Candidate Tracks

A single candidate or postulated track is characterized by a measurement vector or a missed detection for each measurement time. At time n let one such track, the l^th track, be denoted $\mbox{\boldmath$\theta$ }_n^l$. The vector of measurements for this track is

$${\cal Z} ( \mbox{\boldmath$\theta$ }_n^l ) = \{ {\bf z}_{1,j_1 (l)} , {\bf z}_{2,j_2 (l)}, \cdots , {\bf z}_{n,j_n (l)} \}$$ (1)

where the subscript j_m (l) is the measurement index at time mfor the l^th track. We account for the possibility of a missed detection by letting j_m (l) range from 1 to J_m +1, with j_m (l) = J_m +1 indicating a missed detection. Note that a ${\bf z}_{m,J_m +1}$ is not an actual measurement, but indicates a missed detection at time m, and is a notational convenience. There are $L_n' = \prod_{m=1}^{n} (J_m + 1)$ of these candidate tracks: $\mbox{\boldmath$\theta$ }_n^l ; l=1,2, \cdots, L_n^'$.

For a given candidate track $\mbox{\boldmath$\theta$ }_n^l$, measurement noise is assumed additive, Gaussian and temporally white. The trajectory and measurements are assumed to evolve in time according to the state/measurement equations ²

$${\bf x}_{m+1}^l = {\bf\Phi}_m {\bf x}_m^l + {\bf w}_m^l$$ (2)

$${\bf z}_{m,j_m (l)} = {\bf H}_m {\bf x}_m^l + {\bf u}_m^l$$ (3)

where at time m and for candidate track $\mbox{\boldmath$\theta$ }_n^l$, ${\bf x}_m^l$ is the state vector, and ${\bf\Phi}_m$ and ${\bf H}_m$ are the state transition and output matrices respectively. The ${\bf w}_m^l$and ${\bf u}_m^l$ vectors are zero mean, mutually independent, white and Gaussian with known covariance matrices ${\bf Q}_m^l$ and ${\bf R}_m^l$ respectively.

A Kalman filter can be applied to this track to smooth the measurements, to provide minimum variance state vector estimates, and to compute the innovations for the measurement sequence ${\cal Z} ( \mbox{\boldmath$\theta$ }_n^l )$. The Kalman filter equations are:

Kalman Prediction Equations

 

$$\hat{\bf x}_{m+1,m} {\bf\Phi}_{m} \hat{\bf x}_{m,m}$$ =

$$\hat{\bf z}_{m+1,m} {\bf H}_{m+1} \hat{\bf x}_{m+1,m}$$ =

$${\bf v}_{m+1} {\bf z}_{m+1} - \hat{\bf z}_{m+1,m}$$ =

$$\hat{\bf x}_{m+1,m+1} \hat{\bf x}_{m+1,m} + {\bf G}_{m+1} {\bf v}_{m+1}$$ = (4)

Kalman Gain Equations

 

$${\bf\Xi}_{m+1,m} {\bf\Phi}_{m+1} {\bf\Xi}_{m,m} {\bf\Phi}^{\mbox{T}}_{m+1} + {\bf Q}_{m+1}$$ =

$${\bf S}_{m+1} {\bf H}_{m+1} {\bf\Xi}_{m+1,m} {\bf H}^{\mbox{T}}_{m+1} + {\bf R}_{m+1}$$ =

$${\bf G}_{m+1} {\bf\Xi}_{m+1,m} {\bf H}^{\mbox{T}}_{m+1} {\bf S}^{-1}_{m+1}$$ =

$${\bf\Xi}_{m+1,m+1} {\bf\Xi}_{m+1,m} - {\bf G}_{m+1} {\bf H}_{m+1} {\bf\Xi}_{m+1,m}$$ = (5)

where ${\bf G}$ is the Kalman gain matrix, ${\bf\Xi}$ is the measurement error's covariance matrix, and dependence on the track index l is not shown for notational convenience.

The measurement innovations sequence $\{ {\bf v}_{1,j_1 (l)} , {\bf v}_{2,j_2 (l)}, \cdots , {\bf v}_{n,j_n (l)} \}$ and corresponding covariance matrices $\{ {\bf S}_{1,j_1 (l)} , {\bf S}_{2,j_2 (l)}, \cdots , {\bf S}_{n,j_n (l)} \}$ generated in the Kalman filter computation, are to be used in optimum track estimation. Note that a ${\bf v}_{m,j_m (l)}$corresponding to ${\bf z}_{m,J_m +1}$ is not an actual innovations, but is a result of a missed detection at time m, and again is a notational convenience. In such a case, the Kalman state is just the predicted state (i.e. no measurement is available to update the state).