# Acoustic Simulation for Performance Evaluation of Ultrasonic Ranging Systems
## 2 Simulation Setup  
The acoustic simulation software Field II, which works in the MATLABTM environment, exploits the concept of spatial impulse responses [21; 22; 23] to compute the acoustic field. The linear systems theory is the base for the obtaining of the ultrasound field for both the pulsed and continuous wave cases. In fact, by applying to the transducer an excitation in the form of a Dirac delta function, the emitted ultrasound field at a specific point in space as a function of time is obtained using the spatial impulse response. In a second step, the field generated by an arbitrary excitation is found by the convolution of the spatial impulse response with the excitation signal. Thanks to the use of the linear systems theory, any kind of excitation can be considered. The name of this technique, \"spatial impulse response\", derives from the fact that the impulse response is a function of the relative position in the 3D space of the computed acoustic field point, with respect to the transducer [24]. In detail, in order to carry out the numerical simulation, the entire surface of the transducer is divided into small rectangles. The smaller the rectangles' size, the better the field approximation. In practice, the distance from the center of each rectangle to the field point under computation must be very large compared to the size of the rectangles. As a rule of thumb, the element size must be much smaller than the wavelength of the used signal. Every rectangular element can be seen as a rectangular piston, with a known and exact solution for the impulsive response [23]. The emission of a spherical wave by each of the small elements is combined with the impulsive responses due to each element at each desired field point [24]. To date, Field II is the only available and reliable acoustic simulator that is not based on the finite element modeling (FEM) approach (e.g., ANSYS, COMSOL, etc.). The FEM approach is computationally too onerous when dealing with spaces hundreds of times longer than the typical wavelength considered (less than a couple of cm in the band 20-50 kHz), as in the present case. The number of nodes would be huge and the calculation very heavy. Instead, the spatial impulse response approach used by Field II requires the calculation to be carried out only in the points considered and not on all nodes of a mesh extended to the whole considered environment. This makes the simulation for large spaces very efficient and entirely feasible. The simulation presented is just an example to show the potential of the proposed approach. The simulation setup includes diffractive acoustic phenomena, with the possibility to modify the shape and dimensions of the transducer and the emitted signal. Additionally, it allows us to test any ranging or positioning technique that is intended to be applied. In the simulations that follow, the acoustic field and the effectiveness of cross-correlation ranging in the space corresponding to a typical \(4\times 4\times 3\) m\({}^{3}\) room [25] will be examined. In particular, the simulation results are computed on a grid of points belonging to the vertical section A of the room volume schematized in Figure 1. The boundary lines represent the extension of the room, but walls, the ceiling or floor are not considered, since the simulation is conducted as if the emission were in free space, as specified in the Introduction. The used transducer is a disk placed at the center of the ceiling, in position x = 0, y = 0, and z = 0, emitting towards the floor of the room. The transducer is immersed in the air and its central frequency is 40 kHz at a temperature of 20 \({}^{\circ}\)C, a pressure of 1 atm, and a relative humidity of 55% [26; 27]. Air absorption linearization shows an acceptable error in the atmospheric conditions present in a normal room with RH between 50% and 60% and for the frequency range considered [26].