6.2 SMC–Intensity Filter Using a Decomposition of a Likelihood Function 123
6.2.2 Likelihood decomposition for multipath measurements
In the following, the output of the ray tracer for a fixed OS location, based on the database of the urban environment and specific radio channel parameters is consid-ered. The ray tracing prediction is modeled according to [HWLW03], which is based on [HWLW99], [WHL99], [HWLW03] and the fundamentals given in [Gla89] For a given AoAϕ∈[−π, π] the set of multipaths predicted by the ray tracer is defined by Mϕ≡ {fj: [0,∞)→R2|j∈ {1, ..., p}}, (6.44) wherepdenotes the number of predicted multipaths for the particular AoAϕandfj
defines a specific multipath, given as a function of the range. Several multipaths which correspond to one AoA can occur due to resolution conflicts and the possibility that an interaction point is modeled as a radially emitting point source. In the following, an arbitraryj∈ {1, ..., p}is chosen and one multipathfj is considered. It is assumed thatfj is continuously differentiable and bijective on its image. Let therefore denote gj the inverse offj, that is ,gj≡fj−1. Thus, iffj andgj are given, Equation (6.38) can be used to define the respective likelihood function.
Unfortunately as mentioned before, an explicit definition offjcannot be assumed in general. Therefore, neither its Jacobian nor its inverse are known and hence (6.38) cannot be evaluated as in (6.41) for the bearings–only case. To obtain a general
Figure 6.7:Each multipath between the OS and the MS is assumed to propagate linearly in between its interaction points. Boxes around grid points, which are proportional to the covariance matrix of the decomposed likelihood function limit the allowed region of particles (black dots) for each multipath c2014 IEEE.
decomposed likelihood function, which is linear in target space according to [SW09], an additional approximation has to be made. Therefore, instead of considering the explicit form of fj the output of the ray tracer is considered. Additionally to the AoA and RToA for a given MS–OS combination, the ray tracer provides the points of interaction for each multipath.
Therefore, let
Ij,0, ..., Ij,N∈R2 (6.45)
be the interaction points of the multipathfj, whereIj,0 denotes the position of the OS andIj,N denotes the position of the MS. Then, there exist
rj,0, ..., rj,N>0 (6.46) such that
fj(rj,i) =Ij,i (6.47)
for alli∈ {0, ..., N}(see Figure 6.7). Note, thatrj,0= 0 by definition. Furthermore, let
αj,0≡ϕ (6.48)
αj,i≡arctan
Ij,i+1,y−Ij,i,y
Ij,i+1,x−Ij,i,x
, (6.49)
i= 0, ..., N−1 be the angles defined by the multipath at the interaction points. Then fori∈ {0, ...N−1}the functionfj,i: [0, rj,i+1−rj,i]→R2 is defined by
fj,i(r)≡Ij,i+ rcos(αj,i) rsin(αj,i)
!
(6.50) and
J[fj,i](r)≡ cos(αj,i) −r sin(αj,i) sin(αj,i) r cos(αj,i)
!
, (6.51)
denotes the corresponding Jacobian. Note that the set{fj,1, ..., fj,N−1}approximates the ray tracing prediction function by assuming a linear propagation of the radio chan-nel in between the interaction points. Hence, the likelihood function corresponding tofj∈ Mϕand a target state (x, y)T ∈R2can be approximated by
p(ϕ|j,(x, y)T)≈
N−1
X
i=0
Z ri+1−ri 0
rN
Ij,i+ rcos(αj,i) rsin(αj,i)
!
x y
!
, J[fj,i](r)ΣJT[fj,i](r)
dr, (6.52) where the event j denotes that the multipath defined by fj was chosen. Due to marginalization with respect to the different multipaths belonging to a specific AoA ϕ∈[−π, π] the likelihood function of the bearing measurementϕgiven the hypothet-ical position of the MS (x, y)T ∈R2is given by
p(ϕ|(x, y)T)≡
p
X
j=1
p(ϕ|j,(x, y)T)≈
p
X
j=1 N−1
X
i=0
Z rj,i+1−rj,i 0
rN
Ij,i+ rcos(αj,i) rsin(αj,i)
!
x y
!
, J[fj,i](r)ΣJT[fj,i](r)
dr.
(6.53) 6.2.2.2 Implementation Issues
Specific issues concerning the implementation of the proposed likelihood decomposi-tion are explained in the following.
Determination of Multipaths To evaluate the likelihood function defined in (6.53), multipaths have to be determined for a particular AoA measurementϕ ∈ [−π, π].
Therefore, the ray tracer prediction is used. First, by computing the received multi-paths for all possible MS positions (on a grid–cell) and a fixed OS, the database of all possible multipaths in the field of view is generated and sorted with respect to the
AoA. Each entry of the database represents a multipath. Additionally to the AoA and the RToA (with respect to its MS position) also the interaction points characterize a specific multipath.
Given the database of all predicted multipaths the paths corresponding to a specific AoA measurementϕneed to be determined for the evaluation of the likelihood func-tion defined in (6.53). Since each interacfunc-tion point emits radially an electromagnetic wave, it is possible that several multipaths belong to the same AoA measurementϕ.
Depending, on the geometry this fact implies that it might become impossible to ob-tain a reasonable tracking result, due to a large number of falsely chosen multipaths.
Furthermore, the computational effort increases dramatically if a large number of predicted multipaths belongs to the same AoA measurement, since for each path the integral from Equation (6.53) has to be evaluated. Thus, the RToA is utilized to re-duce the number of predicted multipaths for a specific AoA measurement. Therefore, the likelihood function from (6.53) is approximated by
p((ϕ, τ)T|(x, y)T)≈
˜ p
X
j=1 N−1
X
i=0
Z rj,i+1−rj,i 0
rN
Ij,i+ rcos(αj,i) rsin(αj,i)
!
x y
!
, J[fj,i](r)ΣJT[fj,i](r)
dr, (6.54) where ˜pdenotes the predicted multipaths that are chosen according to the measured AoA ϕand RToA τ. Several rules for choosing multipaths out of the database for a given measurement (ϕ, τ)T ∈[−π, π]×R+ are possible. Note that the assignment problem, which is inherently present when studying BML scenarios, shows up here. In this work, first the paths with the best match for the AoAϕare chosen. Afterwards, out of these the multipath with the closest RToA to the measurement τ is selected.
However, a problem for the multipaths with RToA = 0 arises. Due to the fact that the ray tracer computes the wave propagation pointwisely per hypothetical emitter position, the RToA cannot be used to distinguish multipaths when it is equal to 0.
Due to this reason, measurements that possess an RToA of 0 are not used for the localization.
Drawing of Particles New particles have to be created in the initialization step and for the detection of newborn targets in each iteration. The easiest way of distributing particles is to use a uniform distribution over the FOV. In [RCV10] the authors pro-pose to create newborn particles around the measurements from the previous iteration to avoid a high number of additional particles in scenarios with a high probability of detection. In terms of BML this means that particles are distributed along the received multipaths from the previous iteration. Furthermore, context information from the ray tracer database can be used to draw particles only in regions, where
targets are observable.
The database of the ray tracer includes context information about the urban environ-ment in terms of a city and building map. It consists out of a finite set of grid points, since the ray tracer prediction is done pointwisely per OS–MS combination. If there are no multipaths received by the OS for a specific MS position, the respective grid–
point of the ray tracer databases possesses the empty set. This information can be used to find out, where possible targets can be observed by the localization algorithm.
Thus, the allowed particle positions with respect to the context information are the grid points of the ray tracer database which possess at least a set of one multipath.
In the following, these positions are referred to as valid street points. The implemen-tations of the standard and generalized PHD intensity filters studied in the following sections always use only valid street points for the prediction and initialization of new particles.
By applying for each received measurement (except the one with RToA = 0) the approach defined in Section 6.2.2.2, a predicted multipath is chosen out of the ray tracer database. Afterwards, the OS, the interaction points and the MS are linearly connected. The connection is discretized, where the distance between two grid points is given byd >0. Possessing a set of grid points for each multipath, boxes with side lengthdinto the tangential direction and a side length proportional to the covariance of the likelihood function defined in (6.54) into the normal direction are generated.
The center of these boxes are the respective grid points (see Figure 6.7). The union of all boxes for one measurement then limits the region of allowed particle positions per measured multipath. Hence, the set of allowed particle positions for a specific measurement is given by the street points which fall into the union of boxes of the chosen multipaths.
Approximation of the Likelihood Function Since the absolute path length is not observable in BML scenarios, a marginalization with respect to the range is needed for the evaluation of the likelihood function defined in (6.54). Therefore, an integral of Gaussian kernels (which are linear in target space) over the distance between two interaction points has to be computed for all interaction points of the chosen multi-paths. Since the integral defined (6.53) cannot be evaluated analytically [SW09] it has to be approximated. In this work the integral is approximated by step functions centered at the grid points of the discretized multipath, with support lengthd.