Figure 1. The BS, MS and scatterer geometry for

the ROS scattering model.

BS ;

Figure 2. BS,

MS and scatterer geometry for the DOS scattering model.

Figure 3. Scattering geometry for the Gaussian

model

A. Ring of Scatterers Model

In the ROS

model shown in Fig. I, the scatterers are located on a ring centered about the

MS with a radius Rt.. The angular

distribution of the transmitted (or, equivalently, arriving) signal from the MS

is uniformly distributed on 0, 27. In addition, the scatterers are uniformly

distributed around the ring. The true distance between MS and BSi is Ri while

measured propagation distance (i.e., range or TOA measurement) of the jth

multipath component is l zj for a single reflection off the ring.

B. Disk of Scatterers Model

In the DOS

model, the scatterers are located an a solid cir- cular disk of fixed radius Rd with

the MS at the center as shown in Fig. 2. The distance to a scatterer from the

MS, rpos, is uniformly distributed in the range 0, Rd, and the angle 0 is

uniformly distributed in the range 0, 27r.

C. Clipped Gaussian Scattering Model

Fig. 3

illustrates the geometry for the Gaussian scattering model. While the Gaussian

distribution has tails that extend infinitely (i.e., 0 < rgaim < 00), we
consider the case where the Gaussian distribution is clipped so that the area
of support of the clipped Gaussian pdf comprises a circle of radius R 9
(i.e., 0 < rgauss < R9) about the MS.
Consider a
system consisting of Nt anchors, Nr reference nodes and Nu unknown nodes. The
positions of the anchors are known and denoted by Ti with i ? {1, ? , Nt}. The positions of
the reference nodes are also known and denoted by Rj with j ?{1, ? , Nr}. The positions of
unknown nodes are to be estimated and denoted by Uk with k ? {1, ? , Nu}. Here, Ti, Rj and Uk are all 1× D vectors, where
D is the dimension of the coordinates of the environment of interests.
Two scenarios
of applications can be implied by the system model for estimating the unknown
nodes' positions. In the first scenario, the anchors are functioning as TXs;
and the reference nodes and the unknown nodes are functioning as RXs. In the
second scenario, the anchors are functioning as RXs; and the reference nodes
and the unknown nodes are functioning as TXs. The two scenarios are dual
problems and it is sufficed to discuss just one of them. We will consider in
this paper the first scenario.
It is assumed
that the TXs (anchors) will transmit sounding signals sequentially and the
TOA's of the first arrivals of these sounding signals at each RX (a reference
or unknown node) can be measured by the RX. The propagation environment of
interests is assumed to be very complicated and a lot of first arrivals
measured by the RXs are not due to
LOS rays. Moreover, the environment information is assumed not known and
consequently we cannot tell whether a particular measured TOA is due to a LOS
ray or not. Thus, neither the conventional TOA or TDOA-based localization
schemes nor the ray- shooting localization schemes can be employed here for
estimating the unknown nodes' locations. Fortunately, what can be employed is a
fingerprinting-type approach because it does not need any environmental
information and does not depend on a specific propagation mechanism.
A
fingerprinting-type approach usually consists of two steps: the offline step
and the online step. During the offline step, we need to find the TOA
measurements rj at the jth reference
node position, Rj , for every j ? {1, ? , Nr} . Here, rj is a 1 × Nt vector contains the
TOA's of the signals transmitted by the Nt anchors. Note that Nt could be a
small number and {Rj} could be non-uniformly distributed in the area of
interests. Having collected all the TOA measurements at the reference points,
we need to find a model which can best describe the relationship between the
positions Rj and their TOA measurements rj. This model is written as a vector
function, denoted by R? †(r ).
Usually, the
TX locations {Ti} are not needed in establishing the function † (?). Once † (?)
is found, the offline step is done. During the online step, we need to firstly
establish the TOA measurements uk at the kth
unknown node' position, Uk, for every k ? {1, ? , Nu}. Here, uk is a 1 ×
Nt vector represents the TOA's of the signals transmitted by the Nt anchors.
Then Uk is simply estimated by Uk ? †(uk).
Please note
that rj (or uk) can be any measurable quantities that depend on the position
vector Rj (or Uk). For example, the measurement quantities can be RSSI or TDOA
and the fingerprinting procedure stated above still holds.