During the calibration stage of the system, the BTS Initialisation Model operates concurrently with the network monitoring subsystem and the GPS sampling process. The network monitor
runs continuously to detect RSS variations and cell handovers, whereas the sampling process is controlled by the user to request GPS position data at discrete time intervals. When the
position fix is returned by the GPS module, it is combined with the CID and RSS measurements to create a position sample. While the mobile terminal is moving, samples are
collected relative to the serving BTS and stored in the list of samples. Using the observation
model described in the previous section, the BIM estimates the position of BTS landmark recursively each time a new sample is obtained. When a handover is detected, the list of
samples is handled by a fingerprinting process to produce a list of location fingerprints. The latter is then associated with the state of the landmark in order to create a landmark database
entry that will be stored in the map.
As stated previously, the BTS Initialisation Model is a static model of the extended Kalman
Filter maintaining the state of the serving BTS. The state vector at time instant k is parameterised by the Cartesian coordinates and has the form:
[ ] (4.6)
and are the easting and northing coordinates of the target base station within ellipsoidal
orthographic coordinate system. Subscript b denotes that the state only contains the coordinates of the base station. As the latter is a fixed point in the Cartesian plane, the BIM
105 the same as the posteriori estimate from the previous step (k-1). The state model is shown
below.
̂ (4.7)
Similarly, the time update of the state covariance is:
(4.8)
In this state model, the posterior state is estimated using the current measurement obtained at time k. The measurement consists of the sample collected from the current position of the
mobile terminal during the survey of the base station transmitter. The sample at time k has the following form:
( ( * * (4.9)
Where
is the cell identity code identifying the serving BTS communicating with the
mobile terminal at time k.
is the GPS position fix measured at time k in latitude and longitude
coordinates. It is important to note that these geodetic coordinates are converted to
Cartesian easting and northing coordinates and respectively in the global
east north frame F.
is the received signal power measured at the MT from the GPS ground-truth.
Initially, there is no prior knowledge about the location of the target BTS until the first GPS sample is taken by the mobile terminal. The state estimate is therefore initialised using
106
̂ +ω (4.10)
ω is a vector used to initialise the BTS state to coincide with a position within a certain distance from the GPS sample. The state is not initialised to coincide with the first GPS
samples as it would lead to a zero distance between the mobile and the BTS, which violates the definition of the log-distance observation model expressed in (4.4).
The covariance matrix associated with the landmark state estimate is the following diagonal matrix:
[ ] (4.11)
The error in the initial state estimate is assumed to coincide with the typical error of the Cell Id proximity sensing method. This error is relative to the size of the cell covered by the BTS transmitter. As far as the BIM is concerned, the BTS position is the state to be estimated and
the observation model expressed in 4.4 is rewritten as:
( ) (4.12)
Note from the observation model expressions (4.4) and (4.12) that the state of the mobile
terminal is omitted from the model function . Indeed, in this case, the MT position is not the state to be estimated as it is measured by the GPS module as part of the sample denoted in
(4.9). It is considered by the system as a perfect measurement as the GPS uncertainty is
negligible compared to the error in the BTS state estimate.
The BTS initialisation model is a sequential Kalman Filter which updates the state of
the BTS landmark using a single RSS measurement. Thus, the measurement of (4.12) is a scalar which is assumed uncorrelated with previous measurements. The noise is
107 assumed to be Gaussian distributed with mean zero and variance R. As a result, the
observation model function h denoted in 4.12 lends itself to the propagation model equation of (4.4) to predict the RSS measurement ̂ as follows:
̂ (√( ̂ ) ( ̂ ) ) (4.13)
When a new measurement sample is collected at time k, the BTS initialisation model uses the coordinates of the a priori state estimate ̂ and ̂ equation (4.13) to predict the
path loss measurement ̂ in order to proceed to the EKF measurement update stage. Subtracting the predicted measurement from the real measurement yields the measurement
innovation , which represents the contribution of the measurement to update the state estimate.
̂ (4.14)
Following the EKF approach, the BTS initialisation model relies on the linear approximation
of the observation model around the a priori state estimate . As only one RSS measurement is processed at a time, the Jacobian of the predicted measurement function is the
following row matrix:
* ̂ ̂
̂ ̂ +
(4.15)
The partial derivatives of the predicted observation with respect to the landmark’s eating and
108 ̂ ̂ ( ̂ ) ( ̂ ) ( ̂ ) (4.16) ̂ ̂ ( ̂ ) ( ̂ ) ( ̂ ) (4.17)
The constant in expressions (4.16) and (4.17) is denoted below:
(4.18)
The constructed Jacobian matrix is then used to compute the innovation variance and the filter gain as shown below:
(4.19)
(4.20)
Finally, the measurement innovation and the Kalman Filter gain are used to produce the a posteriori estimate of the state and its associated covariance as follows:
(4.21)
(k) (4.22)
When the network monitoring subsystem detects a handover to a new BTS, the state
estimate produced by the BIM is used to initialise the BTS landmark within the map. As explained in Section 4.2.3, the initialisation consists of creating a Landmark Database Entry
within the database representing the initial map of landmarks. The CCS system starts a new iteration of the BTS initialisation model to estimate the position of the new serving BTS after
109 GPS sample as shown in (4.10). The state is then updated using the EKF equations as more
samples are received until the next handover and so on. If the CID of the newly observed BTS matches that of a landmark already stored in the database, the state of the matching landmark
entry is extracted from the database. It is then integrated into the state vector so that the EKF resumes the estimation using new measurements.
As stated previously, the initialisation of a BTS does not only consist of estimating the position of the landmark. The database entry representing the initialised landmark associates
mobile location fingerprints to the position of the BTS. These location fingerprints are the very same samples that were used during the BIM calibration phase to initialise the BTS
landmark. This fingerprint association process is performed by the BTS initialisation model to prepare the system for the Network Localisation Stage.