Predictive Angle Tracking Algorithm Based on Extended Kalman Filter

In the case of moving sources, various target angle tracking algorithms have been proposed and reported in the literature for multiple narrow-band targets. Yang and Kaveh proposed an iterative adaptive eigen-subspace method in conjunction with the multiple signal classification (MUSIC) algorithm to track the DOA angles of multiple targets (Yang & Kaveh, 1988). Due to the data association problem caused by multi-target tracking, the adaptive MUSIC method fails to track targets when they are moving close to each other. Although the method proposed by Sword, et al. (1990) can avoid the data association problem, errors are accumulated in each iteration, making it unable to track targets that are mutually close. Due to the nature of prediction-correction filtering process, Kalman filter (KF) can reduce estimation errors and avoid the data association problem when applied to angle tracking, as stated in several references (Javier & Sylvie, 1999; Yang, 1995; Park, et al. 1994). Rao, et al. (1994) proposed to estimate DOA angles of targets using the maximum likelihood method and feeding the results to a KF. However, it is assumed that the signal powers of the targets are all different, making the algorithm impractical. Javier and Sylvie (1999) suggested to estimate target angles using the projection approximation subspace tracking algorithm with deflation (PASTd) (Yang, 1995) and a Newton-type method (for MUSIC spectrum) for the use in the KF. It has lower computational load and better tracking performance than Rao’s algorithm, but still exhibits poor tracking success rate at low signalto-noise ratios (SNRs). Park, et al. (1994) proposed an approach, which utilizes predicted angles obtained from Sword’s method. The approach also uses the constrained least-squares criterion to confine the dynamic range of angles. The choice of relevant parameters is empirical and is not suitable for various scenarios of different moving speeds and SNRs. Besides, the tracking performance degrades seriously with an increasing number of crossing targets. Later on, to improve Park’s method, Ryu, et al. (1999, 2002) suggested to obtain the angle innovations of the targets from a signal subspace, instead of the sensor output covariance matrix, via projection approximation subspace tracking (PAST) algorithm (Yang, 1995). Chang, et al. (2005) modified Park’s algorithm by incorporating a spatial smoothing (Shan et al., 1985) technique to overcome multipath interference, and also coherent signalsubspace (CSS) (Wang & Kaveh, 1985) processing for tracking wideband targets. All of the above algorithms are based on the sample covariance matrix or signal subspace made with

multiple snapshots of data from a sensor array.However, they all fail to track multiple targets when only a single snapshot measurement is available between two consecutive time steps during the tracking process, because DOA estimation using subspace-based approach requires sample covariance matrix or signal subspace with a rank of more than one.
In the case of a single snapshot measurement within each time increment, tracking multiple targets becomes feasible if the sensor array output is directly used as the measurement data in the extended Kalman filter (EKF) (Kong & Chun, 2000).However, the EKF is an approximate nonlinear state estimation technique with first-order linearization accuracy, and is suitable for the tracking problem since the measurement model is nonlinear in terms of the angles (states) to be estimated.The algorithm proposed by Kong and Chun (2000) exhibits low tracking success rate when targets approach near the points of intersection.The reason for this weakness is the EKF can be difficult to tune and often gives unreliable estimates if the system nonlinearities are severe.

Tracking algorithm
For tracking non-stationary targets efficiently and effectively, the predictive angle tracking algorithm based on extended Kalman filter (PAT-EKF) is presented.In the proposed algorithm, the sensor array output is used as measurement data in EKF, since the measurement model is nonlinear in terms of angle estimates.Using the predicted angles, the PAT-EKF algorithm modifies Park's method to obtain angular innovation, from which the angle estimates are updated (smoothed) via Kalman gain.

Data model
In the data model, M targets moving (for tracking) in a plane are considered, which contain an array of L sensors (or hydrophones).The sensor positions are assumed to be known, and it takes them to be placed uniformly on a line with spacing of d between two adjacent sensors (abbreviated as ULA), measured in the unit of wavelength λ.The motion of the targets is assumed to be at constant angular speed in the presence of Gaussian disturbance, and is observed every , measured clockwise with respect to y axis, denote the DOA angle of the mth target at time t.Assuming that these targets are located in the far field and their radiated signals are narrowband with a common angular frequency ω 0 , the output of the lth isotropic sensor at time t is then where () is the signal transmitted by the mth target at time t, n l (t) is a complex Gaussian white noise with zero mean and variance 2 n σ , which is uncorrelated with the target signals, and τ lm is the difference in time delays of the mth target reaching the first (reference) sensor and the lth sensor.Suppose that there are K measurements (snapshots) that are taken for each increment T, and the time increment is sufficiently small allowing us to approximate the target as stationary.
Figure 1 shows the sensor array and source configurations in 2-D space.

PAT-EKF algorithm
First, it depicts the discrete-time state (process) model for the target motion.For each time index k, it defines the state vector for the mth target as ( ) ( ) ( ) , consisting of its DOA and angular speed.The mth target motion can lead to the process equation (Park et al., 1994) (1 ) ( ) ( ) where the process noise vector w m (k) is assumed to be Gaussian distributed with zero mean and covariance

Q
Assume that the motion of each target is mutually independent.By defining the composite state vector as 1 () () , . . ., () , the system dynamics is governed by the process model The process noise vector w(k)= 1 () , . . ., () reflects the random modeling error, which is Gaussian distributed with zero mean vector and covariance 1 (,, ) The matrices F and Q are all block diagonal.Although the process equation is a linear model, the measurement model of ( 2) is a vector nonlinear function of the target DOAs (and thus, of the target state vectors as well), which can be restated as where n(k) is complex Gaussian noise process with the known covariance 2 n σ I , and is assumed to be uncorrelated with the process noise w(k).Assuming that a uniform linear array of L sensors with a half wavelength of inter-element spacing d is deployed, the partial derivative (Jacobian) matrix of the measurement model ( 6) is given by 1 By augmenting the real and imaginary parts of each complex matrix H m (k), it has the composite real matrix of dimension 2L×2M

HH H HH
which can be expressed as For k=1,2,…, the proposed tracking algorithm can be summarized in the following four steps.
Step 1. Prediction of angles The prediction of the state vector and its covariance matrix can be obtained from the existing estimates by the equations The first element of each state vector ˆ(| 1 ) m kk − x is the predicted estimate ˆ(| 1 ) m kk θ − of θ m (k).The predicted direction matrix A(k|k-1) can be obtained by (3) using ˆ(| 1 ) m kk θ − for θ m (k).From (2), the predicted output of the sensor array becomes and can be obtained once Step 2. Computations of the angle innovation After time T, a new array output is observed and the direction matrix can be written as where δA(k) is the error matrix, which can be derived, according to (Sword et al., 1990), as (1 ) Thus, the residual array output Δr(k) can be obtained and written as Note that the first row vector of δA(k) in ( 14) is a null vector.To reduce the computation, the null vector allows us to define a (L−1)×1 vector ∆r  , which is obtained by removing the first element of Δr in ( 16).By substituting ( 15) into ( 14), ∆r  can be represented by (dropping k temporarily) In ( 17), the (L−1)×M matrix B is as suggested in Park's algorithm, will be used to constrain the absolute values of innovations for the cases of nearby targets, where L is a weighting matrix with diagonal form.
Step 3. Estimation of the angles The estimated angle can be obtained as Furthermore, ˆ() m k θ and ˆ() m sk are substituted into ( 7) and ( 8) to update the matrix () k H .
Step 4. Smoothing the estimated angles Since the state vector is real-valued, it formulates the state estimation equation as where () k ∆r =[Δr R Δr I ] T is a real vector; Δr R and Δr I are the real and imaginary parts of Δr(k) from ( 16).K(k) is the Kalman Gain matrix, given by The covariance matrix of ˆ(|) kk x is given by The proposed PAT-EKF algorithm requires the number of 7LM 2 +16L 2 M+LMK real multiplications, whereas the Park's and Kong's algorithms require, respectively, the numbers of 3LM 2 +K(3L 2 +LM) and 5M 3 +10LM 2 +8L 2 M+LMK real multiplications (K is the number of snapshots).Table 1 shows the comparison of computational complexity among these algorithms for M=3, L=8 and different number of snapshots.It is evident that the PAT-EKF algorithm has lower computational complexity than the Park's algorithm for K ≥ 30, where K ≥ 30 is often needed for acceptable tracking performance.Although the computational complexity is higher than the Kong's algorithm, the proposed method has much better performance as demonstrated by the simulations.

PAT-EKF algorithm for tracking targets in 3-D space
The PAT-EKF algorithm is now extended to track narrow-band targets in 3-D space, where the system of sensor array and source configurations is shown in Figure 2, where s m (k) is the signal transmitted by the mth target, of which φ m and θ m are the azimuth and elevation respectively.ρ m is the range from the mth target to the first (reference) sensor in the uniform linear array.As explained later, the number of sensors L must satisfy the condition L ≥ 3M+1, where M is the number of targets.All the targets can be located in the near field or far field.In the following formulations, far-field targets are treated.The output of the lth sensor for the kth sampling interval can be expressed as From the array and source configurations shown in Figure 2, τ lm (k) can be expressed as where (x l , y l , z l ) is the lth sensor position relative to the reference sensor.Here x l =(l−1)d, y l =0, and z l =0.The location coordinate of the mth signal source is given by sin cos For the use of the EKF, it redefines the state vector for the mth target as in 3-D space.By augmenting the real and imaginary parts of each complex matrix H m (k), it has the composite real matrix of dimension 2L×6M 1 1 ( ( ),..., ( )) () ( ( ),..., ( ))

HH H HH
which can be expressed as with the following equations For the 3-D PAT-EKF algorithm, the recursive equations of ( 9)-( 13), ( 16) and ( 20)-( 22) remain unchanged and the recursive equations of ( 14)-( 15) and ( 17)-( 19) need to be changed as stated in the following context.
Let δρ(k), δφ(k), and δθ(k) be the unknown innovations of ρ(k), φ(k), θ(k) respectively, from time k-1 to time k.The (l,m) element of δA(k) can be derived as There is one limiting condition, i.e., L−1 ≥ 3M, under which the L−1 linear equations are used for solving 3M unknown variables.
Figure 3 shows typical sample run for crossing tracks, all based on a single snapshot of data vector (K=1) at SNR=10dB of each target.The PAT-EKF and Park's algorithms exhibit much better tracking capability than Kong's algorithm (Kong & Chun, 2000) especially at the cross points in the trajectory.
Two moving targets are tracked over an interval of 20s with T=1 in 3-D space.During each T interval, K(=160) snapshots of sensors data are generated.Figure 4 shows the tracking performances of the 3-D PAT-EKF algorithm for the combinations of range, elevation, and azimuth at 3dB of SNR.In Figure 4, dot represents the true angle and line represents the tracked angle.The 3-D PAT-EKF algorithm is very effective in tracking the targets in 3-D space, even at low SNR.
Note that the PAT-EKF algorithm is excluded for performance comparison, simply because it fails to track the angle of the first signal source for the simulation example illustrated in Figure 5.In this example, the trajectory of the first signal source reveals its significantly changing behavior of angles.This also indicates that tracking capability of the PAT-EKF algorithm is rather limited when there exists a signal source with significant rates of angle variation.

Concluding remarks
This chapter has presented the PAT-EKF algorithm for tracking multiple targets.The proposed algorithm modified Park's algorithm by using the sensor array output vector rather than the sample covariance matrix and incorporating EKF instead of KF.These modifications allow the proposed algorithm to lower computational load, and also improve the tracking success rate particularly at lower snapshots.The PAT-EKF algorithm is then extended to track the azimuth, elevation, and range of multiple targets in 3-D space.Through computer simulations, the effectiveness of each proposed algorithm has been demonstrated.The drawback of the PAT-EKF algorithm is that it fails to track any target with a significant rate of angle variations.

Acknowledgment
This work was supported by National Science Council of Taiwan under contracts NSC 99-2923-E-022-001-MY3.

=
Assume that all the signal sources are narrowband with a common angular frequency ω.In vector-matrix notation, the received output vector of the sensor array is r(

Table 1 .
Computational complexity comparison for M=3, L=8 and different K values.