This section presents one complete recursion cycle for augmented state GMM-ITS (AS GMM-ITS). At each scan, the augmented state is smoothed, and the results are later used in Section 5.1 to determine the smoothed target state at a fixed lag of *N* using the proposed method.

The GMM-ITS algorithm [4] details are omitted, and only its application to the augmented state is discussed in order to minimize the complexity at this stage. Let *k*
_{
τ
}=*k*−*N*+*r*, (0≤*r*≤*N*) be the variable to address each scan in the smoothing interval. Each measurement in each scan not selected by the already established track initializes a new track [5].

At scan *k*−*N*, the track state is initialized as

$$ {}p\left({\mathbf{x}}_{k-N}|\chi_{k-N},\mathbf{Z}^{k-N}\right)=\mathcal N\left({{\mathbf{x}}_{k - N} ;\hat {\mathbf{x}}_{k - N|k - N},{\mathbf{P}}_{k - N|k - N}} \right). $$

((18))

where \(\left [ {{\hat {\mathbf {x}}_{k - N|k - N}},{{\mathbf {P}}_{k - N|k - N}}} \right ]\) is the filtered target trajectory state and its error covariance matrix updated at scan *k*−*N*. The track trajectory state is approximated by a single Gaussian component *c*
_{
k−N
} at initialization. Here, the probability of the component *c*
_{
k−N
} is \({\xi _{k-N}^{c_{k-N}}}=1\).

### 4.1 State augmentation

At scan *k*
_{
τ
}=*k*−*N*(*r*=0) in the smoothing interval, the track trajectory state and its associated error covariance matrix \(\hat {\mathbf {X}}_{k - N|k-N}^{\text {AS}}\) and \({\mathbf {P}}_{k - N|k-N}^{\text {AS}}\) are augmented, respectively, as

$$ {}\hat {\mathbf{X}}_{k - N|k-N}^{\text{AS}} = \left[ {\begin{array}{*{20}c} {\hat {\mathbf{x}}_{k - N|k - N} }\! &\! {\hat {\mathbf{x}}_{k - N-1|k -N} } \ldots {\hat {\mathbf{x}}_{k - 2N|k - N}} \\ \end{array}}\right]^{T} $$

((19))

and

$$ {}{\mathbf{P}}_{k - N|k-N}^{\text{AS}} = \left[{\begin{array}{*{20}c} {{\mathbf{P}}_{k - N|k - N}} & - & - & - \\ -& {{\mathbf{P}}_{k - N - 1|k - N}} & - & - \\ - & - & \ddots & - \\ - & - & - & {{\mathbf{P}}_{k - 2N|k - N}} \\ \end{array}} \right]. $$

((20))

The superscript AS stands for augmented state, where the (−) sign on the right hand side of (20) corresponds to cross covariance terms of state elements of the augmented state, not detailed here for reasons of clarity. The augmented state propagation matrix is

$$ \textit{\textbf{F}}^{\text{AS}} = \left[ {\begin{array}{*{20}c} {{\mathbf{F}}_{k - 1}} & {\textbf{0}_{n,n}} & \mathbf{\cdots} & {\textbf{0}_{n,n}} \\ {\mathbf{I}_{n}} & {\textbf{0}_{n,n}} & \mathbf{\cdots} & {\textbf{0}_{n,n}} \\ \mathbf{\ddots} & \mathbf{\ddots} & \mathbf{\cdots} & \mathbf{\ddots} \\ {\textbf{0}_{n,n}} & \mathbf{\cdots} & {\textbf{I}_{n}} & {\textbf{0}_{n,n}} \\ \end{array}} \right]. $$

((21))

The augmented plant noise covariance matrix is

$$ \textit{\textbf{Q}}^{\text{AS}} = \left[ {\begin{array}{*{20}c} {{\mathbf{Q}}_{k - 1}} & {\mathbf{0}_{n,n}} & \mathbf{\cdots} & {\mathbf{0}_{n,n}} \\ {\mathbf{0}_{n,n}} & {\mathbf{0}_{n,n}} & \mathbf{\cdots} & {\mathbf{0}_{n,n}} \\ \mathbf{\ddots} & \mathbf{\ddots} & \mathbf{\cdots} & \mathbf{\ddots} \\ {\mathbf{0}_{n,n}} & \mathbf{\cdots} & {\mathbf{0}_{n,n}} & {\mathbf{0}_{n,n}} \\ \end{array}} \right]. $$

((22))

The augmented measurement matrix with respect to the position component of measurement becomes

$$ \mathbf{H}^{\text{AS},{p}} = \left[ {\begin{array}{*{20}c} {\mathbf{H} }_{k}^{p} & {\textbf{0}_{m,n\times (N)}} \\ \end{array}} \right], $$

((23))

where

$$ {\mathbf{H}}_{k}^{p} = \left[ {\begin{array}{*{20}{c}} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \end{array}} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}. $$

((24))

The linearized augmented measurement coefficient matrix with respect to the Doppler component of measurement becomes

$$ {\mathbf{H}^{\text{AS},{w}}} = \left[ {\begin{array}{*{20}{c}} {{\mathbf{H}^{{w}}}}(\mathbf{{x}}_{k})&{{{\mathbf{0}}_{1,n \times (N)}}} \end{array}} s\right], $$

((25))

where **I**
_{
n
} is an *n*-dimensional identity matrix, and **0**
_{
n,n
} and **0**
_{
m,n
} are matrices of zeros, where *n* is the order of the target state vector, and *m* is the order of the position measurement vector. Terms **w**
_{
k−1}, **v**
_{
k
}, *F*
_{
k−1}, and *Q*
_{
k−1} are defined in Section 2.1. The order of matrices *F*
^{AS} and *Q*
^{AS} is ((*N*+1).*n*×(*N*+1).*n*), while *H*
^{AS,p} has dimensions of (*m*×(*N*+1).*n*), and *H*
^{AS,w} has dimensions of (1×(*N*+1).*n*).

### 4.2 State prediction

At any scan *k*
_{
τ
}=*k*−*N*+*r*, (1≤*r*≤*N*) inside the smoothing interval, the predicted augmented state conditioned on the component \({c_{k_{\tau }-1}}\) is obtained by standard Kalman filter

$$ \begin{array}{l} {}\left[ {\hat {\textit{\textbf{X}}}_{k_{\tau} |k_{\tau} - 1}^{\text{AS},{c_{k_{\tau} - 1}} },{\textit{\textbf{P}}}_{k_{\tau} |k_{\tau} - 1}^{\text{AS},{c_{k_{\tau} - 1} }}} \right]\\= {\textbf{KF}}_{\mathrm{P }}\left(\begin{array}{l} \hat {\textit{\textbf{X}}}_{k_{\tau} - 1|k_{\tau} - 1}^{\text{AS},{c_{k_{\tau} - 1} }},{\textit{\textbf{P}}}_{k_{\tau} - 1|k_{\tau} - 1}^{\text{AS},{c_{k_{\tau} - 1} }}, {\textit{\textbf{F}}}^{\text{AS}},{\textit{\textbf{Q}}}^{\text{AS}} \end{array}\right), \\ \end{array} $$

((26))

where **KF**
_{P} represents the standard Kalman filter prediction. The probability density function (PDF) for the predicted augmented state and error covariance at scan *k*
_{
τ
} is

$$ \begin{array}{l} {}p\!\left(\! {{\textit{\textbf{X}}}_{k_{\tau} }^{\text{AS}} |{c_{k_{\tau}-1}},\chi_{k_{\tau}},\textbf{Z}^{k_{\tau} - 1}} \!\right) = \mathcal{N}\left(\begin{array}{l} {\textit{\textbf{X}}}_{k_{\tau} }^{\text{AS}} ;\hat {\textit{\textbf{X}}}_{k_{\tau} |k_{\tau} - 1}^{\text{AS},{c_{k_{\tau} - 1} }} \!, {\textit{\textbf{P}}}_{k_{\tau} |k_{\tau} - 1}^{\text{AS},{c_{k_{\tau} - 1} }} \end{array} \right). \\ \end{array} $$

((27))

### 4.3 Measurement selection and likelihood calculation

To reduce the computational requirements, a subset of validated measurements from all the measurements received by the sensor is selected at each scan *k*
_{
τ
} in the smoothing interval. A gating procedure[20] is used to select the validated measurements for each measurement component *g* corresponding to each track component \({c_{k_{\tau } - 1}}\). A gating test (28) is applied to the position component (\({\textbf {y}_{k_{\tau },i}^{g,{p}}} \)) of each measurement received at each scan in the smoothing interval.

To simplify the notations in the rest of the paper, denote \({s^{p}} = \left \{ {\text {AS},g,{c_{{k_{\tau }} - 1}},p} \right \}\).

$$ \begin{array}{l} \left({\textbf{y}_{k_{\tau},i}^{g,{p}} - \hat{\textbf{y}}_{k_{\tau} |k_{\tau} - 1}^{{{c_{k_{\tau} - 1} }},p}} \right)^{T}\!\!\! \left(\!{{\textit{\textbf{S}}}_{k_{\tau},i }^{s^{p}} }\! \right)^{- 1} \!\! \left({\textbf{y}_{k_{\tau},i}^{g,{p}} - \hat{\textbf{y}}_{k_{\tau} |k_{\tau} - 1}^{{c_{k_{\tau}-1},p}}} \right)\! \le \!\kappa\,. \\ \end{array} $$

((28))

where

$$ \hat{\textbf{y}}_{k_{\tau} |k_{\tau} - 1}^{{c_{k_{\tau} - 1} },p} ={{{{\textit{\textbf{H}}}}^{\mathrm{\text{AS}},p}}}{\hat {{\textit{\textbf{X}}}}_{{k_{\tau} }|{k_{\tau}} - 1}^{\text{AS},{c_{k_{\tau} - 1} }}} $$

((29))

and

$$ {{{\textit{\textbf{S}}}}_{k_{\tau},i }^{s^{p}} }={{\textit{\textbf{H}}}}^{\text{AS},{p}} {{\textit{\textbf{P}}}}_{k_{\tau} |k_{\tau} - 1}^{\text{AS},{c_{k_{\tau} - 1} }} \left({{{\textit{\textbf{H}}}}^{{\text{AS},{p}} }}\right)^{T}+\textbf{R}_{k_{\tau},i}^{g,{p}}\,\,\,, $$

((30))

where *κ* is the selection threshold. In simulated two-dimensional surveillance, *κ* is selected as 13.3, which corresponds to the gating probability *P*
_{g}=0.99. Each *i*-th validated measurement is represented by \({\textit {\textbf {z}}}_{k_{\tau },i}\) and has \(G_{k_{\tau },i}^{s}\) validated components. Hereafter, the parameters \(\left (\textbf {y}_{k_{\tau },i}^{g},\textbf {R}_{k_{\tau },i}^{g},\gamma _{k_{\tau },i}^{g},G_{k_{\tau },i}^{g}\right)\) are attributed to the selected measurement \(\textit {\textbf {z}}_{k_{\tau },i}\). At any scan *k*
_{
τ
} in the smoothing interval, the measurement likelihood of the selected measurement \({\textit {\textbf {z}}}_{k_{\tau },i}\) becomes

$$ {p_{{k_{\tau} },i}} = p\left({{{\textit{\textbf{z}}}_{{k_{\tau} },i}}|{\textbf{Z}^{{k_{\tau}} - 1}}} \right) = \sum\limits_{g = 1}^{G_{{k_{\tau} }}^{s}} {\gamma_{{k_{\tau} }}^{g}} p_{{k_{\tau} },i}^{g}\,\,\,\,, $$

((31))

where \(p_{{k_{\tau } },i}^{g}\) is the likelihood of measurement component \(\textbf {y}_{k_{\tau },i}^{g}\)

$$ p_{{k_{\tau} },i}^{g} = p\left({{\textbf{{y}}}_{{k_{\tau} },i}^{g}|{\textbf{Z}^{{k_{\tau}} - 1}}} \right) = \sum\limits_{{c_{{k_{\tau}} - 1}}} {\xi_{_{{k_{\tau}} - 1}}^{{c_{{k_{\tau}} - 1}}}} p_{{k_{\tau} },i}^{g,{c_{{k_{\tau}} - 1}}}, $$

((32))

where \(p_{{k_{\tau } },i}^{g,{c_{{k_{\tau }} - 1}}}\) is the likelihood of measurement component \(\textbf {y}_{k_{\tau },i}^{g}\) with respect to track component \(c_{k_{\tau }-1}\)

$$ p_{{k_{\tau} },i}^{g,{c_{{k_{\tau}} - 1}}} = \left\{ {\begin{array}{*{20}{c}} {p_{{k_{\tau} },i}^{g,{c_{{k_{\tau}} - 1}},{p}}p_{{k_{\tau} },i}^{g,{c_{{k_{\tau}} - 1}},{w}}};&{\textbf{y}_{{k_{\tau} },i}^{g,{c_{{k_{\tau}} - 1}}} \in {\textbf{Y}}_{{k_{\tau} },i}^{{c_{{k_{\tau}} - 1}}}}\\\\ 0\,\,;&{\textbf{y}_{{k_{\tau} },i}^{g,{c_{{k_{\tau}} - 1}}} \notin {\textbf{Y}}_{{k_{\tau} },i}^{{c_{{k_{\tau}} - 1}}}} \end{array}} \right., $$

((33))

where \(\textbf {Y}_{{k_{\tau } },i}^{{c_{{k_{\tau }} - 1}}}={\left \{ {\textbf {y}_{{k_{\tau } },i}^{g,{c_{{k_{\tau }} - 1}}}} \right \}_{g = 1:G_{{k_{\tau } },i}^{s}}}\), \(p_{{k_{\tau } },i}^{g,{c_{{k_{\tau }} - 1}},{p}}\) is the likelihood of the position component of measurement \(\textbf {y}_{k_{\tau },i}^{g}\), and \(p_{{k_{\tau } },i}^{g,{c_{{k_{\tau }} - 1}},{w}}\) is the likelihood of the Doppler component conditioned on the position component of measurement

$$ \begin{array}{l} p_{{k_{\tau} },i}^{g,{c_{{k_{\tau}} - 1}},p} = \frac{1}{{{P_{g}}}}\mathcal{N}\left(\! {\textbf{y}_{{k_{\tau} },i}^{g,{c_{{k_{\tau-1}} }}}; \textbf{\^ y}_{{k_{\tau} }|{k_{\tau-1}}}^{{c_{{k_{\tau-1}} }},p},{{\textit{\textbf{S}}}}_{{k_{\tau} },i}^{s^{p}}}\! \right), \end{array} $$

((34))

where \({{\textit {\textbf {S}}}}_{{k_{\tau } },i}^{s^{p}}\) is as defined in (30).

The augmented track component is updated by the measurement position component using standard Kalman filter update represented by **KF**
_{U}

$$ \begin{array}{l} {}\left[ {\hat{{\textit{\textbf{X}}}}_{{k_{\tau} }|{k_{\tau} },i}^{s^{p}}{\kern 1pt}, {{\textit{\textbf{P}}}}_{{k_{\tau} }|{k_{\tau} },i}^{s^{p}}} \right] \\= {\textbf{KF}}_{\mathrm{U}}\left({\textbf{y}_{{k_{\tau} },i}^{g,{p}}{\kern 1pt},\textbf{{R}}_{{k_{\tau} },i}^{g,{p}},\hat {{\textit{\textbf{X}}}}_{{k_{\tau} }|{k_{\tau}} - 1}^{\text{AS},{c_{{k_{\tau}} - 1}}}{\kern 1pt}, {{\textit{\textbf{P}}}}_{{k_{\tau} }|{k_{\tau}} - 1}^{\text{AS},{c_{{k_{\tau}} - 1}}},{{{\textit{\textbf{H}}}}^{\text{AS},{p}}}} \right). \end{array} $$

((35))

The likelihood of the decorrelated Doppler component conditioned on the position component of measurement \(\textbf {y}_{k_{\tau },i}^{g}\) is calculated in (36). A transformation is applied to augmented state \(\hat {{\textit {\textbf {X}}}}_{{k_{\tau } }|{k_{\tau } },i}^{s^{p}}\) using transformation matrix *T*
^{ι} to provide only the filtered state \(\hat {\textit {\textbf {x}}}_{{k_{\tau } }|{k_{\tau } },i}^{g,{c_{{k_{\tau }} - 1}},p}\) (the predicted Doppler mean at current scan *k* requires only the filtered state in (8–13)) for the calculation of Doppler likelihood of the *i*-th measurement at current scan *k*. To simplify the notation in the rest of the paper, denote \({s^{w}} = \left \{ {\text {AS},g,{c_{{k_{\tau }} - 1}},w} \right \}\).

$$ p_{{k_{\tau} },i}^{g,{c_{{k_{\tau}} - 1}},{w}}\!\! =\! \mathcal{N}\left(\!{w_{{k_{\tau} },i}^{g};\hat w_{k_{\tau},i}^{g},\,{{\textit{\textbf{S}}}}_{{k_{\tau} },i}^{s^{w}}}\! \right), $$

((36))

where

$$ \hat w_{k_{\tau},i}^{g}={h^{{w}}}\left({\textit{\textbf{T}}^{\iota}}{\hat {{\textit{\textbf{X}}}}_{{k_{\tau} }|{k_{\tau} },i}^{s^{p}}}\! \right) $$

((37))

$$ {\textit{\textbf{T}}^{\iota}}=\left[ {\begin{array}{*{20}{c}} {{\textbf{I}_{n}}}&{{\textbf{0}_{n, n \times (N)}}} \end{array}} \right] $$

((38))

and

$$ \begin{array}{l} {{\textit{\textbf{S}}}}_{{k_{\tau} },i}^{s^{w}} = {{\textit{\textbf{H}}}}_{{k_{\tau} },i}^{s^{w}}{{\textit{\textbf{P}}}}_{{k_{\tau} }|{k_{\tau} },i}^{s^{p}}{\left({{{\textit{\textbf{H}}}}_{{k_{\tau} },i}^{s^{w}}} \right)^{T}} + {\sigma_{w}^{2}}, \end{array} $$

((39))

where

$$ {{\textit{\textbf{H}}}}_{{k_{\tau} },i}^{s^{w}} = {{{\textit{\textbf{H}}}}^{\text{AS},{w}}}\left(\hat {{\textit{\textbf{X}}}}_{{k_{\tau} }|{k_{\tau} },i}^{s^{p}} \right). $$

((40))

### 4.4 State update

The decorrelated Doppler measurement component \(w_{k_{\tau },i}^{g}\) is used to update the track using standard extended Kalman filter **EKF**
_{U}

$$ \begin{array}{l} \left[ {\hat {{\textit{\textbf{X}}}}_{{k_{\tau} }|{k_{\tau} },i}^{\text{AS},g,{c_{{k_{\tau}} - 1}}}{\kern 1pt} {{\textit{\textbf{P}}}}_{{k_{\tau} }|{k_{\tau} },i}^{\text{AS},g,{c_{{k_{\tau}} - 1}}}} \right] \\ ={\textbf{EKF}}_{\mathrm{U}}\left({w_{{k_{\tau} },i}^{g}{\kern 1pt},{\sigma_{w}^{2}},\hat {{\textit{\textbf{X}}}}_{{k_{\tau} }|{k_{\tau} },i}^{s^{p}},{{\textit{\textbf{P}}}}_{{k_{\tau} }|{k_{\tau} },i}^{s^{p}},\hat w_{k_{\tau},i}^{g}},{{{\textit{\textbf{H}}}}^{\text{AS},{w}}} \right). \end{array} $$

((41))

At current scan *k*
_{
τ
}, a new track component is formed \(c_{k_{\tau }}^{-}\!\!:\!\!\{i,g,c_{k_{\tau }-1}\}\). Then, the a posteriori augmented track trajectory state PDF at scan *k*
_{
τ
} is a mixture of augmented track trajectory state estimates with respect to each new component. The Gaussian PDF of each augmented trajectory state with respect to each new track component \(\bar {c}_{k_{\tau }}\) is

$$ \begin{array}{l} p\left({ {{\textit{\textbf{X}}}}_{{k_{\tau} }}^{\text{AS}}|{{\bar c}_{{k_{\tau} }}},{\chi_{{k_{\tau} }}},{{{\textbf{{Z}}}}^{{k_{\tau} }}}} \right) = \mathcal{N}\left({\hat {{\textit{\textbf{X}}}}_{{k_{\tau} }}^{\text{AS}}; \hat {{\textit{\textbf{X}}}}_{{k_{\tau} }|{k_{\tau}}}^{\text{AS},{{\bar c}_{{k_{\tau} }}}},{{\textit{\textbf{P}}}}_{{k_{\tau} }|{k_{\tau} }}^{\text{AS},{\bar c_{{k_{\tau}} }}}} \right), \end{array} $$

((42))

where

$$ \begin{array}{l} {}\left[ {\hat {{\textit{\textbf{X}}}}_{{k_{\tau} }|{k_{\tau} }}^{\text{AS},{{\bar c}_{{k_{\tau} }}}}{\kern 1pt} {{\textit{\textbf{P}}}}_{{k_{\tau} }|{k_{\tau} }}^{\text{AS},{\bar c_{{k_{\tau}} }}}} \right] = \left\{ {\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {\hat {{\textit{\textbf{X}}}}_{{k_{\tau} }|{k_{\tau}} - 1}^{\text{AS},{c_{{k_{\tau}} - 1}}}}&{{{\textit{\textbf{P}}}}_{{k_{\tau} }|{k_{\tau}} - 1}^{\text{AS},{c_{{k_{\tau}} - 1}}}} \end{array}} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ;}&{i = 0}\\ {\left[ {\begin{array}{*{20}{c}} {\hat {{\textit{\textbf{X}}}}_{{k_{\tau} }|{k_{\tau} },i}^{\text{AS},g,{c_{{k_{\tau}} - 1}}}}&{{{\textit{\textbf{P}}}}_{{k_{\tau} }|{k_{\tau} },i}^{\text{AS},g,{c_{{k_{\tau}} - 1}}}} \end{array}} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ;}&{i > 0} \end{array}} \right.. \end{array} $$

((43))

The fixed lag smoothed trajectory state in Section 5 requires the Gaussian sum of all the track components to obtain the augmented track state at scan *k*
_{
τ
} in the smoothing window using (44) and (45).

$$ \hat {{\textit{\textbf{X}}}}_{{k_{\tau} }}^{\text{AS}} = \sum\limits_{{\bar c_{{k_{\tau} }}}} {\xi_{_{{k_{\tau}} }}^{{{\bar c}_{{k_{\tau} }}}}} \hat {{\textit{\textbf{X}}}}_{{k_{\tau} }|{k_{\tau} }}^{\text{AS},{{\bar c}_{{k_{\tau} }}}}, $$

((44))

$$ {{\textit{\textbf{P}}}}_{k_{\tau} }^{\text{AS}} =\sum\limits_{\bar c_{k_{\tau}} } {{\xi_{_{{k_{\tau}} }}^{{{\bar c}_{{k_{\tau} }}}}}\!\left\{ \begin{array}{l} {{\textit{\textbf{P}}}}_{{k_{\tau} }|{k_{\tau} }}^{\text{AS},g,{\bar c_{{k_{\tau}} }}}+ \left[ {\hat {{\textit{\textbf{X}}}}_{{k_{\tau} }|{k_{\tau} }}^{\text{AS},{{\bar c}_{{k_{\tau} }}}}- \hat X_{k_{\tau} }^{\text{AS}}} \right] \\ \left[ {\hat {{\textit{\textbf{X}}}}_{{k_{\tau} }|{k_{\tau} }}^{\text{AS},{{\bar c}_{{k_{\tau} }}}} - \hat {{\textit{\textbf{X}}}}_{k_{\tau} }^{\text{AS}}} \right]^{T} \\ \end{array} \right\}}. $$

((45))

The component probability \({{\bar c}_{{k_{\tau } }}}\) is updated as

$$ \xi_{_{{k_{\tau} }}}^{{{\bar c}_{{k_{\tau} }}}} = \frac{{\xi_{_{{k_{\tau}} - 1}}^{{c_{{k_{\tau}} - 1}}}}}{{{\Lambda_{k_{\tau}}}}}\left\{ {\begin{array}{*{20}{c}} {1 - {P_{\mathrm{d}}}{P_{\mathrm{g}}}{\kern 1pt} {\kern 1pt} ;}&{i = 0}\vspace{0.2cm }\\ {\frac{{{P_{\mathrm{d}}}{P_{\mathrm{g}}}}}{{{\rho_{{k_{\tau} },i}}}}\gamma_{{k_{\tau} },i}^{g}\,\,p_{{k_{\tau} },i}^{g,{c_{{k_{\tau}} - 1}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ;}&{i > 0} \end{array}} \right.. $$

((46))

The likelihood ratio at scan *k*
_{
τ
} is defined as

$$ {\Lambda_{{k_{\tau} }}} = 1 - {P_{\mathrm{d}}}{P_{\mathrm{g}}} + {P_{\mathrm{d}}}{P_{\mathrm{g}}}\sum\limits_{i = 1}^{{m_{{k_{\tau} }}}} {\frac{{{p_{{k_{\tau} },i}}}}{{{\rho_{{k_{\tau} },i}}}}}. $$

((47))

The track augmented trajectory state \({\hat {\textit {\textbf {X}}}}_{k_{\tau }}^{\text {AS}}\) and its augmented error covariance matrix \({{\textit {\textbf {P}}}}_{k_{\tau }}^{\text {AS}}\) provide filtered and smoothed trajectory states \(\left \{{\textbf {\textit \^{x}}}_{{k_{\tau } }|{k_{\tau } }},{{{\textbf {\textit \^{x}}}}_{{k_{\tau }} - 1|{k_{\tau } }}}, \ldots,{{\textbf {\textit \^{x}}}}_{k_{\tau }} - N|{k_{\tau } } \right \}\) and their corresponding error covariance matrices \(\left \{ {{{{\textit {\textbf {P}}}}_{{k_{\tau } }|{k_{\tau } }}},{{{\textit {\textbf {P}}}}_{{k_{\tau }} - 1|{k_{\tau } }}}, \ldots,{{{\textit {\textbf {P}}}}_{{k_{\tau }} - N|{k_{\tau } }}}} \right \}\) at scan *k*
_{
τ
} in the smoothing interval.

The target existence state at scan *k*
_{
τ
} updates as [15]

$$ \begin{array}{l} {}P\left\{ {{\chi_{{k_{\tau} }}}|{\textbf{Z}^{{k_{\tau} }}}} \right\} \\=\frac{{{\Lambda_{{k_{\tau} }}}}}{{1 \!- \! P\left\{ {{\chi_{{k_{\tau} }}}|{\textbf{Z}^{{k_{\tau}} - 1}}} \right\} + {\Lambda_{{k_{\tau} }}}P\left\{ \! {{\chi_{{k_{\tau} }}}|{\textbf{Z}^{{k_{\tau}} \! - \! 1}}} \right\}}}P \!\left\{ \! {{\chi_{{k_{\tau} }}}|{\textbf{Z}^{{k_{\tau}} \!- \! 1}}} \!\right\}\!\!. \end{array} $$

((48))