POSTERIOR CRAMÉR-RAO BOUNDS ANALYSIS FOR PASSIVE TARGET TRACKING 1

Transcription

1 Vol5 No JOURNAL OF ELECRONICS(CHINA) January 8 POSERIOR CRAMÉR-RAO BOUNDS ANALYSIS FOR PASSIVE ARGE RACKING Zhang Jun Zhan Ronghui (School of Electronic Science and Engineering, National Univ of Defense echnology, Changsha 473, China) Abstract For the problem of deterministic parameter estimate, the theoretical lower bound of estimate error is the Cramér-Rao bound; while for random parameter, the lower bound of estimate error is generally termed by Posterior Cramér-Rao Bound (PCRB) Under the bacground of passive tracing where the target s state can be seen as a time-varying random parameter, PCRB of the state estimate error is analyzed in this paper, and the relation between PCRB and varied condition is also fully investigated using different simulation eamples he presented analytical method provides a theoretical base for performance assessment of all inds of suboptimal estimate algorithms used in practice Key words Passive target tracing; Posterior Cramér-Rao Bound (PCRB); Nonlinear filtering; State estimate CLC inde N84 DOI 7/s I Introduction Passive localization system has been paid much attention in the field of modern electronic warfare because of its inherent advantages such as strong hidden ability, wide applicability and long-distance sensing Generally, the range information is not available in passive system and the target s state can only be estimated through measurements Obviously, passive target tracing is in essence the problem of nonlinear filtering, and the main tas is to see optimal estimation algorithm However, the desired optimal algorithm may be too complicated to implement, for this reason, all inds of suboptimal algorithms, such as nonlinear least square estimator, pseudolinear estimator and nonlinear filters, are widely investigated in practical use For deterministic parameter estimate, the well nown Cramér-Rao Lower Bound (CRLB) gives an indication of performance limitation, so it is usually used to determine whether improved performance requirements are realistic or not for any suboptimal algorithm For the problem of passive target tracing, the error lower bound has been analyzed Manuscript received date: June, 6; revised date: October, 6 Communication author: Zhan Ronghui, born in 978, male, PhD student School of Electronic Science and Engineering, National University of Defense echnology, Changsha 473, China zhanrh@nudteducn in Refs[,] with the assumption that the velocity of target is ideally constant Under this assumption, the state of the target is an unnown but deterministic parameter So essentially, this error lower bound is nothing but the conventional CRLB for deterministic parameter estimate In practical application, however, the assumed condition of ideally constant velocity can not be met because the state of target is subject to random disturbances (modeled as random noise), and the state has to be treated as a random parameter Accordingly, the resulted error lower bound is referred to as Posterior Cramér-Rao Bound (PCRB) [3,4] his paper aims to analyze the achievable tracing accuracy in passive localization system by investigating PCRB, thus to construct a theoretical base for performance evaluation of large number of eisting suboptimal algorithms II PCRB Analysis for Nonlinear Estimation Consider the following nonlinear equations f ( +, w ) () z+ h+ ( +, v + ) () where and z are the state and measurement at instant with dimension n and m, respectively; w and v denote the mutually independent process noise and measurement noise, respectively; f and h represent the nonlinear functions

2 ZHANG et al Posterior Cramér-Rao Bounds Analysis for Passive arget racing 85 Assume { } i X i i and { } i Z zi i are accumulative state sequences and measurement sequences up to time, then the joint probability density function of X and Z at instant can be represented as p p( X, Z ) p( ) p( z ) p( )(3) i i j j i j Let JX ( ) be the n n information matri of X derived from Eq(3) and J be the inverse of the n n right-lower bloc of [ JX ( )] then the matri J will provide a lower bound on the mean square error of state If X is decomposed as X [ X ], then the information matri JX ( ) can be written as [5] A B J( X ) B C X E{ log p } { log } E p X X (4) X E{ log p } { log } E p where E{ log p } E{ log p } E{ log p } where the superscript is the transpose symbol; and denote the operators of the first and second-order partial derivatives, respectively; ie, α β β α and α α α α n According to the inversion theorem of bloc matri (see Appendi A), it is not difficult to now that J C B A B (5) hus, computation of the n n matri J involves calculation of the inverse of n ( ) n ( ) matri A Clearly, the computational compleity is very high as grows, and therefore a recursive calculation equation is more preferable Note that X+ [ X + ] at instant +, so p + and JX ( + ) can be derived as p+ p( X+, Z+ ) p( X, Z, +, z+ ) p( X, Z) p( + X, Z) p( z+ +, X, Z ) p( X, Z ) p( ) p( z ) (6) X + X + X + X + X + + E p + E p + E p + B C + D D + X E{ log p } { log } { log } E p E p D D JX ( ) { log } { log } { log } D E{ log p ( + )} (8) D (9) + E{ log p( )} + E p D E{ log p( )} E{ log p( )} D [ D ] { log ( )} () z () Similarly, with the inverse of the n n right-lower bloc of [ JX ( + )] replaced by the information submatri J, + it follows from Eq(5) and Eq(7) that A B J+ D D B C + D D D D ( J + D ) D () A he above equation gives a description for recursive computation of the information matri in general nonlinear system, and PCRB of state estimate satisfies the inequality P E[( ) ( ) ] J III System Model for Passive racing B (7) Consider the problem of passive localization and tracing in yplane, - the discrete state evolution equation of target motion can be represented as + F + w (3) where, [ y y ] is the state vector; F diag{ Fc, Fc} is the state transition matri and Fc, is the measurement time interval; w is the zero mean process noise with covariance 4 3 Q diag{ Q, Q }, /3 / Q q 3 / he measurement equations are dependent

3 86 JOURNAL OF ELECRONICS (CHINA), Vol5 No, January 8 upon the desired measurement parameters If the bearing and Doppler frequency are included, then according to Ref[6] θ tan ( / y) (4) yy + f f (5) c + y If additional measurement parameter (bearing rate) is taen into consideration, then y y θ (6) + y Accordingly, the nonlinear measurement equations can be written in compact version as [7] z h( ) + v θ + vθ f + v f (7a) z h( ) + v θ + vθ θ v f v + θ + f (7b) where v is the zero mean Gaussian noise with covariance R IV PCRB Calculation for Passive racing For system equations of Eq(3) and Eq(7), it is not difficult to derive from Eqs(8)~() that D F Q F (8) D F Q [ D ] (9) D Q + E{ h ( ) R h( )} () y ( )/( ) / r r c λr / r h( ) 3 / r ( y r cy)/( λr ) y / r 4 3 y ( )/ ( )/( ) / r ay b r r c λr y / r / r h( ) 4 3 / r ( a by)/ r ( yr cy)/( λr ) + / y / r r (a) (b) where λ is the wavelength of received signal, and r + y, a y, b y, c + yy Further, according to matri inversion lemma (see Eq(A) of Appendi A), the recursion of information matri can be represented as J ( Q + FJ F ) + + E h h { ( ) R ( )} () Generally, the initial information matri is calculated from the a priori probability density function p( ) If the state is Gaussian and ~ N ( ;, P), then J can be recursively com- puted from the initial condition J P V Simulations For the sae of simplicity, the target s motion is restricted in yplane - he measurement interval is s and the variance of process noise is q 4 in the first three eamples he observer is assumed to be fied at the origin and the initial position of the target is (, y ) (8m,6m) With the a priori covariance matri P diag{4e8,e4,4e8,e4}, PCRB, denoted as [σ P σv σpy σ Vy ] sqrt(diag{ P }), is calculated under different simulation conditions For each simulation, 5 Monte-Carlo runs are performed Eample PCRB Let {, } i M θ i fi i and 3 {,, } i M θ i θ i fi i be the measurement consisting of two and three parameters; also assume that the target s initial velocity is (, y ) ( m/s, m/s) and the wavelength is λ m (this value is supposed to be constant throughout the paper) he results are plotted in Fig and Fig when the measurement of bearing, bearing rate and Doppler frequency are 3mrad, 3mrad/s and Hz, respectively From Fig and Fig, it can be seen that a two-parameter set (including bearing and Doppler frequency) ehibits long convergent time and high tracing errors However, when the parameter bearing rate is included, the performance can be

4 ZHANG et al Posterior Cramér-Rao Bounds Analysis for Passive arget racing 87 greatly improved his is true for both position and velocity estimate Now, assume that the target has an initial velocity of (, y ) ( m/s,m/s), and still the three-parameter set is used But in this eample, the measurement are different and are assumed to be {3mrad, 3mrad/s, Hz}, {5mrad, 5mrad/s, Hz} and {mrad, mrad/s, 3Hz}, denoted by E, E and E, 3 respectively he results are plotted in Fig5 and Fig6 Fig PCRB of position error Fig PCRB of velocity error Eample PCRB under different heading angles In this eample, the three-parameter set is used and the measurement are the same as those in Eample, and 3 are used to denote different trajectories with corresponding initial velocity of (, y ) (m/s,m/s), (, y ) ( m/s,m/s) and (, y ) ( m/s, m/s), respectively, ie, the initial heading angles are 9, 35 and 8 respectively he results are shown in Fig3 and Fig4 Fig3 PCRB of position error under different trajectories Fig4 PCRB of velocity error under different trajectories As clearly illustrated in Fig3 and Fig4, different trajectories lead to different PCRBs of position error and velocity error under the same measurement precision, which is not lie the case of active target tracing he reason behind this phenomenon is that in passive system, the tracing performance is dependent upon the observability Weaer system observability usually leads to slower convergence speed and higher tracing errors, and vice versa Eample 3 PCRB Fig5 PCRB of position error Fig6 PCRB of velocity error As can be seen clearly in Fig5 and Fig6, with the growth of measurement error, the PCRBs of tracing error increase simultaneously and the convergence speed becomes slower and slower So, high precision measurement of the parameters is the basic premise for accurate target tracing Eample 4 PCRB under different process noise o further investigate the effect of process noise on state estimate, in this eample, different noise variances are used to compare the resulting PCRBs Again, the initial velocity of the target is assumed to be (, y ) (m/s,m/s) and the measurement precision of the three-parameter set is {3mrad, 3mrad/s, Hz} Specifically, V, V, and V 3 are used to represent the different variances, with q 4, 6 and 5, respectively he results are illustrated in Fig7 and Fig8 Fig7 PCRB of position error under different noise variances Fig8 PCRB of velocity error under different noise variances

5 88 JOURNAL OF ELECRONICS (CHINA), Vol5 No, January 8 It should be noted that for illustrative convenience, PCRBs of position error in Fig7 are only plotted partially (4 s), but this will not affect the conclusion of the simulation eample From Fig7 and Fig8, it is seen that on the whole, an increase of process noise will lead to higher PCRBs, both for position error and velocity error However, a careful comparison will find that the strength of process noise has less significant effect on the PCRBs of position error than other factors (discussed in the previous eamples), which means that the process noise is not the main factor affecting the target s position estimate accuracy, though it may have great effect on the velocity estimate error VI Conclusions In this paper, the general PCRB for the problem of nonlinear state estimate has been analyzed, and the calculation equations for PCRB have been derived from the application bacground of passive target tracing Four typical simulation eamples are provided to investigate the PCRBs of tracing error under different conditions From these results, it is found that the measurement parameter set, the relative geometry (between target and observer) and the measurement are three main factors which have important effect on the state estimate accuracy Specifically, both high measurement precision and strong observability will lead to lower PCRB; besides, the inclusion of bearing rate shows the obvious advantage in quicening convergence speed and reducing estimate errors By comparison, the strength of process noise has less significant effect on PCRBs of position estimate error, but it is still a main factor to affect the velocity estimate accuracy he PCRB in passive target tracing is the theoretical lower bound of estimate error that is reachable through optimal estimate technique, so it is meaningful for performance assessment of different suboptimal tracing algorithms he analytical method presented in the paper is also suitable for active target tracing Appendi A Inversion theorem of bloc matri: A B M C D M A + A DE CA A DE E CA E E D CA B Matri inversion lemma: ( A+ BCD) (A) A A B( DA B C ) DA (A) + References [] Gong Xiangyi he Study of challenge technology of single observer passive location and tracing using frequency rate and differential direction of arrival [PhD dissertation], National University of Defense echnology, Changsha, 4, 3 34 龚享铱 利用频率变化率和波达角变化率单站无源定位与跟踪的关键技术研究 [ 博士学位论文 ], 国防科学技术大学, 长沙, 4 年, 3 34 [] K Becer hree-dimensional target motion analysis using angle and frequency measurements IEEE rans on Aerospace and Electronic Systems, 4(5), 84 3 [3] P ichavsý Posterior Cramér-Rao bounds for adaptive harmonic retrieval IEEE rans on Signal Processing, 43(995)5, 99 3 [4] N Bergman Recursive Bayesian estimation: Navigation and tracing applications [PhD dissertation], Linöping University, Sweden, 999, [5] P ichavsý, C H Muravchi, and A Nehorai Posterior Cramér-Rao bounds for discrete-time nonlinear filtering IEEE rans on Signal Processing, 46(998)5, [6] Sun Zhongang, Zhou Yiyu, and He Liin Active and passive location techniques with single/multiple observers Beijing, National Defence Industry Press, 996, 孙仲康, 周一宇, 何黎星 单多基地有源无源定位技术 北京, 国防工业出版社, 996 年, [7] Li Zonghua Study on passive location and tracing technology for moving emitter using non-maneuvering single observer [PhD dissertation], National University of Defense echnology, Changsha, 3, 8 李宗华 无机动单站对运动辐射源的无源定位跟踪技术研究 [ 博士学位论文 ], 国防科学技术大学, 长沙, 3 年, 8