## Figure (i.e., 0 < rgaim < 00),

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.

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