Return on Investment (roi) for Multi-Technology son juan Ramiro, Mark Austin and Khalid Hamied

A.2. Observed Time Differences (OTDs)

OTDs are reported by UEs to the Radio Network Controller (RNC) for Active Set management every time an MR message is sent. According to the Third Generation Partnership Project (3GPP) [6], an OTD sent from a given UE that reports sector i can be expressed in chips² as:
T_M[i] = Q + R[i] − T_P[i] (A.1)
where T_M[i] is ranged between [0,38399], Q is the internal time reference of the UE, R[i] is the synchronization reference at sector i, and T_P[i] is the propagation time, which is proportional to the distance d[i] between the UE and sector i as follows:

_P 1 1 _i ² (y_i − y₀)² (A.2) T [ ]i = ⋅d i[ ] = ⋅ (x − x₀) +
ρ ρ
where _r is the chip length in meters, and [x_i , y_i] and [x₀ , y₀] are respectively the sector and UE locations.

A.3. Algorithm Description

OTDs are proportional to the propagation distance between the UE and the reported sector, and hence can be used for geo-location of events. Details are described in the following paragraphs.

A.3.1. Geo-Location of Events

OTDs reporting the same sectors within a site (or BTS) do not provide any new linearly independent equation because the distance between those sectors and the UE is exactly the same. Therefore, it is possible to simplify the problem by grouping MRs in terms of sites as follows:
T_NM[i_S] = T_M[i] + 256 • T_cell[i] (A.3)
where T_NM[i] is the normalized OTD, i_S is the site to which sector i belongs, and T_cell is a known network parameter that relates the synchronization reference of the different sectors within a site.
Considering that the internal time reference of the UE is unknown, relative values must be taken, so that the relative distance between sites i_S and j_S regarding a given UE can be written as:
d[i_S, j_S] = r • (T_NM[ j_S, i_S] + R[i_S , j_S]) (A.4) where the operator X[a,b] = X[a] − X[b].
Appendix A: Geo-Location Technology, UMTS 265
Assuming that the synchronization reference of each site is known, the location of the UE [x₀, y₀] can be derived by solving the following system of nonlinear equations:
⎛d[1,2] = ρ⋅(T_NM [2,1]+ R[1,2])⎞ (A.5)
⎜ _⋮ ⎟
⎜⎝d[1,n] = ρ⋅(T_NM [n,1]+ R[1,n])⎟⎠
where n is the number of different reported sites. Firstly a Taylor series [7] is applied for linearizing, and then the system of equations is solved using the well-known Recursive Least Squares (RLS) method [8]. It is worth highlighting that at least three different sites, that is, n ≥ 3, must be reported in a given MR for the system of equations to be solvable. In case there are only two sites reported, the location of the UE could be calculated using Propagation Delay (PD) measurements, which report the distance to the call-setup sector, so that an extra equation could be added to Equation (A.5) as below:
T_PD[ ]i _S d i[ ]_S (A.6)
where T_PD[i_S] is the PD between the UE and the call-setup sector associated to site i_S. The use of PD is limited by the fact that it is reported with a granularity of three chips, and that it can be used only close enough in time to the call establishment.