In the standard PHD filter, it is presumed that a target produces at most one measurement. Due to the multipath effect of OTHR, a single target may generate several measurements. Therefore, the measurement model of the MP-PHD filter proposed in this paper is different from the standard PHD filter. In this section, we first introduce the RFS measurement model for OTHR and then derive the update equation of the MP-PHD filter based on the theory of FISST.
3.1 The RFS measurement model for OTHR
At each time k, a finite set of measurements of OTHR, denoted by \( {Z}^k=\left\{{z}_{k,1},{z}_{k,2},\cdots {z}_{k,{N}_k}\right\} \), where \( {z}_{k,1},{z}_{k,2},\cdots {z}_{k,{N}_k} \) are the received measurements at time k and N
k
is the number of measurements. Since the measurements include several detections from targets and clutter, the collection of measurements of OTHR can be modeled by RFS
$$ {Z}^k={\varTheta}_{k,1}\left({x}_k\right)\cup {\varTheta}_{k,2}\left({x}_k\right)\cup {\varTheta}_{k,3}\left({x}_k\right)\cup {\varTheta}_{k,4}\left({x}_k\right)\cup {\varGamma}_k $$
(9)
where Θ
k,i
(x
k
), i = 1, ⋯, 4 denotes the measurement originated from the ith propagation path and Γ
k
denotes the RFS of clutter. It is presumed that conditional on x
k
, Θ
k,i
(x
k
), i = 1, ⋯, 4 and Γ
k
are independent RFSs.
3.2 The update equation of MP-PHD filter in OTHR
Note that both the MP-PHD filter and the standard PHD filter recursion require two steps: prediction and update. In the following subsection, we only derive the update equation according to the above RFS measurement model since the prediction step of the MP-PHD filter is identical to the standard PHD filter.
In ref. [17], the probability generating functional (PGFL) of update equation for the multitarget Bayes filter can be written as
$$ {G}_{k\Big|k}\left[h\right]=\frac{\frac{\delta F}{\delta {Z}_k}\left[0,h\right]}{\frac{\delta F}{\delta {Z}_k}\left[0,1\right]} $$
(10)
where F[g, h] is two-variable PGFL as
$$ F\left[g,h\right]={\displaystyle \int {h}^X}\cdot {G}_k\left[g\Big|X\right]\cdot {f}_{k\Big|k-1}\left(X\Big|{Z}^{\left(k-1\right)}\right)\delta X $$
(11)
$$ {G}_k\left[g\Big|X\right]={\displaystyle \int {g}^Z\cdot {f}_k\left(Z\Big|X\right)\delta Z} $$
(12)
where f
k|k − 1(X|Z
(k − 1)) is the predicted multitarget distribution, f
k
(Z|X) is the multitarget likelihood function and h
X is defined by h
X = 1 when X = ∅ and h
X = ∏
x ∈ X
h(x) otherwise. The update PHD filter can be given by
$$ {D}_{k\Big|k}(x)=\frac{\delta {G}_{k\Big|k}}{\delta x}\left[1\right] $$
(13)
Therefore, the update equation for MP-PHD filter can be derived by the following procedures:
-
1)
Derive the PGFL G
k
[g|X] by using the OTHR measurements and exploiting Eq. (11) to derive a closed form of F[g, h]
-
2)
Derive the derivatives of F[g, h] by using Eq. (10) to derive a closed form of G
k|k
[h]
-
3)
According to Eq. (11), derive a closed-form update for the MP-PHD filter in OTHR
In the following part of this subsection, we will present the details of the derivation. First, we derive the PGFL G
k
[g|X] according to the OTHR measurements in Eq. (9). Since Θ
k
(x) = Θ
k,1(x) ∪ Θ
k,2(x) ∪ Θ
k,3(x) ∪ Θ
k,4(x), the PGFL G
k
[g|x] of Θ
k
(x) is
$$ \begin{array}{l}{G}_k\left[g\Big|x\right]={\displaystyle \prod_{i=1}^4{G}_{k,i}\left[g\Big|x\right]}={\displaystyle \prod_{i=1}^4\left(1-{p}_{D,k}(x)+{p}_{D,k}(x){p}_{g,i}(x)\right)}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}={\left(1-{p}_{D,k}(x)\right)}^4+{\left(1-{p}_{D,k}(x)\right)}^3{p}_{D,k}(x)\left({\displaystyle \sum_{i=1}^4{p}_{g,i}(x)}\right)+{\left(1-{p}_{D,k}(x)\right)}^2{p}_{D,k}^2(x)\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\left({\displaystyle \sum_{i=1}^3{\displaystyle \sum_{j=i+1}^4{p}_{g,i}(x){p}_{g,j}(x)}}\right)+\left(1-{p}_{D,k}(x)\right){p}_{D,k}^3(x)\left({\displaystyle \sum_{i=1}^2{\displaystyle \sum_{j=i+1}^3{\displaystyle \sum_{m=j+1}^4{p}_{g,i}(x){p}_{g,j}(x)}}}{p}_{g,m}(x)\right)\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}+{p}_{D,k}^4(x)\left({\displaystyle \sum_{i=1}{\displaystyle \sum_{j=2}{\displaystyle \sum_{m=3}{\displaystyle \sum_{n=4}{p}_{g,i}(x){p}_{g,j}(x){p}_{g,m}(x){p}_{g,n}(x)}}}}\right)\end{array} $$
(14)
where p
g,i
(x) = ∫g(z)g
k,i
(z|x)dz, p
D,k
(x) is the detection probability, and g
k,i
(z|x) is the likelihood of the ith propagation path. Consequently, if we abbreviate q
D,k
(x) = 1 − p
D,k
(x), Eq. (11) can be summarized as follows:
$$ \begin{array}{c}F\left[g,h\right]={e}^{\lambda c\left[g\right]-\lambda}\cdot {G}_{k\Big|k-1}\left[h\right({q}_{D,k}^4+{q}_{D,k}^3{p}_{D,k}\left({\displaystyle \sum_{i=1}^4{p}_{g,i}}\right)+{q}_{D,k}^2{p}_{D,k}^2\left({\displaystyle \sum_{i=1}^3{\displaystyle \sum_{j=i+1}^4{p}_{g,i}{p}_{g,j}}}\right)\\ {}+{q}_{D,k}{p}_{D,k}^3\left({\displaystyle \sum_{i=1}^2{\displaystyle \sum_{j=i+1}^3{\displaystyle \sum_{m=j+1}^4{p}_{g,i}{p}_{g,j}{p}_{g,m}}}}\right)+{p}_{D,k}^4\left({\displaystyle \sum_{i=1}{\displaystyle \sum_{j=2}{\displaystyle \sum_{m=3}{\displaystyle \sum_{n=4}{p}_{g,i}{p}_{g,j}{p}_{g,m}{p}_{g,n}}}}}\right)\left)\right]\end{array} $$
(15)
where c[g] = ∫g(z)c
k
(z)dz and G
k|k − 1[h] = ∫h
X
f
k|k − 1(X|Z
(k − 1))δX.
To derive a closed-form update equation for the multipath multitarget PHD filter, we assume that the predicted multitarget distribution is a Poisson process:
$$ {G}_{k\Big|k-1}\left[h\right]={e}^{\mu s\left[h\right]-\mu } $$
(16)
where s[h] = μ
− 1∫h(x)D
k|k − 1(x)dx and μ = ∫D
k|k − 1(x)dx. Thus, Eq. (15) can be written as
$$ \begin{array}{c}F\left[g,h\right]= \exp \Big(\lambda c\left[g\right]-\lambda +\mu s\left[h{q}_{D,k}^4\right]+\mu s\left[h{q}_{D,k}^3{p}_{D,k}\left({\displaystyle \sum_{i=1}^4{p}_{g,i}}\right)\right]+\mu s\left[h{q}_{D,k}^2{p}_{D,k}^2\left({\displaystyle \sum_{i=1}^3{\displaystyle \sum_{j=i+1}^4{p}_{g,i}{p}_{g,j}}}\right)\right]\\ {}+\mu s\left[h{q}_{D,k}{p}_{D,k}^3\left({\displaystyle \sum_{i=1}^2{\displaystyle \sum_{j=i+1}^3{\displaystyle \sum_{m=j+1}^4{p}_{g,i}{p}_{g,j}{p}_{g,m}}}}\right)\right]+\mu s\left[h{p}_{D,k}^4\left({\displaystyle \sum_{i=1}{\displaystyle \sum_{j=2}{\displaystyle \sum_{m=3}{\displaystyle \sum_{n=4}{p}_{g,i}{p}_{g,j}{p}_{g,m}{p}_{g,n}}}}}\right)\right]-\mu \Big)\end{array} $$
(17)
Next, we deduce the formula for \( {\scriptscriptstyle \frac{\delta F}{\delta Z}}\left[0,h\right] \). Note that we set g = 0 because the formula for \( {\scriptscriptstyle \frac{\delta F}{\delta Z}}\left[g,h\right] \) is very cumbersome without setting it.
Lemma 1.
The derivatives of F[g, h] is given by the formula
$$ \frac{\delta F}{\delta Z}\left[0,h\right]=F\left[0,h\right]\cdot {\displaystyle {\prod}_Z\cdot {\displaystyle \sum_{\wp \angle Z}{\displaystyle \prod_{W\in \wp }{d}_W\left[0,h\right]}}} $$
(18)
where the notation ‘℘ ∠ Z’ is shorthand for “℘ partitions Z into cells W”, ∏
Z
= ∏
z ∈ Z
λc(z),
$$ {d}_W\left[0,h\right]=\left\{\begin{array}{c}\hfill 1+\mu s\left[h{q}_{D,k}^3{p}_{D,k}\left({\displaystyle \sum_{i=1}^4{\ell}_{z_1,i}}\right)\right]\kern16em ifW=\left\{{z}_1\right\}\hfill \\ {}\hfill \mu s\left[h{q}_{D,k}^2{p}_{D,k}^2\left({\displaystyle \sum_{i=1}^3{\displaystyle \sum_{j=i+1}^4{\displaystyle \sum_{a=1}^2{\displaystyle \sum_{\begin{array}{l}b=1\\ {}a\ne b\end{array}}^2{\ell}_{z_a,i}{\ell}_{z_b,j}}}}}\right)\right]\kern11em ifW=\left\{{z}_1,{z}_2\right\}\hfill \\ {}\hfill \mu s\left[h{q}_{D,k}{p}_{D,k}^3\left({\displaystyle \sum_{i=1}^2{\displaystyle \sum_{j=i+1}^3{\displaystyle \sum_{m=j+1}^4{\displaystyle \sum_{a=1}^3{\displaystyle \sum_{\begin{array}{l}b=1\\ {}a\ne b\end{array}}^3{\displaystyle \sum_{\begin{array}{l}c=1\\ {}a\ne c\\ {}b\ne c\end{array}}^3{\ell}_{z_a,i}{\ell}_{z_b,j}{\ell}_{z_c,m}}}}}}}\right)\right]\kern5em ifW=\left\{{z}_1,{z}_2,{z}_3\right\}\hfill \\ {}\hfill \mu s\left[h{p}_{D,k}^4\left({\displaystyle \sum_{i=1}{\displaystyle \sum_{j=2}{\displaystyle \sum_{m=3}{\displaystyle \sum_{n=4}{\displaystyle \sum_{a=1}^4{\displaystyle \sum_{\begin{array}{l}b=1\\ {}a\ne b\end{array}}^4{\displaystyle \sum_{\begin{array}{l}c=1\\ {}a\ne c\\ {}b\ne c\end{array}}^4{\displaystyle \sum_{\begin{array}{l}d=1\\ {}a\ne d\\ {}b\ne d\\ {}c\ne d\end{array}}^4{\ell}_{z_a,i}{\ell}_{z_b,j}{\ell}_{z_c,m}{\ell}_{z_d,n}}}}}}}}}\right)\right]\kern2em ifW=\left\{{z}_1,{z}_2,{z}_3,{z}_4\right\}\hfill \end{array}\right. $$
(19)
and where
$$ {\ell}_{z,i}(x)=\frac{g_{k,i}\left(z\Big|x\right)}{\lambda c(z)}\begin{array}{cc}\hfill \hfill & \hfill i=1,\cdots, 4.\hfill \end{array} $$
(20)
See Appendix 1 for the proof of Lemma 1.
Lemma 2.
The multipath posterior PGFL G
k|k
[h] is given as
$$ {G}_{k\Big|k}\left[h\right]={F}_0\left[h\right]\cdot \frac{{\displaystyle {\sum}_{\wp \angle {Z}_k}{\displaystyle {\prod}_{W\in \wp }{d}_W\left[0,h\right]}}}{{\displaystyle {\sum}_{\wp \hbox{'}\angle {Z}_k}{\displaystyle {\prod}_{W\in \wp \hbox{'}}{d}_W\left[0,1\right]}}} $$
(21)
where
$$ {F}_0\left[h\right]= \exp \left(\mu s\left[\left(h-1\right){q}_{D,k}^4\right]\right) $$
(22)
and d
W
[0, h] is the Eq. (
19
).
See Appendix 2 for the proof of Lemma 2.
According to Lemmas 1 and 2, we can obtain the MP-PHD filter update equation in Proposition 1 as follows:
Proposition 1.
The update equation for the MP-PHD filter is
$$ {D}_{k\Big|k}\left(x\Big|{Z}^{(k)}\right)\cong {L}_{Z_k}\left(x\Big|{Z}^{\left(k-1\right)}\right)\cdot {D}_{k\Big|k-1}\left(x\Big|{Z}^{\left(k-1\right)}\right) $$
(23)
where
$$ {L}_{Z_k}\left(x\Big|{Z}^{\left(k-1\right)}\right)={q}_{D,k}^4(x)+{\displaystyle \sum_{\wp \angle {Z}_k}{\omega}_{\wp }}\cdot {\displaystyle \sum_{W\in \wp}\frac{b_W}{d_W}} $$
(24)
here
$$ {\omega}_{\wp }=\frac{{\displaystyle {\prod}_{W\in \wp }{d}_W}}{{\displaystyle {\sum}_{\wp \hbox{'}\angle {Z}_k}{\displaystyle {\prod}_{W\in \wp \hbox{'}}{d}_W}}} $$
(25)
where
$$ {d}_W=\left\{\begin{array}{c}\hfill 1+{D}_{k\Big|k-1}\left[{q}_{D,k}^3{p}_{D,k}\left({\displaystyle \sum_{i=1}^4{\ell}_{z_1,i}}\right)\right]ifW=\left\{{z}_1\right\}\hfill \\ {}\hfill {D}_{k\Big|k-1}\left[{q}_{D,k}^2{p}_{D,k}^2\left({\displaystyle \sum_{i=1}^3{\displaystyle \sum_{j=i+1}^4{\displaystyle \sum_{a=1}^2{\displaystyle \sum_{\begin{array}{l}b=1\\ {}a\ne b\end{array}}^2{\ell}_{z_a,i}{\ell}_{z_b,j}}}}}\right)\right]ifW=\left\{{z}_1,{z}_2\right\}\hfill \\ {}\hfill {D}_{k\Big|k-1}\left[{q}_{D,k}{p}_{D,k}^3\left({\displaystyle \sum_{i=1}^2{\displaystyle \sum_{j=i+1}^3{\displaystyle \sum_{m=j+1}^4{\displaystyle \sum_{a=1}^3{\displaystyle \sum_{\begin{array}{l}b=1\\ {}a\ne b\end{array}}^3{\displaystyle \sum_{\begin{array}{l}c=1\\ {}a\ne c\\ {}b\ne c\end{array}}^3{\ell}_{z_a,i}{\ell}_{z_b,j}{\ell}_{z_c,m}}}}}}}\right)\right]ifW=\left\{{z}_1,{z}_2,{z}_3\right\}\hfill \\ {}\hfill {D}_{k\Big|k-1}\left[{p}_{D,k}^4\left({\displaystyle \sum_{i=1}{\displaystyle \sum_{j=2}{\displaystyle \sum_{m=3}{\displaystyle \sum_{n=4}{\displaystyle \sum_{a=1}^4{\displaystyle \sum_{\begin{array}{l}b=1\\ {}a\ne b\end{array}}^4{\displaystyle \sum_{\begin{array}{l}c=1\\ {}a\ne c\\ {}b\ne c\end{array}}^4{\displaystyle \sum_{\begin{array}{l}d=1\\ {}a\ne d\\ {}b\ne d\\ {}c\ne d\end{array}}^4{\ell}_{z_a,i}{\ell}_{z_b,j}{\ell}_{z_c,m}{\ell}_{z_d,n}}}}}}}}}\right)\right]\kern08em ifW=\left\{{z}_1,{z}_2,{z}_3,{z}_4\right\}\hfill \end{array}\right. $$
(26)
$$ {b}_W=\left\{\begin{array}{c}\hfill {q}_{D,k}^3{p}_{D,k}\left({\displaystyle \sum_{i=1}^4{\ell}_{z_1,i}}\right)\kern27.5em ifW=\left\{{z}_1\right\}\hfill \\ {}\hfill {q}_{D,k}^2{p}_{D,k}^2\left({\displaystyle \sum_{i=1}^3{\displaystyle \sum_{j=i+1}^4{\displaystyle \sum_{a=1}^2{\displaystyle \sum_{\begin{array}{l}b=1\\ {}a\ne b\end{array}}^2{\ell}_{z_a,i}{\ell}_{z_b,j}}}}}\right)\kern20.5em ifW=\left\{{z}_1,{z}_2\right\}\hfill \\ {}\hfill {q}_{D,k}{p}_{D,k}^3\left({\displaystyle \sum_{i=1}^2{\displaystyle \sum_{j=i+1}^3{\displaystyle \sum_{m=j+1}^4{\displaystyle \sum_{a=1}^3{\displaystyle \sum_{\begin{array}{l}b=1\\ {}a\ne b\end{array}}^3{\displaystyle \sum_{\begin{array}{l}c=1\\ {}a\ne c\\ {}b\ne c\end{array}}^3{\ell}_{z_a,i}{\ell}_{z_b,j}{\ell}_{z_c,m}}}}}}}\right)\kern14.9em ifW=\left\{{z}_1,{z}_2,{z}_3\right\}\hfill \\ {}\hfill {p}_{D,k}^4\left({\displaystyle \sum_{i=1}{\displaystyle \sum_{j=2}{\displaystyle \sum_{m=3}{\displaystyle \sum_{n=4}{\displaystyle \sum_{a=1}^4{\displaystyle \sum_{\begin{array}{l}b=1\\ {}a\ne b\end{array}}^4{\displaystyle \sum_{\begin{array}{l}c=1\\ {}a\ne c\\ {}b\ne c\end{array}}^4{\displaystyle \sum_{\begin{array}{l}d=1\\ {}a\ne d\\ {}b\ne d\\ {}c\ne d\end{array}}^4{\ell}_{z_a,i}{\ell}_{z_b,j}{\ell}_{z_c,m}{\ell}_{z_d,n}}}}}}}}}\right)ifW=\left\{{z}_1,{z}_2,{z}_3,{z}_4\right\}\hfill \end{array}\right. $$
(27)
See Appendix 3 for the proof of Proposition 1.