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The MBS system facilitates accurate 3D position computation. Since the MBS is a network of tightly synchronized beacons, trilateration can be done using pseudoranges determined from time‐stamped TOA measurements from the beacons and the beacon coordinates available from the beacon data.

The range equation in 3D space from the receiver to the transmitter is given by

(39.1)

The location of the transmitters is given by (x_i, y_i, z_i), and the unknown location of the mobile units is given by (X, Y, Z) in some local coordinate frame. The pseudorange measurement has a receiver time bias additive term as well, so that the usual pseudorange measurement equation can be written as

(39.2)

where c is the speed of light, and Δt corresponds to the receiver time bias. Traditionally, a minimum of four pseudorange measurements would be required for 3D trilateration to solve for the four variables: X, Y, Z, and receiver time bias. In a terrestrial network, estimating the Z coordinate through trilateration is error prone due to limited VDOP. When the z‐axis is available through barometric techniques, a minimum of three pseudorange measurements is sufficient for 3D trilateration.

There is another aspect of trilateration that is quite different for a terrestrial system of beacons when compared to a GPS satellite system. In a GPS system, traditionally, the trilateration problem is linearized to a weighted least squares (WLS) problem. The linearization works well due to the large distance of the satellites relative to the receiver. In the terrestrial system, the case of when a receiver is close to a beacon has to be carefully considered. In such conditions, locally linearized algorithms can suffer from position divergence. In general, the best estimate of the receiver location when altitude aiding is available can be obtained as the set of (X,Y,Z,Δt) that minimizes the objective function

Figure 39.15 Sample channel responses from MBS beacons.

Figure 39.16 Channel spread statistics.

(39.3)