Integrated Detection and Tracking via Closed-Loop Radar with Spatial-Domain Matched Illumination

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1 Integrated etecton and Trackng va Closed-Loop Radar wth Spatal-oman Matched Illumnaton Pete Nelsen and Nathan A. Goodman epartment of Electrcal and Computer Engneerng, The Unverst of Arzona 30 E. Speedwa Blvd, Tucson, Arzona 857 USA Abstract We develop a framework for closed-loop detecton and trackng of targets. The framework s based on a Baesan representaton that assgns probabltes to potental realzatons of the radar channel. In ths case, dfferent realzatons are characterzed b dfferent number and locatons of targets present. The Baesan channel representaton can then be used to customze a transmt llumnaton pattern. The probablstc representaton s updated based on receved measurements and Kalman-based predcton of the states of possble targets. Smulaton results from a tral eperment of the closed-loop sstem are provded. I. INTROUCTION Adaptve and knowledge-based (KB) sgnal processng focus on mprovng radar performance through advanced sgnal processng at the receve end of the sstem. However, rather than develop transmsson waveforms and sgnalprocessng technques ndependentl, t s useful to consder the mplementaton and performance of closed-loop radar sstems that possess an adaptve radar transmtter that reacts dnamcall to the propagaton envronment and to prevousl receved data. Ths tpe of sstem, termed cogntve radar n [] or an actve-testng sstem n [-5], s based on a probablstc representaton of the possble realzatons of the radar propagaton channel. Rather than operate based on a predefned pattern for search, acquston, and track, the adaptve transmtter of a cogntve radar sstem contnuall nterrogates the envronment n order to reduce uncertant assocated wth the dfferent channel hpotheses. Thus, the probablstc model gudes the sstem s actve nterrogatons. In [6], we presented a closed-loop framework for radar target dentfcaton. The essental components of the framework ncluded ) an ensemble of possble propagaton channels (n [6], the ensemble was comprsed of a set of known target mpulse responses); ) a probablstc ratng assgned to each alternatve n the ensemble; 3) waveform desgn strateges for detectng the true hpothess as effcentl as possble; 4) a Baesan update to the probabltes wth each data collecton; and 5) a termnaton crteron. In [6], we used sequental hpothess testng to determne when to termnate the eperment, and two dfferent matched waveform technques were compared. The frst waveform technque was an etenson of waveform desgn based on mutual nformaton [7]. The second technque was based on SNR consderatons as suggested n [8-9]. In ths paper, we consder spatal-doman applcaton of the mutual-nformaton-based matched waveform technque n order to develop and test a closed-loop mplementaton of target detecton and trackng. Through the equvalence of angle of arrval wth spatal frequenc, we develop a probablstc representaton of potental target locatons n ( k, k, ω ) space. Ths probablstc representaton can then be converted nto a two-dmensonal spectral varance functon ( k and k ) upon whch the waterfllng [7] operaton can be appled to fnd a matched transmt beampattern. However, snce the target model ncludes oppler shft ( ω ), potental targets can move between the tme when the ensemble s updated wth a data collecton and the tme n the future when the net transmsson wll occur. Thus, we also use a Kalman-based predcton step to antcpate the status of the probablstc channel representaton at the tme when the transmsson wll occur. The result s a sstem that performs full ntegrated search and track functons. Furthermore, the sstem scans and shapes ts transmt beam not accordng to a pre-defned or fed tmelne, but accordng to the uncertant of the probablstc model. The radar sgnal model s descrbed n Secton II, the spatal doman matched llumnaton strateg s descrbed n Secton III, and other detals necessar for closed-loop operaton are descrbed n Secton IV. Smulated results are presented n Secton V, and we make our conclusons n Secton VI. II. SIGNAL MOEL Our goal n ths paper s to demonstrate concepts of closedloop radar such as adaptve llumnaton based on probablstc channel representaton. Therefore, we emplo a smplfed sgnal model that gnores range resoluton, ground clutter, and jammng. As far as range resoluton, one can assume that the radar sstem transmts a ver narrowband or contnuous-wave sgnal. We make ths assumpton on the sgnal model n order to be clear that we are performng spatal-doman spectral shapng, not temporal waveform shapng. Temporal waveform shapng was the subject of [6], and jont spato-temporal waveform shapng wll be the subject of future work. Let a radar sstem llumnate a radar channel wth a narrowband sgnal. The geometr, whch s depcted n Fg., s smlar to a look-down geometr for detectng movng targets on the Earth s surface ecept for the aforementoned assumpton on ground clutter. The prmar aes for the sstem s antenna arra elements are the - and -drectons /08/$ IEEE 546 RAAR 008 Authorzed lcensed use lmted to: The Unverst of Arzona. ownloaded on Aprl 5, 009 at 6: from IEEE Xplore. Restrctons appl.

2 k measurements. Samples are also collected at P tme nstants, and the relatve phase of all MNP measurements are stacked nto a space-tme steerng vector a ( k, k, ω). The total receved sgnal vector s then (, ) q q q, q, q αq ω v S k k a k k. (3) Net, allowng for Q targets, the measurement vector due to all targets for a sngle transmsson s Q q ( q) ( q) ( q) ( q) ( q) (, ) αq,, ω z S k k a k k + w (4) Fgure. Geometr of the radar sgnal model. Element poston n the -drecton produces a measured phase shft dependent on spatal frequenc component k. Element poston n the -drecton enables measurement of the k spatal frequenc component. In a look-down geometr, k and k would be related to azmuth and elevaton angles, respectvel. Moreover, look-down geometres are often defned such that elevaton angle unquel defnes target range. Although our sstem has no range resoluton, the relatonshp between k and target range mples that oppler (range rate) measurements can be related to k rate, or k. Usng a farfeld appromaton, the relatonshp between k and target oppler s a constant. Hence, defnng postve oppler shft to mean relatve moton toward the radar sstem, we have ω βk () where β s a postve constant. On transmt, the antenna elements act as a phased-arra n order to shape the transmt beam. Let the two-dmensonal (voltage) pattern of the transmt beam be denoted b S( k, k ). Note that use of a captal S as well as k and k emphasze that the transmt beam pattern s a spatalfrequenc-doman representaton of the transmt waveform. In other words, (, ) S k k can be found from a two-dmensonal dscrete-tme Fourer Transform of the ampltude and phase of each antenna element. On receve, the antenna arra acts as a mult-channel sstem such that the output of each antenna element s measured. Let the q th target estng wthn the llumnaton pattern of the radar be characterzed b a comple reflecton coeffcent α q, spatal frequenc coordnates ( q) ( q) k and k, and a oppler shft ω ( q). Hence, the receved sgnal at the (m,n) th antenna and p th tme sample due to the q th target s,, (, ) ep ( q q q q q ) αq m n ω p v m n p S k k j k + k t. ata samples are collected over M antennas n the -drecton and N antennas n the -drecton for a total of MN spatal k () where w s a vector of addtve whte Gaussan nose. The measurement model n () clearl ndcates that the sgnal reflected from a target and ncdent on the sstem s a (,, ) q q q three-dmensonal snusod wth frequences k k ω. Therefore, a frequenc-doman representaton of the sgnal component s where (,, ω) (, ) (,, ω) V k k S k k H k k (5) Q ( q) ( q) ( q) (,, ω) αqδ,, ω ω H k k k k k k (6) and ( k, k, ) q δ ω s a three-dmensonal delta functon. The data-doman space-tme data are then obtaned b the transformaton k, k, ω (,, ) (,, ) z V k k ω a k k ω dk dk dω + w, (7) whch b substtuton of (5) and (6) and applcaton of the sftng propert becomes (4). The functon H( k, k, ω ) s the transfer functon of the radar channel, whch wll be used n the net secton to help defne the adaptve transmt beampattern. From (4) we see that the transmt pattern sgnfcantl affects the receved sgnal. In partcular, the transmt pattern controls the SNR of an sgnals reflected from targets, whch n turn affects the sstem s ablt to detect the target and estmate ts parameters. For the closed-loop radar paradgm, SNR affects the sstem s ablt to update the probablstc ratngs of the dfferent channel alternatves. For the applcaton descrbed n ths paper, dfferent channel alternatves consst of dfferent combnatons of the number k, k, ω space. and locaton of targets n ( ) III. SPATIAL-OMAIN MATCHE ILLUMINATION In ths secton, we summarze the nformaton-based matched llumnaton technque of [7] and then descrbe how we appl the technque to the desgn of transmt beamformng for closed-loop radar. Let h () t be a random process that can be thought of as an ensemble of mpulse responses. We wll assume that all of the sample functons of h () t have fnte energ and are causal 547 Authorzed lcensed use lmted to: The Unverst of Arzona. ownloaded on Aprl 5, 009 at 6: from IEEE Xplore. Restrctons appl.

3 mpulse responses. If we further assume that h () t s a Gaussan random process, then we can fnd the waveform that mamzes the mutual nformaton between the ensemble of mpulse responses and the receved waveform. Let the waveform have fnte energ E, be confned to the tme nterval T < t < T, and be essentall bandlmted such that most of ts energ s contaned wthn the frequenc band f T s. The nformaton-mamzng waveform under these constrants has the magntude-squared spectrum defned b [7] k ω S ( f ) σ nt ma 0, A f σ H ( f ) Ts 0 f > Ts (8) Fgure. screte target parameter (hpothess) space. k where T s the nterval durng whch the receved sgnal s observed and the quantt σ H ( f ) s called the spectral varance and s defned b { { } } E E σ H f H f H f. (9) The spectral varance functon quantfes the uncertant n the ensemble of target transfer functons at frequenc f. The constant A n (8) enforces the fnte-energ constrant b solvng Ts σ nt E ma 0, A df σ H ( f ). (0) Ts The soluton n (8) s obtaned b performng the waterfllng acton [7] on the functon σnt σ H ( f ). In [6], we ntegrated matched llumnaton technques wth sequental hpothess testng n order to effcentl determne the true mpulse from among a fnte number of known alternatves. Although the ensemble n [6] was not Gaussan, we computed a spectral varance accordng to σ M M H m m m m m m ( f ) P H ( f ) P H ( f ) () where P m and Hm ( f ) were, respectvel, the probablt and transfer functon assocated wth the m th hpothess. It was found n [6] that waterfllng on the spectral varance functon n () produced waveforms that performed ver well n that applcaton. Hence, n ths paper, we propose to compute spatal llumnaton functons, also known as the transmt beampattern, b waterfllng on a spatal-spectral varance functon, σ H ( k, k). Ths functon can be obtaned n a smlar manner as () b usng the channel transfer functon defned b (6) for varous channel hpotheses. Each tme an observaton s made, the probablt functon descrbng the lkelhood of the dfferent hpotheses s updated, whch results n a new spatal-spectral varance functon and new transmt beampattern for the net observaton. Before ths spatal-doman waterfllng can occur, however, some addtonal detals must be addressed. Frst, n ths applcaton there trul are an nfnte number of hpotheses. Snce the target parameters are contnuous, there are an nfnte number of values the can take even f the are lmted to wthn a fnte range. Furthermore, there are also man permutatons of the radar channel that depend on the number of targets present. In order to compute a spectral varance n the stle of (), the number of hpotheses must be lmted to a reasonable number. Second, the transfer functon defned n Secton II above s a three-dmensonal functon of k, k, and ω, et a spatal-onl llumnaton pattern can onl control llumnaton n the k and k dmensons, not the ω dmenson. Hence, a threedmensonal channel transfer functon wth an nfnte number of possble realzatons must be reduced to a two-dmensonal spectral varance functon wth a manageable number of possble realzatons. Frst, we propose to reduce the number of hpotheses b dvdng the three-dmensonal target parameter space nto a three-dmensonal hpothess grd. Let the normalzed target parameters k, k, and ω each be confned to the nterval [ ππ, ] and let the parameter volume be dvded nto cells as depcted n Fg.. For the moment, the dmensons of the cells are determned b mamum sstem resoluton, whch s controlled b the sze of the antenna arra and the tme duraton, T, of a sngle observaton nterval. Each cell n the volume represents a dfferent combnaton of target locaton (k and k ) and oppler shft. Furthermore, each cell s assgned a probablt that quantfes the current sstem understandng of whether or not a target s present n that cell. Unfortunatel, even for a smplfed, low-resoluton sstem wth, sa, 0 cells n each dmenson, there are cells that ma or ma not possess a target. The number of 548 Authorzed lcensed use lmted to: The Unverst of Arzona. ownloaded on Aprl 5, 009 at 6: from IEEE Xplore. Restrctons appl.

4 channel hpotheses s even larger, but the problem can be smplfed b notng that the spectral varance of each (k,k ) cell can be calculated separatel from the others. In other words, for a gven (k,k ) cell, we onl need to consder the dfferent hpotheses along the oppler dmenson. Furthermore, we propose to treat the decson of target presence n the ndvdual cells of the target volume as ndependent. Wth ths assumpton, the total varance n a (k,k ) cell s the sum of the varances n the dfferent oppler bns of that cell. Consder the th oppler bn for a partcular (k,k ) cell n the volume of Fg., and let the current probablt of target presence n that cell be P( ; k, k ). Suppose that f a target s present n that bn, that ts reflecton coeffcent s α. Accordng to (6), f the target s present, the transfer functon value for that cell s equal to the target s reflecton coeffcent. There are two hpotheses for that cell a target s ether present or not. Substtutng these two hpotheses nto the fnte-hpothess spectral varance calculaton of (), the varance for that cell becomes ( ) α ( ) σ Pk ;, k + Pk ;, k 0 P( k k) α + P( k k) ( ;, ) α ( ;, ) α ;, ;, 0 P k k P k k. () Then, under the assumpton that the decsons n dfferent cells are ndependent, the total varance for a partcular (k,k ) bn s N ( k, k ) Pk ( ;, k ) Pk ( ;, k ) (3) σ α α where N s the number of cells n the oppler dmenson. Our proposed sstem apples the spectral varance calculaton of (3) to each (k,k ) bn and then performs the waterfllng operaton of the spectral varance functon to determne the adaptve transmt power pattern for the net llumnaton. IV. AITIONAL CLOSE-LOOP ETAILS In the prevous sectons, we have descrbed our radar sgnal model and the proposed technque for matchng the transmt llumnaton pattern to the probablstc representaton of the radar channel. We have taken the mutual-nformaton-based matched waveform of [7] and adapted t to our scenaro where our hpotheses do not form a Gaussan ensemble. However, two requrements of our cogntve radar sstem have not et been addressed. Frst, we have not descrbed how to update the probabltes assocated wth the cells n the target volume. Second, the fact that the target s allowed to possess oppler shft mples that the target s movng. In the followng subsectons, we deal wth these two ssues and then summarze the sstem. A. Baesan Probablt Updates To update the probabltes assocated wth each target cell, we appl Baes rule. Let z be the space-tme data observed due to the radar sstem s th transmsson, and H 0 and H be the null and target-present hpotheses of a partcular target cell. Accordng to Baes rule, the probablt that a target s present n a gven cell after observaton of z s ( z ) P H ( z ) ( z ) p( z ) P H P H (4) where the lowercase p denotes a probablt denst functon. The probablt that a target s not present s ( z ) P H 0 ( z 0) ( 0 z ) p ( z ) P H P H. (5) The denomnator n (4) and (5) s dffcult, f not mpossble, to evaluate. However, snce the denomnator s the same for both hpotheses, t essentall serves as a scalng factor such that the sum of the two probabltes s unt. In practce, rather than evaluate the denomnator, we nstead evaluate the numerator n (4) and (5), then scale the results for a total probablt of one. One nce feature of the Baesan probablt update s that the current probablt depends on the prevous probablt, whch n turn depends on the probablt before that, and so on. Hence, all that has been learned b the sstem from pror measurements s retaned n the recursve probablt calculaton, even f the measurements themselves are not stored. Ths feature of state-space models s noted n []. B. Probablt Predcton ue to Target Moton The second requrement that stll needs to be addressed s how to deal wth targets n moton. oppler-shfted targets mples that target moton s present, and our sgnal model specfcall relates oppler shft to the target s rate of change n k. After measurng z, the sstem updates the probablt n each hpothess cell accordng to (4) and (5). However, f the target s movng, t ma have moved nto a dfferent cell b the tme the transmsson occurs. Ths causes the probabltes updated wth measurement z to be out of date b the tme the (+) th transmsson occurs. Therefore, the probabltes must be propagated forward n tme to the pont when the net transmsson wll occur. Changng target parameters and Baesan channel representaton suggest the relevance of a Kalman flter-based approach. We use the Kalman predcton equatons to update the target parameter state and covarance, whch can then be used to determne the probablt that the target wll be wthn a partcular target parameter cell at some tme n the future. Frst, t would be neffcent to propagate the probablt n ever sngle target parameter cell. Instead, we perform a soft detecton step to narrow the number of possble channel hpotheses to those wth a reasonable current probablt of havng a target present. An probablt update from (4) that 549 Authorzed lcensed use lmted to: The Unverst of Arzona. ownloaded on Aprl 5, 009 at 6: from IEEE Xplore. Restrctons appl.

5 eceeds a certan threshold s declared to be a potental target. We then perform a mamum lkelhood search wthn the assocated target parameter cell. Ths provdes a fner estmate of the potental target s true parameters. Suppose the soft detecton occurs after the th observaton. The estmates are placed nto a state vector accordng to T u ˆ ˆ ˆ k k k. (6) Furthermore, we can also ntalze the covarance assocated wth the current target state. Let ths covarance matr be M. The lnear target moton model results n the predcted target state at the net transmsson ( Δ T seconds later) accordng to u Au (7) + Begn: Unform Pror Probablt n all Target Parameter Cells Calculate Spectral Varance (3) Spatal oman Waterfllng Illumnate Scene/ Collect ata (4) Update Cell Probabltes (4),(5) No where 0 0 A 0 T Δ. (8) 0 0 After applng the Kalman predcton equatons, the target state s assumed to be a three-dmensonal Gaussan random vector wth mean Au and predcted covarance M + AM + A R g. (9) where R g s the covarance of the nput nose process that models target maneuverablt. Therefore, the probablt that ths soft-detected target falls wthn a partcular cell of the target parameter space shown n Fg. s equal to the ntegral of the pdf of the random state vector over the volume of the cell. Most of the probablt wll occur n the cell where the target s predcted state vector les. Other nearb cells wll also have some probablt that ths same target s present, dependng on the qualt of the parameter estmaton and the maneuverablt of the target. Also, b (8) we see that the k state s not coupled wth the other two parameters. Hence, the ntegraton can be broken nto a one-dmensonal ntegraton for k and a two-dmensonal ntegraton for k and k. In summar, an target parameter cells that are not llumnated b the transmt beam pattern retan ther probabltes for the net spectral varance calculaton. An llumnated cells are updated wth the Baesan equatons of (4) and (5). Fnall, f after the Baesan update an cell eceeds a pre-determned threshold for soft detecton, the potental target state n that cell s estmated and propagated forward to the net transmsson. C. Closed Loop Summar Fgure 3 summarzes the operaton of the closed-loop sstem for ntegrated detecton and trackng. The scenaro o An Cells Eceed the Soft etecton Threshold? Yes Estmate Parameters and Propagate Probabltes Forward (7),(9) Fgure 3. Closed-loop radar flow dagram. begns wth a volume of target parameters cells, each wth an ntal (low) probablt that a target s present. Snce the probablt dstrbuton s unform, the frst spectral varance and waveform pattern wll be flat. From then on, each tme data are collected, the cell probabltes are updated and checked for the presence of soft detectons. If no soft detectons are present, the net spectral varance s calculated based on the probablt updates. If a soft detecton s obtaned, relevant cells are updated b propagatng the estmated target state forward to the net transmsson tme and ntegratng the predcted pdf. V. SIMULATION RESULTS We now present snapshot results from smulaton of the proposed closed-loop sstem. The parameters for the smulaton were as follows. The antenna arra was a 0 b 0 rectangular grd, and the number of tme samples was also 0. Therefore, the total number of measurements taken n a sngle llumnaton was 000. A sngle target was added, and the target cell probabltes were all ntalzed to 0.0. Fgure 4 shows snapshot mages of the probablt map over spatal frequences for the zero-oppler bn where the target was placed, as well as the adaptve llumnaton pattern. From left to rght n Fgure 4, each row shows ) the probabltes n the (k,k ) cells of the target oppler bn just pror to waterfllng. For all llumnatons after the frst, ths s also the probablt map after the possble predcton step; ) the llumnaton pattern that results from the probablt map n ); and 3) the probablt map after updatng wth measurements. 550 Authorzed lcensed use lmted to: The Unverst of Arzona. ownloaded on Aprl 5, 009 at 6: from IEEE Xplore. Restrctons appl.

6 Tmt # Post Predcton Probablt Transmt Power Pattern Post Measurement Probablt has wdened enough that the sstem needs to put power on the target to update t agan. Although not shown, the sstem contnues to alternate naturall between llumnatng and not llumnatng the target cell. Tmt #57 Tmt #56 Tmt #35 Tmt #0 VI. CONCLUSIONS We have descrbed a framework for a closed-loop, cogntve radar sstem that adapts ts transmt beam n order to detect and track targets accordng to a probablstc representaton of the radar channel rather than a fed pattern of nterleaved search and track sweeps. Several steps are requred, but the end result s a sstem that compromses between searchng for new targets and updatng estng target tracks. The compromse at an gven tme depends on the radar sstem s current state of understandng of the channel, whch n turn depends on the results from pror llumnatons. Fgure 4. Probablt and llumnaton pattern snapshots at fve dfferent llumnaton tmes. One can see that at the frst teraton, the probablt map pror to llumnaton s constant, whch reflects constant pror probablt of target presence. After the frst llumnaton, the probablt map for that oppler bn s no longer constant. At the tenth llumnaton, the (3,3) cell from the upper left (whch s the correct spatal locaton of the target) s showng some lkelhood of target presence, but the llumnaton pattern s stll farl constant over the spatal bns. At the 35 th llumnaton, the probablt of the true target locaton hasn t changed much from the tenth llumnaton, but the llumnaton pattern shows that target presence has been deemed ver unlkel n several cells, whch s wh the receve no llumnaton power. At the 56 th llumnaton, the (3,3) cell s strong enough to be declared a target. In fact, the target parameters have been measured well enough and the target maneuverablt s low enough that the sstem does not need to put power on the target cell. Instead, power s dstrbuted to other cells where target presence s most lkel. In absolute terms, target presence n these cells s unlkel, but relatve to other cells and the target track, these cells that receve power n the 56 th llumnaton are the most uncertan. Fnall, n the 57 th llumnaton, the target track covarance REFERENCES [] S. Hakn, Cogntve radar: a wa of the future, IEEE Sg. Proc. Mag., vol. 3, no., pp , Jan [].R. Fuhrmann, Actve-testng survellance sstems, or, plang twent questons wth a radar, n Proc. th Annual Adaptve Sensor and Arra Processng (ASAP) Workshop, MIT Lncoln Laborator, Lengton, MA, Mar. -3, 003 [Cd-Rom] [3].R. Fuhrmann, Actve-testng survellance for multple target detecton wth composte hpotheses, n Proc. 003 IEEE Workshop on Statstcal Sgnal Proc., pp , Oct [4].R. Fuhrmann, One-step optmal measurement selecton for lnear Gaussan estmaton problems, n Proc. 007 Waveform verst and esgn Conf., pp. 4-7, June 007. [5].R. Fuhrmann and G. San Antono, Kalman flter and etended Kalman flter usng one-step optmal measurement selecton, n Proc. 007 Waveform verst and esgn Conf., pp. 3-35, June 007. [6] N.A. Goodman, P.R. Venkata, and M.A. Nefeld, Adaptve waveform desgn and sequental hpothess testng for target recognton wth actve sensors, IEEE J. Sel. Topcs n Sg. Proc, vol., no., pp. 05-3, June 007. [7] M.R. Bell, Informaton theor and radar waveform desgn, IEEE Trans. Info. Theor, vol. 39, no. 5, pp , Sept [8] J.R. Guerc and S. U. Plla, Theor and Applcaton of Optmum Transmt-Receve Radar, IEEE 000 Internatonal Radar Conference, Washngton, C, pp , Ma 8-, 000. [9].A. Garren, M.K. Osborn, A.C. Odom, J.S. Goldsten, S.U. Plla, and J.R. Guerc, Enhanced target detecton and dentfcaton va optmsed radar transmsson pulse shape, IEEE Proc. Radar, Sonar and Navgaton, 48(3), pp , June Authorzed lcensed use lmted to: The Unverst of Arzona. ownloaded on Aprl 5, 009 at 6: from IEEE Xplore. Restrctons appl.