POSERIOR CRAMÉR-RAO BOUND FOR RACKING ARGE BEARING Kristine L. Bell and Harry L. Van rees Center of Excellence in C 3 I George Mason University Fairfax, VA 22030-4444, USA bell@gmu.edu, hlv@gmu.edu ABSRAC he posterior Cramér-Rao bound on the mean square error in tracing the bearing, bearing rate, and power level of a narrowband source is developed. he formulation uses a linear process model with additive noise and a general nonlinear measurement model, the measurements are the sensor array data. he joint Bayesian Cramér-Rao bound on the state variables over the entire observation interval is formulated and a recursive bound on the state variables as a function of time is derived based on the nonlinear filtering bound developed by ichavsy et al (1998) and analyzed by Ristic et al (2004). he bound is shown to have the same form as when the measurements are bearing and power estimates with variance equal to the deterministic Cramér-Rao bound for a single data snapshot. he bound is compared against simulated performance of the maximum a posteriori penalty function (MAP-PF) tracing algorithm developed in Zarnich et al (2001). 1. INRODUCION We consider the problem of bounding the mean square error (MSE) performance in tracing the bearing, bearing rate, and power level of a narrowband source using observations from a linear array. A recursive form of the posterior or Bayesian Cramér-Rao bound (BCRB) [1] is derived based on the recursive nonlinear filtering bounds developed in [2] and further analyzed in [3]. he bound is shown to have the same form as when the observations are bearing and power estimates with variance equal to the deterministic Cramér- Rao bound (CRB) for a single data snapshot. he bound is compared against simulated performance of several tracing algorithms. his paper is organized as follows. Section 2 contains a review of Bayesian [1] and hybrid CRBs [4], and the BCRB for nonlinear filtering [2], along with some special cases his research was sponsored by the Defense Advanced Research Projects Agency/Special Programs Office (DARPA/SPO) under Grant #HR0011-04-1-0061. used in this paper. In Section 3, the linear process model and nonlinear array data observation model are presented, and the recursive bound is derived. Section 4 contains simulation results and Section 5 contains a summary. 2. BCRB FOR NONLINEAR FILERING 2.1. Bayesian Cramér-Rao Bound he BCRB [1] provides a lower bound on the MSE matrix for random parameters. Let z denote an n 1 vector of observations, and denote an r 1 vector of random parameters to be estimated. Let p() denote the a priori probability density function (pdf) of, p(z ) denote the conditional pdf of z given, and p(z, ) = p(z )p() denote the joint pdf of z and. We use the notation E z, } to denote expectation with respect to p(z, ), E z } to denote expectation with respect to p(z ), and E } to denote expectation with respect to p(). Let ˆ(z) denote an estimate of which is a function of the observations z. he estimation error is ˆ(z) and the MSE matrix is ] ] } Σ = E z, [ˆ(z) [ˆ(z). (1) he BCRB C provides a lower bound on the MSE matrix Σ. It is the inverse of the Bayesian information (BIM) J, Σ C J 1, (2) the matrix inequality indicates that Σ C (or equivalently Σ J 1 ) is a positive semi-definite matrix. Let η ϕ be the m n matrix of second-order partial derivatives with respect to the m 1 parameter vector ϕ and n 1 parameter vector η, 2 2 ϕ 1η 1 ϕ 1η 2 2 ϕ 1η n 2 2 η ϕ = ϕ 2 η 1 ϕ 2 η 2 2 ϕ 2 η n........ (3). 2 2 ϕ mη 1 ϕ mη 2 2 ϕ mη n.
he BIM for is defined as J = E z, ln p (z, ) }. (4) his may be expressed as a sum of two terms, J = J D + J P, (5) J P is the a priori information matrix J P = E ln p () }, (6) and J D is the contribution to the information from the data. It is the expected value of the standard Fisher information matrix ϑ() with respect to the a priori pdf p(), J D = E z, ln p (z ) } = E ϑ()}, (7) ϑ() = E z ln p (z ) }. (8) 2.2. Hybrid Cramér-Rao Bound In some cases, the parameter vector contains some parameters which are random variables, and some which are nonrandom, deterministic parameters, r =, (9) r denotes the random components and d denotes the deterministic components. he estimator ˆ(z) is assumed to be unbiased in the deterministic components, and the MSE matrix is defined as in (1), however the expectation is only with respect to the random components r. he hybrid information matrix [4] has the same form as in (5), but with expectation only with respect to r in J D, d J D = E r ϑ()}, (10) and non-zero elements only in the upper-left bloc of J P reflecting the prior information for the random components, } ] [E J P = r r r ln p ( r ) 0. (11) 0 0 Note that the upper-left bloc of the total hybrid information matrix J is the BIM for the random parameters and the lower left bloc is the standard FIM for the deterministic parameters. 2.3. Nonlinear Filtering Model and BCRB Following [2], the general state space model is x +1 = f (x, v ) (12) z = h (x, w ), (13) x is the r 1 state vector at time, z is the n 1 observation vector at time, v } and w } are independent white noise sequences, and f ( ) and h ( ) are nown nonlinear functions. he initial state x 0 is assumed to have nown pdf p(x 0 ). he relationships in (12) and (13) determine the conditional pdfs p(x +1 x ) and p(z x ). Define the collection of state vectors and observations up to time as: X = [ x 0 x 1 x ] (14) Z = [ z 1 z 2 z ]. (15) he joint pdf of the ( + 1)r 1 vector X and the r 1 vector Z is given by p(z, X ) = p(x 0 ) p(z j x j ) p(x i x i 1 ). (16) j=1 i=1 We wish to estimate the state vector X from the observations Z. he BIM and the BCRB are ( + 1)r ( + 1)r matrices. he lower right r r bloc of C, which we denote as C is the BCRB for estimating x, and it s inverse is the BIM J = C 1. In [2], the BIM is shown to follow the recursion J +1 = Ω ( D 12 ) ( J + D 11 D 12 + Γ +1, (17) D 11 = E x x x ln p (x +1 x ) } (18) D 12 x = E x +1 x ln p (x +1 x ) } (19) x Ω = E x +1 x +1 ln p (x +1 x ) } (20) Γ +1 = E z,x x +1 x +1 ln p (z +1 x +1 ) }. (21) he recursion is initialized with J 0 = E x x 0 x 0 ln p (x 0 ) }. (22) Note that the expectation in (21) is with respect to the joint pdf p(z +1, X +1 ). his expectation can first be taen with respect to the conditional pdf p(z +1 X +1 ) and then with respect to the marginal pdf p(x +1 ), i.e. Γ +1 = E x ϑ +1 } (23) ϑ +1 = E z x x +1 x +1 ln p (z +1 x +1 ) }. (24) Note that ϑ +1 is the standard Fisher Information Matrix (FIM) for estimating the state vector x +1 based on the observations z +1. he recursion can be used in the hybrid parameter case, defining the expectations in (18)-(21) only with respect to the random components, and defining J 0 analogously to (11).
2.4. Linear AWGN Process In the special case the state process model is linear with additive white Gaussian noise (AWGN), we have v N (0, Q ). hen and the recursion has the form J +1 = Q 1 x +1 = F x + v, (25) D 11 = F Q 1 F (26) D 12 = F Q 1 (27) Ω = Q 1, (28) ( J + F Q 1 F F Q 1 Q 1 F +Γ +1. (29) Applying the matrix inversion lemma as in [3], we get J +1 = ( Q + F J 1 F + Γ+1. (30) 2.5. Linear AWGN Process and Observations If the observation model is also linear with AWGN, we have w N (0, R ). hen z = H x + w, (31) Γ +1 = ϑ +1 = H +1R 1 +1 H +1, (32) and the recursion has the form [3]: J +1 = ( Q + F J 1 F Defining Υ + H +1 R 1 +1 H +1. (33) = Q + F J 1 F (34) G +1 = Υ H +1 ( H+1 Υ H +1 + R +1, (35) the BCRB obeys the recursion C +1 = J 1 +1 = Υ G +1 H +1 Υ, (36) which is the same as the recursion for the Kalman filter MSE matrix [2],[3]. 3. BCRB FOR RACKING BEARING he model consists of a moving target radiating a narrowband signal that is received by an N-element linear array, as shown in Figure 1. Let d n denote the position of the nth element of the array and let u = cos( ) denote the bearing of the target at time. he N 1 array response vector to a signal from bearing u has the form v(u ) = [ e j 2π λ d1u e j 2π λ d2u e j 2π λ d N u ], (37) λ is the wavelength of the narrowband signal. d 1 d 2 d 3 d N Fig. 1. arget/array geometry. 3.1. Nonlinear Array Data Observation Model At the array, the complex envelope of the observations has the form z = s v(u ) + w, (38) s is a complex random signal snapshot with power E[s s ] = α, and w is a complex vector of uncorrelated sensor noise samples with E[w w H ] = σ2 wi. he source signals and noise are assumed to be sample functions of independent zero-mean complex Gaussian random processes. It is assumed that the snapshots are sufficiently spaced that the observations are independent from snapshot to snapshot. he signal power, α }, is assumed to be a sequence of unnown, deterministic parameters. he noise power σ 2 w is assumed to be constant and nown. he array data z is then jointly complex Gaussian with zero mean and covariance matrix K (u, α ) = α v(u )v H (u ) + σ 2 wi, (39) and the pdf of the array data is given by p(z ; u, α ) = exp z H K 1 (u }, α )z π N. (40) K (u, α ) he standard 2 2 FIM for estimating the (nonrandom) parameters u and α from z has the form [5],[6]: ϑ = ϑu u ϑ u α, (41) ϑ α u ϑ α α γ 2 ϑ u u = 2Λ d (42) (1 + γ ) ϑ u α = 0 (43) ϑ α α = N 2 σ 4 w (1 + γ ) 2, (44)
and Λ d = d = 1 N ( ) 2 2π 1 λ N N ( dn d ) 2 n=1 (45) N d n (46) n=1 γ = N α σw 2. (47) Note that the FIM only depends on α and not u, and that the off-diagonal term ϑ u α is zero. 3.2. Linear Process Model We wish to trac the target bearing u and bearing rate u. Let the two-dimensional vector u be defined as u u =. (48) u We assume the motion of the target is described by a linear, nearly constant velocity model with random acceleration [3], [7]. Let t denote the time interval from to +1. he model is u +1 = F u u + B u n, (49) F u = B u = 1 t (50) 0 1 [ 1 ] 2 t2. (51) t he random acceleration n is a zero mean white Gaussian noise process with variance σn. 2 his model has the form of (25) with F = F u, v = B u n, and Q = Q u, Q u is the singular matrix [ 1 ] Q u = σnb 2 u B u = σn 2 4 t4 1 2 t3 1 2 t3 t 2. (52) We assume the initial state is Gaussian with nown mean and covariance, u 0 N (µ u, Σ u ). Since the observations depend on the unnown signal power α, it must also be included in the vector of state variables to be traced. We define the three-dimensional state vector x as x = u = α u u. (53) α We assume the power is non-random and constant, i.e, α +1 = α, (54) the initial value α 0 is an unnown, non-random parameter. Note that this can be written as α +1 = F α α + B α n, (55) with F α = 1 and B α = 0. his also has the form of (25) with F = F α, v = B α n, and Q = Q α Q α = σ 2 nb α B α = 0. hen, the entire state vector process model is x +1 = Fx + Bn, (56) F = B = Fu 0 = 1 t 0 0 1 0 (57) 0 F α 0 0 1 1 Bu 2 = t2 t, (58) B α 0 and the process noise covariance matrix is given by 1 Qu 0 Q = = σ 2 4 t4 1 2 t3 0 1 0 Q n 2 t3 t 2 0. (59) α 0 0 0 3.3. BCRB he state space model described in the previous two sections is a hybrid model with random component u and non-random component α. It has a linear AWGN process model and a general, non-linear measurement model. he BIM recursion is given by (30) with Q = Q defined in (59), F = F defined in (57), and Σu 0 J 0 =. (60) 0 0 he matrix Γ +1 is given by Γ +1 = E u ϑ +1 }, (61) ϑ +1 is the 3 3 FIM for estimating u +1, u +1, and α +1 from z +1, and the expectation is with respect to p(u +1 ). he FIM has the form ϑ u u ϑ u u ϑ u α ϑ u u 0 ϑ u α ϑ = ϑ u u ϑ u u ϑ u α = 0 0 0, ϑ α u ϑ α u ϑ α α ϑ αu 0 ϑ αα (62) the zero entries occur because the observations do not depend on u and we have dropped the subscript on α because it is constant. Defining the selection matrix 1 0 0 H =, (63) 0 0 1
we can write (62) as ϑ = H ϑu u ϑ u α H = H ϑ H, (64) ϑ αu ϑ αα ϑ is the FIM for u and α, given in (41)-(47). he FIM ϑ does not depend on u, therefore expectation in (61) is trivial and Γ +1 = ϑ +1 = H ϑ+1 H. (65) he BIM recursion is then given by J +1 = ( Q + FJ 1 F + H ϑ+1 H. (66) Note that this has the same form as the linear observation 1 model form in (33) with R +1 = ϑ +1. he same bound can be obtained using an observation model in which z is a two-dimensional vector of noisy estimates of u and α with mean equal to the true values and covariance matrix equal to the inverse of the FIM ϑ, i.e. z = Hx + w, (67) ( ) 1 w N 0, ϑ. Furthermore, since the off-diagonal terms ϑ u α are zero for this problem, we can write Γ +1 in partitioned form [ H Γ +1 = u ϑ u+1u H ] +1 u 0 0 H, (68) αϑ α+1 αh α H u = [ 1 0 ] and H α = 1. he partitioning of the terms related to u and α in (57), (59), (60), and (68) results in a partitioning of J +1, J u J +1 = +1 0 0 J α, (69) +1 J u +1 and Jα +1 may be calculated from separate recursions. he recursion for J u +1 is given by ( J u +1 = Q u + F u [J u ] 1 F u + H u ϑ u+1 u H +1 u, (70) with J u 0 = Σ 0. he recursion for J α +1 has a similar form with J α 0 = 0 and simplifies to J α +1 = J α + ϑ αα = ( + 1)ϑ αα, (71) which is the standard Fisher information for estimating α from + 1 data snapshots. 4. SIMULAION RESULS We compare the bound to simulated tracing performance in a scenario in which the array was a uniform linear array 15 20 Bearing (u) BCRB Sim 40 Bearing Rate (u dot) 45 6 4 Power (α) 2 ime Fig. 2. Hybrid CRB for tracing bearing, bearing rate, and power compared to performance of Kalman filter tracer using synthetic estimates. with half-wavelength spacing and N = 10 elements. he initial bearing was Gaussian with mean µ u = 0.1 and standard deviation σ u = 0.03. he initial bearing rate was Gaussian with µ u = 0 and standard deviation σ u = 0.0004. he signal power α was constant at 20 db and the noise power was constant at 0 db. he random acceleration had standard deviation σ n = 0.00001. he hybrid CRB was calculated for =1 to =500, and is plotted in Figures 2-4. he bounds for u and u decrease rapidly at first and then level out, while the bound for α decreases linearly. he bound is compared against the performance of three tracing algorithms. In Figure 2, the observations were synthetic estimates of u and α, each drawn from a Gaussian distribution with variance equal to the inverse of ϑ u u and ϑ αα, respectively. his corresponds to the linear observation model in (67). he standard Kalman filter (KF) tracer was used to compute the trac estimates, and the MSE averaged over 100 Monte Carlo trials. he tracer achieved the bound in this simple but unrealistic scenario. In Figures 3 and 4, the observations were actual array data. In Figure 4, state estimates were obtained from a tracer which computed the maximum lielihood estimates (MLEs) of u and α based on the single observation z, and used these as measurements in the Kalman filter. MSE estimates were averaged over 100 Monte Carlo rials. his tracer did not achieve the bound for the bearing or bearing rate because there were many outliers among the single snapshot MLEs for the bearing u. he signal power bound was achieved in spite of the outliers. In Figure 4, state estimates were obtained from the sequential version of the MAP-PF tracer [8],[9] for single source bearing estimation. he MAP-PF tracer computes penalized MLEs which restrict the single snapshot MLE
15 20 Bearing (u) BCRB Sim 40 Bearing Rate (u dot) 45 6 4 Power (α) 2 ime 15 20 40 Bearing (u) BCRB Sim 45 6 4 Bearing Rate (u dot) Power (α) 2 ime Fig. 3. Hybrid CRB for tracing bearing, bearing rate, and power compared to performance of Kalman filter tracer using MLEs from single snapshot array data. Fig. 4. Hybrid CRB for tracing bearing, bearing rate, and power compared to performance of MAP-PF tracer using penalized MLEs from single snapshot array data. to a region near the current state estimate and outliers are greatly reduced. MSE trac estimates were averaged over 100 Monte Carlo rials. his tracer came much closer to achieving the bound because the variance of the penalized MLEs was much closer to the single snapshot CRB. 5. SUMMARY he recursive hybrid CRB on the mean square error in tracing the bearing, bearing rate, and power level of a narrowband source was derived based on the nonlinear filtering bound developed in [2]. he bound was shown to have the same form as when the measurements were bearing and power estimates with variance equal to the deterministic CRB for a single data snapshot. he bound was compared against simulated performance of a KF tracer using synthetic estimates, a KF tracer using single snapshot MLEs, and the MAP-PF tracer using penalized single snapshot MLEs. he KF tracer using synthetic estimates achieved the bound, however the tracers using actual array data suffered performance degradation due to outliers in the bearing estimates. he MAP-PF tracer, which uses penalized MLEs to reduce outliers came significantly closer to the bound than the KF tracer using MLEs. 6. REFERENCES [1] H. L. Van rees, Detection, Estimation, and Modulation heory, Part I, New Yor, NY: John Wiley and Sons, 1968. [2] P. ichavsy, C. H. Muravchi, and A. Nehorai, Posterior Cramér-Rao Bounds for Discrete-ime Nonlin- ear Filtering, IEEE rans. Signal Processing, vol. 46, no. 5, pp. 1386-1396, May 1998. [3] B. Ristic, S. Arulampalam, and N. Gordon, Beyond the Kalman Filter, Boston, MA: Artech House, 2004. [4] Y. Rocah and P.M. Schultheiss, Array Shape Calibration Using Sources in Unnown Locations-Part I: Near-Field Sources, IEEE rans. Acoust., Speech, Signal Process, vol. ASSP-35, pp. 286299, March 1987. [5] A. J. Weiss and B. Friedlander, On he Cramér Rao Bound for Direction Finding of Correlated Signals, IEEE ransactions on Signal Processing, vol. 41, no. 1, pp. 495-499, January 1993. [6] H. L. Van rees, Optimum Array Processing: Detection, Estimation, and Modulation heory, Part IV, New Yor, NY: John Wiley and Sons, 2002. [7] Y. Bar-Shalom, X. R. Li, and. Kirubarajan, Estimation with Applications to racing and Navigation. New Yor, NY: John Wiley and Sons, 2001. [8] R. E. Zarnich, K. L. Bell, and H. L. Van rees, A Unified Method for Measurement and racing of Multiple Contacts from Sensor Array Data, IEEE rans. Signal Processing, vol. 49, no. 12, pp. 2950-2961, Dec. 2001. [9] K. L. Bell, MAP-PF Position racing with a Networ of Sensor Arrays, Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP 05), Vol. 4, pp. 849-852, March 2005.