## 4 Fingerprinting-Based Positioning with Spatial Side Information
### Discretized Area Fractions with Tiles
collected at each sensor in the 2D sensor array, there is a vector of fingerprints associated with each sub-tile. An example of tiles and sub-tiles is depicted in Figure 9a. The tiles depicted in the figure have a square shape; however, these could have any other shape if desired. In the context of this work, we will work only with square tiles. We assume a uniform size for the sub-tiles within the tiles (or sensors within the 2D sensor array). The side length of a sub-tile is indicated by \(S_{S}\). The size of a tile, and thus the 2D sensor array, is measured by the number of sub-tiles per side of the tile (that is, the number of sensors per side of the 2D sensor array), \(N_{SS}\), for a given side length of a sub-tile. The side length of a tile is indicated by \(S_{T}\), with \(S_{T}=N_{SS}\times S_{S}\). For a square tile, there are a total of \(N_{ST}\) sub-tiles in a tile, with \(N_{ST}=N_{SS}\)2. The parameters and variables related to the tiles and sub-tiles (and correspondingly associated with the 2D sensor array) are summarized in Table 5. Finally, we assume that the whole positioning area is covered by an integer total number of tiles, \(N_{T}\). That is, it is assumed that the tiles cannot be divided into smaller fractions. Footnote 2: The 2D sensor array is defined as the 2D sensor array, and the 2D sensor array is defined as the 2D sensor array. Tiles covering the whole positioning area uniformly, with a contiguous arrangement as a grid and non-overlapping, as depicted in Figure 9a, are referred to as _fixed tiles_. Fixed tiles are tiles that have associated fingerprints used in the training phase. Thus, the tiling of the positioning area with fixed tiles, in a contiguous arrangement as a grid, is used only in the training phase. Such a grid-like arrangement is an aid in developing the pattern-matching methods. In the testing phase, when one intends to find a position, the grid-like arrangement of fixed tiles is unknown. In the testing phase, for a random position within the whole positioning area, not constrained to the positions of the fixed tiles from the tiling process, the 2D sensor array will describe a tile that, in general, it is not aligned with the fixed tiles of the grid-like arrangement. A tile described by the 2D sensor array at a random position within the positioning area is referred to as a _moving tile_. In other words, moving tiles are the products of the samples generated using the 2D sensor array at random positions. In the context of this work, moving tiles will be generated at random positions to benchmark different pattern-matching methods in the testing phase. An example of a moving tile is shown in Figure 9b. It is assumed that a moving tile always keeps the same orientation as the fixed tiles in the grid-like arrangement described by the fixed tiles, or in other words, it is not rotated. Finally, to estimate a position during the testing phase, an algorithm implementing a pattern-matching method will use as its input the fingerprints collected from the 2D sensor array, acting as a moving tile, and compute the estimated position of the 2D sensor array. The pattern-matching method is previously trained in the training phase with information from the fixed tiles. It is noted that in the scope of our work, in the training phase, we choose to discretize the positioning area with a contiguous arrangement of non-overlapping fixed tiles as a grid. From one perspective, it could be thought that such discretization does not offer