IEEE Systems, Man and Cybernetics Magazine - October 2020 - 30

projecting the height of the RSSI values, as shown in Figure 5. Note that the model does not directly take into
account physical boundaries for RF.
If the path around the region of interest is not closed
Model Generation
or
nearly closed, the models will have jagged points of
To generate a model for the RSSI, the closed point map is
inflection or points of focus. The example model has a
segmented using Delaunay triangulation prior to
central point of focus. This model was
chosen for illustrative purposes to emphasize how the method can work, even with
Tests for Model Validity and Stability
Model 1 Model 2 Model 3 Model 4
nonideal measurements.
For ease of inspection, the four models
60
are overlaid upon the present physical
boundaries in addition to the triangula40
tion. As depicted in Figure 6, the models
are
valid for regions inside of the meaModel 1
20
surement points. One can also see that
the central point adds inflection to the
0
Model 2
model of BSSID-1.
-20
It should be noted that these are empirically produced models based upon the
Model 3
-40
measurements. The models use a linear
interpolation between points, creating a
-60
smooth contour for RSSI drop off between
the ground-truth points of inflection. This
Model 4
-80
approach makes it far less computationally intensive than simulating the interaction
Model Differences = Column - Row
of RF at each boundary, making it possible
to generate the models on simple singleFigure 7. Displaying the differencing matrix allows for one to visually
board computers.
inspect the regions where the models have zero difference. If the
-1.5 ≤ Model Differences ≤ 1.5

the measurement points may be wrong or a physical difference exists in situ.

three models do not have enough measurement diversity at any
given point, the model will be prone to singularities, or the given
BSSID will be co-located with another that was modeled. Thus, the
measurement routine will fail.

RSSI Location Bands
Model 1 Model 2 Model 3

Model 4
2.5
2
1.5
1
0.5

Model 2

0
-0.5

Model 3

-1

-1.5 ≤ RSSI ≤ 1.5

Model 1

-1.5
Model 4

Figure 8. For a given set of RSSI measurements, it becomes

-2
-2.5

convenient to display them as bands of convergence. One can see
how much any given model influences the final solution based upon
the difference map between the two models.

30

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE O ctober 2020

Model Validity Checks
To validate the four models against each
other, a differencing method is employed
and demonstrated in Figure 7. This allows
for the four models to be rapidly compared
for regions where, when advanced to live
positioning data, the models suggest that a
measurement set is valid for more than
one position.
The original models are along the edge
of the difference matrix. The difference
between a model and itself is, by definition,
zero. This populates the diagonal with zero
matrices. If the matrix were unaltered, the
area above and below the diagonal would
be symmetric.
The regions below the diagonal represent the full representation of the difference between the model matrices. A large
positive or negative value in the difference
between two models means there is a high
likelihood that the models present a
unique solution for positioning when they
are overlaid.
The entries above the diagonal were edited to highlight the regions within ±1.5 dB of



IEEE Systems, Man and Cybernetics Magazine - October 2020

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