y v y Fig.. Reconstructed vectors using mtching pursuit t SNR=.d () position estimtes (b) velocity estimtes. Fig. plots s function of the SNR nd we cn clerly see tht PDN outperforms MP. remins bove for much lower SNR for PDN when compred with MP. When < some of the non-trget sttes dominte the reconstructed vector. Since PDN outperforms MP for ll further simultion results we shll use only PDN. We used independent Monte Crlo runs to generte these results. Now we demonstrte the dvntges of hving dptive energy lloction. We ssume we hve estimtes of the trget RCS vlues from the estimtion of the previous processing intervl. We pply the optimiztion principle described in section IV. As we see from Fig. 4 the dptive energy lloction gives significnt improvement in performnce. The vlue of is higher when compred with the equl energy trnsmission. Even t n extremely low SNR of.d the vlue of remins greter thn for the proposed energy lloction scheme. Also t n SNR of.d nerly doubles while using the energy lloction scheme. VI. CONCLUSION We proposed sprse modeling pproch to estimte the positions nd velocities of multiple trgets using distributed MIMO rdr. We demonstrted the ccurte reconstruction of //$6. IEEE 6

5 FOR STUDENT PAPER COMPETITION p x p y Δ PDN MP SNR (in d) Fig.. Performnce metric for mtching pursuit nd bsis pursuit denoising s function of SNR v x 9 9 v y Δ Optiml Energy Alloction Equl Energy Alloction Fig.. Reconstructed vectors using bsis pursuit de-noising t SNR=.d () position estimtes (b) velocity estimtes SNR (in d) the trget stte vectors. We introduced new metric to nlyze the performnce of the system. Also we proposed n optiml dptive energy lloction mechnism to further enhnce the performnce. We show this improvement using simultions. REFERENCES A. M. Himovich R. S. lum nd L. J. Cimini MIMO rdr with widely seprted ntenns IEEE Signl Process. Mg. vol. 5 pp. 6 9 Jn. 8. J. Li nd P. Stoic MIMO rdr signl processing. Hoboken NJ: John Wiley & Sons Inc. 9. S. Gogineni nd A. Nehori Polrimetric MIMO rdr with distributed ntenns for trget detection IEEE Trns. Signl Process. vol. 58 pp Mr.. 4 J. Li nd P. Stoic MIMO rdr with colocted ntenns IEEE Signl Process. Mg. vol. 4 pp. 6 4 Sep S. Gogineni nd A. Nehori Polrimetric MIMO rdr with distributed ntenns for trget detection in Proc. 4rd Asilomr Conf. Signls Syst. Comput. Pcific Grove CA Nov R. rniuk nd P. Steeghs Compressive rdr imging in IEEE Rdr Conference oston MA Apr. 7 pp C.-Y. Chen nd P. P. Vidynthn Compressed sensing in MIMO rdr in 4nd Asilomr Conference on Signls Systems nd Computers Pcific Grove CA Oct. 8 pp Y. Yo A. P. Petropulu nd H. V. Poor Compressive sensing for MIMO rdr in IEEE Interntionl Conference on Acoustics Speech nd Signl Processing Tipei Apr. 9 pp. 7. Fig. 4. Performnce metric with nd without dptive energy lloction. 9 Y. Yu A. P. Petropulu nd H. V. Poor MIMO rdr using compressive smpling IEEE Jour. of Selected Topics in Signl Proc. vol. 4 pp Feb.. M. Hurtdo J. J. Xio nd A. Nehori Trget estimtion detection nd trcking: A look t dptive polrimetric design IEEE Signl Process. Mg. vol. 6 pp. 4 5 Jn. 9.. Friedlnder Wveform design for MIMO rdrs IEEE Trns. Aerosp. Electron. Syst. vol. 4 pp. 7 8 Jul. 7. S. Gogineni nd A. Nehori Compressive sensing for MIMO rdr with widely seprted ntenns IEEE Trns. Signl Process. submitted. S. G. Mllt nd Z. Zhng Mtching pursuits with time-frequency dictionries IEEE Trns. Signl Process. vol. 4 pp Dec S. S. Chen D. L. Donoho nd M. A. Sunders Atomic decomposition by bsis pursuit SIAM Review vol. 4 pp Mr.. 5 M. Grnt nd S. oyd. (9 Jun.) CVX: Mtlb softwre for disciplined convex progrmming (web pge nd softwre). Online. Avilble: boyd/cvx 6. (8) Grph implementtions for nonsmooth convex progrms recent dvnces in lerning nd control ( tribute to M. Vidysgr) V. londel S. oyd nd H. Kimur editors pges 95- lecture notes in control nd informtion sciences Springer. Online. Avilble: boyd/grph dcp.html //$6. IEEE 7