Adaptive Waveform Design in Rapidly-Varying Radar Scenes

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1 Adaptive Waveform Design in Rapidly-Varying Radar Scenes YingLi,WilliamMoran,SandeepPSira,AntoniaPapandreou-Suppappola,andDarrylMorrell SenSIPCenter,DeptofElectricalEngineering,ArizonaStateUniversity,Tempe,AZ TheUniversityofMelbourne,Parkville,Vic3010,Australia Zounds,Inc,Mesa,AZ DepartmentofEngineering,ArizonaStateUniversity,PolytechnicCampus,Mesa,AZ Abstract We consider a waveform-agile sensing algorithm for designing transmitted waveforms in rapidly-varying radar scenes to improve target detection performance Specifically, we first track the scattering function of rapidly-varying sea clutter in low signal-to-clutter ratios(scrs) at each burst by estimating the clutter s space-time covariance matrix Simultaneously, we schedulethewaveformtobetransmittedinthenextburstby minimizing the sea clutter influence based on the estimated clutter statistics The effectiveness of our waveform-agile sensing approach is demonstrated by detecting a moving target in heavy sea clutter using configured waveforms, and then comparing the resulting performance to that of detecting the target using fixedparameter linear frequency-modulated waveforms I INTRODUCTION Waveform diversity and design has recently been successfully used in radar applications, such as target detection and tracking, to improve radar performance Increasing the accuracy of target detection and tracking can be of great importance in secure navigation, military and coastal security operations, and maritime rescue As a result, waveform-agility can be pursued in real applications, especially with new developments in radar hardware and technologies that make it possible to design the transmitted waveform on-the-fly When detecting or tracking a moving target in low SCR, waveform design becomes a challenging problem, especially for fast varying radar scenes Although approaches to this problem under slowly-varying conditions have been proposed[1],[2], situations in which the radar scene varies quickly present significant additional difficulties Inthiswork,weproposeawaveformdesignmethodthat adaptively chooses the parameters of the phase-modulated (PM) signal to be transmitted at the next time instant in ordertominimizetheeffectsoflowscrclutterinrapidvarying radar scenes The method exploits a formulation of the space-time representation of the clutter scene in the scattering function domain This formulation is vectorized to obtain a dynamic system characterization for detecting a target in heavy sea clutter Our method first estimates the spacetime covariance matrix of the sea clutter using the multiple particle filter sequential Monte Carlo method[3] This method is chosen in order to overcome an inherent dimensionality problem Then, an optimization procedure is used to find the PM waveform parameters that minimize the effect of the clutter and improve detection The effectiveness of this method isdemonstratedbydetectingamovingtargetinahighsea clutter scenario The paper is organized as follows In Section II, we describe the rapidly-varying radar scene and radar returns, and Section III provides the estimation method of the clutter space-time covariance matrix In Section IV, we propose the waveformagile algorithm for target detection in rapidly-varying clutter, and we provide simulations illustrating the performance of the algorithm in Section V II RAPIDLY-VARYING RADAR SCENE CARACTERIZATION In a rapidly-varying radar scene, we consider a radar operatingat f s zpulserepetitionfrequency(prf)thattransmits aburstof Kpulsesineachdwell,asshowninFigure1The length N s pulse s n [i], i = 0,1,,N s 1,istransmitted repeatedly K times throughout the nth dwell Fig 1 Rapidly-varying radar scene transmitting K pulses in each dwell A Scattering matrix Weconsiderthe kthpulse, k = 1,,K,andthe mth rangebin, m = m 0,m 1,,m Mn 1atthe nthdwellthen, the corresponding complex reflectivity coefficients of the aggregatescatterersontheseasurfacearedenotedby x n [m,k] Thecoefficientsformthereflectivitymatrix B n thatisgiven 2009 International WD&D Conference /09/$ IEEE

2 by 2 3 x n[m 0, 1] x n[m 0, 2] x n[m 0, K] x n[m 1, 1] x n[m 1, 2] x n[m 1, K] B n= x n[m Nv 1, 1] x n[m Nv 2, 1] x n[m Nv 1, K] where m 0 isthelowestrangebininthevalidationgateat dwell n,andthevalidationgateatdwell nhas M n rangebins Thereflectivitymatrixhasdimensions N v K,where N v = M n + N s 1Therange(fasttimeordelay)increasesdown the columns, and the transmitted pulses(slow time) increase acrosstherowsof B n Thescatteringmatrix A n isdefinedwithelementsthatare obtained by taking the short-time Fourier transform of the elementsof B n intheslow-timedirection, A n [m,l] = 1 K B n [m,k]e j2πkl/k K k=1 where l [ K 1 2,, K 1 2 ]Then, A n = B n D (1) where D is the discrete Fourier transform matrix Thus, A n contains the range-doppler description of the complex reflectivitiesin B n Thescatteringmatrix A n providestherange-dopplerdescriptionofthescatterercomplexreflectivitiesinmatrix B n ThisisdemonstratedinFigure2Eachelementin A n describes the states of the scatterers In each range bin, along the slow-time, static scatterers are represented in the middle column; the first (K 1)/2 elements represent the negative Doppler shifts and are related to the scatterers moving away from the sensor with different velocities; the last (K 1)/2 elements represent the positive Doppler shifts and are related to the scatterers moving toward the sensor with different velocities According to the characteristics of sea clutter, most scatterers reside around the middle of each row(corresponding to zero Doppler) B Observation and dynamic models InFigure1,thereturnfromeachdwellissampledat f b ztoyield y n [k,m] = y n (k,mt b ), k = 1,,K, m = m 0,m 1,,m Mn 1,where T b = 1/f b isthefastsampling intervalthen,theradarreturnatthe nthdwellisgivenby: y n [k,m] = N s 1 i=0 x n [k,m i]s n [i] + v n [k,m] (2) where v n [k,m]isadditivewhitegaussiannoisenotethatwe haveassumedthatanidenticalpulse s n [i],i = 0,1,,N s 1 is transmitted throughout each dwell 1 X [ m, K ] K 1 0 X[ m, ] K 1 X[ m 1, ] 1 Nv [ ] 2, K XmNv 1 2 Fig 2 Scattering matrix of sea clutter and the evolution of the scatterers Thetransmittedsignalmatrix P n canbeexpressedasan M n N v matrixgivenby s n [0] s n [1] s n [N s 1] 0 0 s n [0] s n [1] s n [N s 1] P n = 0 0 s n [N s 2] s n [N s 1] Then, using the transmitted signal matrix and the radar returns in(2),wecanformthe M n Kobservationmatrix Y n as Y n = P n B n + V n (3) ere, V n isthe M n Knoisematrixatdwell nwithelements v n [m,k] We assume that the scatterers and the target move with constant velocity As the scatterers move, some of them that do not stay in the middle columns of A n will move out of the validation gate, and some range-doppler cells will become empty As shown in Figure 2, the solid parallelogram representsthescatteringmatrixatthe (n 1)thdwell,and the dashed-dotted rectangular represents the scattering matrix atthe nthedwellthecellsmovingoutofthematrixwill be populated according to the adjacent elements in the same column(same Doppler shifts), which have constant velocity To better describe the evolution of the scattering matrix, an evolution matrix F is introduced, that describes the movement ofthescatterersandthepopulationoftheemptycellsthe new scatterer cells that are populated into the scattering matrix are described using exponential weighted summations of the complex reflectivities in the immediate neighborhood of these cells, as illustrated in Figure 2 This procedure will continue along each column until all empty cells are populated We vectorizethematrix A n toform a n =vec(a n )bystacking thecolumnsof A n fromlefttorighttoformalength KN v vector The evolution of the vectorized scattering matrix is given by a n = Fa n 1 + w n (4) 2009 International WD&D Conference /09/$ IEEE

3 Thezero-mean,complexGaussiannoise w n hascovariance Q n, and F incorporates the scatterer movements between dwells and populates the range-doppler cells that move into the validation gate The matrix F is a KN v KN v block-diagonalmatrixwhen l = (K 1)/2,, 1,the blockmatrixisgivenby 2 l 1 e l α (2 l 1 l + 1)e (Nv+ l 1)α e α e Nvα F l = When l = 1,,(K 1)/2,theblockmatrixis 0 0 F l = 0 1, e Nvα e α (2 l 1 l + 1)e (Nv+l 1)α 2 l 1 e lα andwhen l = 0,thescatterersarenotmoving,andtheblock matrixisan N v N v identitymatrix I Nv Basedontherelationshipbetweenthereflectivitymatrix B n andthescatteringmatrix A n in(1),theobservationmatrix Y n in(3)canalsobewrittenas Y n = P n A n D 1 + V n Using thematrixpropertyvec(gzl) = (L G)z,where G, Z, and Larethreearbitrarymatrices, z =vec(z),and denotes the Kronecker product[4], the vectorized observation is then given by y n = (D P n )a n + v n = ˇP n a n + v n (5) where y n = vec(y n ), ˇPn = D P n,and denotesthe ermitian operator Thus, Equations(4) and(5) represent the dynamic state and observation equations in a state space representation where a n istheunknownstateand, a n,thevectorizedscattering matrix A n,isrelatedtotheoriginalreflectivitymatrix B n of the clutter using(1) III ESTIMATION OF SEA CLUTTER STATISTICS As we will see in Section IV, the waveform-agile detection algorithm highly depends on the accurate estimation of sea clutter statistics In this section, we discuss the clutter estimation algorithm introduced in[5] Let Σ an bethecovariancematrixof a n Thedynamicmodel in(4)canalsobedescribedas: Σ an =E[a n a n ] = FΣ an 1 F + Q n, (6) where Q n is the covariance matrix of w n Similarly, the covariance matrix of the vectorized observations in(5) can be extended to: Σ yn = ˇP n Σ an ˇP n + R n, (7) where R n isthecovariancematrixof v n Using(7),wecan obtain p(y n Σ an )toupdatethefilternotethat Q n in(6)and R n in(7)areassumedtobewishartdistributed;thisfollows fromthecovarianceofthemultinormalsamples w n and v n Also,as A n and B n arerelated,onceweestimate Σ an,we canalsoobtain Σ bn = (D I Nv )Σ an (D I Nv ) Particle filtering(pf) is a sequential Monte Carlo method that uses sampling to approximate probability density functions It provides a good approximation to Kalman filtering when the system models are nonlinear or non-gaussian[6], [7] owever, when the dimensionality of the state space is large, as in ourcase, a huge set of particles is needed to provide sufficient support and the computational complexity becomes prohibitive As discussed in [3], multiple particle filtering(mpf) can be used to overcome this dimensionality problem as follows If the dynamic and measurement equations ofasystemcanbewrittenas α n = f n (α n 1,ω n 1 )and β n = h n (α n,γ n ),where α n isahigh-dimensionalsystem statevectorattimestep n, f n and h n are(possiblynonlinear) functions,and ω n and γ n arerandomvectorsusingthempf approachin[3], α n isdividedinto Lsubvectors,givenby α n = [α T 1,n α T 2,n α T L,n ]T Each α l,n, l = 1,2,,L, is estimated using a different PF As a result, L PFs are runningsimultaneously,andthestatevector α n isputtogether accordingly at each time n In order to take advantage of Bayesian techniques to solve the system equations, we vectorize the dynamic and observation models in(6) and(7) Specifically, we obtain Σ an = (F F) Σ an 1 + Q n (8) Σ yn = ( ˇP n ˇP n ) Σ an + R n, where Σ an =vec(σ an ), Σyn =vec(σ yn ), Qn =vec(q n ) and R n =vec(r n )Aftervectorization,thedimensionality of Σ an isgivenby Ξ = (KN v ) 2 Thevalueof Ξcanbe quitelarge,evenifweconsiderasmallnumberofpulses Forexample,ifweuse K = 9pulsesand M n = 10range bins,thenevenifwereducethesignallengthto N s = 6, weobtain N v = M n + N s 1 = 15andthus Ξ = 18,225 This implies that we need to estimate an unknown dynamic state whose dimensionality is 18,225; this high dimensionality prevents direct implementation of particle filtering or Kalman filtering, even if the transformations are linear We therefore apply the MPF approach discussed above[3] In(8),theevolutionmatrix F Fisblockdiagonal, F 1 F 0 0 F F = 0 F 2 F F K F and F k isdefinedinsectioniithestructureof F Fleads to a natural decomposition of the dynamics of the state vector 2009 International WD&D Conference /09/$ IEEE

4 into K independent subsystems, Σ an = [ Λ T 1,n Λ T 2,n Λ T K,n whereeachvector Λ k,n, k = 1,,K,hasdimension KN 2 v Asaresult,wecanusetheMPFwith L = KPFsapplied simultaneously, one for each of the K subsystems For the kth subsystem, the estimation of this segment of the current state can be obtained using the dynamic and measurement models Λ k,n ] T = (F k F)Λ k,n 1 + V k,n Σ yn = ( ˇP n ˇP n ) Σ an + R n Forthe kthpf,theweightofthe ithparticle, i = 1,,M s can be updated according to w (i) k,n p(σ yn Λ i (i) w(i) k,n Λ k,n)p(λ (i) k,n Λ(i) ˆΛ k,n 1 k,n 1 ) k,n 1 π k (Λ (i) k,n Λ(i) k,n 1, ˆΛ k,n 1, Σ yn ) where Λ k,n = [ˆΛ T 1,n ˆΛT ˆΛ T k 1,n k+1,n ˆΛT k,n] T, ˆΛ T j,n = Ms i=1 w(i) j,n 1 Λ(i) j,n, j k, and π k( ) is the importance density that is chosen here to be the prior density [6] The observation used in the kth PF has a complex Gaussian distribution with zero-mean and covariance matrix Σ yn IV WAVEFORM DESIGN ALGORITM In order to improve the target detection performance in rapidly-varying radar scenes, we propose to design the transmit waveforms at each time instant Specifically, instead of transmitting the same type of pulse at all dwells, the transmitted waveform is designed at each dwell to minimize a cost function in terms of the estimated clutter covariance matrix Figure 3 demonstrates the proposed waveform design algorithmatthebeginningofthealgorithm, K pulsesofa linear frequency-modulated(lfm) chirp with fixed parameters istransmittedatthefirst(n = 1)dwellUsingtheobservations and the method described in Section III, the covariance matrix (ˆΣ an )ofthefast-varyingseaclutterisestimatedusingthis estimate, we predict the covariance matrix for the next time stepusing(6)thatis,weobtain Σ an = F ˆΣ an F Wethen design the next waveform to transmit by minimizing a cost functionthatisaimedtoreducetheeffectoftheclutterthis waveformisthentransmittedatthenexttimestep,andthe observations are used to estimate the clutter statistics at this new time step The waveform design algorithm steps are also summarized in Table I Thewaveformweuseinthedesignalgorithmateachtime step nisaunimodularpmwaveformgivenby s n (t;pm) = exp(jψ(t)), 0 t T d, wherethephasemodulationcanbeexpandedintermsofan orthogonalsetofbasisfunctionsas ψ(t) = N i=1 θ iψ i (t), where { 1, (i 1) T t i T ψ i (t) = 0, otherwise Σ ~ ˆA 1 Σ~ A 2 Σ ~ ˆA 2 Σ A ~ n Fig 3 Depiction of waveform design algorithm ere, T d and Tarethepulsedurationandsamplinginterval, respectively Inordertomitigatetheeffectoftheclutter,wechoose the waveform at time step n that minimizes the cost function givenbythetraceof ˇP n Σan ˇP n [2],[8]Recallthat ˇPn is relatedtothetransmittedwaveformmatrix P n in(3)andthus tothetransmittedsignalalso,thecovariancematrix Σ yn of thevectorizedobservationsin(5)isrelatedto ˇP n Σ an ˇP n in (7),where a n isthevectorizedscatteringmatrixoftheclutter Thus, minimizing this covariance matrix results in minimizing the effect of the clutter on the target detection The waveform selection algorithm is thus given by ˆΣ ~ A n s n(t;pm) =arg min s n(t;pm) trace( ˇP n Σan ˇP n ) (9) where Once the optimal waveform is transmitted, the target is detected using the generalized likelihood ratio test(glrt) [9]Withrespecttorangebin j,let 0 denotethehypothesis thatonlyclutterispresent,andlet 1 denotethepresenceof bothclutterandtargettheglrtdetectordecides 1 if Λ GLRT j = ln p(y n,j 1, Σ an ) p(y n,j 0, Σ > γ, (10) an ) where γ isathresholdthatachievesadesiredfalsealarm probability and can be calculated using(7) V SIMULATIONS In our simulations, we compared the detection performance ofamovingtargetinheavyseaclutterbytransmittingfixed LFM chirps and designed PM waveforms under different SCR scenariosweused K = 9pulsestotransmitineachdwell, andthevalidationgatesizewas M n = 11Thelengthofthe transmittedwaveformtransmittedwas N s = 6FortheMPF, weused L = 9PFs,andeachPFused50particlesThesimulated sea clutter was generated using a compound-gaussian model[10]; at each time step, 200 Monte Carlo simulations were used to obtain the space-time covariance matrix of the 2009 International WD&D Conference /09/$ IEEE

5 TABLE I WAVEFORM DESIGN ALGORITM DESCRIPTION db 65 db Attimestep n = 1, At the beginning of the dwell: transmit K LFM chirp pulses with fixed parameters Attheendofthedwell: Estimatetheseacluttercovariancematrix, ˆΣan for n = 2 : N D, Calculatethepredictedcovariancematrix, Σan,basedonthe previous estimate using(6) Design the transmitted waveform by: s n(t;pm) =arg min s n(t;pm) trace( ˇP n Σan ˇP n ) At the beginning of the dwell: Transmitthedesignedwaveform s n (t;pm) Attheendofthedwell: GLRT detection Estimate the covariance matrix at this time step Probability of detection db using designed waveforms using fixed LFM chirps Probability of false alarm Fig 4 Detection performances of the waveform design approach compared with using fixed waveforms under different SCR values reflectivity vector The range bin size and Doppler resolution were chosen such that, from one dwell to another, the scatterer inthe lthcolumn, l = (K 1)/2,,0,,(K 1)/2, ofthescatteringfunctionmovedby lbinsthetargetwas assumedtomovewithaknownconstantvelocityof20m/s Target returns from all pulses and range bins were combined to form the observation vector The amplitudes of the target returns were sampled from a zero-mean, complex Gaussian process with known variance Based on the estimated space-time covariance matrix of the sea clutter using the MPF method, the GLRT detector was used to detect the range bin location of the target, utilizing each pulse of each burst The PM waveforms transmitted at each dwell were designed by minimizing the cost function in (9) following Section IV The receiveroperating characteristict(roc) curves are shown in Figure 4 for57db,-65db,and-70dbscrvaluesforcomparison, the detection performance of fixed LFM chirp waveforms is alsoshowninfigure4thesewaveformsusedachirprate of1,000gz/sandtimeduration saswecan see, the waveform design approach increased the detection performance by about 5 db VI CONCLUSION We proposed a waveform-agile target detection algorithm for the difficult scenarios of rapidly-varying radar scenes We first estimated the space-time covariance matrix of the sea clutter using multiple particle filtering, and then we discussed the waveform design algorithm to adaptively design the transmitted signal at each time step The proposed technique minimizes a cost function to reduce the influence of the clutter on detecting a moving target We demonstrated the increased performance of our new approach using simulations and compared it with using fixed linear frequency-modulated waveforms REFERENCES [1] B Friedlander, A subspace method for space time adaptive processing, IEEE Transaction on Signal Processing, vol 53, pp 74 82, January 2005 [2] S P Sira, D Cochran, A Papandreou-Suppappola, D Morrell, W Moran, S oward, and R Calderbank, Adaptive waveform design for improved detection of low-rcs targets in heavy sea clutter, IEEE Journal on Selected Topics in Signal Processing, pp 56 66, June 2007 [3] PMDjuric,TLu,andMFBugallo, Multipleparticlefilering, in IEEE International Conference on Acoustics, Speech, and Signal Processing, June 2007, pp [4] J R Magnus and Neudecker, Eds, Matrix Differential Calculus with Applications in Statistics and Econometrics Wiley, 1999 [5] YLi,SPSira,BMoran,SSuvorova,DCochran,DMorrell,and A Papandreou-Suppappola, Adaptive sensing of dynamic target state in heavy sea clutter,, in IEEE Int Workshop on Computational Advances in Multi-Sensor Adaptive Processing, December 2007, pp 9 12 [6] ADoucet,NdeFreitas,andNJGordon,Eds,SequentialMonte Carlo Methods in Practice Springer, 2001 [7] MSArulampalam,SMaskell,NGordon,andTClapp, Atutorial on particle filters for online nonlinear/non-gaussian Bayesian tracking, IEEE Transactions on Signal Processing, vol 50, pp , 2002 [8] S Sira, D Cochran, A Papandreou-Suppapola, D Morrell, W Moran, and S oward, A subspace-based approach to sea clutter suppression for improved target detection, in Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, October 2006, pp [9] S M Kay, Fundamentals of Statistical Signal Processing: Detection Theory Prentice-all, 1993, vol 2 [10] KDWard,CBaker,andSWatts, Maritimesurveillanceradar,PartI: Radar scattering from the ocean surface, IEE Proc F: Communications, Radar and Signal Processing, vol 137, no 2, pp 51 62, International WD&D Conference /09/$ IEEE