## 4 Fingerprinting-Based Positioning with Spatial Side Information
### Discretized Area Fractions with Tiles  
The fingerprint samples measured using the 2D sensor array correspond to the sampling of a small area equivalent to the size of the 2D sensor array. Thus, each sample returned by the 2D sensor array can be interpreted as a small discretized area fraction of the whole area in which the positioning is performed. In the training phase, to sample the fingerprints for the whole _positioning area_, the 2D sensor array is moved to different positions to collect fingerprints. When sampling at contiguous positions with spacing shorter than the side length of the 2D sensor array, some of the fingerprint samples measured at one position will overlap with the fingerprint samplestaken at another position. This is not wrong; the use of the fingerprints is ultimately decided by the pattern-matching algorithm in use. Here, we will control the collection of fingerprints so that there is no overlapping between samples. Such an approach has the objective of retaining the same density of samples used per ground truth position sampled and, in addition, using this methodology for a pattern-matching method based on classification. We consider a case in which the whole positioning area is divided into perfect area fractions of sizes equivalent to that of the 2D sensor array, as shown in Figure 9. Figure 9a,b represents the positioning areas. The positioning areas in the figures are divided into 12 area fractions equivalent to a given size of the 2D sensor array. The thick lines delimit boundaries equivalent to the size of the 2D sensor array, and the thin lines subdivide these into sizes equivalent to the space covered by each sensor in the 2D sensor array. The different colors inside the squares delimited by the thin lines represent a vector of fingerprints for each position associated with a sensor in the 2D sensor array, coded into the three color channels Red-Green-Blue (RGB). Such color coding is an aid in depicting that the fingerprints vary with the position and with some other components, like noise-like variations in the signal providing the fingerprints. Dividing the whole positioning area into perfect area fractions can ideally be achieved by sampling contiguous parts of the whole area; moving the 2D sensor array to the next position, without overlapping; and repeating this process until the whole area is covered. For example, in Figure 9, the whole positioning area can be sampled by placing the 2D sensor array within the 12 area fractions depicted in the figure. In practice, this is difficult to achieve; however, by sampling at the positions where the samples of the sensors overlap, an equivalent result can be achieved by post-processing the sampled fingerprints. Assuming uniform statistics for the fingerprints, the fingerprints at overlapping subdivisions can be discarded. The discretized area fractions mentioned above are called _tiles_ in the context of this work. So, in Figure 9, the positioning areas are actually divided into 12 (fixed) tiles. We assign these the name tiles due to the resemblance of dividing the whole positioning area into area fractions with the grid pattern outlined by square tiles. Thus, the process of dividing the positioning area into tiles is referred to as _tiling_. It is recalled that there is a direct relationship between a tile and the 2D sensor array. The 2D sensor array is actually the one sampling the whole positioning area into area fractions equal to its size and generating the tiles. Then, each sensor in the 2D sensor array is analogous to a subdivision in the tile, which we call a _sub-tile_. It is assumed that the sub-tile is the smallest granularity for discretizing the positioning area and sample positions. Since a vector of fingerprints is  
Figure 9: Positioning area divided into 12 fixed tiles and a moving tile. Fingerprints color-coded for different positions. The side length of the sub-tile is \(S_{S}\); the side length of the tile is \(S_{T}\). (**a**) Details of a fixed tile. (**b**) Details of a moving tile.