## 3 Literature Review and Related Work
### Fingerprinting-Based Positioning Implementing Fingerprint Images Processed Using CNNs
In these articles, it is observed that the larger the number of receiving antennas and thus side information points, the lower the positioning error. Articles labeled with _Yes_, without parentheses, in the last column of Table 3 correspond to articles proposing the use of spatial side information on the positioning device side and implementing a 2D CNN. These articles depart from simply using a receiver with multiple antennas, as is the case for entries labeled with _(Yes)_, and therefore need to be differentiated. Each one of these articles are discussed in detail next. We note that, from our perspective, two of the entries labeled with _Yes_ make use of spatial side information in 2D, namely [120] and [133]. In [120], a transmitter and a receiver implementing a 32-element planar antenna array are tested for positioning in the 60 GHz frequency range. The receiver composes a beam covariance matrix from 36 beam patterns, which is processed as a fingerprint image using a 2D CNN. The authors discuss the beam patterns over the azimuth and elevation angles, suggesting the possibility of positioning in 2D in a vertical plane. In [130; 131], geomagnetic fingerprints are collected by walking along a path. Geomagnetic fingerprints are differentiated in 3D using a magnetometer, thus producing three sequences, with one for each dimension. In [130], each sequence is arranged into a square matrix and converted into an image representation. Three images, with one for each dimension, are stacked to form a tensor, which is interpreted as a single image with three channels. This final image is input into a 2D CNN. In [131], each sequence is transformed into a square matrix through a recurrence plot transformation. The resulting matrices from the three dimensions are stacked to form a tensor, which is input into a 2D CNN. In both cases, if there is spatial side information in 2D, it is not preserved in relation to the displacements in 2D. Then, the sequences have adjacent side information relative to the previous and next position, which are interpreted as contributions that are projected into only one dimension. In [133], the authors propose dividing the whole positioning area into a grid of squares. Each square has a vector of CSI amplitudes associated with it. The authors collect fingerprints by moving in the scenario for a finite number of steps, associated with a sliding window. The fingerprints are then arranged into a so-called sub-window, which is actually a 2D portion of the grid that covers the whole positioning area. Thus, the arrangement of the fingerprints seems to take into account the correlation of the CSI of the neighbor grid squares in 2D. The proposed method exploits these correlations to produce a better position estimate than that which would be obtained without this side information. The positioning resolution is in the order of the size of the squares in the grid. In [153], RSS fingerprints are collected as a time-series of spatially distributed fingerprints of 5 to 40 time samples by moving a positioning device along a path. The fingerprints at each time sample are arranged as vectors. Then, the vectors for all of the time samples are arranged into a matrix. The matrix is then interpreted as an image and processed using a 2D CNN. In [154], the RSS fingerprints measured at the current and previous position are considered. The fingerprints in these two positions are used as the input in the training and testing phases. Vectors of the RSSs for the current and previous position are arranged as matrices and then converted into images to be processed using a 2D CNN. In the model proposed, each image is first pre-processed by a series of 2D convolution layers and then processed by a layer concatenating the two sources to finally produce a position estimate. From one perspective, this arrangement could be interpreted as a time-series of fingerprints with only two samples.