Sensor management for PRF seletion in the tra-before-detet ontext Fotios Katsilieris, Yvo Boers, and Hans Driessen Thales Nederland B.V. Haasbergerstraat 49, 7554 PA Hengelo, the Netherlands Email: {Fotios.Katsilieris, Yvo.Boers, Hans.Driessen} @nl.thalesgroup.om Abstrat We onsider the pulse repetition frequeny (PRF) seletion problem for target traing in the ontext of trabefore-detet. Two sensor management riteria are proposed for solving the PRF seletion problem. From the information theoreti lass, a Kullba-Leibler divergene based riterion is employed and from the tas based lass, a riterion that is based on the ovariane matrix of the posterior density. The proposed sensor management riteria sueed in resolving the ambiguities, avoid hoosing a PRF that would plae the target in a blind zone and they both produe an unexpeted but interesting aspet in the results. That is, they an prevent ambiguities from reappearing due to the motion model without being expliitly designed to handle this unpredited problem. I. INTRODUCTION The use of PRF during the operation of a radar auses problems suh as blind zones, target range and veloity ambiguities that the use of a single PRF annot resolve [1]. The lassial solution to the PRF seletion problem is to perform PRF staggering (or PRF jittering) [1], meaning the repetitive (or random) use of at least 3 different PRFs. Even though this approah resolves the range and veloity ambiguities, it has ertain disadvantages. During the staggering (or jittering), the target an be plaed in a blind zone aused by one or more PRFs. This is undesirable beause the orresponding measurement is wasted. Furthermore, these methods are ad-ho solutions and not optimal in any sense. In [], the authors use an evolutionary algorithm that selets 3 out of 8 or 9 PRFs to be transmitted within one dwell time for target searh purposes. On the ontrary we will try to use one PRF per dwell and solve the ambiguities on dwell to dwell basis for target traing purposes. In [3], simulated annealing is applied to obtain a medium PRF (3 out of 8) set by minimizing the range and veloity blind areas in order to ahieve an improved blind zone map. In [4], the maximization of the mutual information is used as a riterion for PRF seletion and it is ompared to random and oprime PRF seletion. We use the maximum expeted Kullba-Leibler divergene whih is equivalent and we ompare it to another adaptive ovariane-riterion. For addressing the PRF seletion problem, we loo into the target traing system and its goal, meaning the estimation of F. Katsilieris is also a PhD student at the Dept. of Applied Mathematis of the Univ. of Twente, Enshede, the Netherlands. the posterior probability density of the target states (position, veloity et.) onditioned on the reeived measurements and possibly the extration and presentation to the operator a proper point estimate [5]. Given the goal of the traing system, it an be seen that there are two lasses of riteria that are diretly related to the quantities produed by the traing system. The information theoreti lass whih taes into aount the probability density of the target states and the tas based lass whih taes into aount a derivative quantity, most ommonly a point estimate suh as the minimum variane estimate. From the first lass, we propose employing the maximum expeted Kullba-Leibler divergene riterion and from the seond lass we propose employing the minimum trae of the expeted ovariane matrix of the posterior density. The motivation for hoosing these two riteria is that we want to find optimal and not ad-ho solutions to the PRF seletion problem. Furthermore, there is an ongoing disussion on whether information theoreti or tas-based riteria should be used and what is the pratial interpretation of the information theoreti riteria. A more elaborate disussion on the later subjet an be found at [6]. The ontributions of the approah presented in this paper are: a systemati, riterion-based manner to address the PRF seletion problem the appliation of two types of sensor management riteria and their implementation by means of a partile filter the illustration of the advantages of the proposed riteria by means of a realisti simulated example The rest of the paper is organized as follows. In setion II the problem under onsideration is desribed and in setion III the system desription is given. The proposed solution is desribed in setion IV and in setion V the simulation results are presented. Finally, in setion VI the onlusions are disussed and some open questions are also presented. II. PROBLEM FORMULATION We onsider a senario a target has to be traed by a radar and the radar an utilize several PRFs, of whih only one an be used at eah time of transmission. The fat that the radar needs to transmit pulses with a given frequeny auses the following problems [1]:
blind (range) zones exist the target annot be deteted. This happens beause the radar antenna annot reeive any ehoes while transmitting a pulse. range ambiguities exist due to the PRF. Assume for example that there have been transmitted n pulses and then the radar starts reeiving an eho. How an it be sure from whih pulse the eho was reeived and therefore exatly the target is? veloity ambiguities exist beause it is not possible to diretly measure the pulse duration differene due to the Doppler effet. For this reason, the phase differene between the transmitted and the reeived pulses is measured. Obviously, the phase shift is subjet to a modulo π operation and therefore aliasing an happen. onfliting PRF requirements for resolving range and veloity ambiguities. In order to avoid the range ambiguities, low PRFs have to be used but in order to avoid the veloity ambiguities, high PRFs have to be used. The system under onsideration an be mathematially desribed by the following (disrete time) state and measurement equations: s = f(s 1, w 1 ) (1) z = h(s, P RF, v ) () s 0 p(s 0 ) (3) = 1,,... is the time index, s = [x v x y v y ρ ] T R 5 is the 5 dimensional state of the system desribing the position and veloity of a target in Cartesian oordinates and the amplitude of its eho, w is the 5 dimensional proess noise with probability density p w (w ), P RF is the hosen PRF at time, z R 3 is the reeived radar measurement, meaning the refleted power level of the target in the N r N d N b sensor ells, N r, N d, N b are the number of range, Doppler, and bearing ells respetively, v is the 3 dimensional measurement noise with probability density p v (v ), s 0 is the initial state of the system with probability density p(s 0 ). The vetor and possibly non-linear funtion f( ) : R 5 R 5 desribes the dynamis of the system. Similarly, the vetor and possibly non-linear funtion h( ) : R 5 R 3 desribes how the measurement z is related to the system state s and the hosen PRF P RF. The onsidered problem amounts to finding the optimal, in the sense of the proposed riteria, sequene of P RF of the pulses to be transmitted. The hosen sequene of PRFs will then be used for solving the attahed filtering problem of determining the posterior probability density funtion p(s Z, U ) that desribes the inemati properties and the amplitude of the target. By Z = {z 1,..., z } the measurement history is denoted and by U = {P RF 1,..., P RF } the hosen PRF history. A. Dynamial model III. SYSTEM SETUP A target with simple dynamis will be onsidered and therefore a linear Gaussian nearly onstant veloity motion model [7] will be employed: : F = Σ = s +1 = f(s, w ) = F s + w (4) 1 T 0 0 0 0 1 0 0 0 0 0 1 T 0 0 0 0 1 0 0 0 0 0 1 w N (µ, Σ) b x T 3 /3 b x T / 0 0 0 b x T / b x T 0 0 0 0 0 b y T 3 /3 b y T / 0 0 0 b y T / b y T 0 0 0 0 0 b ρ and b x = b y are the power spetral densities of the aeleration noise in the x y diretion, T is the sampling time, µ = [0 0 0 0 0] T is the mean of the Gaussian noise and b ρ is the variane of the inrement in amplitude. B. The role of PRF in the radar measurement model Firstly, the hoie of PRF affets the maximum unambiguous range (r fold ) and veloity (d fold ), see (5). If the range (or veloity) of the target is higher than r fold (or d fold ) then the radar annot be sure what is the orret range (veloity) of the target beause any target return from r + n P RF would give the same measurement, r (0, P RF ) and n = 0, 1,.... A similar relationship holds for the veloity domain. Seondly, the range and veloity resolution ( r, d) depend on the hosen PRF, the pulse ompression fator P CR and the number of transmitted pulses n P, see (6). Thirdly, the length of the blind zones (r blind ) depends on the pulse width and the loation of the blind veloities (d blind ) depends on the hosen PRF and the wavelength of the waveform arrier, see (7) n = 0, 1,,.... r fold = P RF, d fold = λ P RF (5) r = P CR P W, d = λ P RF (6) n P r blind = P W, d blind = n λ P RF (7) By using the equations for r fold and r blind we an derive an expression for the blind zones the target annot be deteted: [ r n, P RF ] n + P W, n = 0, 1,... (8) P RF
r is the distane between the radar and the target at time. C. Measurement model The onsidered appliation deals with traing a target in the tra-before-detet ontext. This means that the reeived measurements are not thresholded in order to obtain plot measurements. On the ontrary, all the N r N d N b sensor ells are onsidered. We will follow the approah presented in [8] with the differene that in the onsidered senario there is only one target and no target birth or death. In eah ell, the measurement will be: z ijl (s, P RF ) = z ijl A, (s, P RF ) = A h A (s, P RF ) + v (9) z ijl A, (s, P RF ) is the omplex amplitude data of the target in the ell ijl, A = ρ e iϕ is the omplex amplitude of the target, ϕ (0, π), h A (s, P RF ) is the refletion form and v is omplex Gaussian noise with zero mean and ovariane σ. The refletion form h A (s, P RF ) is given by: h A (s, P RF ) = e (r i r ) R (d j d ) D (b l b ) B (10) i = 1,..., N r, j = 1,..., N d, l = 1,..., N b, R = ( r ), D = ( d ), B = ( b ) are onstants related to the size of a range, a Doppler and a bearing ell respetively. r, d, b are the range, Doppler and bearing resolutions of the radar and r = ( x + y mod P RF ( mod ) (11) d = ṙ = x v x + y v y x + y (1) b = artan(y /x ) (13) are the apparent target range and Doppler and its bearing, is the speed of light and λ is the wavelength of the waveform arrier. These measurements, onditioned on the states s of the target, are assumed to be exponentially distributed and therefore the lielihood funtion p(z ijl s, P RF ) will be: p(z ijl s, P RF ) = 1 with P = ρ and h ijl P (s, P RF ) = µ ijl e 1 ) µ ijl zijl (s,p RF ) (14) µ ijl = E[z ijl (s, P RF )] = P h ijl P (s, P RF ) + σ (15) [ ] h ijl A (s, P RF ) = e (r i r ) R (d j d ) D (b l b ) B (16) As it an be notied from (15,16) and (11,1,13) the reeived measurement depends both on the target states (position, veloity, amplitude) and on the PRF that is hosen. Therefore, z ij = v, h ij (s, P RF, v ), if no target in ell ij or (8) is true if target in ell ij and (8) is false (17a) (17b) h ij (s, P RF, v ) is given by (9). This means that if we are not areful, we might even hoose a PRF that puts the target in a blind zone and therefore maes the target undetetable. This is espeially important in the tra-before-detet ontext, the targets usually have low SNR and no measurements should be wasted. IV. PROPOSED SOLUTION We propose solving the desribed target traing problem by employing sensor management riteria for hoosing the best PRF and the reursive Bayesian estimation theory for reursively estimating the posterior density p(s Z, U ). A. Reursive Bayesian estimation In the reursive Bayesian estimation ontext, given a probability density funtion p(s 1 Z 1, U 1 ), the predition step is performed using the Chapman-Kolmogorov equation: p(s Z 1, U 1 ) = p(s s 1 ) p(s 1 Z 1, U 1 ) ds 1 (18) p(s s 1 ) is usually determined by the inematis model of the target. Then the preditive density p(s Z 1, U 1 ) is updated with the reeived measurement z using Bayes rule p(s Z, U ) = p(z s, P RF ) p(s Z 1, U 1 ) p(z Z 1, U ) p(z s, P RF ) is the lielihood funtion and (19) p(z Z 1, U ) = p(z s, P RF ) p(s Z 1, U 1 ) ds (0) is a normalizing onstant whih in pratie does not have to be alulated if we employ a partile filter. Therefore, it will hold that p(s Z, U ) p(z s, P RF ) p(s Z 1, U 1 ) (1) and (18,19) an be easily approximated using a standard SIR partile filter [5] with N partiles s i and orresponding weights q i : {s i, q i }, i = 1,..., N () suh that the approximation onverges to the true posterior distribution p(s Z, U ) as N, see [9].
B. Sensor management riteria As it is disussed in the introdution, riteria from two lasses will be used. From the information theoreti lass, the maximum expeted Kullba-Leibler divergene will be used. From the tas-based lass, the minimum trae of the expeted ovariane matrix of the posterior density will be used. 1) Maximum expeted Kullba-Leibler divergene: The maximum expeted Kullba-Leibler divergene and the minimum onditional entropy are theoretially equivalent for sensor management purposes [6] but the KL-based riterion has lower omputational omplexity and this is why it is hosen, see the partile approximations in [10] and [11]. The Kullba Leibler divergene between two densities q(s) and p(s) is given by ( ) q(s) KL[q(s) p(s)] = q(s) log ds (3) p(s) As suggested in [10] for example, the maximum expeted KL divergene between the preditive and the simulated posterior density an be used for hoosing the best PRF P RF. The sensor management riterion would then be: P RF = arg max P RF E Z {KL[q(s) p(s)]} (4) q(s) = p(s z, Z 1, P RF, U 1 ) (5) p(s) = p(s Z 1, U 1 ) (6) and z is the simulated measurement using s and P RF. We use a partile approximation of the expeted KL divergene similar to [10]. In the following formulas, z p will denote the simulated measurement at time, using s p with weight q p 1 and P RF. E Z [KL(p(s z, Z 1, P RF, U 1 ) p(s Z 1, U 1 ))] = p(z s, P RF ) p(s Z 1, U 1 ) ( ) p(z s, P RF ) log ds dz p(z Z 1, U 1, P RF ) P { ( q p p(z p 1 log sp, P RF )} ) ˆp M (z p Z (7) 1, U 1, P RF ) p=1 ˆp M (z p Z 1, U 1, P RF ) = M { q m 1 p(z p sm, P RF ) } m=1 (8) This evaluation is repeated K times (1 time per PRF) and then the PRF that gives the highest value in (7) is hosen. In (7) and (8), M denotes the number of partiles used within the riterion and P is the number of simulated measurements. The omputational omplexity of this riterion is O(MP K) and the orresponding omputational omplexity of the equivalent onditional entropy would be O(M P K). ) Minimum trae of the expeted ovariane matrix: If we assume that the unertainty about the target s attributes (position, veloity and amplitude) is suffiiently represented by the mean and the ovariane of the orresponding probability density funtion, then it is intuitive to hoose a riterion that selets the PRF whih leads to the minimum trae of the expeted ovariane matrix of the posterior density. The ovariane of a probability density funtion p(s) of a random variable S is given by: Cov[p(s)] = (s µ s )(s µ s ) T p(s) ds (9) s µ s = s p(s)ds is the expeted value of S. The orresponding sensor management riterion would then be: P RF = arg min P RF tr[e Z {Cov[p(s z, Z 1, P RF, U 1 )]]} (30) We approximate (30) as follows: E Z [Cov(p(s z, Z 1, P RF, U 1 ))] = p(z Z 1, P RF, U 1 ) (s µ s )(s µ s ) T p(s z, Z 1, P RF, U 1 ) ds dz { P M } q 1(s m m µ p s )(s m µ p s ) T p=1 q p 1 µ p s = m=1 s p(s z p, Z 1, P RF, U 1 ) ds (31) M q m s m (3) n=1 s m p(s z p, Z 1, P RF, U 1 ) (33) whih is evaluated using z p in (18) and (19). We remind to the reader that z p denotes the simulated measurement at time, using s p with weight qp 1 and P RF. The updated weight q m is evaluated using the simulated measurement zp in (14) followed by a normalization step. Again, this evaluation is repeated K times (1 time per PRF) and then we hoose the PRF that gives the lowest value in (30). In (31) and (3), M denotes the number of partiles used within the riterion and P is the number of simulated measurements. The omputational omplexity of the ovariane based riterion is O[(M + 1)P K] whih is between O(MP K) and O(M P K).
TABLE I THE CHOSEN PRFS, PULSE WIDTHS (P W ), PULSE COMPRESSION FACTORS (P CR) AND NUMBER OF PULSES (n P ). PRF PW PCR n P Hz se 10 6 1.4 53 0.013 3 4 18.9 0.036 8 5 15.1 0.045 10 5.5 13.7 0.05 11 3.5 3. 0.1 47 Fig.. The expeted KL divergene between the preditive and the posterior density for eah PRF. The PRFs that put the target in a blind zone result in a lower KL divergene and therefore they are not hosen. Fig. 1. The senario onsidered in our simulations. V. SIMULATIONS In the simulated senario, the radar is assumed to be at the origin of the axes. The target to be traed starts at = 0 from [x true 0, y0 true ] = [74., 74.] m and moves with onstant veloities vx true = vy true = 300 m/s for 60 se towards the radar. Its SNR is assumed to be 11 db. The hosen PRFs along with the orresponding pulse widths (P W ), pulse ompression fators (P CR) and number of transmitted pulses (n P ) are shown in Table I. They were hosen suh that the range and veloity resolutions and the duty yle (P RF P W ) are onstant. These onditions mae sure that no PRF is favored due to better resolution or more overed area. Fig. 1 depits the senario under onsideration in our simulations. In Fig. 1 the position of the radar, the trajetory of the target and the blind zones aused by eah PRF an be seen. The N = 10 4 partiles are initially distributed uniformly suh that: r 0 [0, 115] m, b 0 [0.75, 0.85] rad, d 0 [ 500, 0] m/s SNR 0 [4, 16] db ρ 0 = σ 10 SNR/10 [1.5849, 6.3096] W atts ϕ 0: is onsidered random and does not affet the results We assume that we want to tra a highly maneuverable target, suh as a fighter or a missile, and therefore we use high proess noise. For the dynamial model we use: b x = b y = 400 m /s 4 and b ρ = 10 3 W atts T = 1 se and = 1,..., 60 se The parameters for the measurement model are: λ = 0.03 m, = 3 10 8 m/s and σ = 1/ beam width b = 0.1 rad 5.7 o r, d aording to (11, 1) and Table I Due to the high omputational load involved in our experiments, we only hoose 100 out of the N = 10 10 3 partiles and we simulate 1 measurement per hosen partile for the evaluation of the riteria, meaning 100 measurements in total. The hoie of the 100 partiles is performed by a multinomial resampling step. This proedure has to be repeated 5 times beause we employ 5 different PRFs. Aording to the notation in [10], we use M = 100 partiles, P = 100 simulated measurements (1 from eah partile) and K = 5 (5 PRFs) for evaluating the riteria. Fig. and 3 show a harateristi example of the obtained sensor management results for the two riteria. In Fig., higher KL divergene represents better PRF hoie and therefore, the PRFs that would put the target in a blind zone are avoided beause they lead to lower KL divergene. On the ontrary, in Fig. 3, lower trae of the ovariane matrix represents better PRF hoie and therefore, the PRFs that would put the target in a blind zone are avoided beause they lead to higher trae of the ovariane matrix. Fig. 4 and 5 show the sequene of the hosen PRFs produed by the two riteria. It an be notied that the highest PRF is preferred. This an be explained by the fat that a high proess noise is used for traing a highly maneuverable target. This leads to ambiguities being reated at every time instane in the veloity domain and therefore the highest PRF must be hosen for resolving them. The aforementioned explanation was verified by a new set of experiments with lower proess noise the medium PRFs were also hosen, provided that they would not plae the target in a blind zone. VI. CONCLUSIONS The proposed riteria were found to produe similar results in the sense that both riteria resolve the ambiguities and avoid hoosing a PRF that would plae the target in a blind zone. Furthermore, both riteria produed an unexpeted but interesting result, namely the highest PRF would be preferred
Fig. 3. The trae of the expeted ovariane matrix of the posterior density for eah PRF. The PRFs that put the target in a blind zone result in a higher ovariane and therefore they are not hosen. Fig. 4. The sequene of the hosen PRFs by KL-based riterion. Notie that the highest PRF is preferred. by both riteria unless its seletion would lead to plaing the target in a blind zone. The seletion of the high PRF depends on the maneuverability of the target to be traed, as verified by a seond series of experiments traing targets with lower maneuverability did not lead to high PRF preferene. The fat that both riteria managed to detet that the motion model reates problems in the traing proess by onstantly introduing veloity ambiguities and they managed to tale this problem by hoosing the highest PRF when neessary is an extra advantage over the lassial solutions. PRF staggering would address this problem every 5 se, when the lowest PRF Fig. 5. The sequene of the hosen PRFs by the ovariane-based riterion. Notie that the highest PRF is preferred. would be used, and PRF jittering would address it at random time instanes. We would also lie to point out that the obtained results are also appliable when plot measurements are used instead of the unthresholded measurements. An interesting extension to the presented solution would be to also inlude lutter effets both in Doppler and range. We would also lie to explore the behavior of the riteria in a maneuvering and/or multi-target senario. Another interesting and partially open question is to explore how the results obtained by the KL based riterion perform in the ontext of the ovariane based riterion and vie versa. Aording to the disussion in [11] about the representation of unertainty in unimodal distributions, we expet the aforementioned omparison to indiate that the riteria produe very similar results. A more pratial explanation is that the riteria have similar behaviors beause both resolve the ambiguities and try to avoid the blind zones. ACKNOWLEDGMENTS The researh leading to these results has reeived funding from the EU s Seventh Framewor Programme under grant agreement n o 38710. The researh has been arried out in the MC IMPULSE projet: https://mimpulse.isy.liu.se The authors would also lie to anowledge Mélanie Boquel (Thales Nederland B.V.) for providing the tra-beforedetet algorithm [1]. REFERENCES [1] M. I. Solni, Introdution to Radar Systems, 3rd ed. MGraw-Hill Siene/Engineering/Math, 00. [] C. Alabaster, E. Hughes, and J. Matthew, Medium PRF radar PRF seletion using evolutionary algorithms, IEEE Transations on Aerospae and Eletroni Systems, vol. 39, no. 3, pp. 990 1001, July 003. [3] S. Ahn, H. Lee, and B. Jung, Medium PRF set seletion for pulsed Doppler radars using simulated annealing, in Proeedings of IEEE Radar Conferene (RADAR), may 011, pp. 90 94. [4] C. Kenyon and N. Goodman, Range-Doppler ambiguity mitigation via losed-loop, adaptive PRF seletion, in International Conferene on Eletromagnetis in Advaned Appliations (ICEAA), sept. 010, pp. 608 611. [5] B. Risti, S. Arulampalam, and N. Gordon, Beyond the Kalman filter. Arteh House, 004. [6] E. Aoi, A. Baghi, P. Mandal, and Y. Boers, A theoretial loo at information-driven sensor management riteria, in Proeedings of the 14th International Conferene on Information Fusion, vol. 1, 011, pp. 1180 1187. [7] M. G. Rutten, B. Risti, and N. J. Gordon, A omparison of partile filters for reursive tra-before-detet, in Proeedings of the 8th International Conferene on Information Fusion, 005, pp. 169 175. [8] Y. Boers and H. Driessen, A partile filter multi target tra before detet appliation, in Proeedings of IEE Radar, Sonar and Navigation, 004, pp. 351 357. [9] X.-L. Hu, T. Shon, and L. Ljung, A basi onvergene result for partile filtering, IEEE Transations on Signal Proessing, vol. 56, no. 4, pp. 1337 1348, april 008. [10] A. Douet, B. Vo, C. Andrieu, and M. Davy, Partile filtering for multi-target traing and sensor management, in Proeedings of the 5th International Conferene on Information Fusion, 00, pp. 474 481. [11] Y. Boers, H. Driessen, A. Baghi, and P. Mandal, Partile filter based entropy, in Proeedings of the 13th International Conferene on Information Fusion, 010. [1] M. Boquel, H. Driessen, and A. Baghi, A partile filter for TBD whih deals with ambiguous data, in Proeedings of the IEEE Radar Conferene, 01.