## 4 Fingerprinting-Based Positioning with Spatial Side Information
### Two-Dimensional Sensor Array  
We propose using a two-dimensional array arrangement of sensors, suitable for measuring fingerprints that are physically adjacent in space, as shown in Figure 2. A 2D sensor array makes it possible to collect fingerprints with spatial side information as desired. A set or vector of fingerprints is collected at each sensor in the 2D sensor array.  
We formulate a conjecture based on the idea that upon increasing the number of spatial points at which the fingerprints are measured, the amount of information should increase, and therefore, a better estimate of the position should be achieved. This conjecture will be studied through simulations in the following sections.  
An actual description of how the 2D sensor array should be built it is out of our scope. The goal is to test whether there is an improvement in the positioning performance, in terms of reducing the positioning error, given the assumption that the 2D sensor array exists.  
In this work, we focus on fingerprints based on the signal strengths of radio signals. Therefore, this 2D sensor array could be either a 2D arrangement of multiple antennas with the suitable complementing hardware or a 2D arrangement of multiple independent receivers or sensor nodes.  
Also, an arrangement that produces the same output could be devised, for example, by physically scanning using one sensor the area equivalent to that of the 2D sensor array. An alternative approach could combine the movement of more than one sensor to generate a matrix of fingerprints analogous to that obtained by the 2D sensor array--for example, a linear array of sensors or a linear antenna array arranged perpendicular to the direction of movement. In the context of our work, we assume the availability of a 2D sensor array that instantaneously returns the RSS values for each sensor or array element.  
The proposal of such a 2D sensor array is not far from the current state-of-the-art. Arrays of multiple antennas are currently in use with massive MIMO technology [53], with the antennas generally arranged into 1D linear arrays. Instead of the typical 1D linear array used in MIMO, in full-dimension MIMO, also known as 3D MIMO, it it proposed for the antennas to be arranged into 2D planar arrays [205,206]. For example, a planar array of \(8\times 8\) antennas, with half-wavelength spacing, for a carrier frequency of 2.6 GHz would have a size of approximately 462 mm \(\times\) 462 mm. From the perspective of our study, the challenge is that these antenna arrays have mainly been considered on the base station side of the wireless network. In contrast, our proposal requires such an array of antennas on the positioning device side.  
Research on urban navigation utilizing a circular antenna array mounted into a vehicle is reported in [202]. The use of such an antenna array is aligned with the proposed implementation of a 2D sensor array on the device side, although in this case, it is implemented using a circular antenna array.  
Intuitively, one may expect that the bigger the sensor array, with more sensors and thus more side information, the smaller the positioning error. The main concern is that the size of such a 2D sensor array should be suitable for real practical use. Possibly, a 2D sensor array with the characteristics required for positioning with spatial side information would not be suitable for use as a hand-held positioning device. However, it should have a size suitable for industrial applications--for example, with a size in the order of that of industrial robots, warehouse trolleys, and moving (autonomous) machinery. Thus, we aim for a 2D sensor array with an approximate side length possibly ranging from 300 mm to 1000 mm.