Simplification of the Dynamic Cramér-Rao Bound for Target Tracking in Clutter A simplified dynamic Cramér-Rao lower bound (CRLB) is derived for the general case of target tracking in clutter with non-IID measurement noise components, discussed recently in [3]. The recent work [3] derived the dynamic Cramér-Rao lower bound (CRLB) for target tracking (state estimation) in clutter. It was shown that the effect of the clutter (i.e., measurement origin uncertainty) was quantified by a scalar information reduction factor (IRF) when the measurement noise components are independent and identically distributed (IID) and by an information reduction matrix when the measurement noise components are not IID The purpose of this note is to show that in the general case of target tracking with non-IID components of the measurement noise vector, the effect of the clutter is still quantified by a scalar IRF–the matrix IRF in [3] was unnecessary. Assume the original target measurement equation is z(k)= H(k)x(k)+ w(k) (1) where x is the state of the target of interest and the covariance of original measurement noise w(k), assumed zero-mean white Gaussian, is the general positive definite matrix R. It was shown in [3] that the state estimation Bayes information matrix (BIM), whose inverse is the dynamic CRLB, has the recursive form given below in (2). Note that the dynamic state estimation falls within the Bayesian estimation framework with the state being modeled as (a sequence of) random variables. This is in contrast to the “unknown constant” model of parameter estimation used in the Fisher framework [1]. Consequently, we use the term BIM, rather than FIM (Fisher information matrix), as in [2]. 1 The BIM recursion given in [3] is J n+1 = Q ¡1 + H T R ¡1 Q 2 (P d , ¸, R, V)R ¡1 H ¡ Q ¡1 F (J n + F T Q ¡1 F ) ¡1 F T Q ¡1 (2) where Q 2 is the information reduction matrix, which depends on the target detection probability P d , the 1 In [3] the term FIM was used (unfortunately). Manuscript received March 14, 2008; revised April 22, 2010; released for publication July 15, 2010. IEEE Log No. T-AES/47/2/940859. Refereeing of this contribution was handled by D. Blair. 0018-9251/11/$26.00 c ° 2011 IEEE spatial density ¸ of the false measurements or clutter (modeled as a spatial Poisson process), the target measurement noise covariance matrix R, and the observation space volume V. In view of the spatial Poisson process model for the false measurements, they are uniformly distributed in the volume V and their number has a Poisson distribution with parameter ¸V. Consider the transformation of the original measurements, z L (k)= R ¡1=2 z(k) (3) which yields IID components. The measurement equation after this transformation becomes z L (k)= H L (k)x(k)+ w L (k) (4) where H L (k)= R ¡1=2 H(k) and w L (k) is zero mean Gaussian with identity covariance matrix. The measurement z (k) can also be a false alarm or clutter because in target tracking one usually does not know whether a measurement is from the target of interest or not. In the latter case, z (k) is assumed uniformly distributed in the (original) observation volume V. It can be easily shown that the transformed measurement z L (k) is still uniformly distributed in the transformed volume V L = V=jR ¡1=2 j. This is because the Jacobian of the transformation (3) is a constant, although the space (domain of the distribution of z L ) has been scaled by the determinant of the transformation. Therefore, after this transformation the problem setting is the same as that in the case with IID measurement noise in [3]. Consequently, the BIM J n (whose inverse yields the dynamic CRLB), is given in the general case of arbitrary R, by the recursion J n+1 = Q ¡1 + q 2 (P d , ¸, R L , V L )H T L R ¡1 L H L ¡ Q ¡1 F(J n + F T Q ¡1 F ) ¡1 F T Q ¡1 : (5) Comparing (2) and (5), one can see that after the transformation (4) of the original measurements, the information reduction effect due to measurement origin uncertainty can be always captured by a scalar IRF, as opposed to the information reduction matrix that according to [3] was apparently necessary. YAAKOV BAR-SHALOM Dept. of Electrical and Computer Engineering University of Connecticut U-2157 Storrs, CT 06269 XIN ZHANG United Technologies Research Center East Hartford, CT 06108 PETER WILLETT Dept. of Electrical and Computer Engineering University of Connecticut U-2157 Storrs, CT 06269 E-mail: (willett@engr.uconn.edu) CORRESPONDENCE 1481