## C.2 Position Calculation Types

03.713GPPFunctional descriptionLocation Services (LCS)Release 1999Stage 2TS

The location estimate is performed by a Position Calculation Function (PCF) located in the MS or in the network. With the same network architecture, MS functions, LMU functions and measurement inputs the PCF can be based on one of two possible types of E-OTD location calculation; known as ‘hyperbolic’ and ‘circular’.

The hyperbolic type is introduced in section (a) below followed by a brief description of the circular type in section (b).

a) Hyperbolic Type

There are three basic timing quantities associated with this type of E-OTD location calculation:

– Observed Time Difference (OTD). This means the time interval that is observed by a mobile station (MS) between the reception of signals (bursts) from two different Base Transceiver Stations (BTS) in the cellular network. A burst from the BTS 1 is received at the moment t1, and a burst from the BTS 2 is received at the moment t2. Thus the OTD value in this case is: OTD = t2 – t1. If the two bursts arrive exactly at the same moment, then OTD = 0.

– Real Time Difference (RTD). This means the relative synchronization difference in the network between two BTSs. If the BTS 1 sends a burst at the moment t3, and the BTS 2 at the moment t4, the RTD between them is: RTD = t4 – t3. If the BTSs transmit exactly at the same time that means that the network is synchronized and there is no need to calculate RTDs, hence RTDs = 0.

– Geometric Time Difference (GTD). This is the time difference between the reception (by an MS) of bursts from two different base stations due to geometry. If the length of the propagation path between the BTS 1 and the mobile station is d1, and the length of the path between the BTS 2 and the MS is d2, then GTD = (d2 – d1) / **, where ** is the speed of radio waves. If both BTSs are exactly as far from the MS, GTD = 0.

The relationship between these three quantities is:

OTD = RTD + GTD.

OTD is the quantity measured by the mobile station to be located. RTD is a quantity related to the network (BTSs). GTD is a quantity related to the geometry of the situation (positions of the mobile and BTSs). GTD is the actual quantity that is useful for location purposes, since it contains information about the position of the MS. If only OTD values are known, no location can be calculated, thus also RTD values must be known.

The MS location estimate can be computed in the MS or by the network depending on implementation. Whichever method is used the MS location estimate is calculated from the GTD (as calculated from the measured OTD and known or measured RTD) based on the fact that the possible location for the MS observing a constant GTD value (d2 – d1 = constant) between two BTSs is a hyperbola. The MS can be located in the intersection of two hyperbolas obtained with three base stations and two GTDs. If more GTDs are available the possible location area can be reduced.

Figure C.1: E-OTD location (hyperbolic)

The dashed line represents the determined GTD, i.e., represents a constant difference in distance to two BTSs. The measurement result is not exact, thus the gray area represents the area of uncertainty for the MS based on that OTD measurement. The black area at the intersection of the hyperbolas is the calculated most likely location for the MS.

b) Circular Type

The E-OTD Circular location calculation type does not measure time differences at the MS and LMU between the receipt of signals from pairs of BTSs. Rather, it measures the arrival time of those signals individually.

There are five quantities associated with the circular type of E-OTD:

– The Observed Time at the MS (MOT) at which a signal arrives from a BTS. This is a time measured against the MS’s internal clock.

– The Observed Time at the LMU (LOT) at which a signal arrives from a BTS. This is a time measured against the LMU’s internal clock. In general there will be a time offset ** between the MS’s internal clock and the LMU’s internal clock.

– The geometrical Distance from MS to BTS (DMB).

– The geometrical Distance from LMU to BTS (DLB).

These quantities are related by:

DMB – DLB = ** (MOT – LOT + **)

in which ** is the speed of the signals (speed of radio waves) and there will be one such equation for each BTS. Since there are three unknown quantities (MS position *x, y* and clock offset **) at least three BTSs are required to solve for the MS location and the unknown clock offset **. This is the same total number of BTSs as is required for the hyperbolic type of E-OTD. The position of the MS is defined by the intersection of circles centred on the BTSs common to observations made by the MS and LMUs, hence the notation ‘circular’ as the E-OTD type as shown in Figure C.2 below.

Figure C.2: E-OTD location (circular)

The hyperbolic and circular types differ in the relationship between the MS measurement error margin and the geographic location of the MS relative to BTSs. In all other respects the implementation is identical.