## 6 Results of the Simulations
### Observations for Cases with Spatial Side Information and Aggregation in the Fingerprint Domain
#### 6.8.1 Lower Bound in Performance for the Smallest 2D Sensor Array Considered  
From the results obtained using pattern-matching methods 3-MultFingIn-PosOut and 4-MultFingIn-ClassOut, we observe in Figures 19 and 20 the following trend. This trend indicates that on starting with a small number of sub-tiles and then increasing the amount of side information by adding more sub-tiles, that is, increasing \(N_{SS}\), the error distance decreases until the \(N_{SS}\) that produces the best result with the minimum error distance (the best \(N_{SS}\)). Then, the results when selecting a small 2D sensor array can be used to define a lower bound in the performance. Here, because the number of NN parameters grows with \(N_{SS}\), the lower bound in performance exists in the context of obtaining the worst error distance using the smallest number of NN parameters. This means that selecting a bigger 2D sensor array will result in a smaller positioning error up to the \(N_{SS}\) that produces the best result. So, the smallest 2D sensor array considered is regarded to be that producing the worst result with the smallest number of sub-tiles, \(N_{SS}\), that can be obtained using these methods. Figure 26: Comparison of the best results (continuous lines) and the results with the smallest 2D sensor array (dashed lines) for the pattern-matching method 4-MultFingIn-ClassOut with different side lengths of the sub-tiles \(S_{S}\). For the smallest 2D sensor arrays considered, we selected equivalent tile sizes consisting of the following numbers of sub-tiles: \(N_{SS}=6\), with \(S_{S}=30\) mm, and \(N_{SS}=5\), with \(S_{S}=50\) mm and \(S_{S}=70\) mm. This selection was based on practical observations. The first observation is the quadratic growth in the number of tiles required to cover the whole positioning area, as a function of the inverse of the side length of the tiles. The smaller the tiles, the greater the number of tiles required to cover the positioning area, and the greater the number of reference positions associated with the fingerprint samples that needs to be collected. Thus, operating using a large number of small tiles to cover the positioning area adds overheads in terms of the time and costs required to collect the fingerprint samples in a real scenario. Another aspect to consider is that the greater the number of tiles, the greater the number of tile-classes required in pattern-matching method 4-MultFingIn-ClassOut. This translates to selecting an NN with a suitable representational capacity for the number of output classes. The second observation is related to pattern-matching method 4-MultFingIn-ClassOut and the convolution operation. In general, there is a requirement for the minimum size of the tensor input to a CNN such that it is suitable for actually performing the convolution operation. In method four, the tensor input to the CNN is a tensor of fingerprints. Then, the number of sub-tiles in a tile must be greater than the convolution kernel combined with the stride size. In pattern-matching method 4-MultFingIn-ClassOut, we used a convolution kernel of \(3\times 3\) and a stride size of 1; therefore, we adopted a minimum number of sub-tiles per side of the tile of \(N_{SS}=5\). Now, it is noted that the worst error distance with the smallest number of NN parameters considered (the smallest \(N_{SS}\) considered, with \(N_{SS}=5\) and 6) reduced the positioning error in distances ranging from 1.60 to 1.75 m, for the three \(S_{S}\) considered, to that in the case without spatial side information using pattern-matching method 1-SingFingIn-PosOut, measured at the 95 th percentile in Figure 21. This is equivalent to an order of 4.6 to 9.7 of the side lengths of the 2D sensor array. This last observation is an attempt to normalize in the 2D sensor array sizes the gain obtained using the different 2D sensor array sizes, under the assumption that the size of the 2D sensor array defines the positioning resolution for the samples.