# Simulating Signal Aberration and Ranging Error for Ultrasonic Indoor Positioning
## 2 Field II and Simulation Setup  
In this section, the operating principle of the Field II simulator is briefly outlined, and the simulation configuration is described in detail. The acoustic field simulator Field II [22] employs the concept of spatial impulse responses [23; 24; 25]. The ultrasound field for both the pulsed and continuous wave cases is found using the linear systems theory. The spatial impulse response gives the emitted ultrasound field at a specific point in space as a function of time, when the transducer is excited by a Dirac delta function. In a second step, the field generated by any kind of excitation is found by convolving the spatial impulse response with the excitation function. Since the linear systems theory is used, any excitation can be considered. The impulse response is a function of the position relative to the transducer, hence the name spatial impulse response of the technique [26]. Briefly, the transducer surface is divided into small rectangles, introducing a transducer surface and field approximations that are as much smaller as the elements into which the transducer surface is divided are smaller. The approximation is reduced by using small rectangles, where the distance to the field point is large compared to the size of the rectangles. Typically, the element size is much smaller than the wavelength of the signal to be simulated. Each of the rectangular elements is considered a rectangular piston, of which the exact solution for the impulsive response is known [25]. A spherical wave is emitted by each of the small elements, and the impulsive responses due to each element are added together at each desired field point [26]. In the simulations that follow, the aim is to examine the acoustic field and the effectiveness of an established ranging technique based on the correlation in a typical \(4\times 4\times 3\) m\({}^{3}\) room [27]. In particular, the simulation results will be examined on a grid of points belonging to a vertical section (see Section A of the room volume, Figure 1) and on a horizontal section halfway between the floor and the ceiling (see Section B of the room volume, Figure 1). The transducer is a circular planar and is placed in the center of the ceiling, in position \(x=0\), \(y=0\), and \(z=0\), and emits towards the floor of the room. The transducer is immersed in the air, and a linearized air absorption (slope 39.3 dB/m-MHz, constant term \(-0.262\) dB/m, i.e., about 0.917 dB/m @ 20 kHz and 1.703 dB/m @ 50 kHz) has been assumed around 40 kHz, corresponding to a transducer central frequency of 40 kHz at a temperature of 20 \({}^{\circ}\)C, a pressure of 1 atm, and a relative humidity of 55% [28; 29]. The emitted and received signals at all points in the space depends on the shape and size of the emission surface (i.e., on the aperture D) of the transducer. In this work, a circular plane transducer was considered, with acoustic properties similar to those of the most commonly used transducersfor positioning applications--for example, the typical piezoelectric transducer Murata MA4054S (D = 9.9 mm) or Pro-wave 328 ST/R160 (D = 13.1 mm) [30; 31]. The emission disk is divided into a certain number of rectangles. In particular, square elements with sides 0.125 mm by 0.125 mm were used for all the simulations that follow. The transducer element size chosen in this work is a good compromise between the accuracy of the solution and computational resources engaged in the simulation. In Figure 2, for displaying purposes, in order to visualize the single elements, the dimension of the mathematical elements is 1 mm by 1 mm. The signal used for the simulations is a linear chirp with a bandwidth of 30-50 kHz and 5.12-ms duration [10; 27]. For simulation purposes, the signal was sampled at f\({}_{\text{S}}\) = 1 MHz. The acoustic field was calculated in a set of points in the space for the duration of a time window compatible with the complete reception of the signal itself [26].