Let X̄_k denote any unbiased estimator of the vector X_k (unknown and random); the covariance of X̄_k has a lower bound that is expressed as follows [6]: (1) C k = E [ ( X ^ k − X k ) ( X ^ k − X k ) T ] ≥ J k − 1where J_k is referred to as the Fisher information matrix (FIM). The inverse of the FIM J k − 1 is the PCRLB. The inequality in (1) means that the difference C k − J k − 1 is a positive semi-definite matrix.

Tichavsky et al. [9] provided a Riccati-like recursion to calculate the FIM J_k for the general nonlinear filtering problem. The parameters to be estimated are contained in the state vector X_k, where k denotes the time step. At each time step, the sensor obtains one measurement vector Z_k. The general form of the dynamic and measurement model is: (2) X k + 1 = f k ( X k , q k ) (3) z k = h k ( X k , r k )where f_k( ) and h_k( ) are (in general) nonlinear functions, and q_k and r_k are the dynamic and measurement noise, respectively, which are assumed to be independent and white processes (i.e., sequences of mutually independent random variables or vectors). The FIM is then computed recursively as: (4) J k + 1 = D k 33 − D k 21 ( J k + D k 11 ) − 1 D k 12 + J Z ( k + 1 )where: (5) D k 11 = E { − Δ x k x k ln   p ( X k + 1 | X k ) } (6) D k 12 = E { − Δ x k x k + 1 ln   p ( X k + 1 | X k ) } (7) D k 33 = E { − Δ x k + 1 x k + 1 ln   p ( X k + 1 | X k ) } (8) J Z ( k + 1 ) = E { − Δ x k + 1 x k + 1 ln   p ( z k + 1 | X k + 1 ) }and Δ ϕ θ is a second-order partial derivative operator defined as: (9) Δ ϕ θ = ∇ ϕ ∇ θ Twhere: (10) ∇ ϕ = [ ∂ δ ϕ 1 , … , ∂ δ ϕ n ] Tis a first-order partial derivative operator with respect to the parameter vector φ. Throughout this article E_φ will denote mathematical expectation with respect to φ. In Equation (4)J_z (k + 1) denotes the measurement contribution to the FIM. The PCRLB is then the inverse of the FIM calculated recursively through Equation (4) and the initial target distribution covariance C₀ (the initial FIM is the inverse of the initial target distribution covariance, i.e., J 0 = C 0 − 1).

If the dynamic and measurement noise are additive Gaussian, i.e.: (11) X k + 1 = f k ( X k ) + q k ,       q k ∼ N ( 0 , Q k ) (12) z k = h k ( X k ) + r k ,         r k ∼ N ( 0 , R k )where N (μ,Σ) denotes the Gaussian distribution with mean μ and covariance Σ, the recursive formula is simplified to: (13) J k + 1 = J Z ( k + 1 ) + Q k − 1 − Q k − 1 E [ F k ] ( J k + E [ F k T Q k − 1 F k ] ) − 1 E [ F k T ] Q k − 1and: (14) J z ( k + 1 ) = E X k + 1 [ H k + 1 T R k + 1 − 1 H k + 1 ]where F_k and H_k are the Jacobians of the nonlinear functions f_k (x_k) and h_k (x_k), i.e.: (15) F k = [ ∇ x k f k T ( x k ) ] T (16) H k = [ ∇ x k h k T ( x k ) ] T

Furthermore, if the target dynamics are linear (i.e., f_k (x_k) = A_kX_k), then it is straightforward to show that: (17) J k + 1 = ( Q k + A k J k − 1 A k T ) − 1 + J Z ( k + 1 )where J_z (k + 1) is given in (14). Equation (17) also holds if the target dynamics are non-random (i.e., Q_k = 0).

For the first type of extended target tracking models, the dynamic and measurement vectors are both extensions of the ones in the point model with additional states and measurements. The equations are thus also extensions of the point model and are still in the general form. The superscripts “p” and “e” are used in the paper to indicate the “point” and “extended” target tracking models, respectively. Thus, the extended model is expressed as: (18) X k + 1 e = f k e ( X k e , q k e ) (19) z k e = h k e ( X k e , r k e )where X e = [ X p X n ] and the parameters of the extended target, such as target size and shape, are expressed by the additional state vector Xⁿ. The PCRLB can then be calculated through the general recursive equation in (4). As proven in [7], under certain sufficient conditions, which are satisfied in most extended target tracking applications, the bound of the target centroid dynamics is always smaller than that of the point model: (20) Γ [ ( J k e ) − 1 ] < ( J k p ) − 1where Γ (·) is a function to obtain the N S p × N S p left-upper sub-matrix and N S p is the dimensionality of the state vector X^p. Furthermore, the conclusion still holds in cluttered environments [8].

In the second type of extended target tracking framework, the state space and the dynamic equation of the extended target model are in the general form described by (2). However, a high resolution sensor can obtain multiple measurements at each time step. It is assumed that m_k independent measurements generated from the target, denoted as Z_k ≜ {Z_k (i):i = 1,2,…,m_k}, are observed by the sensor. Each of the measurements is distributed according to the known spatial extent model p_z (Z_k (i) | X_k), also in the measurement equation form: (21) Z k ( i ) = h k ( X k , r k ( i ) ) ,       i = 1 , … , m kwhere r_k (i) are i.i.d. stochastic vectors and the total number of measurements m_k is unknown and random. The overall probability density is then: (22) p ( Z k | X k , m k ) = ∏ i = 1 m k p z ( Z k ( i ) | X k )

For the spatial distribution model of the extended target, the number of sensor measurements per frame is unknown and random. Thus, the recursive calculation of the PCRLB cannot be applied directly. Referencing the ideas for calculating the PCRLB in the case of a single point target in a cluttered environment [10,12,13], the general form of the measurement contribution is given, and an extended information reduction factor (EIRF) approach is proposed to calculate the PCRLB. The measurement contribution at time k given that there are m_k measurements at that time is thus: (23) J Z ( k : m k ) = E { − Δ X k X k   ln   p ( Z k | X k , m k ) } = E { − Δ X k X k   ln   [ ∏ i = 1 m k p z ( Z k ( i ) | X k ) ] } = ∑ i = 1 m k E { − Δ X k X k   ln   p z ( Z k ( i ) | X k ) } .

It is noticed that for each i (1 ≤ i ≤ m_k), Z_k (i) is i.i.d. to p_z (z_k | X_k), then: (24) E { − Δ X k X k ln   p z ( Z k ( i ) | X k ) } = E { − Δ X k X k   ln   p z ( z k | X k ) }for all i (1 ≤ i ≤ m_k). It then follows from (23) and (24) that: (25) J Z ( k : m k ) = m k E { − Δ X k X k   ln   p z ( z k | X k ) } .

The following expression is defined as the measurement contribution in the conventional case (where just one measurement originates from p_z (z_k | X_k) at time k): (26) J Z 1 ( k ) = E { − Δ X k X k   ln   p z ( z k | X k ) }

Thus: (27) J Z ( k : m k ) = m k J Z 1 ( k )

The conclusion above obeys the usual intuition that the measurement uncertainty is reduced by multiple i.i.d. measurements (for the case that m_k > 1).

If the dynamic and measurement noise are additive Gaussian (described by (11) and (12)), the measurement contribution is then written as follows [using (14) and (27)]: (28) J Z ( k : m k ) = m k E X k [ H k T R k − 1 H k ]

Equation (25) indicates that the measurement uncertainty brought by multiple target generated measurements is generalized to a single multiplier that equals to the number of measurements at the corresponding time. The overall measurement contribution can then be calculated as a weighted sum of the conditional ones, which is referred to as the extended information reduction factor (EIRF) approach. The overall measurement contribution is formulated as: (29) J Z ( k ) = E m k [ J Z ( k : m k ) ] = ∑ m k p ( m k ) J Z ( k : m k ) = ∑ m k p ( m k ) · m k · J Z 1 ( k ) = E [ m k ] · J Z 1 ( k ) = E [ m k ] · E { − Δ X k X k   ln   p z ( z k | X k ) }where p(m_k) is the probability that m_k measurements are obtained by the sensor and E[m_k] is the mathematical expectation of the number of measurements at time step k. Note that the EIRF (which is equal to E[m_k]) can be either greater or less than one. Hence, the information might be either reduced or enlarged. For a common extended target tracking scenario, the mean number of measurements is usually much greater than one, so that the information is enlarged and therefore the PCRLB is decreased. Hence the estimation performance might be improved through the multiple sensor measurements.

Because the probability distribution of m_k is usually prior information in a specific sensor and target scenario, the value of the measurement contribution can easily be calculated using (29). Then, the FIM J_k can be calculated through the recursion in (4) together with the initial FIM J 0 = C 0 − 1. The corresponding EIRF bound is denoted as: (30) PCRLB ( EIRF ; k ) = J k − 1

If the dynamic and measurement noise are additive Gaussian (described by (11) and (12)), the overall measurement contribution is then: (31) J Z ( k ) = E [ m k ] E X k [ H k T R k − 1 H k ]

Furthermore, if the target dynamics are linear (i.e., f_k (x_k) = A_kX_k), the recursion of J_k is then simplified to (17), with J_z (k) calculated by (31).

The first example is tracking an extended target whose shape is modeled as a stick using an equation-extension model for extended target tracking, which is described in Section 3.1. As shown in Figure 1, the sensor is located at (x₀, y₀) on a 2-D plane, and the state vector of the target is X^e = (x, ẋ, y, ẏ, l), where l is the target length and the superscript “e” denotes the extended model. The target is moving with nearly constant velocity (NCV), and the direction of the velocity is assumed to be along the stick. The dynamic model is: (32) X k + 1 e = F k e X k e + q k e = [ 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 ] X k e + q k ewhere T is the time interval between sensor measurements and q k e is the Gaussian dynamic noise with zero mean and covariance Q k e.

The sensor obtains not only the conventional target position measurements, such as the distance and azimuth angle of the target centroid, but also the extended measurements that describe the target extent. The measurement model is: (33) z k e = h k e ( X k e ) + r k e = [ ρ ( X k e ) θ ( X k e ) L ( X k e ) W ( X k e ) ] + r k pwhere [ ρ ( X e ) θ ( X e ) ] = [ ( x − x 0 ) 2 + ( y − y 0 ) 2 arctan y − y 0 x − x 0 ] is the conventional part of the measurement vector, denoting the distance and azimuth angle of the target centroid, and [ L ( X e ) W ( X e ) ] = [ l cos φ l sin φ ] is the extended part of the measurement vector, denoting the down-range and cross-range extent of the target. The parameter is the angle between the line of sight (LOS) and the target velocity vector, denoted as the VLOS angle: (34) φ = arctan [ x ˙ ( y − y 0 ) − y ˙ ( x − x 0 ) x ˙ ( x − x 0 ) + y ˙ ( y − y 0 ) ] .The recursive computation of the FIM (13) can be applied here directly with the calculation of the measurement contribution in (14). The Jacobian H k e in (14) is a 4 × 6 matrix. The non-zero elements of the first two rows are: (35) H k e [ 1 , 1 ] = X k e ( 1 ) [ X k e ( 1 ) ] 2 + [ X k e ( 3 ) ] 2 ,         H k e [ 1 , 3 ] = X k e ( 3 ) [ X k e ( 1 ) ] 2 + [ X k e ( 3 ) ] 2 (36) H k e [ 2 , 1 ] = − X k e ( 3 ) [ X k e ( 1 ) ] 2 + [ X k e ( 3 ) ] 2 ,     H k e [ 2 , 3 ] = X k e ( 1 ) [ X k e ( 1 ) ] 2 + [ X k e ( 3 ) ] 2

The first four elements of the third and fourth row of the Jacobian H k e are calculated by: (37) H k e ( 3 , j ) = ∂ L ( X k e ) ∂ X k e ( j ) = − X k e ( 5 ) · sin φ k · ∂ φ k ∂ X k e ( j ) , H k e ( 4 , j ) = ∂ W ( X k e ) ∂ X k e ( j ) = X k e ( 5 ) · cos φ k · ∂ φ k ∂ X k e ( j ) ,     j = 1 , … , 4and ∂ φ k ∂ X k e ( j ) can be calculated through (34): (38) [ ∂ φ k ∂ X k e ( 1 ) ∂ φ k ∂ X k e ( 2 ) ∂ φ k ∂ X k e ( 3 ) ∂ φ k ∂ X k e ( 4 ) ] = [ X k e ( 3 ) [ X k e ( 1 ) ] 2 + [ X k e ( 3 ) ] 2 − X k e ( 4 ) [ X k e ( 2 ) ] 2 + [ X k e ( 4 ) ] 2 − X k e ( 1 ) [ X k e ( 1 ) ] 2 + [ X k e ( 3 ) ] 2 X k e ( 2 ) [ X k e ( 2 ) ] 2 + [ X k e ( 4 ) ] 2 ]

The last elements of the third and fourth row of the Jacobian H k e are: (39) [ H k e ( 3 , 5 ) H k e ( 4 , 5 ) ] = [ ∂ L ( X k e ) ∂ X k e ( 5 ) ∂ W ( X k e ) ∂ X k e ( 5 ) ] = [ cos   φ k sin   φ k ]

In this simulation example, the sensor is static and located at the origin of the coordinate system, i.e., (x₀, y₀) = (0, 0). The target moves with initial velocity v₀ = 10 in the direction with the initial VLOS angle Ø₀ = 20°. The initial position of the target is (15,000, 10,000), and the initial length is l₀ = 50. The initial FIM is J₀ = {[diag (20,3,20,3,1)]²}⁻¹, and the covariance of the state noise is Q_k = diag (3, 0.1, 3, 0.1, 1). The time interval between sensor measurements is T = 1, and the measurement noise is zero-mean white Gaussian noise with standard deviations: σ_ρ = 5, σ_θ = 0.1°, σ_L = 3, and σ_W = 3. All the parameter units are in the metric system.

Because the target dynamics are random (with non-zero dynamic noise) in the simulation scenario, the calculation of the measurement contribution using (14) requires the evaluation of the mathematical expectation of H k + 1 T R k + 1 − 1 H k + 1 with respect to X_k+1. A sampling scheme is used here. From the initial target state and the dynamic model, multiple target state sequences are generated, and the corresponding measurement contribution is computed. The overall measurement contribution is then computed as an average of the measurement contributions conditional on each state sequence. In this simulation, 10,000 state sequences were sampled to approximate the measurement contribution. The comparison of the bounds of target centroid dynamics (position and velocity) using both the extended and point target models is shown in Figure 2. The numerical result also confirms the theoretical development in [7] that the bound of the extended model is always smaller than that of the point model because the three sufficient conditions proposed in [7] are satisfied in the tracking models discussed here. The improvement in the bound of the extended target tracking model is a result of the fact that the measurements of the target extent are directly dependent on the target centroid dynamics (position and velocity). Figure 2(a) shows that the new measurements of the target extent are of relatively small importance for the estimation of target position, especially in the early stage of tracking. However, the PCRLBs of the velocities [as shown in Figure 2(b)] of the extended target model in both the x and y directions decrease sharply upon the arrival of the measurements (the initial values of the FIM of the extended and point models are equivalent). In the conventional point target tracking framework, the velocity is estimated only through the target centroid positions, and the uncertainty of the velocity is not reduced until the second frame of measurements (so the bound of the velocity in the first scan is no less than the initial covariance). However, the information on the target velocity is carried with the target extent measurements in the extended target model. Thus, the uncertainty of the velocity is significantly reduced with the arrival of the first scan of sensor measurements. This advantage could greatly benefit the tracking systems for defense, especially for velocity-sensitive ones, such as anti-missile systems. The early and precise sensing of the target velocity will substantially improve the power of defense surveillance systems.

Figure 3 illustrates the PCRLBs of the target velocity (in the direction of the x axis) and target length for different values of σ_L (the effect of σ_L on the PCRLB curves for the target position is negligible and therefore not shown here). In Figure 3, the parameters except σ_L are set to the same values as before, and the curves of PCRLB corresponding to σ_L = 0.1, σ_L = 3, and σ_L = 100 are presented. The numerical result unsurprisingly coincides with the intuition that the performance bound is improved by the accurate measurement of the target extent. Furthermore, Figure 3(b) shows that the estimation bound of the target length benefits from the accuracy of the target extent measurement more evidently than that of the target velocity [which is shown in Figure 3(a)] because the relationship between the target length and extent is more direct in this sensor-target geometry (Ø₀ = 20°). The influence of the sensor-target geometry on the estimation bound is reported below.

Next, the impact of the VLOS angle on the PCRLB is analyzed. The effect of φ on the estimation bounds of the position and velocity are negligible and not shown, but the impact of φ on the PCRLB curves for the target length also depends on the accuracy of the target extent measurements. Figure 4 shows the PCRLB curves for the initial VLOS angles Ø₀ = 0°, 45°, and 90° for four combinations of the values of σ_L and σ_W (corresponding to the case of good and poor accuracy in the measurement of the down-range/cross-range extent).

In Figure 4(a) (down-range/cross-range extent are both accurate) and Figure 4(d) (down-range/cross-range extent are both inaccurate), the bounds corresponding to all initial VLOS angles are equivalent because the information provided by the down-range/cross-range extent is symmetric. In Figure 4(b), the bound for Ø₀ = 0° (180°), is dramatically smaller than that for Ø₀ = 90°. The superior performance bound is a result of the target orientation being along the direction with the best measurement accuracy.

A similar interpretation can be proposed for Figure 4(c). From this result, it follows that, for a moving sensor platform, it is possible to design an optimal movement trajectory that minimizes the PCRLB for the target length. When the down-range extent is more accurate than the cross-range extent, the optimal trajectory should make the VLOS angle approach 0°/180°; when the down-range extent is less accurate than the cross-range extent, it should approach 90°.

The following simulation example is presented to illustrate the performance bounds for tracking an extended target that can be resolved as multiple point features at each sampling time by a high resolution sensor. The simulation scenario discussed below is similar to Example 1 discussed in Section 5.1. The observer is located at (x₀, y₀) on a 2-D plane. The target is moving with nearly constant velocity (NCV), and the dynamic model is of the conventional form: (40) X k + 1 = A k X k + q k ,   q k ∼ N ( 0 , Q k )where X = (x, ẋ, y, ẏ)^T, A k = diag ( [ 1 T 0 1 ] , [ 1 T 0 1 ] ) and T is the time interval between measurements. The measurement model is: (41) Z k ( i ) = h k ( X k ) + r k ( i ) ,         r k ( i ) ∼ N ( 0 , R k ) ,       i = 1 , … , m kwhere m_k is the number of measurements, and: (42) h k ( X k ) = [ ρ ( X k ) θ ( X k ) ] = [ ( x k − x 0 ) 2 + ( y k − y 0 ) 2 arctan y k − y 0 x k − x 0 ]includes the distance and azimuth angle of the target. As described in Section 2.2, r_k (i) are i.i.d. stochastic vectors. The number of target measurements per frame m_k is Poisson distributed, and the mean number is λ_T. Thus, the probability that m_k measurements originating from the target are observed is: (43) Pr ( m k ) = λ T m k m k ! e − λ Tand the PCRLB can be calculated recursively by the EIRF method described in Section 4.2. The overall measurement contribution (given in (29)) is then: (44) J Z ( k ) = E [ m k ] J Z 1 ( k ) = λ T J Z 1 ( k )where: (45) J Z 1 ( k ) = E X k [ H k T R k − 1 H k ]and the expression for H_k is a 2*4 sub-matrix of H k e given in Section 4.1: (46) H k [ X k ( 1 ) [ X k ( 1 ) ] 2 + [ X k ( 3 ) ] 2 0 X k ( 3 ) [ X k ( 1 ) ] 2 + [ X k ( 3 ) ] 2 0 − X k ( 3 ) [ X k ( 1 ) ] 2 + [ X k ( 3 ) ] 2 0 X k ( 1 ) [ X k ( 1 ) ] 2 + [ X k ( 3 ) ] 2 0 ]

The method of calculating the mathematical expectation with respect to the state vector in (45) is the same as that described in Example 1.

In the simulation, the sensor is also static and at the origin of the coordinate system, while the target moves with initial velocity (v_x0, v_y0) = (10, 15) starting from the initial position (15000,10000). The initial FIM is J₀ = {[diag (20, 3, 20, 3)]⁻²}⁻¹. The covariance of the state noise is Q_k = diag (3, 0.1, 3, 0.1), and the measurement noise is zero-mean white Gaussian noise with standard deviations of σ_ρ = 5 and σ_θ = 0.1°. The mean number of measurements per frame is λ_T = 2.5, and the time interval is T = 1. All the parameter units are in the metric system. The estimation bounds calculated by the EIRF approach for the position and velocity along both the x and y axes are shown in Figure 5. Equation (28) indicates that the estimation performance bound is affected by the number of measurements originating from the target. In the EIRF methodology, the overall impact is manifested as a constant scalar, the mean number of measurements per frame. The PCRLB curves (calculated by the EIRF approach) for the target position and velocity in the direction of the x axis for various values of the Poisson intensity (the mean number of measurements per frame) λ_T is shown in Figure 6(a) and (b), respectively. The curve corresponding to λ_T = 1 is equivalent to the point target tracking bound, and, therefore, the simulation results indicate that the large quantity of measurements decreases the estimation bound, therefore possibly improving the estimation accuracy.